Economic Dynamics : Theory, Games and Empirical Studies [1 ed.] 9781608767618, 9781604569117

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ECONOMIC DYNAMICS: THEORY, GAMES AND EMPIRICAL STUDIES

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

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ECONOMIC DYNAMICS: THEORY, GAMES AND EMPIRICAL STUDIES

CHESTER W. HURLINGTON

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EDITOR

Nova Science Publishers, Inc. New York

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All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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New York

CONTENTS

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Preface

vii

Chapter 1

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio Gustavo A. Marrero and Alfonso Novales

Chapter 2

Lévy Processes and Option Pricing by Recursive Quadrature Gianluca Fusai, Giovanni Longo, Marina Marena and Maria Cristina Recchioni

31

Chapter 3

Labor Market Imperfections and Endogenous Business Cycles Alvaro Rodriguez

59

Chapter 4

Analyzing Economic Policy Using High Order Perturbations Michael Ben-Gad

77

Chapter 5

Invariance in Economic Dynamics and the Sustainable Development Issue Vincent Martinet and Gilles Rotillon

99

Chapter 6

Endogenous Capital Utilization and Depreciation under Capital Maintenance in the AK Growth Model J. Aznar-Márquez and J.R. Ruiz-Tamarit

121

Chapter 7

Social Status, the Spirit of Capitalism, and the Term Structure of Interest Rates in Stochastic Production Economies Liutang Gong, Yulei Luo and Heng-fu Zou

135

Chapter 8

On the Investment-Uncertainty Relationship Kit Pong Wong

157

Chapter 9

Realized Volatility and Correlation Estimators under Non-Gaussian Microstructure Noise Amir Safari, Wei Sun, Detlef Seese and Svetlozar Rachev

171

1

vi Chapter 10

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Index

Contents Why Increased Knowledge Does not Necessarily Improve Trading Success: A Monte-Carlo Simulation Jürgen Huber and Michael Kirchler

199

209

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PREFACE Economic dynamic theory includes dynamic games, dynamic general equilibrium theory, and empirical studies. The following topics are subsumed: business cycles, asset pricing, search models, intergenerational issues, fertility, financial systems. This new book presents the latest research from around the globe. In endogenous growth settings, long-run growth effects are important for welfare, but they should not be the only consideration for policy evaluation. In Chapter 1, welfare effects along the first periods of the transition following a fiscal policy reform are found to be of opposite sign to long-run effects. Hence, fully characterizing the transitional dynamics is crucial when characterizing the effects of downsizing public investment, an important fiscal policy issue in industrialized economies. Starting from a standard fiscal position and a benchmark parameter calibration, the authors show that downsizing public investment improves welfare under either capital or gross income taxes, provided public capital is not very productive. On the other hand, downsizing is found to improve welfare with independence of the tax system considered for high levels of the unproductive public expenditure/output ratio, or for low values of the elasticity of intertemporal substitution, the discount factor and/or the public capital elasticity in the aggregate technology. Additionally, for high levels of the output elasticity of private capital, downsizing is shown to be optimal under the less distorting taxes, but not under gross income and capital income taxes. In Chapter 2 the authors present a quadrature method for pricing discretely monitored pathdependent options when the dynamics of the underlying assets are described by Lévy processes. The literature mainly deals with the continuous monitoring case and assumes that the log-returns are normally distributed. The contribution of the present paper consists in providing a flexible computational method that allows to deal with the discrete monitoring rule that is common for exotic options and general enough to deal with non-Gaussian models. Since it is well known that the convergence of the discrete monitoring price to its continuous monitoring counterpart is very slow (Broadie et al. (1997, 1999)), fast and accurate pricing methods for the discrete monitoring case become particularly important. Also, effective alternatives to the Black-Scholes world, such as Lévy processes, are becoming increasingly popular, Cont and Tankov ([8]). The hybrid computational method proposed combines numerical inversion of the characteristic function and Gaussian quadrature extending the approach proposed by Tse et al. (2001) and Sullivan (2000) to the case of exponential Lévy

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viii

Chester W. Hurlington

processes. The higher accuracy of Gaussian quadrature provides accurate results up to a very large number of monitoring dates, much larger than the number of monitoring dates usually considered in the existing literature. The authors present an error analysis for the numerical method proposed and they provide detailed numerical examples for plain vanilla options, discrete barrier options (with a single or a double barrier), and Bermudian options for a variety of Lévy processes, such as the Gaussian, CGMY, Kou Double- Exponential, Merton Jump-Diffusion. Chapter 3 examines the consequences of introducing labor market imperfections in a rather conventional macro model. In spite of the fact that all agents behave in reasonable way and enjoy perfect foresight, the model allows for a conflict between investment and savings decision that can result in instability. It is found that wage rigidity preserves the stability of economies that were initially stable. Economies that were completely unstable benefit from the introduction of friction: they either become stable or they approach a stable periodic orbit. It is shown that the cycle generated by the model is consistent with procyclical movements in real wages, the rate of capacity utilization and the price level. Chapter 4 demonstrates the use of high order general perturbations to analyze policy changes in dynamic economic models. The inclusion of high moments in approximating the behavior of dynamic models is particularly necessary for welfare analysis. I apply the method of general perturbations to the analysis of permanent changes to a flat rate tax on the return to capital in the context of the standard Ramsey optimal growth model. Reliance on simple linearizations or quadratic approximations are adequate for generating impulse responses for the variables of interest or the welfare analysis of small policy changes. However when considering the welfare implications of sizable policy changes, the failure to include higher moments can lead not only to quantitatively serious inaccuracies, but even to spurious welfare reversals. In Chapter 5, the authors examine how the Economic Dynamics Theory can be used to address the sustainable development issue. Sustainability is related to economic dynamics and intergenerational equity concerns. The authors first describe how the environmental concerns have been taken into account in the neoclassical approach of the optimal economic growth. In such a framework, the Hamiltonian of the optimization problem is interpreted as a sustainability indicator. The authors examine how these results are consistent with the way the sustainability issue is described as a requirement to conserve utility level through time. Sustainability is then achieved when genuine saving is positive, “greening” the Net National Product. After that, the authors use the invariance approach (Noether’s theorem) to exhibit the conditions for conservation laws in economic dynamics to exist. If something is preserved along an optimal economic path, it can be interpreted as a representation of sustainability. The authors discuss how the existence conditions are related to the economy’s characteristics (time preference and technology). Moreover, these conservation laws are linked to time and state variables transformations in the invariance theory. The authors relate these transformations to endogenous changes in the economy (environmental preferences and technological progress) and show that the conserved quantities can be interpreted as modified Hamiltonians, defining new sustainability account systems. In the traditional exogenous growth model, the decision about how much to save is based on the comparison between costs and benefits of a higher consumption today rather than tomorrow. This endogenously decided savings are automatically channelled to investment, which plays a passive role. The basic model assumes full utilization of the installed capital,

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Preface

ix

and capital depreciation determined exogenously as a constant fraction of the capital stock. These assumptions, however, do not conform to available data. Observed facts show that firms do not always decide to use all the installed capital, and also that they choose the value of the depreciation rate. Actually, the depreciation rate depends on the internal resources devoted to repair and maintenance of the capital stock, which deteriorates either through the use in the production process or simply through the natural process of ageing. Maintenance reduces the depreciation of capital, investment is subject to adjustment costs, and the degree of capital utilization affects itself the maintenance activity. As explained in Chapter 6, despite of the empirical evidence, the depreciation rate has been usually regarded as an exogenous and constant parameter. According to the neoclassical growth theory, this one affects negatively the long-run level of the variables and the short-run rates of growth. In the new theory of endogenous growth, the constant depreciation rate also affects negatively the long-run rate of growth. But the authors are interested in the depreciation rate as an endogenous variable, and this leads us to explore simultaneously the determinants of both depreciation and growth. Consequently, in this chapter the authors study an extended version of the one-sector AK growth model introducing adjustment and maintenance costs. Agents are allowed to under-use the installed capital and to vary the depreciation rate. The model is analyzed using particular functional forms and it is solved in closed-form. The authors find that adjustment and maintenance costs (efficiency) reduce (increases) investment, depreciation, capital utilization as well as the rate of growth. Impatience reduces the rate of growth but increases depreciation and utilization. Depreciation and utilization depend negatively on the rate of population growth. Chapter 7 studies capital accumulation and equilibrium interest rates in stochastic production economies with the concern of social status. Given a specific utility function and production function, explicit solutions for capital accumulation and equilibrium interest rates have been derived. With the aid of steady-state distributions for capital stock, the effects of fiscal policies, social-status concern, and stochastic shocks on capital accumulation and equilibrium interest rates have been investigated. A significant finding of this paper is the demonstration of multiple stationary distributions for capital stocks and interest rates with the concern of social status. Chapter 8 examines the investment-uncertainty relationship in a canonical real options model. The authors show that the critical lump-sum payoff of a project that triggers the exercise of the investment option exhibits a U-shaped pattern against the volatility of the project. This is driven by two opposing effects of an increase in the volatility of the project: (i) the usual positive effect on option value, and (ii) a negative effect on option value due to the upward adjustment in the discount rate. The authors further show that such a U-shaped pattern is inherited by the expected time to exercise the investment option. Thus, for relatively safe projects, greater uncertainty may in fact shorten the expected exercise time and thereby enhance investment. This is in sharp contrast to the negative investment-uncertainty relationship as commonly suggested in the extant literature. Realized volatility and correlation estimators suffer from microstructure noise, resulting in biased and imprecise estimators. This suggests that estimators do not converge for highfrequency levels, where noise especially exists. To solve the problem of noise, some approaches have been suggested in literature. In particular, the subsampling and averaging approach works well. Moreover, the realized volatility literature usually assumes Gaussian microstructure noise despite the fact that noise in real world financial markets does not follow

x

Chester W. Hurlington

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a Gaussian process. In Chapter 9 suggests what the authors believe to be a more realistic microstructure noise processes. The fractional stable noise that the authors suggest is themost realistic process compared to other noise processes investigated in our simulations. Therefore, realized volatility and correlation estimators should be unbiased and converge faster under this type of noise process. Empirically, some of the estimators exhibit heavy tails and some exhibit dynamic behaviors. This is especially so in the case of absolute based realized correlations which exhibit negative asymmetry in the dependence structure between minute by minute frequency data of CAC and FTSE stock indices. In Chapter 10 the authors investigate the widely held belief that success in the stock market can largely be attributed to the information underlying the trading decisions. The authors present results from a Monte-Carlo simulation of a financial market with asymmetrically informed traders whose information is based on future dividends. The authors observe a J-shaped distribution of returns: while the best informed can outperform all others, average informed traders have lower returns than the worst informed.

In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 1-30

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 1

T RANSITIONAL DYNAMICS AND W ELFARE E FFECTS OF THE P UBLIC I NVESTMENT /O UTPUT R ATIO∗

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Gustavo A. Marrero1† and Alfonso Novales2‡ 1 Depatamento de Fundamentos del An´alisis Econ´omico, Universidad de la Laguna 2 Departamento de Economia Cuantitativa, Universidad Complutense, Campus de Somosaguas, 28023 Madrid, Spain

Abstract In endogenous growth settings, long-run growth effects are important for welfare, but they should not be the only consideration for policy evaluation. In this paper, welfare effects along the first periods of the transition following a fiscal policy reform are found to be of opposite sign to long-run effects. Hence, fully characterizing the transitional dynamics is crucial when characterizing the effects of downsizing public investment, an important fiscal policy issue in industrialized economies. Starting from a standard fiscal position and a benchmark parameter calibration, we show that downsizing public investment improves welfare under either capital or gross income taxes, provided public capital is not very productive. On the other hand, downsizing is found to improve welfare with independence of the tax system considered for high levels of the unproductive public expenditure/output ratio, or for low values of the elasticity of intertemporal substitution, the discount factor and/or the public capital elasticity in the aggregate technology. Additionally, for high levels of the output elasticity of private capital, downsizing is shown to be optimal under the less distorting taxes, but not under gross income and capital income taxes.

Keywords: Transitional dynamics, endogenous growth, distorting taxes, public investment, simulation methods. JEL Classification: E0, E6, O4 ∗

The authors acknowledge helpful comments received from Jordi Caball´e. Financial support from the Spanish Ministry of Education (through DGICYT grant no. SEJ2006-1435) is gratefully acknowledged. † E-mail address: [email protected] ‡ E-mail address: [email protected]

2

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1.

Gustavo A. Marrero and Alfonso Novales

Introduction

Reconsidering the fiscal role of governments is a central issue on economic policy in industrialized economies, and there is a wide consensus on the potential welfare gains from downsizing policies relative to the observed size of public sector in most industrialized economies. Public resources could be at a minimum classified as either expenditures that may directly affect productivity (infrastructure, public firms investment, etc.), or expenditures that do not have such effect (health, social security, law and order, defense, etc.) [Musgrave (1997)]. It is a standard characteristic in developed economies that governments face a level of mostly precommitted expenditures of the second kind because of previous social policies, and this paper characterizes the welfare-maximizing productive public expenditures/output ratio in a stylized endogenous growth model with private and non-congested public capital. We do not question how the level of unproductive public expenditure size is determined which,1 as in Cassou and Lansing (1998, 1999) and Marrero and Novales (2005, 2007), we assume to be exogenous.2 The analysis is performed under alternative tax systems, since welfare effects of a public investment policy could well be different, as suggested by Burgess and Stern (1993). This paper relates to the vast literature on fiscal policy in dynamic settings. Starting with Lucas (1990), Jones and Manuelli (1990) and Barro (1990), endogenous growth models have become a standard environment to analyze the incidence of fiscal policies. Additionally, the link between public expenditures and the private production process was emphasized in Ratner (1983) and Aschauer (1989). Barro (1990), Futagami et al. (1993), Glomm and Ravikumar (1994, 1997, 1999), Turnovsky (1996, 2000), Aschauer (2000), Chen (2006), Marrero (2005, 2008), all discuss the optimality of productive public expenditures in growth models. Barro (1990) shows that the growth-maximizing public investment ratio coincides with the one maximizing welfare. In contrast, most other authors conclude that the welfare-maximizing ratio is strictly lower than the one maximizing growth. Futagami et al. (1993) point out that the reason for this discrepancy is the existence of transitional dynamics, which arises because public capital does not fully depreciate every period. Alternatively, in Glomm and Ravikumar (1994), public and private capital fully depreciate and thus the economy does not display transition. However, optimal predictions then differ from Barro (1990) because tax revenues are converted into infrastructure with some lag. Turnovsky (2000) extends the Barro (1990) model of productive government expenditure by introducing an elastic labor supply, and analyzes fiscal policy effects in the form of changes in public expenditures under diferent modes of tax financing. However, the economy always lies on its balanced growth path, so there is no possibility of analyzing dynamic effects of such policies. Discrepancies from Barro then come from the presence 1 See Chen (2006) for a detailed analysis on this issue. This author stresses that certain economic factors (mostly exogenous) that alter the marginal utility of private consumption relative to the marginal utility of public consumption, would affect the size of public consumption. 2 In fact, this is not an unrealistic assumption. A current government must often take as given some items included in public consumption, such as public wages, interest payments on public debt and bureaucratic or administrative disbursements, since they were approved before hand, possibly by previous governments. As a percentage of GDP, these public expenditure concepts are far from zero and have even increased over time in most developed countries. In addition, the political cost of cutting down these items could be high, even if their levels are above their optimum values.

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Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio

3

of alternative tax systems and because leisure is a choice variable. Marrero (2008) reexamines the optimal choice of public investment in a more general framework, which allows for long-lasting capital stocks, a lower depreciation rate for public capital than for private capital, an elasticity of intertemporal substitution that differs from unity and the need to finance a non-trivial share of public, unproductive services in output. Under an income tax system, he showed that all this factors might be relevant in determining the optimal public investment policy. In this paper we consider a framework with elastic labor supply which shows a non trivial transitional dynamics. We study, simultaneously, the sensitivity of the welfaremaximizing public investment/output ratio to alternative tax scenarios and the effect of public expenditure policies on welfare. In dynamic settings, an optimal policy can be described in terms of a trade-off between initial and future consumption. In general, the optimal policy induces a sacrifice in initial welfare (by reducing initial consumption and/or leisure), with a faster accumulation of capital along the former periods of the transition. This accumulation allows for an eventual faster growth in output, investment and consumption, compensating the initial loss of utility. Our contribution is twofold: i) to study in detail the relationship between this welfare trade-off and the tax system in an extended Barro-type framework with transitional dynamics, and ii) to characterize and interpret the magnitude of the welfare trade-off, which happens to be tax-dependent. A complex framework like this easily becomes analytically intractable, so numerical techniques are needed to characterize the transitional dynamics. However, computing a numerical solution is not easy either, since the variables in the economy experience growth in steady-state. To avoid this problem, approximated equilibrium conditions obtained from the model written in stationary ratios are combined with the equilibrium equations for the level of the variables [similar to Novales et al. (1999)]. In computing welfare-maximizing policies, careful attention is given to the transitional dynamics associated with how the economy responds to such optimal policy. We compute the associated path of variables such as consumption, leisure, public and private investment and output and, finally, we characterize the time path for utility. In addition to lump-sum taxation, four tax scenarios are alternatively considered: taxes on total income, gross capital income, labor income and private consumption. Under each tax system, a constant tax rate is chosen so that, in steady-state equilibrium, total public expenditures are entirely financed with the selected tax system, and lump-sum becomes zero. For simplicity, we assume that any possible deficit along the transition to steady-state will be financed with lump-sum transfers, while any surplus will be translated into positive transfers being assigned to the private sector. We start by characterizing the welfare-maximizing public investment/output ratio, emphasizing the effect of such policy on the dynamic properties of welfare and the main macroeconomic variables. We relate the welfare-maximizing ratio to the initial level of 7% considered in the benchmark setting, and we discuss whether a downsizing in public investment is welfare improving, as well as the dependence of this analysis on the tax system considered. After that, we perform a local sensitivity analysis for the structural parameters in the economy to understand which are the main determinants of the welfare-maximizing policy. The robustness of conclusions obtained in the previous section to changes in parameter values is also discussed.

