This book focuses on three essential elements of agricultural supply chains: Planting and Growing, Processing and Sellin
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Table of contents :
Preface
Acknowledgments
Contents
Contributors
Part I Planting and Growing
1 Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits
1.1 Introduction
1.2 Model
1.3 Optimal Allocation Policy
1.4 Heuristics
1.4.1 Proposed Policy: OnePeriod Lookahead
1.4.2 Benchmarks
1.5 Numerical Study
1.5.1 Calibration
1.5.2 Simulation Setup
1.5.3 Results
1.6 Future Research
References
2 Agricultural Production Planning Under YieldDependent Cost and Price
2.1 Introduction
2.2 Literature Review
2.3 The Model and Analysis
2.4 Empirical Analysis
2.5 Conclusions
References
3 Mechanisms for Effective Sharing of Agricultural Water Between HeadReach and TailEnd Farms
3.1 Introduction
3.2 A SurfaceWater Allocation Problem for a Farming Community
3.2.1 The Centralized Problem
3.2.2 A Naive Decentralized Mechanism
3.3 Improved Decentralized Pricing Mechanisms for Water Distribution
3.3.1 A SingleLane, SingleCrop System
3.3.1.1 Socially Optimal Schemes
3.3.2 Reward and Premium Charts for the Entire Farming Community
3.4 A Special Case: RiskNeutral Farmers
3.5 Concluding Remarks
References
4 Portfolio Management Issues in the Commercial Seed Industry: A Modeling Framework and Industry Implementation
4.1 Introduction to the Commercial Seed Market
4.1.1 Chapter Focus
4.2 Portfolio Management with Sequential Production
4.2.1 Four Contextual Features of Seed Corn Market
4.2.2 Sequential Production
4.3 Mathematical Development
4.3.1 Problem Formulation
4.3.2 Solution
4.3.3 Mathematical Properties of Solution
4.4 Implementation and Benefits
4.4.1 Implementation
4.4.2 Benefits
4.5 Conclusion
References
Part II Processing and Selling
5 Procurement Management in Agricultural CommodityProcessing
5.1 Introduction
5.2 Literature Review
5.3 The Role of Demand Correlation in Agricultural Processing with Fixed Proportions Technology
5.3.1 Model Description and Assumptions
5.3.2 The Optimal Strategy
5.3.3 Analysis
5.4 The Impact of Spot Price Uncertainty on Contract Selection in Agricultural Processing
5.4.1 Model Description and Additional Assumptions
5.4.2 The Optimal Strategy
5.4.3 Analysis
5.5 Conclusion and Future Research
References
6 The Influence of YieldDependent Trading Costs on Pricing and Production Planning Under Supply Uncertainty
6.1 Introduction
6.2 Literature Review
6.3 The Model
6.4 Model Analysis
6.4.1 The Case of No Trading of the Fruit
6.4.2 Incorporating the Trading Option
6.4.3 The Impact of YieldDependent Trading Costs
6.4.3.1 The Impact on Expected Profit
6.4.3.2 The Impact on Pricing and Production Decisions
6.4.3.3 The Impact of Less Than Perfect Correlation Between Trading Costs and Yield
6.5 Value of Fruit Futures
6.6 Conclusions
References
7 Capacity Management in Agricultural Commodity Processing
7.1 Introduction
7.2 Literature Review
7.3 Model Description
7.4 Characterization of the Optimal Strategy
7.4.1 Periodic Input Processing and Output Inventory Decisions
7.4.2 Capacity Investment Decisions
7.5 Performance of Heuristic Capacity Investment Policies
7.6 Conclusion
References
8 A Prescriptive Model for Selling Wine Futures to Mitigate Quality Uncertainty
8.1 Introduction
8.1.1 The Impact of Barrel Scores on Wine Futures Decisions
8.2 Literature Review
8.2.1 Pricing and Quantity Decisions Under Uncertainty
8.2.2 Advance Selling Mechanisms
8.2.3 Wine Tasting and Wine Analytics
8.3 The Model
8.3.1 Quality Indicator
8.3.2 Consumer Utilities and the Demand for Wine Futures
8.3.3 Winemaker's Optimization Problem
8.4 Analysis
8.4.1 The Impact of Consumers' and Winemaker's Risk Preference
8.4.2 The Impact of Barrel Score
8.4.3 The Impact of Consumer Heterogeneity
8.5 Numerical Illustrations
8.6 Summary
References
9 Wine Analytics: Futures or Bottles?
9.1 Introduction
9.2 Literature Review
9.3 The Model and Its Analysis
9.3.1 Empirical Foundation for the Model
9.3.1.1 Models 1A and 1B: Futures Price Evolution
9.3.1.2 Models 2A and 2B: Bottle Price Evolution
9.3.1.3 Functional Forms for the Analytical Model
9.3.2 The Model
9.3.3 Analysis
9.4 Financial Benefits from Our Proposed Model
9.5 Conclusions
References
Part III Government Interventions
10 Implications of Farmer Information Provision Policies: Heterogeneous Farmers and Market Selection
10.1 Introduction
10.2 Literature Review
10.3 Model Description
10.4 Analysis: Farmer's Market Selection and Farmer's Profit
10.4.1 Farmer's Threshold Market Selection Rule Under Provision Policy δ
10.4.2 Farmer's Profit Function Under Provision Policy δ
10.4.3 Farmers' Expected Total Profit Under Provision Policy δ
10.5 Analysis: Comparisons of Provision Policies
10.5.1 Benchmark Provision Policies
10.5.2 Partial Intensity Policy (Fρ)
10.5.3 General Policy (Kρ)
10.5.3.1 Numerical Examples
10.5.3.2 Perceived Unfairness and Nominal Fees
10.6 Conclusion
References
11 Agricultural Market Information: Economic Value and Provision Policy
11.1 Introduction
11.2 Economic Value of Agricultural Market Information
11.2.1 Impact of Information on Farmer's Welfare
11.2.1.1 Impact of Private Signal on Farmer's Welfare
11.2.1.2 Impact of Public Signal on Farmer's Welfare
11.2.1.3 Impact of Signal Precision on Price Variation
11.2.2 Nonidentical Private Signal Precisions
11.3 Optimal Information Provision Policy
11.3.1 Farmers' Equilibrium Analysis
11.3.2 Central Planner's Optimal Information Provision Policy
11.4 Conclusion
References
12 Knowledge Sharing Among Smallholders in DevelopingEconomies
12.1 Introduction
12.2 Related Literature
12.3 Model Preliminaries
12.3.1 Base Case: Farmer's Effort Decision When There Is No Knowledge Sharing
12.4 Knowledge Sharing and Learning
12.4.1 Learning Process
12.4.2 A Coordinated System: Sharing Decisions Made by a Coordinator
12.4.3 A Decentralized System: Sharing Decisions Made by Individual Farmers
12.4.3.1 A Farmer's Best Response Function in a Decentralized System
12.4.3.2 PureStrategy Nash Equilibrium
12.4.3.3 Motivations for Sharing/Nonsharing in a Decentralized System
12.4.4 The Efficiency of Voluntary Sharing
12.5 Income Inequality and Reward Mechanisms
12.5.1 Impact of Increasing Knowledge Sharing on Income Inequality
12.5.2 Reward Mechanisms
12.5.2.1 Two Ineffective Mechanisms
12.5.2.2 A QuotaBased Reward Mechanism
12.6 Improving the Learning Effectiveness Factor h
12.7 Extensions
12.7.1 When Farmer's Knowledge Follows a General Distribution
12.7.2 When Each Farmer's HighQuality and LowQuality Outputs Are Stochastic
12.7.2.1 The Productive Factor β Follows the Binomial Distribution
12.7.2.2 Farmer k's HighQuality Output Is βk x + εk
12.7.3 When a Farmer's Output Capacity Is Not Fixed and the Markets Are Not Separate
12.8 Concluding Remarks
References
13 Policy Interventions for an AgricultureBased Malaria Medicine Supply Chain
13.1 Introduction
13.2 Related Literature
13.3 Model
13.3.1 Overview
13.3.2 Equilibrium Condition
13.3.3 Two Models of PriceDependent Demand
13.3.4 Performance Measures
13.4 Analysis
13.5 Summary of Implications for Policy Makers
13.6 Conclusions
References
14 The Impact of Crop Minimum Support Price on Crop Production and Farmer Welfare
14.1 Introduction
14.2 Literature Review
14.3 Model Preliminaries
14.4 A Model of the CreditBased MSP Scheme
14.4.1 Farmers' Crop Selection and Production Decisions
14.4.2 Impact of MSPs on Farmers' Production Decisions
14.4.3 Impact of MSPs on Farmers' Profits and Surplus
14.5 Conclusions
References
15 Input vs. OutputBased Farm Subsidies in Developing Economies: Farmer Welfare and Income Inequality
15.1 Introduction
15.2 Literature Review
15.3 Model Preliminaries
15.3.1 Farmers' Planning Problem
15.3.2 OutputBased and InputBased Subsidy Schemes
15.3.3 The Government's Subsidy Scheme
15.4 Equilibrium Analysis
15.4.1 InputBased Subsidy Scheme
15.4.1.1 The Farmers' Planting Decisions
15.4.1.2 The Government's InputBased Subsidy Level Decision
15.4.2 OutputBased Subsidy Scheme
15.4.2.1 The Farmer's Planting Decision
15.4.2.2 The Government's OutputBased Subsidy Level Decision
15.5 InputBased versus OutputBased Subsidy Schemes
15.6 Discussion
15.6.1 The Impact of Subsidy Schemes on the Aggregate Level Performance
15.6.2 A Combined Subsidy Scheme
15.6.2.1 The Farmers' Planting Decisions
15.6.2.2 The Government's Subsidy Level Decision
15.6.3 When the Yield Rate Is Uncertain
15.7 Conclusion
References
Springer Series in Supply Chain Management
Onur Boyabatlı Burak Kazaz Christopher S. Tang Editors
Agricultural Supply Chain Management Research Operations and Analytics in Planting, Selling, and Government Interventions
Springer Series in Supply Chain Management Volume 12
Series Editor Christopher S. Tang University of California Los Angeles, CA, USA
Supply Chain Management (SCM), long an integral part of Operations Management, focuses on all elements of creating a product or service, and delivering that product or service, at the optimal cost and within an optimal timeframe. It spans the movement and storage of raw materials, workinprocess inventory, and finished goods from point of origin to point of consumption. To facilitate physical flows in a timeefficient and costeffective manner, the scope of SCM includes technologyenabled information flows and financial flows. The Springer Series in Supply Chain Management, under the guidance of founding Series Editor Christopher S. Tang, covers research of either theoretical or empirical nature, in both authored and edited volumes from leading scholars and practitioners in the field – with a specific focus on topics within the scope of SCM. Springer and the Series Editor welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Ms. Jialin Yan, Associate Editor, Springer (Germany), email: [email protected]
More information about this series at http://www.springer.com/series/13081
Onur Boyabatlı • Burak Kazaz • Christopher S. Tang Editors
Agricultural Supply Chain Management Research Operations and Analytics in Planting, Selling, and Government Interventions
Editors Onur Boyabatlı Lee Kong Chian School of Business Singapore Management University Singapore
Burak Kazaz Whitman School of Management Syracuse University Syracuse, NY, USA
Christopher S. Tang Anderson School of Management University of California Los Angeles (UCLA) Los Angeles, CA, USA
ISSN 23656395 ISSN 23656409 (electronic) Springer Series in Supply Chain Management ISBN 9783030814229 ISBN 9783030814236 (eBook) https://doi.org/10.1007/9783030814236 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The world is facing a new global hunger crisis. Even before the COVID19 pandemic ravaged the world, IMF (International Monetary Fund) reported that geopolitical conflicts, climate shocks, and economic downturns have caused acute hunger among 135 million people worldwide in 2019. While the UN World Food Program continues to provide food to feed people with food insecurity, there is a global concern about food sufficiency due to increasing demand for food and decreasing supply of natural resources. Due to population growth, longevity resulting from better health care, and increasing consumption caused by economic development in developing countries such as China and India, the demand for Earth’s resources is expected to continue to grow. For instance, the development of infrastructure and housing to support population growth has depleted the amount of arable land for cultivation. According to the World Bank, the arable land per person has declined gradually over time. To produce more food to feed the world, agriculture has never been more important. Agriculture is an essential sector for all economies in terms of employment, food availability and safety, and domestic and international trade. For decades, most agricultural economists applied macroeconomic theory in decisions pertaining to the optimization of food production and distribution. However, a few researchers used microeconomic theory to examine how individual farmers respond to market information, incentive pricing mechanisms, and different market structures in the trade of agricultural goods. Examining challenges in agricultural supply chain operations through the lens of microeconomic theory is imperative because it can enable policymakers and social enterprises to develop and design market information provision policy, incentive contracts, and market structures for improving farmer and consumer welfare. By developing better policies, poor farmers can improve their earnings by growing more food for the world, making supply meet demand. In general, the enterprises in a traditional agricultural supply chain can be depicted in the following figure. The enterprises include growers who sell agricultural products to processors, who then sell these products distributors, eventually
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reaching consumers. The government provides information provision and policy support to all these entities in the agricultural supply chain. This book contains a collection of stateoftheart supply chain models that examine three essential elements of agricultural supply chains: planting and growing, processing and selling, and government interventions. This book is divided into three parts, each of which contains several chapters prepared by various contributing authors. These three parts can be described as follows: A. Planting and Growing The chapters in this part examine the following research questions: How should a farmer dynamically allocate planting of farmland among different crops when these crops have rotation benefits across growing seasons? What is the impact of farm yielddependent acquisition cost and selling price on an agricultural producer’s production planning decisions? How to obtain a socially optimal distribution of the available surface water to a farming community under uncertainty in rainfall? What are the production planning challenges faced by commercial seed firms in their effort to produce and sell seed corn to farmers who are located in the continental US? B. Processing and Selling As farmers harvest their crops, these crops are converted into multiple outputs in processors. In the context of agricultural industries, these processors face unique challenges, and as a response to these challenges, the chapters in this part intend to examine the following research questions: How should a processor manage its procurement using contract and spot markets when processing a single input leads to multiple outputs in proportions as commonly observed in agricultural industries? What is the role of yielddependent nature of trading costs for a processor in making pricing and production planning decisions under supply uncertainty? In the context of oilseeds industries, how to choose the right processing and storage capacities to invest when a processor periodically manages its input processing and output inventory while facing uncertainties both in input and output prices? In the context of wine industry, what is the use of advance selling as a form of operational flexibility to a winemaker in mitigating wine’s quality rating risk? How should a wine distributor allocate budget between bottled wine and wine futures in the presence of weather and market uncertainties?
A schematic view of an agricultural supply chain
Preface
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C. Government Interventions Because farmers in developing countries often receive limited education and have smaller means to invest in better seeds, fertilizers, and farming equipment, they are left to their own devices. Given limited resources, it is very difficult for them to break the poverty cycle. Therefore, it is important to examine the government’s role in alleviating farmer poverty. In this part, various chapters attempt to examine the role of different government interventions in the context of a variety of agricultural industries and answer the following research questions: Should the government offer help via market information? What should a government offer to improve farmer welfare? Input and/or output subsidies? Price subsidies? Other government policy interventions? Market information? Knowledge sharing and learning platforms? It is a privilege for us to have this opportunity to work with all contributing authors to compile this collection of recent research work on agricultural supply chains. We hope this book stimulates researchers and inspires policymakers to improve the whole sector to ensure food sustainability and food security for the world. Singapore Syracuse, NY, USA Los Angeles, CA, USA April 2021
Onur Boyabatlı Burak Kazaz Christopher S. Tang
Acknowledgments
Facing the challenges caused by the COVID19 pandemic, we would like to express our sincere appreciation to all contributing authors for their time, dedication, and effort to share their cuttingedge research with us. We are equally grateful to Dr. Mirko Janc for typesetting each page and each chapter beautifully and expeditiously. Of course, we are responsible for any errors that may occur in this book. Last but not least, this book is dedicated to all farmers around the world: their crops nourish our body and inspire our soul.
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Contents
Part I Planting and Growing 1
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Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Onur Boyabatlı, Javad Nasiry, and Yangfang (Helen) Zhou
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Agricultural Production Planning Under YieldDependent Cost and Price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burak Kazaz
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Mechanisms for Effective Sharing of Agricultural Water Between HeadReach and TailEnd Farms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Milind Dawande, Srinagesh Gavirneni, Mili Mehrotra, and Vijay Mookerjee Portfolio Management Issues in the Commercial Seed Industry: A Modeling Framework and Industry Implementation . . . . Saurabh Bansal and Mahesh Nagarajan
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Part II Processing and Selling 5
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Procurement Management in Agricultural Commodity Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Onur Boyabatlı and Bin Li
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The Influence of YieldDependent Trading Costs on Pricing and Production Planning Under Supply Uncertainty . . . . . . . . . . . . . . . . . . Burak Kazaz and Scott Webster
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Capacity Management in Agricultural Commodity Processing . . . . . . . 103 Onur Boyabatlı and Jason (Quang) Dang Nguyen
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A Prescriptive Model for Selling Wine Futures to Mitigate Quality Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Tim Noparumpa, Burak Kazaz, and Scott Webster xi
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Wine Analytics: Futures or Bottles?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Mert Hakan Hekimo˘glu, Burak Kazaz, and Scott Webster
Part III Government Interventions 10
Implications of Farmer Information Provision Policies: Heterogeneous Farmers and Market Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 159 ChenNan Liao, YingJu Chen, and Christopher S. Tang
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Agricultural Market Information: Economic Value and Provision Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Xiaoshuai Fan, YingJu Chen, and Christopher S. Tang
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Knowledge Sharing Among Smallholders in Developing Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Shihong Xiao, YingJu Chen, and Christopher S. Tang
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Policy Interventions for an AgricultureBased Malaria Medicine Supply Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Burak Kazaz, Scott Webster, and Prashant Yadav
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The Impact of Crop Minimum Support Price on Crop Production and Farmer Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Prashant Chintapalli and Christopher S. Tang
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Input vs. OutputBased Farm Subsidies in Developing Economies: Farmer Welfare and Income Inequality . . . . . . . . . . . . . . . . . . . 265 Christopher S. Tang, Yulan Wang, and Ming Zhao
Contributors
Saurabh Bansal Smeal College of Business, The Pennsylvania State University, University Park, PA, USA Onur Boyabatlı Lee Kong Chian School of Business, Singapore Management University, Singapore YingJu Chen School of Business and Management and School of Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong Prashant Chintapalli Indian Institute of Management Bangalore, Bengaluru, India Milind Dawande Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX, USA Xiaoshuai Fan Division of Information Systems & Management Engineering, School of Business, Southern University of Science and Technology, Shenzhen, China Srinagesh Gavirneni Johnson Graduate School of Management, Cornell University, Ithaca, NY, USA Mert Hakan Hekimo˘glu Lally School of Management, Rensselaer Polytechnic Institute, Troy, NY, USA Burak Kazaz Whitman School of Management, Syracuse University, Syracuse, NY, USA Bin Li Economics and Management School, Wuhan University, Wuhan, China ChenNan Liao College of Management, National Taiwan University, Taipei, Taiwan Mili Mehrotra Gies College of Business, University of Illinois UrbanaChampaign, Champaign, IL, USA
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Vijay Mookerjee Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX, USA Mahesh Nagarajan Sauder School of Business, University of British Columbia, Vancouver, BC, Canada Javad Nasiry Desautels Faculty of Management, McGill University, Montreal, QC, Canada Jason (Quang) Dang Nguyen Ivey School of Business, Western University, London, ON, Canada Tim Noparumpa Chulalongkorn Business School, Chulalongkorn University, Bangkok, Thailand Christopher S. Tang Anderson School of Management, University of California Los Angeles (UCLA), Los Angeles, CA, USA Yulan Wang Faculty of Business, The Hong Kong Polytechnic University, Hong Kong, Hong Kong Scott Webster W. P. Carey School of Business, Arizona State University, Tempe, AZ, USA Shihong Xiao The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Prashant Yadav Center for Global Development, Washington, DC, USA Technology and Operations Management, INSEAD, Fontainebleau, France Ming Zhao School of Economics and Management, University of Electronic Science and Technology of China, Chengdu, China Yangfang (Helen) Zhou Lee Kong Chian School of Business, Singapore Management University, Singapore
Part I
Planting and Growing
Chapter 1
Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits Onur Boyabatlı, Javad Nasiry, and Yangfang (Helen) Zhou
1.1 Introduction The objective of this chapter is twofold. First, to study how a farmer should dynamically allocate farmland between two crops when the crops have rotation benefits across growing seasons. Second, to develop a practically implementable heuristic allocation policy and examine its performance in comparison with other heuristic policies commonly suggested in the literature. The chapter is based on our companion paper (Boyabatlı et al. 2019). We focus on the same two crops as in our companion paper, corn and soybeans, the two most planted crops in the USA, which together account for 55.5% of total acres harvested in 2014 (USDA 2015a) with an estimated total market value of $92 billion in the same year (USDA 2015b). Corn is an important input for many agricultural and industrial products (such as animal feed and ethanol); soybeans are the main source of animal feed and second largest input of vegetable oil in the world. Both crops compete for the same farmland in the USA, as their growing time windows coincide (between late March and June). The farmland allocation of corn and soybeans exhibits the following two unique challenges. First, as shown in Fig. 1.1, the revenues for both crops are uncertain in each growing season. This is because both crop revenues depend on factors such as the harvest volume as well as sale price at the end of the growing season: Harvest volume is uncertain due to unpredictable weather, and onslaught
O. Boyabatlı () · Y. (Helen) Zhou Lee Kong Chian School of Business, Singapore Management University, Singapore email: [email protected]; [email protected] J. Nasiry Desautels Faculty of Management, McGill University, Montreal, QC, Canada email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_1
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Fig. 1.1 Yearly corn and soybean revenues (in $/acre) in Iowa from 1980 to 2013 based on data reported by USDA
of pests and diseases throughout the growing season (Kazaz and Webster 2011); sale price is uncertain as it is linked to the prevailing price at the regional exchange (spot) markets (Goel and Tanrısever 2017). Second, there are crop rotation benefits across growing seasons, that is, it is more profitable to grow a crop on rotated farmland (where the other crop was grown) than on nonrotated farmland (where the same crop was grown). As highlighted in Hennessy (2006), these benefits include increasing revenue—by improving the soil structure and breaking the pest cycles— as well as decreasing farming costs—by requiring less fertilizers (due to improved soil structure) and pesticides (due to lower pest populations). Specifically, rotation between soybeans and corn can increase the corn revenue due to soybeans fixing the nitrogen content of soil which is desirable for corn growth and decrease corn farming cost due to requiring applying less fertilizer (nitrogen). We consider a stochastic multiperiod model in which a farmer decides how to allocate farmland between corn and soybeans in each growing season (period) to maximize the total expected profit over a finite planning horizon. In each season these allocation decisions are made in the presence of revenue uncertainty of each crop as well as the crop rotation benefits. The base model in Boyabatlı et al. (2019) assumes that crop rotation benefits carry through for one growing season and, thus, the farmland allocation decision in each season is affected by only the allocation in the previous season and not the earlier seasons. In this chapter we focus on
1 Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits
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a generalized version of this model where the rotation benefits carry through for two seasons. We characterize the optimal dynamic allocation policy in this general model and show that it extends the structure of the optimal policy characterized in our companion paper. To develop a practically implementable heuristic allocation policy and examine its performance in comparison with other heuristic policies commonly suggested in the literature we focus on a special case of our model where the rotation benefit carries over for only one growing season. This is a reasonable assumption for modeling cornsoybeans rotation (Hennessy 2006), which is the focus of our numerical study that we use for comparative study of the heuristic policies. We propose a oneperiod lookahead policy, which we can characterize in closed form based on the optimal policy structure. We examine its performance with those commonly suggested heuristic policies in a numerical study with models calibrated to data obtained from United States Department of Agriculture (USDA) and extant resources. We show that our proposed oneperiod lookahead policy outperforms all other heuristic policies and provides a nearoptimal performance. This chapter contributes mainly to the literature on farm planning, particularly to that on farmland allocation, which has received considerable attention in both the operations management and agricultural economics fields. See, e.g., Glen (1987), Lowe and Preckel (2004) and Ahumada and Villalobos (2009) for a review of papers that study deterministic crop planning problems. The majority of papers in this stream that incorporate uncertainty use singleperiod models (where crop rotation benefits are irrelevant) and examine the interplay between the farmland allocation decision and other operational features, for example, the penalties associated with cash flow variability (Collender and Zilberman 1985), government price support for crops (Alizamir et al. 2019), rainfall uncertainty (Maatman et al. 2002), and irrigation management (Huh and Lall 2013). As emphasized by Lowe and Preckel (2004), crop rotation benefits across growing periods are an integral part of the farmland allocation problem. Only a few papers consider these benefits and study the dynamic farmland allocation problem under uncertainty, and they mostly suggest heuristic allocation policies and evaluate their performance numerically. Among these, Taylor and Burt (1984) build a stochastic dynamic program for a farmer to decide when to grow wheat or lay the farmland fallow. Based on this formulation, they develop a heuristic allocation policy and numerically study its performance using a calibration representing a typical wheat farmer in Montana. Cai et al. (2013) also use a stochastic dynamic programming formulation but focus on a farmer’s allocation decision between corn and soybeans. They numerically compare the performance of different heuristic allocation policies and show that growing corn in the entire farmland provides the best performance. Different from these two papers, we consider the case where farmers face significant uncertainty for their crop revenues in a growing period and crop rotation benefits across growing periods. This is the first work that characterizes the optimal dynamic allocation policy under uncertainty in the presence of multiyear crop rotation benefits. The paper closest to our work is Livingston et al. (2015), who study the farmland allocation between corn and soybeans in a multiperiod framework. Other than considering crop allocation in a stochastic dynamic program in each period as we do,
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they also consider fertilizer application rate and hence fertilizer cost uncertainty. As we focus on farmland allocation, we do not consider fertilizer application decision nor the farming cost uncertainty. However, they do not analytically characterize the optimal solution and resort to a numerical study to compute the farmer’s optimal actions. Calibrating their model to a typical farmer in Iowa, they suggest that the farmer should implement a rotationbased allocation policy. The remainder of this chapter is organized as follows. We describe the model in Sect. 1.2 and characterize the optimal allocation policy in Sect. 1.3. We discuss the heuristic policy that we propose and its benchmark heuristics in Sect. 1.4. We examine the performance of all heuristics in comparison with the optimal policy in a numerical study in Sect. 1.5. We discuss future work in Sect. 1.6.
1.2 Model We consider a farmer who allocates a single acre of farmland among two cash crops (corn and soybeans) in each growing season to maximize the expected total profit over a finite number of growing seasons, denoted by T . However, different from the base model in Sect. 1.3 in Boyabatlı et al. (2019), we consider a twoseason (rather than oneseason) crop history, that is, the farmland allocation decision in each growing season is affected by the allocations in the previous two seasons. Decision Variables To capture this feature of a twoseason crop history, we define a set of decision variables which differ from the base model in Boyabatlı et al. i,j i,j (2019). In particular, let αt ∈ [0, 1] and βt ∈ [0, 1] for i, j ∈ {c, s} denote the proportion of farmland allocated to corn and soybeans, respectively, in time period (growing season) t on which crop i was grown in period t − 2 and crop j was grown i,j i,j in period t − 1. By definition, i j (αt + βt ) = 1. For notational convenience, . . c,c s,c c,s s,s we define α t = αt , αt , αt , αt and β t = βtc,c , βts,c , βtc,s , βts,s to denote the corn and soybean allocations in period t, respectively. Revenue Uncertainties The allocation decisions α t and β t are made in the presence of revenue uncertainty of each crop. We denote by r˜tc and r˜ts , respectively, the uncertain corn and soybean revenue per acre in period t. We assume that r˜ t = (˜rtc , r˜ts ) follow correlated stochastic processes with Markovian property, that is, the distribution of future revenues can be fully characterized by the revenue realizations in the current period. Crop Rotation Benefits As discussed in Sect. 1.1, the crop rotation benefits are attributed to increasing crop revenues and decreasing farming costs. Because there is a twoseason crop history, the rotation benefits are defined based on what was grown on the farmland in the last two periods. To capture the revenueenhancing crop rotation benefits, we assume that the uncertain revenue per acre of crop j ∈ {c, s} in period t is
1 Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits
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j
i. r˜t if it is grown on farmland where the same crop j was grown in the previous two periods, j j ii. (1 + b1 )˜rt if it is grown on farmland where the same crop j was grown in the previous period and the other crop (−j ) was grown in period t − 2, and j j iii. (1 + b2 )˜rt if it is grown on farmland where the other crop (−j ) was grown in the previous period and the same crop j was grown in period t − 2, and j j iv. (1 + b3 )˜rt if it is grown on farmland where the other crop (−j ) was grown in the previous two periods. Here, case i. represents the nonrotated farmland (where the same crop was grown in the previous two seasons), whereas the other three cases represent the rotated farmland (where the other crop was grown in any of the previous two seasons). Consistent with practice, we assume that revenueenhancing rotation benefit based on the period t − 1 allocation is larger than the benefit based on the period t − 2 j j allocation, that is, b2 > b1 > 0, and the revenueenhancing crop rotation benefit is the largest on the farmland where the other crop was grown in the previous two j j seasons, that is, b3 > b2 . To capture the costreducing crop rotation benefits, we assume that the unit farming cost of crop j is i. ωj if it is grown on farmland where the same crop j was grown in the previous two periods, j ii. (1 − γ1 )ωj if it is grown on farmland where the same crop j was grown in the previous period and the other crop (−j ) was grown in period t − 2, and j iii. (1 − γ2 )ωj if it is grown on farmland where the other crop (−j ) was grown in the previous period and the same crop j was grown in period t − 2, and j iv. (1 − γ3 )ωj if it is grown on rotated farmland where a fallow crop was grown in the previous period. j
j
j
Similarly, we assume γ3 > γ2 > γ1 > 0. The special case where the crop rotation history is only one year, as considered in Boyabatlı et al. (2019), can be captured by j j j j j j setting b1 = 0, b2 = b3 = bj , γ1 = 0, and γ2 = γ3 = γ j for j ∈ {c, s}. Formulation We formulate the farmer’s problem as a finite horizon stochastic dynamic program. In each period t ∈ [1, T ], the sequence of events is as follows: (i) At the beginning t, the farmer observes the corn allocations c,c ofs,cperiod c,s s,s α t−1 = , the soybean allocations β t−1 = αt−1 , αt−1 , αt−1 , αt−1 c,c s,c c,s s,s c , rs ) βt−1 , βt−1 , βt−1 , βt−1 , and corn and soybean revenues r t−1 = (rt−1 t−1 i,j
from period t − 1. The farmer then chooses the corn allocations αt and the i,j i,j i,j soybean allocations βt , where i j (αt + βt ) = 1, constrained by the available farmland where crop i was grown in period t −2 and crop j was grown c,s s,s s,s c,s in period t−1, that is, (i) βts,s +αts,s = βt−1 +βt−1 , (ii) βts,c +αts,c = αt−1 +αt−1 , c,s c,s s,c c,c c,c c,c s,c c,c (iii) βt +αt = βt−1 +βt−1 , and (iv) βt +αt = αt−1 +αt−1 . For example, the first constraint ensures that the sum of the proportion of farmland allocated to corn and soybeans in this period where soybeans were grown in the previous
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two periods, that is, αts,s + βts,s , equals the actual proportion of farmland where soybeans were grown in the previous two periods, which is given by c,s s,s βt−1 + βt−1 . (ii) At the end of period t, the corn and soybean revenues r˜ t = (˜rtc , r˜ts ) are realized and the farmer collects the revenues from the crop sales. The farmer’s immediate payoff in period t ∈ [1, T ] is given by L(α t , β t  α t−1 , β t−1 , rt−1 ) = αtc,c + (1 + b1c )αts,c + (1 + b2c )αtc,s + (1 + b3c )αts,s Et [˜rtc ] + βts,s + (1 + b1s )βtc,s + (1 + b2s )βts,c + (1 + b3s )βtc,c Et [˜rts ] − αtc,c + (1 − γ1c )αts,c + (1 − γ2c )αtc,s + (1 − γ3c )αts,s ωc − βts,s + (1 − γ1s )βtc,s + (1 − γ2s )βts,c + (1 − γ3s )βtc,c ωs ,
(1.1)
where Et [.] denotes the expectation operator conditional on the available information at time t, that is, Et [.] = E[.r t−1 ]. In Eq. 1.1, the first line of the righthand side corresponds to the total expected revenue from growing corn in period t. It is the sum of expected revenue from growing corn on four different farmlands: farmland where corn was grown in the last two periods (which does not experience any revenueenhancing rotation benefit), farmland where soybeans were grown in period t − 2 and corn was grown in the previous period, farmland where corn was grown in period t − 2 and soybeans were grown in the previous period, and farmland where soybeans were grown in the last two periods (which experiences the largest revenueenhancing rotation benefit). The second line of the righthand side in Eq. 1.1 denotes the total expected revenue from growing soybeans in period t in a similar fashion. The last two lines of the righthand side in Eq. 1.1 denote the total farming cost incurred from growing corn and soybeans in period t, respectively. These expressions capture the costreducing rotation benefit experienced for each crop based on the four different farmlands. Let Vt (α t−1 , β t−1 , r t−1 ) for t ∈ [1, T ] denote the optimal value function from period t onward given α t−1 , β t−1 , and r t−1 , which satisfies Vt (α t−1 , β t−1 , r t−1 ) = max L(α t , β t  α t−1 , β t−1 , r t−1 ) + Et Vt+1 (α t , β t , r˜ t ) α t ,β t
c,s s,s s,s c,s + βt−1 , βts,c + αts,c = αt−1 + αt−1 , s.t. βts,s + αts,s = βt−1 s,c c,c s,c c,c βtc,s + αtc,s = βt−1 + βt−1 , βtc,c + αtc,c = αt−1 + αt−1 , i,j i,j i,j i,j 0 ≤ αt ≤1, 0 ≤ βt ≤ 1, (αt + βt ) = 1,
(1.2)
i∈{c,s} j ∈{c,s}
with a boundary condition VT +1 (·) ≡ 0. The farmer’s optimal total expected profit over the entire planning horizon is given by V1 (α 0 , β 0 , r 0 ), where α 0 , β 0 , and r 0
1 Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits
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denote the observed corn allocations, soybean allocations, and crop revenues at the beginning of the planning horizon, respectively.
1.3 Optimal Allocation Policy We now solve for the farmer’s optimization problem stated in Eq. 1.2 and characterize the optimal allocation decisions and the optimal value function in period t ∈ [1, T ]. For this purpose, we first define the following recursive operators: . c,c Ktc,c (rt−1 ) = max − ωc + Et [˜rtc + Kt+1 (˜rt )],
c,s (˜rt )] , − (1 − γ3s )ωs + Et [(1 + b3s )˜rts + Kt+1 . c,c (˜rt )], Kts,c (rt−1 ) = max − (1 − γ1c )ωc + Et [(1 + b1c )˜rtc + Kt+1 c,s (˜rt )] , − (1 − γ2s )ωs + Et [(1 + b2s )˜rts + Kt+1 . s,c (˜rt )], Ktc,s (rt−1 ) = max − (1 − γ2c )ωc + Et [(1 + b2c )˜rtc + Kt+1 s,s (˜rt )] , − (1 − γ1s )ωs + Et [(1 + b1s )˜rts + Kt+1 . s,c (˜rt )], Kts,s (rt−1 ) = max − (1 − γ3c )ωc + Et [(1 + b3c )˜rtc + Kt+1 s,s − ωs + Et [˜rts + Kt+1 (˜rt )] ,
i,j
i,j
with KT +1 (r T ) = 0 for i, j ∈ {c, s}. Here, Kt (r t−1 ) denotes the expected marginal profit of farmland in the remaining planning horizon (from period t onward) where crop i was grown in period t − 2 and crop j was grown in period t − 1. Consider, for example, Ktc,c (r t−1 ). It is given by the maximum profit from two options: (1) growing corn in period t and optimally using the farmland in c,c the remaining periods (which yields the expected marginal profit Et [Kt+1 (r˜ t )]) and (2) growing soybeans in period t and optimally using the farmland in the c,s remaining periods (which yields the expected marginal profit Et [Kt+1 (r˜ t )]). For the first option, because corn was grown in the previous two periods, growing corn in this period does not accrue any rotation benefits. For the second option, because corn was grown in the previous two periods, growing soybeans in this period accrues the largest rotation benefits, captured by b3s and γ3s . For notational convenience, we define . c,c (r˜ t )], Ct(0) = −ωc + Et [˜rtc + Kt+1 (1)
Ct
. c,c = −(1 − γ1c )ωc + Et [(1 + b1c )˜rtc + Kt+1 (r˜ t )],
. s,c Ct(2) = −(1 − γ2c )ωc + Et [(1 + b2c )˜rtc + Kt+1 (r˜ t )], (3)
Ct
. s,c = −(1 − γ3c )ωc + Et [(1 + b3c )˜rtc + Kt+1 (r˜ t )],
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. s,s = −ωs + Et [˜rts + Kt+1 (r˜ t )],
(1)
. s,s = −(1 − γ1s )ωs + Et [(1 + b1s )˜rts + Kt+1 (r˜ t )],
St St
. c,s St(2) = −(1 − γ2s )ωs + Et [(1 + b2s )˜rts + Kt+1 (r˜ t )], (3)
St (0)
(1)
. c,s = −(1 − γ3s )ωs + Et [(1 + b3s )˜rts + Kt+1 (r˜ t )], (2)
(3)
(0)
(1)
(2)
(3)
where Ct < Ct , Ct < Ct , St < St , and St < St Using this notation, the recursive operators can be rewritten as: (0)
(3)
(1)
(2)
(2)
(1)
by construction.
Ktc,c (r t−1 ) = max{Ct , St }, Kts,c (r t−1 ) = max{Ct , St },
(1.3)
Ktc,s (r t−1 ) = max{Ct , St }, Kts,s (r t−1 ) = max{Ct(3) , St(0) }. Let (ˆ·) be associated with the optimal decision.
Proposition 1 In period t ∈ [1, T ], the optimal allocation is characterized by the following cases where, in each case, the unlisted decision variables equal 0: (0)
(2)
c,c s,c (i) If Ktc,c (rt−1 ) = Ct and Ktc,s (rt−1 ) = Ct , then
αtc,c = αt−1 + αt−1 , c,s c,c s,c s,c c,s s,s s,s s,s c,s
αt = βt−1 + βt−1 ,
αt = αt−1 + αt−1 , and
αt = βt−1 + βt−1 . (2) (0)
ts,s = β s,s + β c,s , β
s,c (ii) If Kts,c (rt−1 ) = St and Kts,s (rt−1 ) = St , then β t−1 t−1 t =
tc,s = β c,c + β s,c , and β
tc,c = α s,c + α c,c . α c,s + α s,s , β t−1
t−1
(2)
t−1
t−1
(2)
t−1
t−1
s,c c,c αtc,s = βt−1 + βt−1 , (iii) If Ktc,s (rt−1 ) = Ct and Kts,c (rt−1 ) = St , then
s,s c,s s,s
tc,c = α s,c + α c,c , β
ts,c = α c,s + α s,s , and
β α = β + β . t t−1 t−1 t−1 t−1 t−1 t−1 (0)
(0)
s,c c,c αtc,c = αt−1 + αt−1 , (iv) If Ktc,c (rt−1 ) = Ct and Kts,s (rt−1 ) = St , then
c,s s,s c,s s,c c,c
ts,s = β c,s + β s,s .
αts,c = αt−1 + αt−1 , βt = βt−1 + βt−1 , and β t−1 t−1 (0)
(1)
(3)
αtc,c = (v) If Ktc,c (rt−1 ) = Ct , Ktc,s (rt−1 ) = St , and Kts,s (rt−1 ) = Ct , then
s,c c,c s,c c,s s,s c,s s,c c,c s,s c,s s,s αt−1 +αt−1 ,
αt = αt−1 +αt−1 , βt = βt−1 +βt−1 , and
αt = βt−1 +βt−1 . (3) (1) (0) c,c s,c s,s c,c
(vi) If Kt (rt−1 ) = St , Kt (rt−1 ) = Ct , and Kt (rt−1 ) = St , then βt = s,c c,c c,s s,c c,c c,s s,s
ts,s = β c,s +β s,s . αt−1 +αt−1 , βt = βt−1 +βt−1 ,
αts,c = αt−1 +αt−1 , and β t−1 t−1 (3)
(1)
(1)
(vii) If Ktc,c (rt−1 ) = St , Ktc,s (rt−1 ) = St , Kts,c (rt−1 ) = Ct , and (3)
tc,c = α s,c + α c,c , β
tc,s = β s,c + β c,c , Kts,s (rt−1 ) = Ct , then β t−1 t−1 t−1 t−1 s,c c,s s,s s,s c,s s,s
αt = αt−1 + αt−1 , and
αt = βt−1 + βt−1 . (2) (1) (3)
tc,c = (viii) If Kts,c (rt−1 ) = St , Ktc,s (rt−1 ) = St , and Kts,s (rt−1 ) = Ct , then β s,c c,c c,s s,c c,c s,c c,s s,s s,s c,s s,s αt−1 +αt−1 , βt = βt−1 +βt−1 , βt = αt−1 +αt−1 , and
αt = βt−1 +βt−1 . (3) (1) (2) c,c s,c c,s c,c
(ix) If Kt (rt−1 ) = St , Kt (rt−1 ) = Ct , and Kt (rt−1 ) = Ct , then βt = s,c c,c s,c c,c c,s s,s c,s s,s αt−1 +αt−1 ,
αtc,s = βt−1 +βt−1 ,
αts,c = αt−1 +αt−1 , and
αts,s = βt−1 +βt−1 .
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The intuition behind Proposition 1 is similar to the intuition behind Prop. 8 of Boyabatlı et al. (2019). In particular, the optimal allocation decisions are characterized based on which of the two options, growing corn or soybeans in period t (and optimally using the farmland in the remaining periods), is the most profitable option on the farmland where crop j was grown in the previous period and crop i was grown in period t − 2, as captured by the recursive operators i,j Kt (r t−1 ) for i, j ∈ {c, s} given in Eq. 1.3. For example, consider the first case presented in Proposition 1. When Ktc,c (rt−1 ) = Ct(0) and Ktc,s (rt−1 ) = Ct(2) , (0) (1) (2) (3) (0) (1) (2) (3) because Ct < Ct , Ct < Ct , St < St , and St < St by definition, (1) (3) s,c s,s we have Kt (r t−1 ) = Ct and Kt (r t−1 ) = Ct . In other words, growing corn is the most profitable option regardless of which crop was grown in the previous two periods. Therefore, the whole farmland is optimally allocated to corn, that is, i,j c,c s,c c,c s,c c,s s,s αt = 1 where
αtc,c = αt−1 +αt−1 ,
αtc,s = βt−1 +βt−1 ,
αts,c = αt−1 +αt−1 , i j
i,j s,s s,s c,s
+β (and, thus, βt = 0 for i, j ∈ {c, s}). Consider another and
αt = β t−1
t−1
example: what is the optimal allocation when Ktc,c (rt−1 ) = St(3) , Ktc,s (rt−1 ) = St(1) , (1) (3) Kts,c (rt−1 ) = Ct , and Kts,s (rt−1 ) = Ct (case vii in Proposition 1)? In this case, growing corn is the most profitable option on the farmland where soybeans were grown in period t − 2 regardless of the crop grown in period t − 1, while growing soybeans is the most profitable option on the farmland where corn was grown in period t −2 regardless of the crop grown t −1. Therefore, the total optimal in period i,s i,s corn allocation in period t is given by i (αt−1 + βt−1 ), whereas the total optimal i,c i,c soybean allocation in period t is given by i (αt−1 + βt−1 ). The other cases are characterized in a similar fashion. When the crop history is only one year, as discussed in Boyabatlı et al. (2019), the optimal allocation policy has a simple structure: when the farmer optimally chooses to grow both crops each crop is only grown on the farmland where the other crop was grown in the previous year. When the crop history is two years the optimal allocation policy has a more complex structure. In particular, when the farmer optimally chooses to grow both crops it is not necessary that each crop is only grown on the farmland where the other crop was grown in the previous two years (see, for example, case vii in Proposition 1). The characterization of the optimal total expected profit from period t onward follows the same structure as the characterization in Prop. 8 of Boyabatlı et al. (2019): Proposition 2 The optimal value function from period t onward is given by Vt (α t−1 , β t−1 , r t−1 ) c,j c,j s,j j,c s,j j,s (αt−1 + αt−1 )Kt (rt−1 ) + (βt−1 + βt−1 )Kt (rt−1 ) , = j ∈{c,s} i,j
j ∈{c,s}
where Kt (r t−1 ) for i, j ∈ {c, s} is given by Eq. 1.3.
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The optimal total expected profit from period t onward is given by the product of the proportion of farmland allocated to each crop in the previous period where crop c,j s,j c,j s,j j was grown in period t − 2 (αt−1 + αt−1 for corn and βt−1 + βt−1 for soybeans) j,c
and its corresponding expected marginal profit as given in Eq. 1.3 (Kt (r t−1 ) for j,s corn and Kt (r t−1 ) for soybeans). When the crop history is only one year, that j j j j j j is, b1 = 0, b2 = b3 = bj , γ1 = 0, and γ2 = γ3 = γ j for j ∈ {c, s}, it c,c s,c c,s follows that Kt (r t−1 ) = Kt (r t−1 ) and Kt (r t−1 ) = Kts,s (r t−1 ). In this case, c,j s,j denoting corn allocation in period t − 1 with αt−1 = j ∈{c,s} ((αt−1 + αt−1 )) c,j s,j and soybean allocation in period t − 1 with βt−1 = j ∈{c,s} ((βt−1 + βt−1 )), and using βt−1 = 1 − αt−1 , it is easy to establish that the optimal value function above is identical to the optimal value function presented in Prop. 8 of Boyabatlı et al. (2019).
1.4 Heuristics As seen in Proposition 1, though the optimal policy has a neat structure, it is still complex and may not be suitable for implementation in practice. Therefore, we discuss heuristic policies that are easier to use. From this section onward we focus on a special case of our model where the rotation benefits carry through for only one growing season. This is consistent with the base model in Sect. 3 in Boyabatlı et al. (2019). Moreover, a oneseason crop rotation history is a reasonable assumption for modeling cornsoybeans rotation (Hennessy 2006), which is the focus of our numerical study in the next section. We first leverage the optimal structure in Proposition 1 to design a oneperiod lookahead policy, which we analytically characterize in closed form in Sect. 1.4.1. In Sect. 1.4.2 we discuss benchmark heuristics that are commonly used in practice and literature.
1.4.1 Proposed Policy: OnePeriod Lookahead This policy is based on the optimal policy structure in Proposition 1, and its allocation in period t is obtained by maximizing the sum of the expected profits of this period and period t + 1. In order to characterize this oneperiod lookahead policy in closed form, we impose additional structure on the crop revenue processes and model their evolution using a singlefactor bivariate meanreverting process. Specifically, corn and soybean revenues at time τ , r τ = (rτc , rτs ), evolve as follows:
1 Dynamic Crop Allocation in the Presence of TwoSeason Crop Rotation Benefits
drτc = κ c (ξ c − rτc )dτ + σ c d W˜ τc ,
13
(1.4)
drτs = κ s (ξ s − rτs )dτ + σ s d W˜ τs ,
where κ j > 0, ξ j , and σ j > 0 for j ∈ {c, s} represent the meanreversion rate, meanreverting level, and volatility, respectively; (d W˜ τc , d W˜ τs ) denote the increment of a standard bivariate Brownian motion with correlation ρ > 0, which we empirically verify for corn and soybean in Sect. 1.5. This continuous time revenue process gives the revenue at the discrete decision time epochs (i.e., when farming decisions are made) conditional on the revenue realization in the previous period. c , r s ), r˜ = (˜ Specifically, given r t−1 = (rt−1 rtc , r˜ts ) evolve as t t−1 c r˜tc = e−κ rt−1 + (1 − e−κ )ξ c + σ c c
c
s r˜ts = e−κ rt−1 + (1 − e−κ )ξ s + σ s s
s
1 − e−2κ c z˜ , 2κ c c
s 1 − e−2κ
2κ s
(1.5)
z˜ s ,
where (˜zc , z˜ s ) follow a standard bivariate Normal distribution with correlation ρ. That is, E[˜rt r t−1 ] = e−κ rt−1 + (1 − e−κ )ξ j , j
j
j
j
1 − e−2κ = (σ j )2 , 2κ j j
j VAR[˜rt r t−1 ]
1 − e−(κ +κ ) c s ρσ σ , κc + κs c
COV[˜rtc , r˜ts r t−1 ] =
(1.6)
s
where VAR and COV denote variance and covariance, respectively. With this characterization, we next show an identity in the following proposition, which is then used to analytically characterize the oneperiod lookahead policy. Proposition 3 Under the bivariate Normal distribution specified in Eq. 1.6,
j (−j ) uμ1 + v − uμ1 − v + v}] = + v) λ
(−j ) j uμ1 + v − uμ1 − v (−j ) + (uμ1 + v) λ
(−j ) j uμ1 + v − uμ1 − v , (1.7) + λφ λ
j E1 [max{u˜r1
(−j ) + v, u˜r1
j (uμ1
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where (·) and φ(·) denote the cumulative distribution function and the prob. ability density function of the standard Normal distribution, respectively, λ = j (−j ) j . j j . j u2 VAR1 +u2 VAR1 −2uu COV1 , and μ1 = E[˜r1 r 0 ], VAR1 = VAR[˜r1 r 0 ], . c s and COV1 = COV[˜r1 , r˜1 r 0 ] follow from Eq. 1.6 with t = 1. Finally, we obtain the allocation of the oneperiod lookahead policy in period t ∈ [1, T ] from Eq. 5 of Proposition 2 in our companion paper (which as mentioned is a special case of Proposition 1 in this chapter) by substituting t = 1 (t − 1 = 0, t + 1 = 2) with an arbitrary t (t − 1, t + 1), with the identity in Proposition 3.
1.4.2 Benchmarks We choose the following benchmarks, which are easily implementable in practice, as all of them have a closedform characterization of their periodic allocation decisions. Always Rotate The farmer grows crops only on rotated farmland in each period. Always Rotate (Monoculture) The farmer grows only one of the crops in each period and rotates to the other in the next period where the crop in the first period is determined by maximizing the total expected profits. Myopic The allocation in each period is determined by maximizing the expected profits only in the current period, that is, ignoring all future cash flows. We measure the performance of any heuristic H by its percentage profit loss . H ∗ 100: H = 1 − V1H (α0 , r 0 )/V1 (α0 , r 0 ), where V1 (α0 , r 0 ) and V1H (α0 , r 0 ) denote the total expected profit over the planning horizon under the optimal and heuristic allocation policy, respectively.
1.5 Numerical Study We examine the performance of all heuristics in a numerical study. Instead of using a latticebased approach (as in Sect. 5 of Boyabatlı et al. 2019), we use a simulation approach. We first discuss in Sect. 1.5.1 the data calibration, in Sect. 1.5.2 the simulation setup, and in Sect. 1.5.3 the numerical results.
1.5.1 Calibration Our numerical experiments use publicly available data from USDA complemented by data reported in other academic studies.
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Calibration for Revenue Process Parameters We estimate the yearly revenue ($/acre) of each crop as the product of its yield (bushel/acre) and sales price ($/bushel), where we obtain yield data and sales prices for each crop as reported by USDA for farmlands in Iowa. We adjust the obtained crop revenue data based on the consumer price index from United States Department of Labor. We then remove from these crop revenues the effect of rotational benefits to obtain crop revenue without these benefits. We use the resulting corn and soybean revenue to calibrate the discretetime (yearly) version of the singlefactor bivariate meanreverting process as specified in Eq. 1.5. We use the seemingly unrelated regression (Zellner 1962) to estimate κ j , ξ j , and σ j (for j {c, s}) as the error terms are correlated. Calibration for Other Operational Parameters We use the data presented in Iowa State University extension and outreach report1 to calibrate the (variable) farming cost ωj and the costreducing crop rotation benefit γ j for crop j ∈ {c, s}. We also adjust these (nominal) data based on the consumer price index. Because there is no costreducing rotation benefit of soybean reported in the literature, we assume the variable cost for the farmer growing soybean after soybean is the same as the variable cost for the farmer growing soybean following corn. To estimate the farming cost ωc and ωs , we take the average of the cost data for corn following corn and soybean following soybean. To estimate the costreducing crop rotation benefit for corn, we compare the average cost for corn following corn and corn following soybean. We estimate α0 by using the fraction of the total farmland allocated to corn in Iowa in 2014.
1.5.2 Simulation Setup For numerical computation, we follow the standard procedure in the literature and discretize the continuous revenue process in Eq. 1.4 to a lattice (see Sect. 5.2.1 in our companion paper for the details). To compute the relevant performance measures we first generate 10,000 sample paths of corn and soybean revenue pairs in Eq. 1.4. On each sample path we map the realized corn and soybean revenues to the nearest node on the lattice and use the parameters from that node for computation. Using this approach, we compute the relevant performance measures along each sample path and average across all sample paths. We compare the expected profits under any two policies (heuristic or optimal policy). As standard in the literature, we use the same 10,000 sample paths for computing the expected profit under each policy.2 After taking the difference between the profits under these two policies on each sample path, we then use a 1 http://www.extension.iastate.edu/agdm/cdcostsreturns.html. 2 Using the same sample paths is a variance reduction technique called Common Random Numbers,
and it is standard in simulation in comparing the performance of two systems.
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onesample twotailed Student ttest with the null hypothesis that the difference between the expected profits under these two policies is zero. Paralleling the analysis in Devalkar et al. (2011), the simulation analysis is carried out around the baseline scenario while considering different planning horizons, where Tˆ ∈ {5, 10, 15, 20, 30, 40, 50, 100}. In summary, the simulation analysis is carried out in our baseline scenario by considering eight different planning horizons.
1.5.3 Results The main finding of Boyabatlı et al. (2019) (as reported in Sect. 5.2.4) is that the oneperiod lookahead policy not only outperforms all the heuristic policies considered but also provides a nearoptimal performance. We now examine whether this finding continues to hold in our simulation analysis. Does the OnePeriod Lookahead Policy Outperform All the Heuristic Policies Considered? We examine whether the difference between the profit under the oneperiod lookahead policy and the profit under each of the other heuristic policies is statistically significant. Table 1.1 summarizes the comparison between the oneperiod lookahead policy and the myopic policy. In particular, the mean and the standard deviation of the expected profit under each heuristic obtained from 10,000 sample paths as well as the corresponding pvalue from the Student ttest are reported for different planning horizons. We observe that, for each planning horizon, the mean expected profit is higher under the oneperiod lookahead policy and the profit difference is statistically significant at a significance level of 1% (we reject the null hypothesis with a significance level of 1%—that is, a confidence level of 99%—and accept the alternative hypothesis that the difference between the expected profits under these two policies is not zero). We conduct a similar comparative analysis between the oneperiod lookahead policy and the other two heuristic policies. Table 1.2 summarizes the comparison with the always rotate policy and
Table 1.1 Oneperiod lookahead policy versus myopic policy Horizon T Oneperiod lookahead policy Myopic policy (in years) Expected profits Standard deviation Expected profits Standard deviation pvalue 5 1287.4 394.19 1282.1 392.12 0, or alternately, when Yield > Demand/NA Acres. Furthermore, this threshold of yield depends on the NA decision itself. We handle these issues as follows. We expand the objective function in the NA problem and specify the expected profit in two partitions of NA yield—one
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in which the yield is high and the SA production is not required, and the second one in which the SA production is required and follows the linear policy. We also introduce a constraint that transforms the sequential formulation into a simultaneous formulation. After this step, the expected profit from the NA acreage has an analytically tractable expression. This expression is continuous and differentiable (for continuous yield distributions) and can be optimized using any gradient based routine, such as the one in Excel Solver. This analytical expression for the problem for one hybrid then lends to an aggregation to the portfolio level problem. To this end, we first sum the expressions of expected profit from all hybrids at NA acreages. We then reimpose the parent seed constraints that are functions of these acreages. The summation operations in the objective function and the parent seed availability constraints are linear operations, and therefore preserve the tractability of the single hybrid problem. This complete formulation can be solved using standard optimization routines. Finally, we discuss the extension of this solution to the case when the demand for a hybrid is uncertain and is represented with a continuous probability distribution. The linear policy for SA production is not optimal for uncertain demand in general, nevertheless, it is naturally appealing to use in practice. The optimal linear policy for the uncertain demand case is obtained in two steps. First, using the quadrature one would discretize the demand distribution into three or more scenarios and their corresponding probability weights as prescribed by the quadrature results. In our numerical experimentation we have found three scenarios to perform well. Second, one would modify the linear policy to the form SA Acres = a − b × (NA Acres × NA Yield) where the parameters a and b depend on the hybrid seed parameters including yield distribution as well as demand distribution. This solution is integrated in the NA problem as before, obtaining a policy to implement in SA.
4.3.3 Mathematical Properties of Solution A major strength of the solution approach is that it is easy to implement in standard optimization environments such as Excel and can support real time decision making. The tractability of the SA solution (the linear policy) also lends to several desirable properties. A major reason for firms to adopt sequential production strategy was that it would allow them to start with a lower acreage in NA (as compared to the case of single production when a hybrid could be produced only once). At first blush it would appear that this reduced NA usage should reduce the firm’s overall production cost from the combined NA and SA production. However, the SA production is nearly 50% more expensive and as such it is not clear if a firm’s total production cost in sequential production will be lower. One can show using some lattice results that provided the profit margin on a hybrid is sufficiently high, on average the production
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cost is indeed lower than in single production. From a managerial perspective, this result provides the insight that while sequential production always results in an increase in expected profit, the average cost of production may increase or decrease based on the profit margin of a product. The linear policy also has some analytically desirable properties when the demand is uncertain. We discuss three such properties here. The proofs for these properties are available from the authors. The first property is that on average, the firm’s profit from using the linear policy will be at least 50% of the amount it would make when following the true optimal SA policy. This result is independent of problem parameters such as cost and revenue values and demand and yield distributions. Second, the linear policy converges to the optimal policy uniformly (independent of parameter values), and thirdly, the linear policy converges to the optimal policy as service levels increase. Overall, these results provide a validation for a use of the linear policy in practical decision making.
4.4 Implementation and Benefits We implemented the optimization protocol described above at Corteva between 2014–2015. We next describe this implementation as well as the benefits experienced at the firm.
4.4.1 Implementation We implemented the theory developed in the form of two modules. The first module was spreadsheet based and was hosted on cloud. The spreadsheet contained a list of all potential hybrids that could be offered in the market during the coming year. It also contained information of which hybrids were targeted towards the same market; the firm would need to choose one of these hybrids. The cloud location facilitated a collection of the parameter values for these hybrids. Typically, the R&D and field production would provide estimates of yield distributions for the hybrids and the availability of parent seeds and production cost, respectively; the marketing team would provide aggregated estimates of demand; and the sales team would provide the price (and price elasticity information where applicable). Many of these functions were dispersed geographically, and the cloudbased module provided a convenient mode for information collection. This information was then pulled into a second module. This module housed the portfolio optimization routine based on the solution described earlier. This module was used to support real time decision support at the firm. During the annual crop planning meeting, this module would be first used to generate the optimal acreage plan for NA and SA based on the availability of parent seeds. Subsequently, managers would do a whatif analysis to understand the implications of deviating
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from the normative plan. Many times changes were considered to the prescribed plan to account for strategic factors, e.g., increasing the availability of hybrid seeds in a specific market where the firm had experienced a decline in market share recently; or remove certain hybrids from the portfolio in order to contain product proliferation. For such instances, the module provided a direct estimate of the cost (or equivalently loss of profit) from the changes. The final plan provided information to various functions of the firm: It provided an estimate of the earnings and cost from the seed corn business to the finance arm; The R&D function obtained an estimate for which parents were being used in the plan—this information was important in order to maintain a genetic diversity in the hybrids offered in the market; the marketing function obtained an estimate of the likely service levels in the field during the next year’s selling cycle.
4.4.2 Benefits The firm experienced monetary as well as nonmonetary benefits from the implementation of the decision support tool. The monetary benefits had at least two sources, (i) better selection of hybrid seeds based on portfolio considerations, and (ii) better allocation of parent seed inventories. The firm seeks to continuously develop better hybrids to offer in individual markets in order to stay competitive. Often times, there may be two or more candidate hybrids for a target market that differ on their yield distributions. The optimization protocol provides an objective way to select the best candidate hybrid in such situations. As an example, consider two hybrids, A and B, that can be offered in a market. The mean and standard deviation of the yield distribution of hybrid A are equal to 100 bags/acre and 30 bags/acre; for seed B these values are 104 bags/acre and 34 bags/acre. Hybrid B provides more output from land on average but also has a larger uncertainty. The status quo at the firm before our engagement would have considered both these seeds to be in the same “ballpark.” In contrast, using the optimization protocol described earlier, one can provide a more nuanced recommendation. For example, for one such combination of seeds the net profit from each of the two candidates as a function of market prices is shown in Fig. 4.4. This figure shows that if the target selling price in the market is higher than $115, the firm should select hybrid A, otherwise it should choose hybrid A. The second source of benefit is more efficient use of parent seeds. Recall that the inventory of many parent seeds tends to be limited during the NA decision. When faced with such shortages a common practice at the firm was to equally allocate the shortage of the parent seed among all derivative hybrid seeds. This allocation heuristic tends to be suboptimal, in general. As an example, consider the problem instance shown in Table 4.1, where three parent seeds 1, 2, 3 provide three hybrids A, B, C. The male/female role of the parent seeds is shown in the center of the table. The female (F) role means that 0.8 bags of that parent will be required per acre to produce the hybrid; the male (M) role requirement is 0.2 bags/acre. Without any
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Fig. 4.4 Illustration for better hybrid selection using optimization protocol
Table 4.1 Illustrative problem for parent seed allocation
constraint on parent seed inventories, it is optimal to produce the hybrids A, B, C on 200, 400, and 600 acres, respectively. This plan requires 160 bags, 240 bags, and 800 bags of parents 1, 2, and 3, respectively. However, only 140 bags, 200 bags, and 610 bags of parents 1, 2, 3, respectively are available. Table 4.2a and b show the results of using the heuristic of allocating the shortages equally among hybrids. In Step 1, we first focus on parent 3. It is used to produce hybrids B and C. We reduce the acreage of both these hybrids by 120 acres each (to 280 acres and 480 acres) to bring down the parent seed requirement to 608 bags inline with the availability of 610 bags. At this plan, parent seed 2 is still in shortage, as shown in Table 4.2a. In the next step we address this shortage (when accounting for M and F needs of 0.2 and 0.8 bags/acre). Table 4.2b shows that at end of this step, the resultant production plan meets all parent seed constraints. However, parent seeds 1 and 3 are not used in full now, even though they had a shortage in the first place. The total acreage across the three hybrids is equal to 936 acres. In contrast the optimization protocol described earlier provides a better usage of the hybrids, as shown in Table 4.3. In this solution the inventories of the three parent seeds are used in full, with a total acreage of 950 acres. Overall, analysis at the firm showed that with the use of the optimization protocol, the total NA acreage for the seed corn portfolio increased. This increase was not undesirable. As a first order effect, this increase led to a decrease in SA acreage which was nearly 50% more costly on a per acre basis. Furthermore, the analysis showed that managers may have overrelied in the past on SA acreage, essentially using it as a planned activity than a reactive activity which was the
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Table 4.2 a and b: Heuristic based allocation
Table 4.3 Optimization based allocation
initial purpose of developing the SA production capability. Overall, a use of the two modules increased the firm’s profit by 2–3% annually, which translated into double digit millions in US dollars. Based on the monetary benefits the firm made a successful business case for an internal analytics division. The firm also experienced several nonmonetary managerial benefits. Various functions that participated in the seed corn business used the inputs and outputs of the two modules as a basis for recommendations or explanations for decisions. The decision making process during the annual crop planning meeting was perceived to be more competent and objective.
4.5 Conclusion Commercial seed firms face competition from each other in a large global market. They also face the expectations from the farming community as well as society at large for providing the raw material for staple crops. Recent technology advances have facilitated this process by providing better varieties of seeds and a better understanding of other inputs to maximize the yields of seeds in fields. Numerous avenues exist to further this process. The longterm weather conditions in the USA
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and elsewhere are changing, with average temperatures increasing systematically. This in turn is likely to change local moisture and pestilence rates in yet unknown ways. Developing a longterm R&D program that can be effective in this evolving environment is a challenging yet fruitful direction for future efforts. A second dimension is to use better analytics methods to improve the production and utilizations of commercial seeds. Recent developments have sought to use machine learning methods (such as those explored in Syngenta Analytics Competition at INFORMS) and expert judgments in statistical fashion to quantify the likely yield scenarios of hybrids. These quantifications provide inputs to prescriptive tools to optimize the production, storage, and distribution of seeds. This is a promising area of research with likely significant and immediate impacts on food supply chains.
References Bansal S, Dyer JS (2020) Planning for enduser substitution in agribusiness. Oper Res 68(4):1000– 1019 Bansal S, Nagarajan M (2017) Product portfolio management with production flexibility in agribusiness. Oper Res 65(4):914–930 Bansal S, Gutierrez GJ, Keiser JR (2017) Using experts’ noisy quantile judgments to quantify risks: theory and application to Agribusiness. Oper Res65(5):1115–1130 Bansal S, Lowe TJ, Jones PC (2020) Case article—Suncrest AgriBusiness Company: exploiting the flexibility of backup capacity. INFORMS Trans Educ (forthcoming) https://pubsonline. informs.org/doi/10.1287/ited.2019.0232ca Cong L, Ran FA, Cox D, Lin S, Barretto R, Habib N, Hsu PD, Wu X, Jiang W, Marraffini LA, Zhang F (2013) Multiplex genome engineering using CRISPR/Cas systems. Science 339(6121):819–823 Fortune Business Insights (2020) Commercial seed market size, share, and industry analysis. Accessed at: https://www.fortunebusinessinsights.com/industryreports/infographics/ commercialseedmarket100078 Howard PH (2009) Visualizing consolidation in the global seed industry: 1996–2008.Sustainability 1(4):1266–1287 Hubbard K (2019) The sobering details behind the latest seed monopoly chart. Accessed at https:// civileats.com/2019/01/11/thesoberingdetailsbehindthelatestseedmonopolychart/ Jones PC, Lowe TJ, Traub RD, Kegler G (2001) Matching supply and demand: the value of a second chance in producing hybrid seed corn. Manuf Serv Oper Manage 3(2):122–137 Jones PC, Lowe TJ, Traub RD (2002) Matching supply and demand: the value of a second chance in producing seed corn. Rev Agricult Econ 24(1):222–238 Jones PC, Kegler G, Lowe TJ, Traub RD (2003) Managing the seedcorn supply chain at Syngenta. Interfaces 33(1):80–90 Papier F (2016) Supply allocation under sequential advance demand information. Oper Res 64(2):341–361
Part II
Processing and Selling
Chapter 5
Procurement Management in Agricultural Commodity Processing Onur Boyabatlı and Bin Li
5.1 Introduction The objective of this chapter is to develop a theoretical basis for understanding the tradeoffs facing a commodity processing firm in the choice of alternative arrangements for sourcing a primary input, where this input gives rise to multiple outputs in fixed proportions. Fixed proportions production technology is relevant for several agricultural industries in practice. For example, in the wheat industry, wheat seeds are processed (by grounding and sieving) to produce wheat bran and coarse powder flour. In the sugar industry, sugarcane is processed (by grinding and heating) to produce the white table sugar and the crystalized sugar particles that can be further processed into animal feed. In the cocoa industry, cocoa beans are processed (by cleaning, roasting, and grinding) to produce cocoa butter and cocoa powder. Despite its prevalence, as highlighted by Chen et al. (2013b), the proportional production technology has received limited attention in the operations management community. Motivated by the semiconductor and the chemical industries, the papers in this stream consider a coproduction system, where one single production run gives rise to multiple outputs with random yields. Since the yield uncertainty is a major issue, majority of these papers assume the demands of outputs are deterministic to simplify the analysis, and thus, they remain silent on the impact of the demand uncertainty. However, given the volatility of the business environment
O. Boyabatlı () Lee Kong Chian School of Business, Singapore Management University, Singapore email: [email protected] B. Li Economics and Management School, Wuhan University, Wuhan, China email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_5
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today, including the agricultural industries, one of the main challenges for the multiproduct firms is the demand uncertainty. A vast stream of papers studies the procurement management in multiproduct firms under demand uncertainty focusing on the flexible production technology. With the flexible production technology, the firm can adjust the production volume of multiple outputs based on their demand realizations where each output requires processing of a single input. This literature establishes that the multiproduct firms with flexible technology benefit from a lower demand correlation due to increasing demandpooling value (Mieghem and Rudi 2002). With the fixed proportions production technology, processing a single input generates multiple outputs at the same time, and thus, the firm is unable to adjust the production volume for one output without affecting the others. Therefore, it is an open question how demand correlation would affect the firms with fixed proportions technology. Our first objective is to answer this question. Our analysis and results presented in this chapter that are related to this objective are adapted from Boyabatlı (2015). Besides the uncertain output demands, the spot price uncertainty of the primary input is also a critical factor for procurement management in agricultural industries as the spot prices show considerably variability (Meyer 2013). In addition to spot procurement, processing firms in agricultural industries use longterm procurement contracts. These contracts may take different forms in terms of pricing and delivery requirement. As highlighted by Kleindorfer and Wu (2003), the quantity flexibility contract is the most commonly used contract form in practice. A quantity flexibility contract specifies the volume of input reserved in advance of the spot market. The actual delivery volume is decided within this reserved quantity on the day. In practice, a common strategy is to use a single contract for procurement. Because there are multiple contracts available, the firm needs to choose which contract to use. Our second objective is to examine the optimal contract selection problem and how input spot price uncertainty shapes this selection. To achieve these objectives, we model a processing firm who procures an input, produces, and sells two outputs in fixed proportions of this input in a single period. The firm can source the input from the spot market and from a quantity flexibility contract in advance of the spot market. The outputs can be sold to demand markets at fixed prices. In the first stage, the firm chooses the contract volume to reserve to maximize its expected profit in the presence of spot price and demand uncertainties. In the second stage, after the uncertainties are realized, the firm decides the volume of input to exercise from the contract, the volume of spot procurement, and the production quantity of each output within its available yield through fixed proportions technology. With this model, we first analyze the impact of demand correlation on the contract procurement decisions and the optimal expected profit. To generate sharper managerial insights, we consider a single quantity flexibility contract available for procurement. We identify the interplay between the impact of demand correlation on the optimal contract volume and the contract parameters: when the exercise price of the contract is sufficiently high, a higher contract market dependence better combats the increasing demand correlation. However, a lower contract market dependence
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better combats the same when the exercise price is sufficiently low. We also find that a higher demand correlation is beneficial because it reduces the demand mismatch in the output markets facilitating the effective usage of the same input for both outputs under the fixed proportions technology. We next investigate the firm’s optimal contract selection strategy in the presence of spot price uncertainty and how this strategy is affected by the changes in spot price uncertainty. To generate sharper managerial insights, we assume that the output demands are deterministic. We consider two quantity flexibility contracts: one with a lower reservation price and the other with a lower exercise price. We characterize a contract index in closed form that determines the optimal contract choice. We identify that, different from the impact of expected spot price—that is, a higher expected spot price increases the reliance on the contract with lower exercise price, a higher spot price variability does the same only when the expected spot price is low and the additional exercise price of the other contract is sufficiently high. Otherwise, a higher spot price variability increases the reliance on the contract with the lower reservation price. The remainder of this chapter is organized as follows: Sect. 5.2 surveys the related literature and discusses the contribution of our work. Sect. 5.3 characterizes the optimal processing and contract decisions as well as the role of demand correlation. Sect. 5.4 derives the optimal contract selection strategy and investigates the impact of spot price uncertainty. We conclude in Sect. 5.5 with a discussion of future research directions.
5.2 Literature Review This chapter contributes to two streams of literature: (i) supply management in multiproduct firms and (ii) contract procurement in the presence of spot markets. As highlighted by Chen et al. (2013b), in the literature on supply management in multiproduct firms, fixed proportions technology has received limited attention. Motivated by the semiconductor industry, a stream of papers analyzes coproduction systems where multiple outputs are produced in a single production run. The standard coproduction problem foresees different grades or quality levels of output, where yields for these different grades are typically random. The problem of contracting for inputs (e.g., wafer starts in semiconductor manufacturing) when facing demand schedules for each of the grades has some similarities to the processing problem considered in this chapter. However, the primary focus in the coproduction literature is on the production quantity and the allocation of the realized production output to the product demands. Therefore, the majority of papers in this stream assume deterministic demands to simplify the analysis. In a few exceptions, Hsu and Bassok (1999), Rao et al. (2004), and Ng et al. (2012)
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investigate the structural properties of the optimal solution and further propose heuristic solutions under stochastic demand in a newsvendor setting. Under a similar setting, Tomlin and Wang (2008) solve for the optimal production, pricing, allocation decisions and analyze the value of different operational flexibilities. All of these papers remain silent on the impact of demand correlation. This chapter also contributes to the literature on contract procurement in the presence of spot markets. The papers in this stream examine different operational decisions for the commodity processors in various industries. These studies capture the idiosyncratic features of different commodity markets, including those for energy (Wu et al. 2002), iron ore (Chen et al. 2013a), gasoline (Goel and Gutierrez 2011) as well as commodity markets associated with such agricultural industries as olives (Kazaz 2004) and citrus fruit (Kazaz and Webster 2011). All these papers consider only one single contract available for the firm to make the contractual arrangement. In addition, within this stream of literature, only a few papers capture the fixed proportions technology (see, for example, Li et al. 2020). Closest to our model, Boyabatlı et al. (2011) analyze the optimal procurement, processing and production decisions of a meatpacker in beef supply chains, where the meatpacker processes fed cattle to produce two beefproducts in fixed proportions. The meatpacker’s procurement portfolio consists of spot procurement and a single quantity commitment contract whose price is benchmarked on the prevailing spot price. In the literature on contracting in the presence of spot markets, several papers analyze the firm’s optimal contracting decision in the presence of multiple contracts in different contexts such as electronics (Wu and Kleindorfer 2005; Pei et al. 2011), semiconductors (Martínezde Albéniz and SimchiLevi 2005), electricity (Anderson et al. 2017), and natural gas (Abada et al. 2017). We refer the readers to Xu et al. (2020) for a review of the recent papers in this area. Different from these papers that focusing on the optimal sourcing decisions, we study a different operational issue faced by the agriprocessor—that is, selecting one single contract among the available contracts. We characterize the optimal contract selection strategy for the agriprocessor with fixed proportions technology in the presence of spot market. Our analysis provides new insights on the impact of spot price uncertainty on the optimal contract selection strategy.
5.3 The Role of Demand Correlation in Agricultural Processing with Fixed Proportions Technology We describe the basic model and relevant assumptions (Sect. 5.3.1) and derive the optimal processing and contract volumes (Sect. 5.3.2). We then examine the impact of demand correlation on the firm’s optimal contract decision and expected profit (Sect. 5.3.3).
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5.3.1 Model Description and Assumptions We use the following notation and conventions throughout the chapter: A realization of the random variable y˜ is denoted by y. Bold face letters represent column vectors of the required size and represents the transpose operator. The expectation operator and probability function are denoted by E[·] and Pr(·), respectively. We have (x)+ = max(x, 0). The monotonic relations (increasing, decreasing) are used in the weak sense unless otherwise stated. We use the subscript j ∈ {1, 2} to denote the index of the final product. All the proofs are available from the authors upon request. We consider a firm that procures and processes a single commodity input to produce two final products in fixed proportions so as to maximize its expected profit in one selling season. We model the firm’s problem in a twoperiod framework: In stage 1, the firm makes the contract decision in the presence of spot price and demand uncertainties; in stage 2, the firm decides its processing and production volumes after the uncertainties are realized. For the procurement decision, we consider two sources: the contract and the spot market. A typical contract specifies the volume of input reserved by the firm in advance of the spot market. On the spot day, the firm decides the volume of input to be delivered within the reserved capacity. In particular, we consider a quantity flexibility contract that is characterized by the unit reservation price β > 0 paid in advance of the delivery, and the unit exercise price b > 0 paid upon delivery on the spot day. Let Q denote the volume of input reserved from the contract in advance of the spot market. On the spot day, the firm can also buy input from the spot market at the prevailing spot price S. Additionally, the contracted input can be sold to the spot market on the day at a unit price S(1 − t) where 0 ≤ t ≤ 1. Here, t denotes the transaction cost associated with spot sale. We assume that S˜ has a continuous distribution with bounded expectation μS and standard deviation σS . ˜ denote its cumulative distribution function. For sensitivity analysis in Let F (S) Sect. 5.4.3, we assume S˜ to follow a Normal distribution. For the processing and production decisions, the firm processes z units of input at a unit cost of ω regardless of where the input is sourced from (either from reserved contract or from spot market). Each unit of processed input yields a1 and a2 units of output 1 and output 2, respectively, in fixed proportions (where a1 + a2 ≤ 1). The available output j can be converted into the final product j by incurring a production cost cj ≥ 0. After the production, the firm can salvage product j from sj ≥ 0 if there is no unsatisfied demand Dj ; otherwise, the unit is sold at a unit price pj ≥ max(cj , sj ). We normalize the penalty costs for unsatisfied demands to zero for brevity. The positive penalty costs can be easily incorporated without affecting our results structurally. Since processing one unit of input yields aj units of product j , the firm has a unit sales revenue aj (pj − cj ) when there is unsatisfied demand; otherwise, it collects a unit salvage revenue aj (sj − cj )+ for each unit of processed input. The demands for the final products are stochastic, correlated and ˜ = (D˜ 1 , D˜ 2 ). D ˜ is a bivariate random variable with a continuous are denoted by D distribution that has bounded expectation (μ1 , μ2 ) with covariance matrix , where
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jj = σj2 for j ∈ {1, 2}, 12 = 21 = ρσ1 σ2 and ρ denotes the correlation ˜ coefficient between D˜ 1 and D˜ 2 . For sensitivity analysis in Sect. 5.3.3, we assume D to follow a bivariate Normal distribution. Throughout our analysis, we assume that the distributions of input spot price S˜ and product demands D are statistically independent. We refer the reader to Boyabatlı (2015) for the analysis with general model in which these distributions are ˜ − t) − b)+ ]; correlated. For the quantity flexibility contract, we assume β > E[(S(1 otherwise, the firm optimally reserves an infinite volume of input due to profitable spot resale. For processing, we assume ω ≥ a1 (s1 − c1 )+ + a2 (s2 − c2 )+ ; that is, ω is sufficiently high such that processing is not profitable when both products are salvaged. Otherwise, the firm optimally buys infinite volume of input from the spot market when the spot price is sufficiently low. Let . h3 = a1 (p1 − c1 ) + a2 (p2 − c2 ) − ω, . h2 = a1 (p1 − c1 ) + a2 (s2 − c2 )+ − ω, . h1 = a1 (s1 − c1 )+ + a2 (p2 − c2 ) − ω
(5.1)
denote the unit processing margins under different demand realizations. In particular, h3 denotes the unit processing margin when both markets have unsatisfied demand; h2 (h1 ) is the margin when only market 1 (2) has unsatisfied demand. Without loss of generality, we assume h1 ≤ h2 . To rule out uninteresting cases, we assume h3 > b; otherwise, the contract is never exercised on the day for processing. We also assume that ω is sufficiently low such that all three unit processing margins are positive. This is not a critical assumption for our analysis, but it simplifies the exposition dramatically.
5.3.2 The Optimal Strategy In this section, we describe the optimal solutions for the firm’s operational decisions, including optimal processing and contract volumes. We solve the firm’s problem using backward induction starting from stage 2. In stage 1, the firm reserved Q units of input. In stage 2, the firm observes the input spot price S and demand realizations (D1 , D2 ). Constrained by the reserved capacity Q, the firm optimally decides the processing volume z, how to source this volume from reserved capacity and spot market, and the spot resale of the contracted input. Since the stage 2 decisions can be rewritten as a singlevariable optimization problem over the processing volume z, we define (z) as the stage 2 profit as a . function of z. Let π denote the optimal profit; that is, π = max0≤z≤Q (z). The stage 2 profit for a given z is characterized by . (z) = − min(z, Q) max (min (S, b) , S(1 − t)) − (z − Q)+ S + Q(S(1 − t) − b)+
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Dj + . + aj (sj − cj ) z − aj +
(5.2)
In Eq. 5.2, the first line is the total procurement cost and the second line is the sales revenue from both products minus the processing cost. The total procurement cost is given by the sum of the procurement cost for the first min(z, Q) units and the same for the remaining (z − Q)+ units minus the spot sale revenue when the input volume Q is profitably sold. For 0 ≤ z ≤ Q, when the spot sale is profitable (i.e., S(1 − t) ≥ b), it is also cheaper to source from the contract than the spot market. Therefore, the unit procurement cost is the opportunity cost of not selling this input to the spot market (i.e., S(1 − t)). When the spot sale is not profitable (i.e., S(1 − t) < b), the unit procurement cost is given by the cost of the cheapest source (sourced either from the contract at a cost of b, or from the spot market at a cost of S). Combining these two scenarios, the unit procurement cost is given by max (min (S, b) , S(1 − t)). For the processing volume exceeding Q, the input can only be sourced from the spot market, and the unit procurement cost is S. The stage 2 objective function is piecewise linear and concave in Q. Moreover, since we have assumed ω ≥ a1 (s1 − c1 )+ + a2 (s2 − c2 )+ , the firm never processes up to Q when there is no unsatisfied demand in both markets. Therefore, the optimal solution occurs at the breakpoints {0, D1 /a1 , D2 /a2 } and min Dj /aj , Q for j ∈ {1, 2}. Proposition 1 characterizes the optimal processing volume z∗ in stage 2. Proposition 1 Given contract volume Q and spot price S and demand realizations (D1 , D2 ), the optimal processing volume z∗ is given by
z∗ =
⎧ 0 ⎪ ⎪ ⎪ D[1] ⎪ ⎪ ,Q min a[1] ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ min D[2] , Q a[2]
D[1] ⎪ ⎪ a[1] ⎪ ⎪ ⎪ D[2] ⎪ max min a[2] ,Q , ⎪ ⎪ ⎪ ⎪ ⎩ D[2] a[2]
if h3 ≤ max (min (S, b) , S(1 − t)) if h[1] ≤ max (min (S, b) , S(1 − t)) ≤ h3 ≤ S if max (min (S, b) , S(1 − t)) ≤ h[1] ≤ h3 ≤ S D[1] a[1]
if h[1] ≤ max (min (S, b) , S(1 − t)) ≤ S ≤ h3
(5.3)
if max (min (S, b) , S(1 − t)) ≤ h[1] ≤ S ≤ h3 if S ≤ h[1] ,
where [1] denotes the index of the product with min (D1 /a1 , D2 /a2 ) and [2] denotes the other; h[1] and h3 are the unit processing margins as defined in Eq. 5.1. The optimal processing volumes are determined by the magnitude of unit processing margins and unit sourcing costs. The sourcing cost takes two forms, max (min (S, b) , S(1 − t)) and S, incorporating spot procurement and resale options. Consider, for example, the second last case in Eq. 5.3. The condition h3 ≥ S implies that it is profitable to process the input from spot market (and reserved capacity) when both markets have unsatisfied demand, so the firm can profitably process at least D[1] /a[1] units of input (beyond which the unit processing margin becomes h[1] ). The condition max (min (S, b) , S(1 − t)) ≤ h[1] ≤ S indicates that it is also profitable to process the contracted input solely when
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only market [2] has unsatisfied demand, so the firm can profitably process at least min D /a , Q units of input. Putting them together, we have [2] [2] z∗ = max min D[2] /a[2] , Q , D[1] /a[1] in this case. In stage 1, the firm decides the optimal contract volume Q∗ in the presence of ˜ uncertainties so as to maximize the expected profit spot price S˜ and demand D . ˜ ˜ V (Q) = E[π(Q; S, D)] − βQ. Proposition 2 The optimal contract volume is given by Q∗ = 0 if β ≥ −b + μS (1 − t) + G(b, h3 ). Otherwise, Q∗ solves ∂V /∂QQ∗ = 0, where ∂V D˜ 1 D˜ 2 = −β − b + μS (1 − t) + Pr > Q, > Q G(b, h3 ) ∂Q a1 a2 D˜ 1 D˜ 2 + Pr > Q, ≤ Q G(b, max(b, h2 )) a1 a2 D˜ 1 D˜ 2 + Pr ≤ Q, > Q G(b, max(b, h1 )) a1 a2 D˜ 1 D˜ 2 + Pr ≤ Q, ≤ Q G(b, b), a1 a2 . ˜ λ), b) − S(1 ˜ − t)  S˜ ≤ λ/(1 − t)] for b ≤ λ, and and G(b, λ) = E[max(min(S, h1 ≤ h2 ≤ h3 as given in Eq. 5.1. In the first order condition −β − b denotes the marginal contract procurement cost, μS (1 − t) denotes the expected spot resale revenue whereas the remaining terms characterize the expected processing revenue in excess of the spot resale. In particular, b λ λ/(1−t) . ˜ − t)) dF (S) ˜ + t S˜ dF (S) ˜ + (λ − S(1 ˜ − t)) dF (S) ˜ G(b, λ) = (b − S(1 0
b
(5.4)
λ
denotes the expected processing revenue in excess of the spot resale with the unit processing margin λ and the exercise price b. When S ≤ b, the contract is optimally not exercised, and thus, the firm receives back b which is deducted as a part of the marginal cost. When b < S ≤ λ, the firm can profitably process both contracted input and spotsourced input. Exercising the contract (which is cheaper) yields an opportunity gain of not sourcing from spot market (i.e., S). When λ < S ≤ λ/(1 − t), only the contracted input can be profitably processed, so the value of the additional unit of contracted input is λ. At stage 2, the processing revenue takes different forms as it depends on the demand and spot price realizations. To delineate the intuition behind the processing revenue expressions, we focus on the first term of the first order condition in Proposition 2. When both product demands are high such that D1 /a1 > Q
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and D2 /a2 > Q at stage 2, the firm profitably exercises the contract (due to h3 > b) and sells both products and generates a unit processing margin of h3 . The corresponding expected processing margin G(b, h3 ), multiplied by the probability of its occurrence; that is, Pr (D1 /a1 > Q, D2 /a2 > Q), is part of the expected processing revenue expression in excess of the spot resale in the first order condition.
5.3.3 Analysis In this section, we investigate the impact of demand correlation on the firm’s optimal contract volume and optimal expected profit. We also examine whether there exist structural differences for the impact of demand correlation based on the contract type. To this end, we study how the sensitivity results are impacted by the exercise price b of the contract. Proposition 3 For b ≥ h1 , ∂Q∗ /∂ρ ≥ 0. For b < h1 , there exists a unique bˆ ∈ ˆ In this case, [0, h1 ) such that ∂Q∗ /∂ρ ≤ 0 if b ≤ bˆ and ∂Q∗ /∂ρ ≥ 0 if b ≥ b. bˆ > 0 if G(0, h3 ) < G(0, h2 ) + G(0, h1 ), and bˆ = 0 otherwise, where G(b, λ) is as defined in Proposition 2. Proposition 3 showcases that the impact of demand correlation on the firm’s optimal contract procurement strategy critically depends on the contract type. In particular, when the exercise price is sufficiently high, the firm is better off by increasing its contract market dependence as a response to the increasing demand correlation. However, the firm may benefit from a lower contract market dependence as a response to the same when the exercise price is sufficiently low. We now explain the intuition behind this result. With a higher demand correlation, it is more likely that when the demand in one market is high (low), the demand in the other market is also high (low). Therefore, a higher demand correlation ρ increases the probability of unsatisfied demand in each market (with a processing margin h3 ) due to a higher probability of high demand in both markets, favoring a higher contract volume. It also decreases the probability of unsatisfied demand in low demand market and no unsatisfied demand in high demand market (with a processing margin h1 or h2 ) due to demand synchronization of two markets, favoring a lower contract volume. Therefore, the impact on Q∗ is determined by comparing the increase in the expected marginal revenue of contracting based on the h3 case with the decrease in the same based on the h1 and h2 cases. When b ≥ h1 , the firm optimally does not exercise the contract when the processing margin is h1 , and thus, the h1 case plays no role in the firm’s optimal procurement decision. The increase in the probability of h3 case outweighs the decrease in the probability of h2 case, increasing Q∗ . When b < h1 , under the condition given in Proposition 3, Q∗ decreases in ρ when the exercise price b is sufficiently small. This condition is satisfied, for example, when there is no spot sale (t = 1) and ω = a1 (s1 − c1 )+ + a2 (s2 − c2 )+ . It is easy to establish that the condition in Proposition 3 is never satisfied in the absence of spot market access, and thus, Q∗ increases in ρ for any b in the absence of spot market.
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Proposition 4 For a given Q, ∂V (Q)/∂ρ ≥ 0. This comparative static result holds when Q is adjusted to the optimal path. The common intuition prevalent in the academic literature on multiproduct firms argues that increasing demand correlation is detrimental for the firm’s profitability (see, for example, Mieghem and Rudi 2002). Interestingly, Proposition 4 demonstrates that a multiproduct firm with fixed proportions technology benefits from an increasing demand correlation. With a lower demand correlation, when the demand for one product is high, it is more likely that the demand for the other is low. Therefore, the firm with fixed proportions technology can enjoy a sales revenue only in one of the markets. With a higher demand correlation, when the demand for one product is high, it is more likely that the demand for the other is also high. Therefore, the firm can enjoy sales revenues in both markets. In summary, a higher correlation decreases the demand imbalance in the product markets, and facilitates the effective usage of the fixed proportions technology.
5.4 The Impact of Spot Price Uncertainty on Contract Selection in Agricultural Processing In previous section, we study the role of demand correlation with fixed proportions technology in a singlecontract setting. In this section, we investigate the impact of input spot price uncertainty on the firm’s contract selection decision in a twocontract setting. We describe the additional modelling assumptions (Sect. 5.4.1) and derive the optimal contract selection strategy (Sect. 5.4.2). We then examine the impact of spot price uncertainty—including the expected spot price and spot price variability—on this optimal strategy (Sect. 5.4.3).
5.4.1 Model Description and Additional Assumptions Different from Sect. 5.3.1, in stage 1, the firm first chooses one contract out of the two available contracts to source from before deciding its contract volume. We consider two quantity flexibility contracts that are indexed with superscript i = {1, 2}. To avoid uninteresting cases, we assume b1 ≤ b2 and β 1 ≥ β 2 without loss of generality; that is, the first contract has a lower exercise price but a higher reservation price. Let k denote the index of the selected contract, (β k , bk ) denotes its parameters, and Qk denotes its volume. As a result of the optimal contract selection decision, choosing either contract 1 or 2 is optimal. Our analysis focuses on determining conditions under which . each selection strategy is optimal. To this end, we define β = β 1 − β 2 ≥ 0 . as the additional reservation price of the first contract; and b = b2 − b1 ≥ 0 as the additional exercise price of the second contract. For a given b1 and β 2 ,
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we characterize the optimal selection strategy regions within (b , β ) space. We analyze the impact of input spot price uncertainty on the optimal selection strategy by analyzing its impact on these regions. Throughout this section, to simplify the analysis we assume that product demands in both markets are deterministic and there is no spot resale possibility; i.e., t = 1. To avoid uninteresting cases we also assume h3 > bi for i = {1, 2}; otherwise, contract i is never exercised on the day for processing.
5.4.2 The Optimal Strategy In this section, we describe the optimal decisions for the firm. We solve this problem using backward induction. In stage 1, the firm selected the contract and its volume. Recall that k is the index of the selected contract. The firm’s stage 2 problem is a special case of the singlecontract case analyzed in Sect. 5.3. In particular, with deterministic demands (D1 , D2 ), the optimal processing volume z∗ at the realized spot price S can be adapted from Eq. 5.3 by setting t = 1 and substituting Q with Qk and (β, b) with (β k , bk ). ∗ In stage 1, the firm optimally selects contract k and its volume Qk to reserve with respect to input spot price S˜ uncertainty so as to maximize the expected profit. Proposition 5 characterizes the optimal contract selection strategy. ∗
Proposition 5 The optimal contract volume Qk is given by
Q
[X] ∗
=
⎧ ⎪ 0 ⎪ ⎨D
[1]
a[1] ⎪ ⎪ ⎩ D[2] a[2]
if β [X] + b[X] ≥ G(b[X] , h3 ), if G(b[X] , h3 ) > β [X] + b[X] ≥ G(b[X] , max(h[1] , b[X] )), if G(b[X] , max(h[1] , b[X] )) > β [X] + b[X] ,
where [X] denotes the contract with the higher value of −β i − bi + G(bi , h3 ); [1] denotes the index of the product with min (D1 /a1 , D2 /a2 ) and [2] denotes the other. Proposition 5 characterizes a unique index for each contract where the contract with the higher index is chosen in the optimal solution. The index −β i − bi + G(bi , h3 ) for contract i corresponds to the expected marginal profit from its first contracted unit (when both product markets have unsatisfied demands). Once the optimal contract choice is made, its optimal volume to reserve (if any) is given by the ordering of the total marginal cost β + b and the two expected marginal revenue expressions—that is, one when both product markets have unsatisfied demands, i.e., G(b, h3 ), and the other when the product market with lower demand does not have any unsatisfied demand left, i.e., G(b, max(h[1] , b)). We next analyze the optimal sourcing policy within (b , β ) region.
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Fig. 5.1 The optimal contract selection strategy of the firm
. ¯β = ¯ b (≤ h3 − b1 ) to be the unique G(b1 , h3 ) − b1 − β 2 and Corollary 1 Let . 1 2 1 ¯ b , h3 ) = β +b + ¯ b . There exists a unique T = G(b1 , b1 + solution to G(b + β 1 b ) − b for a given b ≥ 0 such that using the first contract is optimal when ¯ β ); using the second contract is optimal when β > T for β ≤ min(Tβ , β ¯ b ). b ∈ [0, ¯ β denotes the largest value such that the first contract is costFor a given β 2 , effective to be used in the absence of the second contract. Similarly, for a given b1 , ¯ b denotes the largest value such that the second contract is costeffective to be used in the absence of the first contract. The threshold Tβ denotes the additional reservation price of the first contract for a given b such that the firm is indifferent between the two contracts. Figure 5.1 illustrates the optimal contract selection in Corollary 1. It can be proven that the threshold Tβ is a concave increasing function of b .
5.4.3 Analysis In this section, we analyze the impact of expected spot price μS and spot price variability σS on the optimal contract selection strategy. To this end, we investigate how μS and σS affect the threshold Tβ in Fig. 5.1. For example, when an increase in one of these parameters increases Tβ , contract 1 region in Fig. 5.1 expands. Therefore, we conclude the increase of this parameter favors the contract with the lower exercise price (first contract in our model). Proposition 6 ∂Tβ /∂μS > 0. It follows from Proposition 6 that a higher expected spot price always favors the contract with the lower exercise price. A higher μS increases the expected spot procurement cost and, thus, increases the reliance on the contracted input on the spot day. Since b1 < b2 by assumption, for a given spot price distribution there is a higher likelihood of using the first contract than the second contract.
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Proposition 7 There exists a unique μˆ S ∈ b1 , (h3 + b1 )/2 such that (i) when μS ≥ μˆ S , ∂Tβ /∂σS ≤ 0; (ii) when μS < μˆ S , ∂Tβ /∂σS < 0 for b ∈ [0, 2(μS − b1 )+ ) and ∂Tβ /∂σS ≥ 0 ¯ b ). for b ∈ [2(μS − b1 )+ , The impact of spot price variability σS on the contract selection decision critically depends on the expected spot price μS . A higher spot price variability favors the contract with the lower exercise price (first contract in our model) only when μS is sufficiently low; otherwise, it favors the contract with the lower reservation price (second contract in our model). As observed from Eq. 5.4, the marginal revenue of each contract at stage 2 depends on the spot price within a processing window (b < S ≤ λ, using the notation in Eq. 5.4)—that is, when it is profitable to use the spot procurement, but the contract procurement is less expensive. A higher σS increases the likelihood of low and high spot price realizations. With a low (high) μS , high (low) spot price realizations appear in this window, and a higher σS increases (decreases) the expected value of the window. Since b1 < b2 , this window is wider with the first contract, and a change in the expected value of this window impacts the first contract to a larger extent. Therefore, with a low (high) μS , a higher σS increases (decreases) the expected marginal revenue of the first contract more, and thus, the first (second) contract is favored.
5.5 Conclusion and Future Research This paper provides insights on the optimal procurement decisions of a commodity processing firm in agricultural industries. As commonly observed in these industries, we consider a firm that uses a single input to produce multiple outputs in fixed proportions. As summarized in the Introduction, we provide insights on (i) the role of demand correlation with fixed proportions technology and (ii) the impact of spot price uncertainty on the firm’s contract selection problem when facing two quantity flexibility contracts: one with a lower exercise price and the other with a lower reservation price. There are a number of limitations to the present study that lead to open research questions. First, we assume there are only two contacts in the contract selection problem. Although the analysis will be more complex, we expect our main results to continue to hold in an mcontract world. Second, in the contract selection model, we assume product demands are deterministic. Considering the stochastic demands would be an interesting avenue for future research. In this case, we expect that optimal contract selection decision cannot be characterized based on a singlecontract index. Third, we assume that the contracts are fully reliable; that is, all the capacity reserved in the contract market can be used later on the spot day. In agricultural markets, the reserved capacities may correspond to the harvest, and may be prone to yield risks due to weather conditions and diseases. It would
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be interesting to analyze the impact of yield uncertainty on the optimal contract procurement as well as the optimal contract selection in our setting. Fourth, we only consider the input spot market in our model. It could be interesting to consider the output spot market in the future study, following the examples of Plambeck and Taylor (2013) and Boyabatlı et al. (2017). Fifth, our focus is on a single contract selection out of the available contracts. In practice, it is not uncommon for the firms to use a portfolio of procurement contracts in commodity processing (e.g., Martínezde Albéniz and SimchiLevi 2005). Therefore, it would be practically relevant to investigate the optimal procurement from multiple contracts for a multiproduct firm with fixed proportions technology in the presence of spot markets. Finally, we focus on a partial equilibrium model where the contract parameters are exogenous. This is a reasonable assumption for shortterm planning considered in this paper. In medium to longterm, suppliers can adjust the contract terms with respect to the changes in the uncertainties. Analyzing the impact of endogenous contract terms on the sourcing portfolio in an equilibrium setting is a potential avenue for future research, following the examples of Wu and Kleindorfer (2005) and Pei et al. (2011) who provide some results for the case of a singleproduct firm.
References Abada I, Ehrenmann A, Smeers Y (2017) Modeling gas markets with endogenous longterm contracts. Oper Res 65(4):856–877 Martínezde Albéniz V, SimchiLevi D (2005) A portfolio approach to procurement contracts. Prod Oper Manage 14(1):90–114 Anderson E, Chen B, Shao L (2017) Supplier competition with option contracts for discrete blocks of capacity. Oper Res 65(4):952–967 Boyabatlı O (2015) Supply management in multiproduct firms with fixed proportions technology. Manage Sci 61(12):3013–3031 Boyabatlı O, Kleindorfer PR, Koontz SR (2011) Integrating longterm and shortterm contracting in beef supply chains. Manage Sci 57(10):1771–1787 Boyabatlı O, Nguyen J, Wang T (2017) Capacity management in agricultural commodity processing and application in the palm industry. Manuf Serv Oper Manage 19(4):551–567 Chen Y, Xue W, Yang J (2013a) Technical note optimal inventory policy in the presence of a longterm supplier and a spot market. Oper Res 61(1):88–97 Chen YJ, Tomlin B, Wang Y (2013b) Coproduct technologies: product line design and process innovation. Manage Sci 59(12):2772–2789 Goel A, Gutierrez GJ (2011) Multiechelon procurement and distribution policies for traded commodities. Manage Sci 57(12):2228–2244 Hsu A, Bassok Y (1999) Random yield and random demand in a production system with downward substitution. Oper Res 47(2):277–290 Kazaz B (2004) Production planning under yield and demand uncertainty with yielddependent cost and price. Manuf Serv Oper Manage 6(3):209–224 Kazaz B, Webster S (2011) The impact of yielddependent trading costs on pricing and production planning under supply uncertainty. Manuf Serv Oper Manage 13(3):404–417 Kleindorfer PR, Wu DJ (2003) Integrating longand shortterm contracting via businesstobusiness exchanges for capitalintensive industries. Management Sci 49(11):1597–1615
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Li B, Boyabatlı O, Avcı B (2020) Economic and environmental implications of biomass commercialization in agricultural processing. SMU Working Paper Meyer G (2013) Agricultural traders in a sweat over us drought. https://www.ft.com/content/ 018967b460ba11e2a31a00144feab49a. Accessed 29 Sept 2020 Mieghem JAV, Rudi N (2002) Newsvendor networks: inventory management and capacity investment with discretionary activities. Manuf Serv Oper Manage 4(4):313–335 Ng TS, Fowler J, Mok I (2012) Robust demand service achievement for the coproduction newsvendor. IIE Trans 44(5):327–341 Pei PPE, SimchiLevi D, Tunca TI (2011) Sourcing flexibility, spot trading, and procurement contract structure. Oper Res 59(3):578–601 Plambeck E, Taylor T (2013) On the value of input efficiency, capacity efficiency, and the flexibility to rebalance them. Manuf Serv Oper Manage 15(4):630–639 Rao US, Swaminathan JM, Zhang J (2004) Multiproduct inventory planning with downward substitution, stochastic demand and setup costs. IIE Trans 36(1):59–71 Tomlin B, Wang Y (2008) Pricing and operational recourse in coproduction systems. Manage Sci 54(3):522–537 Wu DJ, Kleindorfer PR (2005) Competitive options, supply contracting, and electronic markets. Manage Sci 51(3):452–466 Wu DJ, Kleindorfer PR, Zhang JE (2002) Optimal bidding and contracting strategies for capitalintensive goods. Eur J Oper Res 137(3):657–676 Xu J, Gürbüz MC, Feng Y, Chen S (2020) Optimal spot trading integrated with quantity flexibility contracts. Prod Oper Manage 29(6):1532–1549
Chapter 6
The Influence of YieldDependent Trading Costs on Pricing and Production Planning Under Supply Uncertainty Burak Kazaz and Scott Webster
6.1 Introduction Due to the high fluctuations in the crop supply, agribusinesses typically perceive the price of fruit in the open market as random. The Ayvalık Chamber of Commerce in Turkey, the region that provides more than 70% of the country’s production of olive oil, reports that yielddependent purchasing cost of olives is influential in the Turkish olive oil industry. Kazaz (2004) is the first to introduce the relationship between lower crop yields and the higher purchasing cost of olives as “yielddependent.” Other agricultural products experience a similar behavior; wine grapes, for example, also have yielddependent cost structure: the purchasing cost of Australian wine grapes was averaging at $100 per ton in 2006 when there was an oversupply; this price was even lower than the cost of growing the fruit. The cost of the same wine grapes was five to thirty times higher in the next year due to a long drought that reduced the crop supply by more than 50%: wine grape prices ranged between $500 and $3,000 per ton in 2007 according to the The New York Times on April 17, 2008. The fluctuations in yields of citrus, olive, and grapes necessitate the introduction of a yielddependent trading cost structure into pricing and production planning decisions. Our study examines a firm’s combined decisions of price setting and production planning while operating under yielddependent trading cost and revenue (generated from selling the fruit in the open market). We compare our findings with models that employ static cost parameters. We find that agribusinesses operate in an
B. Kazaz () Whitman School of Management, Syracuse University, Syracuse, NY, USA email: [email protected] S. Webster W. P. Carey School of Business, Arizona State University, Tempe, AZ, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_6
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increased level of risk with lower expected profits due to the yielddependent nature of the purchasing costs. Firms in the agricultural industry counter the uncertainty in cost and crop yields by employing three methods. First, they can lease farm space in order to grow fruit in anticipation of reducing the future purchasing costs. Leasing is less costly in expectation than purchasing costs and it widely utilized in agricultural settings. The lease is determined by the number of trees (or vines), and the producer is responsible for the cost of growing the fruit and maintaining the farm space, including pruning, stem cutting, weed control, and insect and disease management. Second, the firm can trade fruit in the open market. It can purchase additional fruit from other growers when the realized supply is insufficient, or it can sell some or all of its fruits in the open market when it obtains excess fruit supply. We consider the fact that the unit purchasing cost and the unit spotselling revenue of the fruit will change with the amount of realized supply. It is important to note here that agricultural businesses cannot use inventories strategically to battle supply uncertainty. Olives need to be pressed within 48 hours of collection in order to achieve the highest quality olive oil. Similar observations can be made for the production of fresh orange juice and wine where the fruit has to be squeezed immediately after harvest to obtain the highest quality output. Third, the firm can set prices in a way to reduce the risks stemming from yield fluctuations and cost uncertainty. The Ana Gıda Group owns more than half of Turkey’s extra virgin olive oil market share, and the firm’s 56% market share provides it with the ability to set prices. Our study highlights the value that can be gained from purchasing fruit futures. There are futures for the finished products like olive oil, frozen and concentrated orange juice, and fine wine; however, futures contracts do not exist for fruits like olives, oranges, or grapes. We find that a riskneutral firm cannot increase its expected profit through the use of fruit futures. A sufficiently riskaverse firm, however, would benefit from fruit futures. It is necessary to point out that the same riskaverse firm would not purchase fruit futures when the trading costs are defined in a static manner. As a result, fruit futures provide value to a riskaverse firm under yielddependent trading costs.
6.2 Literature Review The literature that examines production planning decisions under supply uncertainty commonly assumes that price is exogenous, e.g., Yano and Lee (1995); GrosfeldNir and Gerchak (2004); Bollapragada and Morton (1999), and Rajaram and Karmarkar (2002). Supply uncertainty in the context of multiple suppliers is examined in Tomlin (2009); Tomlin and Wang (2005); Dada et al. (2007), and Federgruen and Yang (2008). This literature assumes prices are exogenous; our paper, on the hand, incorporates an endogenous pricing decision into production planning decisions. Kazaz (2004) is the first publication that introduced the concept of yielddependent purchasing cost of fruit. Our study departs from Kazaz (2004) in several
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ways: (1) Kazaz (2004) assumes the selling price is exogenous and we consider it as endogenous to the model, (2) this study extends Kazaz (2004) by examining a riskaverse agricultural firm and the impact of having a fruit futures market, (3) the revenue from selling the fruit in the open market is also yielddependent in our study, and (4) we analyze less than perfect correlation between the yield of fruit and prices in the open market. PriceSetting Newsvendor Problem (PSNP) examines endogenous prices under demand uncertainty. Examples of these studies include Petruzzi and Dada (1999, 2001); Federgruen and Heching (1999, 2002), and Kocabıyıko˘glu and Popescu (2011). These studies do not consider supply uncertainty. Li and Zheng (2006) incorporate endogenous pricing into inventory replenishment under supply and demand uncertainty; however, the selling price is determined before supply uncertainty is realized in their model setting. Moreover, when the firm makes an emergency purchase, it occurs at static sots; thus, it does not consider yielddependent trading costs. Their model is also limited to riskneutral setting. Tang and Yin (2007) also study pricing under supply uncertainty; their demand function is linear in price and supply uncertainty is restricted to a discrete uniform distribution. Our study enhances these earlier publications by incorporating continuous and arbitrary distributions that define supply uncertainty, by generalizing the pricedependent demand function, by considering the firm’s opportunity to sell the crop in the open market, by featuring the yielddependent trading costs, and by incorporating risk aversion and the use of fruit futures. Kazaz and Webster (2015) also examine optimal pricesetting quantity decisions for a riskaverse firm operating under supply and demand uncertainty; however, their study ignores the yielddependent cost structure. Tomlin and Wang (2008) also consider endogenous prices under supply uncertainty in the context of two products where one can be substituted for a lower quality output. Their study does not consider yielddependent cost structure. Agricultural problems without yielddependent cost structure and/or endogenous pricesetting behavior are examined in other publications such as Burer et al. (2009) and Huh and Lall (2008).
6.3 The Model Our model is a twostage stochastic program. In the first stage, the firm determines the amount of farm space to be leased, denoted Q, at a unit cost of leasing cl . Randomness in supply is described as a stochastically proportional yield using multiplicative random error term, denoted with u, ˜ where u is a realization, and g(u) is the pdf defined on a support [ul , uh ] with a mean u = E[u] ˜ and a variance σ 2 . At the end of stage 1, the firm observes a crop yield of Qu. In stage 2, the firm makes the following four decisions in order to maximize its profit: (1) the selling price, denoted p, (2) the amount of realized supply from the leased farm space (internal growth) to be converted to the final product, denoted qi , (3) the amount of fruit to buy from other growers, denoted qb , and (4) the amount of fruit to sell in the open
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market, denoted qs . The sum of the converted supply and the fruit sold in the open market cannot exceed the realized crop yield, and thus, we have qi + qs ≤ Qu. The firm then presses the fruit in order to obtain the final product; cp denotes the unit processing cost and the firm incurs the cost of cp (qi + qb ). In stage 2, the firm may purchase additional fruit and increase its production amount. We denote the unit cost of buying additional fruit as yielddependent and express it as b(u); the firm then incurs b(u)qb . The firm can also sell its fruit in the open market at a yielddependent unit revenue, denoted s(u), and the firm earns s(u)qs . We assume that there is a positive spread, defined as δ(u) = b(u)−s(u) > 0 for all u, between the unit purchasing and selling costs of fruit in the open market. This prohibits an arbitrage opportunity and the firm cannot make additional profits by buying and selling crop at the same time. We make no assumptions regarding the shape of the yielddependent costs except that both b(u) and s(u) are decreasing in yield u (we use the terms increasing/decreasing and positive/negative in their weak sense throughout this study). We also do not assume that expected trading costs are equal to their costs at the expected yield. In our study, E[b(u)] ˜ = b(u) and E[s(u)] ˜ = s(u). Our conclusions are robust because the findings hold under convex, concave, and other forms of yielddependent fruit trading cost functions. We consider that the demand d(p) is decreasing in price and there exists a unique inverse p(d). We assume that the revenue p(d)d is strictly concave implying that 2p (d)+p"(d)d < 0. Our model considers that the remaining finished product at the end of the selling season is salvaged at a unit revenue of s2 where s2 ≤ cp + s(uh ). The model can be expressed as follows: uh Stage 1: max E[(Q)] = −cl Q + P (Q, u)g(u) du. Q≥0
Stage 2:
Given Q and u, P (Q, u) =
ul
max
(p,qi ,qb ,qs )≥0 qi +qs ≤Qu
π(p, qi , qb , qs Q, u),
where π(p, qi , qb , qs Q, u) = p min{(qi + qb ), d(p)} − cp (qi + qb ) − b(u)qb + s(u)qs + s2 ((qi + qb ) − d(p))+ .
(6.1)
From s2 ≤ cp + s(uh ) and p (d) < 0, we have qi + qb = d(p), and from ∂π/∂qs = s(u) > 0, we have qi + qs = Qu in the secondstage optimal solution. Therefore, the secondstage problem can be rewritten as follows: P (Q, u) =
max
(p,qi )≥0 qi ≤min{d(p),Qu}
π(p, qi Q, u),
where π(p, qi Q, u) = (p − cp − b(u))d(p) + (b(u) − s(u))qi + s(u)Qu.
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6.4 Model Analysis 6.4.1 The Case of No Trading of the Fruit We begin the analysis with the variant of the problem that uses deterministic supply; we replace the supply random variable u˜ with its mean u. Thus when the firm leases Q units of farm space, and it realizes a crop yield of Qu. In the second stage, due to the lack of a trading option (buying or selling) of the fruit (i.e., qb = qs = 0), the firm converts the entire crop to the final product. In this setting, the selling price clears the production, and therefore, we obtain d(p) = qi = Qu. The firststage objective is then (Q) = p(Qu) − ((cl /u) + cp ) Qu. Remark 1 (a) The optimal amount of farm space to be leased, denoted by Q0 , under deterministic supply satisfies p(Q0 u)u + p (Q0 u)Q0 (u)2 = cl + cp u; (b) the optimal deterministic profit, denoted by (Q0 ), is (Q0 ) = −p (Q0 u)(Q0 u)2 . We next examine the firm’s objective function under supply uncertainty: E[(Q)] = −(cl + cp u)Q + = (Q) −
uh
p(Qu)Qug(u)du ul
uh
[p(Qu) − p(Qu)]Qug(u)du.
(6.2)
ul
Proposition 1 (a) The firststage objective function is concave in Q, and the optimal amount of farm space to be leased satisfies
uh
[p(Qu)u + p (Qu)Qu2 ]g(u) du = cl + cp u;
(6.3)
ul
(b) the optimal profit is E[π(Q∗ )] = −
uh
p (Q∗ u)(Q∗ u)2 g(u) du
ul
and is less than its deterministic equivalent; and
(6.4)
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(c) if 3p d) + p (d) d ≤ 0, then the optimal amount of farm space to be leased is less than that of the deterministic supply, i.e., Q∗ < Q0 .
Proof The proof follows from Kazaz and Webster (2011).
Using a linear demand function d(p) = a−bp, the optimal amount of farm space to be leased and the optimal profit expressions under deterministic and stochastic supply can be expressed as follows: [a − b((cl /u) + cp )] Q0 < Q0 , = 2 2u(1 + cv ) 1 + cv 2 2(u + σ 2 ) 2 [a − b((cl /u) + cp )]2 u2 (Q0 ) ∗ 2 1 + cv = E[(Q∗ )] = = [Q u] < (Q0 ), 2 2 b 1 + cv 2 4b(u + σ ) Q∗ =
[a − b((cl /u) + cp )]u 2
=
where cv = σ/u is the coefficient of variation of supply uncertainty. The above expressions do not depend on the form of the pdf of supply uncertainty. Moreover, both the mean and the variance terms influence the optimal amount of farm space to be leased and the optimal profit. Preserving the mean supply, the above inequality shows that both the optimal amount of farm space to be leased and the optimal expected profit decrease in cv.
6.4.2 Incorporating the Trading Option We next examine the flexibility to trade the fruit crop in the open market with the ability to purchase additional fruit (i.e., qb ≥ 0) and the option of selling fruit supply (i.e., qs ≥ 0). We present the analysis for the buying and selling options independently. Let us set qb = 0 in Eq. 6.1. The unit selling revenue s(u) is salvage revenue, but it is the yielddependent. The firm would only sell its fruit when the realized crop supply is high. When the realized yield is low, the firm sets a higher selling price of its product in order to compensate for the lack of production. This leads to a threshold for a maximum production level for each realization of u, where the remaining crop would be sold in the open market. We denote this threshold by T S(u) and its value can be obtained by setting Qu = 0. As a result, the objective function in Eq. 6.1 becomes a maximization over a single variable qi , and the optimal value of qi provides T S(u). Proposition 2 The threshold for the amount of final product to be produced is T S(u) = −[p∗ − cp − s(u)]d (p∗ )
(6.5)
and is increasing in u. Proof The proof follows from Kazaz and Webster (2011).
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We next set qs = 0 in Eq. 6.1 and analyze the fruit buying flexibility. Recall that b(u) is the yielddependent and the firm would buy additional fruit when the realized crop supply is low. The threshold for the buying option, denoted T B(u), can be obtained by setting qi = 0. The objective function in Eq. 6.1 becomes a maximization problem over a single variable qb . Proposition 3 The threshold for the amount of fruit to be purchased in the open market is T B(u) = −[p∗ − cp − b(u)]d (p∗ ) and is increasing in u. Proof The proof follows from Kazaz and Webster (2011).
We next combine the flexibilities from buying and selling fruit in the open market conditions. It is important to observe that T B(u) < T S(u) as a consequence of b(u) > s(u). Proposition 4 For a given realized yield of Qu, the optimal decisions for the selling price, the amount of crop yield to be converted to finished product, the amount to purchase from other growers, and the amount to sell in the open market are
p∗ =
(qi∗ , qb∗ , qs∗ ) =
⎧ ⎪ ⎪ ⎨p(T B(u)) if Qu ≤ T B(u)
p(Qu) if T B(u) ≤ Qu ≤ T S(u) ⎪ ⎪ ⎩p(T S(u)) if Qu ≥ T S(u) ⎧ ⎪ if Qu ≤ T B(u) ⎪ ⎨(Qu, T B(u) − Qu, 0) (Qu, 0, 0) ⎪ ⎪ ⎩(T S(u), 0, Qu − T S(u))
if T B(u) ≤ Qu ≤ T S(u) if Qu ≥ T S(u),
and the optimal secondstage profit is π(p∗ , qi ∗ , qb ∗ , qs ∗ Q, u) ⎧ ⎪ ⎪[p(T B(u)) − cp − b(u)]T B(u) + b(u)Qu if Qu ≤ T B(u) ⎨ = [p(Qu) − cp ]Qu if T B(u) ≤ Qu ≤ T S(u) ⎪ ⎪ ⎩[p(T S(u)) − c − s(u)]T S(u) + s(u)Qu if Qu ≥ T S(u). p Proof The proof follows from Kazaz and Webster (2011).
We next examine and define the following three sets of yield realizations for each value of Q: B(Q) = {u : T B(u) > Qu, u ∈ [ul , uh ]}
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N(Q) = {u : T B(u) ≤ Qu ≤ T S(u), u ∈ [ul , uh ]} S(Q) = {u : T S(u) < Qu, u ∈ [ul , uh ]}. We can see that B(Q), N(Q), and S(Q) are a partition of [ul , uh ], i.e., B(Q) ∪ N (Q)∪S(Q) = [ul , uh ] and B(Q)∩N(Q) = N(Q)∩S(Q) = B(Q)∩S(Q) = ∅. The set B(Q) contains values of u where it is optimal to buy fruit in the open market, the set N (Q) contains values of u where it is optimal to not trade fruit in the open market, and the set S(Q) contains values of u where it is optimal to sell fruit in the open market. It is important to highlight that, in the deterministic demand version of Kazaz (2004) with yielddependent exogenous price model, we do not have the “no trade” region of N(Q) in the optimal policy structure. We incorporate the optimal secondstage decisions from above into Eq. 6.2. E[(Q)] = −cl Q +
[[p(T B(u)) − cp − b(u)]T B(u) + b(u)Qu]g(u)du B(Q)
[(p(Qu) − cp )Qu]g(u)du
+
N (Q)
[[p(T S(u)) − cp − s(u)]T S(u) + s(u)Qu]g(u) du.
+ S(Q)
(6.6) Proposition 5 The firststage objective function in Eq. 6.6 is concave in Q. Proof The proof follows from Kazaz and Webster (2011).
The expected profit in Eq. 6.6 is a convex combination of the expected profits from three policies: buy additional fruit from the open market, do not trade fruit, and sell fruit in the open market. The concavity of the objective function is obtained without limiting the pdf of supply uncertainty. The next proposition shows that the value of the trading option is decreasing in the spread between the yielddependent purchasing cost and selling revenues of ˆ the fruit. For any b(u) and s(u) functions, let b(u) = b(u) + /2 and sˆ (u) = ˆ s(u) − /2. Replacing b(u) and s(u) in Eqs. 6.5 through 6.6 with b(u) and sˆ (u), we let E[∗ ()] denote the optimal expected profit with the option of trading in the open market and E[N T ∗ ] denote the optimal expected profit in the absence of trading flexibility. The value gained from the trading option is then VT∗ () = E[∗ ()] − E[∗N T ]. Proposition 6 The value of trading in the open market, VT∗ (), is decreasing in . Proof The proof follows from Kazaz and Webster (2011).
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6.4.3 The Impact of YieldDependent Trading Costs We numerically illustrate the impact of yielddependent cost and revenue structure on a firm’s decisions and profits by using cost and demand data from two olive oil producers in Turkey. One firm uses a convexdecreasing yielddependent purchasing cost, and the other firm uses a linearly decreasing yielddependent cost. The yielddependent cost functions are defined as s(u) = αγ − βγ uγ − δ/2 and b(u) = αγ − βγ uγ + δ/2, where γ ∈ {0, 1, 0.5, 0.25} and δ ∈ {2, 3, 4}. The case of γ = 0 represents the static open market trading costs. The case of γ = 1 represents the case when the yielddependent cost function is linearly decreasing in the yield. The other two values of γ , 0.50 and 0.25, reflect different degrees of convexity in b(u) and s(u); convexity increases as γ approaches 0 in these settings. We assume u˜ is uniform on [0, 1]. Table 6.1 provides the values of αγ and βγ . We set the values of αγ and βγ such that E[s(u)] ˜ + δ/2 = E[b(u)] ˜ − δ/2 = 7.09 for all γ . We use the demand function and the cost parameters provided Firm 1 and set the demand as d(p) = 270,000 − 9,000p, the leasing cost per unit as cl = 2.93, and the processing cost per unit as cp = 2.97.
6.4.3.1
The Impact on Expected Profit
We report the firm’s optimal decision for the amount of farm space to be leased in Table 6.2. The table also reports expected profit under various values of the spread (δ) and different trading cost functions (static, linear, and convex). As shown in Table 6.2, a yielddependent cost structure reduces profits of agricultural businesses, i.e., E[(Q∗ δ, 0 < γ ≤ 1)] < E[(Q∗ δ, γ = 0)]. The inequality holds for two reasons. First, at low realizations of supply, the firm prefers to purchase additional fruit from the open market. Under a yielddependent cost structure, however, the unit purchasing cost is higher under low realizations, and thus the firm pays more for the additional fruit. Second, when the realized supply is high, the firm prefers to sell some of its crop in the open market. Under a yielddependent cost structure, the unit revenue is smaller at higher realizations of the yield. These reasons also explain why the optimal amount of leased farm space under a yielddependent cost structure is less than that under static costs, i.e., Q∗δ, 0 cl /u. Otherwise, there would be no businesses that would lease farm space to grow fruit and sell in the open market because expected profit would be negative. However, if E[s(u)] ˜ > cl /u and the firm has static trading costs, then the expected profit is increasing in leased farm space, resulting in an infinite amount of initial investment. In contrast, under a yielddependent cost structure, investment in leased farm space exhibits diminishing returns leading to the finite optimal space decisions in Table 6.2.
6.4.3.2
The Impact on Pricing and Production Decisions
If cost is static, then the firm’s pricing and production decisions do not depend on supply uncertainty when the firm is in the trading mode. Pricing and production decisions are only influenced by the yield and are decreasing in the yield when the firm is not trading in the open market. Under yielddependent costs, the optimal selling price is decreasing and the optimal production quantity is increasing continuously. This is due to two reasons. First, the firm is leasing much less farm space. Second, buying cost is convex decreasing in the yield, resulting in a significant reduction in the purchasing cost after the lowest values of u. Consequently, the buying cost is cheaper under the yielddependent cost structure for some values of u. Finally, under static cost, the firm sells in the open market over a larger interval of realized u. Aside from leasing a smaller amount of farm space, this result is because the selling revenue is significantly larger for the static cost than the yielddependent cost curve in this region. Under the yielddependent cost structure, the firm relies heavily on buying additional fruit from the open market.
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The Impact of Less Than Perfect Correlation Between Trading Costs and Yield
In the preceding analysis, the fruit trading costs and random yield are perfectly correlated, i.e., the trading cost is solely determined by the realization of u. ˜ Suppose trading costs are not perfectly correlated with u. ˜ It can be shown that the structure of the firststage profit function remains the same. The optimal expected profit increases when shifting from perfect correlation to less than perfect correlation.
6.5 Value of Fruit Futures In this section we analyze the value of fruit futures for mitigating supply risk. Presently, there is no futures market for fruit (e.g., olives, oranges, and grapes). Suppose that a futures market exists for fruit at a price cf equal to the expected buying cost, i.e., cf = E[b(u)]. ˜ Setting cf = E[b(u)] ˜ is the lower limit on the price in a viable futures market (i.e., corresponding to the case where expected profit of selling futures short is zero). Let Qf denote a firststage decision on the amount of fruit futures purchased at the beginning of the growing season at price cf . Then, the revised model is Stage 1:
max E[(Q, Qf )]
Q,Qf ≥0
=
uh
(P (Q, Qf , u) − cl Q − cf Qf )g(u) du.
(6.7)
ul
Stage 2:
Given Q, Qf , and u, P (Q, Qf , u) =
max
(p,qi ,qb ,qs )≥0 qi +qs ≤Qu+Qf qi +qb ≤d(p)
π(p, qi , qb , qs Q, Qf , u).
where π(p, qi , qb , qs Q, Qf , u) = (p − cp )(qi + qb ) − b(u)qb + s(u)qs
(6.8)
At the beginning of second stage, the amount of fruit available is Qu + Qf . We note that Propositions 2 through 4 continue to hold for the secondstage problem, except that Qu is replaced with Qu + Qf . Proposition 7 The firststage objective function in Eq. 6.7 is jointly concave in Q and Qf . Proof The proof follows from Kazaz and Webster (2011).
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Proposition 8 E[(Q, 0)] ≥ E[(Q, Qf )] for any Q, Qf ≥ 0.
Proof The proof follows from Kazaz and Webster (2011).
Proposition 8 shows that fruit futures do not improve the firm’s profitability in a riskneutral setting but can add value under a riskaverse objective function. We model risk aversion with a concave utility function U (x) from profit x where U (x) > 0 and U (x) ≤ 0. The secondstage problem is unaffected by the introduction of a riskaverse utility function (i.e., utility maximization requires that the deterministic secondstage profit be maximized). Thus, the riskaverse model can be described as follows: Stage 1:
max E[U ((Q, Qf ))]
Q,Qf ≥0
=
uh
U (P (Q, Qf , u) − cl Q − cf Qf )g(u) du.
ul
(6.9)
Stage 2:
Given Q, Qf , and u, P (Q, Qf , u) =
max
(p,qi ,qb ,qs )≥0 qi +qs ≤Qu+Qf qi +qb ≤d(p)
π(p, qi , qb , qs Q, Qf , u).
Proposition 9 The objective function in Eq. 6.9 is jointly concave in Q and Qf .