4

Gustavo A. Marrero and Alfonso Novales

The rest of the paper is organized as follows: the model economy is presented in section 2; in section 3, the competitive equilibrium is described and a procedure to solve for the dynamics of the level of the variables is proposed; in section 4, we characterize the welfaremaximizing public investment/output ratio; in section 5, we carry out a local sensitivity analysis; finally, in section 6 the paper is closed by setting the main conclusions and possible extensions.

2.

The Environment

There are three different economic agents: a continuum of firms indexed by i ∈ [0, 1], households and a government, who cares only about fiscal policy.

2.1.

Firms

A large number of identical firms, indexed by i ∈ [0, 1], produce the single consumption good in the economy. Each firm rents the same amount of private inputs from households (private capital, k˜t , and labor, ˜lt ) to produce y˜t units of output. The total amount of physical ˜ t , is taken as a proxy for the index of knowlcapital used by all the firms in the economy, K ˜ g , affects edge available to each firm [as in Romer (1986)]. Additionally, public capital, K t the production process of all individual firms. Except for these externalities, the private production technology is a standard Cobb-Douglas function presenting constant returns to scale in the private inputs and increasing returns in the aggregate. For any firm,  θ g ˜ θk K ˜g ˜ t, K ˜ g ) = F ˜l1−α k˜tα K y˜t = f (˜lt , k˜t , K , θg , α ∈ (0, 1), θk ≥ 0, t t t t

(1)

 θ g ˜ 1−α K ˜ α+θk K ˜g , Y˜t = F L t t t

(2)

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where α is the economic share of capital, θg and θk are, respectively, the elasticities of output with respect to public capital and the knowledge index, and F is a technological scale factor, common to each firm. Since firms are identical, from (1), aggregate output, Y˜t , is produced according to,

˜ t is aggregate labor. where L During period t, each firm pays the competitive-determined wage w ˜t on the labor it hires and the rate rt on the capital it rents. The dynamic problem faced by firms turns out to be static at each point in time, ˜ t, K ˜ g) − w M ax f (˜lt , k˜t , K ˜t ˜lt − rt k˜t , t

{lt ,kt }

and optimality leads to the usual marginal productivity conditions:  θ g y˜t Y˜t ˜g ˜ θk K =α =α , rt = αF ˜lt1−α k˜tα−1 K t t ˜t K k˜t  θ g Y˜t y˜t ˜ t = (1 − α)F ˜l−α k˜tα K ˜ θk K ˜g W = (1 − α) = (1 − α) , t t t ˜lt ˜t L

(3) (4)

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio

5

where we have used the fact that each firm treats its own contribution to the aggregate capital stock as given, rents the same amounts of the private inputs and produces the same amount of output. Aggregate private capital stock evolves according to, ˜ t+1 = (1 − δ k )K ˜ t + I˜tk , K

(5)

where I˜tk is gross private investment.

2.2.

Households

The representative consumer chooses the fraction of time to spend as leisure, ht . She is the owner of physical capital, and allocates her resources between consumption, C˜t , and investment in physical capital, I˜tk . The price of the single consumption commodity and the time endowment of households are both normalized to one, and zero population growth is assumed every period. Decisions are made each period to maximize the discounted aggregate value of the time separable utility function, ∞ X

max

∞ {C˜t ,ht ,K˜ t+1 }t=0 t=0

β t U (C˜t , 1 − ht )

(6)

subject to her resource constraint in every period h



˜t + K ˜ t+1 + T˜t ≤ W ˜ t ht (1 − τ w ) + K ˜ t 1 − δ k + rt 1 − τ k (1 + τ c )C

i

˜t ≥ 0K ˜ t+1 ≥ 0 and ht ∈ [0, 1] , C (7)

and to the transversality condition, that places a limit on the accumulation of capital,

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˜ ˜ t+1 ∂u(Ct , ht ) = 0. lim β t K t→∞ ∂ C˜t

(8)

˜ t+1 denotes the stock of physical capital at the end of time t, with K ˜ 0 > 0, τ k , where K w c τ and τ are the flat tax rates on capital income, labor income and private consumption, respectively, and T˜t is a net transfer made by households to the public sector, which may be either positive or negative; β ∈ (0, 1) is the discount factor and U (C˜t , ht ) is a ∁2 mapping, strictly concave, increasing in C˜t and (1 − ht ) and satisfying Inada conditions.3 U (C˜t , ht ) is particularized to be of the constant elasticity of substitution family [King and Rebelo (1988)], h i1−θ C˜tρ (1 − ht )1−ρ −1 U (C˜t , ht ) = , ρ ∈ [0, 1], if θ > 0 and θ 6= 1, (9) 1−θ U (C˜t , ht ) = ρ ln C˜t + (1 − ρ) ln(1 − ht ), ρ ∈ [0, 1], θ = 1, where 1/θ is the constant elasticity of intertemporal substitution of private consumption and ρ denotes the importance of consumption relative to leisure in utility. 3

Corner solutions in (6)-(8) are avoided, and restrictions (7) and (8) must hold with equality for utility to be maximized.

6

Gustavo A. Marrero and Alfonso Novales

Optimality conditions are standard: the consumption-saving decision, (10), the consumption-leisure choice, (11), C˜t+1 C˜t ρ 1−ρ

(  ) 1   i 1−ρ(1−θ) 1 − ht+1 (1−ρ)(1−θ) h = β , (10) 1 − δ k + rt+1 1 − τ kt+1 1 − ht =

C˜t (1 + τ ct ) , ˜t (1 − ht ) W

(11)

˜ t+1 > 0, ht ∈ (0, 1) and the transversality condition the budget constraint (7), C˜t > 0, K (8).

2.3.

The Public Sector

The public sector collects distorting and non-distorting taxes to finance its total current expenditures, divided into productive public expenditures, I˜tg , and unproductive public expenditures, C˜tg . For any tax system considered, I˜tg = κi Y˜t , C˜tg = κc Y˜t ,

(12) (13)

where κi is the policy instrument and κc is assumed to be exogenous and fixed. Once κi is chosen, we assume it remains constant forever. Public capital accumulates according to ˜ g, ˜ g = I˜g + (1 − δ g )K K t t t+1

(14)

where δ g ∈ (0, 1) is the public capital depreciation factor, which we consider to be different from that of private capital. The government is not allowed to issue any debt, and only flat taxes are considered.4 The budget constraint must balance every period,

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˜ t − τ k A˜t rt . T˜t = C˜tg + I˜tg − τ c C˜t − τ w ht W

(15)

In the simulation exercise, four tax scenarios are alternatively considered: taxing only total gross income, gross capital income, labor income or private consumption.

3.

The Equilibrium, the Calibration, the Simulation and the Government Problem

In this section, we define the competitive equilibrium and the balanced growth path and characterize the transitional dynamics of variables in levels. 4 Focusing on taxes as the public finance instrument has empirical justification: Taxation represents more than 80% of total public revenue in industrialized economies. Additionally, a state-owned industry could be treated in the similar way as taxes [see section 4 of Burgess and Stern (1993)] and printing money to finance public deficit is not permitted in developed countries. Allowing for public debt would extend the model beyond the scope of the paper.

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio

3.1.

7

The Competitive Equilibrium and the Balanced Growth Path

˜ 0, K ˜ g > 0, the competitive equilibrium is a set of alloStartingnfrom an initial state, K 0 o∞ ˜ t, K ˜ t+1 , I˜tk , Y˜t , C˜ g , K ˜ g , I˜g cations C˜t , ht , L , a set of prices p˜ = {rt , w ˜t }∞ t t+1 t t=0 and a t=0  n o∞  ˜ t+1 }∞ , such that, given p˜ and π ˜ : (i) {C˜t , ht , K fiscal policy π ˜= κ ¯ i , η, τ k , τ c , T˜t t=0 t=0 ∞ ˜ t+1 , L ˜ t} maximize households’ welfare [satisfying (7), (8), (10) and (11)]; (ii) {K t=0 sat˜ isfy the profit-maximizing conditions [(2)-(3)], and Kt accumulates according to (5); (iii) ˜ g , I˜g }∞ evolve according to (12)-(14); (iv) the budget constraint of the public {C˜tg , K t+1 t t=0 sector, (15), and the technology constraint (2) to produce Y˜t holds; finally, (iv) markets clear every period, ˜ t = ht , L Y˜t = C˜t + C˜tg + I˜tk + I˜tg .

(16) (17)

A balanced growth path (bgp) is defined as an equilibrium path along which aggregate variables either stay constant or grow at a constant rate. Hereinafter, variables with bar “−” denote values along the bgp. Jones and Manuelli (1997), among others, have shown that cumulative inputs must present constant returns to scale in the private production process for the existence of a steady-growth equilibrium (i.e., α + θg + θk = 1). Additionally, rt must be constant and high enough for the equilibrium to display positive steady-growth. From now on, we will focus on the special case: α + θg + θk = 1. Under these conditions, ˜ t, K ˜ g , C˜ g and T˜t must it is easy to show from the equilibrium conditions, that Y˜t , C˜t , K t t all grow at the same constant rate along the bgp, denoted by γ¯ hereinafter, while bounded variables, such as tax rates, rt and ht , must be constant. Particularizing condition (10) to a bgp equilibrium, a positive long-term growth rate is achieved whenever n h io 1 1 − β(1 − δ k ) 1−ρ(1−θ) − 1 > 0 ⇔ r¯ > . γ¯ = β 1 − δ k + (1 − τ k )¯ r (1 − τ k )β

(18)

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However, even though γ¯ will then be positive, it cannot get so high that it allows households to follow a chain-letter action [(8) must hold on the bgp], ¯ (1−ρ)(1−θ) K ˜ 0 (1 + γ¯ ) β t (1 + γ¯ )t ρ(1 − h) = 0 ⇔ β(1 + γ¯ )ρ(1−θ) < 1, h i1−ρ(1−θ) t→∞ C˜0 (1 + γ¯ )t lim

(19)

which ensures that time-aggregate utility (6) remains finite.

3.2.

Alternative Tax Scenarios

Four alternative single-tax scenarios are considered, with taxes on either total gross income, gross capital income, labor income or private consumption. In each case, the tax rate is chosen so that total public expenditures, C˜tg + I˜tg , can be entirely financed along the bgp

8

Gustavo A. Marrero and Alfonso Novales

with the single tax being considered. Thus, T˜t = 0 along this equilibrium path:
¯W ¯ t + r¯K ¯ t ⇔ τ = κi + κc , I¯tg + C¯tg = τ h ¯ t ⇔ τ k = (κi + κc ) /α, I¯g + C¯ g = τ k r¯K t

t

¯W ¯ t ⇔ τ w = (κi + κc ) / (1 − α) , I¯tg + C¯tg = τ w h I¯tg + C¯tg = τ c C¯t ⇔ τ c = (κi + κc ) (Yt /Ct ),

(20) (21) (22) (23)

where we have imposed τ k = τ w = τ in (20), the tax rate applied to total income. By combining (15), (3) and (4), we obtain T˜t = 0 every period under either output, capital or labor tax systems. Hence, it is only under consumption taxes that T˜t may be different from zero along the transition. Regarding the alternative tax systems, we point out that (i) given κi and κc , tax rates are time-invariant; (ii) any change in κi affects contemporaneously the tax rate being used as fiscal instrument; (iii) that change is more than proportional under either capital, labor or consumption taxes, while being proportional under income taxes; (iii) τ k , τ w and τ must be between zero and one, while τ c cannot be so large that C˜t might ever become negative.

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3.3.

Benchmark Calibration

The economy is calibrated following the standards in the literature. Wherever needed, we have chosen ratios of variables or parameter values so as to approximate industrialized economies during the eighties, considering the time unit to be one quarter. An annual after¯ of 1/3, and an annual growth tax net capital rate of return of 6%, an average labor share, h, rate of 2.5%, are all replicated in steady-state under the benchmark calibration. Private capital depreciates at a 10% annual rate, hence δ k = .025. We assume public capital depreciates at a lower rate of 5% , so that δ g = .0125.5 For the instantaneous utility function, Mehra and Prescott (1985) suggest a relative risk aversion parameter between 1.0 and 2.0, and we choose θ = 1.20 . The elasticity of private capital, α, takes a standard value of .36, hence θg + θk = .64. The empirical literature discussing the productive nature of public capital shows controversial conclusions, different data sources and econometric techniques leading to rather different estimations of θg in a Cobb-Douglas production function.6 In the benchmark calibration, we set θg = .20, but we will perform a sensitivity analysis in section (5.), allowing for θg to vary between .06 and .30. Two different public expenditure concepts are considered: Schuknecht and Tanzi (1997) is used to set the public consumption-to-output ratio, κc , equal to .17. Calibration of the 5 Auerbach and Hines (1987) estimated a depreciation rate in the U.S. of 0.137 for equipment and 0.033 for structures. Since private capital includes a larger share of equipment than public capital, the estimated depreciation rate for private capital is expected to be larger. Ai and Cassou (1995) found support for this in the form of an estimated δ g of just over half that of δ. 6 Aschauer (1989) and Munnell (1990) estimate high values of θg , equal to 0.39 and 0.34, respectively. Accounting for non-stationarity in the data, Lynde and Richmond (1992) and Ai and Cassou (1995) obtain lower but still significant estimates: the former get θg = 0.2 using time series techniques, while the latter estimate ϕ between 0.15 and 0.2, using a GMM method. In a more recent paper, Shioji (2001) uses dynamic panel techniques to estimate the elasticity of output with respect to infrastructure to be somewhere around 0.1 and 0.15. On the other hand, papers by Holtz-Eaking (1994), Hulten and Schwab (1991) and Tatom (1991), among others, put that estimate very close to zero. Sturm et al. (1997) offer a selective review of these empirical studies. See also Cazzavilan (1993), Munnell (1992) and Glomm and Ravikumar (1997).

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio

9

Table 1. Benchmark calibration Taxes Income Capital Labor Consumption

θk .44 .44 .44 .44

α .36 .36 .36 .36

θg .20 .20 .20 .20

F .30 .59 .24 .24

δk .025 .025 .025 .025

¯ h .33 .33 .33 .33

β .996 .996 .996 .996

θ 1.20 1.20 1.20 1.20

ρ .35 .34 .36 .35

κc .17 .17 .17 .17

κi .07 .07 .07 .07

δg .012 .012 .012 .012

τi .24 .67 .38 .52

Note: Each row shows the benchmark calibration under each tax system for θ g = .20. They all ¯ = .33. The reproduce a rate of growth γ ¯ = .62%, an after-tax capital rate of return of .015 and h i column for τ shows the tax rate under each of the four tax rules.

initial level of the public investment ratio, κi , is less evident. In international accounts of fiscal policy variables, public investment generally includes just central government activities, leaving aside local expenditures and public enterprises. Easterly and Rebelo (1993) try to correct for this deficiency. They estimate levels of κi in the eighties of .07 for New Zealand, .11 for Portugal, .07 for Australia, .08 in Japan, .02 in U S, among others. Barro and Sala-i-Martin (1995) set total consolidated public investment to be about 30% of total investment. For industrialized economies, this leads to an average of κi about 7% in the eighties. We use .07 as benchmark value for κi . In section 5., we will show that the welfaremaximizing public investment/output varies only slightly with the level of κi , which will have important qualitative implications. Finally, using steady-state equilibrium conditions for the model in ratios, β, ρ and F are chosen accordingly. In general, these values vary with θg and with the tax system considered. Table 1 shows calibrated values for all parameters, conditional on κi = .07 and θg = .20. Among the alternative tax systems, the main difference refers to the scale factor F . As expected, F must take a higher level to replicate the 2.5% annual growth rate under capital income taxes than under less distorting tax scenarios.

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3.4.

Simulating the Competitive Equilibrium

The competitive equilibrium cannot be solved analytically, so a numerical solution is required. Unfortunately, computing a numerical solution is far from trivial, since the variables in the economy experience growth in steady-state. In that setting, it is fairly simple to solve for stationary ratios, but level variables are needed to analyze welfare issues. The normalized level of a variable Z˜t , Zt = Z˜t /(1 + γ¯ )t , will grow at a zero rate along the bgp, but the steady-state value of Zt is not well defined, so standard numerical methods applied directly to normalized variables cannot be used either. In the Appendix we describe a procedure that uses the dynamics of stationary ratios to recover the equilibrium path for normalized level variables, starting from a given initial state of the economy.7 7

Novales et al. (1999) describes an alternative method to solve for the dynamics of level variables in endogenous growth models.

10

Gustavo A. Marrero and Alfonso Novales

3.5.

Solving the Government’s Problem

The public sector maximizes welfare of the representative household along the competitive equilibrium. The policy instruments are κi and the associated tax rate, and the government commits itself to the announced policy. The economy is assumed to start on the bgp associated to the benchmark calibration, with an initial public investment/output ratio of 7% and a public capital stock, K0g , of 100.8 A standard search method is used to numerically handle this control problem. Given the tax system, the initial state (K0 , K0g ) and a level of κi : (a) (27)-(32) is solved for the bgp and the level of γ¯ is obtained; (b) the process described in the previous subsection allows us to recover time series for Ct and ht ; (c) the utility of the representative consumer is evaluated9 ) ( t  1−θ ∞ X β(1 + γ¯ )ρ(1−θ) Ctρ (1 − ht )1−ρ βt − ; (24) 1−θ 1−θ t=0

(d) the process is repeated for any feasible level of κi , and the one maximizing (24) will be the welfare-maximizing choice. To evaluate the infinite sum in (24), a truncated version with t∗ periods is used, where ∗ t is chosen so that equilibrium time series are close enough to the bgp.10 For each policy, ∗ time series {Ct , ht }tt=0 are used to estimate welfare up to period t∗ : ∗

t X

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t=0

( t  1−θ ) β(1 + γ¯ )ρ(1−θ) Ctρ (1 − ht )1−ρ 1 . − 1−θ (1 − β) (1 − θ)

(25)

After period t∗ , the economy is considered to be close enough to the bgp associated to the implemented policy. Therefore, according to (24), since β(1 + γ¯ )ρ(1−θ) < 1 [by (19)], the term  t   ∞ ¯ 1−ρ 1−θ X β(1 + γ¯ )ρ(1−θ) Ctρ∗ (1 − h) (26) 1−θ ∗ t=t +1  t∗ +1   ρ β(1 + γ¯ )ρ(1−θ) 1−ρ 1−θ ¯   = Ct∗ (1 − h) (1 − θ) 1 − β(1 + γ¯ )ρ(1−θ) approximates aggregate utility after period t∗ , which is added-up to the numerical value obtained from (25).11

¯g . The welfare-maximizing policy is shown to be invariant The initial state is K0g = 100 and K0 = 100/k to this choice. 9 The range of feasible lifetime utility values for (24) is bounded because: (i) the paths of Ct and ht converge to bounded limits, since the numerical procedure imposes the competitive equilibrium to be on the stable manifold; (ii) γ¯ is bounded from above by (19). In addition, the single period utility function is continuous and strictly concave, and the choice set is convex, conditions that ensure the existence of at most one interior solution to the problem of maximizing (24). 10 ∗ ¯ < 10−3 , with t∗ < 1500, due to computational restrictions. t is chosen so that Xt∗ − X 11 Notice that (25) and (26) must be computed simultaneously, because t∗ and Ct∗ depend on the whole transitional dynamics up to period t∗ . 8

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 11

4.

The Public Investment Policy under Alternative Tax Scenarios

In this section we discuss the transitional dynamics and its relevance for correctly characterizing the welfare-maximizing public investment/output ratio. An optimal policy will generally imply a sacrifice in initial welfare (by reducing initial consumption and/or leisure), with a faster accumulation of capital along the former periods of the transition. This accumulation allows for an eventual faster growth in output, investment and consumption, compensating the initial loss of utility. We examine in detail the relationship between this welfare trade-off and the tax system characterizing and interpreting the magnitude of the welfare trade-off, which happens to be tax-dependent. The theoretical analysis is supplemented by numerical results, focusing on the dependence of the welfare-maximizing policy with respect to the tax system and the main structural parameters. Moreover, to make the welfare analysis clear, we compare the welfare dynamics in the following situations: (i) when the κi -ratio is chosen to maximize long-run growth, κi∗ , (ii) when the κi -ratio is chosen to maximize welfare over 4 periods, κisr , and finally (iii) when κi takes the value that maximizes welfare along the entire transition, κi+ . By comparing the obtained optimal ratio with the initial benchmark of 7%, we can discuss whether downsizing improves welfare, and its dependence on the tax system can be evaluated. Notice that the utility paths are not comparable among the alternative tax scenarios considered, since their calibrated scale factors, F , are different. However, ratios of variables are unaffected by scale factors and hence, the resulting welfare-maximizing public investment ratios under alternative tax scenarios can be compared.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

4.1.

Maximizing Steady-State Growth

As in a Barro-type setting, changes in κi may have two different effects on γ¯ : (i) a higher κi has a positive effect on output, since public capital is productive, and (ii) a negative effect, because an increment in κi leads to a parallel rise in the corresponding distorting tax rate, accordingly with (20)-(23). Hence, an inverted U-shaped relationship between γ¯ and κi should be expected. The growth-maximizing public investment ratio, κi∗ , must equalize, in the margin, these two opposite effects. First, notice that the government also uses distorting taxes to finance unproductive expenses, which has an additional negative effect on growth. Secondly, the endogeneity of labor supply makes consumption and labor income taxes to also have adverse effects on growth, although they should be expected to be small. ¯ as a function of κi , for the Figure 4..1 shows the long-run savings rate,12 γ¯ , C/Y and h benchmark economy with θg = .20 and under the four alternative tax systems considered. The domain of κi is restricted to satisfy conditions (18) and (19) on growth, in addition to ¯ < 1. On the other hand, figure 4..2 shows the relationship between κ ∗ C¯ > 0 and 0 < h i and θg , for θg between .06 and .30 under the alternative tax scenarios. For the benchmark calibration, Figure 4..1 shows the inverted U-shaped relationship between γ¯ and κi only under capital, labor and income taxes. Precisely, under these tax 12

The savings rate, st , is equal to in the bgp.

˜ t+1 −K ˜t K , ˜t Y

˜ t , s¯ = γ¯ /M ¯ as in Barro (1990). Dividing this expression by K

12

Gustavo A. Marrero and Alfonso Novales Steady State under Income Taxes 1

0.06 0.05

0.8

0.04 0.6 0.03 0.4 0.02 0.2

0.01

0

0

0

0.07

h

0.14

0.16

0.21

0.28 Ig/Y

Growth Rate(%)

0.35

C/Y

0.42

0.49

Saving Rate

Steady State under Labor Income Taxes 1.6

0.14

1.4

0.12

1.2

0.1

1 0.08 0.8 0.06 0.6 0.04

0.4

0.02

0.2 0

0 0

0.07

0.14

0.21

0.28 0.320.35

0.42

C/Y

Saving Rate

0.49

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Ig/Y h

Growth Rate(%)

(a): Steady-state under alternative tax scenarios ¯ C/ ¯ Y ¯ and γ Note: The left scale corresponds to h, ¯ in percent while the right one refers to the savings rate. The x-axis shows the steady-state public investment/output ratio, I g /Y = κi . Structural parameters are such that, for an initial κi of .07, the annual ¯ after-tax capital rate of return is 6%, h=1/3 and γ ¯ =.62%. Public capital elasticity is .20. Growth-maximizing values of κi are .16, .04, .32 and .51 under total income, capital income, labor income and consumption taxes, respectively.

Figure 1. Continued on next page.

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 13 Steady State under Capital Income Taxes 0.04

1

0.035 0.8

0.03 0.025

0.6

0.02 0.4

0.015 0.01

0.2

0.005 0

0.038 0.035

0

h

0

Ig/Y

0.07

Growth Rate(%)

0.105

C/Y

Saving Rate

Steady State under Private Consumption Taxes 2.4

0.16

2.1

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1.8

0.12

1.5

0.1

1.2

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0.04

0.3

0.02

0 0

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0.14

0.21

0.28

0.35

0.42

0.49

0.51

0

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Ig/Y h

Growth Rate(%)

C/Y

Saving Rate

(b): Steady-state under alternative tax scenarios ¯ C/ ¯ Y ¯ and γ Note: The left scale corresponds to h, ¯ in percent while the right one refers to the savings rate. The x-axis shows the steady-state public investment/output ratio, I g /Y = κi . Structural parameters are such that, for an initial κi of .07, the annual ¯ after-tax capital rate of return is 6%, h=1/3 and γ ¯ =.62%. Public capital elasticity is .20. Growth-maximizing values of κi are .16, .04, .32 and .51 under total income, capital income, labor income and consumption taxes, respectively.

Figure 1.

14

Gustavo A. Marrero and Alfonso Novales

public investm ent ratio (%)

49 42 35 28 21 14 7 0 0.05

0.10

0.15

tax_y

tax_k

Tg 0.20 tax_h

0.25

0.30

tax_c

Note: Figures 4.2 shows the growth-maximizing public investment ratio, κi∗ , as function of the elasticity of public capital, under income taxes ( tax y), capital income taxes (tax k), labor income taxes (tax h) and private consumption taxes (tax c ). Structural parameters are such that, in all cases, the annual after-tax capital rate of return is 6%, ¯ h=1/3 and γ ¯ =.62%, with an initial public investment ratio of .07 (the solid line.)

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Figure 2. Public investment ratio maximizing steady-state growth. scenarios, the positive influence of public capital dominates when κi is low enough (the upward part of the curve). Thus, an increase in κi , when κi is initially below κi∗ (3.8%, 31.8% and 16.5%, under taxes on total income, labour or capital income, respectively) would produce faster growth. On the other hand, levels of κi above κi∗ would lower the long-run saving rate, inducing a negative relationship between γ¯ and κi . Under consumption taxes, the disincentive effect on savings of an increase in the tax rate is not large enough to compensate the positive impact of κi , and the relationship between γ¯ and κi is positive and monotone, although strictly concave. Several additional comments regarding the growth-maximizing public investment ratio are of interest. ¯ due to a tax raise is very small, First, under income taxes, the negative impact on h since the tax burden is shared between labor and capital income. Hence, κi∗ is in that case only slightly lower than θg (1 − κc ), as obtained in section V in Barro (1990) and Marrero and Novales (2005, 2007) under a perfectly inelastic labour supply. Higher levels of θg and lower values of ρ in the calibration would surely magnify the impact on labor supply of an increase in κi , making κi∗ to depart from θg (1 − κc ). For the benchmark calibration, as θg moves between .06 and .30, κi∗ goes from 4.97% to 24.8% (see figure 4..2) while

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 15 θg (1 − κc ) is between 4.98% and 24.9%. Second, as expected, the tax system on capital income is the one that better neutralizes the positive effect of public capital on growth. An increment in κi raises the associated tax rate more than proportionally [see (21)]. Hence, savings and growth are strongly discouraged above a level of κi well below θg (1 − κc ). Consequently, κi∗ goes from 1.2% to just 5.8% as θg changes between .06 and .30 (figure 4..2). ¯ only under labor Third, an increment in κi has a significant (and negative) impact on h income taxes, and just for high levels of κi [figure 4..1]. Consequently, for the mentioned range of θg , κi∗ falls between 22.3% and 34.5%, far above the levels obtained under the previous tax scenarios (figure 4..2). Finally, even though the effect on labor along the transition is significant under consumption taxes (as it will be seen latter), the steady-state effect is less relevant. Hence, κi∗ is around 50.0%, the maximum feasible value, which drives C/Y down to a value close to zero. Clearly, this should be expected to be very different from the welfare-maximizing policy.

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4.2.

Short-run Welfare Effects

Downsizing public investment would reduce the ratio κi below its initial level of 7%, leaving a higher fraction of additional resources available to the private sector, which could be reassigned as additional consumption and savings.13 Additionally, the single tax rate being used to generate revenues will be permanently reduced. Figure 4..3 shows the percent initial impact on working hours and public investment, for the benchmark economy with θg = .20 and for alternative levels of κi . The domain of κi is restricted so that conditions (18) and (19) hold, together with Ct > 0 and 0 < ht < 1 for all period t. Figure 4..3 shows that, with independence of the tax system considered, the crowdingout of private consumption by public investment is shown to be less than perfect, since the initial impact on private investment is different from zero for changes in κi . A cut in κi accompanied of a reduction in the tax rate on either labor income or capital income will initially produce an incentive to work or save, respectively, affecting consumption and saving decisions. Under consumption taxes, the cut in the associated tax rate is immediately transmitted as an incentive to work [by (11)],14 which has a positive effect on the net return to capital and hence on short-term savings and private investment. Under capital income taxes, saving becomes increasingly more attractive than consuming as the tax rate declines. Consequently, a strong enough public investment downsizing policy might take the substitution effect to the point of being more important than the direct income effect on private consumption, becoming optimal for households to initially reduce consumption and accumulate more assets (as is shown in figure 4..3 (a) under capital income taxes for κi < .04). This will be more evident for low levels of the public capital elasticity, θg , for which the income effect of downsizing on consumption is lower. Under all other tax systems considered, private consumption initially increases with reductions in κi , the higher impact being obtained under consumption and labor income taxes. 13

For illustrative purposes, only the short-run welfare impact of downsizing policies is discussed. A symmetric reasoning could be made for upsizing policies. 14 Even though this incentive is lower than under labor income taxes.

16

Gustavo A. Marrero and Alfonso Novales

Figure 4..4 shows the public investment ratio achieving the maximum discounted utility over 4 periods following the policy intervention. This ratio is denoted by κisr hereinafter, and it is shown in the figure as a function of θg , θg ∈ [.06, .30], under the four alternative tax scenarios. Under capital income taxes, a strong reduction in κi has a negative short-run welfare effect, since private consumption, as well as leisure, initially declines (see figure 4..3). As a consequence, under capital income taxes, if the government only cares about the very short-run (i.e., 4 periods), the best strategy is to rise κi above the benchmark 7% when θg is higher than .16, the level of κisr being between 2.9% and 10.0% for the range of θg considered. Under the remaining tax scenarios, downsizing initially increases private consumption but it also produces a decline in leisure, and the global impact on short-run welfare is unclear. Initial effects on main macroeconomic variables are very similar under labor income and private consumption taxes, and in figure 4..4, the two lines coincide. For the benchmark economy, κisr falls between .93% and 5.5% under these two tax systems, as θg takes values between .06 and .30. Hence, under these two tax systems, a certain amount of downsizing is always preferable, since it induces higher initial levels of private consumption without encouraging work excessively, which improves short-run welfare. Finally, under income taxes, κisr falls between 1.3% and 8.3% as θg changes in the range considered. Downsizing enhances short-run welfare in this case only for θg below .26.

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4.3.

The Welfare-Maximizing Policy

In Barro (1990), the public investment ratio that maximizes welfare, denoted by κi+ hereinafter, is equal to κi∗ , the public investment ratio maximizing steady-state growth. Futagami et al. (1993) point out to the lack of transition among steady-states being a crucial feature behind this result, while Glomm and Ravikumar (1994) point out that if current public capital is considered as the productive factor instead of current public investment, there exists a one-period gap between tax revenues and its positive effect on output, inducing κi+ to be strictly lower than κi∗ . Precisely, these two forces contrary to the result of Barro (1990) are present in this model, implying κi+ to be strictly lower than κi∗ , with independence of the tax system considered [for this result, see also Marrero and Novales (2005, 2007) and Marrero (2008)]. Under each of the four tax scenarios considered, we have obtained contrary results when characterizing short-run effect on welfare and the long-run effect on growth of a public investment ratio. In general, downsizing public investment below the 7% benchmark improves steady-state growth under capital income taxes, but not under the alternative tax systems. However, the opposite happens when trying to maximize short-run welfare. Therefore, it is essential to analyze the entire transition to find out the time-invariant public investment ratio that equilibrates welfare loses and gains in the short- and the long-run. Figure 4..5 shows κi+ as a function of θg under the four tax systems. From these figures and the results previously discussed in section 4.2., several characteristics of the welfaremaximizing public investment policy can be pointed out: 1. The direct relationship between κi+ and θg is independent of the tax system considered, since the positive effect of public capital on growth and welfare increases with

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 17

Initial Impact on Private Consumption (%) 7

0

3.5

0

-20

-3.5

-40 -7 0

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tax_c

Initial Impact on Private Investment (%) 85

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120

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0.095

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0.07

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0.21

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0.35

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Public investment-to-output ratio tax_y

tax_k

tax_h

tax_c

(a): Initial impact of main macroeconomic variables Note: The Y-axis shows the percent change of the variable between its initial level and the level attained one period after the policy intervention, under income ( tax y), capital income (tax k), labor income (tax h) and private consumption taxes (tax c ). The new level of κi is shown in the X-axis. Initial public investment ratio is .07 (where curves intersect) and elasticity of public capital is .20. Structural parameters ¯ are such that the annual after-tax capital rate of return is 6%, h=1/3 and γ ¯ =.62%.

Figure 3. Continued on next page.

18

Gustavo A. Marrero and Alfonso Novales Initial Impact on working hours (%) 15 0 -15 7.5

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0 -2.5

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0 .0 35

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0.14 0.21 0.28 0.35 0.42 Public investment-to-output ratio tax_y

tax_k

tax_h

0.49

0.56

tax_c

Initial Impact on Public Investment (%) 600

450

300

150

0

-150 0

0.07

0.14

0.21

0.28

0.35

0.42

0.49

0.56

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Public investment-to-output ratio tax_y

tax_k

tax_h

tax_c

(b): Initial impact of main macroeconomic variables Note: The Y-axis shows the percent change of the variable between its initial level and the level attained one period after the policy intervention, under income ( tax y), capital income (tax k), labor income (tax h) and private consumption taxes (tax c). The new level of κi is shown in the X-axis. Initial public investment ratio is .07 (where the lines intersect) and elasticity of public capital is .20. Structural parameters ¯ are such that the annual after-tax capital rate of return is 6%, h=1/3 and γ ¯ =.62%.

Figure 3.

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 19

public investment ratio (%)

10.5 8.75 7 5.25 3.5 1.75 0 0.05

0.10 tax_y

0.15 T g tax_k

0.20 tax_h

0.25

0.30

tax_c

Note: Figure 4..4 shows the public investment ratio maximizing welfare over 4 periods, κisr , as function of the elasticity of public capital, under income taxes ( tax y), capital income taxes (tax k), labor income taxes (tax h) and private consumption taxes (tax c ). Structural parameters are such that, in all cases, the annual after-tax capital ¯ rate of return is 6%, h=1/3 and γ ¯ =.62%, with an initial public investment ratio of .07 (the solid line).

Figure 4. Public investment ratio maximizing welfare over 4 periods.