Proof The proof follows from Kazaz and Webster (2011). Proposition 10 If b(u) and s(u) are ≥ E[U ((Q, Qf ))] for any Q, Qf ≥ 0.
static,
Proof The proof follows from Kazaz and Webster (2011).
then
E[U ((Q, 0))]
Proposition 10 shows that, under static trading costs, the firm does not benefit from fruit futures regardless of the firm’s degree of risk aversion. However, as shown below, fruit futures can add value under a yielddependent cost structure. We employ a constant absolute risk aversion (CARA) utility function, U (x) = 1 − e−rx with the risk aversion coefficient r = 0.1. In our numerical illustrations, we compute the profit in our utility function in currency units of 100,000 (e.g., the firm realizes 50% of the maximum possible utility with realized profit of approximately 700,000). We use the same cost parameters described in Sect. 6.4.3. Table 6.3 reports the optimal lease quantity Q∗ , the optimal futures Q∗f , the optimal fruit commitment at time zero, and the corresponding value of the utility function. Results are reported in the presence and absence of fruit futures. The following three factors influence the optimal decisions:
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1. The shape of yielddependent cost curve (γ ) When the yielddependent cost definition switches from a linear to a convex form (e.g., from γ = 1 to γ < 1), the firm increases futures and reduces farm space. Note that, for a given value of spread δ, the trading costs of fruit are different as γ changes, while the cost of buying futures remains the same. The cost of fruit in the open market is much higher at significantly low yield realizations under a convex cost function compared to a linear cost function. Thus, the firm reduces the negative effects of yielddependent purchasing cost of the fruit by increasing futures. In sum, the firm trades off leased farm space for an increase in fruit futures. 2. Spread (δ) An increase in the spread between the buying cost and the selling revenue of fruit results in increased farm space and fewer futures. As spread increases, the relative cost of buying fruit in the open market becomes more expensive, which in turn causes an increase in the unit cost of futures. Thus, higher values of spread make the leasing option relatively more preferable over futures and fruit trading options. 3. Degree of Risk Aversion (r) Futures do not always add value under risk aversion. A firm operating under yielddependent trading costs must be sufficiently riskaverse in order to benefit from fruit futures. For example, if r ≤ 0.04 instead of r = 0.1 (i.e., the firm is less riskaverse), then Q∗f = 0 and it is never optimal to purchase futures. We next describe how a futures market affects the expected fruit commitment at time zero and the optimal farm lease quantity. Early Fruit Commitment Early fruit commitment is higher in the presence of a fruit futures market. This is evident from comparing Q∗ u + Q∗f (futures market) with Q∗ uQf = 0 (no futures market) in Table 6.3. While increased convexity in the yielddependent cost structure leads to reduction in the expected fruit commitment in the absence of fruit futures, it causes an increase in the initial expected fruit commitment in the presence of fruit futures. Farm Lease Quantity The riskaverse lease quantity is less than the riskneutral lease quantity for all instances. This is evident from comparing column 7 in Table 6.3 (Q∗ Qf = 0) with column 3 of Table 6.2. The results mirror the behavior of the classic newsvendor model where risk aversion leads to lower order quantities relative to the optimal riskneutral quantity (see, Eeckhoudt et al. 1995). Agrawal and Seshadri (2000) find the same behavior for a pricesetting newsvendor under multiplicative random demand. However, in contrast with the newsvendor literature, the optimal lease quantity for a riskaverse firm can be larger than the optimal lease quantity for a riskneutral firm. For example, at δ = 3, γ = 1, and uniform random yield, we have Q∗RA > Q∗RN when ul > 0.15, e.g., at ul = 0.3, Q∗RA = 217,740 and Q∗RN = 212,662. When ul is low, the benefits of trading (i.e., the opportunity to purchase additional fruit from the open market) put a downward pressure on the optimal lease amount. Unlike the setting where uncertainty comes from demand,
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Table 6.3 The optimal investments in the farm space to be leased, fruit futures, fruit commitment at time zero, and the utility function value in the presence versus the absence of fruit futures under various values of spread and different trading cost functions as presented in Kazaz and Webster (2011) δ
γ
Q∗
Q∗f
Q∗ u+ Q∗f
U [(Q∗ , Q∗f )]
Q∗  Qf = 0
U [(Q∗  Qf = 0)]
Q∗ u Qf = 0
3 3 3 4 4 4
0.00 1.00 0.50 0.00 1.00 0.50
143,143 65,248 57,163 141,869 81,906 67,081
0 50,201 56,308 0 37,646 49,425
71,572 82,826 84,890 70,935 78,599 82,965
0.56822 0.55296 0.55370 0.57872 0.54188 0.54157
143,143 117,034 79,079 141,869 122,605 116,820
0.56822 0.54220 0.53334 0.57872 0.53512 0.52451
71,572 58,517 39,539 70,935 61,302 58,410
the initial investment amount for a firm facing supply risk is not monotone when the objective function switches from riskneutral to riskaverse. We conclude with the following observations: (1) yielddependent trading costs present a riskier environment for agricultural businesses compared to static costs; (2) the agricultural firm’s supply risk increases with the convexity of yielddependent trading costs, (3) a sufficiently riskaverse firm may benefit from a futures market, (4) in setting where the firm benefits from a futures market, the consumers benefit as well because the larger expected fruit commitment translates into a lower expected prices for the consumer, and (5) in contrast to the newsvendor literature, the firm’s quantity decision can increase in risk aversion.
6.6 Conclusions We study the impact of a yielddependent cost structure on the pricing and production planning decisions of an agricultural firm that operates under supply uncertainty. It is common in agricultural industry to lease farm space in order to grow fruit, but leasing farm space introduces supply uncertainty with random yield. After harvesting the crop, the firm can buy additional fruit (or sell fruit) in the open market. Traditional modeling approaches are generally inspired by repetitive manufacturing activities where the unit cost of acquiring additional raw materials/ingredients does not depend on random supply—they are static costs. However, random yield causes agricultural businesses to operate under yielddependent costs. The unit purchasing cost of fruit changes from one year to the next, and its value is impacted by the realized supply in the region. Open market prices are higher (lower) when yield is low (high). We show that the expected profit is concave in leased farm space. Our conclusions are robust as they are derived under general pricedemand functions without limitations on the probability distribution of supply uncertainty. Moreover, we make
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no assumptions on the shape of the yielddependent cost functions, i.e., they do not need to be convex or concave. Our analysis yields three main conclusions. First, incorporating the yielddependent cost structure into the problem has a profound impact on optimal investment in farm space and on expected profit. Agricultural businesses that commonly operate under yielddependent trading costs face high supply risk compared to firms that operate under static costs. Using data available from a Turkish olive oil producer, we show that the agricultural firm leases a smaller amount of farm space under a yielddependent cost structure compared with static costs (approximately half the size). Moreover, the firm’s expected profit is lower. In essence, the benefits from the options of buying and selling fruit in the open market under the yielddependent cost structure are muted compared with the static cost scheme. The firm purchases additional fruit when it realizes a lower supply, and these costs are higher under a yielddependent cost structure compared to static cost. On the flipside, the firm prefers to sell some of its fruits when it obtains a high crop supply, and the selling revenue of fruit in the open market is also lower under a yielddependent cost structure compared to static cost. We find that the cost of ignoring the yielddependent cost structure increases tremendously with decreasing values of the spread in the open market buying and selling price. Second, we identify conditions under which an agricultural firm may benefit from future futures. A riskneutral firm does not benefit from fruit futures. However, a riskaverse firm can use fruit futures to mitigate the risk of high unit purchasing costs when yield is low. If firms are sufficiently riskaverse, then establishing fruit futures can help agricultural firms mitigate supply risks. We find a riskaverse firm would not purchase fruit futures when operating under static costs, i.e., fruit futures can add value only when the firm is operating under a yielddependent cost structure. Moreover, we show that increasing convexity in the yielddependent cost structure increases investment in fruit futures and reduces investment in the leased farm space. Third, under supply uncertainty, leased farm space does not follow a monotone behavior when shifting from riskneutral to riskaverse. This is in contrast to the classic Newsvendor Problem under demand uncertainty where it is well known that a riskaverse firm commits to a smaller initial quantity than a riskneutral firm. The flexibility to trade fruit in the open market puts a downward pressure on the initial lease quantity. However, a riskaverse firm may still commit to a higher leased farm space than a riskneutral firm even in the presence of the trading flexibility.
References Agrawal V, Seshadri S (2000) Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manuf Serv Oper Manag 2(4):410–423 Blackburn J, Scudder G (2008) Supply chain strategies for perishable products: The case of fresh produce. Prod Oper Manag 18(2):129–137 Bollapragada S, Morton TE (1999) Myopic heuristics for the random yield problem. Oper Res 47(5):713–722
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Bure S, Jones PC, Lowe TJ (2009) Coordinating the supply chain in the agricultural seed industry. Eur J Oper Res 185:354–377 Dada M, Petruzzi N, Schwarz L (2007) A newsvendor’s procurement problem when suppliers are unreliable. Manuf Serv Oper Manag 9(1):9–32 Eeckhoudt L, Gollier C, Schlesinger H (1995) The riskaverse (and prudent) newsboy. Manag Sci 41(5):786–794 Federgruen A, Heching A (1999) Combined pricing and inventory control under uncertainty. Oper Res 47(3):454–475 Federgruen A, Heching A (2002) Multilocation combined pricing and inventory control. Manuf Serv Oper Manag 4(4):275–295 Federgruen A, Yang N (2008) Selecting a portfolio of suppliers under demand and supply risks. Oper Res 56(4):916–936 GrosfeldNir A, Gerchak Y (2004) Multiple lotsizing in production to order with random yields: review of recent advances. Ann Oper Res 126:43–69 Huh WT, Lall U (2008) Optimizing crop choice and irrigation allocation under contract farming. Working paper Jones PC, Lowe T, Traub RD, Keller G (2001) Matching supply and demand: The value of a second chance in producing hybrid seed corn. Manuf Serv Oper Manag 3(2):116–130 Kazaz B (2004) Production planning under yield and demand uncertainty with yielddependent cost and price. Manuf Serv Oper Manag 6(3):209–224 Kazaz B, Webster S (2011) The impact of yielddependent trading costs on pricing and production planning under supply uncertainty. Manuf Serv Oper Manag 13(3):404–417 Kazaz B, Webster S (2015) Technical note – Pricesetting newsvendor problems with uncertain supply and risk aversion. Oper Res 63(4):807–811 Kocabıyıko˘glu A, Popescu I (2011) The newsvendor with pricing: A stochastic elasticity perspective. Oper Res 59(2):301–312 Li Q, Zheng S (2006) Joint inventory replenishment and pricing control for systems with uncertain yield and demand. Oper Res 47(2):183–194 Petruzzi N, Dada M (1999) Pricing and the newsvendor problem: A review with extension. Oper Res 47(2):183–194 Petruzzi N, Dada M (2001) Information and inventory recourse for a twomarket, pricesetting retailer. Manuf Serv Oper Manag 3(3):242–263 Rajaram K, Karmarkar US (2002) Product cycling with uncertain yields: analysis and application to the process industry. Oper Res 47(2):183–194 Tang C, Yin R (2007) Responsive pricing under supply uncertainty. Eur J Oper Res 182:239–255 Tomlin B (2009) The impact of supply learning when suppliers are unreliable. Manuf Serv Oper Manag 11(2):192–209 Tomlin B, Wang Y (2005) On the value of mix flexibility and dual sourcing in unreliable newsvendor networks. Manuf Serv Oper Manag 7(1):37–57 Tomlin B, Wang Y (2008) Pricing and operational recourse in coproduction systems. Manag Sci 54(3):522–537 Yano CA, Lee H (1995) Lot sizing with random yields: A review. Oper Res 43(2):311–334
Chapter 7
Capacity Management in Agricultural Commodity Processing Onur Boyabatlı and Jason (Quang) Dang Nguyen
7.1 Introduction This chapter analyzes the capacity investment decisions of an agricultural processing firm that uses a primary commodity input (e.g., palm fruit, soybean, and coconut) to produce a commodity output (e.g., palm oil, soybean oil, and coconut oil) and a byproduct (e.g., palm kernel, soybean meal, and coconut cake). The results presented in this chapter parallel those from our companion paper, Boyabatlı et al. (2017), but they are obtained using a more general model of the processing firm than our companion paper. We characterize the investment decisions related to input processing capacity and output storage capacity, which are essential to several important oilseed industries (e.g., palm, soybean, rapeseed, sunflower seed, and coconut), as well as the grain industry (e.g., corn and wheat). According to the U.S. Department of Agriculture’s Foreign Agriculture Service Report,1 the production volume of the major crude vegetable oils in the year 2019 corresponds to 210 million tonnes which account for an estimated market value of $270 billion. Processing firms in agricultural industries are exposed to several uncertainties that pose unique challenges for capacity management. First, since both the commodity input and the commodity output are traded on spot markets (Devalkar et al.
1 http://www.fas.usda.gov/psdonline.
O. Boyabatlı () Lee Kong Chian School of Business, Singapore Management University, Singapore email: [email protected] J. (Quang) Dang Nguyen Ivey School of Business, Western University, London, ON, Canada email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_7
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Fig. 7.1 Daily spot prices (in Malaysian ringgit per metric ton) of crude palm oil (CPO) and palm fresh fruit palm bunches (FFBs) in the Malaysian Peninsula for the period of January 2006 to December 2013, as reported by the Malaysian Palm Oil Board. (a) Daily Spot Prices of CPO and (b) Daily Spot Prices of FFB
2011), processing firms face uncertainties in both input and output spot prices. These spot prices are closely linked and exhibit considerable variability, as shown for the palm industry in Fig. 7.1. The uncertainty in these spot prices plays a key role in capacity management because the profit from processing depends on both input sourcing and output selling prices. Moreover, the processor can hold output inventory for sale at a later date in order to benefit from fluctuations in the output spot price, where that inventory can be sourced from inhouse production (Fackler and Livingston 2002) and spot market (Kouvelis and Ding 2013). Second, there is uncertainty in the output yield (extraction rate) from each input, which is driven by several factors that include weather conditions and the extent of pests and diseases during the input’s growing period (Boyabatlı et al. 2020), the harvest timing of the input, and the processing technology used (Chang et al. 2003). It is important to consider this uncertainty in capacity planning because the yield affects the amount of obtained output, and thus, the resulting processing profits. Given these characteristics, our research objective is twofold. First, we seek to understand how an agricultural processing firm should determine the optimal capacity investment portfolio—including the input processing and output storage capacity levels—when facing uncertainties in input and output spot prices as well as in output yield. Second, we examine the performance of the optimal capacity investment policy against heuristic capacity investment policies that are commonly used or potentially can be used in practice. Because these heuristic policies ignore some operational factors (e.g., output yield uncertainty, byproduct revenue) during capacity planning, this performance comparison is instrumental in understanding the criticality of these operational factors for capacity investment to generate valuable managerial insights. To this end, we develop a stylized model of the processing firm’s decisions as a multiperiod optimization problem over a finite planning horizon. In each period, the firm (i) procures an input commodity at its prevailing (input) spot price, (ii) sells the produced output commodity at its prevailing (output) spot price, and (iii) generates
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a fixed marginal revenue by selling the byproduct. Since the commodity output is traded on the spot market, the firm can also procure the output (at its prevailing spot price) for storage to be sold in later periods. At the beginning of the planning horizon, the firm chooses the input processing and output storage capacity levels. In the rest of the planning horizon, the firm periodically makes decisions about the processing volume and output inventory while constrained by these capacity levels. More specifically, in each period, the processing volume is chosen with respect to output yield uncertainty, and the output inventory level is chosen after this uncertainty is realized. Using this model, we characterize the firm’s optimal decisions in closed form. We identify three optimal capacity investment strategies. When the investment cost of processing capacity relative to storage capacity is sufficiently high, the firm invests in a storagedominating portfolio, where the storage capacity is strictly greater than what is required for production (with full utilization of processing capacity) under all yield realizations. When that relative cost is sufficiently low, the firm invests in a processingdominating portfolio, where the processing capacity is analogously strictly greater than what is required for production (with full utilization of storage capacity) under all yield realizations. In all other cases, the firm invests in a mixed portfolio. We find that higher investment cost of one capacity type decreases the level of other capacity type. This result is important from a managerial perspective because it underscores the need to ensure that the effects of each capacity type’s investment cost are evaluated in a holistic fashion. To address our second research objective, we conduct numerical experiments using realistic instances. We focus on the palm industry, which is one of the most important agricultural industries in South East Asia, with Malaysia and Indonesia being the two largest players in the global market. We calibrate our model using publicly available data from the Malaysian Palm Oil Board (MPOB), complemented by proprietary and publicly available data from the Malaysian palm oil processing firms (mills). Our numerical results suggest that the heuristic policy that sets storage capacity at the level required for full utilization of processing capacity under the maximum yield is nearly optimal. We also find that if the output yield uncertainty is ignored in capacity planning, assuming the average yield is better than assuming the maximum yield, as often done in practice. Another important finding is that although byproduct revenue constitutes a small portion of the total revenues of the firm, ignoring it during capacity planning could significantly reduce the firm’s profits. The chapter proceeds as follows. Sect. 7.2 surveys the related literature and discusses the contributions of our work. Sect. 7.3 describes the general model and the basis for our assumptions, and Sect. 7.4 derives the optimal strategy. Using a model calibration based on the palm industry, we compare the performance of the optimal capacity investment policy against heuristic policies in Sect. 7.5. Sect. 7.6 concludes with a discussion of future research directions.
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7.2 Literature Review This chapter studies the capacity investment decisions of a processing firm in the context of agricultural industries. In the operations management (OM) literature, there is a vast amount of research that studies capacity investment decisions in processing environments (for a review, see Van Mieghem 2003), but not in the context of agricultural industries. In the context of these industries, there is a stream of papers that examines processing planning for a variety of products including olives (Kazaz 2004), citrus fruit (Kazaz and Webster 2011), and processed food (Mehrotra et al. 2011). Another stream of studies focuses on addressing the yield uncertainty in agricultural processing, including Behzadi et al. (2017), Boyabatlı et al. (2020), and Serhatli et al. (2020). A recent emerging body of research examines the impact of government policy interventions to encourage agricultural innovation and support farmers’ profitability (see, e.g., Shi et al. 2019; Akkaya et al. 2021; Alizamir et al. 2019). These studies, however, do not consider capacity investment decisions which, as discussed in the Introduction section, face new challenges due to the idiosyncratic features of agricultural industries. In short, the capacity investment decisions of agricultural processing firms constitute an important lacuna in the OM literature. Because the majority of agricultural products are commodities, the literature most relevant to our paper is the OM research on commodity processing. The papers in this field examine operating decisions (e.g., processing and inventory) of a commodity processing firm in a variety of models. These studies capture the idiosyncratic features of different commodity markets, including those for energy, including natural gas, petroleum, and electricity (Secomandi 2010b; Lai et al. 2011; Dong et al. 2014; Zhou et al. 2019; Trivella et al. 2021; Glenk and Reichelstein 2020), electronic equipment (Pei et al. 2011), metals (Plambeck and Taylor 2013), and semiconductors (Kleindorfer and Wu 2003), as well as commodity markets associated with such agricultural industries such as beef (Boyabatlı et al. 2011), cocoa (Boyabatlı 2015), corn (Goel and Tanrısever 2017), and soybean (Devalkar et al. 2011). Because the focus of these papers is on operating decisions, they either assume (often implicitly) abundant processing and storage resources or consider fixed capacity levels for these resources. Our contribution consists of examining how these capacity levels are chosen, describing how they are affected by uncertainty in commodity spot prices, and comparing the profits from using the optimal capacity investment policy with those from using a heuristic policy. In the rest of this section, we discuss the differences between this chapter and our companion paper, Boyabatlı et al. (2017). Boyabatlı et al. (2017) examine a similar capacity investment problem with a special focus on the oilseeds industries. In these industries, the quality of crude vegetable oil decreases not only with metal (e.g., iron) contamination, which occurs when the storage facility is not lined with suitable protective coating, but also with solidification and fractionation, which occurs if the storage facility cannot maintain a specific temperature. This inferior quality results in a revenue loss for
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the quantity of output exceeding the storage capacity. Processing firms in practice often minimize quality issues related to storage conditions by planning operations in such a way that the entire volume of output goes through the storage facility after processing. As such, Boyabatlı et al. (2017) consider a model in which the revenue loss due to excess production is prohibitively high and so the firm adopts a policy of no excess production. We build on their model but relax this assumption to consider a more general model setting that allows for excess production. Due to the relaxation of this assumption, we extend their characterization of the optimal capacity investment portfolio.
7.3 Model Description We use the following mathematical representation throughout the chapter. Boldface letters represent row vectors. The monotonic relations are used in the weak sense. Subscript t denotes period t, superscript I denotes inputrelated parameters and decision variables, while superscript O (B) denotes the parameters and variables related to the output (byproduct). A realization of the random variable x˜ is denoted by x, whereas the expectation operator is denoted by E. We use (u)+ = max(u, 0). We consider an agricultural processing firm that produces a commodity output and a byproduct using a commodity input. The firm’s objective is to maximize its expected total (discounted) profit over a finite planning horizon by choosing input processing and output storage capacity investment levels at the beginning of the planning horizon, and subsequent input processing and output inventory volumes periodically subject to the chosen capacity levels. Let K = (K I , K O ) denote the firm’s capacity investment portfolio, where K I is the input processing capacity and K O is the output storage capacity. We assume . that the total capacity investment cost is given by C(K) = β I (K I )2 + β O (K O )2 , which is convex increasing in K. The convexity of the capacity investment costs can also be attributed to limits on production technology and increasing managerial complexity or maintenance cost with additional investment. We assume that the unit procurement cost of the commodity input is given by the prevailing input spot price, whereas the unit sales price of the commodity output is given by the prevailing output spot price. These input and output spot prices are assumed to follow correlated Markovian stochastic processes, i.e., current spot price realizations are sufficient to characterize the distribution of future spot prices. We use a singlefactor, bivariate, meanreverting price process to model how input and output spot prices evolve. Specifically, input and output spot prices at time t, P t = (ptI , ptO ), are modeled as follows: dptI = θ I (p¯ I − ptI ) + σ I d W˜ tI , dptO = θ O (p¯ O − ptO ) + σ O d W˜ tO ,
(7.1)
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where θ j > 0 is the meanreversion parameter, p¯ j is the longterm price level, and σ j is the volatility for j ∈ {I, O}; we use (d W˜ tI , d W˜ tO ) to denote the increment of a standard bivariate Brownian motion with correlation ρ. Because input and output spot prices are often positively correlated in practice, we assume ρ > 0. Note that our structural analysis in Sect. 7.4 does not depend on the detailed specification of these stochastic processes, which is, however, necessary for the numerical studies in Sect. 7.5. We further assume that the firm may also procure commodity output (from the spot market) at the prevailing output spot price for the purpose of storage to be sold at a later period, and the output storage incurs a perperiod unit holding cost of h. Turning now to the processing of inputs, we consider a perperiod unit processing cost c > 0. The production yields of the commodity output and the byproduct for each unit of the processed input are, respectively, given by a and a B . We assume that the byproduct is immediately sold at a fixed unit price pB . Therefore, . c = c − a B pB captures the effective processing cost, which can be negative if the byproduct revenue is sufficiently high. We assume that the output yield a˜ is uncertain. Moreover, a˜ has the following characteristics: (i) it is independent and identically distributed across periods, (ii) it is statistically independent of the spot price processes, and (iii) it follows a Bernoulli distribution: a = a l with probability q ∈ [0, 1] and a = a h with probability 1 − q for 0 < a l < a h ≤ 1 − a B , where the last inequality follows because the overall output yield cannot exceed 1.2 It follows that a¯ = qa l + (1 − q)a h represents the average output yield. The storage capacity K O affects processing activities because output is placed in the storage facility before being dispatched from the plant. This means that profitability declines if the output yield realized after processing exceeds the available storage capacity. To capture this phenomenon, we assume that the excess output volume (beyond the available storage capacity) is sold to the spot market and generates a marginal sales revenue that is strictly less than the output spot price. That is, the firm incurs a loss (relative to the prevailing spot price) when the excess output is sold. We assume that this loss is given by the sum of a unit cost d ≥ 0 and 1−α proportion of the prevailing output spot price; here α ∈ [0, 1). The term d may represent the cost associated with process interruption due to retrieving the excess output from facility or the cost of using temporary storage tanks to handle the excess output. The proportion 1 − α may signify the reduction in spot sale revenue due to the output’s inferior quality as a result of improper storage conditions. As discussed earlier, while this chapter assumes a general value of d (and α), our companion paper Boyabatlı et al. (2017) considers a special case of our model where d → ∞. We formulate the firm’s problem as a finitehorizon stochastic dynamic program. At period t = 0, the firm chooses the perperiod processing capacity K I (in units of, say, metric tons of input per day) and the storage capacity K O (metric tons of
structural analysis in Sect. 7.4 also holds for a general discrete distribution of a˜ with more than two realizations.
2 Our
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output), which are then fixed in all subsequent periods. In each period t ∈ [1, T ], the sequence of events is as follows: 1. At the beginning of period t, the input and the output spot prices Pt = (ptI , ptO ) are realized; the firm then decides on the input processing volume zt within the processing capacity level K I , accounting also for the output inventory level st−1 carried from period t − 1. 2. The output yield a is observed, which determines the available output volume; the output yield beyond the available storage capacity K O − st−1 is then sold to the spot market at a lower marginal sales revenue than the spot price. After observing the available output volume, the firm decides on the output inventory level st within the storage capacity K O . Required inventory that is not provided by the available output volume is procured from the spot market; the output volume that is not stored is sold to the spot market. The firm’s immediate payoff in period t ∈ [1, T ] is given by . L(zt , st  st−1 , Pt ) = −ptI zt − czt + Ea˜ (αptO − d)(az ˜ t − (K O − st−1 ))+ − ptO (st − (st−1 + min(az ˜ t , K O − st−1 )))+
+ ptO (st−1 + min(az ˜ t , K O − st−1 ) − st )+ − hst .
(7.2)
In Eq. 7.2, the first two terms capture the effective processing and procurement costs, while the remaining terms capture the cash flows resulting from the realized output yield. Within these cash flows, the first term denotes the spot sale revenue for the excess output yield beyond the available storage capacity. The second term denotes the spot procurement cost for the inventory level beyond the available output volume, and the third term denotes the spot sale revenue for the available output volume that is not stored. The last term denotes the inventory holding cost incurred. Note that the output spot procurement does not occur if there is any excess output yield after processing. Let Vt (st−1 , Pt ) for t ∈ [1, T ] be the optimal value function from period t onward given st−1 and Pt ; this function satisfies Vt (st−1 , Pt ) = s.t.
max
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zt ≤ K I , s t ≤ K O ,
with boundary condition VT +1 (sT , PT +1 ) = 0 and initial inventory level s0 = 0, where δ ∈ [0, 1] is the discounting factor and Et [·] is our shorthand notation for E[·  Pt ].3 comparison with the formulation given in our companion paper, the constraint zt ≤ (K O − st−1 )/a h does not exist because the no excess production assumption is relaxed in this chapter.
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At period t = 0, the firm chooses K = (K I , K O ) after observing P0 and thus incurs the capacity investment cost C(K) = β I (K I )2 + β O (K O )2 . It follows that the firm’s optimal expected total (discounted) profit over the planning horizon is given by ∗ = maxK≥0 δE0 [V1 (0, P˜ 1 )] − C(K).
7.4 Characterization of the Optimal Strategy Using a similar intuition as employed by Boyabatlı et al. (2017), the firm’s immediate payoff in period t ∈ [1, T ], as given by Eq. 7.2, can be decoupled into two components: . Lpr (zt  st−1 , Pt ) = −(ptI + c)zt + Ea˜ ptO (az ˜ t + st−1 )
− ((1 − α)ptO + d)(az ˜ t − (K O − st−1 ))+ ,
. Lsc (st  Pt ) = (ptO + h)st , where the subscripts “pr” and “sc” refer (respectively) to “processing return” and “storage cost”. This decoupling suggests that one can disentangle sameperiod processing and inventory decisions as independent. In particular, the firm first decides on the processing volume and sells the entire output (including the inventory carried over from the previous period) to its spot market, which generates the processing return Lpr (zt  st−1 , Pt ); and then does the firm choose the output inventory level st to (re)stock from the spot market, incurring the sourcing and storage cost Lsc (st  st−1 , Pt ). While the inventory decision affects only the subsequent period’s processing decision through limiting the available storage capacity, the processing decision is independent from any other decisions. It follows that the inventory decision in period t − 1 can be grouped with the processing decision in the subsequent period t, allowing us to rewrite the optimization problem given by Eq. 7.3 as independent twostage optimization problems, demonstrated in Fig. 7.2. Note that Lsc (·  PT ) = 0 (because inventory is not needed in period T ) and so the optimal value function in period t ∈ [1, T − 1] is given by
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! δ τ −t Et Gτ (P˜ τ ) ,
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 st , P˜ t+1 )
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(7.5) For expositional purposes, we define the following terms that are used in the remainder of this chapter. Let pmt denote the processing margin (per input), smt the storage margin (per output), and let the loss due to excess production in period t: . ¯ tO , pmt = −ptI − c + ap . O smt = −ptO − h + δEt [p˜ t+1 ], . let = (1 − α)ptO + d.
(7.6)
The processing margin pmt is given by the difference between the output spot sale revenue per expected output yield, a, ¯ and the sum of input spot procurement and unit processing costs. The storage margin smt is given by the difference between the expected spot sale revenue in the subsequent period and the spot sourcing and storage costs. The loss due to excess production let captures the reduction in the marginal spot sale revenue from the prevailing spot price when the output yield after processing exceeds the available storage capacity.
7.4.1 Periodic Input Processing and Output Inventory Decisions To derive the optimal solution for Eq. 7.5, we first characterize the optimal process∗ (s , P ing volume zt+1 t t+1 ) for a given output inventory level st in Proposition 1. All technical proofs are available upon request. ∗ (s , P Proposition 1 The optimal processing volume zt+1 t t+1 ) is given by
⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨min K O h−st , K I ∗ Oa zt+1 (st , Pt+1 ) = t ⎪ , KI min K a−s ⎪ l ⎪ ⎪ ⎪ ⎩ I K
if pmt+1 ≤ 0, if 0 < pmt+1 ≤ a h (1 − q) let+1 , if a h (1 − q) let+1 < pmt+1 ≤ a¯ let+1 , if pmt+1 > a¯ let+1 .
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The optimal processing volume is characterized by the effective processing margin per input, which depends in turn on the sufficiency of the storage capacity K O − st under each yield realization; if that capacity is insufficient, then the firm incurs a loss let+1 . Hence pmt+1 , pmt+1 − a h (1 − q) let+1 and pmt+1 − a¯ let+1 denote the effective processing margin when K O − st is sufficient (respectively) under both yield realizations, under the low yield realization only, and under neither yield realization. Processing is unprofitable when pmt+1 ≤ 0; otherwise, ∗ (s , P zt+1 t t+1 ) is determined by the smallest positive effective processing margin. Suppose, for instance, that 0 < pmt+1 ≤ a h (1 − q) let+1 ; then it is profitable to process if K O −st is sufficient under both yield realizations but not if it is insufficient under the high yield realization. We conclude that the firm optimally processes up to (K O − st )/a h unless constrained by the processing capacity K I . By Proposition 1, the optimal expected processing profit L∗pr (st , Pt+1 ) is given by L∗pr (st , Pt+1 )
O I K − st = + min K , (pmt+1 )+ ah
+ K O − st + KI − (pmt+1 − a¯ let+1 )+ al
O K O − st I K − st − min K + min K I , , al ah + · pmt+1 − a h (1 − q) let+1 . O st pt+1
(7.7)
We now turn to the optimal inventory decision, which by Eq. 7.5 is the solution to max 0≤st
≤K O
− Lsc (st  Pt ) + δEt [L∗pr (st , P˜ t+1 )] ,
(7.8)
where Lsc (st  Pt ) = (ptO + h)st and L∗pr (st , Pt+1 ) is as given in Eq. 7.7. Proposition 2 The optimal output inventory level st∗ (Pt ) is characterized by ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎨K O − a h K I + st∗ (Pt ) = + ⎪ K O − al K I ⎪ ⎪ ⎪ ⎩ O K
if smt ≤ 0, if 0 < smt ≤ Yt , if Yt < smt ≤ Yt + Zt ,
(7.9)
if smt > Yt + Zt ,
O ] (as given by Eq. 7.6) and where smt = −ptO − h + δEt [p˜ t+1
+ + ! . δ # mt+1 − p , Yt = h Et p # mt+1 − a h (1 − q) $ let+1 a
(7.10)
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+ + ! . δ # mt+1 − a h (1 − q) $ . Zt = l Et p let+1 − p # mt+1 − a¯ $ let+1 a Recall that smt denotes the storage margin per output. If smt ≤ 0, then it is not profitable to hold inventory. If smt > 0, then st∗ (Pt ) is determined by the tradeoff between the storage margin smt and the opportunity cost of holding inventory (i.e., holding inventory reduces the subsequent period’s processing volume since there will be less free storage capacity). In particular, Yt (Yt + Zt ) denotes this opportunity cost when the subsequent period’s processing volume is constrained under the high yield realization only (under both yield realizations). If Yt < smt ≤ Yt + Zt , for instance, then it is profitable to hold inventory when the subsequent period’s processing volume is constrained under the high yield realization but not when that volume is constrained under both yield realizations. As a result, the firm + stores up to K O − a l K I . We can rewrite the optimal expected profit Gt (Pt ) for the twostage problem in period t, as given in Eq. 7.5, by substituting the optimal inventory level st∗ (·) ∗ (s , P in the optimal processing volume zt+1 t t+1 ) for a given inventory level st , as characterized by Proposition 1, pmt+1 − a¯ $ let+1 )+ K I + max(smt , Yt + Zt ) min a l K I , K O + Gt (Pt ) = Et (# max(smt , Yt ) min(a h K I , K O ) − min(a l K I , K O ) O h I + + sm+ t (K − a K ) .
(7.11)
For K I units of the input processing capacity, the firm earns at least a unit processing margin (pmt+1 − a¯ let+1 )+ when the output is sold directly to the spot market without going through the storage capacity in the subsequent period (if doing so is profitable). Recall that the marginal sales revenue in this case is strictly lower than the marginal spot sales revenue due to the loss let+1 . Furthermore, this sales revenue is guaranteed regardless of the storage capacity in the subsequent period, and thus, it is available under all yield realizations, i.e., the unit margin is given by (pmt+1 − a¯ let+1 )+ for all K I units of the processing capacity. For the first a l K I units of the output storage capacity, the firm faces the tradeoff between holding inventory this period—which has a unit profit of storage margin per output smt —versus processing in the subsequent period. Storing these units constrains the subsequent period’s processing volume under both yield realizations and thus incurs a unit opportunity cost Yt + Zt . Therefore, marginal revenue of these capacity units is given by the maximum from the two options. Following a similar intuition, the marginal revenue for the next storage capacity units up to a h K I is given by the maximum of smt and the unit opportunity cost Yt since storing these units constrains the subsequent period’s processing volume only under high yield realization. For the remaining (K O − a h K I ) units of the storage capacity, holding inventory this period does not limit the subsequent period’s processing volume. Therefore, marginal
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revenue of these capacity units is given by the storage margin (when it is profitable to hold inventory).
7.4.2 Capacity Investment Decisions In this section, we provide the optimal solution for the firm’s capacity investment decisions. At period t = 0, the firm observes P0 and chooses the capacity portfolio K = (K I , K O ) to maximize its expected profit over the entire planning horizon: maxK≥0 V (K) − C(K), where C(K) = β I (K I )2 + β O (K O )2 represents . the capacity investment costs and V (K) = δE0 [V1 (0, P˜ 1 )] captures the expected profit for a given capacity portfolio K. It follows from ! Eq. 7.4 that V1 (0, P1 ) = T −1 τ −1 ˜ max0≤z1 ≤K I Lpr (z1  0, P1 ) + τ =1 δ E1 Gτ (Pτ ) and that the expected profit for a given K can therefore be written as V (K) = M 1 K I + M 2 min(a l K I , K O ) + M 3 [min(a h K I , K O ) − min(a l K I , K O )] + M 4 (K O − a h K I )+ , where T + . t M 1 = E0 # mt − a¯ $ δ p let , t=1
T −1 . t ˜ ˜ δ max s% mt , Yt + Zt , M 2 = Y0 + Z0 + E0 . M 3 = Y0 + E0 . M 4 = E0
T −1
T −1 t=1
t=1
˜ δ max s% mt , Yt , t
δ t s% m+ . t
(7.12)
t=1
The terms pmt , smt , and let are as given in Eq. 7.6 and Yt and Zt are as given in Eq. 7.10. In Eq. 7.12, M 1 (M 4 ) denotes the expected marginal revenue of the processing capacity K I (of the storage capacity K O ) in the absence of the other capacity type. In particular, M 1 is the total expected processing profit when the output is sold to the spot market with marginal sales revenue of αptO − d under all yield realizations in each period or, in other words, when the effective processing margin is given by pmt − a¯ let . Similarly, M 4 denotes the total expected storage profit over the entire planning horizon; this term is relevant for the storage capacity + units K O − a h K I , which have no effect on the processing activities. In contrast,
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M 2 and M 3 capture the expected marginal revenue from using both K I and K O for production capacity over the entire planning horizon. Because holding inventory in each period may limit the subsequent period’s processing volume, this production capacity can be used either for holding inventory or for processing. So in each period, the expected marginal revenue is given by the maximum of the storage margin smt and the relevant expected processing benefit per output. For the first min a l K I , K O capacity units, the relevant processing benefit is given by Yt +Zt . Yet because holding inventory constrains the subsequent period’s processing volume only under the high yield realization, the processing benefit for the other min(a h K I , K O ) − min(a l K I , K O ) capacity units is given by Yt . Since K O is unoccupied at the beginning of the planning horizon, the first period’s processing volume is not constrained by inventory, and so only the processing benefit per output (Y0 + Z0 or Y0 ) is relevant. ∗
∗
Proposition 3 The optimal capacity portfolio K∗ = (K I , K O ) is characterized by
∗
∗
(K I , K O ) =
⎧ M +a h (M 3 −M 4 )+a l (M 2 −M 3 ) M 4 ⎪ if β ∈ 1 , ⎪ 1 I O ⎪ 2β 2β ⎪
⎪ ⎪ M +a h M +a l (M −M ) a h M 1 +a h M 3 +a l (M 2 −M 3 ) ⎪ 1 3 2 3 ⎪ if β ∈ 2 , ⎪ ⎪ 2β I +2(a h )2 β O 2β I +2(a h )2 β O ⎪ ⎨ M 1 +a l (M 2 −M 3 ) M 3 , 2β O 2β I ⎪ ⎪ ⎪ l l l ⎪ ⎪ M 1 +a M 2 , a M 1 +a M 2 ⎪ ⎪ 2β I +2(a l )2 β O 2β I +2(a l )2 β O ⎪ ⎪ ⎪ ⎪ M ⎩ 1 , M2 I O 2β 2β
if β ∈ 3
if β ∈ 4 if β ∈ 5 ,
where β = (β I , β O ) and M i for i ∈ {1, . . . , 4} is as given in Eq. 7.12, and " a h M 1 + a h (M 3 − M 4 ) + a l (M 2 − M 3 ) βI . 1 = β : O ≥ , (7.13) β M4 a h M 1 + a h (M 3 − M 4 ) + a l (M 2 − M 3 ) βI . > O 2 = β : M4 β h l a M 1 + a (M 2 − M 3 ) , ≥ M3 " a h M 1 + a l (M 2 − M 3 ) a l M 1 + a l (M 2 − M 3 ) βI . 3 = β : > O ≥ , M3 β M3 " a l M 1 + a l (M 2 − M 3 ) βI al M 1 . 4 = β : > O ≥ , M3 β M2 βI al M 1 . . 5 = β : O < β M2
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The optimal processing and storage capacity levels are characterized by the ratio of the expected marginal revenue of an additional capacity unit to its marginal investment cost. The marginal investment cost of each capacity type is given by 2β j ∗ ∗ for j ∈ {I, O} unless K O = aK ˆ I , where aˆ ∈ {a l , a h }, in which case that cost is given by 2β I +2(a) ˆ 2 β O . The expected marginal revenue of each capacity type takes different forms based on the relative processingtostorage capacity investment cost . ratio, which we define as η = β I /β O . To delineate the intuition, we fix β I and discuss how the optimal portfolio changes as β O increases. First, if β O is sufficiently low (i.e., if β ∈ 1 ), then there is excess storage capacity—that is, this capacity is strictly larger than what is required for production (with full utilization of processing capacity) under both yield realizations. Hence there is no production benefit to having additional storage capacity, and in this case the expected marginal revenue of storage capacity is given by the total expected storage profit M 4 . Because there is excess storage capacity, an additional unit of processing capacity can be used for output production; thus its marginal revenue is given by the sum of the individual revenue M 1 and the joint revenue, which is M 2 − M 4 for the first a l units and M 3 − M 4 for the next (a h − a l ) units. Hence the expected marginal revenue of processing capacity is given by M 1 + a h (M 3 − ∗ M 4 )+a l (M 2 −M 3 ). Second, as β O increases, we see that K O decreases and there is no excess storage capacity in the optimal solution. As a result, M 4 has no effect on the expected marginal revenue of either capacity type. For instance, if β ∈ 2 , ∗ ∗ then K O = a h K I ; here the expected marginal revenue of processing capacity is the same as in the β ∈ 1 case except that M 4 is not deducted. As β O increases further, the expected marginal revenue of processing capacity is given by the sum ∗ of the relevant joint revenue (which is different in each region because K O is O decreasing) and the individual revenue M 1 . Finally, if β is sufficiently high (i.e., if β ∈ 5 ), then the processing capacity is strictly larger than what is required for production (with full utilization of storage capacity) under both yield realizations; hence this capacity’s expected marginal revenue depends only on M 1 . Since in this case an additional unit of storage capacity can be used for production under both yield realizations, this capacity’s expected marginal revenue is given by M 2 . Corollary 1 outlines three distinct strategies for determining the optimal investment in capacity. ∗ ∗ . Corollary 1 Let γ ∗ = K O /K I denote the optimal capacity ratio. (i) If β ∈ 1 , then γ ∗ > a h . In this case, K∗ is characterized by the “storage∗ dominating” portfolio and ∂K j /∂β −j = 0 for j ∈ {I, O}. ∗ l (ii) If β ∈ 234 , then γ ∈ [a , a h ]. In this case, K∗ is characterized by the ∗ “mixed” portfolio and ∂K j /∂β −j ≤ 0 for j ∈ {I, O} with strict inequality holding for β ∈ 24 . (iii) If β ∈ 5 , then γ ∗ < a l . In this case, K∗ is characterized by the “processing∗ dominating” portfolio and ∂K j /∂β −j = 0 for j ∈ {I, O}. When the processing capacity cost relative to the storage capacity cost is sufficiently high, the firm invests in a storagedominating portfolio whereby the storage
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capacity is strictly greater than what is required for production (with full utilization of processing capacity) under both yield realizations. When this relative cost is sufficiently low, the firm invests in a processingdominating portfolio whereby the processing capacity is strictly larger than what is required for production (with full utilization of storage capacity) under both yield realizations. Otherwise, the firm invests in a mixed portfolio. With the mixed portfolio, higher investment cost of one capacity type (weakly) decreases the level of the other capacity type. This is an important result for managers because it establishes that the firm should evaluate holistically the effects of each capacity type’s investment cost.
7.5 Performance of Heuristic Capacity Investment Policies In the previous section, we characterized the firm’s optimal capacity investment policy. In this section, we conduct numerical experiments to compare the performance of this optimal policy against a variety of heuristic capacity investment policies, focusing on the profit loss due to employing the heuristic policy. We define this . loss as hp = [(∗ − (Khp ))/∗ ], where ∗ is the optimal expected profit (as given by Proposition 3) and (Khp ) is the expected total profit evaluated with the I , K O ), which is chosen by the heuristic policy (hp). capacity portfolio Khp = (Khp hp We consider the following heuristic policies in our analysis. Heuristics Based on Ignoring the Production Yield Uncertainty We consider two heuristic policies in which the firm does not take into account production yield uncertainty in capacity planning. In the deterministic yield (maximum) or DYM heuristic, capacity planning is based on the maximum possible yield; this policy is most often implemented by palm oil mills in practice. Let KDY M denote the optimal capacity investment with this policy, which can be obtained from Proposition 3 by replacing a l with a h in all the relevant expressions. In the deterministic yield (average) or DYA heuristic, capacity planning is based on the average yield. Let KDY A denote the optimal capacity investment with this policy, which can be obtained from Proposition 3 by replacing a l and a h with a¯ in all the relevant expressions. NoByproduct (NB) Heuristic Under this heuristic, the firm does not account for byproduct revenue when planning for capacity. The optimal capacity investment with this policy, KN B , is obtained from Proposition 3 by substituting the effective processing cost c = c − a B pB for c. High Yield–Balanced Portfolio (HYBP) Heuristic Under this heuristic, the firm chooses its capacity investment portfolio by fixing K O = a h K I . The optimal capacity investment under this policy is given by I KH Y BP =
M 1 + a h M 3 + a l (M 2 − M 3 ) 2β I + 2(a h )2 β O
and
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O KH Y BP
a h M 1 + a h M 3 + a l (M 2 − M 3 ) = . 2β I + 2(a h )2 β O
Our companion paper Boyabatlı et al. (2017) investigates the profit loss due to employing these heuristic policies using numerical experiments calibrated for the palm industry. This chapter builds on their numerical calibration with the addition of two new parameters (α and d) to capture the revenue loss associated with excess production. Therefore, the actual numerical results in this chapter are different from our companion paper. However, as we discuss below, these results lead to similar observations with our companion paper. We first summarize the parameters that represent our baseline scenario. Except for α and d, the other parameters presented in Table 7.1 are calibrated by Boyabatlı et al. (2017), where “RM” denotes the Malaysian ringgit (currency) and “mt” denotes metric ton (equal to 1,000 kg or about 1.1 US tons) in Table 7.1. Using the parameters summarized in Table 7.1, the optimal capacity investment ∗ in our baseline scenario is given by the storagedominating portfolio with K I = Table 7.1 Summary of parameters representing the baseline scenario Notation θI pI σI θO pO σO ρ p0I p0O a¯ ah c h pB aB βI βO r T α d
Description Meanreversion parameter for input spot price process Longterm input spot price level Volatility of input spot price Meanreversion parameter for output spot price process Longterm output spot price level Volatility of output spot price Correlation between input and output spot prices Initial input spot price Initial output spot price Average output yield Maximum output yield realization Unit processing cost Unit holding cost Unit byproduct revenue Byproduct yield Input capacity investment cost coefficient Output capacity investment cost coefficient Annual compound interest rate (for the calculation of the discount factor δ = (1 + r)−1/250 ) Length of the planning horizon Proportion of output spot price salvaged from excess production Unit cost of excess production
Baseline value 0.00345 532.75 8.60 0.00437 2689.87 39.08 0.734 528.5 RM per metric ton 2,570.5 RM per metric ton 19.72% 20.37% 40 RM per metric ton 1 RM/day per metric ton 1,510.70 RM per metric ton 5.53% 75 0.25 10% 1, 250 weekdays 0.7 0 RM per metric ton
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∗
858.91 mt/day and K O = 1,653.66 mt. In this scenario, the optimal expected profit is 56,012,483.86 RM over the fiveyear (1250 weekdays) planning horizon. In order to compare the performance of the optimal capacity investment policy with that of heuristic policies, we extend our numerical instances around the baseline (which also allows us to assess the sensitivity of our results to several key parameters). In particular, for η = β I /β O , we consider η ∈ [−30%, 30%] of the baseline value ηˆ = 300 in 10% increments. We also consider the maximum yield a h ∈ [20.37%, 22.37%] in 0.05% increments as well as holding costs h ∈ {0.5, 1, 2} and interest rates r ∈ {0%, 10%, 20%}. Finally, we consider α ∈ {0.5, 0.7, 0.9} and d ∈ {0, 250}. Altogether we evaluate 1890 numerical instances. In theory, the optimal capacity investment policy can take additional forms (as described in Proposition 3). Yet we observe only storagedominating portfolios (i.e., γ ∗ = ∗ ∗ K O /K I ≥ aH ) and high yield–balanced portfolios (i.e., γ ∗ = aH ) in all of our numerical instances. Table 7.2 summarizes the percentage profit loss hp × 100 incurred under each of the considered heuristic policies using a classification of the numerical instances based on the optimal capacity investment policy (storagedominating or high yield–balanced). Table 7.2 suggests that our main observations are consistent with those reported by Boyabatlı et al. (2017). We now summarize the main insights (detailed discussions about the intuition of these insights can be found in Sect. 5.3 of our companion paper). 1. The High Yield–Balanced Portfolio (HYBP) Is the Best Performing Heuristic Policy and It Provides a NearOptimal Performance Table 7.2 shows that the HYBP heuristic does not only outperform the other heuristics but also results in a nearoptimal profit level. In particular, even in numerical instances where η is high enough to result in positive profit loss (because the optimal policy is storagedominating), the average profit loss with this heuristic is only 0.57%.
Table 7.2 Performance of heuristic capacity investment policies in the palm industry, where DYM = deterministic yield (maximum) heuristic, DYA = deterministic yield (average) heuristic, NB = nobyproduct heuristic, and HYBP = high yield–balanced portfolio heuristic. For each of these heuristics, the boldface values report the average percentage loss observed in the relevant numerical instances while the other values report the minimum and the maximum percentage loss observed. K∗ is storagedominating when η > a h M 1 + a h (M 3 − M 4 ) + a l (M 2 − M 3 ) /M 4 , whereas it is high yield–balanced when a h M 1 + a h (M 3 − M 4 ) + a l (M 2 − M 3 ) /M 4 > η ≥ a h M 1 + a l (M 2 − M 3 ) /M 3 Optimal policy (% instances) K ∗ is storagedominating K ∗ is high yieldbalanced
Percentage loss (%) = hp × 100 DYM DYA NB 66.73 0 63.76 (6.92, 159.16) (0, 0) (61.16, 64.99) 72.39 11.46 64.95 (7.63, 159.20) (1.04, 23.42) (64.29, 65.01)
HYBP 0.57 (0, 3.48) 0 (0, 0)
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2. If the Firm Chooses to Ignore Output Yield Uncertainty in Capacity Planning, It Is Better to Represent the Yield Using the Average Yield Rather Than the Maximum Yield We observe that the profit loss under the DYM heuristic is greater than the corresponding loss under the DYA heuristic in all our numerical instances. 3. It Is Essential to Account for the Byproduct Revenue During Capacity Planning Although byproduct revenue constitutes a small portion of the total revenues of an agricultural processing firm, ignoring the byproduct revenue during capacity planning significantly reduces (at least by 61.6% in our numerical instances) the firm’s profits.
7.6 Conclusion This chapter contributes to the operations management literature on commodity processing by studying the capacity investment decisions of a processing firm in the context of agricultural industries. The papers in this literature either (often implicitly) assume abundant processing and storage resources or consider fixed capacity levels for these resources. We consider a multiperiod optimization problem and characterize the optimal processing and output storage capacity levels and the periodic processing and inventory decisions in closed form. As summarized in the Introduction section, we provide insights on the structure of the optimal capacity investment policy and the performance of the optimal capacity investment policy in comparison with heuristic policies that have practical and theoretical significance. In our computational study throughout Sect. 7.5, we used a calibration based on the palm industry to study the impact of employing heuristic capacity investment policies. Although the magnitude of the impact can be different, we expect our results to be structurally the same for the other oilseed industries and grain industries. This is because the characteristics of the palm processing environment that drive our results are common in these other industries. For example, processing capacity is significantly more expensive than the storage capacity, and thus, the processing firm would typically invest in storagedominating portfolio. Another common feature is that the processing margin would be low when the byproduct revenue is not accounted for, because of the commodity nature of both the input and output, the processing margin is low when the byproduct revenue is not accounted for. Nevertheless, future research may recalibrate our model to represent a processing firm in a different agricultural industry. Relaxing the assumptions made on the processing environment gives rise to a number of interesting areas for future research. First, we assume a fixed byproduct price. This is a reasonable assumption for some of the agricultural industries but is a limitation for some others. For example, in the soybean industry, the byproduct of soybean processing is the meal, and because it is a commodity, it has an uncertain price correlated with the input and the output prices. Second, we focus on expected profit maximization and do not consider the risk associated with the profit. Such
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risk considerations can be incorporated in our model by using real option valuation techniques, by imposing a utility function to the decision maker or by imposing risk constraints to the decision problem. Third, we assume that the marginal procurement cost and the marginal sales revenue of the output are given by the prevailing spot price of this commodity. Analyzing the case where there is a spread between these two prices due to deadweight transactions costs, as considered in Kazaz and Webster (2011), should prove to be an interesting problem for future research. In this case, the processing and inventory decisions cannot be characterized in closed form, and thus, the optimal capacity investment decisions can only be evaluated numerically.
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Chapter 8
A Prescriptive Model for Selling Wine Futures to Mitigate Quality Uncertainty Tim Noparumpa, Burak Kazaz, and Scott Webster
8.1 Introduction In this study, we examine the risks stemming from uncertainty in quality perceptions of fine wine production. The process of making fine wine begins immediately after the harvest. The quality of the grapes is influenced by the climatic conditions during the growing season. After harvest, grapes are sorted, destemmed, and pressed to produce the wine. After a short fermentation process, Bordeauxstyle wines go through a long aging process in oak barrels. This barrel aging takes approximately 18 to 24 months. This is a long period of time for a winemaker to tie up the cash in the liquid whose quality is uncertain when it is bottled. Our study considers quality uncertainty as the quality perception established by wine tasting experts. Harvest in the Northern hemisphere takes place in September and October months for Bordeauxstyle wines. Approximately eight months after harvest, a wine tasting expert visits the winery and takes samples from barrels. The
T. Noparumpa Chulalongkorn Business School, Chulalongkorn University, Bangkok, Thailand email: [email protected] B. Kazaz () Whitman School of Management, Syracuse University, Syracuse, NY, USA email: [email protected] S. Webster W. P. Carey School of Business, Arizona State University, Tempe, AZ, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_8
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same wine critic provides a review and gives a barrel sample score often out of 100 points.1 This score is referred to as the barrel score. French winemakers combat quality uncertainty by selling their wine in advance. After the barrel score is established by the influential wine critics, the winemaker determines the amount of wine to be sold in advance of bottling in the form of wine futures contracts. The winemaker also determines the price of these wine futures contracts. The buyer of wine futures is promised to receive the wine at the end of the aging process when the wine is bottled. When the wine completes its aging process and gets bottled, it is reviewed by the same tasting experts again. The tasting expert provides another score often referred to as the bottle score. The bottle score is not necessarily equal to the barrel score as the aging process can lead to a higher or lower quality wine. After observing the tasting experts’ barrel scores, the winemaker sells some (or all) of its wine in the barrel in the form of wine futures. It is important to highlight that selling wine in the form of futures contracts is a form of operational flexibility that assists riskaverse wine producers in mitigating quality risk stemming from the unknown and random bottle score. The winemaker converts her inventory partially to muchneeded cash prior to the distribution and retailing stage. More importantly, she transfers the quality perception risk to the buyers of wine futures contracts. Our study develops a formulation, a multinomial logit model, that examines the winemaker’s two critical decisions under quality uncertainty stemming from random bottle scores. After observing the barrel score, the model evaluates consumers’ utilities gained from purchasing the wine in the form of wine futures, from waiting and purchasing the wine when it is bottled and the utility from not purchasing the wine. After incorporating these utilities, the model assists the winemaker in making the two decisions described earlier: The amount and price of wine futures. The practice of selling wine in the form of wine futures has been a long practice in France. Since the seventeenth century, highly soughtafter French wines have been sold in advance while the wine is aging in barrels. This advance selling mechanism is referred to as “en primeur.” This term describes that the wine is purchased prior to being bottled but it is also used as a reference to the time period when tasting experts meet with various chateaus in order to sample and review their wines. Today, fine wine futures can be bought and traded via an online electronic exchange called the London International Vintner’s Exchange, or the Livex. Livex allows wine producers, merchants, and investors to buy and sell fine wine both as futures and as final bottled product. Wine production has traveled from the Old World (e.g., Italy, France, Spain) to the New World (e.g., Australia, South Africa, Argentina and Chile, and the United States) and fine wine production efforts increased dramatically in these new wine regions. Fine wine producers from the New World recently began to adopt the same en primeur system and sell their
1 Some
critics in the UK offer tasting scores out of 20 points. Our methodology is robust in the sense that it does not depend on the scale used by the tasting experts; the method can be easily adjusted for this scale.
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wines in the form of wine futures contracts prior to the bottling process. This action enables winemakers to (1) recuperate much of the needed cash in advance and (2) transfer the future quality risk stemming from the random bottle score to the buyers of the futures contracts.
8.1.1 The Impact of Barrel Scores on Wine Futures Decisions Before presenting a model, it is appropriate to demonstrate the impact of barrel scores established by tasting experts in winemakers’ two critical decisions associated with the quantity and price of wine futures. Let us consider vintage t of a Northern hemisphere winemaker which means that its grapes are harvested in September and October of year t. Tasting experts from various publications such as The Wine Advocate, The Wine Spector and Decanter would visit the winery in March and April of year t + 1. These tasting experts would publish their barrel scores in May of year t + 1. Recall that the wine continues to age in the barrel for another year and it is not bottled until the summer of year t + 2. Noparumpa et al. (2015, 2018) provide empirical evidence regarding the impact of barrel scores in winemakers’ futures decisions. Noparumpa et al. (2015) use the barrel scores of the most influential wine critic of the wine industry: Robert Parker Jr. of the Wine Advocate. Noparumpa et al. (2018) extend the analysis by incorporating the barrel scores of the most widely distributed magazine publication: The Wine Spectator. The barrel and bottle scores of these two magazines use a scale of 100 points. Their analysis considers the following 12 chateaus from Bordeaux: Angelus, Cheval Blanc, Clos Fourtet, Cos d’Estournel, Ducru Beaucaillou, Duhart Milon, Evangile, Leoville Poyferre, Mission Haut Brion, Pavie, Pichon Lalande, and Troplong Mondot. The data set includes vintages from 2006 to 2011 from the same chateaus. The wine futures prices and bottled wine prices are collected from Livex. Table 8.1 demonstrates the impact of barrel scores on wine futures prices. It shows that the barrel scores established by the Wine Advocate and Wine Spectator are both influential in determining the prices for wine futures. The regression results presented in Table 8.1 show the impact of barrel scores on wine futures prices is at the highest statistical significance of 1%. The findings are consistent between the barrel scores established by the Wine Advocate and Wine Spectator. Table 8.1 Summary of regression results from Noparumpa et al. (2015, 2018) demonstrating the impact of barrel scores established by the Wine Advocate and Wine Spectator on wine futures prices Parameter Intercept Barrel score Adjusted R2
Wine Advocate Coefficient −2.52 × 10−16 0.071 0.50
(pvalue) (1) (2.03 × 10−12 )
Wine Spectator Coefficient 3.63 × 10−16 0.841 0.70
(pvalue) (1) (2.18 × 10−20 )
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Table 8.2 Summary of regression results from Noparumpa et al. (2015, 2018) demonstrating the impact of barrel scores established by the Wine Advocate and Wine Spectator on the proportion of wine sold in the form of wine futures Parameter Intercept Barrel score Adjusted R2
Robert Parker Jr. Coefficient −2.13 × 10−16 0.709 0.50
(pvalue) (1) (3.17 × 10−12 )
Wine Spectator Coefficient 3.48 × 10−16 0.73 0.53
(pvalue) (1) (2.16 × 10−13 )
Noparumpa et al. (2015, 2018) illustrate the impact of barrel scores in determining the proportion of wine that is sold in the form of wine futures. For the same data set described above using the 12 chateaus and their vintages from 2006 till 2011, these two publications present the influence of barrel scores on the quantity of wine sold in the form of futures. Table 8.2 demonstrates the impact of barrel scores on the proportion of the wine sold as futures. It shows that the barrel scores established by the Wine Advocate and Wine Spectator are both influential in determining the quantity of wine that is sold in the form of wine futures. The regression results presented in Table 8.2 show the impact of barrel scores on the proportion of wine sold in the form of wine futures is at the highest statistical significance of 1%. The findings are consistent between the barrel scores established by the Wine Advocate and Wine Spectator. Tables 8.1 and 8.2 establish the strong statistical evidence regarding the important role of the barrel scores on the winemakers’ two important decisions: The price of wine futures contracts and the proportion of wine to be sold in advance in the form of wine futures. As a result of these empirical findings, it is essential to examine and determine the optimal futures allocation decisions for a winemaker who intends to maximize the expected profits while minimizing the quality risks stemming from random bottle scores. This study provides answers to the following research questions: 1. How should a winemaker allocate the production between futures and retail distribution in the presence of an uncertain bottle rating? 2. What is the impact of risk aversion and market characteristics on the winemaker’s decision regarding futures quantity and price? 3. How does the value a future market for a winemaker depends on the characteristic of the winemaker and the market?
8.2 Literature Review There is a vast literature examining risk mitigation methods in agriculture. The related literature that investigates operational flexibilities as a risk mitigation approach can be divided into three primary segments: Pricing and quantity decisions
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under uncertainty, advance selling mechanisms, and wine tasting and its influence on wine analytics.
8.2.1 Pricing and Quantity Decisions Under Uncertainty In supply chain and operations management literature, there is a growing list of publications that examine pricing and quantity decisions under uncertainty in the context of agricultural production operations. These studies employ a PriceSetting Newsvendor Problem (PSNP) framework in their modeling choices. Examples include Jones et al. (2001), Kazaz (2004), and Kazaz and Webster (2011, 2015). These studies extend the PSNP problems to account for the supply uncertainty. Our modeling approach in this paper departs from these publications as we use a multinomial logit model that accounts for consumer preferences.
8.2.2 Advance Selling Mechanisms Advance selling mechanisms are widely examined in the literature that examined the interface of marketing and operations management. Examples include Gale and Holmes (1992, 1993), Shugan and Xie (2000, 2005), Xie and Shugan (2001), Fay and Xie (2010), Boyacı and Özer (2010), Tang and Lim (2013) and Cho and Tang (2013). These studies demonstrate that advance selling can be used as a tool to help a firm mitigate fluctuations in demand. Our study extends this stream of research to apply in the wine industry.
8.2.3 Wine Tasting and Wine Analytics There is a stream of literature that examines wine economics and wine analytics. These studies focus on the influence of weather conditions in understanding wine prices and wine production. Ashenfelter (2010) and Ashenfelter et al. (1995) focus on the impact of climatic conditions on the quality and price of aged wines. In addition to the literature on the influence of weather, there is a growing list of publications that examine wine tasting and expert opinions on the perception of quality and wine prices: Ali et al. (2008), Ashenfelter and Jones (2013), Stuen et al. (2015), Bodington (2015), and Olkin et al. (2015) and Masset et al. (2015). Hekimo˘glu and Kazaz (2020) present a predictive analytics method in estimating the true value of wines using weather, market and tasting expert reviews. Hekimo˘glu et al. (2017) develop a prescriptive analytical model for a wine distributor’s portfolio selection problem under weather and market uncertainty. Kazaz (2020) collate predictive and prescriptive analytical models featuring wine futures used by
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winemakers and distributors/importers. Our study extends the literature by offering a prescriptive model that benefits winemakers in determining the amount of wine to be sold in the form of wine futures and the price of these wine futures contracts.
8.3 The Model In this section, we introduce a multinomial logit model to help a winemaker in her decisions regarding the amount of wine to be sold as futures and the price of wine futures. Before proceeding with the model, let us examine the timing of decisions made by a winemaker. Figure 8.1 presents a depiction of time epochs and the decisions to be made. At the time of harvest t0 , a winemaker obtains the amount of wine to be produced and sold for that vintage; the total amount of wine production is denoted Q. Eight to ten months after harvest, corresponding to the time epoch t1 , tasting experts visit the winery and establish the barrel score denoted s1 . In essence, the barrel score is provided before the winemaker determines her critical decisions regarding wine futures. The winemaker’s decision regarding the amount of wine to be sold as futures is qf and its value cannot exceed the total amount of wine production; therefore, qf ≤ Q. The winemaker’s other critical decision is the price of wine futures; the price is denoted pf . The remaining wine that is not sold in the form of futures will be sold in retail. The amount of wine reserved for retail sales is denoted qr and its value is equal to (Q − qf ). At time epoch t2 , the aging process gets completed and the wine is bottled. At this time, the tasting experts provide a second score known as the bottle score. However, this bottle score is random at time epoch t1 when the winemaker is making the critical decisions regarding wine futures. the remaining wine is bottled and sent to expert reviewers for blind tasting. We denote
Fig. 8.1 Timeline of events and the decisions made by the winemaker during the aging process
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the random bottle score s˜2 . We assume that the expected value of the random bottle score is equal to the already established barrel score; thus, E[˜s2 s1 ] = s1 . We express the variance in the random bottle score as V [˜s2 s1 ] = σ 2 . We denote the retail price of bottled wine in time epoch t2 by pr (˜s2 s1 ) because it is a function of the random bottle score s˜2 for a given barrel score of s1 .
8.3.1 Quality Indicator In the proposed model, the random bottle score s˜2 is derived from standardized random variable z˜ that is independent of s1 and is expressed as s˜2 = s1 + z˜ σ . The expected value and variance of z˜ are defined as E[˜z] = 0 and V [˜z] = 1, respectively. The realization of random variable z˜ is denoted with z; the pdf and the cumulative distribution function (cdf) of z˜ are expressed as g(z) and G(z), respectively. To reflect the market condition where the retail price of bottled wine pr is influenced by the bottle rating s2 without loss of generality, the bottle price of wine can be normalized to be equivalent to the realized bottle rating, i.e., pr (s2 ) = s2 . As a result, the expected retail price at time epoch t1 can then be expressed as equivalent to the barrel score s1 , i.e., E[pr (˜s2 s1 )] = s1 .
8.3.2 Consumer Utilities and the Demand for Wine Futures To reflect the idiosyncratic nature of the consumers in the wine futures market, Noparumpa et al. (2015) offer a random demand function that is based on the consumers’ utility of purchasing wine as futures, retail, or not purchasing wine at all. We express the utility from purchasing the wine in the form of wine futures with Uf , the utility from purchasing the wine during the retail sales as Ur , and the utility from not purchasing the wine as U0 . At time epoch t1 , the random utility associated with above decision by the consumer are as follows: Uf = Vf − pf = θ s1 + εf − pf , Ur = θ E[˜s2 s1 ] + εr − θ E[pr (˜s2 s1 )] = εr , U0 = ε0 , where εf , εr , and ε0 are i.i.d Gumbel random variables with zero mean and a scale parameter β. From above, the random utility of consumer purchasing wine as futures Uf is presented as a surplus between the valuation and the price of wine futures. To incorporate the risk preference of the consumer, a riskadjusted discount rate, denoted θ , is derived via the conditionalvalueatrisk (CVAR) framework. The valuation of wine futures by an average consumer, vf , is equal to the conditional
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expected value of bottle score discounted to time t1 at the riskfree rate. Therefore, at time t1 , the valuation of a future by a random consumer is presented as Vf = vf + εf = θ s1 + εf , where εf is a random variable with E[εf ] = 0. For a consumer who chooses not to purchase a future at time t1 , there are two additional options: Either purchase the wine at retail price pr (˜s2 s1 ) = s˜2 , or do not purchase at all. For the consumer who purchases the wine at retail, the utility at t1 is equivalent to the differences between the expected valuation and the expected price discounted by the riskadjusted discount rate. Lastly, the utility of not purchasing at all is zero. From the formulation of random utility above, a multinomial logit (MNL) modeling framework can be used in order to derive the probability of futures purchase. The probability that a consumer will purchase wine futures is equivalent to the probability of the random utility of future purchase being higher than both of the utilities from purchasing the wine at retail and from not purchasing the wine at all. Through this MNL framework, the demand for futures, denoted qf (pf ), can be derived by multiplying the market size, denoted M(s1 ), with the probability that the consumer will value purchasing futures to be higher than both retail purchase and not purchasing it. Therefore, qf (pf ) = M(s1 )P [Uf > max{Ur , U0 }] = M(s1 )
e(θs1 −pf )/β . 2 + e(θs1 −pf )/β
(8.1)
Inverting Eq. 8.1 enables the winemaker to determine the price of wine futures: pf (qf ) = θ s1 + β ln
M(s1 ) − qf . 2qf
(8.2)
It should be highlighted that the market size M(s1 ) is nondecreasing function of barrel score s1 , i.e., M (s1 ) ≥ 0. This definition reflects the real world nature of the hype effect from higher barrel scores. Specifically, higher barrel scores create a larger market size.
8.3.3 Winemaker’s Optimization Problem At the time of harvest, alternatively at time epoch t1 , the winemaker faces the risk of selling a bottled wine at an uncertain retail price in the future. To incorporate the winemaker’s risk preference into the optimization model, a winemaker’s riskadjusted discount factor, denoted by φ, is introduced into the model. This parameter describes the winemaker’s degree of risk aversion. With the higher degree of
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uncertainty in the bottle price, a more riskaverse winemaker would prefer the revenue in t1 to be more valuable. This is reflected in the smaller value of φ. Therefore, the winemaker’s expected profit can be expressed as (qf ) = qf pf (qf ) + φE[pr (˜s2 s1 )](Q − qf )
M(s1 ) − qf = qf (θ − φ)s1 + β ln + φs1 Q. 2qf
(8.3)
In Eq. 8.3, the firm term qf pf (qf ) reflects the winemaker’s earning from futures sales. The second term, φE[pr (˜s2 s1 )](Q − qf ) represents the winemaker’s retail profits discounted to time epoch t1 with the winemaker’s riskadjusted discount rate φ. Thus, the winemaker’s problem can be expressed as a maximization of profits from futures sales and the expected retail sales, i.e., ρ ∗ = max (qf ). qf ≤Q
Using the concavity property of the profit maximization problem with MNL demand presented by Aydin and Porteus (2008) and in the case of a nested demand MNL presented in Li and Huh (2011). Noparumpa et al. (2015) show that the above constrainedoptimization problem above is concave. Thus, there exists a unique optimal solution for the winemaker’s decisions pertaining to the futures price, the futures quantity and her profit. Proposition 1 presents these optimal decisions. The closedform expressions for the optimal decisions change depending on whether the winemaker has sufficient wine production. Let us define the ratio of total wine production to the market size as α o = Q/M(s1 ). The derivations utilize α o and the Lambert W function, denoted W (z), as in Corless et al. (1996), where W (z) returns the value of w that satisfies z = wew . Proposition 1 The optimal amount of wine to be sold as futures, the optimal future price and the optimal profit are as expressed in Table 8.3. Proof (of Proposition 1) The proof follows from the proof presented in Noparumpa et al. (2015). Table 8.3 summarizes the findings of the Proposition 1 from Noparumpa et al. (2015) to show the closedform expression of the optimal wine futures allocation, optimal wine futures price and optimal profit. It must be noted here that the value of α o corresponds to the optimal fraction of the futures market where consumers prefer buying the wine in the form of wine futures; its value is equivalent to αo =
e(θ−φ)s1 /β−W (e
(θ−φ)s1 /β /(2e))
2e + e(θ−φ)s1 /β−W (e
(θ−φ)s1 /β /(2e))
.
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Table 8.3 Optimal amount of wine futures to be sold, optimal wine futures price and optimal profit of the winemaker
8.4 Analysis We begin our analysis with the insights that can be gained from problem parameters. In a market for fine wine, the consumer is more of an “affluent” segment and thus is less sensitive to risk. On the other hand, each individual winemaker, especially the “smaller” boutique wineries are more concern with risks, is more riskaverse than the consumer, leading to θ > φ. On the consumer side, the riskadjusted discount rate θ is derived from the CVAR framework, where the lower value of θ corresponds to a higher degree of risk aversion. The analysis leads to three main insights that illustrate the influence of increasing values in (1) consumers’ and winemaker’s risk preferences, (2) barrel score, and (3) the degree of consumer heterogeneity on the winemaker’s optimal decisions. Proposition 2 The following results show the impact of an increase in a parameter on optimal values: Proof (of Proposition 2) The proof follows from the proof presented in Noparumpa et al. (2015).
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Condition
Increase
Supply constraint is not binding α∗ qf∗ pf∗ ρ∗
Supply constraint is binding α∗ qf∗ pf∗ ρ∗
θ >φ θ >φ
cv γ θ −φ θ φ Q s1 β
↓ ↓ ↑ ↑ ↓ — ↑ ↓
— — — — — ↑ ↓ —
↓ ↓ ↑ ↑ ↓ — ↑ ↓
↓ ↓ ↓ or ↑ ↑ ↓ or ↑ — ↑ ↓_↑
↓ ↓ ↓ or ↑ ↑ ↓ or ↑ ↑ ↑ ↓_↑
— — — — — ↑ — —
↓ ↓ ↓ or ↑ ↑ — ↓ ↑ ↓ or ↑
↓ ↓ ↓ or ↑ ↑ — ↑ ↑ ↓ or ↑
Key: ↑ = increase, ↓ = decrease, ↓_↑ = decrease, then increase, — = no change, ↓ or ↑ = both possible. When M(s1 ) = M for all s1 then: 1 —, 2 ↓, 3 ↑.
8.4.1 The Impact of Consumers’ and Winemaker’s Risk Preference Proposition 2 illustrates that as consumers become less riskaverse, reflected with a higher value of θ , the optimal allocation of wine allocated for sales as futures, the optimal price of futures, and the optimal profit, all increases. It also shows that as a winemaker becomes more riskaverse, with lower values of φ is smaller or similarly an increasing value of θ − φ, a winemaker tends to allocate more wine to be sold as futures. Such scenario is reflected in many real world examples where smaller wineries with higher concern for risks tend to allocate more wine to be sold as futures.
8.4.2 The Impact of Barrel Score The comparative statics study into the effect of increasing the barrel score s1 illustrates that the optimal number of cases of wine reserves to be sold as futures qf∗ , and the optimal proportion of wine futures market that chooses to purchase futures α ∗ increase with higher values of barrel score s1 . This implies that, while an increase in barrel scores may provide an incentive for the winemaker to sell wine as bottle at a higher retail price, but due to the riskaverse nature, the winemaker would prefer to receive cash early in advance and thus would allocate more wine to be sold as futures.
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8.4.3 The Impact of Consumer Heterogeneity The degree of consumer heterogeneity is presented as the dispersion rate, β, from the Gumbel distribution. The lower the value of β represents the scenario in which the consumers have similar preferences towards purchasing wine as futures. The higher value of β represents the wine market that is less homogenous in term of consumers’ preferences on purchasing wine as futures. Proposition 3 (a) The optimal futures price pf∗ and the expected profit ρ ∗ are convex in β, and (b) the optimal futures price switches from decreasing behavior to an increasing behavior before the optimal expected profit, i.e., βpf ≤ βρ . Proof The proof follows from the proof presented in Noparumpa et al. (2015).
Proposition 3 shows that, the optimal price of futures pf∗ and optimal profit ρ ∗ are initially decreasing as β increases. Then as β becomes sufficiently large, the optimal futures price and the optimal profit start increasing again. Furthermore, Proposition 3 illustrates that optimal futures price switches from a decreasing behavior to an increasing behavior before the optimal profit, i.e., βpf < βρ . As a result of this finding, the effect of consumer heterogeneity on the winemaker’s optimal decision can be categorized into three regions as shown in Fig. 8.2. In Fig. 8.2, Region 1 represents a more homogenous consumers market, β < βpf . When consumer heterogeneity begins to increase, consumers with high willingness to pay begin to leave the market, and thus, the winemaker lowers the price of futures and allocates less wine for futures resulting in a lower profit. On the other hand, in Region 3, the market is filled with heterogeneous consumers, β > βpf . In this instance, the winemaker can take advantage of this situation by allocating a small amount of wine to be sold as futures to consumers that have very high willingness pay and charging them a much higher price of futures, resulting in an increasing profitability. This result highlights the potential positive influence of consumer heterogeneity on the market for fine wine. This finding is particularly interesting as it differs from traditional marketing belief that firms will be better off in a market that is more homogenous. Traditional marketing literature illustrates that monopolistic firm can benefit from bundling customers to create a more homogenous market (Carlton and Perloff 2010; Varian 2009). The analysis of Region 3 presented above can be used to explain the nature of the market condition during the economic crisis of the late 2000s to the early 2010s. Due to the economic crisis in traditional market for fine wine, namely Europe, North America, and the booming Asian of economy – the market for fine wine can be seen as more heterogeneous, with high willingness to pay consumers in the new market. As a result, Bordeaux wineries reduced their allocation of wine futures and increased their future prices to take advantage of the few buyers with high willingness to pay. In doing so, the Bordeaux wineries could increase their profitability while reserving more of their wine to be sold at retail once the world economy recovered. Region 2 corresponds to the market environment where the heterogeneity among the consumers is not sufficiently high for the winemaker to fully take advantage
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Fig. 8.2 The effect of increasing consumer heterogeneity on the optimal amount of wine allocated for sales as futures, optimal price of futures and optimal profit for 2008 Cheval Blanc (2008 vintage of Cheval Blanc with parameters s1 = 96, M(s1 ) = 5070.74, Q = 4,165, θ = 0.9726, and φ = 0.8692) as presented in Noparumpa et al. (2015)
of the consumers who have higher willingness to pay, i.e., βpf < β < βρ . In this scenario, the winemaker increases the price of wine futures to accommodate the consumers with high willingness to pay. However, this increase in price cannot cover the loss from the larger portion of consumers with lower willingness to pay, resulting in a decrease in profitability.
8.5 Numerical Illustrations To support the analytical model presented and to highlight the effectiveness of wine futures as a tool for mitigating quality risks, this section provides a predictive
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analytics study that investigates the optimal allocations, pricing decisions and profitability derived from the prescriptive analytical model. The study uses the rating scores and production data of twelve Bordeaux wineries of various sizes and one smaller artisanal U.S. winery. To estimate the consumer adjusted discount rate θ and the degree of consumer heterogeneity β, a bivariate model, pf = θ s1 + βx, where x = ln[(1/2)[M(s1 )/qf − 1]], was used to explain the wine futures price in Eq. 8.2. Through the regression analysis, the consumer riskadjusted discount rate is estimated to be equal to 0.9726, and the degree of consumer heterogeneity is estimated to be 23.5275. It must be noted that the estimated consumer riskadjusted discount rate presented suggests that the consumers in the wine market exhibit a very small degree of risk aversion when compared to the riskneutral consumers, whose θ was calculated to be 0.9756. To estimate each of the winemaker riskadjusted discount rate φ, the Capital Asset Pricing Model (CAPM) is employed, and thus φ = (1 + rf + γ (rm − rf ))−1 . From the CAPM model, the market return rm and winemaker’s risk measure γ , where γ = COV(rj , rm )/V (rm ), are estimated using the market return of Livex 100 index from 2006 to 2013. Table 8.4 summarizes the impact of wine futures allocation as a percentage of total production, α, on profitability of the twelve Bordeaux wineries used in the study. We use ρ to describe the percentage difference between a winemaker having the ability to sell wine as futures and the case when the futures market does not exist. This study illustrates that from 2006 to 2011, the ability to sell wine as futures can generate on average an additional 10.10% of profitability for the Bordeaux winemakers. In addition to profits improvement, this empirical study also shows that the percentage of wine that should be allocated as wine futures ranges from 12.30% to a 64.03% with an average of 27.65%. Furthermore, the results presented in Table 8.4 further support the effect of winemaker’s risk preference on the optimal allocation decision and profitability. For a winemaker with lowrisk aversion such as Pavie (φ = 0.97247639), the amount of wine allocated for sales as futures 2001–2011 ranges from as low as 2.08% to a maximum of 31.45% with an average of 12.30%. This results in an average increase in profitability of only 6.27%. On the other hand, wineries that are more riskaverse such as Troplong Mondot (φ = 0.83791897) would allocate on average of 42.55% of production for sales as futures, increasing profitability by 16.11% on average. It must be noted that the relationship presented in Table 8.4 does not take into account other factors that may influence the allocation decision. An example of this can be seen with Evangile (φ = 0.85688923), which received a barrel score of 98 points in 2009, and as a result allocated 100% of their production quantity for sale as futures. This result is consistent with the analysis of the analytical model that suggests that a winemaker would allocate more wine for sales as futures with an increasing barrel score. We next examine the financial impact of wine futures for boutique winery in the United States: Heart and Hands Wine Company. The winemaker’s riskadjusted discount rate φ is estimated to be 0.76595, the consumer riskadjusted discount rate as 0.99659, and the degree of consumer heterogeneity β is equal to 10. The
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Table 8.4 A summary of the results pertaining to the allocation of wine to be sold as futures and its impact on winemaker’s profitability as presented in Noparumpa et al. (2015) Winemaker Angelus Cheval Blanc Clos Fourtet Cos d’Estournel Ducru Beaucaillou Duhart Milon Evangile Leoville Poyferre Mission Haut Brion Pavie Pichon Lalande Troplong Mondot
φ Min α 0.96936308 06.90 0.86918809 08.59 0.88701179 09.48 0.87673835 04.96 0.88961788 14.23 0.79816123 14.80 0.85688923 21.40 0.90829830 06.94 0.94221522 17.19 0.97247639 02.08 0.84258235 06.40 0.83791897 10.86 Weighted average
Max α 049.35 071.25 045.85 043.74 065.63 047.05 100.00 038.90 080.46 031.45 049.09 068.99
Avg α 18.71 39.26 29.18 21.15 39.45 26.24 64.03 23.45 38.04 12.30 29.20 42.55 27.65
Min ρ 2.16 3.23 3.38 1.78 4.88 6.47 7.84 2.43 5.59 1.17 2.53 4.22
Max ρ 14.45 24.81 15.56 14.84 22.06 19.03 41.81 12.74 24.33 14.75 18.54 25.41
Avg ρ 05.71 13.91 10.21 08.21 13.53 10.92 22.70 07.91 11.80 06.27 11.01 16.11 10.10
parameters presented here resemble the market conditions for a smaller U.S. winery that is more riskaverse than the larger and more established Bordeaux wineries. Heart and Hands has a small but avid group of “cult” followers, and thus, consumer heterogeneity is reflected by a smaller of β. On average Heart and Hands should allocate 55.03% of their wine as futures; it would increase profits by 13.87%. This result highlights the important role that wine futures play in assisting smaller U.S. wineries in mitigating quality risks and increasing their profitability.
8.6 Summary This study examines the benefits of wine futures as a form of operational flexibility to mitigate quality rating risks in the wine production process. The proposed prescriptive analytical model incorporates the risk preferences of both the winemaker and the consumers. A multinomial logit model is used to formulate the random demand function that is derived via the consumers’ utility of purchasing wine as futures, retail, or not purchasing at all. Solutions to the optimization problem is presented as closedform expressions. A comparative statics study yields insights into the factors that influence the winemaker’s optimal allocation and pricing decisions. Key conclusions can be drawn from the influence of the barrel score, risk preferences, and the degree of consumer heterogeneity, on the optimal allocation of futures, futures pricing decision and the corresponding profitability. The study illustrates that, when the market for wine is filled with heterogonous consumers, a winemaker can increase her profitability by charging a higher price of futures to the consumers with high willingness to pay. This finding contradicts traditional marketing belief that emphasizes the benefits of a more homogenous
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market condition. Our numerical illustrations demonstrate that selling wine as futures increases profits by 10.10% in Bordeaux with a significantly higher impact for the boutique and artisanal winemakers in the USA.
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Noparumpa T, Kazaz B, Webster S (2015) Wine futures and advance selling under quality uncertainty. Manuf Serv Oper Manag 17(3):411–426 Noparumpa T, Kazaz B, Webster S (2018) Wine futures: Pricing and allocation as levers against quality uncertainty. In: Consigli G, Stefani S, Zambruno G (eds) Handbook on recent advances in commodity and financial modelling – Quantitative methods in banking, finance, insurance, energy and commodity markets. Springer, p 113–139 Olkin I, Lou Y, Stokes L, Cao J (2015) Analyses of winetasting data: A tutorial. J Wine Econ 10(1):4–30 Shugan SM, Xie JH (2000) Advance pricing of services and other implications of separating purchase and consumption. J Serv Res 2(3):227–239 Shugan SM, Xie JH (2005) Advance selling as a competitive marketing tool. J Res Market 22(3):351–373 Stuen ET, Miller JR, Stone RW (2015) An analysis of wine critic consensus: A study of Washington and California wines. J Wine Econ 10(1):47–61 Tang CS, Lim WS (2013) Advance selling in the presence of speculators and forward looking consumers. Prod Oper Manag 22(3):571–587 Van Mieghem JA, Dada M (1999) Price versus production postponement: Capacity and competition. Manag Sci 45(12):1631–1649 Varian H (2009) Intermediate economics. WW Norton Company, New York, NY Xie J, Shugan SM (2001) Electronic tickets, smart cards, and online payments: When and how to advance sell. Market Sci 20(3):219–243
Chapter 9
Wine Analytics: Futures or Bottles? Mert Hakan Hekimo˘glu, Burak Kazaz, and Scott Webster
9.1 Introduction This chapter features a wine distributor’s budget allocation decision between bottled wine and wine futures. Every September, chateaus (i.e., winemakers) harvest grapes and produce wine. For fine wines, there is a lengthy barrelaging process which takes about 18–24 months before the wine gets bottled. Around the first May following the harvest, chateaus start selling the wine in the form of wine futures while it is still aging in barrels. This practice is especially common for Bordeaux fine wines and is referred to as the en primeur campaign. Approximately a year after the beginning of the en primeur campaign, the wine gets bottled to be sold for retail. The owners of the futures contracts also receive their bottled products. To have a better understanding of the timeline of events, let us consider a 2013 vintage fine Bordeaux wine. The grapes are harvested in September 2013, the chateau starts to sell wine futures around May 2014, and the bottled wine becomes available after May 2015. Therefore, a wine distributor has two newly released products to choose from the same chateau in May 2015: Bottled wine of the 2013 vintage and the wine futures of the 2014 vintage. Hekimo˘glu et al. (2017) investigate this allocation problem by utilizing a combination of empirical
M. H. Hekimo˘glu Lally School of Management, Rensselaer Polytechnic Institute, Troy, NY, USA email: [email protected] B. Kazaz () Whitman School of Management, Syracuse University, Syracuse, NY, USA email: [email protected] S. Webster W.P. Carey School of Business, Arizona State University, Tempe, AZ, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_9
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analysis and mathematical modeling. The results from empirical analysis are used in developing the price functions for wine futures and bottled wines in the mathematical model. Our model examines the role of two factors influencing wine prices: (1) Growing season temperatures, and (2) market conditions. A warmer grape growing season is known to yield a betterquality vintage. Moreover, an interesting phenomenon regarding the impact of the temperature is that a warmer growing season for a new vintage has a negative impact on the futures price of the previous vintage. In addition to the temperature, another important factor influencing the wine prices is the market conditions. London International Vintners Exchange (Livex) operates a global marketplace for fine wine trade. The Livex Fine Wine 100 Index (shortly, Livex 100) captures the marketwide effects driving the wine prices. Livex 100, quoted as the industry benchmark by Bloomberg and Reuters, represents the price movements of the world’s most soughtafter 100 bottled wines. Bordeaux fine wine prices show significant fluctuations in the earlier stages of its lifespan. This complicates a wine distributor’s investment decision as its profit margins become uncertain. Should a distributor trade wine futures? If so, how should they allocate their budget between wine futures and bottled wines? There is a tradeoff such that wine futures exhibit greater price uncertainty compared to bottled wine, however, futures are more liquid as they can be easily traded without requiring any physical product shipment. Our study addresses this problem by first conducting an empirical analysis, which serves as a foundation for the mathematical model, and then examines the mathematical model to solve the distributor’s allocation problem. The empirical analysis finds that the futures prices are influenced by both weather and market conditions whereas bottled wine prices are impacted by the market conditions only. This finding confirms the industrywide perception about wine futures that they are riskier than bottled wines in terms of price uncertainty. Not surprisingly, several wine distributors prefer trading bottled wines only due to that perception. However, the mathematical model shows that a distributor should always allocate some of its budget to wine futures. Furthermore, the study estimates the potential financial benefits that a distributor can gain by using the investment strategies suggested by the mathematical model.
9.2 Literature Review Understanding the wine prices has been a popular topic in the economics literature. Ashenfelter et al. (1995) and Ashenfelter (2008) are two seminal papers that study how Bordeaux wine prices can be explained using weather and age. These studies develop empirical models that show good performance in predicting the prices of mature wines. However, they fail when it comes to predicting young wine prices. Establishing a better knowledge about the factors influencing young wine prices has important practical implications for a wine distributor because distributors typically buy and sell fine wine while they are still young. Hekimo˘glu et al. (2017) fill this
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gap by identifying the fundamental dynamics that drive the evolution of young wine prices. Hekimo˘glu and Kazaz (2020) extend the empirical aspect of Hekimo˘glu et al. (2017) by conducting a comprehensive empirical analysis to identify fair prices for the wine futures. Jones and Storchmann (2001), Lecocq and Visser (2006), Ali and Nauges (2007), Ali et al. (2008), and Ashenfelter and Jones (2013) are other papers which also focus on explaining the Bordeaux wines. Byron and Ashenfelter (1995) and Wood and Anderson (2006) examine Australian wines while Haeger and Storchmann (2006) and Ashenfelter and Storchmann (2010) investigate American wines and German wines, respectively. However, these papers focus on mature wine prices, not young wine prices. Furthermore, our study features a mathematical model yielding investment strategies for the wine distributors. Kazaz (2020) provides a summary of analytical models developed for wine futures. Noparumpa et al. (2015) examine a winemaker’s problem of determining the optimal amount of wine to be sold in the form of wine futures. In their paper, expert tasting scores are utilized as the main driver of the winemaker’s allocation decision. Noparumpa et al. (2015) find that winemakers can increase their profits by selling their wine in the form of futures at the optimal levels suggested by a multinomial logit model. They explain that winemakers benefit from selling wine futures as it allows them to pass the risk of having a poorquality vintage to the wine distributors. An intuitive followup question regarding this finding is then whether a wine distributor is losing money by purchasing the wine in the form of wine futures. Why would a distributor engage in futures trade? Our study shows that wine distributors can still benefit from purchasing futures mainly due to their liquidity advantage over bottled wine. Wine futures is a form of an advance selling mechanism which is widely examined in the operationsmarketing interface. Xie and Shugan (2001), Cho and Tang (2013), Tang and Lim (2013), and Boyacı and Özer (2010) are some notable examples studying the role of advance selling practices in operations management. Fine wines are also considered as longterm investments. Storchmann (2012) provides a comprehensive review about wine economics. Dimson et al. (2015), Jaeger (1981), Burton and Jacobsen (2001), and Masset and Weisskopf (2010) compare the longterm return performance of fine wines to that of financial assets such as treasury bills and equities. Our study departs from these papers as we investigate a distributor’s problem that relies on buying and selling fine wines in relatively short terms. Supply uncertainty is another relevant stream where Yano and Lee (1995), Jones et al. (2001), Kazaz (2004), Kazaz (2011), Boyabatlı et al. (2011), Boyabatlı (2015), Kazaz and Webster (2015), Tomlin and Wang (2008), and Li and Huh (2011) are some notable examples. We model supply uncertainty in the form of quality uncertainty. Weather and market conditions provide signals to the wine industry. Operations management literature has examined the impact of signaling especially on demand; Gümü¸s (2014) is one such example. Our study, on the other hand, focuses on the impact of signaling on price evolution.
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9.3 The Model and Its Analysis This section presents a riskaverse wine distributor’s allocation problem and its analysis: Every May, a wine distributor makes an allocation decision between wine futures (of new vintage) and bottled wine (of previous vintage) from a winemaker. This allocation problem can be generalized as follows: In May of calendar year t, the distributor determines the amount of its budget to be invested in wine futures of vintage t − 1 and bottled wine of vintage t − 2.
9.3.1 Empirical Foundation for the Model Functional forms in the mathematical model for wine prices rely on the empirical findings presented in this section. This section examines how weather and market conditions influence wine futures and bottled wine prices. In May of calendar year t, wine futures of vintage t − 1 and bottled wine of j,t−1 j,t−2 vintage t − 2 from winemaker j are released at the prices of rf 1 and rb1 , respectively. The futures prices in September of calendar year t and May of calendar j,t−1 j,t−1 and rf 3 , respectively. The bottled wine prices year t + 1 are denoted by rf 2 in September of calendar year t and May of calendar year t + 1 are represented by j,t−2 j,t−2 rb2 and rb3 , respectively. After the distributor’s initial allocation decision between futures of vintage t − 1 and bottles of vintage t −2 in May of year t, growing season of the upcoming vintage t takes place until September of year t. In the following subsections, the possible impact of growing season temperatures for vintage t on the prices of vintages t − 1 and t − 2 are analyzed along with the possible impact of market conditions represented by the Livex 100 index. The random variable w˜ t (and its realization wt ) represents the uncertainty in the growing season temperatures for vintage t and the random variable m ˜ t (and its realization mt ) represents the uncertainty in market conditions. Details on the data used in the empirical analysis are available in the online supplement of Hekimo˘glu et al. (2017).
9.3.1.1
Models 1A and 1B: Futures Price Evolution
Standardized futures price of vintage t − 1 from winemaker j in stage i = {1, 2, 3} j,t−1 j,t−1 is computed as sf i = (rf i − μf j )/σf j where μf j and σf j represent the i i i i mean and the standard deviation of the futures price. Average temperature difference between the new growing season (of calendar year t) and the vintage’s own growing season is denoted by wt . A positive wt means that the growing season of new vintage t is warmer than the growing season of vintage t − 1. Percentage change in the Livex 100 index over the new growing
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Table 9.1 Linear regression results demonstrating the impact of weather and market conditions on the evolution of futures prices in Hekimo˘glu et al. (2017). ∗∗∗ denotes statistical significance at 1% j,t−1
Parameter Intercept wt mt Adjusted R2 Observations
Model 1A: sf 2 Coefficient 0.0296 −0.0501 0.0079 0.19 220
j,t−1
− sf 1 tstat 2.85∗∗∗ −4.58∗∗∗ 5.47 ∗∗∗
j,t−1
Model 1B: sf 3 Coefficient 0.0788 −0.1281 0.0223 0.37 220
j,t−1
− sf 2 tstat 4.45∗∗∗ −6.88∗∗∗ 9.01∗∗∗
season is denoted by mt . A positive mt means that the Livex 100 index goes up indicating improved market conditions. The following expressions designate models 1A and 1B which represent the futures price evolution from May to September of year t and from September of year t to May of year t + 1, respectively, where t = {2008, 2009, 2010, 2011, 2012} and j = {1, 2, . . . , 44}: j,t−1 j,t−1 sf 2 = γ0 + γ1 wt + γ2 mt + εj,t − sf 1 j,t−1 j,t−1 sf 3 = η0 + η1 wt + η2 mt + εj,t . − sf 2
(9.1) (9.2)
Table 9.1 shows the regression results pertaining to the above two models. Results of Model 1A show that a warmer growing season for the new vintage has a negative impact on the price evolution of the futures of vintage t − 1 from May to September of year t. This is an interesting finding which can be attributed to the substitutability of the new vintage for the vintage t − 1. It is worth recalling that the vintage t − 1 is still in the phase of barrel aging meaning that its characteristics such as taste and bouquet are not fully developed. As a result, a new vintage with a promising growing season has a negative impact on the futures price evolution of the vintage t − 1. Model 1A results also indicate that improved market conditions have positive effect on the futures price. Model 2A further suggests that the aforementioned effects of weather and market conditions are not completely priced in as of September of year t and those effects continue to influence the futures prices through May of year t + 1.
9.3.1.2
Models 2A and 2B: Bottle Price Evolution
Standardized bottle price of vintage t − 2 from winemaker j in stage i = {1, 2, 3} is j,t−2 j,t−2 computed as sbi = (rbi − μbj )/σbj where μbi j and σbj represent the mean i i i and the standard deviation of the bottle price. Average temperature difference between the new growing season (of calendar year t) and the vintage’s own growing season is denoted by wt . A positive wt means
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Table 9.2 Linear regression results demonstrating the impact of weather and market conditions on the evolution of bottle prices in Hekimo˘glu et al. (2017). ∗∗ and ∗∗∗ denote statistical significance at 5% and 1%, respectively j,t−2
Parameter Intercept wt mt Adjusted R2 Observations
Model 2A: sb2 Coefficient 0.0248 −0.0082 0.0059 0.01 220
j,t−2
− sb1 tstat 1.52 −0.59 2.19∗∗
j,t−2
Model 2B: sb3 Coefficient 0.0187 0.0245 0.0255 0.12 220
j,t−2
− sb2 tstat 0.53 0.82 4.43∗∗∗
that the growing season of new vintage t is warmer than the growing season of vintage t − 2. Percentage change in the Livex 100 index over the new growing season is denoted by mt . A positive mt means that the Livex 100 index goes up indicating improved market conditions. The following expressions designate models 2A and 2B which represent the bottle price evolution from May to September of year t and from September of year t to May of year t + 1, respectively, where t = {2008, 2009, 2010, 2011, 2012} and j = {1, 2, . . ., 44}: j,t−2 j,t−2 sb2 = θ0 + θ1 wt + θ2 mt + εj,t − sb1 j,t−2 j,t−2 sb3 = λ0 + λ1 wt + λ2 mt + εj,t . − sb2
(9.3) (9.4)
Table 9.2 presents the findings related to these models. Growing season weather of the new vintage does not have any statistically significant effect on the bottle price evolution of vintage t − 2. This finding makes sense because vintage t − 2 is bottled which means that it has completed the barrelaging process and features betterestablished characteristics unlike the vintage t − 1 which is still aging in barrels. As a result, the new vintage is not considered as a potential substitute for the bottled vintage t − 2. Model 2A shows that improved market conditions affect the price evolution of vintage t − 2 in a positive way through September of year t and Model 2B suggests that this effect continues to last through May of year t + 1.
9.3.1.3
Functional Forms for the Analytical Model
This section presents the functional forms for wine futures and bottled wine prices based on the empirical results. For notational simplicity, the superscripts j, t − 1, and t − 2 from futures and bottled wine prices, and the subscript t from w and m are dropped. Wine futures price in May of year t is denoted by f 1 . For a given (w, m) realization, the realized futures price in September of year t is f 2 (w, m) and the expected futures price in May of year t + 1 is f 3 (w, m). In line with the results in
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Table 9.1, the futures price satisfies ∂f 3 (w, m)/∂w < ∂f 2 (w, m)/∂w < 0 (i.e., weather has negative impact) and ∂f 3 (w, m)/∂m > ∂f 2 (w, m)/∂m > 0 (i.e., market conditions have positive effect). Bottled wine price in May of year t is denoted by b1 . For a given (w, m) realization, the realized bottle price in September of year t is b2 (m) and the expected bottle price in May of year t + 1 is b3 (m). In line with the results in Table 9.2, the bottle price satisfies ∂b3 (m)/∂m > ∂b2 (m)/∂m > 0 (i.e., market conditions have positive effect).
9.3.2 The Model A twostage stochastic program is used to formulate the distributor’s allocation problem. Distributor determines the amount of investment in the futures of vintage t −1 (denoted x1 ) and bottles of vintage t −2 (denoted y1 ) from a winemaker in May of year t, corresponding to stage 1. Distributor makes this decision constrained by its allotted budget (denoted B) to the winemaker and a valueatrisk (VaR) constraint. Without loss of generality, f 1 and b1 are set to 1 for notational simplicity. In September of year t, corresponding to the beginning of stage 2, the realizations of weather and market random variables are observed by the distributor. Based on these realizations, the futures and bottle prices evolve to f 2 (w, m) and b2 (m), respectively. Expected values of both random variables are normalized to zero, i.e., E[w] ˜ = E[m] ˜ = 0. Their probability density functions are denoted by φw (w) and φm (m) which are defined over the supports of [wL , wH ] and [mL , mH ], respectively. Let = [wL , wH ] × [mL , mH ]. In September of year t (stage 2), the distributor has the recourse flexibility where it determines the amount of futures to buy or sell (denoted x2 ) and the amount of bottles to buy (denoted y2 ). At this stage, the distributor can buy more or sell some of its existing futures mainly because of the liquidity of wine futures. Wine futures can easily be traded by simply transferring the ownership rights through Livex; futures transactions do not require any immediate physical flow of goods. On the other hand, in September of year t, the distributor cannot sell the bottled wine purchased in May of year t due to logistical and legal constraints. In May of year t + 1, corresponding to the end of stage 2, the distributor collects revenues from the futures and bottles by selling them to its customers such as wholesalers, liquor stores, restaurateurs. From September of year t to May of year t + 1, the uncertainties in the futures and bottle prices are represented by z˜ f and z˜ b , respectively. In May of year t + 1, futures are priced at f 3 (w, m) + zf and bottles are priced at b3 (m) + zb where f 3 (w, m) and b3 (m) represent the expected values as of September of year t since E[˜zf ] and E[˜zb ] are set to zero. Our data suggests that if the price increases from May to September of year t, then it generally increases from September of year t to May of year t + 1. The following assumptions are incorporated considering this observation. Note that all price functions f 2 (w, m), f 3 (w, m), b2 (m) and b3 (m) are linear in their arguments.
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If f2 (w, m)f1 , then f3 (w, m)f2 (w, m)for all ∈{>, =, , =, 1.
(9.17)
Note that this assumption is relaxed in Sect. 4 where we present the application of our model using Bordeaux futures and bottled wine prices.
9.3.3 Analysis The details of the derivations, along with the proofs of the technical results, are presented in the online supplement of Hekimo˘glu et al. (2017). The set representing the realization space for (w, ˜ m) ˜ is partitioned as follows: 0 = {(w, m) ∈ : f3 (w, m)/f2 (w, m) = b3 (m)/b2 (m) = 1} 1 = {(w, m) ∈ : f3 (w, m)/f2 (w, m) < 1 and b3 (m)/b2 (m) < 1} 2 = {(w, m) ∈ : f3 (w, m)/f2 (w, m) ≥ max{b3 (m)/b2 (m), 1} \ 0} 3 = {(w, m) ∈ : b3 (m)/b2 (m) ≥ max{f3 (w, m)]/f2 (w, m), 1} ∪ 0}. These subsets represent expected returns of futures and bottles in stage 2. 0 represents the realizations where futures and bottles yield neither gains nor losses. In 1, both futures and bottles yield losses. In 2, futures bring greater profits than bottles whereas, in 3, bottles bring greater profits. The threshold market mτ is defined as b3 (mτ )/b2 (mτ ) = 1 and f 3 (0, mτ )/f 2 (0, mτ ) = 1. The weather threshold wτ (m) is defined as f 3 (wτ (m), m)/f 2 (wτ (m), m) = 1 for m ≤ mτ . Following from Eqs. 9.5, 9.6 and 9.17, the following holds:
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mτ < 0, wτ (m) < 0 for all m < mτ , and wτ (mτ ) = 0.
(9.18)
Furthermore, the following is assumed to eliminate trivial instances of this problem: mτ > mL
and
wτ (mL ) > wL .
(9.19)
The optimal solution to the riskneutral version of stage 2, where the secondstage VaR constraint 9.10 is relaxed, is denoted as (x20 , y20 ), i.e., 0 x2 (x1 , y1 , w, m), y20 (x1 , y1 , w, m) = arg max E[(x1 , y1 , w, m, x2 , y2 , z˜ f , z˜ b )] x2 ,y2
s.t. 9.9, 9.11, 9.12. It can be characterized as follows: ⎧ if (w, m) ∈ 1 ⎪(−x1 , 0) 0 0 ⎨ x2 , y2 = ((B − x1 − y1 ) /f2 (w, m) , 0) if (w, m) ∈ 2 , ⎪ ⎩ (−x1 , (B − x1 − y1 + f2 (w, m) x1 ) /b2 (m)) if (w, m) ∈ 3 (9.20) where the distributor sells all futures purchased in stage 1 in 1, buys more futures in 2 if there is any leftover budget from stage 1, swaps futures purchased in stage 1 for bottles, and buys additional bottles if there is any leftover budget in 3. The following is assumed to ensure that the distributor is better off by buying bottled wines in stage 1 as opposed to saving its entire budget for stage 2; this is consistent with the practices of wine distributors. & ∂E[(x1 , y1 , w, ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )] && > 0. & ∂y1 (x1 ,y1 )=(0,0)
(9.21)
Proposition 1 For any (x1 , y1 ), ∂E[(x1 , y1 , w, ∂E[(x1 , y1 , w, ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )] ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )] ≥ > 0. ∂x1 ∂y1 (9.22) Proposition 1 states that a riskneutral distributor observes greater improvement in its expected profit by making investment in wine futures rather than bottled wines in stage 1. This indicates that the riskneutral distributor would allocate its entire budget in wine futures. This result is driven by the liquidity advantage of futures along with the flexibility of swapping them for bottles in stage 2. Mathematical expressions for the values of liquidity and swapping are available in the online supplement of Hekimo˘glu et al. (2017).
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Proposition 2 When φw (w) and φm (m) follow symmetric pdf, (a) E[(x1 , y1 , w, ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )] increases in σm2 ; (b) E[(x1 , y1 , w, ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )] increases in σw2 if the combined value from liquidity and swapping increases in σw2 . Proposition 2 highlights how the degree of market and weather uncertainties influence the expected profit. The distributor observes greater expected profits as the variance of market conditions increases. On the other hand, the impact of the variance of weather conditions can be in either way. If the combined value of liquidity and swapping increases in higher degrees of weather uncertainty, then the expected profit increases as well. The reminder of this section focuses on the solution to the riskaverse distributor’s problem by reinstating the timeconsistent VaR constraints 9.10 and 9.16. Proposition 3, which presents the optimal stage1 decisions for the riskaverse problem, makes use of the following expressions: x1+ = β/[1 − f2 (wH , mL )] x1V = [β + zbα B]/([1 − f2 (wH , mτ )][1 + zbα ]) y1V = [β − [1 − f2 (wH , mL )]x1V ]/[1 − b3 (mL ) − zbα ] x1s = (β − B[1 − b3 (mL ) − zbα ])/[b3 (mL ) + zbα − f2 (wH , mL )] y1s = (B[1 − f2 (wH , mL )] − β)/[b3 (mL ) + zbα − f2 (wH , mL )] − zf α < β/B
& ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )]/∂y1 &(x ,y )=(0,0) ∂E[(x1 , y1 , w, & 1 1 ∂E[(x1 , y1 , w, ˜ m, ˜ x20 , y20 , z˜ f , z˜ b )]/∂x1 &(x ,y )=(0,0) 1
0 in all conditions. This is a surprising result because wine futures exhibit greater price uncertainty than bottled wines yet it turns out that a riskaverse distributor must always carry some futures in its portfolio. Proposition 4 When φw (w) follows a symmetric pdf and (˜zf , z˜ b ) follow a bivariate normal distribution, ∂E[(x1 , y1 , w, ∂E[(x1 , y1 , w, ˜ m, ˜ x2∗ , y2∗ , z˜ f , z˜ b ] ˜ m, ˜ x2∗ , y2∗ , z˜ f , z˜ b )] ≥ > 0. ∂x1 ∂y1 (9.25) Proposition 4 extends Proposition 3 by relaxing Eq. 9.23, which restricts the degree of uncertainty in stage 2. When stage2 uncertainty is not restricted, (x20 , y20 ) may violate the VaR constraint 9.10. As a result, closedform expressions may not be obtained for (x1∗ , y1∗ ). However, Proposition 4 implies that x1∗ > 0 continues to hold meaning that the distributor should always carry some wine futures.
9.4 Financial Benefits from Our Proposed Model It is mentioned earlier that wine distributors are hesitant in purchasing wine futures due to the lack of knowledge about how futures prices evolve into bottled wine prices. This section demonstrates how much a distributor can benefit from purchasing both futures and bottled wine compared to a benchmark strategy of trading only bottled wines. The distributor determines how much to invest in futures and bottles using its allotted budget for the 44 leading Bordeaux winemakers. Data used in this section is described in the online supplement of Hekimo˘glu et al. (2017). Empirical models presented in Sect. 9.3.1 are calibrated for calendar year t ∈ {2008, 2009, 2010}. Using the coefficient estimates, the distributor’s problem of allocating its budget between futures of vintage t − 1 and bottles of vintage t − 2 for each winemaker j ∈ {1, . . . , 44} independently is solved for calendar year t ∈ {2011, 2012}. The distributor’s tolerable loss is assumed to be 20% of budget (i.e., β = 0.2B), and the effect of varying risk aversion is captured at several α values, i.e.,
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α ∈ {1, 0.20, 0.10}. The case of α = 1 corresponds to a riskneutral distributor, whereas α = 0.20 and α = 0.10 correspond to lowrisk averse and highrisk averse distributors, respectively. While the results do not depend on the choice of budget parameter B, B is set to 10,000 arbitrarily. j, t E[1 (x1∗ , y1∗ )] denotes the optimal expected profit from winemaker j in year t. j, t E[1 (0, y1∗∗ )] denotes the expected profit from the benchmark strategy where the distributor does not trade futures at all, i.e., (x1 , x2 ) = (0, 0). The financial benefit is defined as follows: j, t j, t j, t j, t = E 1 (x1∗ , y1∗ ) − E 1 (0, y1∗∗ ) /E 1 0, y1∗∗ .
(9.26)
Table 9.3 presents the average benefit that the distributor would realize from each j winemaker j , i.e., = (1/2) t (j, t ). Table 9.3 shows that a riskneutral distributor can benefit from investing in wine futures by improving its profits by 21.45%. The largest benefit is recorded for Figeac at 84.78%. On the other hand, there are four winemakers that wine distributor cannot benefit from; this is due to relaxation of Eq. 9.17 which now allows for wine futures and bottled wines to have different expected returns. In case of those four winemakers, expected bottle returns are significantly greater than the expected returns of futures that leads to zero benefit from futures trade. While the impact of risk aversion on the financial benefit is not monotone, the benefit increases on average in risk aversion. This increase stems from the fact that use of futures allows the distributor to maintain a greater total investment in stage 1 compared to the benchmark strategy, i.e., f1 x1∗ + b1 y1∗ > b1 y1∗∗ . As a result, the distributor enjoys benefits of 22.98 and 24.29% under lowrisk aversion and highrisk aversion, respectively.
9.5 Conclusions Our study makes three important contributions. First, it builds an analytical model which is based on an empirical foundation. With the empirical analysis, we confirm a common perception in wine industry that wine futures exhibit greater price risk than bottled wines. In fact, this is the reason why several distributors prefer trading only bottled wines. The analytical model is built in line with this finding. Second, by analyzing the analytical model, we show that, despite being risky, a distributor should always make some investment in wine futures. Our study characterizes the optimal investment strategies for a riskaverse distributor. Third, we demonstrate that a wine distributor can significantly improve its profits by trading wine futures strategically. The financial benefit is 21.45% for a riskneutral distributor and it increases to 24.29% for a highly riskaverse distributor. Our study makes important practical contributions. By shedding light on how futures can be traded despite the volatility in prices, more wine distributors are
Average
Winemaker (j) Angelus Ausone Beychevelle Calon Segur Carruades de Lafite Cheval Blanc Clos Fourtet Conseillante Cos d’Estournel Ducru Beaucaillou Duhart Milon Eglise Clinet Evangile Figeac Fleur Petrus Forts Latour Grand Puy Lacoste Gruaud Larose Haut Bailly Haut Brion Lafite Rothschild Lafleur
Riskneutral 4.45% 48.33% 0.00% 1.88% 37.10% 29.71% 38.92% 10.69% 36.04% 0.00% 10.35% 13.28% 14.48% 84.78% 24.80% 30.24% 25.13% 7.34% 1.38% 9.91% 22.06% 55.74%
j
Riskneutral 21.45%
Lowrisk averj sion 7.40% 53.18% 0.00% 1.88% 51.70% 34.44% 38.96% 5.95% 31.53% 2.30% 8.94% 21.90% 33.16% 76.61% 30.97% 30.24% 26.18% 7.34% 1.38% 11.94% 43.32% 35.73%
Highrisk j aversion 10.00% 54.32% 0.00% 1.88% 56.93% 36.89% 39.30% 5.35% 31.99% 4.33% 12.74% 21.71% 34.81% 74.73% 46.23% 30.24% 27.41% 7.34% 1.38% 14.32% 47.28% 33.29% Lowrisk aversion 22.98%
Winemaker (j) Lagrange St Julien Latour Leoville Barton Leoville Las Cases Leoville Poyferre Lynch Bages Margaux Mission Haut Brion Montrose Mouton Rothschild Palmer Pavie Pavillon Rouge Petit Mouton Petrus Pichon Baron Pichon Lalande Pin Pontet Canet Talbot Troplong Mondot Vieux Chateau Certan
Riskneutral 23.67% 70.13% 18.63% 28.20% 36.72% 20.97% 31.84% 9.50% 14.90% 10.93% 0.00% 24.46% 5.00% 3.69% 21.31% 17.06% 10.29% 5.00% 10.44% 0.00% 32.24% 21.33%
Lowrisk averj sion 23.67% 78.21% 18.63% 24.78% 23.82% 20.97% 50.52% 12.99% 14.07% 20.65% 0.00% 25.99% 5.00% 3.69% 17.63% 17.06% 5.85% 5.12% 10.44% 0.00% 31.29% 29.73% Highrisk aversion 24.29%
j
Highrisk j aversion 23.67% 78.84% 21.58% 25.92% 23.39% 20.97% 53.81% 12.62% 17.98% 22.62% 0.00% 28.53% 5.00% 3.69% 16.70% 17.06% 7.49% 6.04% 10.44% 0.00% 31.21% 31.83%
j j Table 9.3 The average financial benefit = j /44 in Hekimo˘glu et al. (2017) where is the average profit improvement for winemaker j , B = 10,000 and β = 2000; and, α ∈ {1, 0.20, 0.10} for riskneutral, lowrisk aversion, and highrisk aversion, respectively 154 M. H. Hekimo˘glu et al.
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expected to include futures in their investment strategies. Furthermore, this study complements Noparumpa et al. (2015) by showing that wine distributors can also improve their profits via futures trade meaning that both winemakers and distributors can benefit. Wine futures mitigate risks stemming from quality uncertainty at the time bottling according to Noparumpa et al. (2015) and its liquidity justifies the investment in this riskier asset according to Hekimo˘glu et al. (2017).
References Ali HH, Lecocq S, Visser M (2008) The impact of gurus: Parker grades and en primeur wine prices. Econ J 118:158–173 Ali HH, Nauges C (2007) The pricing of experience goods: the case of en primeur wine. Amer J Agricult Econ 89:91–103 Ashenfelter O (2008) Predicting the prices and quality of Bordeaux wines. Econ J 118(529):F174– F184 Ashenfelter O, Ashmore D, Lalonde R (1995) Bordeaux wine vintage quality and the weather. Chance 8(4):7–13 Ashenfelter O, Jones GV (2013) The demand for expert opinion: Bordeaux wine. J Wine Econ 8(3):285–293 Ashenfelter O, Storchmann K (2010) Using a hedonic model of solar radiation to assess the economic effect of climate change: the case of Mosel valley vineyards. Rev Econ Statist 92(2):333–349 Boda K, Filar JA (2006) Time consistent dynamic risk measures. Math Methods Oper Res 63(1):169–186 Boyabatlı O (2015) Supply management in multiproduct firms with fixed proportions technology. Manag Sci 61:2825–3096 Boyabatlı O, Kleindorfer P, Koontz S (2011) Integrating longterm and shortterm contracting in beef supply chains. Manag Sci 57(10):1771–1787 Boyacı T, Özer Ö (2010) Information acquisition for capacity planning via pricing and advance selling: When to stop and act? Oper Res 58(5):1328–1349 Burton BJ, Jacobsen JP (2001) The rate of return on investment in wine. Econ Inquiry 39:337–350 Byron RP, Ashenfelter O (1995) Predicting the quality of an unborn Grange. Econ Record 71:40– 53 Cho SH, Tang CS (2013) Advance selling in a supply chain under uncertain supply and demand. Manuf Service Oper Manag 15(2):305–319 Devalkar S, Anupindi R, Sinha A (2015) Dynamic risk management of commodity operations: Model and analysis. Working paper. http://ssrn.com/abstract=2432067 Dimson E, Rousseau P, Spaenjers J (2015) The price of wine. J Financ Econ 118(2):431–449 Gümü¸s M (2014) With or without forecast sharing: Credibility and competition under information asymmetry. Prod Oper Manag 23(10):1732–1747 Haeger JW, Storchmann K (2006) Prices of American pinot noir wines: Climate, craftsmanship, critics. Agricult Econ 35:67–78 Hekimo˘glu MH, Kazaz B (2020) Analytics for wine futures: realistic prices. Prod Oper Manag 29(9):2096–2120 Hekimo˘glu MH, Kazaz B, Webster S (2017) Wine analytics: Fine wine pricing and selection under weather and market uncertainty. Manufacturing Service Oper Manag 19(2):202–215 Jaeger E(1981) To save or savor: The rate of return to storing wine. J Political Econ 89(3):584–592 Jones G, Storchmann K (2001) Wine market prices and investment under uncertainty: an econometric model for Bordeaux Cru Classes. Agricult Econ 26:114–133
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Jones PC, Lowe T, Traub RD, Keller G (2001) Matching supply and demand: The value of a second chance in producing hybrid seed corn. Manuf Service Oper Manag 3(2):122–137 Kazaz B (2004) Production planning under yield and demand uncertainty. Manuf Service Oper Manag 6(3):209–224 Kazaz B (2020) Wine analytics. To appear in C. Druehl and W. Elmaghraby (eds) Pushing the boundaries: Frontiers in impactful OR/OM research, TutORials in Operations Research, INFORMS Kazaz B, Webster S (2011) The impact of yield dependent trading costs on pricing and production planning under supply uncertainty. Manuf Service Oper Manag 13(3):404–417 Kazaz B, Webster S (2015) Technical note – Pricesetting newsvendor problems with uncertain supply and risk aversion. Oper Res 63(4):807–811 Lecocq S, Visser M (2006) Spatial variations in weather conditions and wine prices in Bordeaux. J Wine Econ 1(2):114–124 Li H, Huh WT (2011) Pricing multiple products with the multinomial logit and nested logit concavity: Concavity and implications. Manuf Service Oper Manag 13(4):549–563 Masset P, Weisskopf JP (2010) Raise your glass: Wine investment and the financial crisis. Working paper Noparumpa T, Kazaz B, Webster S (2015) Wine futures and advance selling under quality uncertainty. Manuf Service Oper Manag 17(3):411–426 Storchmann K (2012) Wine economics. J Wine Econ 7(1):1–33 Tang CS, Lim WS (2013) Advance selling in the presence of speculators and forward looking consumers. Prod Oper Manag 22(3):571–587 Tomlin B, Wang Y (2008) Pricing and operational recourse in coproduction systems. Manag Sci 54(3):522–537 Wood D, Anderson K (2006) What determines the future value of an icon wine? New evidence from Australia. J Wine Econ 1(2):141–161 Xie J, Shugan SM (2001) Electronic tickets, smart cards, and online payments: When and how to advance sell. Marketing Sci 20(3):219–243 Yano CA, Lee H (1995) Lot sizing with random yields: A review. Oper Res 43(2):311–334
Part III
Government Interventions
Chapter 10
Implications of Farmer Information Provision Policies: Heterogeneous Farmers and Market Selection ChenNan Liao, YingJu Chen, and Christopher S. Tang
10.1 Introduction In India,1over 50% of the total workforce depends on agriculture for their livelihood, and yet over 500 million farmers in India are smallholders who earn less than US $560 (US $350) per year (UN Millennium Project 2005). The World Bank reported that farmer’s earnings are hindered by their limited access to relevant and timely information for deciding which crop (paddy or sorghum) to grow during the planting season or which market to sell in during the harvest season. To improve farmers’ total profit, there is a general belief that market information can improve farmer welfare because more accurate and timely information can enable farmers to
1 Acknowledgment:
This chapter is adapted from “Information Provision Policies for Improving Farmer Welfare in Developing Countries: Heterogeneous Farmers and Market Selection”, Manufacturing & Service Operations Management, 21(2):254–270, 2019.
C.N. Liao College of Management, National Taiwan University, Taipei, Taiwan email: [email protected] Y.J. Chen School of Business and Management and School of Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong email: [email protected] C. S. Tang () Anderson School of Management, University of California Los Angeles (UCLA), Los Angeles, CA, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_10
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make informed decisions about crop selection, quantity decision, market selection, etc.2 Despite this general belief, empirical findings on the implications of information services such as RML have been mixed. Specifically, the empirical study by Mittal et al. (2010) provides supportive evidence that RML subscribed farmers enjoyed higher income (5–25% increase). However, Fafchamps and Minten (2012) conducted a controlled randomized experiment in 100 villages in the Indian state of Maharashtra about the price of 5 crops (tomatoes, pomegranates, onions, wheat, and soybeans). Surprisingly, they find that the resulting prices received by RML subscribed farmers and regular farmers are statistically indistinguishable. These empirical studies call for systematic investigations about the value of market information for improving farmer welfare. To examine whether governments/NGOs should disseminate market information as widely (and precisely) as possible, Chen and Tang (2015) investigate a situation in which homogeneous farmers who act as price setters engage in a Cournot competition in a single market for a single crop. They claim that, when farmers have no private signals, providing public signal to all farmers is always beneficial. In this chapter, we relax these key assumptions and find new results. Specifically, we consider the following setting: (1) each smallholder farmer is a price taker in the sense that he has no influence on the market price on the individual level; and (2) farmers are heterogeneous in terms of the inherent preference for growing a certain crop or for selling in a certain market (instead of growing a single crop for a single market). In our model, heterogeneous farmers need to select one of the two markets to sell in (or one of the two crops to grow). We adopt the Hotelling model in which there is a continuous type of infinitesimal farmers located uniformly along a line over [−0.5, 0.5] (cf., Lilien et al. 1992).3 We consider two markets (or two crops) located at both ends of the line that we label as the right market (crop) r and the left market (crop) l. (While our model can be applied to market (or crop) selection, we
2 In Kenya and Mali, an NGO program, Mali Shambani, provides a weekly hourlong radio program
that discusses market price trends, current market prices, farming techniques, etc. This free radio program also allows farmers to ask agricultural questions via phone or short messaging services (SMS); see USAID (2011). Likewise, India’s Ministry of Agriculture provides some market information on its website (www.india.gov.in/topics/agriculture) and mounts a hotline service (Kisan call centers) to provide advisory service to farmers over the phone. Besides the information provided by the government and NGOs, forprofit enterprises such as Reuters Market Light (RML, reutersmarketlight.com) and Nokia Life Tools (NLT) broadcast customized information via SMS messages to farmers who subscribed to their service at a nominal fee since 2007 and 2009, respectively (Tang and Sheth 2013). The provided information includes local crop prices, customized, localized, and personalized weather forecasts, agricultural news, and crop advisory. The reader is referred to Chen and Tang (2015) for details of different services provided by the governments, nongovernmental organizations (NGOs), and forprofit companies such as Nokia Life Tools and Reuters Market Light. 3 We shall extend our analysis to the case when farmer’s location is not uniform but a symmetric distribution at the origin 0 and show that our main results continue to hold.
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shall use the term market selection throughout this chapter to ease our exposition.) In each market i = r, l, we assume that the market price pi is a linearly decreasing function of the total quantity qi to be sold in the market i so that pi = ai −bqi . In this chapter, we consider the case when the government has imperfect signal xi about ai , the intercept of the market price function for each market i = l, r. To improve farmers’ expected total profit, the government needs to decide on the information provision policy (R, ρ), where R ⊂ [−0.5, 0.5] is the range of farmers who receive market signals, and ρ ≤ 1 is the percentage of farmers in R who receive market signals. For any given information provision policy δ = (R, ρ), only ρ percentage of farmers located within the range R will receive the market signals (xl , xr ) about the market prices of market l and r, respectively. Depending on the signals received (or not received), we determine the market selection rule that each farmer will follow in equilibrium. By examining the exante expected total profit of all farmers in equilibrium, we determine the optimal information provision policy δ ∗ = (R ∗ , ρ ∗ ) and establish the following results: 1. There exists a unique threshold τ (δ) associated with any given information provision policy δ = (R, ρ) so that a farmer located at θ ∈ [−0.5, 0.5] who receives market signals will sell in the left market l if θ < τ (δ) , and sell in the right market r otherwise. Also, farmers who receive no signals will follow the threshold market selection rule that has the origin 0 as the threshold. 2. Market signals can improve farmer welfare. 3. To maximize farmers’ total profit, providing market signals to all farmers may not be optimal. 4. To ensure fairness among farmers, the government should offer information to all farmers at a nominal fee so that farmers will adopt the intended optimal provision policy willingly. The first result is intuitive because farmers would prefer to sell in a nearby market unless the market signals indicate the higher selling price in the farther market would outweigh the additional transportation cost. The second result is also intuitive because market signals can enable farmers to make better decisions. The third result is caused by the fact that market signals can entice each farmer to selfishly select the more promising market that would yield a higher profit for himself. However, if too many farmers are influenced by the information, it can create serious negative externality that affects other farmers’ profit.4 Finally, the fourth result reveals that a nominal fee will entice farmers to adopt the intended optimal provision policy willingly without the fear of treating any farmer unfairly. To examine the robustness of these four results, we extend our base model to examine different settings including: when the information is distributed through a forprofit company, when the precision of market information varies, when the
4 Each
smallholder farmer is a price taker on the individual level. However, on the aggregate level, the market price drops as the total production quantity increases.
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markets are correlated, when farmer’s location follows a general distribution, when there is potential information leakage, and when the key objective is social welfare. The remainder of this chapter is organized as follows. Section 10.2 reviews the related literature, and Sect. 10.3 presents the model settings. In Sect. 10.4, we analyze the farmer’s market selection rule in equilibrium and the farmer welfare. Section 10.5 examines different information provision policies and identifies the optimal provision policy. Finally, in Sect. 10.6, we summarize the results and discuss the future extensions.
10.2 Literature Review This chapter is related to an emerging stream of research in socially responsible operations. Sodhi and Tang (2014) propose that a sustainable way to alleviate poverty is to engage the poor as producers or distributors by enabling them with financial, information, and material flows along the supply chain. Recent research articles that examine the implications of disseminating agricultural information via mobile phones or Internet include the following. First, Fafchamps and Minten (2012) and Mittal et al. (2010) find mixed results regarding the benefits of disseminating information through mobile phones. By examining the impact of disseminating market price information via mobile phones in India, Parker et al. (2012) provide empirical evidence about the reduction of geographic price dispersion of crops in rural communities. Chen et al. (2013) investigate the implication of ITC eChoupals, an Internet platform that provides market price and crop advisory information. (In Goyal 2010, the author estimates its impact and finds an increase in the price of soy and a reduced price dispersion.) Cole and Fernando (2012) use a randomized experiment to evaluate the influence of Avaaj Otalo, a mobile phonebased agricultural consulting service/forum, on farmers’ agricultural decisions, including the choice of information and advise sources, the choice of pesticides, and the choice of crops. (See also Chen et al. 2015, which examines the peertopeer interactions among farmers on this kind of platform.) In a “farmer” context (Cournot competition in which farmers’ actions are strategic substitutes), Chen and Tang (2015) and Zhou et al. (2020) also reach similar results as obtained by Morris and Shin (2002) and Cornand and Heinemann (2008) in the economics literature. Specifically, Morris and Shin (2002) find that, when the agents’ actions are strategic complements, public information can increase the welfare in the absence of private information. However, in the presence of private signals (with substantial precisions), public information might be harmful. Instead of reducing signal precision, Cornand and Heinemann (2008) suggest that reducing the number of agents receiving signals might be a better choice. Our chapter differs from the above literature in four ways: (1) we consider the case when all agents’ (farmers’) actions are strategic substitutes instead of strategic complements as considered in Morris and Shin (2002) and Cornand and Heinemann (2008), (2) we consider the case when farmers are heterogeneous
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instead of homogeneous as considered in Morris and Shin (2002), (3) we consider the case when farmers are price takers instead of price setters as considered in Chen and Tang (2015) and Zhou et al. (2020), and (4) we consider the case of two markets instead of a single market. By analyzing a model based on different settings as stated above, we show that it is not necessarily optimal to provide information to all farmers (in terms of farmers’ total profit). Instead, it may be optimal for the government to provide access to a specific subset of farmers. However, this information provision policy might create fairness concerns, and we show that the government can ensure fairness by providing information access to all farmers at a nominal fee so that farmers will adopt the intended optimal provision policy willingly. To examine the robustness of these results, we extend our analysis of our base model to incorporate various issues and we find that our main results continue to hold: it may be optimal for the government to provide information access to a specific subset of farmers.