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θg . This relationship is linear under total income [as in Barro (1990)] and capital income taxes, while being concave under labor income and private consumption taxes (as it is also the case for the growth-maximizing ratio κi∗ ). 2. Figure 4..5 shows that downsizing improves welfare under capital income taxes for any θg , with κi+ falling between 1.0% and 5.0%. The associated lower tax creates an important incentive for private capital accumulation, with a negative initial impact on private consumption and leisure (figure 4..3 shows these initial impacts under capital income taxes and low enough levels of κi ). After a number of periods, the higher accumulation of private capital along the transition allows for private consumption and leisure to increase after a short number of periods. Besides, convergence to steady-state is now faster, and the steady-state rate of growth is higher. Hence, the welfare-maximizing policy favors higher future increases on private consumption and leisure in detriment of lower levels in the first periods after downsizing, and our result shows that the positive medium and long-term effects are more important for welfare than the short-run effects.

20

Gustavo A. Marrero and Alfonso Novales

public investm ent ratio (%)

35

28

21

14

7

0 0.05

0.10 tax_y

0.15 tax_k

Tg 0.20 tax_h

0.25

0.30

tax_c

Note: Figure 4..5 shows the welfare-maximizing public investment ratio κi , as a function of public capital elasticity, under total income ( tax y), capital income (tax k), labor income (tax h) and private consumption taxes (tax c ).

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 5. Welfare-maximizing public investment ratio. 3. On the contrary, under labor income taxes, κi+ is always above 7% (between 10.8% and 28.2% for θg in [.06, .30]. Hence, it is now optimum to initially sacrifice private consumption and private investment in favor of leisure and public investment (see figure 4..3 for κi > .07 under labor income taxes). Production and private consumption start growing faster after a short number of periods. Contrary to the case under capital income taxes, now the engine for the recovery of these variables is the accumulation of public capital in the short-run and along the whole transition, together with a small disincentive on the accumulation of private capital. 4. Under total income taxes, κi+ falls between those obtained under capital and labor income taxes. Consequently, the welfare effect of downsizing on public investment depends on θg . As it is shown in figure 4..5, downsizing raises welfare for θg ∈[.06, .12], κi+ falling between 3.6% and the benchmark 7.0%, while κi+ goes from 7.0% to 19.4% for θg ∈ [.12, .36]. When θg is below .12, public capital is not productive enough to compensate the negative impact on welfare of an increase in the tax rate, and the welfare gain due to a reduction in the tax rate on capital income prevails (recall that downsizing was optimal under capital income taxes). 5. Even though the short-run behavior of the main macroeconomic variables under consumption and labor income taxes is very similar when κi rises above the bench-

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 21 mark 7% , private consumption is greatly discouraged under consumption taxes as the economy approaches its new steady-state, while the opposite happens regarding savings and growth (see steady-state analysis from figure 4..1). Consequently, the difference between short- and long-run welfare effects of a particular public investment policy is more pronounced under consumption taxes. However, for the benchmark calibration and for the range of θg considered, κi+ is higher under consumption than under labor income taxes, falling between 13.9% and 34.9%, although this difference is lower than when comparing the growth-maximizing ratios. As it was the case under labor income taxes, the engine of the optimal behavior under consumption taxes is the strong accumulation of public capital along the transition. 6. The welfare-maximizing ratio is always lower than the one maximizing growth. For θg = .20, the difference is 0.6, 3.9, 7.3 and 21.2 percentage points under taxes on capital income, total income, labor income or consumption, respectively. As already mentioned, this difference is more pronounced under less distorting tax systems specially under consumption taxes, since, under that tax system, the growth-maximizing strategy initially reduces private consumption and leisure to a level close to zero. This section has analyzed the welfare effects of a time-invariant public investment ratio policy, emphasizing the need to account for welfare along the transition as well as the importance of the tax system being implemented. Starting from a 7% public investment/output ratio, certain level of downsizing is welfare-improving under capital income taxes or under total income taxes, provided public capital is not very productive. On the other hand, welfare is lower when downsizing from the initial 7% ratio under labor income or consumption taxes (the less distorting tax systems). As we are about to see in the next section, this relationship between the welfare effect of downsizing and taxes reverses for some parameter values.

5.

Sensitivity Analysis

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The main aim of this section is to clarify the determinants of the welfare-maximizing, timeinvariant public investment ratio, κi+ , and to discuss the robustness of the main findings in the previous section to changes in parameter values.15

15

The sensitivity analysis just considers the set of parameter values in a neighborhood around the benchmark levels, restricting initial growth to be non-negative, Ct > 0, 0 < ht < 1 for all t, and the N P G condition to hold.

22

Gustavo A. Marrero and Alfonso Novales

In our setting, the transitional dynamics is affected by all economic fundamentals and, consequently, so is κi+ . Even tough the competitive equilibrium cannot be analytically characterized, we use the numerical method described above to characterize a one-to-one mapping between κi+ and any structural parameter. We show that parameters of special importance in determining κi+ are: (i) the public capital share, θg , and the discount factor, β, with whom κi+ is positively related, a standard result in this literature; (ii) the unproductive public expenditure ratio, κc , with whom κi+ is negatively related [as in Marrero (2008) and Marrero and Novales (2005, 2007)]; (iii) the inverse of the elasticity of substitution, θ, and the private capital elasticity, α, with whom the relationship of κi+ depends on the tax scenario. We also show that, at least in a neighborhood of the benchmark parameterization,

Optimal Public Investment Ratio: Sensitivity Analysis on the Unproductive Public Expenditure Ratio (gsc)

Optimal Public Investment Ratio: Sensitivity Analysis on the discount factor

0.49

0.49 0.42

optimal public investment ratio

optimal public investment ratio

0.42 0.35 0.28 0.21 0.14

0.35 0.28 0.21 0.14

0.07

0.07 0 0.988

0.99

tax_y

E 0.994

0.992

tax_k

0.996

0

0.998

0

tax_h

tax_c

0.085

0.17

tax_y

0.255

gsc 0.34

tax_k

0.425

0.51

tax_h

0.595

tax_c

(a). Sensitivity analysis: the optimal public investment ratio as a function of β and κc Note: The benchmark calibration sets α=.36, β =.996, δ g =.0125, δ k =.025, θ=1.20,κi =.07 and κc =.17 and θg =.20. For this benchmark parameterization, κi+ is shown as a grey circle for each alternative tax rule.

Optimal Public Investment Ratio: Sensitivity Analysis on private capital productivity

Optimal Public Investment Ratio: Sensitivity Analysis on the inverse of the elasticity of intertemporal substitution 0.35

0.49

optimal public investment ratio

0.28 0.21 0.14 0.07

0.28

0.21

0.14

0.07

0

tax_y

tax_k

T

2 ... 13 ... 24 ... 35 ... 45 ... 56 ... 67 ... 78 ... 89 ... 10 0

1. 2

optimal public investment ratio

0.35

0. 00 1 0. 13 0. 27 0. 40 0. 53 0. 67 0. 80 0. 93 1. 07

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0.42

tax_h

tax_c

0

0.28

0.36

0.44

tax_y

0.52

tax_k

D

0.6

tax_h

0.68

0.76

tax_c

(b). Sensitivity analysis: the optimal public investment ratio as a function of θ and α Note: The benchmark calibration sets α=.36, β =.996, δ g =.0125, δ k =.025, θ=1.20,κi =.07, κc =.17 and θg =.20. For this parameterization, κi+ is shown as a grey circle for each alternative tax rule.

Figure 6. Continued on next page.

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 23 Optimal Public Investment Size: Sensitive Analysis on public capital depreciation rate (delta_g)

0.35

Optimal Public Investment Size: Sensitive Analysis on the private capital depreciation rate (delta_k)

0.35

0.32 0.28

0.28

0.25 0.21

0.21

0.18 0.14

0.14

0.11 0.07

0.07

0.04 0.00

0

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

delta_g tax_y

tax_k

tax_h

0.04

0

0.005

0.01 tax_y

tax_c

0.015 tax_k

delta_k 0.02

0.025 tax_h

0.03

0.035

tax_c

(c). Sensitivity analysis: the optimal public investment ratio as a function of δ g and δ k Figure 6.

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changes in the values of the public and private capital depreciation factors or the initial public investment-to-output ratio just imply slight changes in κi+ . Figure 5.1 shows the relationship between κi+ and β, κc , θ, α, displaying some interesting features. The direct relationship between κi+ and θg arises because the positive effect of public capital on growth and welfare increases with θg . The relationship is found to be linear under total income [as in Barro (1990)] and capital income taxes, being concave under labor income and private consumption taxes. On the other hand, a low β reduces the importance of growth on welfare, since β indicates the relative preference between present and future consumption. A higher slope between κi+ and β is found under labor income and consumption taxes than under capital income taxes. Consequently, κi+ approaches the initial level of 7% for low values of β independently of the tax system, although the ranking of κi+ stays constant under changes in β, the largest κi+ arising under private consumption taxes and the lowest under capital income taxes. A high level of κc implies less resources for the private sector to save and consume. In addition, it amplifies the distortions on the strategies of the private sector because unproductive expenses are also financed with distortionary taxation. Consequently, the negative impact on consumption and leisure of a given public investment/output ratio becomes more harmful for welfare the higher is κc , which explains the negative relationship between κi+ and κc . In absolute value, the slope of this relationship is higher under less distorting taxes. Hence, for large enough values of κc , even if unrealistic, κi+ will even get below the initial level of 7% independently of the tax system considered. The elasticity of intertemporal substitution of private consumption is constant and equal to 1/θ. A smaller marginal rate of substitution between present and future consumption means that households have a higher preference for current consumption, relative to future consumption. Since β < 1, the long-run is less important than the short run for aggregate welfare under these circumstances. Consequently, a public investment ratio that strongly stimulates growth at the same time that significantly reducing private consumption initially, should not be expected to be optimal for high values of θ (low values of 1/θ). In the policy experiment, for θ > 13, κi+ is systematically below 7% and far below the value of the investment ratio maximizing steady growth, with independence of the tax system considered.

24

Gustavo A. Marrero and Alfonso Novales

Finally, the relationship between κi+ and α depends crucially on the tax system considered [see figure 5..1]. Since θk + θg + α = 1, then, for a given value of θg , an increase in α amounts to a decrease in θk . But θk refers to a learning-by-doing externality in the production function, as in Romer (1986). It is well known that this externality leads in equilibrium to infra-accumulation of private capital, suggesting that welfare maximization might then require lower taxes on private capital. Effectively, we show that the relationship between κi+ and α is positive (i.e., a negative relationship between κi+ and θk ) under capital as well as under total income taxes. However, when capital income is not being taxed, changes in the tax rate cannot reduce the learning-by-doing externality and hence, the relationship between κi+ and α is positive under labor and consumption taxes. As a consequence of all that, when the share of private capital is sufficiently high, κi+ may be lower under less distorting taxes, and downsizing might improve welfare under private consumption or under labor income taxes, but not under either capital or total income taxes, a very relevant result for fiscal policy making. Summing-up, several important implications are found in this section: • Under the alternative tax scenarios, the welfare-maximizing public investment/output ratio depends only slightly on its initial value. Consequently, downsizing public investment improves aggregate utility for economies starting with a public investment/output ratio slightly higher than the welfare-maximizing ratio characterized in section 4.3.. For example, for the benchmark economy, if the initial level of κi is higher than 12.6%, 3.3%, 23.5% and 29.6%, respectively under total income, capital income, labor income and private consumption taxes, certain downsizing in public investment would improve welfare; • For economies with high enough levels of κc , or θ, and/or sufficiently low levels of β or θg , the welfare-maximizing public investment ratio is very similar, and generally lower than the benchmark 7%, with independence of the tax scenario considered; • In economies with a high share of private capital in output, κi+ may be lower under labor income or consumption taxes than under capital or total income taxes.

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6.

Conclusions

We have characterized the welfare-maximizing, time-invariant, public investment/output ratio in an endogenous growth model with non-congested public capital and a constant ratio of unproductive public expenses. In the first part of the chapter, the model was calibrated to be in line with industrialized economies in the 80s, under alternative distorting tax scenarios and different values of the share of public capital in output, a rather controversial parameter to evaluate. The economy was assumed to be initially along the balanced path associated to the benchmark calibration, the initial public investment-to-output ratio being 7% (a broad definition of public investment is used, as described in Easterly and Rebelo (1993)). Main findings in relation to this first part are: (a1) Independently of the distorting tax scenario considered, short and long-run welfare

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 25 effects are of opposite sign, and the welfare-maximizing public investment ratio is strictly lower than the growth-maximizing ratio; (a2) These differences are more important under less distorting tax systems (labor income taxes and, specially, consumption taxes); (a3) For the benchmark calibration, a downsizing in public investment improves welfare when taxing only capital income or total income, provided that public capital is not very productive.

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Since the assumed tax scenarios are too extreme to be realistic, these results should not be interpreted as strict policy recommendations. However, they emphasize the importance of considering the transition between steady-states, as well as the relevance of the specific tax system in effect, when characterizing a welfare improving policy intervention. In the second part of the paper, a local sensitivity analysis was performed. The simplicity of previous models implied a simple relationship between the welfare-maximizing time-invariant public investment/output ratio and economic fundamentals, which disappears in the more complex framework used in this paper. As in previous models, the discount factor and the public capital share of output positively affect the welfare-maximizing ratio. In addition, the unproductive public expenditure/output ratio, the private capital share and the elasticity of intertemporal substitution of private consumption also affect significantly the welfare-maximizing ratio. On the other hand, changes in the initial public investment ratio only barely affect the welfare-maximizing policy. In relation to this second part, the main findings are: (b1) Qualitative findings (a1) and (a2) are robust, at least locally, to changes in structural parameters; (b2) Regarding results in (a3), downsizing public investment becomes welfare improving independently of the tax system when the elasticity of intertemporal substitution of private consumption and the output elasticity of public capital are low enough or when the unproductive public expenditure-to-output ratio is sufficiently high; (b3) Moreover, for a high value of the elasticity of intertemporal substitution, the no-Ponzi game condition limits the positive effect on growth of the public investment policy under labor income and private consumption taxes. As a consequence, the public investment ratio can be then only slightly higher than the 7% benchmark, so the welfare-maximizing ratio might be lower under less distorting taxes than under total income taxes; (b4) Finally, when the output elasticity of private capital is sufficiently high, a downsizing in public investment is welfare improving under private consumption and labor income taxes, but not under capital income or under total income taxes.

7.

Appendix: Solving the Dynamics

In this Appendix we show a procedure to solve for the dynamics of the competitive equilibrium. The outline of the procedure is as follows: (i) Redefine competitive equilibrium conditions in terms of stationary ratios, Zt = ˜ t , Vt = K ˜ g /K ˜ t , Mt = Y˜t /K ˜ t and κt = K ˜ t+1 /K ˜ t . After consolidating some equaC˜t /K t tions, competitive equilibrium conditions reduce to a system of 6 equations in rt , ht and the

26

Gustavo A. Marrero and Alfonso Novales

Zt , Vt , Mt , κt ratios: (  ) 1 (1−ρ)(1−θ) h i 1−ρ(1−θ) 1 − ht+1 Zt+1 = β , 1 − δ k + (1 − τ k )rt+1 κt Zt 1 − ht ρ 1−ρ

=

(1 + τ c ) Zt ht , (1 − α)Mt 1 − ht

Mt = Zt + κc Mt + κi Mt + κt − 1 + δ k , rt = αMt , Mt = κt Wt+1 =

(27) (28) (29) (30)

θ F ht1−α Wt g , (1 − δ g ) Wt +

(31) κi Mt .

(32)

Condition (27) comes directly from (10); (28) comes from (11) and Lt = ht ; (29) from combining (1), (7), (15), (12), (13); (30) and (31) come from (3) and (4), respectively; ˜ t in (12) and (14). finally, (32) comes from dividing by K (ii) Pick structural parameters to solve (27)-(32) for the bgp.16 (iii) Log-linearize (27)-(32) around the bgp to solve for the dynamics of stationary ratios [Uhlig (1999)]. Let us denote by W (t) = (ktg ) the single beginning-of-period state variable and by Q(t) the vector of real variables Q(t) = (Zt , Mt , rt , ht , κt ) ,and by w(t) ¯ and Q. ¯ The log-linear and qˆ(t) their log-deviations around their values along the bgp, W approximation to conditions (27)-(32) can then be rewritten more compactly as: 0 = Aw(t + 1) + Bw(t) + C qˆ(t),

(33)

0 = F w(t + 2) + Gw(t + 1) + Hw(t) + J qˆ(t + 1) + K qˆ(t),

(34)

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where those conditions showing dynamics of any variable in Q(t) [(27) in our case] are included in (34), and matrices A, B, C,..., are functions of all structural and fiscal policy parameters. Details for the log-linearization of (27)-(32), together with matrices A, B, C,..., are shown below. (iv) The following log-linear law of motion for Q(t) and W (t) is assumed: w(t + 1) = P w(t),

(35)

qˆ(t) = S x ˆ(t),

(36)

where P and S are free matrices of dimension (1x1) and (5x1). Next, conditions (33)-(34) are directly solved by the undetermined coefficients method, imposing the eigenvalues of P to be inside the unit circle, since otherwise Q(t) and W (t) would present explosive paths. (v) Starting at (K0 , K0g ), we have k0g = K0g /K0 , and values of C0 , Y0 , r0 , h0 and k1 are directly obtained from     C0 K0 0 0 0 0  Y0   0 K0 0 0 0       r0  =  0 0 1 0 0  (37)     Q(0),  h0   0 0 0 1 0  k1 0 0 0 0 1 16

In general, analytical expressions characterizing the bgp ara unavailable, so the existence and uniqueness of the bgp is checked through the numerical computation.