10.3 Model Description Our model is based on the following elements: 1. Heterogeneous Farmers. Following the Hotelling model, there is a continuous type of infinitesimal farmers distributing uniformly over a line [−0.5, 0.5].5 Each farmer can produce up to 1 unit. Each farmer is a price taker: his production quantity is too small to influence the market price. However, the total quantity produced by all farmers is large enough to affect the market price. 2. Two Markets. There are two markets located at the opposite ends of the line: the “left” market l is located at −0.5 and the “right” market r is located at 0.5. 3. Uncertain Market Price. For each market i, i = l, r, the unit market price pi = ai − bqi , where qi is the total quantity to be sold by the farmers in market i and b is price elasticity. Also, ai is the intercept (or market size) so that ai = A + ui (i ∈ {l, r}), where A is the mean value of the intercept and ui ∼ N(0, σ 2 ) represents the uncertainty of the intercept in market i.6 We define α ≡ 1/σ 2 to denote the “intrinsic certainty” of the intercepts. 4. Farmer’s Profit Function Without Market Information. For each farmer located at θ ∈ [−0.5, 0.5], we normalize his unit production cost to 0. However, there is a transportation cost for this farmer to sell in market i that depends on the “distance” between his location θ and the market i that he sells in. By accounting for the uncertain market price in market i, the profit for a farmer located at θ who sells one unit in market i is πˆ 0 (θ, i), where 5 We
shall extend our analysis to the case when the distribution is not uniform but a general symmetric distribution. 6 Following a standard assumption used in the literature, we assume that A is sufficiently large so that the pi is almost always positive (GalOr 1985 and Vives 1984). Also, we assume that ul and ur are independent.
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πˆ 0 (θ, l) = al − bql − t (0.5 + θ ),
and
πˆ 0 (θ, r) = ar − bqr − t (0.5 − θ ).
(10.1)
5. Market Signals. For each market i = l, r, the government has a noisy signal xi about the uncertainty of the market intercept, ui , where xi = ui + i . We assume that i ∼ N (0, s 2 ) and l and r are independent. By using β ≡ 1/s 2 to denote the precision of market signals, E(ui (xl , xr )) = Var(ui (xl , xr )) =
β · xi , α+β 1 , α+β
and
for i = l, r.
(10.2)
Because 1/(α + β) < 1/α = σ 2 , we can conclude that market signals enable farmers to obtain a more accurate forecast about the intercept. 6. Government Information Provision Policy. With the possession of market signals (xl , xr ), we shall focus on a class of provision policy δ that can be specified by two decisions K ∈ [0, 0.5] and ρ ∈ [0, 1] so that ρ percentage of farmers located within [−K, K] will receive the market signals.7 The sequence of events goes as follows. The government first sets provision policy δ = (K, ρ). The market signals (xl , xr ) are realized, and the government disseminates them according to policy δ. For each farmer located at θ , he will use the market signals (xl , xr ) he receives (if any) to select the market to sell in. Once the total quantity to be sold in each market (qi ) is determined, the market price pi = A + (ui (xl , xr )) − bqi is realized in market i for i = l, r, and the farmers’ total profit is also realized. For ease of reference, we provide a summary of our notation in Table 10.1. To examine the value of information and the value of centralized control, we shall examine three benchmark provision policies: 1. Centralized Control Policy (C): To maximize farmers’ total profit, the government uses market signals to assign the market for each farmer to sell in. (Besides policy (C), all other provision policies are implemented under a decentralized system.)
7 We
did consider a more general class of provision policy δ that can be specified by two decisions R ⊂ [−0.5, 0.5] and ρ ≤ 1 so that ρ percentage of farmers located within R will receive the market signals. For tractability, we shall focus on the case when R is symmetric about the origin 0. When R consists of a finite number of closed intervals, we show that each provision policy (R, ρ) is dominated by a corresponding provision policy under which R is a continuous interval so that R = [−K, K], where K ∈ [0, 0.5]. We omit the details here, but the reader is referred to the online supporting material for details. Therefore, it suffices for us to focus on a class of provision policy δ = (K, ρ), where K ∈ [0, 0.5] and ρ ∈ [0, 1] throughout this chapter.
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Table 10.1 Summary of notation Notation pi , ai , and qi A ui α xi β K ρ τ (δ) (δ)
π (δ) (θ; xl , xr ) (or π0 (θ; xl , xr )) w (δ) (xl , xr ) (δ)
(δ) (θ)/0 (θ) W (δ)
Meaning The price, intercept, and aggregate quantity in market i so that pi = ai − bqi , where i ∈ {l, r} The expected value of the intercept in each market The uncertainty of the intercept in market i (ai = A + ui ) The intrinsic certainty of the intercept in each market The signal about ui The precision of each signal The range [−K, K] in which the government provides signals The percentage of farmers located within [−K, K] who receive the market signals The market selection threshold associated with policy δ for farmers with signals The expost expected profit of each farmer at θ with (or without) signals conditional on (xl , xr ) under policy δ The expost expected total profit of all farmers conditional on (xl , xr ) under policy δ The exante expected profit of each farmer at θ with/without signals under policy δ The exante expected total profit of all farmers under policy δ
2. Full Information with Decentralized Control Policy (F1): δ (F1) = (K = 0.5, ρ = 1). All farmers receive signals, and each farmer selects the market that maximizes his profit. 3. No Information Policy (F0): δ (F0) = (K = 0.5, ρ = 0). Farmers receive no signals. The above three benchmark provision policies enable us to examine the value of information and the value of centralized control by comparing the exante expected farmers’ total profit. We also consider two additional provision policies: 1. Full Range with Partial Intensity Policy (Fρ): δ (Fρ) = (K = 0.5, ρ ∈ [0, 1]). Each farmer receives signals with probability ρ, where ρ is selected by the government. 2. General Policy (Kρ): δ (Kρ) = (K ∈ [0, 0.5], ρ ∈ [0, 1]). The government selects both K and ρ so that farmers located over the region [−K, K] will receive signals with probability ρ, and farmers located over [−0.5, −K) ∪ (K, 0.5] will receive no signals. In this case, the optimal provision policy is δ ∗ ≡ (K ∗ , ρ ∗ ) = arg max{W (δ) : δ = (K, ρ), K ∈ [0, 0.5], ρ ∈ [0, 1]}.
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10.4 Analysis: Farmer’s Market Selection and Farmer’s Profit We use backward induction to analyze a Stackelberg game in which the government acts as the leader and the farmers act as followers. Specifically, we first examine the market selection rule that each farmer will adopt in equilibrium. Anticipating the market selection rule, the government determines the optimal provision policy δ ∗ = (K ∗ , ρ ∗ ) that maximizes farmers’ total profit.
10.4.1 Farmer’s Threshold Market Selection Rule Under Provision Policy δ Upon disseminating market signals (xl , xr ) according to policy δ = (K, ρ), we now examine each farmer’s market selection rule. By considering the fact that farmers are price takers and riskneutral, it is immediately clear that each farmer will produce and sell one unit in exactly one market. Therefore, for any provision policy δ = (K, ρ), each farmer located at θ will select the market to sell in that yields the higher expected profit. The comparison of the expected profits between two markets yields the following threshold market selection rule: Lemma 1 For any provision policy δ = (K, ρ), each farmer who is located at θ will adopt the following threshold market selection rule in equilibrium: 1. If the farmer receives signals (xl , xr ), then he will sell in the left market l if θ < τ (δ) (xl , xr ) and sell in the right market r if θ ≥ τ (δ) (xl , xr ), where τ (δ) (xl , xr ) = max
− K, min K,
.
1 β · · (xl − xr ) 2(ρb + t) α + β
(10.3)
2. If the farmer receives no signals, then he will sell in the left market l if θ < 0; otherwise, he will sell in the right market r. To explain Lemma 1, let us consider a farmer who is located at θ ∈ [0, K]. First, if he receives no signals, then both markets have the same expected selling price. To reduce transportation cost, he would prefer to sell his crop in the nearby market r. This explains the second statement. Next, suppose this farmer receives signals that have xl > xr (i.e., the expected selling price in market l is higher than that of market r). Also, suppose xl is sufficiently larger than xr so that the threshold τ (δ) (xl , xr ) ∈ (θ, K]. Then the first statement states that it is more profitable for this farmer to “switch” from selling in market r to market l. This is because the higher selling price to be obtained in market l would outweigh the extra transportation cost to be incurred from switching.
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Therefore, relative to the No Information Policy (F0) that has τ (F0) = 0,8 the number of switchers (i.e., farmers who receive signals and then switch from selling in the nearby market to the farther away market) under policy δ = (K, ρ) is equal to ρ · τ (δ) (xl , xr ) − τ (F0)  = ρ · τ (δ) (xl , xr ). As we shall see, the number of switchers associated with a provision policy will play an important role in explaining some of the results later.
10.4.2 Farmer’s Profit Function Under Provision Policy δ For any provision policy δ, we now determine the expected profit of each farmer. In preparation, let us examine the sales quantity to be sold in each market qi when all farmers follow the threshold rules as stated in Lemma 1 under any policy δ. To do so, let us consider the threshold selection rules as stated in Lemma 1. (For ease of exposition, let us consider the case when market signals satisfy xl > xr so that τ (δ) ∈ (0, K].) In this case, we can describe each farmer’s market selection in equilibrium as follows: (1) a farmer who locates over the region [−0.5, 0) will sell in market l regardless of whether he receives signals or not, (2) a farmer who locates over the region [τ (δ) , 0.5] will sell in market r regardless of whether he receives signals or not, and (3) a farmer who locates over the region [0, τ (δ) ) will sell in market l if he receives signals (with probability ρ) and will sell in market r if he receives no signals (with probability (1 − ρ)). By using the fact that the farmers are located uniformly over [−0.5, 0.5], it is (δ) (δ) easy to check that the sales quantities ql = 0.5 + ρτ (δ) and qr = 0.5 − ρτ (δ) . (δ) (δ) Hence, we can apply Eq. 10.1, ql , and qr to determine each farmer’s (expost) expected profit as follows. First, for a farmer located at θ who receives no signals, his (expost) expected profit can be expressed as9 ⎧ ⎪ E(πˆ 0 (θ, l)(xl , xr )) = A + E(ul (xl , xr )) ⎪ ⎪ ⎪ ⎨ − b(0.5 + ρτ (δ) ) − t (0.5 + θ ), if θ < 0, (δ) π0 (θ ; xl , xr ) = ⎪ E(πˆ 0 (θ, r)(xl , xr )) = A + E(ur (xl , xr )) ⎪ ⎪ ⎪ ⎩ − b(0.5 − ρτ (δ) ) − t (0.5 − θ ), if θ ≥ 0.
(10.4)
Also, we can apply Eqs. 10.3 and 10.4 to determine this farmer’s exante expected (δ) profit (δ) 0 (θ ) ≡ E(xl ,xr ) (π0 (θ ; xl , xr )).
ease of notation, we use τ (F0) ≡ 0 to denote the threshold adopted by all farmers under the No Information Policy. 9 For a farmer who receives no signals, his market selection is based on the comparison between E(πˆ 0 (θ, l)) and E(πˆ 0 (θ, r)), which does not depend on (xl , xr ). However, after he selects the market to sell in, his expost expected profit depends on the market selection of other farmers, which depends on (xl , xr ) via the threshold τ (δ) (xl , xr ). 8 For
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Second, for a farmer located at θ who receives signals (xl , xr ) under policy δ, we can apply Eq. 10.1, ql(δ) , and qr(δ) along with the threshold rule to determine his expost expected profit as ⎧ ⎪ E(πˆ 0 (θ, l)(xl , xr )) = A + E(ul (xl , xr )) ⎪ ⎪ ⎪ ⎨ − b(0.5 + ρτ (δ) ) − t (0.5 + θ ), if θ < τ (δ) , π (δ) (θ ; xl , xr ) = ⎪ E(πˆ 0 (θ, r)(xl , xr )) = A + E(ur (xl , xr )) ⎪ ⎪ ⎪ ⎩ − b(0.5 − ρτ (δ) ) − t (0.5 − θ ), if θ ≥ τ (δ) .
(10.5)
Also, we can use Eqs. 10.3 and 10.5 to determine this farmer’s exante expected profit (δ) (θ ) ≡ E(xl ,xr ) (π (δ) (θ ; xl , xr )). The following lemma examines the properties of a farmer’s exante expected profit under any provision policy δ. Lemma 2 For any policy δ = (K, ρ), the exante expected profit associated with a farmer who is located at θ has the following properties: (δ)
1. If he receives no signals, then his exante expected profit is 0 (θ ) = A−0.5b − (δ) 0.5t + tθ , where 0 (θ ) is increasing in θ . 2. If he receives signals, then his exante expected profit is
(θ ) = (δ)
t α(ρb + t)2 (α + β) 2 β exp − θ π α(α + β) ρb + t β 2(ρb + t)(α + β) − 2tθ  − θ  β ρb α(ρb + t)2 (α + β) 2 β exp − K + π α(α + β) ρb + t β 2(ρb + t)(α + β) − 2ρbK − K , (10.6) β (δ) 0 (θ ) +
where (x) ≡
x −∞
αβ αβ 2 exp − y dy. 4π(α + β) 4(α + β)
Note that (δ) (θ ) is increasing in θ . 3. (δ) (θ ) > (δ) 0 (θ ). (δ) (δ) 4. (θ ) − 0 (θ ) is decreasing in θ . Specifically, when a farmer is located close to the origin so that θ  is small (i.e., he is located far away from either markets), he incurs a higher transportation cost regardless of the market he sells in. As such, his expected profit is lower than other
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farmers who are located farther away from the origin so that θ  is large (i.e., who are located near one of the markets). This explains why the exante expected profit is increasing in θ . Next, by noting that a farmer can use the market signals to make better informed market selection decision, statement 3 of Lemma 2 reveals that each farmer can earn a higher exante expected profit when he receives signals. Finally, because of the inherent transportation cost, farmers who are located near a particular market would sell in the nearby market unless the signals suggest the market price in the farther away market is much higher. This observation explains statement 4 of Lemma 2: market signals are less beneficial to those farmers who are located near a particular market (i.e., when θ  is large).
10.4.3 Farmers’ Expected Total Profit Under Provision Policy δ By using the farmer’s expost expected profit function given in Eqs. 10.4 and 10.5 along with the threshold τ (δ) ∈ [−K, K] associated with policy δ = (K, ρ), we can determine the (expost) expected total profit of all farmers, w(δ) (xl , xr ), where w (δ) (xl , xr ) =
0.5 −0.5
+ρ
(δ)
π0 (θ ; xl , xr )dθ K −K
(δ)
(π (δ) (θ ; xl , xr ) − π0 (θ ; xl , xr ))dθ.
(10.7)
From this, we can determine the (exante) expected total profit of all farmers, W (δ) , where W (δ) = E(xl ,xr ) [w (δ) (xl , xr )].
(10.8)
10.5 Analysis: Comparisons of Provision Policies By examining the (expost) and (exante) expected total profit functions given in Eqs. 10.7 and 10.8, we now examine the implications of those three benchmark provision policies (F0), (F1), and (C), as well as the other two policies (Fρ) and (Kρ).
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10.5.1 Benchmark Provision Policies To begin, recall from Lemma 1 that all farmers who receive no signals will select the market to sell in according to the threshold associated with the No Information Policy, τ (F0) = 0. We now determine the thresholds associated with the other two benchmark policies (C) and (F1). Under policy (F1), δ = (K = 0.5 and ρ = 1). By substituting K = 0.5 and ρ = 1 into Eq. 10.3, we get τ
(F1)
(xl , xr ) = max
β 1 · · (xl − xr ) − 0.5, min 0.5, 2(b + t) α + β
. (10.9)
Next, under policy (C), the government can first use the realized market signals (xl , xr ) and Eqs. 10.4, 10.5, and 10.7 to determine the (expost) expected total profit w (C) (τ ; xl , xr ) associated with any threshold τ by setting K = 0.5 and ρ = 1.10 Then the government determines the threshold τ (C) that maximizes w (C) (τ ; xl , xr ) (i.e., τ (C) (xl , xr ) = arg max{w (C) (τ ; xl , xr ) : τ ∈ [−0.5, 0.5]}), where τ (C) (xl , xr ) = max
− 0.5, min 0.5,
β 1 · · (xl − xr ) 2(2b + t) α + β
. (10.10)
Finally, under the centralized control policy (C), the government asks each farmer to follow the threshold rule according to τ (C) (xl , xr ). By using the fact that τ (F0) = 0 along with Eqs. 10.9 and 10.10, we establish the following lemma: Lemma 3 When xl > xr , 0 = τ (F0) < τ (C) ≤ τ (F1) . When xl < xr , 0 = τ (F0) > τ (C) ≥ τ (F1) . While Lemma 3 is established algebraically, it can be interpreted by using the aforementioned notion of “switchers.” Specifically, Lemma 3 reveals that the number of switchers under policy (F1) is higher than that under policy (C) with centralized control. To explain this result, suppose that xl > xr . Under policy (F1), each farmer uses information to select the market selfishly without any concern of the total profit. Therefore, starting from the origin 0, each farmer with θ > 0 will “switch” from the nearby market r to market l that is farther away until θ = τ (F1) . By doing so, each switcher will earn a higher profit, but the farmers’ total profit may decrease because of the negative externality he exerts on others (due to extra selling quantity in the market that is farther away). By noting the fact that this negative externality is managed carefully under policy (C) when the market selection is centrally controlled by the government, the number of switchers
10 The
government should choose a threshold structure. Otherwise, we can always find a pair of farmers who sell in different markets, such that if they exchange the markets they sell in, famers’ total profit becomes higher.
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associated with policy (C) will be lower than that of policy (F1). This explains why 0 = τ (F0) < τ (C) ≤ τ (F1) when xl > xr . By using the thresholds associated with all three benchmark policies as stated above, we can use Eq. 10.7 to compare the expost expected total profits in the following proposition.11 Proposition 1 For any realized signals (xl , xr ), we can compare the expost expected total profits associated with policies (C), (F1), and (F0) to examine the following issues: 1. Value of Centralized Control: w(C) − w (F1) = (2b + t) · (τ (F1) − τ (C) )2 ≥ 0. 2. Value of Information: w(F1) − w (F0) = t · (τ (F1) − τ (F0) )2 = t (τ (F1) )2 ≥ 0. The first statement of Proposition 1 indicates that, when all farmers receive signals under policies (F1) and (C), the government can improve farmers’ “expost” expected total profit by controlling the market selection of each farmer centrally. Also, this improvement is based on the square of the distance between two thresholds τ (F1) and τ (C) , which is equivalent to the square of the difference in the number of switchers between policies (F1) and (C). The second statement reveals that, relative to the case of no information, providing signals to all farmers can improve farmers’ “expost” expected total profit and this improvement is based on the square of the distance between two thresholds τ (F1) and τ (F0) = 0, which equals to the square of the number of switchers under policy (F1). Because Proposition 1 holds for any realized signals, we can use the “sample path analysis” to argue that controlling farmer’s market selection centrally and providing information to farmers will improve farmers’ “exante” expected total profit. Also, by noting that τ (F1) and τ (C) given in Eqs. 10.9 and 10.10 depend on (xl , xr ), we can use the results as stated in Proposition 1 to compare the exante expected total profits associated with policies (C), (F1), and (F0) by computing W (C) − W (F1) = E(xl ,xr ) [w (C) − w (F1) ] and W (F1) − W (F0) = E(xl ,xr ) [w (F1) − w (F0) ]. Proposition 2 By comparing the farmers’ (exante) expected total profits under three benchmark policies, we can examine the following issues: 1. Value of Centralized Control: W (C) − W (F1) =
β b2 > 0. 2 2(b + t) (2b + t) α(α + β)
The value of centralized control is decreasing in α and t and increasing in β.
11 For
ease of exposition, we shall examine the case when b, t, or α is high enough or β is low enough so that τ (C) and τ (F1) lie within (−0.5, 0.5) almost surely. However, when the thresholds are truncated at ±0.5, the expressions are slightly different, but the qualitative characteristics remain the same.
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2. Value of Information: W (F1) − W (F0) =
β t > 0. 2 2(b + t) α(α + β)
The value of information is decreasing in α and b and increasing in β. Proposition 2 has the following implications. Information provision and centralized control both become more valuable when market conditions are more uncertain (i.e., when α is decreasing) and when signals are more precise (i.e., when β is increasing). Also, recall from Proposition 1 that the value of centralized control is increasing in (τ (F1) −τ (C) )2 . When the transportation cost t becomes higher, farmers are more concerned about transportation cost than the market price when deciding which market to sell in. As such, τ (F1) − τ (C)  becomes smaller. This explains why the value of centralized control is decreasing in t. Finally, when price elasticity b is large, the total quantity to be sold in each market outweighs the impact of market signals on the market price. As such, information has less value when b is large. In summary, we find that controlling farmer’s market selection centrally can improve farmers’ (exante) expected total profit. More importantly, providing market information will improve farmers’ (exante) expected total profit. This result motivates us to examine whether it is optimal to provide information to all farmers. In other words, would the Full Information with Decentralized Control Policy (F1) be the optimal provision policy? We examine this question next.
10.5.2 Partial Intensity Policy (Fρ) To examine whether the government should provide information to all farmers, we now examine policy (Fρ) that generalizes policy (F1). Under policy (Fρ), K = 0.5 and ρ is a decision variable, and policy (Fρ) becomes policy (F1) when ρ = 1. Applying Eq. 10.3 along with the fact that K = 0.5, the threshold corresponding to policy (Fρ) is given as τ (Fρ) (xl , xr ) = max
− 0.5, min 0.5,
.
1 β · · (xl − xr ) 2(ρb + t) α + β
(10.11)
Compare the threshold τ (Fρ) (xl , xr ) under policy (Fρ) along with the threshold τ (F1) (xl , xr ) given in Eq. 10.9, it is easy to check that τ (Fρ) (xl , xr ) ≥ τ (F1) (xl , xr ) when xl > xr . However, by noting that ρ percentage of farmers receive signals who will select the market according to threshold τ (Fρ) (xl , xr ) and (1 − ρ) percentage of farmers receive no signals who will select the market according to threshold τ (F0) = 0, the number of switchers under policy (Fρ) is equal to ρ · τ (Fρ) (xl , xr ) ≤ τ (F1) (xl , xr ) for any realized (xl , xr ). It follows from Proposition 1 and Lemma 3 that farmers’ expost expected total profit under policy (F1) is lower than that of
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under policy (C) because there are too many switchers under policy (F1). Therefore, it is possible for the government to improve farmers’ total profit by selecting ρ < 1 so that the number of switchers under policy (Fρ) is smaller than that of under policy (F1). We first substitute threshold τ (Fρ) into Eq. 10.8 to determine W (Fρ) ; i.e., the exante expected total profit under policy (Fρ). Then we determine the optimal ρ ∗ = arg max{W (Fρ) : ρ ∈ [0, 1]} and establish the following results: Proposition 3 Under the provision policy (Fρ), it is possible to have ρ ∗ < 1. For instance, if b, t, or α is large enough or β is small enough so that τ (Fρ) lies within (−0.5, 0.5) almost surely and if t < b, then ρ ∗ = t/b < 1. Proposition 3 reveals that, to maximize farmers’ exante expected total profit, it may not be optimal to provide information to all farmers. This result is different from the results obtained by Chen et al. (2013) and Morris and Shin (2002) that support distributing information to all farmers. There are two key factors that drive this key result. The first factor is caused by the fact that we consider two potential markets for heterogeneous farmers to sell in. In this case, providing information to more farmers may not improve farmers’ total profit. To elaborate, consider the case when farmers receive no information. In this case, each farmer will sell in the nearby market and the sales quantity in each market is identical (due to symmetry). Now, suppose the government provides information to all farmers. Then each farmer will selfishly choose the market to increase his own profit. Consequently, as stated in the intuition of Lemma 3, it is possible that farmers’ total profit is reduced by the fact that too many farmers selfishly switch to sell in the farther away market without considering the impact on the profit of other farmers. The second factor is caused by the fact that farmers are price takers and they have no control of the market price as individuals. In this situation, when a farmer selfishly selects the market that would yield a higher profit for himself, he creates serious negative externality that affects other farmers’ profit.12 Due to the negative externality, it may not be optimal to provide information to all farmers. Even though our model deals with uncertain market condition and noisy market signals in a different context, our result resembles a wellknown result in traffic equilibrium that is known as the Braess paradox. Specifically, Braess states that, when the drivers choose their route selfishly, the overall system performance can deteriorate by adding one extra road to the network. In summary, Proposition 3 reveals that, when information is disseminated to the entire range [−0.5, 0.5], it may not be optimal to set the intensity level ρ = 1 so that all farmers receive signals. This result makes us wonder if the government should disseminate information to the entire range. We examine this issue next.
12 On
the contrary, in the models considered by Chen et al. (2013) and Morris and Shin (2002), farmers are price setters who can control the market price by selecting their production quantity. Consequently, the negative externality imposed by each farmer is reduced because each farmer’s decision will have direct impact on his own profit.
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10.5.3 General Policy (Kρ) We now examine the general policy (Kρ) under which the government selects the range [−K, K] and the intensity level ρ. To determine the optimal general policy, the government solves the following problem: maxK∈[0,0.5],ρ∈[0,1] W (Kρ) , where W (Kρ) can be determined by using Eqs. 10.3 and 10.8. The following proposition summarizes our findings: Proposition 4 Under the general provision policy (Kρ), the optimal value K ∗ and ρ ∗ are given as follows: 1. The optimal intensity level ρ ∗ = 1. ¯ 0.5} and K¯ 2. The optimal range is specified by [−K ∗ , K ∗ ], where K ∗ = min{K, maximizes W (Kρ) for the case when ρ = 1. Also, K ∗ is increasing in β and decreasing in α and b. Akin to Propositions 3 and 4 suggests that it might not be optimal to provide information to all farmers. However, Proposition 4 states that it is more beneficial (in terms of improving farmers’ exante expected total profit) by limiting information access over a targeted range [−K ∗ , K ∗ ] at a full intensity level ρ ∗ = 1. To explain this result, observe from Lemma 2 that farmers located near the origin would benefit more from receiving signals than those framers who are located far from the origin. This observation implies that, instead of disseminating information to farmers over the entire range [−0.5, 0.5] at a lower intensity level ρ < 1 under policy (Fρ), it is actually better to limit information access to a target range [−K ∗ , K ∗ ] with full intensity ρ ∗ = 1.
10.5.3.1
Numerical Examples
We now present numerical examples based on realistic parameters to examine whether the government should provide information to all farmers or not in different environments. Specifically, we collect the data from Agmarknet (http://agmarknet. dac.gov.in), an online portal that provides actual daily information about the selling price and sales volume of different commodities in different markets in India. As an illustrative example, we analyze the data of onions in the state of Maharashtra in 2015. Specifically, we collect the daily market price and sales volume of onions in several markets within Maharashtra, including Mumbai and Pune. We select a random sample of 5 days in each of the 12 months and analyze all of the available data from all markets in the state of Maharashtra (298 data points in total). For example, on July 7, 2015, the sales volume of onions is 730 tonnes and the price is 1900 Rupees/Quintal in the market of Mumbai. In our sample data set, the average of daily sales volume of onions is 1463 tonnes, ranging from 27 to 5455, and the average of selling price is 2049 Rupees/Quintal, ranging from 750 to 5100. To estimate the values of our model parameters, we regress the selling price on daily sales volume by using a linear regression model: p = A0 − bq + . Using
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the 298 data points, we get the estimation of the demand intercept A0 to be 3118 (Rupees/Quintal) with a standard deviation of 1995. We use this standard deviation as σ (the standard deviation of the demand intercept a in our analytical model) to calculate α = 1/σ 2 (i.e., the intrinsic certainty of the demand intercept a). The estimation of b is 0.73 (Rupees/(Quintal·Tonne)). Instead of using only a specific value of b, we consider two additional values of b, 0.3 (a lower value) and 1 (a higher value). Regarding the transportation cost, we use a reasonable value as 100 Rupees per metric ton as reported in Annamalai and Rao (2003). Finally, regarding signal precision level β, we construct 3 different scenarios: β equals 0.2α, α, and 5α, where α is based on our estimated value above. For these three values of β, the signal can help the farmer to reduce the variance of the demand intercept by 16.7%, 50%, and 83.3%, respectively. These parameters should be normalized to fit our model.13 By considering three different values of b and three different values of β, we compute the optimal range of information provision [−K ∗ , K ∗ ] for each combination of (b, β). Our numerical results are presented in Table 10.2. Observe from Table 10.2 that, for any given signal precision level β, the optimal value K ∗ is decreasing in the price elasticity b. This numerical result corroborates with statement 2 as stated in Proposition 4. This result can be explained as follows. For a crop with higher price elasticity b, the selling price pi = ai − bqi becomes more sensitive toward the total quantity qi (and relatively less sensitive toward the market size ai ) in market i. As such, as revealed from the first statement of Lemma 1 that farmers are less inclined to “switch” from selling in one market to the other when b increases (because the absolute value of the threshold τ (δ) is decreasing in b). This observation is consistent with statement 2 of Proposition 2: the value of information decreases as b increases. Consequently, it is optimal for the government to provide information over a smaller range (i.e., a lower value of K ∗ ). This result has the following implications. For different crops with different price elasticities, the government should consider different information provision policies (over different zones). However, for practical purposes, the government Table 10.2 The optimal range of information provision in different cases
13 In
b (Rupees/(Quintal·Tonne)) 0.3 0.3 0.3 0.73 0.73 0.73 1 1 1
β 0.2α α 5α 0.2α α 5α 0.2α α 5α
K∗ 0.40 0.50 0.50 0.16 0.28 0.37 0.12 0.21 0.27
our model, the total quantity in two markets is normalized to one unit. On the other hand, the transportation cost of shipping one unit of crop from one market to the other is normalized to t.
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should adopt a common information provision policy [−K ∗ , K ∗ ] for all crops even though it is not optimal. While it is not considered here explicitly, our model can be used to determine an effective (common) information provision policy for multiple crops with different price elasticities. Next, for any given price elasticity of a given crop b, Table 10.2 asserts that the optimal value K ∗ is increasing in the signal precision level β. This result is consistent with statement 2 as stated in Proposition 4. Also, when signal becomes more precise, information becomes more valuable (as shown in statement 2 of Proposition 2). Consequently, it is optimal for the government to provide information over a wider range (i.e., a higher value of K ∗ ) when the signal precision level becomes higher. This result has the following implications. When the government has different forecast accuracies for different crops, the government should consider different information provision policies. Specifically, for those crops that the government has more accurate market information (e.g., the government has a more accurate prediction about the impact of climate change on market price about a certain crop), the government should disseminate more widely. A more general rule of thumb can be prescribed as follows: as the government develops a more accurate forecast about market trends of different crops, they should disseminate their information more widely.
10.5.3.2
Perceived Unfairness and Nominal Fees
While the optimal general policy (K ∗ , ρ ∗ ) enables the government to maximize farmers’ exante expected total profit, some farmers may object to this policy because of the perceived unfairness. To elaborate, suppose the government offers information according to the optimal policy δ ∗ = (K ∗ , ρ ∗ ) as given in Proposition 4. First, for any farmer who lies outside the range [−K ∗ , K ∗ ], he will receive no (δ ∗ ) signals and will earn an exante expected profit 0 (θ ). Second, for any farmer who lies within the range [−K ∗ , K ∗ ], he will receive signals and will earn an ex∗ ante expected profit (δ ) (θ ). Therefore, following from Lemma 2, we can trace the exante expected profit of each farmer under the optimal general policy (K ∗ , ρ ∗ ). Because information provision is truncated at −K ∗ and K ∗ , the profit function is discontinuous. Also, farmers located just outside the range [−K ∗ , K ∗ ] would feel being treated unfairly by the government due to “income reversal”: they now earn less than their neighbors who are located just inside the range. To mitigate this perceived unfairness, we show that the government can offer signal access to all farmers at a nominal fee p(K ∗ ) so that the farmers will adopt the intended optimal provision policy willingly, and the resulting farmer’s profit function is a continuous function without income reversal. The way to implement this feebased service is as follows. First, the government announces that each farmer can gain access to market information at a nominal fee p(K ∗ ). Second, recall from Lemma 2 that information is more valuable to farmers who are located near the origin and less valuable to farmers who are located far away from the origin. Therefore, the government can
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set p(K ∗ ) so that (1) farmers who are located within the range (−K ∗ , K ∗ ) will purchase the signals, (2) farmers who are located outside the range [−K ∗ , K ∗ ] will not purchase the signals, and (3) farmers who are located at −K ∗ and K ∗ are ∗ (δ ∗ ) indifferent. The last observation reveals that we can use 0 (θ ) and (δ ) (θ ) given ∗ in Lemma 2 to determine p(K ), where (δ ∗ )
∗
0 (K ∗ ) = (δ ) (K ∗ ) − p(K ∗ ). By charging a nominal fee p(K ∗ ), the government can implement the optimal policy (K ∗ , ρ ∗ = 1), while the resulting farmer’s profit function is a continuous function without income reversal. Also, the fee collected by the government can be used as farm subsidies for all farmers or as a way to recoup some of the investment cost associated with information acquisition and information dissemination. More importantly, relative to the case without information, each farmer who used to earn a lower profit due to his disadvantaged location can now earn a higher profit. Therefore, implementing the optimal policy with a nominal fee enables the government to reduce income inequality without incurring perceived unfairness (in terms of restricted information access and income reversal).
10.6 Conclusion We have presented a model to analyze the government’s information provision policy for the case when there are heterogeneous farmers who need to select one of the two markets to sell in (or select one of the two crops to grow). When farmers are price takers, our analysis indicated information is always beneficial to individual farmers. Specifically, farmers who are located far away from both markets will benefit the most from information access. To maximize farmers’ expected total profit, it may not be optimal for the government to provide information to all farmers. Instead, it is optimal for the government to provide information to all farmers who are located within a targeted range. To implement such an optimal provision policy in a fair manner, we show that the government should impose a nominal fee for signal access. We have considered other extensions including social welfare, information leakage, correlated markets, and a more general distribution of farmers as robustness checks and showed that most results continue to hold. The details are provided in Liao et al. (2019).
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Chapter 11
Agricultural Market Information: Economic Value and Provision Policy Xiaoshuai Fan, YingJu Chen, and Christopher S. Tang
11.1 Introduction The economic development efforts in many developing countries constantly face numerous challenges. Among others, the existence of various intermediaries (middlemen) largely creates barriers for improving efficient materials, financial, and information flows (Sodhi and Tang 2013). The World Bank reported that farmers in India obtain merely 24% of the value of their agricultural products because they lack information about market information. This significant information asymmetry between market participants leads to poor price discovery in the value chain, which often results in economic losses for the farmers. To overcome this information gap, governments, NGOs, and the business sectors are leveraging different information and communication technologies (ICT) to disseminate market information to farmers in emerging markets. For example, with financial support by the World Bank, the Rwanda government developed and launched an agricultural marketplace information service (eSoko) in 2011. This free service provides market information
X. Fan Division of Information Systems & Management Engineering, School of Business, Southern University of Science and Technology, Shenzhen, China email: [email protected] Y.J. Chen School of Business and Management and School of Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong email: [email protected] C. S. Tang () Anderson School of Management, University of California Los Angeles (UCLA), Los Angeles, CA, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_11
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(historical and current market price of over 60 agricultural commodities) to farmers through SMS. Ethiopian government also offers similar services in Ethiopia. In India, the Department of Agriculture and Cooperation launched Kisaan SMS Portal in 2013 to provide free information services to farmers regarding market information, crop advisory, weather information, etc. The accessibility of market information (historical and current market price) enables farmers to make longterm production decision (which crop to cultivate and how much to cultivate) and shortterm selling decision (when to sell, where to sell, etc.). As more information becomes accessible by farmers through public and private channels, we seek to investigate the following questions: will agricultural market information create value for farmers? and what is the optimal way to disseminate information to help farmers? Our primary goals are to investigate the impact of private and public information on the farmers’ behaviors and welfare and explore the optimal information provision policy to enhance farmers’ welfare. Specifically, we examine a situation in which multiple riskneutral farmers engage in a Cournot quantity competition. The future market price is uncertain; thus, the farmers face price uncertainty while making their production decisions.1 Each farmer uses a private signal and a public signal to estimate the future market price first and then decide on the production quantity. By analyzing our Cournot competition game with private and public signals, we determine the unique production quantity of each farmer in equilibrium in closedform expressions. By examining various equilibrium outcomes, we obtain the following results when riskneutral farmers have identical private signal precisions and when the public information is available to all the farmers: 1. Private signals do create economic value that can improve farmers’ welfare. However, this value deteriorates as the public signal becomes available (or more precise). Therefore, having more precise signal can be detrimental when farmers engage in competition. 2. In the presence of private signals, the public signal does not always create value for the farmers. Therefore, despite good intentions, public information can cause harmful effect even when all farmers behave rationally. 3. Private or public signals will always help reducing price variation and this effect is more profound when private signals are absent. After understanding the impact of market information on farmers’ behavior, we explore the optimal information provision policy in order to improve farmers’ social welfare. We find that providing information to a few farmers is optimal, but providing information to all farmers can be detrimental.
1 There
is a onetoone correspondence between the market price and the market demand under the Cournot quantity competition.
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11.2 Economic Value of Agricultural Market Information In this section, we evaluate the economic value of market information by assuming that the public information is available to and accessible by all farmers. Consider n farmers who produce and sell the same crop (e.g., coffee beans, tomatoes, etc.) at a common market. In our model, each farmer i incurs production cost cqi , where c is the unit cost and qi is her production quantity (a decision variable). We use Q ≡ ni=1 qi to denote the total production quantity to be sold at the market in the future. To capture the spirit of Cournot competition, we shall assume that the future market price P (Q) is linearly decreasing in Q. To model future market price uncertainty, we shall assume that P (Q) = a − bQ + u, where a > 0 corresponds to the market potential, b > 0 represents price elasticity, and u represents the future price uncertainty.2 We also assume that u ∼ N(0, σu2 ). For notational convenience, we let α ≡ 1/σu2 , where α measures the “intrinsic certainty” of the future market price so that the market price is less (more) certain when α is small (large). Private Signal In our model, we assume each farmer i is endowed with a noisy private signal xi that represents her best estimation of the future price uncertainty u. This noisy private signal xi is observable by farmer i only; however, different farmers receive heterogeneous signals due to their different sources of information. To model heterogeneity, we assume that the noisy private signal xi takes the following form: xi = u + i , where we assume i ∼ N(0, σf2 ), ∀i. For notational convenience, we use γ ≡ 1/σf2 to denote the precision of each farmer’s estimation so that we can set γ = 0 to capture the case when private signals are not available. Observe that the private signal xi is correlated with the future price uncertainty u and that private signals xi and xj are correlated. For ease of exposition, we shall assume that the noise i is statistically uncorrelated with the price uncertainty u so that Cov(u, i ) = 0). In essence, the private signal xi is an unbiased estimator of the price uncertainty u, and i represents the noise (or the residual uncertainty) of farmer i’s private signal. In addition, we assume these residual uncertainties are uncorrelated across different farmers so that Cov(i , j ) = 0, ∀i, j . Public Signal In addition to private signal xi , each farmer i will receive a noisy public signal y that serves as the publicly known estimate of the future price uncertainty u. We assume this public signal is provided by the government or an NGO free of charge as discussed earlier. However, this public signal is also noisy
2 As
a convention, a is assumed to be large and σu2 is reasonably bounded such that the price stays positive with a sufficiently high probability. This assumption is also adopted in GalOr (1985), Ha et al. (2011), and Li (2002).
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because it contains only information about historical trading prices. For this reason, we model the noisy public signal y as y = u + r , where the noise r ∼ N(0, σr2 ) is independent of u and all n other noise {i }’s. For notational convenience, we use β ≡ 1/σr2 to denote the precision of the public signal so that we can set β = 0 to capture the case when the public signal is not available. (Throughout this chapter, we shall use the terms “signals,” “information,” and “estimates” interchangeably.) Not only is the public signal y correlated with the future price uncertainty u, the public signal y is correlated with the private signal xi for all i. To elaborate, by using the fact that y = u + r , xi = u + i and the definition of α, β, and γ , one can check that Corr(y, xi ) = (1 + α/β)−1/2 · (1 + α/γ )−1/2 . Knowing that the functional form of the market price P (Q) and the associated parameters a and b as well as α, β, and γ are common knowledge,3 the sequence of events proceeds as follows: (1) all farmers receive the public signal y, (2) each farmer i observes his private signal xi , (3) each farmer i decides on his production quantity qi during the planting season by taking y, xi , as well as the rational decisions (i.e., production quantities) of all other farmers into consideration,4 (4) all farmers transport their production quantity Q (in total) to the market during the harvest season for sales, and (5) the actual future price uncertainty u is realized and the market is cleared. Before we investigate the equilibrium behavior of different farmers, let us first examine whether the private signal xi and the public signal y can help farmer i to obtain a more accurate estimate about the price uncertainty u, even though both xi and y are correlated with u. To do so, let us recall from above that u ∼ N (0, σu2 ), xi = u + i , where i ∼ N(0, σf2 ), and y = u + r , where r ∼ N (0, σr2 ). In this case, the random vector (u, xi , y) is multivariate and normally distributed. Specifically, the conditional distribution of price uncertainty (uxi , y) has the following properties:5 Lemma 1 By using the observed private signal xi and public signal y, farmer i can determine the conditional distribution of future price uncertainty (uxi , y), where
3 The
common knowledge assumption regarding α, β, and γ can be justified when farmers utilize past observations to make inference. 4 In the economics literature, Townsend (1983) and Singleton (1987) argue that it is typically optimal for farmers (or participants) to infer the signals (or beliefs) of other farmers and adjust their production quantities accordingly. 5 We can use the same approach to determine farmer i’s estimate about farmer j ’s private signal x j by computing E(xj xi , y) and Var(xj xi , y).
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E {uxi , y} =
σu2 [σr2 xi + σf2 y] σu2 σf2 + σu2 σr2 + σf2 σr2
, and Var {uxi , y} =
183
σu2 σr2 σf2 σu2 σf2 + σu2 σr2 + σf2 σr2
.
By noting that Var {uxi , y} = σu2 σr2 σf2 /(σu2 σf2 + σu2 σr2 + σf2 σr2 ) < σu2 , we can conclude that the farmer can indeed utilize the private and public signals to develop more accurate estimate about future price uncertainty u. This result motivates us to examine whether more accurate estimate will increase farmer’s welfare. In order to examine the impact of private and public signals on farmer’s welfare, we first characterize the equilibrium outcomes associated with the Cournot competition game. To begin, notice that the Cournot competition game involves incomplete information as each farmer privately observes her signal xi . Thus, the appropriate solution concept is Bayesian Nash equilibrium, in which each farmer’s quantity decision is contingent on her private signal xi and the public signal y (Fudenberg and Tirole 1991). We will focus exclusively on a particular type of symmetric linear equilibrium in which the production decision in equilibrium takes the following form: qi (xi , y) = A0 + A1 xi + A2 y,
∀i,
(11.1)
where the coefficients A0 , A1 , and A2 are yet to be determined. Also, we can interpret A0 as the base production quantity, A1 as the response factor associated with the private signal xi , and A2 as the response factor associated with the public signal y. With the symmetric equilibrium solution (Eq. 11.1) in mind, let us now consider the case when farmer i receives the public signal y and private signal xi . Given this information, farmer i needs to determine her best response by solving the following optimization problem: ⎧
⎨ max E P qj + qi qi − cqi qi ≥0 ⎩ j =i
⎫ & ⎬ & & xi , y . & ⎭
(11.2)
By using the fact that P (Q) = a − b( j =i qj + qi ) + u and the symmetric solution concept given in (11.1), we can apply Lemma 1 to show the following proposition: Proposition 1 When n farmers engage in Cournot competition by using individual private signals and the public signal, there exists a unique symmetric linear equilibrium in which each farmer i will produce qi after observing his private signal xi and public signal y, where qi (xi , y) = A0 + A1 xi + A2 y, A0 =
a−c , b(n + 1)
A1 =
γ , b [(n + 1)γ + 2α + 2β]
and
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A2 =
2β . b(n + 1) [(n + 1)γ + 2α + 2β]
For a detailed proof, the readers can refer to Chen and Tang (2015). Proposition 1 has the following implications. First, consider the base case when neither private nor public signal is available, i.e., when β = γ = 0. In this case, all farmers will produce the “base quantity” qi = A0 = (a − c)/(b(n + 1)) in equilibrium, where A0 is the wellknown equilibrium associated with the Cournot competition game under certainty. The comparative statics of the base quantity follow from the classical Cournot competition results: each farmer produces more when the profit margin (a − c) is high, the price elasticity (b) is low, or when the competition is less fierce (i.e., n is small). Second, the closedform expressions for the response factors A1 and A2 given in Proposition 1 enable us to obtain the following comparative statics: ∂A1 ∂A1 ∂A2 ∂A2 ∂A2 ∂A1 < 0, < 0, > 0, < 0, > 0, and < 0. ∂α ∂β ∂γ ∂α ∂β ∂γ The comparative statics displayed above can be explained as follows: (a) when the inherent price certainty α is higher (i.e., the market price is less certain), each farmer will be “less responsive” to the observed signal (private or public), (b) when the public signal is more accurate (i.e., when β is higher), each farmer will be less responsive to his own private signal and more responsive toward the public signal, and (c) when the private signal is more accurate (i.e., when γ is higher), each farmer will be more responsive toward his private signal and less responsive toward the public signal.
11.2.1 Impact of Information on Farmer’s Welfare By using the equilibrium outcomes given in Proposition 1, we now examine the impact of private and public signals on the farmer’s welfare. In preparation, let us first define the ex ante expected welfare of farmer i as follows: ⎧
⎨ Wi ≡ E{xi },y max E P qj + qi qi − cqi ⎩ qi ≥0 ⎩ ⎧ ⎨
j =i
⎫⎫ & ⎬⎬ & & xi , y , & ⎭⎭
where the expectation is taken over all possible realizations of public and private signals. By symmetry, the farmer’s welfare is simply SW ≡ nWi . It is of interest to examine the ex ante expected total production quantity n in equilibrium. Because E(x ) = E(y) = 0, Proposition 1 suggests that i i=1 Exi ,y qi = n i=1 Exi ,y [A0 + A1 xi + A2 y] = nA0 . Hence, we can conclude that the expected total production quantity in equilibrium is independent of private or public signals.
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While the public and private signals (y; x1 , . . . , xn ) do not affect the expected total production quantity in equilibrium, they do affect the ex ante expected welfare of each farmer Wi because it involves the variance of the private and public signals due to the term P ( j =i qj + qi )qi . By using Proposition 1 and Lemma 3, we can establish the following result: Proposition 2 The farmer’s welfare can be expressed as
1 1 1 2 2 1 2 1 SW (β, γ ) = nb A0 + A1 + + A2 + + 2A1 A2 , α γ α β α
(11.3)
where A0 , A1 , and A2 are given in Proposition 1.
11.2.1.1
Impact of Private Signal on Farmer’s Welfare
We now evaluate the value of private signals on farmer’s welfare SW (β, γ ). Because γ = 0 corresponds to the case when private signals are not available, the “value of private signals” can be captured by the term δ(β, γ ) ≡ SW (β, γ ) − SW (β, 0). By using the results stated in Propositions 1 and 2, one can show that (after some algebra) δ(β, γ ) =
α+β +γ nγ · . 2 α+β b[(n + 1)γ + 2α + 2β]
(11.4)
By noting that δ(β, γ ) > 0, we can conclude that the private signal has value because it enables farmers to increase farmer’s welfare in equilibrium. To examine the impact of the precision of the private and public signals, we differentiate δ(β, γ ) with respect to β and γ and obtain the following results: Corollary 1 The value of private signals δ(β, γ ) decreases as the public signal becomes more precise (i.e., β is large). Also, the value of private signals δ(β, γ ) decreases as the private signals become more precise (i.e., γ is large) when 1. the future price certainty is low (i.e., α is small), 2. the public signal is not precise (β is small), or 3. the number of farmers n is large. The first statement can be interpreted as follows. As the public signal becomes more precise (i.e., as β increases), Proposition 1 implies that all farmers are more responsive toward the public signal y (because A2 is increasing in β) and less responsive on the private signal xi (because A1 is decreasing in β) in equilibrium. Consequently, the value of private signals decreases as the public signal becomes more precise. The second statement reveals that, under certain conditions, more precise private signals can hurt farmer’s welfare. To elaborate, observe from Eq. 11.4 that, relative to γ , the value of private signals δ(β, γ ) is dominated by α, β, and n.
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Impact of Public Signal on Farmer’s Welfare
We now evaluate the value of public signal on farmer’s welfare SW (β, γ ). Because the case β = 0 corresponds to the case when the public signal is not available, the “value of public signal” can be captured by the term (β, γ ) ≡ SW (β, γ ) − SW (0, γ ). By using the results stated in Propositions 1 and 2, one can show that (after some algebra) 4nβ{−4α 2 − 2α [2β − (n + 1)(n − 3)γ ] + (n + 1)γ [(n − 3)] β + (n + 1)(n − 2)γ } (β, γ ) = − . b(n + 1)2 [(n + 1)γ + 2α]2 [(n + 1)γ + 2α + 2β] Notice that (β, γ ) can be negative; hence, the public signal can be harmful to the farmers. Also, when private signals are not available (i.e., γ = 0), (β, γ ) can be simplified as (β, 0) = SW (β, 0) − SW (0, 0) =
nβ > 0. b(n + 1)2 α(α + β)
To examine the impact of the precision of the private and public signals, we differentiate δ(β, γ ) with respect to β and γ and obtain the following results: Corollary 2 The value of public signal (β, γ ) has the following properties: 1. When private signals are not available (i.e., γ = 0), the value of public signal is always positive. Also, this value is increasing and concave in the precision of the public signal β. 2. When private signals are available, the value of public signal is not necessarily positive. 3. When private signals are available, the value of public signal is decreasing in the precision of the public signal β when a. farmers’ private signals are sufficiently accurate (γ is large), b. competition is intense (the market size n is large), or c. the intrinsic market certainty is low (α is small). The first statement reveals that, without private signals, the value of public signal is increasing and concave in the precision of the public signal β. This result is due to the fact that, without private signals, all farmers will produce the same quantity qi = A0 +A2 y. Hence, as the precision of the public signal β increases, all farmers would benefit from the more accurate public signal by adjusting the production quantity accordingly. We now interpret the second statement. Unlike the private signal case, the “value of public signal” (β, γ ) is not necessarily positive when private signals are present (i.e., when γ > 0). For example, when α = 1, β = 5, γ = 1/2, and n = 4, the
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value of public signal is (β, γ ) < 0.6 Therefore, although intuitively the public signal enables farmers to make better production decisions, the public signal does not always create value for the farmers in equilibrium. To interpret the third statement, let us numerically show the characteristics of (β, γ ) for the case when a = 3, b = 1, c = 1, α = 1, n = 9, and γ = 0.1. Because SW (0, γ ) is independent of β, it suffices to plot SW (β, γ ) with respect to β. Hence, we can conclude that the value of public signal (β, γ ) = SW (β, γ ) − SW (0, γ ) is not monotone in the precision of the public signal β. Therefore, a more precise public signal can create detrimental effect for the farmers. Hence, as the government or NGOs are planning to provide more public information that is intended to help the farmers, the third statement reveals that a more precise public signal does not always result in higher farmer’s welfare.
11.2.1.3
Impact of Signal Precision on Price Variation
We now examine the impact of private and public signals on the variance of future market price Var(P (Q)), where
n n = Var a + u − b Var P qi [A0 + A1 xi + A2 y] . i=1
(11.5)
i=1
By using Proposition 1 and the fact that α = 1/σu2 , β = 1/σr2 , and γ = 1/σf2 , we get the following proposition: Proposition 3 With the presence of private and public signals, the variance of market price Var{P ( ni=1 qi )} satisfies "
n nA21 ( α1 + γ1 ) + n(n − 1)A21 α1 1 2 = [1−2nb(A1 +A2 )]+b Var P qi . α +2n2 A1 A2 α1 + n2 A22 ( α1 + β1 ) i=1
Also, private or public signals would reduce price variation so that
n n [6(α + β) + (n + 5)γ ] ∂ =− Var P qi < 0, ∂γ [2(α + β) + γ (n + 1)]3 i=1
n 4n [2(n + 2)(α + β) + 3(n + 1)γ ] ∂ =− Var P qi < 0. ∂β (n + 1)2 [2(α + β) + γ (n + 1)]3 i=1
6 There
are other (infinitely many) parameter combinations for which (β, γ ) < 0.
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Therefore, we conclude that private or public signals can help reducing price variation.
11.2.2 Nonidentical Private Signal Precisions We are also interested in the case when farmers have access to different sources of information so that their private signal precisions are nonidentical. In this extension, farmer i ∈ {1, . . . , n} has access to the private signal: xi = u + i ,
where i ∼ N(0, σi2 ).
As before, we use γi ≡ 1/σi2 to denote the precision of farmer i’s private signal. We keep all other model characteristics the same as before so that α = 1/σu2 and β = 1/σr2 . Given private signal xi and public signal y, we can use similar analysis of the multivariate normal distribution as presented in Sect. 11.2 to show that farmer i’s conditional expectations can be expressed as γi β and mi ≡ . α + β + γi α + β + γi (11.6) As expected, ki is increasing in γi and decreasing in β and mi is decreasing in γi and increasing in β so that farmer i’s conditional expectation aboutu depends on the precisions of the private and public signals. Likewise, E xj xi , y = ki xi + mi y, i ∈ {1, . . . , n}. With nonidentical private signal precisions, the equilibrium is necessarily asymmetric. Thus, we adopt the following linear form of farmer i’s production strategy: E {uxi , y} = ki xi + mi y,
where ki ≡
qi = A0i + A1i xi + A2i y,
(11.7)
where A0i is farmer i’s base quantities, A1i measures farmer i’s responsiveness to her private signals, and A2i corresponds to farmer i’s responsiveness to the commonly observed public signal. Applying Eqs. 11.11 and 11.7, the following lemma characterizes the production equilibrium in this asymmetric setting. Lemma 2 When the farmers are endowed with different private signal precisions, the equilibrium outcomes {qi }’s are given in Eq. 11.7, where ∀i = 1, . . . , n, A0i =
a−c , b(n + 1)
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A1i =
189
ki /(2 − ki ) 1 , b 1 + j [kj /(2 − kj )]
(11.8)
and A2i satisfies
1 n 1 A2i = mi 1 − b A1j − ml 1 − b A1j . b n+1 n+1 j =i
l=i
j =l
By considering the equilibrium outcomes q1 and q2 presented in Lemma 2, we obtain the following comparative statics. Corollary 3 A farmer with a more precise private signal is more responsive to her private signal (i.e., A1i > A1j whenever γi > γj ). Moreover, each farmer relies more on her private signal when her own precision increases and the other’s precision decreases (i.e., A1i is increasing in γi and decreasing in γj for all j = i). The first statement of Corollary 3 is intuitive because when a farmer’s private signal is more precise, she shall use this information more confidently to craft her production decision. This result captures the information effect as a farmer’s production decision will be aligned with the true market state if her private signals get more informative/accurate. The second statement of Corollary 3 captures the competition effect: a farmer should scale back her response to her private signal when the competing farmer’s private signal becomes more precise. Next, we can further examine the impacts of public signal precision β on the farmers’ production decisions in equilibrium. For ease of exposition and for tractability, we shall focus our analysis on the twofarmer case. Without loss of generality, we assume that γ1 > γ2 . Thus, farmer 1’s private signal is more precise than farmer 2’s. We shall refer to farmer 1 as the more informed farmer and farmer 2 as the less informed one. In view of the result presented in Lemma 2, it suffices to focus on the differentials of response factors associated with the private and public signals; namely, (A11 − A12 > 0) and (A22 − A21 > 0), respectively. Corollary 4 reveals the impact of public signal on the differentials of response factors. Corollary 4 In the twofarmer case, the differentials of response factors to the private and public signals have the following property: (A11 −A12 ) and (A22 −A21 ) are increasing in β when β is small and are decreasing in β when β is large. When β is small, Corollary 4 reveals that, as the public signal becomes more precise, the response of both farmers toward private or public signals will diverge: farmer 1 will become more responsive to her private signal, while farmer 2 will become more responsive to the public signal. On the contrary, as β becomes large, a more precise public signal will cause the response of both farmers toward private or public signals to converge. Welfare Implications Corollaries 3 and 4 enable us to examine the impact of private and public signals on the farmer’s welfare, where farmer i’s expected payoff can be expressed as
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Wi (β, γ1 , γ2 ) ≡ E{xi },y max E {P (q1 + q2 )qi − cqi  xi , y} . qi ≥0
Because farmers 1 and 2 obtain different expected payoffs, we shall discuss them separately. Proposition 4 In the twofarmer case, a farmer’s expected payoff increases when (1) her private signal becomes more precise or (2) her competing farmer’s private signal becomes less precise. Proposition 4 is intuitive because of the underlying competition between the farmers. Next, we examine the impact of public signal on the farmers’ payoffs. In Sect. 11.2.1, we have already seen that improvement of public signal precision can be detrimental to farmers when the private signal precisions are identical. This is an extreme case in which γ1 = γ2 . By “continuity,” we shall expect similar results. To avoid repetition, we hereby discuss how public signal provision influences the farmers differently with nonidentical precisions. The following proposition summarizes our main findings. Proposition 5 In the twofarmer case, farmer 1 earns a strictly higher expected payoff than farmer 2 (i.e., W1 > W2 ). However, public signal provision mitigates the welfare inequality between the two farmers (i.e., (W1 − W2 ) is decreasing in β). Since farmer 1 is more informed than farmer 2, the first statement of Proposition 5 suggests that this informational advantage leads to a higher expected payoff for farmer 1. However, the second statement reveals that public signal provision alleviates the informational gap between farmers so that the payoff discrepancy between these two farmers is reduced. Finally, we examine the value of public signal. Let us define the value of public signal to farmer i according to i (β, γ1 , γ2 ), where i (β, γ1 , γ2 ) ≡ Wi (β, γ1 , γ2 ) − Wi (0, γ1 , γ2 ),
i = 1, 2.
Note that the two farmers obtain different base payoffs ({Wi (0, γ1 , γ2 )}’s) in the absence of public signal. The next proposition examines the value of public signal. Proposition 6 In the twofarmer case, the value of public signal i (β, γ1 , γ2 ) has the following properties: 1. The public signal is more beneficial to the less informed farmer 2 (i.e., 2 (β, γ1 , γ2 ) > 1 (β, γ1 , γ2 )). 2. The value of public signal for a farmer increases when (1) her private signal becomes less precise or (2) her competing farmer’s private signal becomes more precise. 3. The beneficial gap (2 (β, γ1 , γ2 ) − 1 (β, γ1 , γ2 )) increases as the more informed farmer 1’s private signal becomes more precise (i.e., as γ1 increases) or as the less informed farmer 2’s private signal becomes less precise (i.e., as γ2 decreases).
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As indicated by Proposition 6, since farmer 2’s private signal is less precise, this disadvantage is mitigated by the provision of public signal. Therefore, the public signal creates more value to farmer 2 than to farmer 1. This is in line with the result of Proposition 5: the public signal helps reduce welfare inequality. In a similar vein, since the public signal mitigates the information disadvantage from not observing the competitor’s private signal, the value of public signal is high if a farmer’s own signal provides little information or when the competitor’s signal is very informative.
11.3 Optimal Information Provision Policy In the last section, we have shown that providing public information to all farmers is not always beneficial to farmers. This finding motivates us to examine better information provision policy that aims to enhance farmers’ social welfare. The baseline model in which all n farmers receive public information has been presented in the previous section. We now examine the number of farmers n1 < n with access to public signal y provided by the central planner. For any given provision policy n1 < n, there are two types of farmers: farmers in set I who gain information access and farmers in set J who do not have access. Therefore, I  = n1 and I  + J  = n. Then, the sequence of events in the process of the optimal information design can be described as follows: 1. The central planner determines its information provision policy (n1 ), where n1 = I  and I corresponds to the set of farmers with public signal access. 2. Nature chooses the future market state u. 3. Based on the exogenously given precision parameter β, the central planner sends the public signal y to farmers in set I at the beginning of the production cycle. 4. Each farmer i receives the public signal along with his private signal xi with precision γ and each farmer in set J receives only his private signal xj at the beginning of the production cycle. 5. By using the observed signals, each farmer makes his production decision qi . 6. At the end of the production cycle, each farmer harvests and sells qi units at the market price P = a − b ni=1 qi + u.
11.3.1 Farmers’ Equilibrium Analysis To derive the optimal information provision policy n1 via backward induction, we first characterize each farmer’s equilibrium quantity decision qi when farmer i has access to public signal (i.e., when i ∈ I ) and when he has no access to public signal (i.e., when i ∈ J ). To begin, consider farmer i ∈ I who receives his private signal xi and the public signal y. He can determine his best response qi that maximizes his (ex post) expected profit by solving
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Wi (n1 , β, γ , (xi , y)) = max E P qi + qi ≥0
qi +
qi ≥0
qj qi − cqi
j ∈J
i ∈I,i =i
= max E a − b qi +
qi +
qj
+ u qi
j ∈J
i ∈I,i =i
& & & − cqi & xi , y ,
& & & xi , y &
(11.9)
where P (Q) = a − bQ + u. Similarly, for each farmer j ∈ J who receives only his private noisy signal xj (but not the public signal), he solves Wj (n1 , β, γ , (xj )) = max E P qj + qj ≥0
qj + qi qj − cqj
qj ≥0
& & − cqj && xj .
i∈I
j ∈J,j =j
= max E a − b qj +
& & & xj &
qj +
qi + u qj i∈I
j ∈J,j =j
(11.10)
Observe from Eqs. 11.9 and 11.10 that farmers i and j will use the information they possess to compute the ex post expected profit, which involves various conditional expectations including E {uxi , y}. In preparation, we provide the expressions for these conditional expectations in the following lemma by using the fact that (u, x1 , . . . , xn , y) are multivariate and normally distributed. Lemma 3 Consider a farmer i ∈ I who has access to public signal y and a farmer j ∈ J who has no access to the public signal. 1. By using the observed private signal xi and public signal y available to farmer i ∈ I , farmer i can determine the following conditional expectations: E {uxi , y} = kI xi + mI y E {xt xi , y} = kI xi + mI y,
t = i,
(11.11)
where kI ≡ γ /(α + β + γ ) and mI ≡ β/(α + β + γ ). 2. By using the only observable private signal xj available to farmer j ∈ J , farmer j can determine the following conditional expectations: E uxj = E xt xj = E yxj = kJ xj , where kJ ≡ γ /(α + γ ).
t = j,
(11.12)
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From Lemma 3, we observe that farmer i, i ∈ {I, J }, will use the public signal y (if accessible) and its private signal xi to update its belief on the future price uncertainty u and the other farmers’ private signals xt , t = i. These posterior beliefs will influence farmer i’s estimation on other farmers’ production decisions and also his own production decision qi . Given the asymmetric information structure between farmers in sets I and J , each farmer needs to decide on his best response production quantity by maximizing his ex post expected profit as stated in Eqs. 11.9 and 11.10. Clearly, the best response function qi that solves Eq. 11.9 for each farmer i ∈ I will depend on his private signal xi and the public signal y about the future market price uncertainty u. Similarly, the best response function qj that solves Eq. 11.10 for each farmer j ∈ J will depend only on her observable private signal xj . By applying Lemma 3 and by considering the best response functions that solve Eqs. 11.9 and 11.10 “simultaneously,” we obtain the following result: Proposition 7 There exists a unique Bayesian Nash equilibrium production quantity qi for farmer i ∈ I (or farmer j ∈ J ) that satisfies qi (xi , y) = A0 + A1 xi + A2 y, qj (xj ) = B0 + B1 xj , ∀j ∈ J, A0 = B0 =
∀i ∈ I,
(11.13)
where
a−c , b(n + 1)
γ (2α + γ )(n1 + 1) , b 2β(2α + γ ) A2 = , b γ [(n1 + 1)(2α + γ ) + 2β] B1 = , b
(11.14)
A1 =
and
≡ 4(n1 + 1)α 2 + (n + 1)γ [(n1 + 1)γ + 2β] + 2(n1 + 1)α [(n + 2)γ + 2β] . For a detailed proof, the readers can refer to Zhou et al. (2020). Observe that Proposition 7 is generalization of the result stated in Proposition 1 that examines a special case when n1 = n. Also, it has several implications. In preparation, let us observe from Eq. 11.13 that the coefficients A1 and B1 are the response factors associated with the private signals for group I that has public signal access and group J that has no public signal access, respectively. First, let us consider the case when the public signal is not available (or not informative) so that the corresponding precision parameter β = 0. In this case, it is easy to check from Eq. 11.14 that A1 = B1 and A2 = 0. Hence, both groups of farmers will respond to their private signals in the same manner. Second, when the public signal is available so that β > 0, Eq. 11.14 reveals that B1 > A1 . This implies that, as public information becomes accessible to group
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I , farmers in group J who have no access to the public signal will respond more aggressively to their private signals than those of group I . Furthermore, these response factors (A1 , A2 , B1 ) possess the following properties: Corollary 5 The response factors as stated in Proposition 7 satisfy ∂A1 ∂A2 ∂A1 ∂A1 ∂A2 > 0, < 0, > 0, > 0, < 0, ∂β ∂γ ∂n1 ∂β ∂γ ∂B1 ∂A2 ∂B1 ∂B1 < 0, < 0. < 0, > 0, ∂n1 ∂β ∂γ ∂n1 Corollary 5 illustrates three different effects captured in our base model. We call the first effect the information effect caused by the precision level of public information β. Specifically, when the public signal becomes more precise (i.e., as β increases), each farmer i ∈ I with access to public information will put less weight on her private signal (i.e., A1 decreases in β). Instead, she will put more weight on the public signal (i.e., A2 increases in β). The opposite is true when the private signal precision gets improved (i.e., when γ increases). Second, even though farmers in group J have no access to the public signal, the improvement of public signal precision β has an indirect impact on the production quantity of each farmer j ∈ J . Specifically, as shown in Corollary 5, each farmer j ∈ J will hold back her production adjustment in response to her own private signal (B1 becomes smaller as β increases). We shall refer to this effect as the competition effect: due to the underlying competition among farmers, farmers who have no access to the public signal find that their information disadvantage is aggravated. Consequently, they become more conservative in responding to their own private signals. Third, we examine the effect caused by n1 , i.e., the number of farmers who have access to the public signal. When n1 increases, Corollary 5 reveals that each farmer i ∈ I with access to public information will put less weight on the public signal (i.e., A2 decreases in n1 ). This suggests that spreading public information more widely leads to a crowdingout effect: when n1 increases, the information advantage for each farmer i ∈ I is reduced. However, as n1 increases, the information disadvantage for each farmer j ∈ J who has no public information access is aggravated. Hence, due to the competition effect as stated above, each farmer j ∈ J becomes more conservative in responding to her private signal (i.e., B1 decreases in n1 ).
11.3.2 Central Planner’s Optimal Information Provision Policy By considering the (ex post) expected profit as given in Eqs. 11.9 and 11.10, the (ex ante) expected profit for any farmer i ∈ I is given by
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W I (n1 , β, γ ) = E{xi },y {Wi (n1 , β, γ ; (xi , y))}, and the (ex ante) expected profit for any farmer j ∈ J is given by W J (n1 , β, γ ) = E{xj } {Wj (n1 , β, γ ; (xj )}. Hence, to maximize farmers’ total (ex ante) expected profit, the central planner solves S ∗ = max W (n1 , β; γ ) ≡ max {n1 W I (n1 , β; γ ) + (n − n1 )W J (n1 , β; γ )}. n1 ∈[0,n]
n1 ∈[0,n]
(11.15) By substituting each farmer’s production quantity in equilibrium as stated in Proposition 7 into Eqs. 11.9 and 11.10 and by taking the expectation over xi , xj , and y, we can determine each farmer’s ex ante expected profit: W I (n1 , β, γ ) = E{xi },y {Wi (n1 , β, γ ; (xi , y))}
and
W J (n1 , β, γ ) = E{xj } {Wj (n1 , β, γ ; (xj )} for any given information provision policy (n1 ). Specifically, we get the following lemma: Lemma 4 For any given information provision policy (n1 ), each farmer i ∈ I receives an (ex ante) expected profit:
1 1 1 1 1 W I (n1 , β; γ ) = b A20 + A21 + + A22 + + 2A1 A2 , α γ α β α whereas each farmer j ∈ J receives an (ex ante) expected profit:
1 1 W J (n1 , β; γ ) = b B02 + B12 + . α γ Also, W I (n1 , β; γ ) > W J (n1 , β; γ ). Lemma 4 reveals that public signal provides each farmer i ∈ I a competitive advantage so that they can obtain a higher ex ante (expected profit). By applying Lemma 4, we can determine the farmers’ total (ex ante) expected profit W (n1 , β; γ ) ≡ {n1 W I (n1 , β; γ ) + (n − n1 )W J (n1 , β; γ )}. By considering W (n1 , β; γ ) along with Eq. 11.15, we get the following proposition:
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Proposition 8 The optimal information provision policy n∗1 satisfies n∗1
=
1 0
if V (n, β; γ ) > 0 if V (n, β; γ ) ≤ 0, where
V (n, β; γ ) = 16α 4 +16α 3 (β+3γ )+4α 2 γ (7 + n)β + 12 + 2n − n2 γ +4αγ 2 · (4 + n)β + 5 + 2n − n2 γ − n2 − 2n − 3 γ 3 (β + γ ). By noting that V (n, β, γ ) > 0 if and only if the private signals are imprecise or absent (i.e., when γ is small) or when the farmer market is small (i.e., when n is small), we can interpret Proposition 8 as follows. First, when the private signals are imprecise or even absent (i.e., when γ is small) so that V (n, β, γ ) > 0, the value of the public signal is significant and it is easy to check from Proposition 7 and Corollary 5 that the response factors A1 and B1 are very small so that all farmers are not responsive to their own private signals. That is to say, providing public information to farmers will not largely discount the value of private information. At the same time, the response factor A2 is decreasing in n1 . Due to the “crowdingout” effect as discussed in Corollary 5, each farmer i ∈ I will become less responsive to the public signal as more farmers have access to the public signal (i.e., as n1 increases). Combining this observation, we can check from Lemma 4 that W I (n1 , β; γ ) (i.e., the ex ante expected profit of each farmer i ∈ I ) is decreasing in n1 , while W J (n1 , β; γ ) (i.e., the ex ante expected profit of each farmer j ∈ J ) is increasing in n1 . Consequently, one can check that the total farmer’s welfare W (n1 , β; γ ) = n1 W I (n1 , β; γ ) + (n − n1 )W J (n1 , β; γ ) is decreasing in n1 . Furthermore, the term of V (n, β, γ ) captures the difference between exclusive information provision policy and no information provision policy. Consequently, when V (n, β, γ ) > 0, the exclusive information provision policy n∗ = 1 is optimal. Next, let us consider the case when the private signals are sufficiently precise (i.e., when γ is sufficiently large) or when there are many farmers (i.e., when n is large) so that V (n, β, γ ) ≤ 0. In this case, Proposition 8 exerts that the central planner should abandon the public signal so that n∗1 = 0. This result can be explained as follows. First, when the private signals are sufficiently precise, the public signal does not add much value to farmer i ∈ I who has public information access. At the same time, due to the “competition effect” as explained in Corollary 5, providing the public signal to group I will cause farmer j ∈ J to hold back their production adjustments in response to their private signals (because B1 is decreasing in β). These two effects suggest that the central planner should abandon the public signal. Second, when the farmers’ market n is large, disseminating the public information to a large number of farmers can be detrimental because these farmers will end up using the public signal to “coordinate” their production quantities so that they will all produce more when the public signal is favorable that can cause the market price to drop significantly. To mitigate this “herd effect,” the central planner should limit access of public information to a small number of farmers. However, even when the public signal is only accessible by a small
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number of farmers, the informational disadvantage (due to the “competition effect” as explained in Corollary 5) will cause farmers in group J (i.e., the group without access to public information) to hold back their production adjustments in response to their private signals. To reduce this competition effect, we find that it is optimal for the central planner to simply abandon the public signal by setting n∗1 = 0 in order to maximize the farmer’s total profit W (n1 , β; γ ).
11.4 Conclusion Motivated by the recent efforts in providing farmers in developing countries with free access of agricultural market information, we have presented a parsimonious model as an initial attempt to evaluate the value of public signal in the presence of private signals and strategic farmer behavior. By examining the equilibrium outcomes and farmer’s welfare, we have shown that, when farmers are riskneutral, private signals do create value by improving farmer’s welfare, but this value deteriorates as the public signal becomes available (or more precise). Also, our result revealed that the public signal does not always create value for the farmers in the presence of private signals. Therefore, when farmers engage in competition, having too many signals or too much information can be harmful to the farmers. Moreover, we have shown that private or public signals would reduce price variation. By extending our analysis to the case when farmers have different private signal precisions, we have shown that the public signal can help reducing the welfare inequality between farmers. In cognizant of the fact that providing agricultural market information to all farmers can be hurtful, we have determined an optimal information provision policy that is intended to improve farmers’ welfare. Our analysis revealed that, when the private information is imprecise or when the farmer market is small (i.e., the number n is small), providing information to only 1 farmer (or a few farmers) is optimal; otherwise, no public information should be provided. This “exclusivity” result is caused by the following factors: (1) a wider dissemination of information will induce those farmers who have no access to public information pay less attention to their own private information (competition effect). Hence, both the value of private information and farmer’s total expected profit are “discounted” when public information is widely provided and (2) those farmers with access to public information will put less weight on public information as the number of informed farmers increases (crowdingout effect). That is to say, the value of public information is also decreased when public information is widely provided. In other words, providing information to one farmer (or a few farmers) is the most efficient way to utilize the private and public signals and to enhance farmers’ total (ex ante) expected profit.
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References Chen YJ, Tang CS (2015) The economic value of market information for farmers in developing economies. Prod Oper Manag 24(9):1441–1452 Fudenberg D, Tirole J (1991) Game theory. The MIT Press, Cambridge GalOr E (1985) Information sharing in oligopoly. Econometrica 53(2):329–344 Ha AY, Tong S, Zhang H (2011) Sharing demand information in competing supply chains with production diseconomies. Manage Sci 57(3):566–581 Li L (2002) Information sharing in a supply chain with horizontal competition. Management Sci 48(9):1196–1212 Singleton KJ (1987) Asset prices in a time series model with disparately informed, competitive traders. In: Burnett W, Singleton K (eds) New approaches to monetary economics. Cambridge University Press, Cambridge Sodhi M, Tang C (2013) Supplychain research opportunities with the poor as suppliers or distributors in developing countries. Prod Oper Manage 23(9):1483–1494 Townsend RM (1983) Forecasting the forecasts of others. J Political Econ 91(4):546–588 Zhou J, Fan X, Chen YJ, Tang CS (2020) Information provision and farmer welfare in developing economies. Manuf Service Oper Manage 23:1–266
Chapter 12
Knowledge Sharing Among Smallholders in Developing Economies Shihong Xiao, YingJu Chen, and Christopher S. Tang
12.1 Introduction Smallholders1 are the farmers who own less than two hectares of farmland (Thapa and Gaiha 2011). Approximately 500 million smallholders in developing countries, such as India, China, and Nepal, are trapped in a perpetual cycle of poverty (UN Millennium Project 2005). Reasons for smallholder poverty are varied, and insufficient farming knowledge can be an important cause (Carter and May 1999; Scott 2000), where knowledge is the understanding of skills, techniques, and experience that affects agricultural productivity. Therefore, to alleviate poverty, it is vital to improve smallholders’ knowledge. Various “peertopeer” (P2P) knowledge sharing programs, such as WeFarm, Avaaj Otalo, and the Supply Chain Network Project (SCN), have emerged recently to assist knowledge sharing among smallholders. In
1 This
chapter is adapted from the paper “Knowledge Sharing and Learning among Smallholders in Developing Economies: Implications, Incentives, and Reward Mechanisms,” published in Operations Research, March–April 2020, pp. 435–452, https://doi.org/10.1287/opre.2019.1869.
S. Xiao The Hong Kong University of Science and Technology, Kowloon, Hong Kong email: [email protected] Y.J. Chen School of Business and Management and School of Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong email: [email protected] C. S. Tang () Anderson School of Management, University of California Los Angeles (UCLA), Los Angeles, CA, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_12
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particular, WeFarm, the largest digital network for farmers (WeFarm 2017), is an SMSbased platform that allows farmers to post and respond to queries.2 Despite the fact that 831.8 thousand questions raised by farmers on WeFarm received 1.2 million answers from other farmers (WeFarm 2017), it is unclear about the actual knowledge being shared on such platforms. Intuitively, farmers with high knowledge may have little incentive to share voluntarily, although platforms provide the mechanism to share knowledge. This skepticism is supported by some empirical studies. For instance, Feder et al. (2004) and Kiptot et al. (2006) both survey the sharing behaviors of the farmers who received training from Farmer Field Schools or outside experts. They find that there is no significant sharing from the trained farmers to untrained farmers. However, Feder et al. (2004) and Kiptot et al. (2006) do not examine economic motivations for such nonsharing behaviors of highknowledge farmers. Instead, they attribute the lack of knowledge sharing to the complexity of technical information. Our study aims to provide a theoretical economic framework for sharing/nonsharing motivations among farmers, which could contribute to knowledge diffusion literature in the context of agriculture. Our findings can serve as theoretical hypothesis for empirical studies in the future. The reluctance of sharing among highknowledge farmers also raises an interesting question for mechanism design: What kind of reward mechanisms should be adopted by NGOs to encourage more knowledge sharing among farmers? NGOs have developed various reward mechanisms to encourage knowledge sharing. For example, Rainforest Alliance’s SCN project in Guatemala offers a small number of free phone minutes as a reward to the farmers who share effective management practices for irrigation, composting tips, etc.3 The Cambodian GIZ project “Best Farmer 2012” offers fertilizers to the farmers who exhibit highquality vegetables and share their experience (Hoffmann and Diehl 2014). However, which mechanisms are more efficient in enhancing farmer knowledge and controlling cost remains unclear. Therefore, in this study, we attempt to propose a reward mechanism that is efficient in both knowledge enhancement and cost control. We consider a stylized model in which farmers are heterogeneous in their farming knowledge levels and produce the same crop (e.g., onions). Due to the inherent production process, a proportion of each farmer’s output is of high quality and the rest is of low quality. Farmers are involved in a twostage game: in the first stage, each farmer decides on how much knowledge to share with others; and in the second stage, each farmer, after learning from others, decides on how much effort to exert in crop production. We establish as a benchmark the “efficient” shared knowledge level that maximizes farmer welfare when every farmer learns from it. We find that 2 The role of WeFarm in facilitating knowledge sharing can be illustrated by the following example.
Protus, a Kenyan farmer, asked through WeFarm, “Please advise on the right spacing for beans.” This request received an answer from another Kenyan farmer, Monica, that “Normal range is 15 cm from plant to plant and 30–40 cm from row to row. . . . For dry places use wider spacing and vice versa for high potential areas. . . .” (WeFarm 2017). 3 The existence of this reward program was confirmed through personal communication with the project manager of Rainforest Alliance.
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the highest knowledge level shared among farmers on a voluntary basis (defined as the voluntary shared knowledge level) is always lower than or equal to the efficient shared knowledge level. The reason is farmers with high knowledge are reluctant to share. This result provides analytical support for introducing incentive programs to promote sharing. We establish a quotabased reward mechanism that induces the efficient shared knowledge level, while allowing NGOs to control costs. Our main findings can be summarized as follows: 1. Highknowledge farmers are reluctant to share their private knowledge. The reason is that the enhancement of knowledge leads to more highquality output and less lowquality output. Consequently, competition in selling the highquality crop is intensified. Due to selfinterest, highknowledge farmers have no economic incentive to share knowledge. This finding provides a plausible economic explanation for the empirical observations of Feder et al. (2004) and Kiptot et al. (2006). 2. Farmers’ “voluntary” knowledge sharing is always inadequate to maximize farmer welfare. The voluntary shared knowledge level is always lower than or equal to the “efficient” shared knowledge level that maximizes farmer welfare. This result is primarily due to the selfinterest of farmers. Highknowledge farmers are not willing to share their knowledge for the interest of the whole farming community. To maximize farmer welfare, it is necessary to develop reward mechanisms to entice farmers to share the right amount of knowledge. 3. A quotabased reward mechanism can entice farmers to share “voluntarily” up to the “efficient” level. By restricting the number of winners (i.e., reward recipients) in a meticulous manner, this quotabased reward mechanism can entice farmers to share up to the “efficient” level even when the amount of the reward is extremely small. The rest of the chapter is organized as follows. Section 12.2 reviews the related literature. Section 12.3 describes the model setting and presents the base case where farmers (implicitly) compete without knowledge sharing. We analyze the coordinated case and the decentralized case of knowledge sharing in Sect. 12.4. By noting that the voluntary shared knowledge level under the decentralized case is suboptimal, we propose a quotabased reward mechanism in Sect. 12.5. Section 12.6 discusses comparative statics. Section 12.7 explores three extensions. Section 12.8 concludes the study. For ease of exposition, all proofs are given in the Appendix.
12.2 Related Literature Knowledge sharing is the focus of knowledge management, and there has been abundant literature on sharing motivation and impact in a corporate environment (e.g., Huber 1991; Nelson and Cooprider 1996; Hendriks 1999; Chai and Kim 2010). In this line of research, knowledge sharing is defined to be different from but related to communication and information distribution (e.g., Huber 1991; Nelson
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and Cooprider 1996; Hendriks 1999). To learn from others, the ability to reconstruct or internalize specific knowledge is a requisite. That is, it takes knowledge to perceive, make sense of, and acquire further knowledge (Hendriks 1999). The characteristics of knowledge sharing and learning in our model are consistent with existing studies. We model continuous knowledge distribution, and a farmer’s postlearning knowledge depends on his/her original endowed knowledge. However, our study differs from this line of research in both context and methodology. We consider a different context (smallholders) in emerging markets, and our research method is based on analytical models instead of empirical studies. This chapter is also related to the information sharing literature, which originates in the existence of information asymmetry among parties in supply chains. In this stream of literature, the key reason for sharing information is to improve demand forecast accuracy so as to boost supply chain efficiency. Information sharing has been extensively investigated by Li (2002), Li and Zhang (2008), Shin and Tunca (2010), etc. Unlike this body of research, our focus is on knowledge sharing. Additionally, in our context, the shared knowledge (not demand information) is intended to enable farmers to increase their productivity. Finally, this chapter belongs to an emerging research stream in the area of socially responsible operations in emerging markets. (A comprehensive review is provided by Tang and Zhou 2012.) Existing studies mainly investigate how to assist farmers in developing economies to increase productivity, get timely market information, and make agricultural decisions. For instance, Fafchamps and Minten (2012), Mittal et al. (2010), and Bhavnani et al. (2008) discuss the potential and limitations of using mobiledelivered messages to enhance farmers’ productivity. An et al. (2015) study whether and how aggregating farmers into cooperatives can enable them to reduce production costs and increase process yield. Jensen (2007), Aker (2010), and Parker et al. (2012) report reductions in geographic price dispersion when market information is provided to farmers via mobile phones. Chen and Tang (2015) verify the values of the public signal and farmers’ private signals in reducing price variation and improving farmer welfare under the setting of uncertain market demand. In response to the ITC eChoupal initiative implemented in rural India, Chen et al. (2013) examine the economic rationale for ITC to provide information and training to noncontracted farmers. Among the socially responsible operations literature, this chapter is closely related to Chen et al. (2015) who examine the incentive for an individual farmer (she) to share her knowledge with others. By considering the case when all farmers compete in a single market and when there is a minimum output requirement (imposed by the buyers), they show that the highknowledge farmer may be willing to share some knowledge, but at a significantly reduced level. This chapter is fundamentally different from Chen et al. (2015) in four aspects: • First, we focus on a more general sharing mode. Chen et al. (2015) consider onetomany knowledge sharing mode in which only one farmer has knowledge to share while others are only learners. This mode of sharing essentially captures a vertical form of learning (i.e., agents learn from the principal). In contrast, we
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consider a manytomany sharing mode in which each farmer can be a knowledge provider and learner at the same time. This captures a horizontal form of sharing and learning, which is the essential characteristic of P2P knowledge sharing platforms. • Second, we cover many key sharing issues that do not arise in the onetomany knowledge sharing context examined in Chen et al. (2015). These issues include which farmers would share voluntarily and how one farmer’s sharing influences others’ sharing decisions. • Third, our results are more general than those of Chen et al. (2015). Although both Chen et al. (2015) and our study demonstrate that voluntary knowledge sharing exists but is suboptimal, the underlying assumptions that lead to this conclusion are different. Their result is based on two key assumptions: (i) the existence of an outside expert who occasionally answers farmers’ questions on the platform and (ii) a minimum output requirement imposed to facilitate market transactions. In contrast, our conclusion is only based on the assumption that farmers are competing in selling two grades of crops. From this perspective, our model and conclusion are more general than those of Chen et al. (2015). • Fourth, in view of the inadequacy of voluntary knowledge sharing, Chen et al. (2015) have not provided a mechanism that can boost sharing. However, we propose a reward mechanism with small implementation cost that could entice farmers to share “voluntarily” up to the “efficient” shared level.
12.3 Model Preliminaries In this section, we describe the setting where a farming community comprises many “infinitesimal” farmers. Without loss of generality, the number of farmers is scaled to 1. To capture the tight landholding constraint facing the smallholders, we assume the output capacity of each farmer is scaled to 1. The assumption of unit capacity is by no means critical to our study. In Sect. 12.7.3, we shall show that our key insights remain the same when a farmer’s output capacity is a variable. Our modeling context can be described as follows. Two Grades of Output The farmers produce a single crop, but the actual output of each farmer is divided into two grades: highquality H and lowquality L. High quality/low quality can be also interpreted as high/lowquality crop. Quality differentiation is common for produce due to inherent production process or market demands. For example, according to the India’s National Horticulture Board (2016), extra large onions are in great demand and are sold at a higher mandi price. As such, extra large onions can be taken as high quality, while small ones are taken as low quality. Because the aggregate number of farmers is scaled to 1, the total output of highquality produce is QH and lowquality produce QL so that QH + QL = 1. For ease of exposition, we consider the case when these two grades of outputs are sold in two separate markets H and L, respectively. However, our key insights
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remain the same when these two grades of outputs are sold in two interrelated markets when a farmer’s capacity is not fixed. (See Sect. 12.7.3 for more details.) For any given production output, we shall assume that the market price is given as PH = AH − QH and PL = AL − αQL . To ensure that PH > PL > 0 for any output of each type, we shall assume that α ∈ [0, AL ] and AH > AL + 1. It is important to note that even though we consider separate markets, we can deal with the case of interrelated markets by using linear transformation.4 Knowledge Level Each farmer is endowed with certain “private” farming knowledge, e.g., the farming techniques, skills, experience, etc. that affect productivity. We assume that the knowledge levels can be ordered numerically (in an increasing order). The assumption of unidimensional knowledge is strong, but it helps us to derive firstorder insights at an initial stage. For ease of exposition, we assume the knowledge levels associated with all farmers are spread uniformly5 over [0, 1]. In Sect. 12.7.1, we shall show that our key results continue to hold for any general distribution over the range [0, 1] under a mild condition, including any beta distribution Beta(a, b) with b > a. Costly Effort For each farmer with knowledge level k, where k ∈ [0, 1], they employ their knowledge and deploy costly effort x to improve the expected yield of highquality produce. In our context, efforts include time, labor, fertilizer, etc., which can be aggregated and abstracted as effort decision x. Specifically, for any knowledge level k and effort level x, the expected highquality output is βkx and the lowquality output is 1 − βkx, where β is a commonly known productivity factor or the expected environmental condition. In Sect. 12.7.2, we shall show that our results continue to hold when the lowquality and highquality outputs are stochastic, i.e., when the productivity factor β is a random variable. Associated with effort level x,
the two markets are not separated so that PH = AH − u1 QH − u2 QL and PL = AL − u3 QL − u4 QH , where u1 > u2 and u3 > u4 . Then, because QH + QL = 1, we can transform the model as follows:
4 Suppose
PH = AH − u1 QH − u2 QL ⇔ PH = AH − u2 − (u1 − u2 )QH
⇔
A
P
−u
2 H , A = H PH = u −u H PH AH − u2 u −u 1 2 = − QH ⇐ 12⇒ PH = AH − QH , u1 − u2 u1 − u2
PL = AL − u3 QL − u4 QH ⇔ PL = AL − u4 − (u3 − u4 )QL
⇔
A −u
P
L , A = L 4 L PL AL − u4 (u3 − u4 )QL PL = u1 −u u −u 2 = − ⇐ 12⇒ PL = AL − αQL . u −u u1 − u2 u1 − u2 u1 − u2 α= 3 4 u1 −u2
Therefore, after some algebra, we can transform the demand functions for nonseparated markets into those for separated markets, and the transformed prices PH and PL are the scaled prices for original prices PH and PL . 5 The vertical uniform model is commonly adopted in the Marketing and OM literature (see, e.g., Lilien et al. 1992).
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the cost is equal to cx 2 /2, where c is the effort cost factor and c > 0. Note that the assumption of quadratic cost (e.g., cx 2 /2) and linear highquality output (e.g., βkx) is in accordance with the law of diminishing marginal returns, which is commonly used in the agricultural study (see, e.g., Debertin 2012). Because knowledge has a direct impact on productivity, a farmer’s sharing knowledge with others will boost others’ productivity. To analyze whether a farmer has an incentive to share her private knowledge with others, we shall investigate the impact of knowledge sharing on an individual farmer’s payoff. We first establish a benchmark case without knowledge sharing (or before the introduction of sharing platforms). Then we study farmers’ sharing behaviors and corresponding impacts after the launch of a sharing platform.
12.3.1 Base Case: Farmer’s Effort Decision When There Is No Knowledge Sharing In this case, each farmer’s knowledge is endowed by nature. The knowledge levels associated with all farmers are spread uniformly in [0, 1]. Each farmer with private knowledge k (farmer k in short), k ∈ [0, 1], has only one decision to make: how much effort x to exert into farming. The optimal effort level depends on the expected prices in the two markets, PH and PL , respectively. Each farmer forms rational expectation regarding the total output of each grade (QH , QL ). By noting that PH = AH − QH and PL = AL − αQL , each farmer of knowledge level k determines his effort level x(k) by solving the following problem: * + π(k) = max βkx AH − QH + (1 − βkx) AL − αQL − cx 2 /2 , x
(12.1)
where the first and second terms correspond to the expected revenues associated with highquality and lowquality outputs, respectively. Also, the last term corresponds to the effort cost. By firstorder condition, the optimal effort with no knowledge sharing is given as x (0) (k) =
βk · AH − QH − (AL − αQL ) . c
Throughout this chapter, we use the superscript y (s) to denote the variable y with respect to the case when the shared knowledge level is s. Specifically, s = 0 corresponds to the case of no knowledge sharing, and s = 1 corresponds to the case when a farmer with the highest knowledge level shares his knowledge fully. Notice that the optimal effort x (0) (k) depends on the price difference between the highquality output and lowquality output; i.e., [AH − QH − (AL − αQL )]. By using the fact that QH + QL = 1 and by letting A¯ = AH − AL + α, we can express the optimal effort of farmer k in terms of QH so that
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x (0) (k) =
βk ¯ · A − (1 + α)QH . c
(12.2)
Given the effort level x (0) (k), the highquality output of farmer k is βkx (0) (k). Hence, the expected total highquality output produced by all farmers is 1 (0) (0) 0 βkx (k)dk, which should be equal to the expectation of QH in equilibrium for the base case so that (0)
QH =
1
βk · x (0) (k)dk =
0
β2 ¯ (0) · A − (1 + α)QH · c
1
k 2 dk =
0
β2 ¯ (0) · A − (1 + α)QH . 3c
By rearranging the terms, Q(0) H is given by (0)
QH =
¯ β 2 A/3 . (1 + α)β 2 /3 + c
(12.3)
(0) ¯ Substituting QH into Eq. 12.2, we have βkx (0) (k) = β 2 k 2 A/((1 + α)β 2 /3 + c). To ensure that the highquality output of a farmer is less than or equal to 1, we assume β 2 A¯ ≤ c + (1 + α)β 2 /3, that is, c ≥ β 2 A¯ − (1 + α)β 2 /3. This assumption is reasonable because to the indigent smallholders, the cost for improving quality is considerable. By substituting Q(0) H into Eq. 12.1, we get the following lemma:
Lemma 1 Without knowledge sharing, the equilibrium outcomes can be described as follows: 1. For each farmer k, k ∈ [0, 1], he exerts effort x (0) (k) as given in Eq. 12.2. (0) ¯ + α)β 2 /3 + c) and the total 2. The total highquality output QH = (β 2 A/3)/((1 (0) (0) (0) (0) (0) lowquality output QL = 1 − QH so that PH = AH − QH and PL = AL − αQ(0) L . 3. For each farmer k, his payoff π (0) (k) can be expressed as π (0) (k) =
β 2k2 ¯ (0) 2 (0) · A − (1 + α)QH + AL − α + αQH . 2c
Also, the farmer welfare W (0) can be expressed as W (0) =
1
π (0) (k)dk =
0
¯ β 2 A¯ 2 c αβ 2 A/3 . + A − α + L 2 2 6[(1 + α)β /3 + c] (1 + α)β 2 /3 + c (12.4)
Lemma 1 can be interpreted as follows. The farmer’s effort x (0) (k) and his payoff π (0) (k) increase in his private knowledge level k so that a farmer with higher knowledge level will exert more effort and earn a higher payoff. Also, in equilibrium, the total highquality output Q(0) H increases in the productivity
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(0)
factor β. The quantities QH and W (0) will serve as benchmarks when we assess the implications of knowledge sharing.
12.4 Knowledge Sharing and Learning We have analyzed farmers’ effort exertion problem in the base case of no knowledge sharing in Sect. 12.3.1. To see how knowledge sharing impacts individual farmers’ payoffs and the farmer welfare, we consider the situation when the NGO launches a knowledge sharing and learning platform (e.g., WeFarm, Avaaj Otalo, etc.). In this case, if a farmer shares his knowledge, he enhances other farmers’ knowledge and productivity. We begin this section by modeling the learning process. Then we compare two systems: (i) a coordinated system where the NGO controls individual farmers’ sharing decisions and (ii) the decentralized system where the NGO has no power on farmers’ sharing decisions, and farmers can share knowledge at their own free will.
12.4.1 Learning Process Before we explore the impact of knowledge sharing on farmers’ productions, let us present a model of learning process. This process is based on two seemingly reasonable assumptions that enable us to obtain tractable results. 1. Learn from the best. Among the knowledge levels shared by farmers, each farmer only learns from the “supremum” of the shared knowledge levels, denoted by t ∈ (0, 1], that is, learning from the best. We refer to the supremum of shared knowledge, t, as the shared level or the highest knowledge shared, interchangeably. 2. Linear learning effect. After learning from the shared level t, each farmer k advances his knowledge to level k (t) , where k (t) = k + h[t − k]+ ,
(12.5)
and h ∈ [0, 1] captures the “learning effectiveness” of the platform so that farmers can advance their knowledge to a higher level as h increases. Since we assume uniform distribution of original knowledge, the density of the advanced knowledge level equals 1/(1 − h) over [ht, t] and equals 1 over [t, 1]. Note that although the assumption of “learning from the best” may appear to be strong, it is reasonable in some realworld applications, such as the Rainforest Alliance’s SCN project and WeFarm. In the former case, Rainforest Alliance chooses the two best shared management practices for irrigation, composting tips,
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etc. Thus farmers will learn from the best practices chosen by Rainforest Alliance. In the latter case, farmers are empowered to “validate users and information,” and “rate content that they receive,” which WeFarm “attaches to key words, locations, and profiles to make sure it is constantly improving and learning in order to get the best quality content possible to our users.” WeFarm “also use[s] lots of filters and processes to make sure the advice being shared is as good as possible,” according to Kenny Ewan, the CEO of WeFarm, in an interview with This is Money.co.uk (2016).
12.4.2 A Coordinated System: Sharing Decisions Made by a Coordinator In a coordinated system, a coordinator (an NGO) has complete information about the initial knowledge level of each farmer and has full control of selecting a farmer with initial knowledge level t to share his knowledge with all other farmers. We consider a twostage game as follows: in the first stage, the coordinator dictates the highest shared knowledge, which is learned by the farmers, and in the second stage, each farmer decides on the farming effort x. To maximize the farmer welfare, which shared level t should the NGO select? The answer to this question depends on the impact of knowledge sharing and learning on the farmer welfare. Suppose the NGO dictates the highest shared level to be t, each farmer k, k ∈ [0, 1], learns from t and enhances his knowledge to k (t) = k + h[t − k]+ according to the learning process given in Eq. 12.5. Each (t) farmer k forms rational expectation on the total highquality output QH , and the (t) total lowquality output QL , and determines his optimal effort level by solving the following optimization problem: (t) (t) π (t) (k (t) ) = max βk (t) x(AH −QH )+(1−βk (t) x)(AL −αQL )−cx 2 /2 . x
(12.6)
By the firstorder condition, the optimal effort level for farmer k after learning from the shared level t is given by x (t) (k (t) ) =
βk (t) ¯ (t) [A − (1 + α)QH ]. c
(12.7)
1 The expected total highquality output produced by all farmers is 0 βk (t) · (t) x (t) (k (t) )dk, which should be equal to the expectation of QH in equilibrium so that (t) QH
= 0
1
βk (t) · x (t) (k (t) )dk
12 Knowledge Sharing Among Smallholders in Developing Economies
=
β2 (t) · [A¯ − (1 + α)QH ] · c
t
2 k + h(t − k) dk +
0
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1
k 2 dk ,
t
where the last step is through substitution of x (t) (k (t) ) given in Eq. 12.7 with k (t) replaced by k + h[t − k]+ . Let M(t) ≡ E[(k ) ] = (t) 2
t
2 k+h(t−k) dk+
0
1
k 2 dk =
t
1 h(1 + h) 3 + t , 3 3
(12.8)
and correspondingly, when there is no knowledge sharing (i.e., t = 0), M(0) = 1 E[k 2 ] = 0 k 2 dk = 1/3. (t) By rearranging the terms, QH can be expressed as (t)
QH =
¯ β 2 AM(t) . (1 + α)β 2 M(t) + c
(12.9)
(t)
By substituting QH into Eq. 12.7 and the farmer’s objective function given in Eq. 12.6, we get the following lemma: Lemma 2 When the highest shared knowledge level equals t, the equilibrium outcomes are given as follows: 1. Each farmer k advances his knowledge to k (t) by Eq. 12.5, and he exerts effort x (t) (k (t) ) as given in Eq. 12.7. (t) ¯ + α)β 2 M(t) + c) and the 2. The total highquality output QH = β 2 AM(t)/((1 (t) (t) (t) (t) (t) total lowquality output QL = 1 − QH so that PH = AH − QH and PL = (t) AL − αQL . 3. Each farmer with advanced knowledge level k (t) earns a profit π (t) (k (t) ), where π (t) (k (t) ) =
β 2 (k (t) )2 (t) (t) · [A¯ − (1 + α)QH ]2 + AL − α + αQH . 2c
(12.10)
Also, the farmer welfare W (t) can be expressed as W (t) =
1
π (t) (k (t) )dk
0
=
¯ β 2 M(t)A¯ 2 c αβ 2 AM(t) . + A − α + L 2 2 2[(1 + α)β M(t) + c] (1 + α)β 2 M(t) + c
(12.11)
Lemma 2 has the following implications. First, when t = 0, M(0) = 1/3, and Lemma 2 reduces to Lemma 1. Second, when t > 0, M(t) increases in t by (t) Eq. 12.8. Hence, the total highquality output Q(t) H increases in t and QL decreases (t) (t) in t, according to Eq. 12.9. Also, PH decreases in t, while PL increases in t.
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Therefore, when the shared level is t, it changes the production dynamics: each farmer k advances his knowledge to k (t) and exerts effort level x (t) (k (t) ) so that the farming community will generate more highquality output and less lowquality output. This reveals that, by advancing farmers’ knowledge levels, the competition in the highquality (lowquality) market is intensified (relieved). As we shall see, the production dynamics associated with knowledge sharing will play an important role in a farmer’s mind when deciding how much knowledge to share. Third, in view of this tradeoff, one wonders if knowledge sharing and learning are beneficial in terms of improving the farmer welfare. By comparing W (t) given in Eq. 12.11 and W (0) given in Eq. 12.4, one can check that W (t) > W (0) for any t > 0. Hence, relative to no knowledge sharing, knowledge sharing is always beneficial because it promotes the farmer welfare. Given knowledge sharing and learning are beneficial, should the NGO select the highest knowledge level t = 1 to share? To answer this question, let us differentiate the farmer welfare W (t) given in Eq. 12.11 with respect to t. It can be shown that W (t) is actually strictly quasiconcave in M(t), which is increasing in t.6 Therefore, we obtain the following result: Proposition 1 The farmer welfare W (t) given in Eq. 12.11 is strictly quasiconcave in t, and its unique maximizer t (c) satisfies
t (c) =
⎧ ⎪ ⎨3 ⎪ ⎩
1
3(A¯ + 2α)c 1 −1 >0 h(1 + h) (1 + α)β 2 (A¯ − 2α)
if c ≤ τ (c) if c > τ (c) ,
where τ (c) = (A¯ − 2α)(1 + α)β 2 (1 + h + h2 )/[3(A¯ + 2α)]. Moreover, the efficient shared level t (c) increases in the effort cost factor c, decreases in the productivity factor β, and decreases in the learning effectiveness factor h. Proposition 1 and the fact that W (t) > W (0) for any t > 0 enable us to conclude that sharing knowledge is always beneficial, but sharing the highest knowledge is not always optimal. Specifically, when producing highquality output is costly (c > τ (c) ), farmers are reluctant to exert effort to increase their highquality output so that the competition in the lowquality market is fierce. To improve the farmer welfare, it is optimal for the NGO to impose high level of knowledge sharing. On the contrary, when producing highquality output is not too costly (c ≤ τ (c) ), farmers have already exerted great effort to produce a fair amount of highquality output. To balance the competition landscape in both markets, the efficient shared level should be moderate.
6 Note
that the proof of quasiconcavity does not rely on the specifics of knowledge distribution, but a mild condition that M(t) increases in t. Most of the lemmas and propositions in this chapter are proved in a similar way that requires no specifics of knowledge distribution. For this reason, the key results in this chapter continue to hold when farmers’ knowledge follows a general distribution.
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In summary, we have determined the equilibrium outcomes for any given shared level t in Lemma 2. Also, we identify the efficient shared level t (c) that maximizes the farmer welfare under a coordinated system. However, in many instances, the NGO does not know the knowledge level of any individual farmer, neither can he force any farmer to share his knowledge. Entitled to share knowledge at their own will, would farmers share up to the “efficient” level? To answer this question, we analyze the decentralized case where farmers can freely choose how much knowledge to share.
12.4.3 A Decentralized System: Sharing Decisions Made by Individual Farmers In the decentralized case, the NGO knows only the distribution of the farmers’ knowledge, but not an individual farmer’s knowledge. The NGO cannot force any farmer to share his knowledge or not, so farmers can choose the sharing levels at their own free will. Given the learning process as described in Sect. 12.4.1, the knowledge sharing platform facilitates a twostage decisionmaking process for each farmer as follows. In the first stage, each farmer k, k ∈ [0, 1], has to decide the knowledge level w to share, where w ∈ [0, k]. That is, each farmer k can share any knowledge level up to k. In the second stage, each farmer decides on the effort level x. In both stages, each farmer has to take other farmers’ decisions into consideration. Because the analysis of the twostage game in a decentralized system is rather complex, we consider two questions before we analyze farmers’ voluntary sharing equilibrium. First, will a farmer with knowledge k share his knowledge voluntarily in a decentralized system given other farmers’ sharing decisions? If yes, what is his optimal knowledge sharing level w∗ ∈ [0, k]? The answers to these questions will enable us to provide an economic justification for certain farmers to share their knowledge fully (i.e., w∗ = k). They also enable us to determine the shared level t (d) in equilibrium for the decentralized case.
12.4.3.1
A Farmer’s Best Response Function in a Decentralized System
To formulate a farmer’s best response function, let us consider the case when all farmers, except the farmer with knowledge level k, have made their knowledge sharing decisions so that the current highest shared knowledge level is t. In this case, each farmer , ∈ [0, 1], has already advanced his knowledge level to (t) = + h[t − ]+ . As farmer k advanced his knowledge to k (t) = k + h[t − k]+ , will he share his knowledge? If k ≤ t, then farmer k’s knowledge sharing is futile. Therefore, it suffices to consider the case when farmer k with k > t needs to decide how much knowledge to share. Specifically, we consider the two settings:
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1. Farmer k with k > t Decides to Share His Knowledge At Level w ∈ [0, t] By doing so, farmer k maintains the current shared level. Each farmer ∈ [0, 1] will use his advanced knowledge level (t) = + h[t − ]+ to decide on his effort level as given in Lemma 2. For farmer k, k > t, his advanced knowledge k (t) = k, and thus his profit π (t) (k) is as given in Eq. 12.10 with k (t) replaced by k. 2. Farmer k with k > t Decides to Share His Knowledge At Level w ∈ [t, k] In this case, by learning from farmer k’s knowledge at level w ∈ [t, k], farmer ∈ [0, 1] can advance their knowledge from (t) = + h[t − ]+ to (w) = + h[w − ]+ . The effort making process of each farmer under this setting is given in Lemma 2 when the highest knowledge shared is w. Farmer k’s profit π (w) (k) is as given in Eq. 12.10 with t replaced by w and k (t) replaced by k. We now determine farmer k’s best response, denoted by w∗ (k, t), where = arg max{π (w) (k) − π (t) (k) : w ∈ [t, k]}. Note that we write the decision space as [t, k] rather than [0, k] for the ease of exposition, as we have already analyzed the strategy of sharing any level in [0, t) in the first setting. We can apply Eqs. 12.9 and 12.10 to show that the profit gain can be rewritten as w ∗ (k, t)
π (w) (k) − π (t) (k) 2 2 2 2 2 β k ¯ β k ¯ (w) (t) 2 (t) = A − (1 + α)Q(w) A − (1 + α)Q + αQ + αQ − H H H H 2c 2c ¯ 2 ¯ 2 (w) (1 + α)β 2 Ak (1 + α)β 2 Ak 1 (t) = QH − QH α − + 2 2 2 (1 + α)β M(w) + c (1 + α)β M(t) + c (w) 1 (t) = QH − QH α − [u(k, w) + u(k, t)] , (12.12) 2
where u(k, z) =
¯ 2 (1 + α)β 2 Ak , 2 (1 + α)β M(z) + c
(12.13)
and u(k, z) increases in k and decreases in z. In particular, u(k, k) increases in k. Therefore, given the initial shared level t, farmer k’s best response is the optimal solution to the following optimization problem: (w) 1 (t) max QH − QH α − [u(k, w) + u(k, t)] . w∈[t,k] 2 (w)
(12.14) (w)
(t)
Recall from Lemma 2 that QH is increasing in w so that the term (QH −QH ) > 0 for any w > t. This observation enables us to determine farmer k’s best response in [t, k] as follows: Lemma 3 For any given initial shared level t and for any farmer k such that k > t, the best response for farmer k is to share his knowledge at level w∗ (k, t), where
12 Knowledge Sharing Among Smallholders in Developing Economies
w ∗ (k, t) =
213
k, if α > 12 [u(k, k) + u(k, t)] t,
if α ≤ 12 [u(k, k) + u(k, t)],
and u(k, z) is given in Eq. 12.13. Lemma 3 reveals that farmer k’s best response is either to share his knowledge in full (i.e., w ∗ (k, t) = k) or to maintain the current shared level (i.e., w∗ (k, t) = t or any level in [0, t] if we consider the whole sharing space). We use Lemma 3 to determine the shared level in equilibrium for the multiperson game next.
12.4.3.2
PureStrategy Nash Equilibrium
We now utilize farmer k’s best response w∗ (k, t) for any initial shared level t as given in Lemma 3 to determine the shared level in equilibrium. Because the term u(k, k) + u(k, t) increases in k, we can conclude that there exists a threshold θ (t) such that the best response for farmer k is to share his knowledge in full if and only if k ≤ θ (t), where θ (t) either satisfies α = 12 [u(θ (t), θ (t)) + u(θ (t), t)] or equals 1. This observation enables us to characterize the highest shared knowledge in the decentralized system. Intuitively, for a farmer who shares the highest knowledge in equilibrium, he is indifferent between sharing in full knowledge and not sharing at all. Consequently, the shared level in equilibrium t (d) , if it is interior, should satisfy α = u(t (d) , t (d) ). By applying Eqs. 12.13 and 12.8, it can be shown that t (d) is the solution to the following cubic equation: 2 2 3 2 ¯ 2 2 1 3 α(1 + α)β (h + h ) · t − (1 + α)β A · t + α[(1 + α)β /3 + c]
= 0.
(12.15)
More formally, we obtain the following result: Proposition 2 There exists a purestrategy Nash equilibrium so that every farmer with initial knowledge in [0, t (d) ) will share his knowledge in full and no farmer with initial knowledge in [t (d) , 1] will share his knowledge. Specifically, if α ≥ u(1, 1), then t (d) = 1; otherwise, t (d) equals the unique positive root in (0, 1) that solves the cubic equation given in Eq. 12.15. Also, t (d) increases in the effort cost factor c, decreases in the productivity factor β, and increases in the learning effectiveness factor h. Note that [0, t (d) ) is not a closed set. Given knowledge in [0, t (d) ) is already shared, although farmer t (d) is indifferent from sharing or not in this continuous model, excluding the point t (d) empowers better prediction of our model for reality. In reality, the number of farmers is discrete, and it is likely that there is no farmer with knowledge exactly at level t (d) . Therefore, a nonclosed sharing set [0, t (d) ) would provide a better prediction of which farmers will share their knowledge in reality.
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Proposition 2 implies that lowknowledge farmers are willing to share knowledge, while highknowledge famers are not. The observation that highknowledge farmers are not willing to share is consistent with the existing field studies conducted by Feder et al. (2004) and Kiptot et al. (2006). The reason for farmers’ sharing/nonsharing behaviors, however, is intriguing, and we explain it more fully by analyzing a farmer’s sharing incentive in Sect. 12.4.3.3. It is noteworthy to point out that, besides the equilibrium stated in Proposition 2, there are infinitely many equilibria in this continuous game. For example, it is also an equilibrium when all farmers with knowledge in [t (d) , 1] share knowledge at t (d) . While there are many other equilibria, the equilibrium stated in Proposition 2 has an additional property: farmers’ strategies are all admissible. Essentially, an admissible strategy (Luce and Raiffa 1957) is one that is not weakly dominated. By showing that no strategy adopted by a player in the equilibrium defined in Proposition 2 is weakly dominated, we get the following result: Proposition 3 The equilibrium defined in Proposition 2 remains a Nash equilibrium when farmers use only admissible strategies.
12.4.3.3
Motivations for Sharing/Nonsharing in a Decentralized System
To analyze a farmer’s sharing incentive, we divide his payoff by markets. Recall the two settings from Sect. 12.4.3.1. In the first setting, farmer k maintains the current shared level and his payoff is given in Eq. 12.10 with k (t) replaced by k. We apply Eq. 12.6 to divide farmer k’s payoff π (t) (k) into two terms so that π (t) (k) = πH(t)−c (k) + πL(t) (k). Here, the first term πH(t)−c (k) = βkx (t) (k)(AH − (t)
QH ) − c(x (t) (k))2 /2, which represents his revenue from the highquality market (t) (t) “net of his effort cost.” The second term πL (k) = (1 − βkx (t) (k))(AL − αQL ), which represents his revenue from the lowquality market. Using the optimal effort x (t) (k) as given in Lemma 2 with k (t) replaced by k, we have β 2k2 ¯ (t) (t) · A − (1 + α)QH · AH + AL − α − (1 − α)QH , 2c (12.16) β 2k2 ¯ (t) (t) (t) A − (1 + α)QH · AL − α + αQH . (12.17) πL (k) = 1 − c (t)
πH −c (k) =
In the second setting in Sect. 12.4.3.1, farmer k elevates the current shared level by sharing knowledge w, where w ∈ [t, k], his payoff becomes π (w) (k). Also, we (w) can express π (w) (k) = πH(w) −c (k) + πL (k) in a similar fashion as in Eqs. 12.16 and 12.17. By comparing the profit functions π (w) (k) and π (t) (k), the “profit gain” that farmer k can obtain from sharing his knowledge at level w ∈ (t, k] is as follows:
12 Knowledge Sharing Among Smallholders in Developing Economies (w)
(t)
(w)
215 (t)
π (w) (k) − π (t) (k) = [πH −c (k) − πH −c (k)] + [πL (k) − πL (k)].
(12.18)
By considering Eqs. 12.16, 12.17, and 12.18, we analyze the impact of farmer k’s sharing at level w ∈ (t, k] on the two quantities: (a) farmer k’s expected gain in the (t) highquality market (i.e., πH(w) −c (k) − πH −c (k)) and (b) farmer k’s expected gain in
the lowquality market (i.e., πL(w) (k) − πL(t) (k)).
Proposition 4 For a farmer with initial knowledge k, where k > t, if he raises the shared level from t to w with w ∈ (t, k], his expected gain in the highquality market (w) (t) πH −c (k) − πH −c (k) is decreasing in his sharing level w, while his expected gain in the lowquality market πL(w) (k) − πL(t) (k) is increasing in w. Recall from our earlier discussion of the impact of knowledge sharing that knowledge sharing can advance certain farmers’ knowledge and can induce farmers to produce more highquality output and less lowquality output. Consequently, knowledge sharing can intensify competition in selling the highquality crop and reduce the competition in selling the lowquality crop. This relationship explains why a farmer’s expected gain in selling the highquality (lowquality) crops decreases (increases). We now use Proposition 4 to explain farmers’ sharing behaviors as indicated in Proposition 2. By sharing knowledge, lowknowledge farmers can reduce competition in selling the lowquality crop. Since most of their outputs are of low quality, lowknowledge farmers benefit from sharing, although at the expense of lower revenue from selling the highquality produce. In contrast, highknowledge farmers produce far more highquality outputs. And hence to avoid introducing more competition to selling the highquality crop, highknowledge farmers do not have economic incentive to share any knowledge.
12.4.4 The Efficiency of Voluntary Sharing We now compare the farmer welfare W (d) in equilibrium for the decentralized case against two benchmarks: the coordinated case (Lemma 2 and Proposition 1) and the no knowledge sharing case (Lemma 1). To this end, we compare the voluntary shared level t (d) in the decentralized case stated in Proposition 2 and the efficient shared level t (c) in the coordinated case as presented in Proposition 1. If we can show that t (c) > t (d) > 0, then we can prove W (c) > W (d) > W (0) because the farmer welfare W (t) given in Eq. 12.11 is strictly quasiconcave in the shared level t. By this approach, we obtain the following result: Proposition 5 By comparing the equilibrium outcomes associated with the no knowledge sharing case and the knowledge sharing case associated with the coordinated system and the decentralized system, we find that (a) the shared level is
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highest in the coordinated case (i.e., t (c) ≥ t (d) ≥ 0) and (b) the farmer welfare is highest in the coordinated case (i.e., W (c) ≥ W (d) ≥ W (0) ). Relative to the no knowledge sharing case, Proposition 5 reveals that knowledge sharing in a decentralized system is beneficial: it can increase the farmer welfare from W (0) in the base case to a higher level W (d) . Moreover, the farmer welfare can increase further to W (c) if the system is in the charge of a coordinator. This observation raises two interesting issues: 1. If one can entice the farmers to increase the shared knowledge from t (d) to t (c) , then the farmer welfare can increase from W (d) to W (c) . However, by doing so, will it increase or reduce income inequality among farmers? 2. In the decentralized system, farmers’ voluntary shared level is suboptimal. To entice them to share up to t (c) willingly, what economical reward mechanisms can be utilized by the NGO? We tackle these two issues in the next section.
12.5 Income Inequality and Reward Mechanisms 12.5.1 Impact of Increasing Knowledge Sharing on Income Inequality Before we develop mechanisms for enticing farmers to increase the shared knowledge from t (d) to t (c) voluntarily, let us first understand the implication of doing so. Specifically, by enticing the farmers to increase the shared knowledge from t (d) to t (c) , will it reduce income inequality among farmers? If the answer is yes, then it can provide another reason, in addition to the farmer welfare, for the NGO to develop reward mechanisms that entice farmers to share the right level of knowledge. We measure income inequality with Gini’s coefficient of concentration (Gini index in short), which is developed by Gini (1912). In preparation, let us compute the Gini index by using the Lorenz curve L(x), where L(x) is the cumulative distribution of payoff generated by all farmers with initial knowledge in [0, x]. Clearly, L(0) = 0 and L(1) = 1. First, when there is no income inequality, L(x) is essentially a 45degree line (i.e., the line of equality) and L(x) = x. Second, due to different knowledge levels among farmers, the corresponding Lorenz curve for the case when the highest knowledge shared is t can be computed as
x
L(t) (x) = 0
π (t) (k (t) ) dk, W (t)
where π (t) (k (t) ) is given in Eq. 12.10 and W (t) is given in Eq. 12.11.
(12.19)
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Fig. 12.1 Graphical representation of the Gini index with the Lorenz curve
By using the line of equality (for the case of no income inequality) and L(t) (x) (for the case where the highest knowledge shared is t and farmers have heterogeneous knowledge levels and payoffs), we can determine the Gini index, denoted as G(t) = A/(A + B) (see Fig. 12.1), where A is the size of the shaded area, and B is the size of the area under Lorenz curve. By noting that the line of equality and L(t) (x) as given above, one can check that 1 A + B = 1/2, and A = 1/2 − 0 L(t) (x)dx. Therefore, the corresponding Gini index G(t) = A/(A + B) for the entire farming community is given as 1/2 −
1
L(t) (x)dx = 1−2 1/2
1
1 x
π (t) (k (t) ) dk dx. W (t) 0 0 0 (12.20) By considering farmer k’s payoff function π (t) (k (t) ) and the farmer welfare function W (t) given in Eqs. 12.10 and 12.11, respectively, we can compute the Gini index G(t) for any given shared level t. Through differentiations and some algebra, we obtain the following result for the case when t (c) < 1, which occurred when c < τ (c) as stated in Proposition 1. (t)
G
=
0
L(t) (x)dx = 1−2
Lemma 4 When the NGO entices farmers to increase the shared knowledge level from t (d) to t (c) , the Gini index given in Eq. 12.20 (i.e., income inequality) will decrease. Also, farmer k’s payoff is increased if and only if k < θ , where θ ∈ (t (d) , t (c) ). Lemma 4 implies that, when the NGO entices farmers to increase the knowledge shared level from t (d) to t (c) , it benefits only those farmers with initial knowledge levels below the threshold θ . This is because, by learning from the higher knowledge at level t (c) (instead of level t (d) ), these farmers can produce highquality crops more efficiently with their enhanced knowledge. By doing so, it will lessen the competitive pressure in the lowquality market, which will benefit these farmers who were producing mainly lowquality crops. On the contrary, those farmers
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with higher knowledge levels will be worse off because a higher shared level will intensify the competition in the highquality market. This explains the second statement. Combine this result with Proposition 5, we can make the following conclusion. By enticing more knowledge sharing up to t (c) , the NGO maximizes the farmer welfare W (t) . At the same time, farmers with lower knowledge levels are better off, while farmers with higher knowledge levels are worse off. Consequently, by shifting some income (implicitly) from those farmers with high knowledge levels to farmers with low knowledge levels, we can reduce income inequality and hence reduce the Gini index.
12.5.2 Reward Mechanisms Recall from Proposition 5 and Lemma 4 that, if the NGO can increase the knowledge shared level from t (d) to t (c) , then he can maximize the farmer welfare and reduce income inequality when c < τ (c) . This result further justifies the need to develop effective reward mechanisms to entice farmers to share up to t (c) . Actually, reward mechanisms for encouraging knowledge sharing are observed in practice. For example, the Supply Chain Network Project implemented in Guatemala offers a small number of free phone minutes as reward to the farmers who share good management practices for irrigation, compost tips, etc.7 The Cambodian GIZ project “Best Farmer 2012” gives out some vegetable fertilizers to the farmers who exhibit their highquality vegetables and share their experience (Hoffmann and Diehl 2014). With so many reward mechanisms, however, it is not clear which one is “effective” in terms of the shared level and the implementation cost. This observation motivates us to analyze the effectiveness of certain reward mechanisms. Our analysis is based on a Stackelberg game theoretical model in which the NGO acts as the leader who sets the reward mechanism first. Then the farmers act as followers by entering a “knowledge competition” among themselves, and each farmer selects his knowledge level to share. In the remainder of this section, we first explain why some seemingly intuitive mechanisms are not effective. Then we present a simple reward mechanism that is effective for two reasons: (a) it can entice farmers to share knowledge up to our desired target, t (c) and (b) it can empower the NGO to control the implementation cost of the reward mechanism.
12.5.2.1
Two Ineffective Mechanisms
We now consider two intuitive and commonly adopted reward mechanisms that are shown below to be ineffective in terms of the shared level or the implementation
7 Such
a reward program is confirmed through personal communication with the project manager.
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cost. In our analysis, we continue to assume that farmers use only admissible strategies under different mechanisms. To begin, let us consider Mechanism 1. Mechanism 1 The NGO offers a reward R1 > 0 to each farmer who shares the highest knowledge. Lemma 5 Under Mechanism 1, any knowledge level t in [t (d) , 1] can be the shared level in some purestrategy Nash equilibrium when farmers use only admissible strategies. Lemma 5 implies that Mechanism 1 is not effective, because there are infinitely many Nash equilibria with different shared levels, ranging from t (d) to t (c) , regardless of the amount of the reward R1 . The ineffectiveness of Mechanism 1 is due to the fact that the best response for a farmer k, k ∈ [t (d) , 1], depends on the highest knowledge shared by others. For instance, although farmer k, k ∈ [t (d) , 1], may always desire a lower shared level, yet given the highest knowledge shared by others is t, where t < k, the best response by farmer k is simply to share t. In this case, how much knowledge is shared among the farmers is greatly dependent on their believes of others’ strategies. Thus it is difficult for the NGO to predict which equilibrium will be formed. Moreover, the NGO cannot manipulate the sharing outcome by setting the value of R1 . We now consider Mechanism 2 to overcome the shortcoming of Mechanism 1, which has no restrictions on the winning sharing levels or the winners. Rather, Mechanism 2 specifies the exact knowledge level that entails a reward. Mechanism 2 The NGO offers a reward R2 > 0 to each farmer who shares knowledge at level t (c) . Lemma 6 Under Mechanism 2, let t2∗ be the shared level in equilibrium when farmers use only admissible strategies. Then t2∗ = t (c) ,
if R2 ≥ π (d) (t (c) ) − π (c) (t (c) )
t2∗ ∈ {t (d) , t (c) }, otherwise.
Essentially, π (d) (t (c) ) − π (c) (t (c) ) is farmer t (c) ’s loss from the markets when the shared level increases from t (d) to t (c) . When R2 is large enough (R2 ≥ π (d) (t (c) ) − π (c) (t (c) )), Mechanism 2 can entice farmers with knowledge in [t (c) , 1] to share knowledge at exactly level t (c) , because it disqualifies knowledge levels but t (c) . When the reward is small (R2 < π (d) (t (c) ) − π (c) (t (c) )), Mechanism 2 may not be effective because farmers with knowledge in [t (c) , 1] may not be willing to share if they anticipate the highest knowledge shared by others is low. To pinpoint the shared level at t (c) in any equilibrium, R2 should be sufficiently high (R2 ≥ π (d) (t (c) ) − π (c) (t (c) )). In this case, there are (1−t (c) ) winners and the implementation cost is at least (1 − t (c) )[π (d) (t (c) ) − π (c) (t (c) )]. This cost can be very high when π (d) (t (c) ) − π (c) (t (c) ) is large, which is yet out of the NGO’s control.
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In conclusion, Mechanism 2 is better than Mechanism 1 because, to some extent, the shared level becomes predictable and controllable to the NGO. Yet, to pinpoint the shared level at t (c) , he has to put up with a high reward R2 . In view of this, one wonders if there is a more effective mechanism whose shared level and implementation cost are both controllable by the NGO.
12.5.2.2
A QuotaBased Reward Mechanism
We now present a reward mechanism that is intended to overcome the shortcomings of both Mechanisms 1 and 2 as follows. For ease of exposition, we focus on the case where t (c) < 1, which occurred when c < τ (c) as stated in Proposition 1. Instead of offering a reward to each farmer who shares knowledge at level t (c) as in Mechanism 2, we propose the following quotabased mechanism: QuotaBased Reward Mechanism Farmers who share their knowledge will be sorted according to their shared knowledge levels. Then the NGO gives out the rewards to these farmers, starting from the highest shared level and working its way down until all 1 − t (c) rewards are exhausted. In the event when the number of farmers in the last group to be considered exceeds the remaining quota of rewards, the NGO uses lottery to break the tie. Proposition 6 Under the quotabased reward mechanism, when farmers use only admissible strategies, all purestrategy Nash equilibria have the same shared level t (c) for any R > 0. Proposition 6 can be interpreted as follows. First, with the “quota” for winners, which is 1 − t (c) , the quotabased mechanism entices the farmers with knowledge in [t (c) , 1] to share higher knowledge, t (c) . In essence, the quotabased mechanism complements Mechanism 1 by setting a restriction on the number of winners. Second, instead of (explicitly) specifying the desired knowledge level to be exactly t (c) , the quotabased mechanism introduces a fierce sharing competition among farmers, which creates a strong motivation for farmers with knowledge in [t (c) , 1] to share at t (c) for any reward R > 0, which is under the control of the NGO. Therefore, our quotabased reward mechanism is effective because it empowers the NGO to achieve welfare optimality with very little cost, as long as the “quota” is well set. Akin to the quotabased reward mechanism as defined here, Avaaj Otalo and the Supply Chain Network Project are offering a limited number of smallvalue rewards, such as free phone minutes to a small number of winners. Thus, our result provides a plausible justification for the restricted number of small rewards offered by NGOs. Our finding is also consistent with the experimental study by Gallus (2016). Gallus (2016) design a similar “quotabased” reward mechanism: a competition on Germen language Wikipage retention among newcomers, a certain number of whom would be awarded a purely symbolic award if they contribute to the Wikipage. The newcomers are informed that there are around 4,000 of them, which is similar to the infinite number of farmers in our model. Gallus (2016) found
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a sizable positive effect of the purely symbolic award on newcomer retention in Wikipage. The findings in Gallus (2016) show the generality of our mechanism.
12.6 Improving the Learning Effectiveness Factor h Due to the low literacy level in emerging markets, many knowledge sharing platforms are now using voicebased system (Avaaj Otalo) or imagebased system (the Supply Chain Network Project) to enhance the learning effectiveness factor h so that farmers can advance their knowledge further because k (t) = k + h[t − k]+ . We now investigate the economic impact of h on the farmers’ output and farmer welfare. Recall from Proposition 2 that the decentralized shared level t (d) increases in h. Hence, farmers’ knowledge gets enhanced further when h increases. This observation enables us to obtain the following result: (d)
Proposition 7 The expected quantity of the highquality output QH and farmer welfare W (d) are increasing in the learning effectiveness factor h. Proposition 7 indicates that a higher learning effectiveness factor h will increase farmer welfare. This result provides a formal justification to support investment in better communication technology for farmers to learn from others more effectively. While Proposition 7 reveals that farmer welfare increases when the NGO redeems its sharing and learning platform by increasing h, we now evaluate the impact of this technology advancement on individual farmers. Suppose the NGO has two potential platforms with learning effectiveness factors h1 and h2 , respectively, and h1 < h2 . Here, platform 1 represents old technology (say, feature phones with text only for sharing), while platform 2 represents new technology (say, (d) smart phones with video, image and voice capabilities for sharing). Let th1 and (d)
th2 denote the corresponding shared levels in equilibrium under the decentralized system. Specifically, th(d) is the solution to Eq. 12.15, where h is replaced by hj for j (d) (d) j = 1, 2. By applying Proposition 2, we can conclude that th1 < th2 . By using this observation, we can compare the farmer’s profit function π (t) (k (t) ) given in (d) (d) Eq. 12.10 for the case when h = h1 , t = th1 , and h = h2 , t = th2 , and we obtain the following result: Lemma 7 When the NGO implements a new sharing and learning platform so that the learning effectiveness factor is increased from h1 and h2 , a farmer with initial knowledge level k will earn a higher payoff under the new platform if and only if , th(d) ). k < θh , where θh ∈ (th(d) 1 2 The intuitive explanation of the result stated in Lemma 7 follows the same logic as explained in Lemma 4 and the result can be explained in the same manner because the effect of improving learning effectiveness is similar to increasing shared knowledge level. We omit the details.