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 27 
 ¯ = (Z0 M0 r0 h0 κ1 )′ , and W1 is where, from (36), Q(0) = exp S ln k0g − ln k¯g + ln Q obtained from (35). (vi) K1 and K1g (the state of the economy next period) are easily recovered from κ0 : K1 =

K0 κ0 and K1g = W1 K1 . 1 + γ¯

(38)

(vii) Time series for normalized variables for successive periods are recovered by going recursively through steps (v) and (vi). Stability of the resulting time series is guaranteed since appropriate stability conditions were implemented when solving (35)-(36).

7.1.

Log-linear Optimal Conditions

Uhlig (1999) proposes a procedure where optimal conditions are log-linearized without need of differentiating. ˆt and yˆt denote
Let x
the variables in log-deviation to their steady¯ and yˆt = ln Yt /Y¯ . X a can be approximated: state, x ˆt = ln Xt /X 

X ¯ X

a

   X ¯ a (1 + aˆ = exp a ln ¯ = exp(aˆ x) ≃ (1 + aˆ x) ⇒ X a ≃ X x). X

(39)

In addition, x ˆyˆ ≃ 0 if variables are close enough to their steady-state values. Log-linearized versions of (27)-(32) are (all variables in log-deviations about steady-state): ¯ ˜rk (1 − τ¯k ) ρ ˜˜θh k ˆ t+1 − h ˆ t ) − θ¯ ( h rˆt+1 = 0, ¯ ¯ R 1−h 1 ˆ zˆt − m ˆt + ¯ ht = 0, 1−h ¯m ¯m ¯m M ˆ t − Z¯ zˆt − κi M ˆ t − κc M ˆ t − (1 + γ¯ )ˆ κt = 0,

zˆt+1 − zˆt + κ ˆt +

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r¯k rˆtk

− αm ¯m ˆ t = 0, ˆ m ˆ t − (1 − α)ht − θg w ˆt = 0 g ¯ ¯ ¯ (ˆ κt + w ˆt+1 ) (1 + γ¯ )W − (1 − δ )W w ˆt − κi M m ˆ t = 0,

(40) (41) (42) (43) (44) (45)

h ˜  i 1 = ln KK˜t+1 − ln(1 + γ¯ ) , ˜θ = 1−ρ(1−θ) , ρ ˜ = (1 − ρ)(1 − θ), t ¯ 1−α W ¯ 1−αθg −1θg , r¯k = αM ¯ θg −1 − δ g , M ¯ = Fh ¯ − δ k , and R ¯ = γ¯ = κi F h
k k k k 1 + (1 − τ¯ )¯ r − τ¯ δ . Tax-rates are constant. Steady-state conditions have been used ¯ , deto simplify these expressions. The endogenous state variable is vˆt = ln Wt − ln W noted in general  terms by x ˆ(t). Endogenous control variables are included in yˆ(t)′ =  ˆ t , rˆk , κ m ˆ t , zˆt , h t ˆ t . From (40)-(45), log-linear conditions can be rewritten:

where κ ˆt

Aˆ xt+1 + B x ˆt + C yˆt = 0,

(46)

Fx ˆt+2 + Gˆ xt+1 + H x ˆt + J yˆt+1 + K yˆt = 0,

(47)

28

Gustavo A. Marrero and Alfonso Novales

where ¯ 0 0 0 0 (1 + γ¯ )W

A =

−1 1  M ¯ (1 − κi − κc ) −Z¯  ¯ C =  −αM 0   1 0 ¯ −κi M 0 F = 0, G = 0, H = 0,  ˜ ˜ ¯ θ¯ r k (1−¯ τ k) θ˜ ρh J = 0 1 1− ¯ ¯ − R h 


′

¯ 0 0 0 −θg −(1 − δ g )W  1 0 0 ¯ 1−h 0 0 −(1 + γ¯ )   , 0 r¯k 0   −(1 − α) 0 0 ¯ 0 0 (1 + γ¯ )W , B=

0



, K=



˜ ¯

θ˜ ρh 0 −1 − 1− ¯ h

0 1




′

,

.

References [1] Ai, C. and S.P. Cassou (1995). “A normative analysis of public capital”, Applied Economics, 27, 1201-1209. [2] Aschauer, D.A. (1989). “Is Public Expenditure Productive?”, Journal of Monetary Economics, 23, 177-200. [3] Aschauer, D.A. (2000). “Do states optimize? Public capital and economic growth”, Annals of Regional Science 34 (3), 343-363. [4] Auerbach, A.J. and J.R. Hines, Jr. (1987). “Anticipated tax changes and the timing of investment”. In Martin Feldstein (ed.), The Effects of Taxation on Capital Accumulation, pp. 163-196. Chicago and London: The University of Chicago Press. [5] Barro, R.J. (1990). “Government Spending in a Simple Model of Endogenous Growth”, Journal of Political Economy, 98 (5), S103-S125.

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[6] Barro, R.J. and X. Sala-i-Martin (1995). “Economic Growth”, Advanced Series in Economics, McGraw-Hill. [7] Burgess, R. and N. Stern, (1993). “Taxation and Development”, Journal of Economic Literature, 31, 762-830. [8] Cassou, S.P. and K.J. Lansing (1998). “Optimal fiscal policy, public capital and the productivity slowdown”. Journal of Economic Dynamics and Control 22, 911-935. [9] Cassou, S.P. and K.J. Lansing (1999). “Fiscal policy and productivity growth in the OECD”. Canadian Journal of Economics 32(5), 1215-1226. [10] Cazzavilan, G. (1993). “Public Capital and Economic Growth in European Countries: A Panel Data Approach”, WP 93.11 (University of Venice, Venice). [11] Chen, B.L. (2006). “Economic growth with an optimal public spending composition”, Oxford Economic Papers, 58, 123-136.

Transitional Dynamics and Welfare Effects of the Public Investment/Output Ratio 29 [12] Easterly, W. and S. Rebelo (1993). “Fiscal Policy and Economic Growth: An Empirical Investigation”, Journal of Monetary Economics, 32, 417-458. [13] Futagami, K., Y. Morita and A. Shibata (1993). “Dynamic Analysis of an Endogenous Growth Model with Public Capital”, Scandinavian Journal of Economics, 95 (4), 607625. [14] Glomm, G. and B. Ravikumar (1994). “Public Investment in Infrastructure in a Simple Growth Model”, Journal of Economic Dynamics and Control, 18, 1173-1187. [15] Glomm, G. and B. Ravikumar (1997). “Productive Government Expenditures and Long-Run Growth”, Journal of Economic Dynamics and Control, 21, 183-204. [16] Glomm, G. and B. Ravikumar (1999). “Competitive equilibrium and public investment plans”, Journal of Economic Dynamics and Control 23, 1207-1224. [17] Holtz-Eakin, D. (1994). “Public-sector capital and the productivity puzzle”, Review of Economics and Statistics 76, 12-21. [18] Hulten, C.R. and R.M. Schwab (1991). “Is there too little public capital?”, Discussion paper, American Enterprise Institute Conference on Infrastructure Nedds. [19] Jones, L.E. and R.E. Manuelli (1990). “A Convex Model of Equilibrium Growth: Theory and Policy Implications”, Journal of Political Economy, 98 (5), 1008-1038. [20] Jones, L.E. and R.E. Manuelli (1997). “The Sources of Growth”, Journal of Economic Dynamics and Control, 21, 75-114. [21] King R.G. and S. Rebelo (1988). “Production, growth and business cycles, II: New Directions”, Journal of Monetary Economics, 21, 309-341. [22] Lynde, C. and J. Richmond (1992). “The role of public capital in production”, The Review of Economics and Statistics 74(1), 37-44.

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[23] Lucas, R.E., Jr. (1990). “Supply-side economics: an analytical review”, Oxford Economic Papers, 42, 293-316. [24] Marrero, G.A. (2005). “An active public investment rule and the downsizing experience in the US: 1960-2000”. Topics in Macroeconomics: Vol. 5 : Iss. 1, Article 9. [25] Marrero, G.A. (2008). “Revisiting the optimal stationary public investment policy in endogenous growth economies”, Macroeconomic Dynamics, in press. [26] Marrero, G.A. and A. Novales (2005). “Growth and welfare: distorting versus nondistorting taxes”, Journal of Macroeconomics, 27, 403-433. [27] Marrero, G.A. and A. Novales (2007). “Income taxes, public investment and welfare in a growing economy”, Journal of Economic Dynamics and Control, 31(10), 33483369.

30

Gustavo A. Marrero and Alfonso Novales

[28] Mehra, R. and E.C. Prescott (1985). “The equity premium: a puzzle”, Journal of Monetary Economics, 15, 145-161. [29] Munnell, A. (1990). “How does public infrastructure affect regional performance?”, New England Economic Review, Sept./Oct., 11-32. [30] Munnell, A. (1992). “Policy watch: infrastructure investment and economic growth”, Journal of Economic Perspectives, 6, 189-198. [31] Musgrave, R. (1997). “Reconsidering the Fiscal Role of the Government”, American Economic Review, 87, 156-159. [32] Novales, A., E. Dom´ınguez, J.J. P´erez and J. Ruiz (1999). “Solving Nonlinear Rational Expectations Models by Eigenvalue-Eigenvector Descompositions”, Chapter 4 in Computational Methods for the Study of Dynamic Economies, R. Marim´on and A. Scott (eds.), Oxford University Press. [33] Ratner, J.B. (1983). “Government Capital and the Production Function for the US Private Output”, Economic Letters, 13, 213-217. [34] Romer, P.M. (1986). “Increasing Returns and Long-Run Growth”, Journal of Monetary Economics, 94 (5), 1002-1037. [35] Shioji, E. (2001). “Public capital and economic growth: a convergence approach”, Journal of Economic Growth 6, 205-227. [36] Schuknecht, L. and V. Tanzi (1997). “Reconsidering the Fiscal Role of the Government: the International Perspective”, American Economic Review, 87, 164-168. [37] Sturm, J.E., G.H. Kuper and J. de Haan (1997). “Modelling government investment and economic growth on a macro level: a review”. In Brakman, S., H. Van Ees and S.K. Kuipers (eds.), Market Behavior and Macroeconomic Modeling. London: Macmillan/St. Martin’s Press.

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[38] Tatom, J.A. (1991). “Public capital and private sector performance”, Federal Reserve Bank of St. Louis Review, May/June 3-15. [39] Turnovsky, S.J. (1996). “Optimal tax and expenditure policies in a growing economy”, Journal of Public Economics, 60, 21-44. [40] Turnovsky, S.J. (2000). “Fiscal policy, elastic labor supply and endogenous growth”, Journal of Monetary Economics, 45, 185-210. [41] Uhlig, H. (1999). “A Toolkit for Analyzing Non-linear Dynamic Stochastic Models Easily”, Chapter 3 in Computational Methods for the Study of Dynamic Economies, R.Marim´on and A. Scott (eds.), Oxford University Press.

In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 31-57

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 2

L E´ VY P ROCESSES AND O PTION P RICING BY R ECURSIVE Q UADRATURE Gianluca Fusai1∗, Giovanni Longo1†, Marina Marena2‡ and Maria Cristina Recchioni3§ 1 Dipartimento SEMeQ, Universit`a del Piemonte Orientale via Perrone 18, 28100 Novara, Italy 2 Dipartimento De Castro, Universit`a di Torino Piazza Arbarello 8, 10122 Torino, Italy 3 Dipartimento di Scienze Sociali “D. Serrani”, Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy

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Abstract In this paper we present a quadrature method for pricing discretely monitored pathdependent options when the dynamics of the underlying assets are described by L´evy processes. The literature mainly deals with the continuous monitoring case and assumes that the log-returns are normally distributed. The contribution of the present paper consists in providing a flexible computational method that allows to deal with the discrete monitoring rule that is common for exotic options and general enough to deal with non-Gaussian models. Since it is well known that the convergence of the discrete monitoring price to its continuous monitoring counterpart is very slow (Broadie et al. (1997, 1999)), fast and accurate pricing methods for the discrete monitoring case become particularly important. Also, effective alternatives to the Black-Scholes world, such as L´evy processes, are becoming increasingly popular, Cont and Tankov ([8]). The hybrid computational method proposed combines numerical inversion of the characteristic function and Gaussian quadrature extending the approach proposed by Tse et al. (2001) and Sullivan (2000) to the case of exponential L´evy processes. The higher accuracy of Gaussian quadrature provides accurate results up to a very large number of monitoring dates, much larger than the number of monitoring dates usually considered in the existing literature. We present an error analysis for the numerical method proposed and we provide detailed numerical examples for plain vanilla ∗

E-mail address: E-mail address: ‡ E-mail address: § E-mail address: †

[email protected] [email protected] [email protected] [email protected]. Ph. n.+39-071-2207066, FAX n.+39-071-2207150

32

Gianluca Fusai, Giovanni Longo, Marina Marena et al. options, discrete barrier options (with a single or a double barrier), and Bermudian options for a variety of L´evy processes, such as the Gaussian, CGMY, Kou DoubleExponential, Merton Jump-Diffusion.

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1.

Introduction

We present a quadrature method for pricing discretely monitored path-dependent options under L´evy processes. Most of the literature deals with the continuous monitoring case under the Black-Scholes (Gaussian) model. However, in the market practice, the discrete monitoring rule is commonly used for exotic options such as barrier options, lookback options and Asian options. Since it is well known that the convergence of the discrete monitoring price to its continuous monitoring counterpart is very slow (Broadie et al. (1997, 1999), Fusai and Recchioni (2007)), the development of fast and accurate pricing methods for the discrete monitoring case is a relevant problem. Moreover, alternatives to the Black-Scholes world, such as L´evy processes, are becoming increasingly popular. In fact, L´evy processes present several appealing features. For example, the L´evy processes overcome the Black and Scholes model since they allow to reproduce skewed and fat-tailed distributions as well as the smirk and the smile effects in the implied volatility. Moreover they are analytically tractable since the characteristic functions of the most common L´evy processes are known with an explicit formula (see Section 2). However, no general analytical solutions are available for the valuation of path dependent options under L´evy processes (Kyprianou et al. (2005)), so that this evaluation requires the use of numerical methods, which include semianalytical approximations using Wiener-Hopf factorization, Monte Carlo algorithms, PIDE approaches, lattices and quadrature methods. Recently the quadrature approach has received some attention in the literature. In Gaussian models, this approach has been introduced by AitSahlia and Lai (1997, 1998), Reiner (2000), Tse et al. (2001), Sullivan (2000), and Andricopoulos et al. (2003). O’Sullivan (2005), Fusai and Recchioni (2007) and Lord et al. (2007) extend the quadrature method to exponential L´evy models. The option price is a conditional expectation in a risk neutral world. More specifically, pricing a path-dependent option with discrete monitoring involves the evaluation of one dimensional nested integrals. Quite naturally, this multidimensional integral can be approximated using a quadrature rule. This approach via a suitable use of a quadrature rule requires only the knowledge of the conditional transition probability density function of the underlying process. In contrast to lattice and finite-difference methods, this quadrature approach avoids distribution errors arising from the time and space discretization of the process dynamics, and improves efficiency, since only the monitoring dates need to be considered. By computing the transition density by the inversion of the characteristic function, O’Sullivan (2005) extends the approach in Andricoupolos et al. (2003) to a wide range of stochastic processes with independent increments, whose conditional characteristic function is known in closed form. Lord et al. (2007) combines quadrature methods with the methods based on Fourier transforms (Carr and Madan (1999)). In our approach, the multidimensional integral is approximated using a recursive quadrature procedure combined with the FFT inversion of the characteristic function. We

L´evy Processes and Option Pricing by Recursive Quadrature

33

exploit the high accuracy of the Gaussian quadrature rule, extending the approach proposed by Tse et al. (2001) and Sullivan (2000) to the case of exponential L´evy processes. The FFT inversion of the characteristic function allows us to compute the transition probability density function on an equally spaced grid. Since Gaussian quadrature requires not equally spaced grid points, we apply an interpolation scheme based on cubic spline. The resulting hybrid method, due to the higher accuracy of Gaussian quadrature over Newton-Cotes rules, provides accurate results up to a very large number of monitoring dates and due to its low computational cost allows to deal with a number of monitoring dates much larger than the
number of dates usually considered in the literature. The computational cost is O N S 2 , where S + 1 in the number of grid points and N is the number of monitoring dates and the convergence of the numerical method proposed is theoretically proved when the Gauss Legendre quadrature rule is used to price discrete monitoring double barrier options. The remainder of this paper is organized as follows. In Section 2. we present the L´evy market model for discrete monitoring path dependent options. In particular we derive a recursive formula to price these path dependent options. In Section 3. we propose a numerical method to price discrete monitoring path dependent options and we present an error analysis when the Gauss Legendre quadrature formula is used to evaluate double barrier knock-out barrier options. In Section 4. we present some numerical experiments showing the accuracy and the efficiency of the algorithm in pricing a few path dependent options. In Section 5. we draw some conclusions. Finally, all proofs are gathered in the Appendix.

2.

The L´evy Market Model for Discrete Monitoring Path Dependent Options

Let the risk-neutral process for the stock price be given by

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St = S0 exp ((r − q − m) t + Lt ) , 0 ≤ t,

(1)

where r is the continuously compound interest rate, q is the rate of a dividend yield, Lt is a L´evy processand m is the so-called compensator chosen to ensure that St is a martingale. The L´evy process is fully identified by its
characteristic exponent (i.e. the logarithm of the characteristic function) ψ (ξ) = ln E eiξL . In Table 1 we give the characteristic exponent for the most common L´evy processes: Gaussian, CGMY, Merton jump-diffusion model and the Kou double exponential model. Then, (see, for example, [15], for further details) we can set m = r − q − ψ(−i), (2) i.e. m depends on the Levy process considered, as shown in Table 2. We are interested in pricing path-dependent options under a discrete monitoring rule. Let T be the option maturity (T > 0) and N be a positive integer, we assume that the monitoring dates are equally spaced, that is we consider the N + 1 distinct points t0 = 0, t1 = ∆, t2 = 2∆, . . . , tN = N ∆, where ∆ = T /N is the size length of the partition of the interval [0, T ]. Note that tN = N ∆ = T is the option maturity. The choice of a uniform partition of the interval [0, T ] is made to keep the exposition simple. In fact, as we explain in the following sections, the numerical method developed here can be formulated in a more general context with slight modifications.

34

Gianluca Fusai, Giovanni Longo, Marina Marena et al. Table 1. Characteristic exponents of some parametric L´evy processes. Model G(σ > 0)

ψ (ξ) 2 − σ2 ξ 2



CΓ (−Y ) (M − iξ)Y − M Y + (G + iξ)Y − GY

CGMY(C > 0, G > 0, M > 0, Y < 2) DE (σ > 0, λ > 0, p∗ > 0, η 1 > 0, η 2 > 0)

− 21 σ 2 ξ 2 + λ

 (1−p∗ )η  ∗ η1 2 + ηp +iξ −1 η 2 +iξ 1   1 2 2

− 12 σ 2 ξ 2 + λ eiξα− 2 ξ

JD(σ > 0, λ > 0, δ > 0)



δ

−1

Table 2. Drift parameter under the risk-neutral measure. m = (r − q) − ψ (−i) 2 r − q − σ2 

Model G CGMY DE JD

r − q − CΓ (−Y ) (M − 1)Y − M Y + (G + 1)Y − GY   ∗) η p∗ η 1 2 + − 1 r − q − 12 σ 2 − λ (1−p η 2+1 η 1 +1 α+ 21 δ 2 1 2 −1 r − q − 2σ − λ e



The log-return on a time interval of length size ∆ is defined by: ln

St+∆ = (r − q − m)∆ + Lt+∆ − Lt , St

(3)

and it has characteristic exponent given by: ψ ∆ (ξ) = (ψ (ξ) + iξψ(−i)) ∆, ξ ∈ (−∞, +∞),

(4)

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Let us consider the standard transformation x = ln (St ) and y = log (St+∆ ) and let us define Vn (x, tn ) to be the option price at the n − th monitoring date At maturity tN = T = N ∆, we have: VN (x, tN ) = F (x) , x ∈ (−∞, +∞), (5) where F is the so-called payoff function. Let Φ(x, y, ∆), x, y ∈ (−∞, +∞), ∆ > 0, be the transition probability density function that we assume to be a function of the difference y − x, that is we have Φ(x, y, ∆) = g (y − x, ∆), where g is the probability density describing the transition from x at time t to y at time t + ∆, ∆ > 0. From the knowledge of the transition probability density function the pricing problem can be written recursively as: Z −r ∆ Vn−1 (x, tn−1 ) = e g (y − x, ∆) Vn (y, tn ) dy, n = N, N − 1, . . . , 1, (6) D

where D is the domain of integration, D ⊆ R, depending on the option type and tn = n∆, n = N, N − 1, . . . , 0, ∆ > 0. If we consider the plain vanilla option, the domain of integration D is infinite. If D is a semi-infinite interval, the above iteration is related to the pricing of discretely monitored single barrier options, i.e. options that die (down-and-out or up-and-out) or start to live (down-and-in or up-and-in) when the underlying asset hits the

L´evy Processes and Option Pricing by Recursive Quadrature

35

boundary. If D is a finite interval, the pricing problem is related to the pricing of discretely monitored double barrier options, i.e., options that die (double knock-and-out option) or start to live (double knock-and-in option) when the underlying asset hits the boundary. Minor modifications of equation (6) make possible to price Bermudian options. As shown in (6), the pricing problem involves a recursive univariate integration: one integral for every monitoring date. Roughly speaking, we compute the integral in (6), using Gaussian quadrature on S + 1 points, with weights w ˜j and nodes xj , j = 0, 1, . . . , S, that is we consider the following approximation of Vn−1 Vn−1 (x, tn−1 ) ≈

S X

wj g (xj − x, ∆) Vn (xj , tn ) ,

(7)

j=0

where wj = e−r ∆ w ˜j , j = 0, 1, . . . , S. Indeed we use formula (7) when the transition probability density function is known in a closed form. When we do not dispose of the closed form of the function g (and this happens for several Levy processes) we can use the representation of the kernel g through its Fourier transform: 1 g (z, ∆) = 2π

Z∞

e−iξz χ(ξ, ∆)dξ,

(8)

−∞

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where χ(ξ, ∆) is the characteristic function (Fourier transform) of the kernel g(z, ∆) as a function of z, so that the variable ξ is the conjugate variable of z. For L´evy processes, we have χ (ξ, ∆) = eψ∆ (ξ) , ξ ∈ (−∞, +∞). (9) The idea of using the inversion of the characteristic function (i.e. of the Fourier transform) to recover the transition probability density and of the quadrature rule to approximate the integrals has been recently introduced in [14]. In the present paper inversion (8) is performed through a fast fractional Fourier transform algorithm, see [3] and [7], providing g on an equally spaced grid. Then we use Gaussian quadrature rules to compute the integral appearing in (6), approximating the transition density function on the Gaussian nodes with a cubic spline. Hence the numerical method is an hybrid QUAD-FFT numerical method [14]. Finally it is worth of noting that the integral appearing in (6) is a convolution type that is a cross correlation of the function g and the function Vn . This fact has been recently exploited by some authors such as, for example, [12]. We do not consider this approach that, roughly speaking, consists of approximating the inverse Fourier transform of the product of the Fourier transforms of the transition probability density g and of the function Vn , n = N, N − 1, . . . , 1. This choice is due to the fact that we want to exploit the application of the Gaussian quadrature rules to approximate the integral appearing in (6) and that we do not want to introduce the error due to the numerical approximation of the Fourier transform of the functions VN and the use of a damping factor (see [14], [6]). In the numerical experience of Section 4. we consider the following contracts: plain vanilla options, single barrier options and double barrier options, allowing the early exercise. Let us give a brief description of the contracts.

36

2.1.

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

Plain Vanilla Options


+ For this class of options the payoff function is given by F (x) = ex − ek , where k is the logarithm of the strike, and D = (−∞, ∞). The recursive relation is given by: Z ∞ −r∆ Vn (x, tn ) = e g (y − x, ∆) Vn+1 (y, tn+1 ) dy, n = N − 1, N − 2, . . . , 0, (10) −∞

and we replace the unbounded interval D with the bounded interval [xmin , xmax ], where xmin and xmax are chosen according to the moment bounds for tail probabilities in [13]: P (log (ST ) ≤ xmin ) ≤ 10−h ;

P (log (ST ) ≥ xmax ) ≤ 10−h

(11)

where the threshold parameter for the truncation error h is set equal to 8. Obviously in this case the option value could be computed using only one step, that is choosing ∆ = T . We use the recursion approach also to test the efficiency of the proposed procedure. In fact, in this case, we dispose of benchmarks that is of very accurate approximations obtained with alternative methods (see for example [6]). Thanks to the application of our procedure on this basic case and to the use of the benchmarks we have indications on the minimum number of quadrature points necessary to achieve a prefixed error.

2.2.

Single Barrier Options


+ For this class of options the payoff function is given by F (x) = ex − ek 1D and D = [l, ∞), where 1D is the indicator function. Here l is a lower log-barrier. The recursive relation is: Z ∞ Vn (x, tn ) = e−r∆ g (y − x, ∆) Vn+1 (y, tn+1 ) dy, n = N − 1, N − 2, . . . , 0, (12) l

and we truncate the upper infinite domain to [l, xmax ] choosing xmax as described in (2.1.).

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2.3.

Double Barrier Options


+ For this class of options the payoff function is given by F (x) = ex − ek 1D and D = [l, u] with u the upper log-barrier. The recursive relation is: Z u Vn (x, tn ) = e−r∆ g (y − x, ∆) Vn+1 (y, tn+1 ) dy, n = N − 1, N − 2, ..., 0. (13) l

We note that in this case we do not truncate the domain so that the error introduced is due only to the quadrature rule adopted and will not be affected by the truncation of the domain.

2.4.

Early Exercise Options


+ For this class of options the payoff function is given by F (x) = ex − ek 1D and we have a early exercise payoff Gn (x) at time tn . The recursive relation is:   Z −r∆ g (y − x, ∆) Vn+1 (y, tn+1 ) dy , Gn (x) , n = N −1, N −2, ..., 0, Vn (x, tn ) = max e D

where D can be chosen as in the previous Subsection 2.1., 2.2., 2.3..

(14)

L´evy Processes and Option Pricing by Recursive Quadrature

3.

37

An Hybrid Quadrature Based Numerical Method

Let us describe in detail the numerical method for pricing discrete monitoring path dependent options. Let us recall that we start approximating the integral in (6) using Gaussian quadrature on S + 1 nodes, xj , j = 0, 1, . . . , S, and weights wj , j = 0, 1, . . . , S, that is we consider the following approximation of Vn−1 at x = xi , i = 0, 1, . . . , S: Vn (xi , tn ) ≈

S X

wj g (xj − xi , ∆) Vn+1 (xj , tn+1 ) , n = N − 1, N − 2, . . . , 0,

(15)

j=0

where we recall that tn+1 − tn = ∆, n = 0, 1, . . . , N − 1 and wj , j = 0, 1, . . . , S are the Gaussian nodes multiplied by a suitable factor depending on the integration domain D. We define the matrix A to be a (S + 1) × (S + 1) whose entries are given by: (A)i,j = g(xj − xi , ∆)wj ,

(16)

using (16) and formula (15) we can define a backward induction scheme to price the options that is: V (x,n) = AV (x,n + 1) , n = N − 1, N − 2, . . . , 0, (17) where the vectors x and V(x, n), n = N − 1, N − 2, . . . , 0, are given by: 

  x= 

x0 x1 .. . xS



  , 



  V (x,n) =  

Vn (x0 , tn ) Vn (x1 , tn ) .. . Vn (xS , tn )



  . 

(18)

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Moreover, due to the fact that we have chosen the same log-price grid at each time step and to the fact that the kernel g is of convolution type, we actually need the density function g (z, ∆) only on the interval [−b, b] , where b = xmax − xmin (see formulae (10), (12), (13)). The basic steps of the algorithm can be summed up as follows: 1. Define the parametric model ψ (ξ) (see Table 1 and Table 2) for log-returns. 2. Using the FRFT algorithm, compute the density function of the log-returns on [−b, b] . 3. Choose the Gaussian quadrature rule according with the integration interval D and interpolate the transition probability density function using the natural cubic splines at the nodes of the quadrature rule assigned, construct the matrix A. 4. From the terminal condition V (x,N ) , start the recursion, that is V(x, n) = AV(x, n + 1), n = N − 1, N − 2, . . . , 0.

38

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

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We note that when the payoff function is not a smooth function on D we can split the integration domain D in subsets in order to have a smooth function in each subset and we can apply the Gauss quadrature formula in each subset. In order to keep the exposition simple we consider the case of smooth integrand functions. Indeed the functions Vn , n = N − 1, N − 2, 1 . . . , 0, defined by the recursion (6), are smooth functions since they are defined by a convolution with a smooth kernel. Let us conclude this section estimating the computational cost of the procedure described in Steps 1, 2, 3. We note that when we evaluate the price of a barrier options on N monitoring dates the computation cost is given by N (S + 1)2 plus the computational cost required to compute the matrix A. When we consider equally spaced monitoring dates the computation of the matrix A is independent of the index of the monitoring dates and depends only on the time size length ∆. Recall that the computation of the matrix A requires the computation of a fast fractional Fourier transform and the construction of a cubic spline interpolating the transition probability function on the grid zk , k = 0, 1, . . . , NF − 1 of the FRFT procedure. The computational cost of the fractional Fourier transform, as mentioned previously, is O(NF log2 NF ) arithmetic operations as NF goes to infinity and the cost of the evaluation of the cubic spline coefficients is O(NF ) arithmetic operations as NF goes to infinity. Finally, the evaluation of the cubic spline that approximates the transition probability density on the Gaussian quadrature nodes xi , i = 0, 1, . . . , S, requires O(S) arithmetic operation as S goes to infinity so that the computational cost of the matrix A is given by O(NF (1 + log2 NF ) + S 2 ), as NF and S go to infinity with S/NF < 1. Hence the computational cost of the numerical procedure proposed in this paper to evaluate a path dependent option with N monitoring dates is O(NF (1 + log2 NF ) + S 2 + N (S + 1)2 )) as NF and S go to infinity with S/NF < 1. Finally, the evaluation of the Bermudan options requires to check the early exercise condition at each time step. Moreover, we note that when ∆ → 0+ , that is the number N of monitoring dates goes to infinity, the density function tends to a singular distribution so that in order to get satisfactory approximations we must choose the number S of the Gaussian quadrature nodes depending on the number N of the monitoring dates and the number NF of the fractional fast Fourier algorithm depending on S, as explained in Section 3.1..

3.1.

Error Analysis

In this subsection, we present the error analysis of our numerical method. All proofs can be found in the Appendix. To keep the exposition simple, we consider European call double barrier options (see formula (2.3.)), we replace the fast fractional Fourier transform with the fast Fourier transform and we take the risk free interest rate equal to zero, (i.e. r = 0). Let us recall that Φ(x, y, ∆) describes the transition from x at time t to y at time t + ∆ and it is a function only of the difference y − x, that is Φ(x, y, ∆) = g(y − x, ∆). We assume that for ∆ > 0 the transition probability density function Φ is sufficient regular to allow the manipulation that follows. We begin the error analysis expressing the function g

L´evy Processes and Option Pricing by Recursive Quadrature

39

through the inverse Fourier transform that is: g(z, ∆) =

1 2π

Z

+∞

dξχ(ξ, ∆)e−iξ z ,

(19)

−∞

where χ(ξ, ∆) is the Fourier transform of the kernel g(z, ∆) (i.e. the characteristic function) with respect to the variable z, that is χ(ξ, ∆) =

+∞

Z

eiξz g(z, ∆) dz.

(20)

|g(y − x, ∆)|dy .

(21)

−∞

Let Ω∆ be the quantity defined by: Ω∆ = sup

Z

u

l≤x≤u l

From equations (19) and (20) it is easy to see that the following inequalities hold: 1 kχ(·, ∆)k1 , ∆ > 0, 2π ≤ kg(·, ∆)k1 , ∆ > 0,

kg(·, ∆)k∞ ≤

(22)

kχ(·, ∆)k∞

(23)

where kg(·, ∆)k∞ and kχ(·, ∆)k∞ denote the L∞ norms of the functions g(z, ∆) and χ(ξ, ∆) as functions of z ∈ (−∞, +∞) and ξ ∈ (−∞, +∞) respectively, and kg(·, ∆)k1 and kχ(·, ∆)k1 denote the L1 norms of the functions g(z, ∆) and χ(ξ, ∆) as functions of z ∈ (−∞, +∞) and ξ ∈ (−∞, +∞) respectively. Using (19) we can approximate g as follows: g(z, ∆) ≈ gb(z, ∆) =

1 2π

Z

+µ 2

−µ 2

dξχ(ξ, ∆)e−iξ z ,

(24)

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for a positive and sufficiently large µ (see Lemma 1) and consequently g(z, ∆) ≈ gb(z, ∆) ≈ ge(z, ∆),

where the function ge(z, ∆) is given by the following formula: ge(z, ∆) =

M −1 1 X β χ(ξ l , ∆)e−iξl z , 2π

(25)

(26)

l=0

where M is a positive integer that is the analogous of NF when we use the fast fractional Fourier transform, β is a positive constant and ξ l , l = 0, 1, . . . , M are suitable nodes that we will make precise in the following of this section. We recall that the pricing of barrier option can be made using the following recursive procedure Z u g(y − x, ∆)Vn+1 (y, tn+1 )dy, n = N − 1, N − 2, . . . , 0, (27) Vn (x, tn ) = l

40

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

starting from the payoff condition: VN (y, tN ) = F (y), where tN is given by tN = N ∆ = T and F is the payoff function. Our procedure approximates formula (27) by: S X

Ven (xk , tn ) =

j=0

wj ge∗ (xj − xk , ∆)Ven+1 (xj , tn+1 ),

(28)

where we recall that xj = l + (u − l)(1 + nj )/2, j = 0, 1, . . . , S, are points belonging to the interval [l, u] obtained translating the Gauss Legendre nodes nj , j = 0, 1, . . . , S and wj , j = 0, 1, . . . , S are the corresponding Gauss Legendre weights multiplied by the product of the discount factor e−r∆ and (u − l)/2. Remind that we have assumed r = 0 so that wj are the Gauss Legendre weights w ˜j , j = 0, 1, . . . , S multiplied by (u − l)/2. Moreover, ge∗ (z, ∆) is the cubic spline that interpolates ge(z, ∆) as a function of z on the uniform grid of the interval [−b, b], b = u−l, that is, on the nodes zj = −b+h j, h = 2b/S, j = 0, 1, . . . , S. In the following of this section we give an upper bound to the following quantity: ǫn,S = sup |Vn (xk , tn ) − Ven (xk , tn )|.