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12.7 Extensions In this section, we evaluate our results under three extensions: (i) a farmer’s knowledge follows a general distribution, rather than uniform distribution, (ii) a farmer’s highquality and lowquality outputs are stochastic, and (iii) a farmer’s capacity is not fixed. We show that the key insights remain valid under any of the three extensions.
12.7.1 When Farmer’s Knowledge Follows a General Distribution Throughout this chapter, we assume that the farmers’ knowledge levels are uniformly distributed over [0, 1]. We now extend our analysis to the case when the farmers’ knowledge follows a continuous distribution function F (·) (and a continuous density function f (·)) over the range [0, 1]. In preparation, let us generalize the function M(t) defined in Eq. 12.8 to the case when the farmers’ knowledge follows a density function f (·) so that ˜ M(t) = E[(k (t) )2 ] =
0
t
[ht + (1 − h)k]2 f (k)dk +
1
k 2 f (k)dk.
(12.21)
t
Observe that Eq. 12.21 can be simplified to Eq. 12.8 when the knowledge is uniformly distributed so that f (·) = 1. ˜ Lemma 8 For any knowledge distribution F (·) over the range [0, 1], M(t) ˜ ˜ increases in both t and h and that 2M(t) > t ∂ M(t)/∂t. Because of Lemma 8, we can prove that under a general distribution F (·), most of our key results continue to hold. This is because all key results are independent of the specific form of the distribution but are hinged upon the condition that ˜ ˜ ˜ M(t) is increasing in both t and h and that 2M(t) > t ∂ M(t)/∂t. Therefore, in view of Lemma 8, we can conclude that all key results continue to hold when the farmers’ knowledge follows a general distribution. Specifically, for any continuous distribution F (·), it can be shown that the farmer welfare W (t) is strictly quasiconcave in the shared level t (as stated in Proposition 1), that the farmers’ incentive to share knowledge continues to exhibit the same characteristics (as stated in Proposition 4), that the voluntary sharing equilibria in the decentralized system continues to possess the same properties (as stated in Propositions 2 and 3), that the sharing behaviors in equilibrium under different reward mechanisms continue to hold (as stated in Lemmas 5 and 6 and Proposition 6), and the changes to individual farmers’ payoffs (as stated in Lemma 4, Proposition 7, and Lemma 7). As it turns out, the result stated in Proposition 5 (i.e., t (d) ≤ t (c) ) hinges upon ˜ ˜ a mild condition: M(0) ≤ 1/2, where M(0) is given in Eq. 12.21 when t = 0
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(i.e., when there is no knowledge sharing). More formally, we have the following proposition: Proposition 8 For any continuous knowledge distribution F (·), t (d) ≤ t (c) if ˜ M(0) ≤ 1/2. ˜ The condition M(0) < 1/2 stated in Proposition 8 is actually mild. To elaborate, ˜ observe from Eq. 12.21 that M(0) = E[k 2 ] = Var[k] + E2 [k]. In this case, the ˜ condition M(0) < 1/2 holds when the expected knowledge of the farmers is low and the variance of the knowledge is also low. For instance, let us consider the general Beta distribution Beta(a, b) with E[k] = a/(a + b) and Var[k] = ab/[(a + b)2 (a + ˜ b + 1)]. In this case, one can check that the condition M(0) ≤ 1/2 holds when b ≥ a, which occurs when more farmers have low knowledge levels (i.e., when the distribution of the farmers’ knowledge is skewed to the right). With Lemma 8 and Proposition 8 and the analysis above, we can conclude that most of our key results continue to hold even when the farmers’ knowledge follows a general continuous distribution over the range [0, 1].
12.7.2 When Each Farmer’s HighQuality and LowQuality Outputs Are Stochastic We consider two scenarios with stochastic outputs: (i) the productivity factor β follows the binomial distribution and (ii) the highquality output for a farmer with knowledge k and effort level x is βkx +k , where k ∼ G[a1 , a2 ], and G is a general probability distribution with support [a1 , a2 ].
12.7.2.1
The Productive Factor β Follows the Binomial Distribution
In this subsection, we extend our base model to the case when β follows the binomial distribution (instead of a fixed value). Specifically, we assume β = βi with probability r and β = βj with probability 1 − r, where 1 ≥ βi > βj ≥ 0 and r ∈ [0, 1]. (t) Given the highest knowledge shared is t, let QH (i) be the expected highquality (t) (t) supply when β is realized as βi and similarly Q(t) L (i), QH (j ), and QL (j ). In this case, farmer k’s optimal effort exertion problem is as follows: (t) (t) π (t) (k (t) ) = max r βi k (t) x[AH − QH (i)] + (1 − βi k (t) x)[AL − αQL (i)] x
+ (1 − r) βj k (t) x[AH − Q(t) H (j )] (t) + (1 − βj k (t) x)[AL − αQL (j )] − cx 2 /2.
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Following similar proofs for the base model, one can check that all the key insights remain valid under this scenario. This is because this stochastic model is a weighted average of two possible scenarios where the value of β differs. Thus the tradeoff and the key insights remain the same as in the base model.
12.7.2.2
Farmer k’s HighQuality Output Is βkx + k
We now consider the case when the highquality output for a farmer with knowledge k and effort x is equal to βkx + k . In this extension, β is a known constant; however, k ∼ G[a1 , a2 ], where G is a general probability distribution with support [a1 , a2 ]. We assume k , ∀k ∈ [0, 1], are independently and identically distributed. To guarantee that the highquality and lowquality outputs are nonnegative, we assume 0 ≤ a < b ≤ 1 − β. Given the highest knowledge shared is t, farmer k’s optimal effort exertion problem is as follows: (t) (t) 2 π (t) (k (t) ) = max E (βk (t) x + k )(AH − Q(t) H ) + (1 − βk x − k )(AL − αQL ) − cx /2 , x
1 (t) (t) (t) (t) where Q(t) H = 0 [βk x (k) + k ]f (k)dk, and QL = 1 − QH . Following similar proofs for the base model, one can check that all the key insights remain valid under this scenario. This is because this stochastic model is not substantially different from the base model but merely intensifies or alleviates the tradeoff between markets.
12.7.3 When a Farmer’s Output Capacity Is Not Fixed and the Markets Are Not Separate In our base model, we have assumed unit capacity of each farmer and separate markets. We now relax these two assumptions in this extension as follows. For a farmer of knowledge k, if he exerts effort x, then his total output is Z + ξ kx, which consists of βkx highquality output and Z + (ξ − β)kx lowquality output. This model reduces to our base model when Z = 1 and ξ = 0. First, consider the case when ξ ≥ β. In this case, one can check that, due to the increasing output for both grades, no farmer would share his knowledge in the decentralized case. However, the reward mechanisms still apply. That is, when there is no farmer sharing, using the same “quotabased” reward mechanism, we could induce them to share knowledge up to the optimal level. Second, consider the case when ξ < β. To model nonseparate markets, we assume that the market price in each market is given as PH = AH − u1 QH − u2 QL and PL = AL − u3 QL − u4 QH , where u1 > u2 ≥ 0 and u3 ≥ u4 ≥ 0. We also assume u1 > u4 and u3 > u2 , that is, the supply of highquality (low
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quality) produce impacts its own price more than that for the lowquality (highquality) produce. Note that when u3 = u4 = 0, the price for lowquality produce is constant. Furthermore, when AL = 0, the lowquality produce is not marketable. Given the highest knowledge shared is t, a farmer’s effort optimization problem becomes (t) π (t) (k (t) ) = max βk (t) x(AH − u1 Q(t) H − u2 QL ) x
(t) (t) + [Z − (β − ξ )k (t) x](AL − u3 QL − u4 QH ) − cx 2 /2 . To ensure that a farmer’s lowquality output is nonnegative, we assume Z is large enough such that Z − (β − ξ )k (t) · x (t) (k (t) ) ≥ 0, ∀k, t. We further assume that the price for highquality produce is always higher than the lowquality ones, that is AH − u2 Z > AL − u3 Z > 0; AH − u1
β β Z > AL − u4 Z > 0, β −ξ β −ξ
(12.22) (12.23)
where constraints in Eqs. 12.22 and 12.23 imply that whether all the output is of low quality or high quality, the price for highquality produce is always higher than that for lowquality produce, and the prices are always positive. With the assumptions above and analysis similar to that for the base model, one can check that the key insights in the base model remain valid. The proofs and analysis for this section are provided in the Appendix.
12.8 Concluding Remarks This chapter represents an initial attempt to examine the issue of knowledge sharing and learning in the context of smallholders. We provide a theoretical framework to examine the economic motivations for knowledge sharing/nonsharing behaviors. We find that highknowledge farmers are reluctant to share, because sharing knowledge will enhance others’ productivity and increase the competition in selling the highquality crop. By noting that farmers’ voluntary shared knowledge level may be inadequate to maximize farmer welfare, we have shown that NGOs can increase farmer welfare if they can entice farmers to share the right level of knowledge. To this end, we have developed a simple quotabased reward mechanism that is effective in enticing farmers to share the right amount of knowledge at a small cost. Our theoretical findings could serve as testable hypotheses for economic motivation of knowledge sharing and reward mechanism design. Because this chapter is an initial attempt to analyze knowledge sharing and learning in the context of smallholders, there are abundant possible extensions to
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enrich our findings. Firstly, we have assumed that the knowledge is unidimensional, based on a single attribute. It is worthwhile to study farmers’ sharing behaviors under the setting of multidimensional knowledge. Secondly, we have assumed that farmers only learn from the highest shared knowledge, which is reasonable in some circumstance. This assumption also helps us to specify the tradeoff involved in sharing, derive a “threshold” sharing equilibrium, and obtain firstorder insights more concisely. However, it is also possible that multiple knowledge exchanges may add to the understanding and learning of farmers. Thirdly, we focused on farmer welfare, which is of key concern for many sharing platforms developed to support needy smallholders. Nevertheless, it is an interesting alternative to take the consumer welfare into consideration and determine the efficient shared level that maximizes the weighted average of consumer welfare and farmer welfare.
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Chapter 13
Policy Interventions for an AgricultureBased Malaria Medicine Supply Chain Burak Kazaz, Scott Webster, and Prashant Yadav
13.1 Introduction According to the World Health Organization (WHO 2012) report, the world experienced 219 million cases of malaria that resulted in more than 660,000 deaths. The devastation primarily occurred in subSaharan Africa (accounting more than 90%), and a large fraction of the deaths corresponded to children under five, pregnant women, and malnourished people. Malaria is still one of the deadliest diseases, requiring the attention of governments, pharmaceutical companies, and aid organizations. Due to the resistance against the drugs such as chloroquine and sulfadoxine pyrimethamine (SP), WHO began to recommend artemisinin combination therapy (ACT) as the primary treatment for uncomplicated Plasmodium falciparum malaria since April 2002. As a result, more than eighty four countries and territories in Africa utilize ACT as the primary treatment against malaria. ACTs are manufactured from a starting material derived from artemisinin which is obtained from a plant called Artemisia annua. ACT is manufactured by combining one of the artemisinin derivatives (artemether, artesunate, or dihydroartemisinin) with another antimalarial B. Kazaz () Whitman School of Management, Syracuse University, Syracuse, NY, USA email: [email protected] S. Webster W.P. Carey School of Business, Arizona State University, Tempe, AZ, USA email: [email protected] P. Yadav Center for Global Development, Washington, DC, USA Technology and Operations Management, INSEAD, Fontainebleau, France email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_13
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compound such as lumefantrine, amodiaquine, or piperaquine. WHO has approved eleven different companies to manufacture ACTs. ACT treatments are often provided free by governments in their local clinics because the people in the nations who need them most cannot afford the treatment. Moreover, a majority of the malariaendemic countries are lowincome countries and have to rely on international donor support to purchase ACTs in the treatment of their population. The donors are international agencies, most notably the Global Fund to Fight AIDS, TB, and malaria and the US President’s Malaria Initiative. Our work helps multilateral agencies and philanthropic organizations in their interventions to affect the availability and price of ACTs by increasing the supply of artemisinin and stabilizing its price. Our study recommends these organizations where to invest their time and effort in order to create the highest positive impact in treating malaria. Artemisia annua is grown in Southeastern Asia: Vietnam, Southeast Asia, and Madagascar. China and Vietnam are the primary growing regions supplying more than 80% of the global supply; the remaining part is grown in East Africa (Shretta and Yadav 2012). Growers of Artemisia annua are often small farmers with growing fields less than one hectare. The plant grows in eight months and its leaves are harvested and dried before extracting artemisinin that is used in the making of the medicine. The yield of leaves per growing area fluctuates significantly from one farmer to another in addition to the fact that it is influenced by climatic conditions such as rainfall. Moreover, the artemisinin content in the leaves fluctuates substantially: it can be as low as 0.1% and as high as 1.2%. Collectively, the yield of artemisinin is uncertain, leading to supply uncertainty in the malaria medicine supply chain. Spar and Delacey (2008) and Spar (2008) show that there are significant fluctuations in the price of the ACT from Novartis called Coartem®. Kindermans et al. (2007) and Schoofs (2008) highlight the fact that the planting hectares also differ from one year to the next. Figure 13.1 shows the relationship between limited supply of artemisinin contributing to the price fluctuations. For example, in 2005, the price of artemisinin increased to $1100/kg, followed by an excess in supply reducing the price to $170/kg in 2007. Our study incorporates the farmers’ expectations from the returns from growing Artemesia annua in comparison to other cash crops such as paddy/rice and corn. It is a wellknown belief that the outside option plays a crucial role in determining whether to grow Artemesia annua or an alternative crop. As a result, it can be easily seen that farmers’ choices can create additional supply and price fluctuations for the malaria medicine. Demand uncertainty in the malaria medicine supply chain is as influential as supply uncertainty. Predicting the demand for ACT is not an easy task (see Kindermans et al. 2007 and Shretta and Yadav 2012). Shretta and Yadav (2012) report that the demand increased dramatically in Kenya after one of the worst droughts in 60 years followed by rains. Steketee and Campbell (2010) and SPS (2012) highlight that Kenya does not have proper diagnostic testing and record keeping capabilities, making it difficult to estimate the demand for treatment. As
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Fig. 13.1 Artemisinin spot prices as presented in Kazaz et al. (2016)
a result, demand uncertainty must be examined in the analysis pertaining ting to the malaria medicine supply chain. Another factor that makes it challenging to estimate the demand for ACT is the price of the treatment. Even though ACT is free in governmentrun clinics, many patients seek treatment in private sector clinics, drug shops, and pharmacies. If the patient’s willingness to pay is lower than the price of the medicine, then he/she purchases treatments that are inefficacious (Arrow et al eds). In recognition of the privately run clinics, a pilot project subsidized the cost of ACT in the private sector in 2009 (Adeyi and Atun 2010); however, the limited time for the pilot study did not yield positive outcome. In conclusion, the price of the ACT treatment continues to be a key barrier for many patients. One of the main challenges in matching supply and demand is the long lead time in the growing and manufacturing process. It takes approximately 14–18 months from planting of Artemisia annua to the production of the ACT treatment (Shretta and Yadav 2012). In order to combat price fluctuations in artemisinin, larger pharmaceuticals sign forward contracts with extractors with a predetermined price and quantity. Smaller manufacturers, on the other hand, lack financial resources to engage in forward contracts; they rely on purchasing artemisinin from the spot market and thus operate under price uncertainty. Given the current challenges in supply and demand for ACTs, organizations such as the Bill and Melinda Gates Foundation, UNITAID, Clinton Health Access
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Initiative (CHAI), Global Fund to fight AIDS, TB and Malaria, and the UK Department for International Development began to explore if certain investments/interventions can improve outcomes in terms of availability and price. One primary intervention is focused on stabilizing artemisinin prices. In 2008, the Clinton Foundation agreed with several Chinese and Indian manufacturers to establish price ceilings in the effort to stabilize ACT prices (Schoofs 2008). A second intervention involves the use of forward contracts. In 2009, UNITAID’s Assured Artemisinin Supply Services (A2S2) provided WHOqualified extractors to receive loanbased prefinancing. The goal was to increase supply and create “fair prices” through loanbased prepayments. Unfortunately, none of these interventions stabilized prices completely (UNITAID 2011; Shretta and Yadav 2012). A third intervention involves the semisynthetic production of artemisinin. Through the financial support from the Bill & Melinda Gates Foundation, a research group at the University of CaliforniaBerkeley and Institute for One World Health has developed a semisynthetic manufacturing process without having to rely on the plantbased production of the ACT treatment (Hale et al. 2007). Commercialscale manufacturing of semisynthetic artemisinin continues to be far from feasibility (Paddon and Keasling 2014; Reuters 2014); thus, it is unlikely to contribute to the elimination of price uncertainty in artemisinin. Moreover, some publications highlight that a larger supply of semisynthetic artemisinin can also contribute to the volatility in artemisinin supply and prices because farmers may exit the growth of the plant faster than the increase in semisynthetic manufacturing capability (Van Noordan 2010; Peplow 2013). This chapter develops a model that examines the effects of available farm space, farmer’s selfinterest, uncertainty in crop yield, uncertainty in demand, and the availability of semisynthetic artemisinin on ACT price and supply. We use field data and investigate the impact of various interventions. We find that interventions that improve average yield, create a supportprice for artemisinin, and a larger but carefully managed supply of semisynthetic artemisinin have the greatest potential to improving supply and reducing the volatility in malaria medicine.
13.2 Related Literature Shretta and Yadav (2012) and Dalrymple (2012) present a comprehensive review of the ACT supply chain. Taylor and Xiao (2014) show that donor organizations should subsidize retail purchases over retail sales in malaria medicine supply chains. Yano and Lee (1995) provide a comprehensive review of the publications that feature supply uncertainty. Rajaram and Karmarkar (2002), Galbreth and Blackburn (2006), and Gupta and Cooper (2005) investigate supply uncertainty in process industries. Tomlin and Wang (2008) and Noparumpa et al. (2015) examine coproduction and pricing flexibilities under supply uncertainty. In agricultural settings, Jone et al. (2001) show the benefits of secondary growing options in mitigating supply uncertainty. Burer et al. (2009) examine coordination decisions under yield
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uncertainty. Blackburn and Scudder (2009) focus on the risk of producing and distributing fresh produce. Kazaz (2004) features the yielddependent cost and revenue structure when operating under yield uncertainty. His study shows the negative implications of ignoring the impact of supplier’s risk aversion on leasing, purchasing, and pricing decisions. Li and Zheng (2006), Tang and Yin (2007), and Kazaz and Webster (2011, 2015) examine joint pricing and quantity decisions under yield uncertainty. Tomlin (2009), Tomlin and Wang (2005), Dada et al. (2007), and Federgruen and Yang (2008) investigate the use of reliable contingency suppliers when facing yield uncertainty. Huh and Lall (2013) examine precipitation uncertainty and its impact on crop choices. Our paper differs from these publications as we examine supply uncertainty in decisions concerning public interests (rather than commercial reasons). Our study makes two contributions to the literature. First, our study follows a unique public–private collaboration perspective by deviating from the common performance measure of firmlevel profit or utility. We use field data and demonstrate the effectiveness of various interventions that would benefit organizations such as UNITAID, CHAI, and the Gates Foundation. Second, our study introduces the examination of supply and demand uncertainty in the context of medicinal agriculture. The study serves as a foundation for other medicinal products such as vaccines, which also experience substantial supply and demand uncertainty.
13.3 Model 13.3.1 Overview We begin our discussion by introducing the two levels in artemisininbased malaria medicine supply chain. Level 1 represents ACT manufacturers and level 2 represents farmers who serve as the suppliers. We treat farmers and extractors as a single unit because our modeling approach can capture their relevant decisions adequately. In our model, suppliers choose between growing Artemisia annua and the best alternative crop. The amount of farm space allotted for the plant increases in the expected value of the artemisinin spot price and decreases in its variance. The volatility of the spot price is impacted by the degree of volatility in the crop yield and in the size of the market. Price is guaranteed for units under forward contract and the forward contract price is aligned with the expected spot price. Artemisinin produced without a forward contract is sold in the open market at an uncertain spot price. We consider that the expected spot price decreases with higher fractions of growing Artemisia annua. As in Kazaz et al. (2016), Fig. 13.2 presents an overview of decisions, processes, and relationships in the malaria medicine supply chain.
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Fig. 13.2 Schematic view of the malaria medicine supply chain
13.3.2 Equilibrium Condition The decision variable q represents the amount of farm space allotted for growing Artemisia annua and P (q) describes the random marketclearing price of artemisinin with moments p(q) = E[P (q)] σP2 (q) = V [P (q)]. We introduce two different models and examine how the probability distribution of P (q) is affected by various parameters. The parameter s denotes the quantity of semisynthetic artemisinin, which is not impacted by yield uncertainty. The expected yield from each unit of farm space is μ2 ; the random organic artemisinin yield can then be expressed as Q = qμ2 Z2 ,
(13.1)
where Z2 is a positive random variable with cdf 2 , mean 1, and variance σ22 . qμ2 describes the expected amount of artemisinin from farming q units of land. The sum of random artemisinin supply and the semisynthetic artemisinin is qμ2 Z2 + s. The mean artemisinin supply is qμ2 + s. Some manufacturers offer forward contracts that specify a price and quantity of artemisinin they will purchase at a future point in time. The parameter α describes the fraction of farmer’s land dedicated to producing artemisinin under a forward contract. The forward contract price is equal to the expected spot price p(q). The parameter c describes the total amount of land owned by farmers. We describe the farmers’ utility from the best alternative crop with Ub which has a cdf of ρb (u) and a mean of μb .
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The utility from producing artemisinin for a representative supplier is described as the product of two terms: (1) expected yield per unit of land and (2) the mean– variance utility per unit of artemisinin ua = μ2 × (p(q) − γ σP2 (q)). The parameter γ ≥ 0 describes the degree of risk aversion: the higher the value of γ , the higher the risk aversion and γ = 0 corresponds to the riskneutral setting. It can be observed that the utility is increasing in average yield (μ2 ) and average price (p(q)) and is decreasing in price variance (σP2 (q)), where the rate of decrease is controlled by riskaversion parameter (γ ). The utility from producing artemisinin under forward contract is μ2 p(q) and there is no variance in the price. The utility from the best alternative associated with a unit of land under forward contract is described by Ub0 . We describe Ub0 as Ub conditioned on the utility of the best alternative being less than the utility of artemisinin under contract with a cdf of Ub0 = Ub Ub ≤ μ2 p(q), which is defined as ρb0 (u) = P [Ub0 ≤ u] = P [Ub ≤ uUb ≤ μ2 p(q)] =
ρb (u) ρb (μ2 p(q))
for all u ≤ μ2 p(q).
(13.2)
We next identify a condition for the value of q in equilibrium. For a given q, the amount of farm space not dedicated to forward contract is q − αq.
(13.3)
And, for a given q, the amount of land that is not dedicated to forward contract with utility of the best alternative no more than the utility of producing artemisinin is cρb (ua ) − αqρb0 (ua ) = ρb μ2 (p(q) − γ σP2 (q)) c −
αq ρb (μ2 p(q))
(13.4)
(see Eq. 13.2). Equilibrium can be obtained by setting Eq. 13.3 equal to Eq. 13.4 and by solving for q. 1−α F (q ∗ ) ≡ c − ρb (μ2 (p(q ∗ ) − γ σP2 (q ∗ ))) = 0. α − q∗ ρb (μ2 p(q ∗ ))
(13.5)
When farmers are risk neutral and γ = 0, Eq. 13.5 reduces to q ∗ = cρb (μ2 p(q ∗ )).
(13.6)
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The above expression indicates that the farm space dedicated to Artemisia annua is the fraction of capacity with utility of the best alternative no more than the expected revenue per unit of farm space. The expression in Eq. 13.3 is increasing in q. For any realization of supply and demand random variables, the spot price is decreasing in q; thus, the expected spot price is decreasing in q: p (q) < 0.
(13.7)
When utility ua does not increase as q increases, i.e., d μ2 (p(q) − γ σP2 (q)) ≤ 0, dq
(13.8)
then Eq. 13.4 is decreasing in q making Eq. 13.8 a sufficient condition for a unique equilibrium.
13.3.3 Two Models of PriceDependent Demand We describe the random ACT market size with M = μ1 Z1 , where μ1 is the expected ACT market size and Z1 is a positive random variable with cdf 1 , mean 1, and variance σ12 . We consider the case when Z1 is independent of the yield random variable Z2 . This is a reasonable assumption reflecting the reality because more than 80% of Artemisia annua growth takes place in Asia and more than 90% of ACT demand occurs in subSaharan Africa. The climatic conditions for the growth of the plant are unrelated with the weather conditions in the regions of the demand. We develop two pricedependent demand models in order to carry out our analyses: M1: d(p) = Mρ1 (p), where ρ1 (p) is the fraction of the market willing to pay price p or more, and M2: d(p) = bp−1 , where the total volume purchased at price p is independent of the market size. Model M1 represents a setting where the market is composed of many individual buyers who purchase ACT only if they are able to pay the market price. Model M2 reflects a setting where the market is composed of a few buyers, NGOs, and international donor organizations, who have a total budget of b, and describes the demand as an isoelastic function. M1 is better suited for countries where patients seek treatment in the private sector. M2 reflects the demand in countries where patients are treated in government or NGOrun health clinics who have a fixed budget. We examine the effectiveness of various interventions under each model independently.
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We next examine the form of the random spot price function P (q) under each model. We investigate the potential efficiency from a pricesupport intervention. For this intervention, donor organizations pay a minimum price of p0 ensuring that the marketclearing price will go below the supportprice p0 . Assuming that the willingnesstopay function ρ1 (p) ∈ [0, 1] is a strictly decreasing function of price, ρ1 (p) can be inverted to determine the random marketclearing price and its moments. This can be accomplished by equating qμ2 Z2 + s to the demand μ1 Z1 ρ1 , and by solving for ρ1 and then inverting ρ1 (p), M1:
qμ2 Z2 + s , 1 , p0 P (q) = max ρ1 −1 min μ1 Z1 qμ2 Z2 + s −1 min p(q) = E max ρ1 , 1 , p0 μ1 Z1 qμ2 Z2 + s 2 −1 σP (q) = V max ρ1 , 1 , p0 . min μ1 Z1
(13.9) (13.10)
Following a similar approach for M2 reveals the following: b , p0 qμ2 Z2 + s b p(q) = E max , p0 qμ2 Z2 + s b 2 σP (q) = V max , p0 . qμ2 Z2 + s
M2:
P (q) = max
(13.11)
13.3.4 Performance Measures We develop performance measures for the manufacturer, society, and the supplier. The expected artemisinin volume in equilibrium would be π1 = E[q ∗ μ2 Z2 ] + s = q ∗ μ2 + s, which reflects the manufacturer’s welfare as well as describing a measure for public health based on the medicine’s availability. Another measure for public health is the fill rate described by the expected fraction of total demand satisfied: β = E[min{(q ∗ μ2 Z2 + s)/μ1 Z1 , 1}]. Note that Ub0 = Ub Ub ≤ μ2 p(q) and its cdf is ρb0 (u) = ρb (u)/ρb (μ2 p(q)). The supplier surplus with αq ∗ units under contract at price p(q ∗ ) is αq
∗
μ2 p(q ∗ ) −∞
ρb (t) dt = αq ∗ (μ2 p(q ) − t) ρb (μ2 p(q ∗ )) ∗
μ2 p(q ∗ )
−∞
ρb (t) dt. ρb (μ2 p(q ∗ ))
The cdf of the utility from the best alternative after removing the units under contract is
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⎧ αq ∗ ρb (u) ⎪ ⎪ ⎨ cρb (u) − ρ (μ p(q ∗ )) b 2 , u ≤ μ2 p(q ∗ ) ρb\0 (u) = ∗ c − αq ⎪ ⎪ ⎩ ρb (u), u ≥ μ2 p(q ∗ ).
(13.12)
Then, the supplier surplus for the units not under contract is (c − αq ∗ )
μ2 (p(q ∗ )−γ σ 2 (q ∗ ))
ρb\0 (t) dt
−∞
= c−
μ2 (p(q ∗ )−γ σ 2 (q ∗ )) αq ∗ ρb (t) dt ρb (μ2 p(q ∗ )) −∞ μ2 (p(q ∗ )−γ σ 2 (q ∗ )) ρb (t) = q ∗ (1 − α) dt. ρb (μ2 (p(q ∗ ) − γ σp2 (q ∗ ))) −∞
The total supplier surplus becomes π2 = q α ∗
μ2 p(q ∗ )
−∞
+ (1 − α) = q∗ +
ρb (t) dt ρb (μ2 p(q ∗ ))
μ2 (p(q ∗ )−γ σ 2 (q ∗ )) −∞
ρb (t) dt ρb (μ2 (p(q ∗ ) − γ σp2 (q ∗ )))
α E[(μ2 p(q ∗ ) − Ub )+ ] ρb (μ2 p(q ∗ ))
1−α ∗ 2 ∗ + E[(μ (p(q ) − γ σ (q )) − U ) ] . 2 b ρb (μ2 (p(q ∗ ) − γ σp2 (q ∗ )))
13.4 Analysis This section presents the impact of changes in parameter values on performance measures described earlier. Table 13.1 presents comparativestatic results for q ∗ and π1 when suppliers are risk neutral. The model is too complex to develop similar directional results for β and π2 when suppliers are risk averse. The following intuitive results are concluded in Table 13.1: (1) the increased number of forward contracts (α) has no impact when suppliers are risk neutral, (2) the budget (b) does not play a role under M1, a higher budget amount results in greater supply under M2, (3) an increase in land dedicated to Artemisia annua (c) increases supply,
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Table 13.1 The impact of parameter values when suppliers are risk neutral according to Kazaz et al. (2016); ↑ = increasing, ↓ = decreasing, – = no change, ↑↓ = direction depends on other parameter values Increase in Demand model M1 α M2 M1 b M2 M1 c M2 M1 μ1 M2 σ1
M1 M2
s
μ2
σ2
μb p0
M1 M2 M1 M2 M1 M2 M1 M2 M1 M2
Change in q ∗
Change in π1
–
–
– ↑
– ↑
↑
↑
↑ –
↑ –
linear d(p), ↓ linear d(p), ↓ max{Z1 } ≤ 2 : concave d(p), ↓ max{Z1 } ≤ 2 : concave d(p), ↓ convex d(p), ↑↓ convex d(p)↑↓ – – linear or concave d(p), ↑ ↓ convex d(p), ↑↓ ↓ ↑↓ ↑↓ ↑ ↑ ↑ concave d(p), ↓ concave d(p), ↓ linear d(p), −− linear d(p), −− convex d(p), ↑ convex d(p), ↑ ↑ ↑ ↓
↓
↑
↑
(4) a bigger market size (μ1 ) leads to a higher supply in M1 but has no effect in M2, (5) a greater degree of market volatility (σ1 ) has no effect on supply in M2, (6) a higher degree of attractiveness in the best alternative to artemisinin (μb ) results in a reduced supply, and (7) a higher value of price support (p0 ) increases supply. We next demonstrate the impact of parameters using numerical illustrations. Parameters σ1 , σ2 , s, α, μ1 , μ2 , and b are estimated using historical data from AS2S and UNITAID and the risk aversion parameter γ as 0.008. We utilize a uniformly distributed willingnesstopay function and describe the utility of the best alternative on the basis of the principle of insufficient reason originally introduced by Pierre Laplace in the 1700s (Luce and Raiffa 1957). We employ a symmetric triangular distribution for Z1 and Z2 in order to capture a central tendency about the mean of 1.
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Table 13.2 Description of functions, random variables, and parameters in our basecase model Units: Space unit = 1000 hectares (H) Artemisinin unit = 1000 kg (K) Functions and random variables Z1 , Z2 ~ symmetric triangular ρ1 (p) = 2 − 0.0032p σ1 = 0.1R K, σ2 = 0.3 K/H Parameters Potential farm space (c) = 80 H Semisynthetic supply (s) = 60 K Risk aversion (γ ) = 0.008 Mean demand (μ1 ) = 240 K Purchase budget (b) = 75,000 D
Currency unit = $1000 (D) Ub ~ uniform μb = 4,800 D, σb = 1500 D Forward contract % (α) = 25% Mean yield (μ2 ) = 10 K/H
Table 13.3 Statistics from the basecase model in equilibrium
Hectares producing artemisinin in 000s (q ∗ ) Average total supply in metric tons (π1 ) Average semisynthetic production as fraction of total Average fill rate (β) Average supplier surplus in $000,000s (π2 ) Average total spend in $000,000s Average artemisinin price in $ per kg (p ) Standard deviation in artemisinin price (σP ) Min and max price per kg (in 10,000 trials)
Demand model M1 17 235 26% 89% $10 $73 $345 46 $313, $505
M2 15 212 28% 85% $8 $75 $373 90 $232, $740
Historical data on price, annual supply, need, and fill rates lead to our estimations of μb , σb , c, and coefficients of linear function ρ1 (p) (Table 13.2). Table 13.3 presents the results from solving the problem for the basecase. We next present the sensitivity of performance to changes in the 11 parameters listed in Table 13.1. Each parameter fluctuates between −50% and +50% of its basecase value; the supportprice p0 = 0 in the basecase. We categorize the impact of parameters into three groups—high, moderate, and low sensitivity: High: average yield (μ2 ), average utility of the best alternative (μb ), and average market size (μ1 ). Moderate: available farm space (c), semisynthetic supply (s), and spend budget (b). Low: risk aversion (γ ), yield variability (σ2 ), demand variability (σ1 ), and forward contract % (α). The high category can be interpreted as firstmoment parameters, the low category is akin to secondmoment parameters. The moderate category is similar to quantity parameters. The highcategory parameters represent the averages of random variables and the lowcategory parameters are connected to volatility. Parameters σ1 and σ2 are direct measures of volatility, and parameters γ and α are important volatility measures for suppliers’ decisions. Parameter γ describes the
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degree to which suppliers care about volatility, and parameter α reflects the fraction of suppliers that are not impacted by the market volatility because of the forward contracts. The moderate category has parameters that are connected to the moments of the random variables.
13.5 Summary of Implications for Policy Makers In this section, we present the most influential interventions for the malaria medicine supply chain along with the most surprising results. Increasing the Number of Forward Contracts does not Have a Substantial Impact The A2S2 initiative was originally developed in 2009 with the intent to increase artemisinin supply so that we can satisfy the ACT demand. It offered a tripartite financing model in which WHOqualified extractors received financing at subsidized rates. The underlying premise behind the lowinterest capital lending was that it would encourage more forward contracts with farmers, yielding a higher artemisinin supply. The RBM/UNITAID/WHO (2011) report identified the impact to be 35% below the original estimations but could not to determine the reasons for this underperformance. After the report, this financing program was terminated. It is valuable to indicate that our estimation of the future contracts might be less than its real impact in the event that there are many suppliers who move in and out of the Artemisia annua growing efforts. Reduction in Demand Uncertainty does not Lead to a Substantial Impact We find that a better forecast leading to a reduction in the uncertainty pertaining to demand fluctuations does not lead to a substantial impact. This particularly holds true when the total budget for malaria medicines is constant and is known to all entities in the supply chain. Even if the budget is not fixed and the quantity purchased varies according to the price offered, better epidemiological forecast leading to reductions in demand uncertainty increases the overall supply only in smaller quantities. A Greater Amount of Semisynthetic Artemisinin Production has Moderate Impact, and the Agricultural and Semisynthetic Supply Sources Need to be Managed Carefully We know that a higher level of semisynthetic artemisinin production increases the total supply, improves the fill rate, and reduces price volatility. Van Noordan (2010) and Peplow (2013) provide an extensive debate on this topic. One would expect a substantial impact from an increase in semisynthetic artemisinin production. Our analysis shows, however, that a 50% increase in semisynthetic production results in only an 8% increase in supply and only a 5% increase in fill rate in both models M1 and M2. Moreover, the supplier surplus decreases by approximately 15%. We also observe some risk that overall artemisinin supply could decrease. It is important to mention that there are additional challenges associated with the increment in the semisynthetic artemisinin production. The pharmaceutical
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companies need to seek FDAtype approvals in multiple regions and this effort can be time consuming and costly. Increasing the Agricultural Crop Yield Makes a Significant Impact The increment in the average crop yield creates two positive outcomes. The output per hectare of land planted increases leading to a secondary impact associated with an increase in the farmer’s utility for producing artemisinin. However, when the average crop yield increases, there is also a dampening negative effect. The higher artemisinin output per hectare results in a larger supply, which in turn puts a pressure to reduce prices, leading to a lower farmer interest in growing Artemisia annua. Compared with other interventions, this negative feedback loop is substantially less impactful under M2. The result stems from the fact that the budget spent under M2 leads to larger reductions in price with improved efficiency. Changes in the agricultural methods employed can lead to an increase in the average crop yield. These include planting methods (less in impact) or in planting higher yield seed varieties (greater impact). Highyield seed varieties can also reduce the uncertainty in yield; however, climatic conditions such as rainfall and weather during growing conditions are substantially more influential in the realized crop yields. It is well known that in the years with excessive precipitation, farmers obtain lower yields of Artemisia annua. Dalrymple (2012) reports that highyield seed varieties can have a substantial impact in artemisinin production. However, field reports in Asia show that farmers continue to resist using these highyield seed varieties. The inertia to switch to the highyield seed varieties might be because of a successful history with the strain of Artemisia annua that is grown for a long period of time. The switch requires the encouragement and support of governments in countries like China and Vietnam. This is in addition to the transaction costs associated with switching to highyield seed varieties. Moreover, there is a chance that a larger yield can lead to a higher market price volatility, creating incentives for some farmers to exit and switch to other crops. A Price Support Makes a Significant Impact It is a wellknown fact that when the market price is too low, farmers do not grow the crop in question. A minimum supportprice, often provided by governments’ or donor organizations’ purchase agreements, provides a sufficient incentive mechanism for farmers to deviate from switching to another crop. A market intervention that is similar to a price support can increase the overall supply of agricultural artemisinin and reduce the associated price volatility. This is observed in M2: a $25 million increase in the purchasing budget leads to a 20% increase in supply. Let us compare this finding with the intervention corresponding to a price support intervention. A supportprice set at $360 also requires an average investment of $25 million; however, the overall supply of agriculturally obtained artemisinin supply increases by 30%. Moreover, this intervention decreases price volatility, specifically its coefficient of variation, by 60% in comparison to 7% associated with the budget increase. It is important to note that implementing a price support has its limitations in implementation. The price support needs to be determined in a sustainable manner
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to create the significant price stability: it should neither lead to excess supply with a high price support nor to being ineffective with lower values of price support. It is important to highlight that artemisinin is not a pure commodity and its price changes with differences in quality as in olive oil (AyvazÇavdaro˘glu et al. 2020). Thus, future research needs to develop a qualitydependent price schedule for an effective implementation of a price support. Other Highly Sensitive Interventions It would be beneficial to try to grow Artemisia annua in other regions of the world. It can be more challenging to influence the attractiveness of alternative crops, e.g., rice, corn, and soy, in order to encourage farmers to make the planting decisions.
13.6 Conclusions In this study, we examined the effects of farm space, manufacturer capacity, farmer’s incentive to plant Artemisia, volatility in Artemisia yield, supply of semisynthetic artemisinin, and demand uncertainty in the malaria medicine market. We determined the directional impact of various interventions on the total supply, fill rate, and price volatility of artemisinin. We test our model with numerical illustrations that use parameter and distribution estimations using field data. Our analysis identifies the most influential interventions that improve the efficiency in the malaria medicine supply chain. We show that a supportprice for agriculturally grown artemisinin, improvements in the average crop yield, and a larger managed supply of semisynthetic artemisinin have the greatest impact in improving the efficiency in the artemisininbased malaria medicine supply chain. It is important to be mindful of the fact that budget and resource constraints can dampen the usefulness of these interventions. Our study highlights the application of analytical tools in agricultural settings that can result in policy development in emergingcountry governments.
References A2S2: Assured Artemisinin Supply System (2012) Woerden. Available at http://www.a2s2.org/ Adeyi O, Atun R (2010) Universal access to malaria medicines: innovation in financing and delivery. Lancet 376(9755):1869–1871 Arrow KJ, Panosian C, Gelband H (eds) (2004) Saving lives, buying time: economics of malaria drugs in an age of resistance. Institute of Medicine, National Academies Press, Washington, DC AyvazÇavdaro˘glu N, Kazaz B, Webster S (2020) Agricultural cooperative pricing of premium product. Working Paper. Syracuse University, NY Blackburn J, Scudder G (2009) Supply chain strategies for perishable products: The case of fresh produce. Prod Oper Manage 18(2):129–137
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Chapter 14
The Impact of Crop Minimum Support Price on Crop Production and Farmer Welfare Prashant Chintapalli and Christopher S. Tang
14.1 Introduction In the agricultural sector, crop minimum support prices (MSPs) are subsidy schemes for governments to achieve two primary goals: (i) to protect farmers from fall in crop market prices and (ii) to improve availability of different essential crops. While there are different MSP schemes, we focus on the case when a government adopts creditbased (also known as deficiency payment) MSP scheme under which the government will not take any possession of the crop; instead, it will provide a monetary credit to the farmers cultivating the crop should the prevailing market price of the crop drop below the prespecified MSP.1
1 Besides creditbased MSP, a common MSP scheme is procurementbased MSP under which a government takes inventory possession by procuring the crop from farmers at the support price and then sells it back in the market later on (Ramaswami et al. 2018). The Indian government implements procurementbased MSPs for staple commodities such as rice. However, the procurementbased MSP scheme is costly to implement because of the logistics for handling the physical transactions, and it can cause the market price to drop when the government sells the crop back to the market. To circumvent this issue, the United States Department of Agriculture distributes those commodities procured under the procurementbased MSP scheme as foreign aid (Thompson 1993).
P. Chintapalli Indian Institute of Management Bangalore, Bengaluru, India email: [email protected] C. S. Tang () Anderson School of Management, University of California Los Angeles (UCLA), Los Angeles, CA, USA email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_14
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The creditbased MSP scheme is also known as the Brannan plan in the United States that was introduced in 1948 (Walton and Rockoff 2013). In India, this creditbased MSP scheme, known as the Bhavantar Bhugtan Yojana (or Price Deficit Financing Scheme), has been recently initiated by the government of Madhya Pradesh.2 Currently, this MSP scheme is implemented for eight crops (mostly oilseeds and legumes) that the Indian government takes no (or very low) possession of (Bera 2017). Relative to other MSP schemes that require the government to manage procurement and disposal of crops, the creditbased MSP scheme is easier to administer especially when the Indian government can leverage its mobile technology to disburse funds to farmers. In this chapter, we model, analyze, and evaluate the efficacy of the creditbased MSP scheme for achieving the intended goals in developing countries. To examine the impact of the creditbased MSP scheme on farmer welfare and the availability of two crops, we use a Stackelberg game theoretical framework to model the underlying dynamics in which the government acts as the leader by setting the MSPs for two different crops (that can be substitutes or complements from consumers’ perspective). Then, farmers (pricetakers) serve as followers who decide on their crop production quantities. We consider the case when market consists of two types of farmers (with heterogeneous production costs): myopic farmers (who make production decisions based on recent market prices) and strategic farmers (who make their decisions based on the anticipated current period’s crop prices after taking all the other farmers’ decisions into consideration). Although, in reality, a crop’s market price can be influenced by its yield uncertainty and changing consumers’ preferences, in our model we assume that crop’s yield uncertainty is negligible and consumers’ preferences do not vary drastically. As Guda et al. (2021) mention, the recent improvements in agricultural infrastructure and usage of high quality inputs along with hightech farming have largely reduced crop yield uncertainties in many developing countries. Moreover, the uncertainty in consumer preferences is low for staple crops like cereals. Also, suppressing these two uncertainties in our model will enable us to primarily focus on the impact of farmers’ decisions, which are driven by their nature (myopic or strategic) in the presence of governmental intervention, on their profits and welfare. We refer the reader to Chintapalli and Tang (2020a) and Chintapalli and Tang (2020b) for a discussion on crop yield and market uncertainties. Our analysis of the equilibrium outcomes yields the following four results: 1. MSP induces production: The total production quantity of a crop is always increasing in the MSP of the crop. 2. Low MSP can hurt farmers’ profits: Compared to the case of no MSP, farmers can be worse off by growing a crop when government sets the crop’s MSP low. 3. MSP can be beneficial: A carefully chosen MSP can result in a Pareto improvement of profits for all farmers (regardless of the crop they grow).
2 Chari (2017) provides the requisite steps for implementing the creditbased MSP scheme in India.
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4. MSP can improve farmer’s surplus: Farmers can attain a higher surplus when government offers MSPs for complementary crops instead of substitutable crops. While the first result is as expected, the second is somewhat unexpected, and the third and the fourth results are encouraging. Hence, compared to the case of no MSP, implementing MSPs can be effective in achieving their intended goals (i.e., farmer welfare and crop availability) only when the MSPs are carefully chosen. Therefore, these results sharpen our understanding about MSPs and they may serve as a reminder for policy makers. This chapter is organized as follows. We review the related literature in Sect. 14.2. We provide model preliminaries in Sect. 14.3 and characterize farmers’ equilibrium decisions in Sect. 14.4. In Sect. 14.4.3, we evaluate the impact of MSPs on farmers’ profits and surplus. We conclude the chapter and discuss interesting future research directions in Sect. 14.5.
14.2 Literature Review The agricultural economics literature on MSPs is vast and we refer the reader to Tripathi et al. (2013) for its comprehensive discussion. Without accounting for the price interactions between crops with MSP support and those without MSP support under procurementbased MSP scheme, Fox (1956) develops an economic model to evaluate the impact of MSPs and finds that MSPs can mitigate the fall in GNP during a recession. Dantwala (1967) finds that, despite the increasing MSPs, crop market prices continue to rise because procurementbased MSPs form a lower bound to market prices (Subbarao et al. 2011; Ramaswami et al. 2018). Chand (2003) presents a qualitative assessment of the illeffects of wheatandricecentric procurementbased MSPs on the Indian economy. Besides the Indian context, Spitze (1978) analyzes the impact of federal policy (The Food and Agriculture Act of 1977) on agriculture in the United States. Dean (1996) describes the Brannan plan (a creditbased MSP) and Innes (1990) examines the impact of Brannan plan on a nonstrategic monopolist farmer. This observation motivated us to examine the efficacy of the creditbased MSP scheme for achieving its intended goals (i.e., farmer welfare and crop availability) in developing countries, in which some farmers are strategic. Recent agricultural OM articles that examine social responsibility and public policy issues include Hu et al. (2019); Alizamir et al. (2019); Guda et al. (2021), and Ramaswami et al. (2018). Hu et al. (2019) focus on the value of strategic farmers and show that a tiny fraction of strategic farmers can stabilize the steady state crop prices. While they also extend their analysis to two crops with independent market prices, they do not consider the impact of MSPs. Alizamir et al. (2019) focus on the impact of federal policy on agriculture industry in the United States. They compare two schemes (Price Loss Coverage (PLC) and the Agriculture Risk Coverage (ARC) programs) with respect to (i) farmers’ welfare, (ii) federal expenditure, and (iii)
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consumer welfare. Recently, Guda et al. (2021) and Ramaswami et al. (2018) examine the role of procurementbased MSPs in emerging economies. All of them assume all farmers have same production cost. Besides the fact that all the aforementioned papers focus on procurementbased MSP scheme and our research studies creditbased MSPs, our work differs from the above papers in the following additional ways. First, we consider small pricetaking farmers with heterogeneous production costs and account for their production decisions and the strategic interactions among them (whereas Innes (1990) studies the case of a monopolist farmer). Incorporating production cost disparities among farmers is crucial because the cultivation cost of crops varies across farmers depending on the local soil, the climatic conditions, and the farming practices the farmers employ. Second, we consider a mixture of two types of farmers: myopic (backwardlooking) and strategic (forwardlooking) farmers. Strategic farmers’ decisions are dependent on market prices which are, in turn, influenced by the decisions of all the farmers. Our model fits well in the context of developing countries where a large portion of the farming communities are myopic: their crop selection and production decisions are purely based on the most recently observed market prices. Third, we analyze a multicrop setting, which Innes (1990) relegates to future research, and we examine the impact of the creditbased MSP scheme on crop prices and production quantities. We consider two crops that can be either substitutes or complements from consumers’ perspective. Hence, by capturing the interaction between two crops in our model, we analyze the simultaneous impact of the MSP of each crop on the production of both the crops, especially when a farmer’s production capacity is limited.
14.3 Model Preliminaries In the context of developing countries, we shall assume that farmers are infinitesimally small and each farmer is a pricetaker. For ease of exposition, we standardize the production capacity of each farmer to 1 unit and the size of the farmer population to 1 so that the total production capacity (for both crops) is scaled to 1. Furthermore, we assume that farmers are risk neutral. Heterogeneous Production Cost We consider two crops (A and B) that farmers can produce at unit costs cA and cB , respectively.3 To capture the production cost differential between the crops and the heterogeneous cost structure across farmers, we treat crop B as a stable crop so that its unit production cost is identical across all farmers and is denoted by c, while the cost of crop A varies among the farmers. We model the unit production cost cA (x) of crop A for a farmer who is located at
3 In our analysis we use the index variable k ∈ {A, B} to refer to a crop and use −k to refer to the crop other than k.
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x ∈ [a, b] as cA (x) = c + x where x ∼ U [a, b] and −c < a 0 b so that some farmers find it cheaper to grow A than B while others find it costlier. Market Price Because the government does not take possession of a crop under creditbased MSP scheme, the crop market prices will not be interfered by the MSPs directly. Therefore, the market price of a crop depends on the total quantity of the crop produced, which is available in the market for sale. By denoting qk as the “total” availability of crop k ∈ {A, B}, the market price of each crop is given by4 pA = ω − qA − β · qB
and
pB = ω − qB − β · qA ,
(14.1)
where β < 1 with β > 0 when crops A and B are substitutes, and β < 0 when they are complements, from the consumers’ perspective5 , and ω represents the market potential. To rule out the case when it is not profitable to grow the stable crop B, we shall assume that ω − 1 c to ensure that pB c. Myopic and Strategic Farmers Among all farmers in the market, a proportion θ ∈ [0, 1] of the farmers is strategic (denoted by S), while the remaining proportion (1 − θ ) is myopic (denoted by M). Myopic farmers are “backwardlooking” in the sense that they make their sowing decisions purely based on the most recent (realized) market prices (pA0 , pB0 ) (by treating them as the point estimates of future prices), while ignoring the actions of all other farmers. The existence of myopic farmers in emerging economies is justified by the farmers’ crop production patterns as explained in NITI Aayog (2016) and Hu et al. (2019). The strategic farmers, on the other hand, are “forwardlooking” in the sense that they make their sowing decisions based on the anticipated equilibrium future market prices, which depend on the actions of all the other farmers (both myopic and strategic). Creditbased MSPs In view of Eq. (14.1), it suffices to focus on the case when the creditbased MSPs mA and mB are bounded between (ω − 1) (the lowest market price of a crop) and (ω − β) (the highest market price of a crop). Hence, the case of no MSP corresponds to the case when we set mA = mB = (ω − 1). Sequence of Events The sequence of events that occurs during one planning cycle is depicted in Fig. 14.1.
4 As
mentioned earlier, we suppress yield and market uncertainties in this chapter in order to focus on how farmers’ decisions affect their welfare in the presence of MSPs (mA , mB ). Deterministic demand functions for commodity products are used in the operations management and economics literature, especially when the focus of analysis is on the impact of production decisions of producers on market prices (Basu 2011; Hu et al. 2019). 5 Note that the demand function in Eq. (14.1) can be obtained by a commonly used consumer utility model U (qA , qB ) = ω(qA + qB ) − (1/2) qA2 + qB2 + 2βqA qB where ∂ 2 U /∂qA ∂qB = −β gives the degree of substitutability (if β > 0) or complementarity (if β < 0) between A and B.
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Fig. 14.1 Sequence of events in a planning cycle
14.4 A Model of the CreditBased MSP Scheme By considering different decisionmaking processes adopted by myopic and strategic farmers as stated in the previous section, we now determine their crop selection and production decisions for any past realized market prices (pA0 , pB0 ) of crops A and B in the presence of the creditbased MSPs (mA , mB ).
14.4.1 Farmers’ Crop Selection and Production Decisions For any given MSPs (mA , mB ), a myopic farmer located at x ∈ [a, b] “believes” that the future crop prices are the same as the most recently realized crop market prices (pA0 , pB0 ). Hence, this farmer will sow crop A if, and only if, the anticipated profit from crop A is higher than that from crop B (i.e., max{pA0 , mA } − cA (x) max{pB0 , mB } − cB (x)), where pA0 and pB0 are known. However, because each strategic farmer “anticipates” that the future crop prices are based on the equilibrium crop prices (pA , pB ) that are determined by the sowing decisions of all farmers, a strategic farmer at x ∈ [a, b] will sow crop A if, and only if, the anticipated profit from crop A is higher than that from crop B (i.e., max{pA , mA } − cA (x) max{pB , mB } − cB (x)), where pA and pB are anticipated in equilibrium. These two observations enable us to show later (in Theorem 1) that, in equilibrium, each farmer of type v ∈ {M, S} will make his crop sowing decision according to a threshold value τ v where
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τ M = max{pA0 , mA } − max{pB0 , mB } and τ S = max{pA , mA } − max{pB , mB }.
(14.2)
Specifically, a type v ∈ {M, S} farmer located at x ∈ [a, b] will grow crop A if x τ v and will grow crop B if x > τ v , for v ∈ {M, S}. In view of this production plan and the fact that x ∼ U [a, b], the total quantity of crop k, k ∈ {A, B}, produced by type v, v ∈ {M, S}, is given by qkv , where qAM = Pr(x τ M ) and
qBM = Pr(x > τ M );
(14.3)
qAS = Pr(x τ S ) and
qBS = Pr(x > τ S ),
(14.4)
where τ v for v ∈ {M, S} is given in Eq. (14.2). By noting that θ proportion of farmers are strategic, the total quantities of crops A and B that are produced and available for sale satisfy: qA = θ qAS + (1 − θ )qAM
and
qB = θ qBS + (1 − θ )qBM .
(14.5)
The following lemma provides a condition ensuring that both crops will be produced by each type of farmer in equilibrium; i.e., qkv > 0 for k ∈ {A, B} and v ∈ {M, S}. Lemma 1 If (1 − β) < min{−a, b}, then the thresholds τ M and τ S given in Eq. (14.2) are interior points of [a, b] so that qkv > 0 for k ∈ {A, B} and v ∈ {M, S}. By noting that the production costs are given as cB (x) = c for all x and cA (x) = c + x with x ∼ U [a, b], and −c < a 0 b (by assumption), Lemma 1 states that when the variance in crop A’s cultivation cost is high (i.e., a < −(1 − β) < 0 < (1 − β) < b), then both crops are always produced by both myopic and strategic farmer segments.6 To rule out the trite case when only one crop is produced and instead to focus on the more practical case, we assume that −c < a < −(1 − β) < 0 < (1 − β) < b, so that both crops will be produced in equilibrium. Hence, Eqs. (14.3) and (14.4) can be rewritten as qAM = Pr(x τ M ) =
τM − a b−a
and
qBM = Pr(x > τ M ) =
qAS = Pr(x τ S ) =
τS − a b−a
and
qBS = Pr(x > τ S ) =
6 Note
b − τM ; b−a
b − τS , b−a
(14.6) (14.7)
that in the case when a > 0, i.e., A is costlier to produce than B for all the farmers, then no myopic farmer will produce crop A if max{pA0 , mA } − max{pB0 , mB } < a. On the other hand, the behavior of strategic farmers is more complicated and is solely determined by the MSPs and the proportion of strategic farmers θ. In this chapter, we focus on the more interesting case when −c < a < −(1 − β) < 0 < (1 − β) < b to ensure interior equilibrium and to draw insights. However, our model can be extended to the case of 0 < a < b with some additional analysis and by considering the boundary equilibria.
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where τ v is given in Eq. (14.2) for v ∈ {M, S}. By considering the crop selection and production decisions of each myopic farmer and each strategic farmer in equilibrium, we can derive the expressions for the thresholds τ M and τ S as follows: Theorem 1 For any given MSPs (mA , mB ), each farmer located at x ∈ [a, b] will grow crop A in equilibrium if x τ v and will grow crop B in equilibrium if x > τ v for v ∈ {M, S}, where the thresholds τ M and τ S and the “aggregate” threshold τ are unique and satisfy τ M = max{mA , pA0 } − max{mB , pB0 },
(14.8)
τ S = max{mA , pA (τ )} − max{mB , pB (τ )},
and
τ = θ τ S + (1 − θ )τ M .
(14.9) (14.10)
Also, in equilibrium, the total quantity of each crop available for sale satisfies qA (τ ) =
τ −a b−a
and
qB (τ ) =
b−τ . b−a
(14.11)
Moreover, the equilibrium market prices satisfy b−τ and b−a τ −a . pB (τ ) = ω − qB (τ ) − βqA (τ ) = ω − 1 + (1 − β) b−a pA (τ ) = ω − qA (τ ) − βqB (τ ) = ω − 1 + (1 − β)
(14.12) (14.13)
Theorem 1 states that, in equilibrium, the production decision of each myopic farmer depends on the threshold τ M and the production decision of each strategic farmer depends on the threshold τ S (via τ ). Thus, in equilibrium, each riskneutral farmer grows exactly one of the two crops, A or B. We refer the reader to Chintapalli and Tang (2020b) for the case when farmers are risk averse in the presence of market and yield uncertainties. Also, observe from Eqs. (14.11), (14.12), and (14.13) that the production quantities qA (τ ) and qB (τ ), and the market prices pA (τ ) and pB (τ ), depend on τ . By combining these observations along with Eqs. (14.10), (14.8), and (14.9), we can conclude that the farmers’ welfare depends on τ , which is an implicit function of (mA , mB ; pA0 , pB0 ). Hence, before examining the impact of MSPs on crop production and farmer welfare, we first study the impact of MSPs on τ next.
14.4.2 Impact of MSPs on Farmers’ Production Decisions First, observe from Eq. (14.8) that the value of the threshold τ M for myopic farmers depends on the relative value of mk and pk0 for k ∈ {A, B}. Hence, from the
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perspective of myopic farmers, an MSP mk is effective if max{mk , pk0 } = mk (and ineffective if mk < pk0 ). It follows from Eq. (14.8) that ∂τ M /∂mk 0 if k = A and ∂τ M /∂mk 0 if k = B, which indicates that the total production of a crop by myopic farmers is increasing in the crop’s MSP. Second, observe from Eq. (14.9) that the value of the threshold τ S is more intricate because it depends on the relative value of mk and pk (τ ) for k ∈ {A, B}, where τ is an implicit function of (mA , mB ; pA0 , pB0 ). Even so, we can establish the relative value of mk and pk (τ ) in the following lemma that will prove useful: Lemma 2 The relative value of mk and pk (τ ) for k ∈ {A, B} can be described as follows. For any given MSP m−k of crop −k, there exists a unique value m ˆ k (m−k ) such that mk pk (τ ) if, and only if, mk m ˆ k (m−k ).7 From the perspective of strategic farmers, MSP mk is termed as high if mk m ˆ k (m−k ) (so that max{mk , pk (τ )} = mk as noted in the above lemma) and termed as low if mk < m ˆ k (m−k ) (so that max{mk , pk (τ )} = pk (τ )). (The second row of Table 14.1 summarizes the definition of low and high MSPs from the perspective of strategic farmers.) By using the notion of high and low MSPs from the perspective of strategic farmers, we can divide MSPs (mA , mB ) into four regions as shown in Fig. 14.2, where region I corresponds to the case when mA and mB are both high, region II corresponds to the case when mA is high and yet mB is low, and so forth. Figure 14.2 illustrates the graphs of m ˆ k (m−k ), k ∈ {A, B}. By combining these observations with Eq. (14.9), we notice that ∂τ S /∂mk depends on whether mk , k ∈ {A, B}, is high or low from the strategic farmers’ perspective. To elaborate, let us consider the case when mA is high and yet mB is low (i.e., region II) so that τ S = mA − pB (τ ) as observed from Eq. (14.9). Then, by considering pB (τ ) given in Eq. (14.13), it is easy to check that ∂τ S 1 − β ∂τ =1− . ∂mA b − a ∂mA At the same time, observe from Eq. (14.10) that
Table 14.1 Sensitivity of τ with respect to the MSPs mA and mB MSP influence Farmer’s perspective Equilibrium is price dominated Equilibrium is MSP dominated Myopic farmer’s perspective MSP mk is ineffective (i.e., mk < MSP mk is effective (i.e., mk pk0 so that max{mk , pk0 } = pk0 ) pk0 so that max{mk , pk0 } = mk ) Strategic farmer’s perspective MSP mk is low (i.e., mk < MSP mk is high (i.e., mk pk (τ ) so that max{mk , pk (τ )} = pk (τ ) so that max{mk , pk (τ )} = pk (τ )) mk )
refer the reader to Chintapalli and Tang (2018) for further details on the functions m ˆ k (m−k ), k ∈ {A, B}.
7 We
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Fig. 14.2 The regions of low and high MSPs
∂τ M ∂τ ∂τ S ∂τ M 1 − β ∂τ + (1 − θ ) =θ + (1 − θ ) =θ 1− . ∂mA ∂mA ∂mA b − a ∂mA ∂mA By rearranging the terms, we get
∂τ = ∂mA
∂τ M ∂mA . θ (1 − β) 1+ (b − a)
θ + (1 − θ ) ·
Using it we can retrieve the expression for ∂τ S /∂mA . By using the same approach for different combinations and different cases of high and low MSPs, we obtain the following results: Corollary 1 The impact of MSPs (mA , mB ) on the aggregate threshold τ and the impact of MSPs (mA , mB ) on the threshold τ S for different regions are given in Table 14.2, where each region corresponds to the case in which MSP mk is high or low for k ∈ {A, B}. Also, the total availability qk of a crop k ∈ {A, B} is always increasing in the MSP mk of crop k but decreasing in the MSP m−k of the other crop. Corollary 1 and Eq. (14.11) highlight the fact that MSP of crop k ∈ {A, B} provides an incentive for farmers to produce a higher total quantity of that crop on an aggregate level; i.e., ∂qk /∂mk 0, which results in a decrease of the market price of crop k, as observed from Eq. (14.1) and the fact that β < 1. Additionally, it can be shown that although for any MSP m−k the total production quantity qkM of
( 0)
1+
2θ (1−β) b−a
1 + θ (1−β) b−a ∂τ M (1 − θ) · ∂m B
−θ + (1 − θ) ·
∂τ M (1 − θ) · ∂m B 1 + θ (1−β) b−a
−θ + (1 − θ) ·
∂τ ∂mB
I: mA is high, and mB is high so that mA pA (τ ), mB pB (τ ) II: mA is high, and mB is low so that mA pA (τ ), mB < pB (τ ) III: mA is low, and mB is high so that mA < pA (τ ), mB pB (τ ) IV: mA is low, and mB is low so that mA < pA (τ ), mB < pB (τ )
IV
III
II
∂τ M ∂mA
∂τ M θ + (1 − θ) · ∂m A 1 + θ (1−β) b−a ∂τ M (1 − θ) · ∂m A 1 + θ (1−β) b−a ∂τ M (1 − θ) · ∂m A (1−β) 1 + 2θb−a
θ + (1 − θ) ·
I
0)
∂τ ∂mA (
MSP Region
Sensitivities
Table 14.2 Sensitivity of τ with respect to the MSPs mA and mB
∂τ M ∂mB
∂τ M ∂mB
− −
1 − β ∂τ b − a ∂mA 2(1 − β) ∂τ b − a ∂mA
−
−
(1−θ )(1−β) ∂τ M b−a ∂mB 1 + θ (1−β) b−a
2(1 − β) ∂τ b − a ∂mB
1+
1 − β ∂τ b − a ∂mB
−1
∂τ S ∂mB
−
)(1−β) ∂τ M 1 − (1−θb−a ∂mA 1 + θ (1−β) b−a
1
∂τ S ∂mA
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crop k produced by myopic farmers is increasing in mk , the total production quantity qkS of crop k produced by strategic farmers is increasing in mk if, and only if, mk is high. Thus, when mk is low but effective, strategic farmers try to counter the actions of myopic farmers who produce more of crop k when mk is increased. We request the reader to refer to Chintapalli and Tang (2018) for details.
14.4.3 Impact of MSPs on Farmers’ Profits and Surplus First, we define the profit functions of individual farmers and the total farmer’s surplus in equilibrium. For any MSPs (mA , mB ), a farmer of type v ∈ {M, S} who grows crop k in equilibrium earns a revenue of max{mk , pk (τ )}, where pk (τ ) for k ∈ {A, B} is as given in Eqs. (14.12) and (14.13). Accounting for the production costs cA (x) = c + x and cB (x) = c for x ∈ [a, b], the profit of a farmer of type v ∈ {M, S} and who is located at x ∈ [a, b] is given by ⎧ ⎪ ⎪ ⎪max{mA , pA (τ )} − cA (x) ⎪ ⎨ = max{m , p (τ )} − (c + x) A A π v (xmA , mB ) = ⎪max{mB , pB (τ )} − cB (x) ⎪ ⎪ ⎪ ⎩ = max{mB , pB (τ )} − c
if x τ v
(14.14)
if x > τ v .
Notice that, even though the production decisions of myopic farmers depend on τ M that hinges on the recent prices pk0 , k ∈ {A, B}, their profits for growing a crop depend on the equilibrium prices pk (τ ) but not the recent prices. By aggregating the individual profits of all the farmers of type v ∈ {M, S}, we b obtain the total surplus of type v farmers as Fv (mA , mB ) = a π v (xmA , mB ) · (b − a)−1 dx, where π v (xmA , mB ) is given by Eq. (14.14). Then, the aggregate farmer surplus is given by F(mA , mB ) = θ · FS (mA , mB ) + (1 − θ ) · FM (mA , mB ).
(14.15)
In fact, it can be easily shown that, for any MSPs (mA , mB ), offering MSPs to more complementary crops improves farmer surplus (see Lemma 6 in Chintapalli and Tang 2018). Next, for any given MSP m−k , by differentiating Eqs. (14.14) and (14.15) with respect to mk and by rearranging the terms, we obtain ∂ π v (xmA , mB ) ∂mk
14 Impact of Minimum Support Prices in the Presence of Strategic Farmers
⎧ ⎧ ⎨ ∂mA ⎪ ⎪ ∂ ⎪ ∂mk ⎪ max{m , p (τ )} = ⎪ A A ⎪ ⎨ ∂mk ⎩ ∂pA (τ ) ⎧ ∂mk = ⎪ ⎨ ∂mB ⎪ ∂ ⎪ ∂mk ⎪ max{m , p (τ )} = ⎪ B B ⎪ ⎩ ∂mk ⎩ ∂pB (τ ) ∂mk
259
if x < τ v , mA m ˆ A (mB ), if x < τ v , mA < m ˆ A (mB ), if x > τ v , mB m ˆ B (mA ), if x > τ v , mB < m ˆ B (mA ), (14.16)
and ∂F ∂ ∂ τ −a b−τ · · = max{mA , pA (τ )} + max{mB , pB (τ )} ∂mk b − a ∂mk b − a ∂mk +
(1 − θ )(τ S − τ M ) ∂τ M , · b−a ∂mk
(14.17)
for k ∈ {A, B} and v ∈ {M, S}8 . We obtain Eq. 14.17 by using Eqs. 14.9 and (14.10). Observe from Eqs. (14.16) and (14.17) that the impact of MSPs on a type v farmer’s profit π v (xmA , mB ) and on the aggregate farmer’s surplus F(mA , mB ) hinges on ∂pk (τ )/∂mk , for k ∈ {A, B}. Also, recall from Eqs. (14.12) and (14.13) that the equilibrium prices are linear functions of the aggregate threshold τ . On combining these observations with the impact of MSPs on the thresholds τ and τ s as presented in Corollary 1 (via Table 14.2), we can fully determine the MSP effects on individual farmer’s profits and aggregate farmer’s surplus (i.e., ∂π v (xmA , mB )/∂mk and ∂F/∂mk as stated in Eqs. (14.16) and (14.17)) by specifying the value ∂τ M /∂mk for k ∈ {A, B}. However, this derivative depends on whether mA and mB are effective (i.e., whether mk ≥ pk0 , for k ∈ {A, B}) or not. Therefore, by combining the four cases stated in Table 14.1 along with the four combinations of effective and ineffective MSPs (from the perspective of myopic farmers), we can further refine those four regions of (mA , mB ) that are shown in Fig. 14.2 into 16 zones. As it turns out that some zones are redundant (see Chintapalli and Tang (2018) for details), there are only 10 zones to be examined and these 10 zones are shown in Fig. 14.3. For example, region I where both the MSPs are high as shown in Fig. 14.2 is now divided into 2 zones Z7 and Z10 in Fig. 14.3, where Z7 corresponds to the zone in which both MSPs are high from the
one should note that for a given farmer segment v ∈ {M, S}, the value of τ v changes if mA is increased or decreased, due to the continuity of τ v . Hence, there will be two categories of farmers when the MSP of crop A is increased by δ(> 0) from mA to mA + δ: (i) those farmers who grow the same crop before and after the increase in mA , and (ii) those who switch the crop they grow, based on the perceived profits from the crops, in the manner as discussed in Sect. 14.4.1. However, it should be noted that since δ → 0, by the definition of a derivative, the measure (or mass) of farmers who switch between the crops is 0, because limδ→0 τ v (mA + δ, mB ) = τ v (mA , mB ); so, the definitions of the sensitivities are as given in Eqs. 14.16 and (14.17).
8 Technically,
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Fig. 14.3 The regions of effective, ineffective, low, and high MSPs
strategic farmer’s perspective and both MSPs are effective from the myopic farmer’s perspective. Likewise, in zone Z10 although both MSPs are high from the strategic farmer’s perspective, MSP mA is effective because mA pA0 , but MSP mB is ineffective because mB < pB0 . We can interpret all remaining 8 zones in a similar manner.9 For each of these 10 zones as shown in Fig. 14.3, we can specify the value ∂τ M /∂mk for k ∈ {A, B}. Also, we can use this value along with the impact of MSPs on the thresholds τ and τ s as presented in Corollary 1 (via Table 14.2) to determine the impact of MSPs on a type v farmer through ∂π v (xmA , mB )/∂mk as stated in Eq. (14.16) and on the farmer’s surplus through ∂F/∂mk as stated in Eq. (14.17). This impact is summarized in Table 14.3.10 We make three interesting observations from Table 14.3: 1. MSP for one crop can benefit farmers who grow the other crop. Observe from Columns 3 and 4 of Table 14.3 that, for v ∈ {M, S}, ∂π v (x)/∂mA > 0 for farmers who are located at x > τ v (i.e., who grow B in equilibrium) and ∂π v (x)/∂mB > 0 for farmers who are located at x < τ v (i.e., who grow A in equilibrium). This crossbenefit is because a higher MSP for crop k will lower the production quantity of crop −k (see Corollary 1) so that farmers who grow crop −k will earn a higher profit due the crop’s higher market price.
9 We
show in Chintapalli and Tang (2018) that it suffices to address the case shown in Fig. 14.3. The other case will be similar to the case illustrated in Fig. 14.3 with A and B interchanged. 10 We provide the detailed derivations and other technical details of Table 14.3 in the Appendix of Chintapalli and Tang (2018).
Grow B x > τv
0 0 0 >0 >0 >0 0 0
0
0
Grow A x < τv
0 0 0 0 0 0 0 0 0 >0
Grow A x < τv
Sign of
1
1
1 1 0 0 0 τv
∂π v (x) ∂mB ∂F ∂mA
0 0 0 >0 >0 >0 >0 < 0 if mA → m ˆ A (mB ) if mA → m ˆ A (mB ) 0
Sign of
∂F ∂mB
>0
>0
>0 >0 0 0 0 < 0 if mB → m ˆ B (mA ) >0 >0
Sign of
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2. An increase in MSP for a crop can hurt its farmers. Observe from Columns 2 and 5 of the table that, when a government offers a crop MSP that is low but effective, it can hurt the farmers who grow the corresponding crop. For example, if a government increases MSP mA within zones 4, 8, and 9, the farmers who grow crop A can be worse off. (Similarly, if a government increases MSP mB in zone 6, farmers who grow crop B can be worse off.) The explanation for this counterintuitive result is as follows. When an MSP is low (i.e., mk < m ˆ k (m−k ) for k ∈ {A, B}) and effective (i.e., pk0 < mk for k ∈ {A, B}), it creates unwarranted frenzy among myopic farmers that causes an excess production of the crop, which cannot be outweighed by the countering actions of strategic farmers. This excessive production of a crop reduces its equilibrium market price, which hurts the revenues of all the farmers growing the corresponding crop. Recall that when MSP mk is low (i.e., mk < m ˆ k (m−k ) so that mk < pk (τ ), the revenue from crop k ∈ {A, B} is given by max{mk , pk (τ )} = pk (τ ), which decreases according to Corollary 1.11 3. Increasing MSP does not always improve farmer welfare. Observe from Columns 6 and 7 that increasing MSP for a crop may not improve the aggregate farmer’s surplus F. For example, some farmers are substantially worse off if government offers a higher MSP mA in zones 8 and 9 or offers a higher MSP mB in zone 6. From Table 14.3 we can observe that ∂F/∂mA < 0 in zones Z8 and Z9 when mA is high (i.e., mA ≈ m ˆ A (mB )) and ∂F/∂mB < 0 in zones Z6 if mB is high (i.e., mB ≈ m ˆ B (mA )). However, offering a high mA always improves farmer’s surplus in zone Z4 because an increase in mA increases the profits of the farmers growing crop B (according to Observation 1 made above), which outweighs the degradation in the profits of the farmers who grow crop A.
14.5 Conclusions In this chapter, we examined the efficacy of creditbased (or deficiency payment) MSPs in the context of emerging economies where farming communities primarily comprise smallscale farmers who are pricetakers. We formulated the problem as a Stackelberg game in which government is the leader and farmers are the followers. Government announces crop MSPs first to which farmers respond through their crop sowing decisions. We considered that a fraction of farmers are strategic and the remaining are myopic. We found that, even though the production of a crop by myopic farmers is increasing in the MSP of the corresponding crop, it need not be true for strategic farmers. Specifically, we showed that the production of a crop by strategic farmers is increasing in the crop’s MSP if, and only if, the MSP is high and effective.
11 This
is akin to the tragedy of the commons where rational actions of individual players lead to an overall societal loss.
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However, we found that, despite this nonmonotonic behavior of strategic farmers in the increasing MSP of a crop, the total production of a crop is always increasing in its MSP. Next, we found that setting MSP of a crop low but effective can be detrimental to the farmers who grow the corresponding crop. Such a low but effective crop MSP will create an unwarranted frenzy among myopic farmers, who increase their production of the crop by a disproportionate amount. This increase in the crop’s production reduces the equilibrium market price of the corresponding crop, thereby hurting all the farmers who cultivate the crop. Hence, care should be taken when choosing a crop’s MSP when the objective is to increase the individual profits of all the farmers who grow the crop. Our research has the following limitations. First, we assumed a closed economy in which there are no imports and exports. Incorporating an open economy will change the dynamics of the problem largely. Second, we did not analyze the impact of procurementbased MSPs in the presence of strategic farmers. Moreover, the problem gets complicated if there is inventory carried over time in a procurementbased MSP scheme. Finally, besides farmers’ decisions that are captured in our model, yield uncertainty is another important factor that could affect the dynamic choice of MSPs. We refer the reader to Chintapalli and Tang (2020a) and Chintapalli and Tang (2020b) for the cases of market and yield uncertainties. The MSP problem becomes very intricate when there is inventory carryover along with yield uncertainty. These limitations provide interesting directions for future research.
References Alizamir S, Iravani F, Mamani H (2019) An analysis of price vs. revenue protection: Government subsidies in the agriculture industry. Manage Sci 65(1):32–49 Basu K (2011) India’s foodgrains policy: An economic theory perspective. Econ Polit Week 46(5):37–45 Bera S (2017) Madhya Pradesh launches new farm scheme to hedge price risks in farming. https://www.livemint.com/Politics/uDdclMv4VKUhGvpSEqtmqL/MadhyaPradeshlaunchesnewfarmschemetohedgepricerisks.html. Accessed 13 April 2018 Chand R (2003) Minimum support price in agriculture: changing requirements. Econ Polit Week 38(29):3027–3028 Chari M (2017) How to support farmers without buying their crops – Madhya Pradesh launches bold new experiment. https://scroll.in/article/854202/howtosupportfarmerswithoutbuyingtheircropsmadhyapradeshlaunchesboldnewexperiment. Accessed 13 April 2018 Chintapalli P, Tang CS (2018) The impact of crop minimum support prices on crop production and farmer welfare. Available at SSRN 3262407 Chintapalli P, Tang CS (2020a) Crop minimum support price versus cost subsidy: farmer and consumer welfare. Working paper Chintapalli P, Tang CS (2020b) The value and cost of crop minimum support price: Farmer and consumer welfare and implementation cost. Manage Sci (forthcoming) Dantwala M (1967) Incentives and disincentives in Indian agriculture. Indian J Agric Econ 22(2):1–26 Dean VW (1996) Why not the Brannan plan? Agric Hist 70(2):268–282
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Fox KA (1956) The contribution of farm price support programs to general economic stability. In: Policies to combat depression, NBER, pp 295–356 Guda H, Rajapakshe T, Dawande M, Janakiraman G (2021) An economic analysis of agricultural support prices in developing economies. Prod Oper Manage (forthcoming) Hu M, Liu Y, Wang W (2019) Socially beneficial rationality: The value of strategic farmers, social entrepreneurs, and forprofit firms in crop planting decisions. Manage Sci 65(8):3654–3672 Innes R (1990) Government target price intervention in economies with incomplete markets. Quart J Econ 105(4):1035–1052 NITI Aayog (2016) Evaluation study on efficacy of minimum support prices (MSP) on farmers. http://niti.gov.in/writereaddata/files/writereaddata/files/document_publication/MSPreport.pdf Ramaswami B, Seshadri S, Subramanian KV (2018) The welfare economics of storagebased price supports. Working paper Spitze R (1978) The food and agriculture act of 1977: issues and decisions. Am J Agric Econ 60(2):225–235 Subbarao D, et al (2011) The challenge of food inflation. Tech. rep. Annual Conference of the Indian Society of Agricultural Marketing, Hyderabad, 22 November 2011 Thompson RL (1993) Agricultural price supports. Concise encyclopedia of economics. https:// www.econlib.org/library/Enc1/AgriculturalPriceSupports.html Tripathi AK, et al (2013) Agricultural price policy, output, and farm profitability—examining linkages during postreform period in India. Asian J Agric Dev 10(1):91–111 Walton GM, Rockoff H (2013) History of the American economy. 12th edn. Cengage Learning, Boston, MA
Chapter 15
Input vs. OutputBased Farm Subsidies in Developing Economies: Farmer Welfare and Income Inequality Christopher S. Tang, Yulan Wang, and Ming Zhao
15.1 Introduction Farmer poverty is a serious concern in developing countries. In India, 50% of workforce participates in the agriculture sector, but the average income of an Indian farmer is below US$5 per day (Sodhi and Tang 2014). Also, more than 40 million smallscale farmers in China live below the national poverty line (Zhao 2018). Two major obstacles in these regions are high input purchase costs and output processing costs. First, due to low income and high distribution costs in developing countries, quality fertilizers and seeds are often not affordable (Kwa 2011; Mare et al. 2010). Without affordable inputs, farmer income in India has declined over the years (Jayan 2017). Second, due to poor infrastructure (roads, storage facilities, etc.) in developing countries, farmers incur high output processing (e.g., harvest, transportation, postharvest handling, etc.) costs. Gedaref (2017) reported that the harvest cost increased by four times in Sudan, which led to significant losses among the farmers.
C. S. Tang () Anderson School of Management, University of California Los Angeles (UCLA), Los Angeles, CA, USA email: [email protected] Y. Wang Faculty of Business, The Hong Kong Polytechnic University, Hong Kong, Hong Kong email: [email protected] M. Zhao School of Economics and Management, University of Electronic Science and Technology of China, Chengdu, China email: [email protected] © Springer Nature Switzerland AG 2022 O. Boyabatlı et al. (eds.), Agricultural Supply Chain Management Research, Springer Series in Supply Chain Management 12, https://doi.org/10.1007/9783030814236_15
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The above observations suggest that it is important to help farmers to reduce their input purchasing costs of seeds and fertilizers or/and output processing costs of transportation and postharvest handling Sodhi and Tang (2011). Two commonly observed farm subsidy programs in developing countries are either input or outputbased subsidy schemes. The inputbased subsidies reduce the purchase cost of certain inputs such as fertilizers and seeds. For example, the governments in Mali, Ghana, and Nigeria provide fertilizers to smallholder farmers at a discounted price (Jayne and Rashid 2013, and Wiggins and Brooks 2010). In India, the government deposits appropriate subsidies into farmers’ bank accounts after they purchased the seeds at market price (Prasad 2016). Unlike the inputbased scheme, the outputbased subsidy scheme is intended to reduce different output processing costs. For example, the Indian government offers the transportation subsidy to defray farmers’ transportation cost of their outputs (Roy 2018), while Thai rice farmers receive a storage cost subsidy of 1500 baht per metric ton (GAIN 2017). Governments in other developing countries such as China, Brazil, Ukraine, and Turkey also implement various outputbased subsidy programs (OECD 2009). Despite the popularity of various input and outputbased subsidy schemes, their impact on farmer welfare, productivity, and income gap are not well understood, especially when farmers have different yield rates due to different farming environments: access to irrigation water, soil composition, farming experience, etc. (World Bank 2012). The existing empirical evidence about the implications of these two schemes have been mixed. Some advocates argue that inputbased subsidies can reduce the income inequality (Darko 2015) and alleviate poverty (Dorward and Chirwa 2011), but Blar et al. (1993) argue the opposite. Regarding outputbased subsidies, Chinyamakobvu (2012) argues that outputbased subsidies can increase productivity, while Ahmed (2011) shows that outputbased subsidies for transportation can be detrimental to farmers due to intensified competition. To gain a better understanding about the implications of the input and outputbased subsidy schemes in terms of farmer welfare, productivity, and income gap, we develop a twostage model in which the government first determines the subsidy scheme (i.e., inputbased or outputbased) and the corresponding subsidy level with an aim to maximize farmers’ welfare. Given the subsidy program, different farmers with different yield rates determine their planting quantities as they engage in quantity competition.1 We first analyze the case when farmers’ different yield rates are deterministic. Our equilibrium analysis reveals that both input and outputbased subsidy schemes can improve the income of each farmer. However, we note these two schemes generate different effects: 1. The inputbased subsidy narrows the income gap between the farmers, but the outputbased subsidy widens this gap. 2. The outputbased scheme outperforms the inputbased subsidy scheme in terms of total farmer income as well as farmer productivity. 1 The
Cournot quantity competition is an appropriate representation of various agricultural product markets such as malting barley and banana (Deodhar and Sheldon 1996; Dong et al. 2006).
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A further comparison of equilibrium outcomes reveals that high and lowyield farmers hold opposite preferences: lowyield farmers prefer inputbased subsidies while highyield farmers prefer outputbased subsidies. Therefore, the selection of a particular subsidy scheme will depend on the government’s ultimate goal. We also consider two extensions. First, we study a “combined” subsidy scheme under which the government offers both input and outputbased subsidies (instead of one of them). From the perspective of the farmers’ total income, we find that input and outputbased subsidies are “complementary.” However, from the perspective of the income gap between farmers, these two schemes are “substitutes.” Also, we find that it is sufficient to offer only one of the subsidy schemes but not both. Second, we examine the case when farmers’ yield rates are uncertain. We show that our key results obtained for the deterministic yield case continue to hold. We also find that reducing the yield uncertainty can improve the total farmer income but at the expense of widening the income gap. This chapter is organized as follows. Section 15.2 briefly reviews the relevant literature. Section 15.3 describes the base model. The equilibrium analysis associated with these two subsidy schemes is presented in Sect. 15.4. In Sect. 15.5, we compare the performance of the two subsidy schemes. We extend our base model in Sect. 15.6. Concluding remarks are provided in Sect. 15.7.
15.2 Literature Review Our study belongs to an emerging research stream that deals with socially responsible operations in developing countries. Various researchers examine different mechanisms for alleviating farmer poverty. Due to poor infrastructure for transportation and information access, farmers face high input purchasing and output processing costs and lack proper information to make their selling decisions. To overcome these challenges, Sodhi and Tang (2014) propose direct purchase from and disseminating market information to farmers. An et al. (2015) examine the implications of farm cooperatives that can help affiliated farmers to reduce purchase costs (via aggregation), improve the process yield (via mutual learning), and reduce selling costs (through disintermediation). More recently, various researchers examine the value of market information as well as farming advisory information. Chen et al. (2013) discover an economic incentive for ITC to provide advisory information to farmers even though these farmers are outside the ITC network. In a similar vein, Chen et al. (2015) examine the strategic behavior for expert farmers who can decide on the knowledge level to share with other farmers over a support hotline in India. Instead of providing private information to farmers through ITC network or hotline, Chen and Tang (2015) examine whether the government should share public information with farmers who may be endowed with private knowledge. They find that in the presence of private knowledge, providing public information can be detrimental to farmers due to the unintended herd effect. Tang et al. (2015) study whether farmers should
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adopt market information or agricultural advice when making production planning decisions. They show that the market information is beneficial to farmers; however, the value of the agricultural advice depends on the upfront investment cost. Liao et al. (2019) examine the case when farmers can select the crop to grow and determine the market to sell in. They show that information provision may not improve the farmers’ total welfare. He et al. (2018) examine the farmers’ incentive to join the informational coalitions in the presence of both private and public market information. Recently, some researchers begin to investigate different agricultural supply chains. For example, Hsu et al. (2017) compare three business models in a milk supply chain consisting of a social enterprise and dairy farmers. They find that the social enterprise prefers the partnership model over both the conventional decentralized model and the independent integrated model when the market size of dairy products is intermediate. Hu et al. (2017) study the impact of strategic farmers on the price fluctuation in agricultural markets. They find that the strategic farmers’ selfinterest can reduce price volatility and benefit all farmers. Zhang and Swaminathan (2018) construct a finitehorizon stochastic dynamic program to investigate smallholder farmers’ optimal seeding policy under rainfall uncertainty. They show that each farmer should plant only when the seed amount is larger than a threshold. They find that the adoption of the optimal time dependent thresholdtype planting schedule could help mitigate the risk of yield drop due to severe climate conditions. Unlike the aforementioned studies, here we focus on the agricultural subsidies. In this research stream, Akkaya et al. (2016a) investigate the impact of the government interventions such as tax reductions and farm subsidies on the farmers’ adoption of sustainable farming practice. In a similar vein, Akkaya et al. (2016b) consider the impact of three types of government interventions—price support, cost support, and yield enhancement—on farmer income, consumer surplus, and government spending. They find that when the total budget is public information, the price support and the cost support yield the same performance. Gida et al. (2016) examine a guaranteed support price scheme under which the government purchases the crop at the predetermined price and sells them to the poor customers at a discounted price. Alizamir et al. (2019) compare the price loss coverage (PLC) program (i.e., subsidizing farmers when the market price is below a certain threshold) with the agriculture risk coverage (ARC) program (i.e., subsidizing farmers when farmers’ revenue is below a certain threshold). They find that the PLC program dominates the ARC program in terms of the farmer revenue, the consumer surplus, and the government’s cost for a large range of parameter values. Different from the abovementioned studies, we make an initial attempt to examine the implications of the input and outputbased subsidy schemes on the farmer income and income inequality.
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15.3 Model Preliminaries We now present a parsimonious model that captures the underlying issues of our study. Consider a situation when two farmers (or cooperatives) who grow a single commodity crop and sell it in a local market.2 Due to the heterogeneity in the endowed resources (water sources, soil quality, farming knowledge), we shall consider the case when the yield rate zi is farmerspecific. (In the base model, we assume that zi is deterministic but we shall extend our analysis in Sect. 15.6 to deal with yield uncertainty.) Without loss of generality, we shall assume that 0 < z1 < z2 < 1. For ease of reference, we shall refer to farmer 1 (farmer 2, respectively) as the lowyield (highyield, respectively) farmer.
15.3.1 Farmers’ Planning Problem At the beginning of a planting season, each farmer i, i = 1, 2, decides on his (input) planting quantity qi , which will generate an output quantity zi qi . Hence, the total sales quantity during the harvest season is equal to 2i=1 zi qi . For a commodity crop, it is reasonable to assume that farmers engage in Cournot competition so that the market price P satisfies3 P = m − z1 q1 − z2 q2 ,
(15.1)
where m represents the market potential. We assume that both farmers incur identical “input” unit purchasing cost c that covers the cost of seeds, fertilizers, and labor. We also assume that both farmers incur identical “output” unit processing cost t that covers the transportation and storage costs. Therefore, given the input cost c, the output cost t, and the market price P (stated in Eq. (15.1)), each farmer i selects his planting quantity qi to maximize his expected income πi (qi ), where: πi (qi ) = (P − t)zi qi − cqi = (a − zi qi − zj qj )zi qi − cqi , a ≡ m − t, i, j = 1, 2, i = j.
and
(15.2)
By considering the best response functions of both farmers simultaneously, we can determine the planting quantities in equilibrium. To ensure that in equilibrium the planting quantity for each farmer is positive so that we can examine the issue of income gap between farmers, we make the following two assumptions:
2 Our
main insights continue to hold when there are more than two farmers. reader is referred to Chen and Tang (2015) for a detailed discussion about the existence of the Cournot competition in the agricultural economics.
3 The
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Assumption 2 1 1 The market potential m is sufficiently high so that m c i=1 zi + t.
>
Assumption 2 The yield rate of farmer 2 is moderately higher than farmer 1 so that z1 < z2 < 2z1 . Assumption 1 ensures that the market potential m is sufficiently high so that the market price P given in Eq. (15.1) is attractive enough to entice both farmers to plant a positive amount. Assumption 2 holds when the variation of yield rates across farmers are moderate. There is empirical evidence that supports Assumption 2: Filho et al. (2010) show that the variation of yield rates across all farmers in Brazil is moderate—the coefficient of variation of the yield rates across all Brazilian farmers is less 60%. For each of the four crops (maize, rice, wheat, and soybean), Ray et al. (2015) show that the coefficient of variation of the yield rates across the world is less than 50%. Let us define the following two terms that will enable us to simplify the exposition: K=
2 1 zi i=1
and
V =
1 1 − . z1 z2
(15.3)
By interpreting 1/zi as the “inefficiency” of farmer i, we can interpret K as the “total inefficiency” of both farmers and V as the “inefficiency disparity” between farmers. By considering K and V as defined in Eq. (15.3), Assumptions 1 and 2 can be simplified as a ≡ m − t > cK and K > 3V , respectively.
15.3.2 OutputBased and InputBased Subsidy Schemes As articulated in the Introduction section, we shall analyze two subsidy schemes: 1. The Inputbased Subsidy scheme intends to defray the input purchasing cost: the government offers a subsidy δ for each input unit (e.g., seeds, fertilizers, etc.) so that the effective unit planting cost is reduced from c to c − δ, where δ ∈ [0, c]. 2. The Outputbased Subsidy scheme aims to reduce the output processing cost: the government offers a subsidy θ for each output unit (e.g., transportation, storage, etc.) so that the effective unit processing cost is reduced from t to t − θ , where θ ∈ [0, t]. In the base model, we consider the case in which the government offers either an inputbased subsidy or an outputbased subsidy. However, in a later section, we shall extend our analysis to the case where a combination of both subsidy schemes is offered.
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15.3.3 The Government’s Subsidy Scheme From the government’s perspective, it will determine the subsidy level (i.e., δ or θ ) to maximize the farmer welfare subject to an earmarked budget B. Here, due to the uniqueness of the agricultural industry, the “farmer welfare” under our setting takes both the farmers’ total income π1 (·) + π2 (·) and the income gap π2 (·) − π1 (·) into consideration, where πi (.) is given in Eq. (15.2). In our context, the farmer welfare denoted as (·) can be expressed as (·) = (π1 (·) + π2 (·)) − α · π2 (·) − π1 (·),
(15.4)
where the parameter α ∈ [0, 1] captures the extent to which the government cares about the income gap between farmers. The sequence of events is defined as follows. First, given an earmarked budget B, the government decides which subsidy scheme to adopt and the corresponding subsidy level; i.e., either the inputbased unit subsidy δ under the inputbased subsidy scheme or the outputbased unit subsidy θ under the outputbased subsidy scheme. Next, given a particular subsidy scheme and the corresponding subsidy level (i.e., δ or θ ), each farmer i (i = 1, 2) decides the planting quantity qi , and sells the harvest quantity zi qi in the market according to the market price P as given in Eq. (15.1). The farmers’ income is then realized. Below, we use the backward induction to determine the optimal subsidy level under each scheme.
15.4 Equilibrium Analysis 15.4.1 InputBased Subsidy Scheme We now analyze the equilibrium outcome associated with the inputbased subsidy scheme (δ) via the backward induction. First, we determine the planting quantities of both farmers for any given unit subsidy δ. Anticipating the farmers’ equilibrium planting quantities, we then derive the government’s optimal subsidy level decision.
15.4.1.1
The Farmers’ Planting Decisions
Given the subsidy δ per unit of the planting quantity, farmer i’s unit planting cost is reduced from c to c − δ. By letting a = m − t, farmer i’s expected income πi (·) as given in Eq. (15.2) can be expressed as πi (qi ) = (a − zi qi − zj qj )zi qi − (c − δ)qi .
(15.5)
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Based on the firstorder condition, farmer i’s best response function for any given qj can be derived as qi (qj ) =
(a − zj qj )zi − c + δ 2zi2
,
i, j = 1, 2, and i = j.
(15.6)
, Since dqi (qj )/dδ = 1 2zi2 > 0, we can conclude that farmer i plants more when the government increases the inputbased unit subsidy δ. By considering the best response functions stated in Eq. (15.6) simultaneously for both farmers, we get: Proposition 1 For any given inputbased subsidy δ, the planting quantity q˜i (δ) and the corresponding income π˜ i (δ) associated with farmer i satisfy q˜i (δ) =
a + (c − δ) · K c−δ − 2 > 0 and π˜ i (δ) = (zi · q˜i (δ))2 , 3zi zi
i = 1, 2.
(15.7)
Furthermore, q˜i (δ) and π˜ i (δ) possess the following properties: 1. The planting quantity q˜i (δ) and the harvest quantity zi q˜i (δ) of farmer i are increasing in δ, while his expected income π˜ i (δ) is increasing and convex in δ. 2. Both the output (harvest) quantity and the expected income of the highyield farmer 2 are larger than those of the lowyield farmer 1; i.e., z2 q˜2 (δ) > z1 q˜1 (δ) and π˜ 2 (δ) > π˜ 1 (δ). 3. The income gap between the highyield farmer 2 and the lowyield farmer 1 (i.e., π˜ 2 (δ) − π˜ 1 (δ)) is decreasing in δ. Statement (a) of Proposition 1 reveals that by defraying the unit planting cost from c to c − δ, the inputbased subsidy provides incentives for both farmers to plant more and harvest more, which in turn causes the market price P to decrease. However, the overall effect of the inputbased subsidy is that it can help each farmer to improve income. This implies that under the inputbased subsidy scheme, the benefit of defraying the farmer’s planting cost outweighs the loss caused by the declining market price P . Statement (b) of Proposition 1 shows that the yield advantage enables the highyield farmer 2 to harvest more and earn more than the lowyield farmer 1, resulting in the income inequality between the farmers. Despite the fact that there exists income inequality, statement (c) reveals that the income gap π˜ 2 (δ) − π˜ 1 (δ) is decreasing in the unit subsidy δ. This indicates that the inputbased subsidy scheme can help reduce the income inequality between the heterogenous farmers. The underlying reason is due to the fact that the cost savings per unit of the output quantity is higher for the lowyield farmer because δ/z1 > δ/z2 . In summary, Proposition 1 has the following implication. Insight 1. An increase of the inputbased unit subsidy δ improves the income for both farmers while reducing the income gap.
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The Government’s InputBased Subsidy Level Decision
Anticipating the farmers’ equilibrium planting decisions stated in Proposition 1, the government determines the inputbased unit subsidy δ to maximize the farmer ˜ welfare (δ) given in Eq. (15.4) subject to an earmarked budget constraint B. By noting from Proposition 1 that π˜ 2 (δ) > π˜ 1 (δ), the government’s problem can be rewritten as: ˜ = (1 + α)π˜ 1 (δ) + (1 − α)π˜ 2 (δ), max (δ) δ≤c
s.t.
δ · (q˜1 (δ) + q˜2 (δ)) ≤ B,
(15.8) (15.9)
where Eq. (15.9) is the budget constraint associated with the inputbased subsidy scheme, and q˜i (δ) and π˜ i (δ) are given in Eq. (15.7), i = 1, 2. By applying statement (a) of Proposition 1, it is immediate that both the farmer ˜ welfare (δ) and the total inputbased subsidy cost (i.e., the lefthand side of Eq. (15.9)) are increasing with the subsidy level δ. Hence, the budget constraint Eq. (15.9) is binding. Proposition 2 If B ≤ acK/3, then the optimal inputbased subsidy level δ ∗ is
2 V 2 −1 V2 1 K2 K + + · − aK + 3c δ = 6 6 2 6 2 2
2
2 K V2 V2 K aK − 3c + + B > 0. + 36 + 6 2 6 2 ∗
(15.10)
Furthermore, δ ∗ decreases in the total inefficiency K but increases in the inefficiency disparity V .4 It follows from Propositions 1 and 2 and Assumption 1 (i.e., a > cK) that the total planting quantity (i.e., q˜1 (δ ∗ )+q˜2 (δ ∗ ) = (2a−(c−δ ∗ )·K)K/6−V 2 ·(c−δ ∗ )/2) is increasing in K and decreasing in V , where K and V are given in Eq. 15.3. Thus, when the “inefficiency disparity” V is fixed, as the “total inefficiency” K increases (i.e., as the yield rate decreases), both farmers will grow more and the government needs to reduce the optimal unit subsidy δ ∗ to ensure the budget constraint remains binding. In the same vein, when the “total inefficiency” K is fixed, as the “inefficiency disparity” V increases (i.e., the yield rates between farmers vary more), the yield rates of both farmers diverge further. In this case, the government can increase the unit subsidy δ ∗ to induce the farmers to plant more without violating the budget constraint.
that the optimal inputbased subsidy level δ ∗ is increasing in B and when B = acK/3, δ ∗ = c. As it is never in the best interest of the government to oversubsidize the farmers (i.e., δ ∗ ≤ c is required), the government’s budget cannot be too large so that B ≤ acK/3. 4 Note
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So far, we have shown that the inputbased subsidy scheme is an effective scheme to entice farmers to plant more, harvest more, and earn more. At the same time, through Proposition 1, we have also shown that the adoption of the inputbased subsidy scheme can help reduce the income gap. Will these results hold for the outputbased subsidy scheme? We shall investigate this issue next.
15.4.2 OutputBased Subsidy Scheme We now consider the outputbased subsidy scheme θ . For each unit of planting quantity, the highyield farmer 2 receives more output subsidies than the lowyield farmer 1 because θ z1 < θ z2 . However, under the inputbased subsidy scheme, both farmers receive the same subsidy δ for each unit of planning quantity. As will be shown later, this observation plays a key role in understanding the difference between the two subsidy schemes.
15.4.2.1
The Farmer’s Planting Decision
For any outputbased unit subsidy θ , the income associated with each harvest quantity of farmer i is P −(t −θ ), where P = m−zi qi −zj qj is given in Eq. (15.1). By letting a = m − t, the expected income πi (·) of farmer i given in Eq. (15.2) can be rewritten as: πi (qi ) = a + θ − zi qi − zj qj zi qi − cqi ,
i, j = 1, 2, and i = j.
(15.11)
By considering the firstorder condition, we can obtain the best response function of farmer i as: qi =
(a + θ − zj qj )zi − c 2zi2
,
i, j = 1, 2, and i = j.
(15.12)
By considering the best response functions simultaneously, we get: Proposition 3 For any given outputbased unit subsidy θ , the planting quantity q˜i (θ ) and the corresponding income π˜ i (θ ) associated with farmer i satisfy q˜i (θ) =
a + θ + cK c − 2 > 0 and π˜ i (θ) = (zi · q˜i (θ))2 , 3zi zi
i = 1, 2.
(15.13)
Furthermore, q˜i (θ ) and π˜ i (θ ) possess the following properties: (a) The planting quantity q˜i (θ ) and the harvest quantity zi q˜i (θ ) of farmer i are increasing in the outputbased unit subsidy θ , while his expected income π˜ i (θ ) is increasing and convex in θ .
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(b) Both the output quantity and the expected income of the highyield farmer 2 are larger than those of the lowyield farmer 1; i.e., z2 q˜2 (θ ) > z1 q˜1 (θ ) and π˜ 2 (θ ) > π˜ 1 (θ ). (c) The income gap between the highyield farmer 2 and the lowyield farmer 1 (i.e., π˜ 2 (θ ) − π˜ 1 (θ )) increases in θ . Statement (a) of Proposition 3 shows that a higher outputbased unit subsidy θ motivates both farmers to plant more, harvest more output, and obtain a higher income. Statement (b) reveals that the yield advantage enables the highyield farmer 2 to harvests more and earns more than the lowyield farmer 1, again resulting in income inequality. These results are consistent with those under the inputbased subsidy scheme stated in Proposition 1. However, statement (c) shows that an increase of the outputbased unit subsidy θ further widens the income gap between the two farmers, which is the opposite of Proposition 1. This opposite result is due to the fact that under the outputbased subsidy scheme, the highyield farmer 2 receives a higher subsidy than the lowyield farmer 1 when planting one unit of the crop (i.e., θ z1 < θ z2 ). Hence, the outputbased subsidy scheme favors the highyield farmer. In summary, Proposition 3 enables us to obtain the following implication. Insight 2. An increase of the outputbased unit subsidy improves the income of both highand lowyield farmers, but it widens the income gap. Overall, Insights 1 and 2 reveal that, although both the input and outputbased subsidy schemes can increase the farmers’ income, the inputbased subsidy scheme seems fairer than the outputbased one. This is in light of the fact that the adoption of the inputbased (outputbased, respectively) subsidy scheme can reduce (widen, respectively) the farmers’ income gap.
15.4.2.2
The Government’s OutputBased Subsidy Level Decision
Anticipating the farmers’ planting quantity decisions stated in Eq. (15.13), the government determines the outputbased unit subsidy θ to maximize the farmer ˜ ) given in Eq. (15.4), which takes both the farmers’ total income and welfare (θ the income gap into consideration subject to a limited budget B. According to Proposition 3, π˜ 2 (θ ) > π˜ 1 (θ ). Hence, the government decides the optimal subsidy level θ to maximize the following problem: ˜ ) = (1 + α) · π˜ 1 (θ ) + (1 − α) · π˜ 2 (θ ), max (θ θ≤t
s.t.
θ · (z1 q˜1 (θ ) + z2 q˜2 (θ )) ≤ B,
(15.14) (15.15)
where the lefthand side of the budget constraint Eqs. (15.15) represents the total amount of the outputbased subsidy the government provides. Based on Proposi˜ ) and the total outputbased subsidy cost (the tion 3, both the farmer welfare (θ lefthand side of Eq. (15.15)) are increasing in the unit subsidy θ . This observation implies that the budget constraint Eq. (15.15) is binding.
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Proposition 4 If B ≤ t · (2m − cK)/3, then the optimal subsidy per unit of the harvest quantity θ ∗ is ∗
θ =
−(2a − cK) +

(2a − cK)2 + 24B . 4
(15.16)
Moreover, θ ∗ increases in the total inefficiency K but is independent of the inefficiency disparity V .5 Proposition 4 has the following implications. From Eq. (15.13), we can easily show that the total harvest quantity z1 q˜1 (θ ∗ ) + z2 q˜2 (θ ∗ ) = (2(a + θ ∗ ) − cK)/3 is increasing in K and independent of V . Consequently, when the “inefficiency disparity” V is fixed, as the “total inefficiency” K increases (i.e., as the yield rate decreases), each farmer’s output decreases so that the government can afford to increase the outputbased unit subsidy θ to ensure that the budget constraint remains binding. However, when the “total inefficiency” K is fixed, the optimal outputbased subsidy level θ ∗ shall be independent of V . A closer look at Propositions 2 and 4 implies that when the farmers are less efficient (reflected by a larger K), they may prefer the outputbased subsidy scheme over the inputbased one; while when the farmers are very heterogenous in their growing abilities (represented by a larger V ), they may prefer the inputbased subsidy scheme over the outputbased one.
15.5 InputBased versus OutputBased Subsidy Schemes So far, we have obtained the equilibrium outcomes associated with both the inputand outputbased subsidy schemes. Next, we shall compare the performance of these two subsidy schemes in terms of farmer i’s income π˜ i (·) and harvest quantity zi q˜i (·), i = 1, 2 as well as the farmers’ total income π˜ 1 (·) + π˜ 2 (·) and the income gap π˜ 2 (·) − π˜ 1 (·). Proposition 5 The performance associated with the inputbased versus the outputbased subsidy schemes can be described as follows: (a) The planting quantity, the harvest quantity and the expected income of the lowyield farmer 1 (highyield farmer 2, respectively) are higher (lower, respectively) under the inputbased subsidy scheme than under the outputbased subsidy scheme.
5 Note that θ ∗
is increasing in B and when B = t · (2m − cK)/3, θ ∗ = t. To avoid oversubsidizing the farmers (i.e., θ ∗ ≤ t is required), the government’s budget cannot be too large (i.e., B ≤ t · (2m − cK)/3). Combining Propositions 2 and 4, to ensure that δ ∗ ≤ c and θ ∗ ≤ t, we assume that B ≤ min{acK/3, t · (2m − cK)/3} hereafter.
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(b) The farmers’ total income is lower under the inputbased subsidy scheme than that under the outputbased subsidy scheme: π˜ 1 (δ ∗ ) + π˜ 2 (δ ∗ ) < π˜ 1 (θ ∗ ) + π˜ 2 (θ ∗ ). (c) The income gap is narrower under the inputbased subsidy scheme than that under the outputbased subsidy scheme: π˜ 2 (δ ∗ ) − π˜ 1 (δ ∗ ) < π˜ 2 (θ ∗ ) − π˜ 1 (θ ∗ ). Proposition 5 implies that the high and lowyield farmers favor different subsidy schemes: the lowyield farmer prefers the inputbased subsidy scheme δ ∗ , while the highyield farmer prefers the outputbased subsidy scheme θ ∗ . This is because the inputbased subsidy δ enables the lowyield farmer 1 to obtain higher cost savings per unit of output because δ/z1 > δ/z2 . The outputbased subsidy θ enables the highyield farmer 2 to collect more subsidy per unit of the planting quantity than the lowyield farmer 1 because θ z2 > θ z1 , thereby enhancing the yield advantage of the highyield farmer relative to the lowyield one. Furthermore, because the lowyield farmer 1 produces more while the highyield farmer 2 produces less under the inputbased subsidy scheme than under the outputbased subsidy scheme, the farmers’ total income under the inputbased subsidy scheme is lower than that under the outputbased subsidy scheme (i.e., π˜ 1 (δ ∗ )+ π˜ 2 (δ ∗ ) < π˜ 1 (θ ∗ )+ π˜ 2 (θ ∗ )). In addition, compared with the inputbased subsidy scheme, although the outputbased subsidy scheme is more effective in improving the farmers’ total income, it also widens the income gap. These results are the natural consequences of Propositions 1 and 3. Below, we summarize the main implications obtained from Proposition 5. Insight 3(a). The highyield farmer prefers the outputbased subsidy scheme, while the lowyield farmer favors the inputbased subsidy scheme. Insight 3(b). The inputbased subsidy scheme is more effective in reducing the income gap, while the outputbased subsidy scheme is more effective in improving the farmers’ total income. In practice, due to the different endowed resources such as water availability and the soil type, farmers in different regions may exhibit different yield rates. Insight 3(a) implies that it is better to implement the inputbased subsidy scheme in the regions where the yield rate is low, while the outputbased subsidy scheme is more suitable to be implemented in a region where the yield rate is high. Furthermore, Insight 3(b) reveals that which subsidy scheme to adopt depends on the government’s ultimate goal: improving the farmers’ total income versus reducing the income gap. These observations yield: Proposition 6 The farmer welfare under the inputbased subsidy scheme is higher ˜ ∗ ) > (θ ˜ ∗ )) if and only than that under the outputbased subsidy scheme (i.e., (δ if α, the weight associated with the income gap, is larger than a threshold α, ˆ where αˆ =
π˜ 1 (θ ∗ ) + π˜ 2 (θ ∗ ) − (π˜ 1 (δ ∗ ) + π˜ 2 (δ ∗ )) . π˜ 2 (θ ∗ ) − π˜ 1 (θ ∗ ) − (π˜ 2 (δ ∗ ) − π˜ 1 (δ ∗ ))
Note that both the optimal inputbased unit subsidy δ ∗ (given in Eq. (15.10)) and the optimal outputbased unit subsidy θ ∗ (given in Eq. (15.16)) are independent
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of the weight parameter α. Combining this with Insight 3(a) indicates that when the government puts a high priority on reducing the income inequality between the farmers (i.e., α > α), ˆ it should adopt the inputbased subsidy scheme rather than the outputbased subsidy scheme.
15.6 Discussion We now extend our analysis by considering the following three settings. First, we examine the impact of different subsidy schemes on the total production quantity and the aggregate yield rate. Second, we shall consider a “combined” subsidy scheme under which the government provides both the input and outputbased subsidies. Third, we extend our analysis to the case when the farmer’s yield rate is uncertain.
15.6.1 The Impact of Subsidy Schemes on the Aggregate Level Performance We now examine the impact of the input and outputbased subsidy schemes on the following aggregate level performance measures: • Aggregate planting quantity I˜(·) := q˜1 (·) + q˜2 (·); ˜ := z1 q˜1 (·) + z2 q˜2 (·); and • Aggregate harvest quantity Q(·) ˜ ˜ • Aggregate yield rate Y (·) := Q(·)/ I˜(·), By using the same approach as before, we get: Proposition 7 The impact of the two subsidy schemes on the aggregate farmer performance are as follows. (a) Under the inputbased subsidy scheme δ, both the aggregate planting quantity ˜ I˜(δ) and the aggregate harvest quantity Q(δ) are increasing in δ, while the aggregate yield rate Y˜ (δ) is decreasing in δ. (b) Under the outputbased subsidy scheme θ , both the aggregate planting quantity ˜ ) are increasing in θ , while the I˜(θ ) and the aggregate harvest quantity Q(θ aggregate yield rate Y˜ (θ ) is decreasing in θ . (c) Both the aggregate planting quantity and the aggregate harvest quantity are larger under the inputbased subsidy scheme than that under the outputbased ˜ ∗ ) > Q(θ ˜ ∗ )). The subsidy scheme, respectively (i.e., I˜(δ ∗ ) > I˜(θ ∗ ) and Q(δ aggregate yield rate is higher under the outputbased subsidy scheme than that under the inputbased subsidy scheme (i.e., Y˜ (θ ∗ ) > Y˜ (δ ∗ )). Proposition 7 shows that both the input and outputbased subsidy schemes ˜ However, can increase the aggregate planting and harvest quantities I˜ and Q.
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both subsidy schemes reduce the system production efficiency in terms of the aggregate yield rate; i.e., Y˜ (δ) and Y˜ (θ ) are decreasing in δ and θ , respectively. By comparing the equilibrium outcomes under the two subsidy schemes, statement (c) of Proposition 7 indicates that both the aggregate planting quantity and the aggregate harvest quantity under the inputbased subsidy scheme are larger than those under the outputbased subsidy scheme, respectively. This is because a unit increase in the inputbased subsidy can enable the farmer i to obtain cost savings 1/zi per output unit, while the cost savings associated with a unit increase in the outputbased subsidy is just a unit per harvest quantity. Therefore, the inputbased subsidy provides a stronger stimulus to increase the farmers’ planting and harvest quantities. However, because the lowyield farmer plants more under the inputbased subsidy scheme than under the outputbased subsidy scheme (see Proposition 5), the aggregate yield rate is lower under the inputbased subsidy scheme (i.e., Y˜ (δ ∗ ) < Y˜ (θ ∗ )). In summary, the outputbased subsidy scheme is more efficient in terms of the aggregate yield rate; and the inputbased subsidy scheme is more effective in enticing farmers to plant more and harvest more.
15.6.2 A Combined Subsidy Scheme We now examine a combined subsidy scheme under which the government provides both the input and outputbased subsidies so that each farmer receives δ per unit of the planting quantity and obtains θ per unit of harvest quantity.
15.6.2.1
The Farmers’ Planting Decisions
Under a combined subsidy scheme (δ, θ ), the input cost is reduced from c to c − δ and the output cost is reduced from t to t − θ . In this case, the expected income πi (.) given in Eq. (15.2) associated with farmer i can be expressed as: πi (qi ) = (a + θ − zi qi − zj qj )zi qi − (c − δ)qi , i,
j = 1, 2, i = j,
(15.17)
where a = m − t. By considering the firstorder condition, we can easily derive farmer i’s best response function as follows: qi (qj ) =
(a + θ − zj qj )zi − c + δ 2zi2
,
i, j = 1, 2, and i = j.
(15.18)
Solving these best response functions simultaneously, we obtain the following equilibrium planting quantity and expected income of farmer i: q˜i (δ, θ ) =
a + θ + (c − δ) · K c−δ − 2 , 3zi zi
and
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π˜ i (δ, θ ) = (zi · q˜i (δ, θ ))2 ,
i = 1, 2.
(15.19)
Note that by letting either a ≡ a+θ or c ≡ c−δ, the above quantities are equivalent to their counterparts associated with the input or the outputbased subsidy scheme, respectively. This implies that the results given in Propositions 1 and 3 still hold under the combined subsidy scheme. To avoid repetition, we omit the details here. Next, we characterize the interactions between the input and outputbased subsidies regarding the above equilibrium outcomes. Proposition 8 Under the combined subsidy scheme, both the expected income of farmer i, π˜ i (δ, θ ), i = 1, 2 and the farmers’ total income, π˜ 1 (δ, θ ) + π˜ 2 (δ, θ ), are “supermodular” in the subsidy levels (δ, θ ) while the income gap, π˜ 2 (δ, θ ) − π˜ 1 (δ, θ ), is “submodular” in (δ, θ ). Proposition 8 implies that the effects of the input and outputbased subsidies on both the income of each farmer and the total farmer income are complementary with each other due to supermodularity. However, their effects on the income gap π˜ 2 (δ, θ ) − π˜ 1 (δ, θ ) are substitutes due to submodularity. Recall that compared with the inputbased subsidy scheme, the outputbased subsidy scheme leads to a higher total income but a larger income gap. Proposition 8 reveals that under the combined subsidy scheme, increasing the inputbased subsidy level can enhance the effect of the outputbased subsidy on the farmers’ total income on the one hand while mitigating its impact on the income gap on the other hand.
15.6.2.2
The Government’s Subsidy Level Decision
Anticipating farmers’ planting quantity decisions as stated in Eq. (15.19), the government determines the subsidy levels δ and θ to maximize the farmer welfare ˜ (δ, θ ) given in Eq. (15.4). Because the government subsidizes δ per unit of the planting quantity and θ per unit of the harvest quantity, the total amount of subsidy under the combined subsidy scheme can be derived as δ · (q˜1 (δ, θ ) + q˜2 (δ, θ )) + θ · (z1 q˜1 (δ, θ ) + z2 q˜2 (δ, θ )). By taking the budget constraint into account, the government solves the following problem: ˜ θ ) = ((π˜ 1 (δ, θ ) + π˜ 2 (δ, θ )) − α(π˜ 2 (δ, θ ) − π˜ 1 (δ, θ )), max (δ,
(15.20)
s.t. δ · (q˜1 (δ, θ ) + q˜2 (δ, θ )) + θ · (z1 q˜1 (δ, θ ) + z2 q˜2 (δ, θ )) ≤ B.
(15.21)
δ,θ
It follows immediately from Proposition 8 that the objective function given in Eq. (15.20) is supermodular. Also, by applying statement (a) as stated in Propositions 1 and 3, we can conclude that the subsidy expense; i.e., the lefthand side of the budget constraint as given in Eq. 15.21, is increasing in δ and θ . By using these two observations, we can show that the budget constraint is binding. Hence, we get:
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Proposition 9 Under the combined subsidy scheme, the budget constraint Eq. (15.21) is binding. When α ≥ αˆ (where αˆ is given in Proposition 6), the ∗ satisfies Eq. (15.10) while the optimal subsidy per unit of the planting quantity δm ∗ ˆ the optimal optimal subsidy per unit of the harvest quantity θm = 0. When α < α, subsidy per unit of the harvest quantity θm∗ satisfies Eq. (15.16) while the optimal ∗ = 0. subsidy per unit of the planting quantity δm Although the effects of the input and outputbased subsidies on the farmer welfare are complementary to each other, Proposition 9 indicates that it is never optimal to provide both the input and outputbased subsidies. Recall that the optimal subsidy levels shall lead to the binding of the budget constraint. And based on Eq. (15.19), we can easily show that the total amount of subsidies δ · (q˜1 (δ, θ ) + q˜2 (δ, θ )) + θ · (z1 q˜1 (δ, θ ) + z2 q˜2 (δ, θ )) is convex in both the input and outputbased unit subsidies δ and θ . This implies that in equilibrium, one of the subsidy ∗ or θ ∗ ) shall be degenerated to zero. Also, Proposition 9 confirms levels (either δm m that the government shall provide the inputbased subsidy if and only if its concern about the farmers’ income inequality is high enough (i.e., when α > α). ˆ
15.6.3 When the Yield Rate Is Uncertain We now extend our analysis to the case in which the farmer’s yield rate is uncertain. Specifically, we assume that the yield rate of farmer i, zi , i = 1, 2, is a random variable with mean E(zi ) = μi and variance Var(zi ) = σi2 , where 0 < μ1 < μ2 < 1. To simplify our analysis, following Tang et al. (2015), we assume that σ1 = σ2 = σ so that the yield rate of farmer 2 is stochastically larger than that of farmer 1. We shall refer to farmer 2 as the highyield farmer and farmer 1 as the lowyield farmer. Analogously, we define K = 1/μ1 + 1/μ2 and V = 1/μ1 − 1/μ2 in this section. Furthermore, to ensure that both farmers plant a positive amount, we assume that both Assumptions 1 and 2 still hold here; that is, a = m − t > cK and K > 3V . By noting that P = m − zi qi − zj qj , a = m − t, and E[zi2 ] = μ2i + σ 2 , i, j = 1, 2 and i = j , farmer i’s expected income under input and outputbased subsidy schemes are as follows: Input
(qi ) = E{(P − t)zi qi − (c − δ)qi } = aμi qi − (μ2i + σ 2 )qi2 − μi μj qi qj − (c − δ)qi , Output (qi ) = E{(P − (t − θ ))zi qi − cqi } πi = (a + θ )μi qi − (μ2i + σ 2 )qi2 − μi μj qi qj − cqi . πi
(15.22)
To avoid repetition, here we omit the detailed analysis. Table 15.1 summarizes the results when the farmers’ yield rates are uncertain. It can be easily shown that the main insights 1, 2, 3(a) and 3(b) under the base model continue to hold here when the farmer’s yield rate is uncertain. Specifically, compared to the outputbased subsidy scheme, the inputbased subsidy scheme is preferred by the lowyield farmer 1 and
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Table 15.1 List of results when the yield rates are uncertain Equilibrium outcomes Inputbased 1. q˜i (δ), μi q˜i (δ), π˜ i (δ), and π˜ 1 (δ)+π˜ 2 (δ) are all increasing in the inputbased unit subsidy δ; Subsidy scheme 2. μ1 q˜1 (δ) < μ2 q˜2 (δ) and π˜ 1 (δ) < π˜ 2 (δ); 3. When σ 2 ≤ (a − cK)μ1 μ2 /a, π˜ 2 (δ) − π˜ 1 (δ) is decreasing in δ; otherwise, it is first increasing and then decreasing in δ. Outputbased 1. q˜i (θ), μi q˜i (θ), π˜ i (θ), and π˜ 1 (θ)+π˜ 2 (θ) are all increasing in the Subsidy scheme outputbased unit subsidy θ; 2. μ1 q˜1 (θ) < μ2 q˜2 (θ) and π˜ 1 (θ) < π˜ 2 (θ); 3. π˜ 2 (θ) − π˜ 1 (θ) is increasing in θ. Equilibrium 1. q˜1 (δ ∗ ) > q˜1 (θ ∗ ), μ1 q˜1 (δ ∗ ) > μ1 q˜1 (θ ∗ ), and π˜ 1 (δ ∗ ) > π˜ 1 (θ ∗ ); 2. q˜2 (θ ∗ ) > q˜2 (δ ∗ ), μ2 q˜2 (θ ∗ ) > μ2 q˜2 (δ ∗ ), and π˜ 2 (θ ∗ ) > π˜ 2 (δ ∗ ); Outcome Comparison 3. π˜ 1 (θ ∗ ) + π˜ 2 (θ ∗ ) > π˜ 1 (δ ∗ ) + π˜ 2 (δ ∗ ) and π˜ 2 (θ ∗ ) − π˜ 1 (θ ∗ ) > π˜ 2 (δ ∗ ) − π˜ 1 (δ ∗ ); 4. π˜ 1 (δ ∗ ) − π˜ 1 (θ ∗ ), π˜ 2 (θ ∗ ) − π˜ 2 (δ ∗ ), π˜ 1 (θ ∗ ) + π˜ 2 (θ ∗ ) − (π˜ 1 (δ ∗ ) + π˜ 2 (δ ∗ )), and π˜ 2 (θ ∗ ) − π˜ 1 (θ ∗ ) − (π˜ 2 (δ ∗ ) − π˜ 1 (δ ∗ )) are all increasing in the budget B; ˜ ∗ ) > (θ ˜ ∗ ) if and only if α is larger than a threshold α. 5. (δ ˇ
less preferred by the highyield farmer 2. Again, the outputbased subsidy scheme outperforms the inputbased subsidy scheme in terms of farmers’ total income, while the inputbased subsidy scheme dominates the outputbased subsidy scheme in terms of lower income gap. Thus, the government prefers the inputbased subsidy scheme if and only if it has a high concern over the farmers’ income inequality (i.e., α > α). ˇ However, relative to the base model, we obtain a different result: increasing the inputbased unit subsidy δ may not reduce the income gap, especially when the yield uncertainty is relatively high and the subsidy level δ is low. This is because a higher yield uncertainty may cause both farmers to produce less, which results in a higher market price. Due to his yield advantage, the highyield farmer 2 then generates more expected revenue per unit of the planting quantity (i.e., μi · P ) than the lowyield farmer 1, as the market price increases. Meanwhile, a higher inputbased unit subsidy reduces the lowyield farmer 1’s cost per unit of the harvest quantity more than that of the highyield farmer 2 (i.e., δ/μ1 > δ/μ2 ), and thus benefits the lowyield farmer more than the highyield one. Consequently, when the yield uncertainty is high and the inputbased subsidy level is low, the inputbased subsidy scheme may benefit the highyield farmer more than the lowyield one, leading to the increase of the income inequality in δ. Next, we numerically examine the impact of the farmers’ yield uncertainty σ 2 on the equilibrium outcomes (Fig. 15.1). Figure 15.1 reveals that as the yield uncertainty σ 2 increases, both the farmers’ total income and the income gap between farmers will decrease under both subsidy schemes. This is because as the yield uncertainty (via σ 2 ) increases, both farmers will plant less because the farmer’s
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Fig. 15.1 The impact of the yield uncertainty on the total farmer income and the income inequality under the two subsidy schemes: μ1 = 0.5, μ2 = 0.7, a = 6, c = 1.6, B = 1. (a) Income gap under the two schemes. (b) Farmers’ total income under the two schemes
expected income given in Eq. (15.22) is decreasing in σ 2 under both schemes. As both farmers reduce their planting quantities, both farmers’ incomes drop but the highyield farmer’s income will drop faster because of a bigger drop in its output as σ 2 increases. Consequently, the income gap reduces as σ 2 increases. In practice, in order to improve farmers’ total income, many developing countries such as India help reduce the farmers’ yield uncertainty by providing the timely weather information to farmers (Rathore and Chattopadhyay 2016). Our numerical results imply that reducing the farmers’ yield uncertainty can indeed improve the farmers’ total income, but it may widen the income gap.
15.7 Conclusion Motivated by mixed empirical evidence about the effects of the input and outputbased subsidy schemes, we analytically examine the performance of these two schemes in terms of individual farmer income, the total farmer income and income gap. We have shown that, while both subsidies can improve the income of each farmer, their impact on the income inequality is opposite: a higher inputbased unit subsidy can reduce the income gap, while a higher outputbased unit subsidy widens such gap. Interestingly, the high and lowyield farmers hold different preferences toward the two schemes: the inputbased subsidy scheme is preferred by the lowyield farmer, while the outputbased subsidy scheme is preferred by the highyield farmer. We show that both the input and outputbased subsidy schemes have their own unique advantages: the outputbased subsidy is more effective in improving the farmers’ total income and production efficiency, while the inputbased subsidy is more efficient in reducing the income gap and increasing the total production
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quantity. Therefore, the government should implement the inputbased subsidy scheme only when it has significant concern over the income gap. When the government provides both types of subsidies, we have shown that the inputbased subsidy can enhance the effect of the outputbased subsidy on the farmers’ total income (as complements) but dampen its effect on the income gap (as substitutes). We then show that it is never optimal for the government to provide both types of subsidies. Last, when the farmer’s yield rate is uncertain, we have shown that the main insights when the yield rates of farmers are predetermined continue to hold in this case. We also numerically show that reducing the yield uncertainty can improve the farmers’ total income but it can widen the income gap. Our study is an initial attempt to examine the implications of the input and outputbased subsidy schemes on the farmer income and income gap. There are many research opportunities for further examination. For example, in this study, we consider the impact of the input and outputbased subsidy schemes on the individual farmers. However, it may be of interest to consider their impact on the agricultural supply chain that involves the smallholder farmers, the middlemen and the agricultural companies. Here, one potential research question is to examine the preference of the different stakeholders along the agricultural supply chain over the two subsidy schemes. In this study, we consider that the farmers grow only one crop. However, in practice, there exist multiple crops that the farmers can plant. Under such setting, the decisions of both the government and the farmers are much more complicated: the farmers shall determine which crop to plant while the government shall determine which crop to subsidize and what type of subsidy scheme to adopt. We leave this analysis as the future research.
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