(29)

0≤k≤S

In order to derive an upper bound for ǫn,S we consider the following inequality: |Vn (xk , tn ) − Ven (xk , tn )| ≤ |Vn (xk , tn ) − V1,n (xk , tn )| + |V1,n (xk , tn ) − V2,n (xk , tn )| + |V2,n (xk , tn ) − Ven (xk , tn )| ,

where V1,n , V2,n are the functions defined by: Z u ge(y − x, ∆)V1,n+1 (y, tn+1 )dy , V1,n (x, tn ) =

(30)

(31)

l

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V2,n (x, tn ) =

S X j=0

wj ge(xj − x, ∆)V2,n+1 (xj , tn+1 ),

(32)

and we estimate upper bounds for the following quantities:

ǫ1,n,S = sup |Vn (xk , tn ) − V1,n (xk , tn )|,

(33)

ǫ1,2,n,S = sup |V1,n (xk , tn ) − V2,n (xk , tn )|,

(34)

ǫ2,n,S = sup |V2,n (xk , tn ) − Ven (xk , tn )|.

(35)

0≤k≤S

0≤k≤S

0≤k≤S

These quantities allow us to find an upper bound of the error in terms of different sources of error:

L´evy Processes and Option Pricing by Recursive Quadrature

41

a error arising from the approximation of the transition density (24) by the fast Fourier transform algorithm; b error arising from the discretisation of the integral in (27) by Gaussian quadrature rules; c error arising from the approximation of the transition density on the Gaussian nodes by cubic spline interpolation (see 28). The error analysis shown in detail in the Appendix is based on the two following results: Lemma 1 Let g(z, ∆) be a nonnegative sufficiently regular function for ∆ > 0, let ε be a positive constant, 0 < ε 0, works. c) We finally remark that one of the main advantages of the computational method proposed consists in the fact that it works also when the integral that defines the option value is not of a convolution type, provided that we know the transition density or the characteristic function. This is not true for other numerical methods such as the QUADFFT that strongly depend on the convolution structure of the transition density.

4.

Numerical Experiments

In this section we propose some numerical experiments that show the speed and the accuracy of the hybrid QUAD-FFT method for pricing barrier options and Bermudan options under the Gaussian model (G), the CGMY model (CGMY), the double exponential (DB) model and jump diffusion (JD) models. In particular, we show the CPU time (in seconds) required to execute the algorithm and the convergence of the method when the number of grid points increases. Finally, under suitable assumptions (see Section 3.1.), we show that the convergence of the hybrid QUAD-FFT method is of the second order. All the numerical experiments have been carried out on a 2.80 GHz Xeon PC, with 1GB RAM and the program code has been written in C. Let us start describing the sets of the parameters used in the experiments. We consider the calibration results reported in [15] page 82 concerning the CGMY model, that is:

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

C = 0.0244,

G = 0.0765,

M = 7.5515,

Y = 1.2945.

(41)

In order to make the option prices comparable across models, we choose the CGMY model as benchmark and we caibrate to it the Gaussian model (G), the double-exponential (DE) model and the Merton (JD) model by minimizing the square integrated difference between the real part of the characteristic function of the CGMY and the real parts of the characteristic functions of the remaining models. That is, when we consider the jump-diffusion model we solve the following three optimization problems:   Z ξmaxX 3   CGM Y  ψ∆ (ξ) ψ JD (ξ) 2 ∆ min Re e j − Re e j dξ, j = 1, 2, 3, (42) θ={σ,λ,α,δ} 0

j=1

where ∆1 = 0.25, ∆2 = 0.5, ∆3 = 1 and ξ max has been chosen equal to 250 and the same procedure has been used to calibrate the pure diffusion (G) and the double-exponential (DE)

L´evy Processes and Option Pricing by Recursive Quadrature

43

Table 3. European call options. RMSE for several choices of the number N of monitoring dates and of the number S of quadrature nodes. S 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600

N 12 12 12 12 50 50 50 50 100 100 100 100 250 250 250 250

G 7.41E-05 3.67E-06 8.89E-06 1.22E-06 1.40E-04 2.49E-05 3.88E-06 4.71E-07 1.48E-04 3.46E-05 6.11E-06 1.00E-06 1.51E-04 3.71E-05 8.90E-06 1.82E-06

CGMY 1.61E-04 2.93E-05 5.15E-06 4.22E-06 3.16E-04 4.02E-05 9.79E-06 5.08E-06 3.26E-01 3.73E-05 1.04E-05 5.99E-06 5.07E-01 2.15E-05 4.94E-06

DE 1.25E-04 1.53E-05 2.26E-05 1.53E-05 1.55E-04 3.98E-05 2.25E-05 2.30E-05 3.80E-05 4.58E-05 2.68E-05 2.85E-05 1.56E-04 4.26E-05 2.63E-05 2.63E-05

JD 4.52E-05 8.72E-05 4.94E-06 8.20E-06 2.19E-04 1.26E-04 1.50E-05 8.98E-06 2.65E-04 6.69E-05 2.26E-06 1.16E-05 4.55E-04 7.73E-05 1.34E-06 1.19E-05

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Table 4. Computational time for several choices of the number N of monitoring dates and of the number S of quadrature nodes. CP UA is the time (seconds) required to compute the matrix A, CP UI is the time (seconds) required for the iteration procedure and CP UT is the overall CPU time (seconds). S 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600

N 12 12 12 12 50 50 50 50 100 100 100 100 250 250 250 250

CPUA 2 3 10 27 1 4 10 28 1 3 10 27 1 3 10 27

CPUI 0 1 3 11 1 3 12 47 1 6 24 96 4 15 61 239

CPU 4 6 15 40 4 9 24 77 4 11 36 125 7 20 73 268

44

Gianluca Fusai, Giovanni Longo, Marina Marena et al. Computational time 6 N=12 N=50 N=100 N=250

5.5 5 4.5

log(cpu)

4 3.5 3 2.5 2 1.5 1

7

7.5

8

8.5

9

9.5

log(S)

Figure 1. Overall computational time versus the number of nodes S for several choices of the number N of monitoring dates. models. A similar calibration procedure has been proposed recently in [4]. The calibrated parameters for the JD model are: σ = 0.126349, α = −0.390078, λ = 0.174814, δ = 0.338796.

(43)

The calibrated volatility parameter for the G model is σ = 0.17801.

(44)

The calibrated parameters for the DE model are: σ = 0.120381, Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

η 1 = 9.65997,

λ = 0.330966, η 2 = 3.13868.

p∗ = 0.20761,

(45) (46)

Finally, in the following numerical experiments, the time to maturity is one year, the risk-free rate is 3.67% per year and the dividend yield is set to 0. In the following experiments we choose N = 1016 . From the point of view of the error analysis, note that NF points in the FRFT algorithm roughtly corresponds to at least 4NF points in the FFT algorithm (see [7]). In the first experiment, we compute European call option prices and compare them with the solution obtained by the Carr-Madan formula in [6]. This case is important since we dispose of a reference value that can be used to compute the errors, and as a consequence to choose the number of quadrature points necessary to achieve a prefixed accuracy. Moreover, the errors relative to the European vanilla call options will provide upper bounds for the errors concerning the single and double barrier options.

L´evy Processes and Option Pricing by Recursive Quadrature

45

Table 5. Single barrier knock out call options. Option prices for several choices of the number N of monitoring dates N and of the number S of quadrature nodes. Time to maturity 1 year, spot price 100, strike 100, barrier 90. S 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600

N 12 12 12 12 50 50 50 50 100 100 100 100 250 250 250 250

G 8.3459832033 8.3459901150 8.3459887598 8.3459883292 8.0803819489 8.0803740847 8.0803775227 8.0803772024 7.9874908185 7.9874890844 7.9874900582 7.9874905549 7.8977207444 7.8977158745 7.8977142145 7.8977144658

CGMY 9.3961758593 9.3961714212 9.3961704140 9.3961702111 9.3403645727 9.3403819645 9.3403806141 9.3403808459 9.3260406631 9.3260503666 9.3260486481 9.3260480658 9.3067669123 9.3148486236 9.3148464687 9.3148455432

DE 9.4061976433 9.4062154319 9.4062144940 9.4062143981 9.3346093555 9.3346119144 9.3346109176 9.3346107128 9.3093069992 9.3092976135 9.3093014208 9.3093012677 9.2846151511 9.2846084567 9.2846062380 9.2846061523

JD 9.3888395858 9.3888568173 9.3888541937 9.3888543414 9.3069671014 9.3069760139 9.3069778483 9.3069774677 9.2773524197 9.2773504169 9.2773534141 9.2773535603 9.2481980508 9.2481959683 9.2481985043 9.2481987872

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Table 6. Knock out double barrier call options. Option prices for several choices of the number N of monitoring dates N and of the number S of quadrature nodes. Time to maturity 1 year, spot price 100, strike 100, lower barrier 90 and upper barrier 110. S 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600

N 12 12 12 12 50 50 50 50 100 100 100 100 250 250 250 250

G 0.2428044531 0.2428042985 0.2428042802 0.2428042999 0.1196515774 0.1196514864 0.1196514755 0.1196514871 0.0922390185 0.0922389437 0.0922389348 0.0922389443 0.0710851142 0.0710850528 0.0710850455 0.0710850533

CGMY 0.6535703627 0.6535700777 0.6535700439 0.6535700803 0.5322870186 0.5322867655 0.5322867355 0.5322867678 0.5029097106 0.5029094652 0.5029094360 0.5029094674 0.4800054767 0.4800052373 0.4800052088 0.4800052394

DE 0.5914408736 0.5914405624 0.5914405255 0.5914405652 0.4289230196 0.4289227557 0.4289227244 0.4289227581 0.3838931313 0.3838928821 0.3838928525 0.3838928843 0.3446568690 0.3446566334 0.3446566054 0.3446566355

JD 0.5725702712 0.5725699590 0.5725699220 0.5725699618 0.4055229156 0.4055226560 0.4055226252 0.4055226583 0.3595522655 0.3595520224 0.3595519935 0.3595520246 0.3196893461 0.3196891181 0.3196890910 0.3196891201

46

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

In Table 3 we show the number of the grid point S, S = 1200, 2400, 4800, 9600, the number of monitoring dates N , N = 12, 50, 100, 250 and the root mean square errors (RMSE) of L option prices for different spot prices, i.e., 

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

RMSES = 

L X e j )2 (Vj − V 0

j=1

L

1/2 

(47)

where L is the number of quadrature points xj such that exi ∈ [0.8ek , 1.2ek ], V j is the option value with log spot prices xj and Ve j0 is the corresponding reference value. We note that the number L depends on the number S of the grid points and the number N of monitoring dates. For example, when N = 250 we have: L = 82 when S = 1200, L = 250 when S = 2400, L = 332 when S = 4800 and, finally, L = 998 when S = 9600. The strike has been set equal to 100. Table 4, from left to right, reports the CPU time (in seconds) required to compute the matrix A (CP UA ), the CPU time needed for the iterations (17) (CP UI ), the CPU time required to execute all the steps of the algorithm (CP UT ). When the number of nodes S is multiplied by a factor of two, the time CP UI is approximately multiplied by a factor of four. This behavior confirms the estimate of the computational cost given above in Section 3., in fact CP UI = N S 2 . Moreover, Table 4 shows that the CP UA is independent of the number N of the monitoring dates, in fact we have CP UA = O(NF (1 + log2 NF ) + S 2 ). Figure 1 shows the log log plot of cpu time CP UT versus the log of the number S of nodes for several numbers of monitoring dates. It reveals that when N is sufficiently large the computational time CP UT increases linearly with constant rate of about 1/4. We focus our attention mainly on path dependent options with a very large number of monitoring dates. Our method gives three digits accurate prices, even when the number of monitoring dates is as high as 250. Obviously to get satisfactory approximations of the option values we must pay a price in term of execution time. In fact to approximate almost singular integrands, as those appearing in the CGMY case, we must choose a large number of nodes. Table 5 presents the numerical results of pricing a single barrier option with barrier 90 and Table 6 shows from left to right the number S of the nodes, the number N of monitoring dates and the call knock out double barrier option values under G model, CGMY model, DE model and JD model. The barriers are set at 90 and 110. We recall that for these options analytical pricing formula are not available. In Table 7 we show the Bermudan put option prices. Finally, Figure 2 shows the optimal exercise frontier for the American put option for different monitoring dates. We notice that for all models as much as N = 50 early exercise dates already provide a two digits approximation of the American price with a larger number of monitoring dates. Therefore Bermudan options, respect to barrier options, appear insensitive to the monitoring frequency: this is due to the early exercise payoff that we obtain when the asset hits the optimal exercise frontier. Finally, Figure 2 shows the optimal exercise frontier for the American put option for different monitoring dates. Table 7 and Figure 2 show that we can approximate the true price using the price of a Bermudan option with a little number of monitoring dates, that can be obtained very fast.

L´evy Processes and Option Pricing by Recursive Quadrature

47

Table 7. Bermudan put options. Option prices for several choices of the numbers N of early exercise dates and of the number S of the quadrature nodes. Time to maturity 1 year, spot price 100, strike 100. S 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600 1200 2400 4800 9600

N 12 12 12 12 50 50 50 50 100 100 100 100 250 250 250 250

G 5.638297794 5.638372179 5.638361606 5.638371895 5.665163349 5.665049015 5.665026651 5.665028862 5.669514558 5.669408365 5.669382920 5.669375659 5.672141717 5.672044468 5.672017421 5.672010931

CGMY 5.901900680 5.901783697 5.901750342 5.901756771 5.921574800 5.921528461 5.921505737 5.921497772 5.890987905 5.920495447 5.920470889 5.920464047 6.046461819 5.919777727 5.919769048

DE 6.235216224 6.235026093 6.235023642 6.235039227 6.258062316 6.257947738 6.257918812 6.257912595 6.263650128 6.263636187 6.263670438 6.263663342 6.263396934 6.263286039 6.263258708 6.263252148

JD 6.258957522 6.258741458 6.258872354 6.258858040 6.284667310 6.284509527 6.284450573 6.284466533 6.288598946 6.288546305 6.288623393 6.288612950 6.290771926 6.291166593 6.291131200 6.291123950

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Table 8. Convergence rate in the Gaussian model. N is the number of early exercise dates and S is the number of the quadrature nodes. S 300 600 1200 300 600 1200 300 600 1200 300 600 1200

N 12 12 12 50 50 50 100 100 100 250 250 250

rS 3.84 5.84 4.89 4.00 3.99 6.06 4.13 4.01 4.23 4.01 4.17

We conclude this section showing the rate of convergence of the proposed computational method on a numerical experiment involving an European call vanilla option under the Gaussian model. The reference values used in this experiment are obtained with the well known Black Scholes formula. In this experiment we choose N = 220 , which

48

Gianluca Fusai, Giovanni Longo, Marina Marena et al. CGMY Model 1 N=12 N=50 N=100 N=250

0.98

Optimal Exercise Price

0.96

0.94

0.92

0.9

0.88

0.86

0

0.1

0.2

0.3

0.4

0.5 0.6 Time to Maturity

0.7

0.8

0.9

1

Figure 2. Optimal exercise price for the American put option in the CGMY model. corresponds to p = 3/2 in Section 3.1.. Table 8 shows the ratio rS = |RM SES − RM SE2S |/|RM SE2S − RM SE4S | where RM SES is the root mean square error given in (47). The ratio is approximately 4, that is the computational method shows a second order convergence. Note that the rate of convergence is strictly related to the relation between the number NF of the nodes used in the fractional Fourier transform and the number S of the Gaussian nodes (see Section 3.1.).

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5.

Conclusions

In this paper we have presented a quadrature method for pricing discretely monitored pathdependent options under L´evy processes. The approach requires only the knowledge of the characteristic function of the underlying process and it combines numerical inversion of the characteristic function, its cubic spline interpolation and a Gaussian quadrature. The resulting hybrid method, due to the higher accuracy of Gaussian quadrature over NewtonCotes rules, provides accurate results up to a
very large number of monitoring dates. The computational cost of our method is O N S 2 , when the number of grid points S + 1 goes to infinity and the number of monitoring dates is N goes to infinity. The convergence of the method is theoretically proved when the Gauss Legendre quadrature rule is used to price discrete monitoring double barrier options. The order of convergence of the method is at most 3, since we use cubic spline interpolation in order to compute the transition density function on the Gaussian nodes. Indeed, convergence of the third order could be achieved by choosing a sufficiently high number of points used for the fast Fourier transform in the numerical inversion of the characteristic function. Numerical examples are presented for plain vanilla options, discrete barrier options and Bermudian options for a variety of

L´evy Processes and Option Pricing by Recursive Quadrature

49

L´evy processes, such as the Gaussian, CGMY, Kou Double-Exponential and Merton JumpDiffusion processes. Since we choose p = 3/2, they show a quadratic convergence of the method. Finally, one of the main advantages of the computational method proposed consists in the fact that it works also when the integral that defines the option value is not of a convolution type, provided that we know the transition density or the corresponding characteristic function. This is not true for other numerical methods such as the QUADFFT that strongly depends on the convolution structure of integrals that define the option prices.

Appendix. Error Analysis. We start evaluating the difference g(y − x, ∆) − ge(y − x, ∆) where ge is defined by formula (26). The inverse Fourier transform in (26) is made using the fast Fourier transform algorithm (eventually the fractional Fourier transform algorithm) as described in Bailey [3]. Note that when we use the fast Fourier algorithm we obtain the value of the function g on a uniform grid. Roughly speaking, this algorithm essentially consists of the use of the mid-point rule. In fact let gb be the function defined by: 1 gb(z, ∆) = 2π

Z

µ/2

dξe−iξ z χ(ξ, ∆), −b ≤ z ≤ b,

(48)

−µ/2

where µ is a suitable constantRthat, since |χ(ξ, ∆)| is L1 integrable on the real axis with re+∞ spect to ξ, i.e. kχ(·, ∆)k1 = −∞ |χ(ξ, ∆)| dξ < ∞, ∆ > 0, we can choose µ sufficiently large in order to satisfy the following inequality: sup |g(z, ∆) − gb(z, ∆)| = Z Z µ 1 ǫ +∞ 2 1 sup e−iξ z χ(ξ, ∆)dξ < , e−iξ z χ(ξ, ∆)dξ − µ 3 2π − −b≤z≤b 2π −∞

−b≤z≤b

2

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

∀µ > µ ˜

(49)

for some positive constants µ ˜ and ǫ. We choose β ≈ 2π/(hM ) where h = 2b/S so that the maximum interval of integration is [−π/h, π/h]. We must choose S in order to satisfy the condition π/h ≥ µ/2. Moreover we impose β ≈ hp , p ≥ 1 that is we choose M = int(2π/hp+1 ) where int(·) denotes the integer part of ·. Hence we have:   M ξl = l − β, l = 0, 1, . . . , M − 1 2 Proof of Lemma 1. We have: g(xj − xk , ∆) =

1 2π

Z

+∞

−∞

e−iξ(xj −xk ) χ(ξ, ∆)dξ .

(50)

50

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

Since χ(ξ, ∆) is L1 integrable on the real axis with respect to ξ, then for any given constant ε > 0 there is a constant µ independent on z such that Z Z µ ε 1 +∞ 2 1 e−iξ z χ(ξ, ∆)dξ < . e−iξ z χ(ξ, ∆)dξ − (51) 3 2π −∞ 2π − µ 2

Let us recall that S, β and M satisfy the following relations: µ < 2π/h, β = 2π/(hM ), M = int(2π/hp+1 ), p ≥ 1 so that for p ≥ 1 equations (48), (50) and (49) imply 1 PM −1 −iξ l z βχ(ξ , ∆)e g(z, ∆) − 2π l l=0 R µ R 1 +∞ −iξ z 1 2 = 2π −∞ e χ(ξ, ∆)dξ − 2π e−iξ z χ(ξ, ∆)dξ −µ 2 µ R 2 −iξ z 1 1 PM −1 −iξ l z χ(ξ , ∆) β e + 2π χ(ξ, ∆)dξ − µ e l l=0 2π − 2 Rµ P 1 2 −iξ z M −1 1 −iξ l z χ(ξ , ∆) ≤ 3ǫ + 2π β e χ(ξ, ∆)dξ − µ e l l=0 2π (52) R−µ 2 R 1 2 −iξ z +∞ −iξ z 1 χ(ξ, ∆)dξ − ≤ 3ǫ + 2π e χ(ξ, ∆)dξ + µ e 2π −∞ −2 R R + πh −iξ z 1 +∞ −iξ z 1 χ(ξ, ∆)dξ − 2π − π e χ(ξ, ∆)dξ + 2π −∞ e h R π 1 + h −iξ z 1 PM −1 −iξ l z χ(ξ , ∆) β e χ(ξ, ∆)dξ − π e l l=0 2π 2π −h 2 1 1 2p ≤ ǫ + 2π CS β (2π/h) = ǫ + 2π CS h 2π/h = ǫ + CS h2p−1 .

Last relation in equation (52) follows from the error formula of the composite mid-point rule (see for example [1]) and the fact that β = hp . This concludes the proof. 

Lemma 3 Let µ, ǫ be as in Lemma 1, and g(z, ∆) be a nonnegative sufficiently regular function for ∆ > 0, let nj , wj , j = 0, 1, . . . , S be the nodes and the weights of the Gauss Legendre quadrature rule introduced above and xj = l+(u−l)(1+nj )/2, j = 0, 1, . . . , S, for p ≥ 1 we have: sup

S X

wj |e g (xj − x, ∆)| ≤ (u − l)(ǫ + (2(u − l))2p−1 CS /S 2p−1 + kχ(·, ∆)k1 ), (53)

l≤x≤u j=0

where CS is the constant defined in formula (37) and kχ(·, ∆)k1 is the L1 norm of the characteristic function as explained above. Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Proof: We have: S X

wj |e g (xj − x, ∆)| ≤

j=0

S X

wj {|e g (xj − x, ∆) − g(xj − x, ∆)| +

j=0

|g(xj − x, ∆)|} . P Using Lemma 1 and the fact that Sj=0 wj = (u − l) in formula (54) we obtain S X

(54)

wj |e g (xj − x, ∆)| ≤ (u − l)(ǫ + 8(u − l)3 CS /S 3 + kg(·, ∆)k∞ )

j=0

≤ (u − l)(ǫ + (2(u − l))2p−1 CS /S 2p−1 + kχ(·, ∆)k1 ). This concludes the proof.

(55)

L´evy Processes and Option Pricing by Recursive Quadrature

51

Lemma 4 Let µ, ǫ be chosen as in Lemma 1, and g(z, ∆) be a nonnegative sufficient regular function for ∆ > 0, then for p ≥ 1 we have: Z u sup dy |˜ g (y − x, ∆)| ≤ Ω∆ + (u − l)((2(u − l))2p−1 CS /S 2p−1 + ǫ), (56) l≤x≤u l

where Ω∆ is the quantity defined in formula (21) and CS is given in (37). Proof: Using Lemma 1 and the definition of Ω∆ we have: Z u |˜ g (y − x, ∆)|dy ≤ Zl u [|˜ g (y − x, ∆) − g(y − x, ∆)| + g(y − x, ∆)] dy ≤ l Z u 2p−1 2p−1 Cµ /S + ǫ) + g(y − x, ∆)dy. (u − l)((2(u − l))

(57)

l

This concludes the proof.  Lemma 5 Let be V2,n (xk , tn ) =

S X j=0

and

wj ge(xj − xk , ∆) V2,n+1 (xj , tn+1 ), k = 0, 1, . . . , S,

(58)

∗ V2,n,S := sup |V2,n (xk , tn )| .

(59)

0≤k≤S

Then we have: −n ∗ ∗ V2,n,S ≤ γN 2,S VN,S ,

n = N − 1, N − 2, . . . , 0,

(60)

∗ = sup where VN,S 0≤k≤S |F (xk )|,

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γ 2,S = (u − l)(ǫ + (2(u − l))2p−1 CS /S 2p−1 + kχ(·, ∆)k1 ) ,

(61)

where CS , ǫ are the constants appearing in formula (53). Proof: Equation (58) implies the following inequality: sup |V2,n (xk , tn )| ≤ sup |V2,n+1 (xk , tn+1 )| sup

0≤k≤S

0≤k≤S

S X

wj |e g (xj − xk , ∆)| .

(62)

0≤k≤S j=0

Using Lemma 3 we have for p ≥ 1: sup |V2,n (xk , tn )| ≤

0≤k≤S

(u − l)(ǫ + 8(u − l)2p−1 CS /S 2p−1 + kχ(·, ∆)k1 ) sup |V2,n+1 (xk , tn+1 )| . 0≤k≤S

(63)

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Gianluca Fusai, Giovanni Longo, Marina Marena et al.

∗ Let γ 2,S be the constant given in (61), and let V2,n,S be given by (59), from equation (63) we obtain ∗ ∗ V2,n,S ≤ V2,n+1,S γ 2,S . (64)

From (64), by recursion we get −n ∗ ∗ ∗ V2,n,S ≤ V2,n+2,S γ 22,S ≤ ... ≤ γ N 2,S VN,S

(65)

∗ = sup where we remind that VN,S 0≤k≤S |F (xk )| = sup0≤k≤S |V2,N (xk , tN )| . 

Lemma 6 Let be V1,n (xk , tn ) =

Z

l

and

u

ge(y − xk , ∆) V1,n+1 (y, tn+1 )dy, k = 0, 1, . . . , S,

(66)

∗ V1,n,S := sup |V1,n (xk , tn )| .

(67)

−n ∗ ∗ V1,n,S ≤ γN 1,S VN,S ,

(68)

γ 1,S = Ω∆ + (u − l)(8(u − l)3 CS /S 3 + ǫ) ,

(69)

0≤k≤S

Then we have: ∗ = sup where VN,S 0≤k≤S |F (xk )|,

and Ω∆ , CS are the constants appearing in formula (56). The proof follows arguing as in Lemma 5 using Lemma 4.  Proof of Theorem 2: Let ǫ1,n,S , ǫ1,2,n,S , ǫ2,n,S be the quantities defined in formulae (33), (34), (35). In order to estimate ǫn,S we use the inequality given in (30), that is: ǫn,s ≤ ǫ1,n,S + ǫ1,2,n,S + ǫ2,n,S .

(70)

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We start evaluating ǫ2,n,S given in (35). We have

S X j=0

V2,n (xk , tn ) − Ven (xk , tn ) =

wj ge(xj − xk , ∆)V2,n+1 (xj , tn+1 ) −

(71) S X j=0

wj ge∗ (xj − xk , ∆)Ven+1 (xj , tn+1 ) .

P Adding and subtracting Sj=0 ge∗ (xj −xk , ∆)V2,n−1 (xj , tn−1 ) in formula (71) and applying the triangle inequality and using the fact that wj > 0, j = 0, 1, . . . , S we obtain: |V2,n (xk , tn ) − Ven (xk , tn )| ≤

S X j=0

S X j=0

wj |e g (xj − xk , ∆) − ge∗ (xj − xk , ∆)| |V2,n+1 (xj , tn+1 )| +

wj |e g ∗ (xj − xk , ∆)||Ven+1 (xj , tn+1 ) − V2,n+1 (xj , tn+1 )| ,

(72)

L´evy Processes and Option Pricing by Recursive Quadrature

53

so that we have: ǫ2,n,S ≤ e2,n+1,S sup

S X

wj |e g ∗ (xj − xk , ∆)| +

0≤k≤S j=0

sup |V2,n+1 (xk , tn+1 )|

0≤k≤S

S X j=0

wj |e g (xj − xk , ∆) − ge∗ (xj − xk , ∆)| .

(73)

Using the properties of the natural cubic splines [16] Lemma 4.2 pag. 55 we have: k˜ g (·, ∆) − g˜∗ (·, ∆)k∞ ≤ v∆ /S 3 ,

(74)

where 4 d g˜ 1 3 v∆ = (u − l) sup 4 (z, ∆) , 3 −b≤z≤b dz

and S X

∗

wj |˜ g (z, ∆)| ≤

wj |˜ g ∗ (z, ∆) − g˜(z, ∆)| +

j=0

j=0

S X

S X

(75)

wj |˜ g (z, ∆)| ≤ γ 2,S + (u − l)v∆ /S 3 .

(76)

j=0

Using Lemma 3 and Lemma 5, and formulae (74) and (76) we derive the following recursive relation:

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−n−1 (u − l)v∆ /S 3 , ǫ2,n,S ≤ (γ 2,S + (u − l)v∆ /S 3 )e2,n+1,S + γ N 2,S

(77)

where γ 2,S is the constant given in formula (61). Using the fact that we have ǫ2,N,S = 0 since when n = N VN (x, tN ) = V2,N (x, tN ) = F (x), equation (77) implies  v∆ N −n ǫ2,N,S + ǫ2,n,S ≤ γ 2,S + (u − l) 3 S N −n−1 v∆ X  v∆ N −n−1−q q (u − l) 3 γ 2,S ≤ γ 2,S + (u − l) 3 S S q=0

(u − l)

v∆ S3

NX −n−1  q=0

γ 2,S + (u − l)

v∆ N −n−1−q q γ 2,S . S3

(78)

Now we estimate the quantity ǫ1,2,n,S given in (34). Let us note that the following

54

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

relation holds: V1,n (xk , tn ) − V2,n (xk , tn ) = Z

u

Z

u

l

l

S X j=0

ge(y − xk , ∆)V1,n+1 (y, tn+1 )dy −

ge(y − xk , ∆)V1,n−1 (y, tn+1 )dy −

S X j=0

S X j=0

wj ge(xj − xk , ∆)V1,n+1 (xj , tn+1 ) −

wj ge(xj − xk , ∆)V2,n+1 (xj , tn+1 ) = wj ge(xj − xk , ∆)V1,n+1 (xj , tn+1 ) +

S X j=0

wj ge(xj − xk , ∆)V2,n+1 (xj , tn+1 ).

(79)

Using formula (79) we obtain the following inequality |V1,n (xk , tn ) − V2,n (xk , tn )| ≤ S X

wj |˜ g (xj − xk )| |(V1,n+1 (xj , tn+1 ) − V2,n+1 (xj , tn+1 )| +

j=0

Z u S X wj ge(xj − xk , ∆)V1,n+1 (xj , tn+1 ) ge(y − xk , ∆)V1,n+1 (y, tn+1 )dy − l j=0

(80)

so that, for ν ≥ 2p − 1 ≥ 1 we have:

ǫ1,2,n,S ≤ ǫ1,2,n+1,S sup

S X

wj |e g (xj − x, ∆)| + v1,2,n /S ν ,

(81)

l≤x≤u j=0

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where the quantity v1,2,n is an upper bound of the constants deriving from Rthe error on u the Gauss Legendre quadrature formula applied to approximate the integral l dy ge(y − xk , ∆)V1,n+1 (y, tn+1 )dy, k = 0, 1, . . . , S (see [1] Chapter 5). In particular since we have assumed a smooth kernel g for ∆ > 0 and ν ≥ 2p − 1 ≥ 1 we have: v1,2,n =

c Sν

 ν Z X  sup

0≤k≤S

j=0

l

1/2 2 j d g˜(y − xk , ∆)V1,n+1 (y, tn+1 ) dy  , dy j

u

(82)

where c is a constant independent of n, ∆, S. Using Lemma 3, it is to see that formula (81) implies the following recursive relation: ǫ1,2,n,S ≤ γ 2,S ǫ1,2,n+1,S + v1,2,n /S ν , ν ≥ 2p − 1 ≥ 1,

(83)

where γ 2,S , v1,2,n are the constants given in formulae (61), (82). Choosing ν = 2p − 1 and using equation (83) we obtain: −n ǫ1,2,n,S ≤ γ N 2,S ǫ1,2,N,S +

1 S 2p−1

NX −n−1 q=0

γ q2,S v1,2,n+q ,

(84)

L´evy Processes and Option Pricing by Recursive Quadrature

55

and it is easy to see that ǫ1,2,N,S = 0 since when n = N we have V1,N (x, tN ) = V2,N (x, tN ) = F (x). Now we estimate the quantity ǫ1,n,S given in (33). We have:

Vn (xk , tn ) − V1,n (xk , tn ) = Z

u

g(y − xk , ∆)Vn+1 (y, tn+1 )dy −

l

Z

u l

ge(y − xk , ∆)V1,n+1 (y, tn+1 )dy .

(85)

Ru Adding and subtracting l g(y−xk , ∆)V1,n+1 (y, tn+1 )dy in formula (85) and applying the triangle inequality we obtain |Vn (xk , tn ) − V1,n (xk , tn )| ≤

Z

u l

Z

u

|g(y − xk , ∆)| |Vn+1 (y, tn+1 ) − V1,n+1 (y, tn+1 )|dy + l

|g(y − xk , ∆) − ge(y − xk , ∆)| |(V1,n+1 (y, tn+1 )|dy.

(86)

Using Lemma 1 and 4 we have: ǫ1,n,S ≤ ǫ1,n+1,S sup

Z

u

|g(y − x, ∆)|dy + v1,n,S /S 2p−1 + v˜1,n,S ǫ,

(87)

l≤x≤u l

where the constants v1,n,S and v˜1,n,S by virtue of Lemma 1 and Lemma 5, are given by: −n−1 ∗ (u − l)CS , n = N − 1, N − 2, . . . , 0, v1,n,S = (2(u − l))2p−1 VN,S γN 1,S

N −n−1 ∗ (u − l), n = N − 1, N − 2, . . . , 0, v˜1,n,S = VN,S γ 1,S

(88) (89)

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and the constant γ 1,S and CS are given in (69) and (37) . It is to see that formula (87) implies the following recursive relation: ǫ1,n,S ≤ Ω∆ ǫ1,n+1,S + v1,n,S /S 2p−1 + v˜1,n,S ǫ,

(90)

where Ω∆ is the constant given in formula (21), v1,n,S and v˜1,n,S are given in formulae (88) and (89) respectively. Using equation (90) we obtain −n ǫ1,n,S ≤ ΩN ǫ1,N,S + ∆

1 S 2p−1

NX −n−1 q=0

Ωq∆ v1,n+q,S + ǫ

NX −n−1

Ωq∆ v˜1,n+q,S ,

(91)

q=0

and it is to see that ǫ1,N,S = 0 since when n = N we have VN (x, tN ) = V1,N (x, tN ) = F (x). This concludes the proof. 

56

Gianluca Fusai, Giovanni Longo, Marina Marena et al.

References [1] K.E. Atkinson, An introduction to Numerical Analysis, John Wiley, New York, 1969. [2] A. D. Andricopoulos, M. Widdicks, P. W. Duck and D. P. Newton, “Universal Option Valuation Using Quadrature Methods,” Journal of Financial Economics, vol. 67, no. 3, pp. 447-471, 2003. [3] D.H. Bailey and P. N. Swarztrauber, “A fast method for the numerical evaluation of continuous Fourier and Laplace transforms”, SIAM Journal on Scientific Computing, vol. 5, no. 15, pp. 1105–1110, 1994. [4] D. Belomestny, and M. Reiß, “Spectral Calibration of Exponential L´evy Models,” Finance and Stochastic, vol. 10, pp. 449-474, 2006. [5] P. Carr, H. Geman, D.B. Madan and M.Yor, “The Fine Structure of Asset Returns: An Empirical Investigation,” Journal of Business, vol. 75, pp. 305-332, 2002. [6] P. Carr and D.B. Madan, “Option Valuation Using the Fast Fourier Transform,” Journal of Computational Finance, vol. 2, no. 4, pp. 61-73, 1999. [7] K. Chourdakis, “Option Pricing Using the Fractional FFT,” Journal of Computational Finance, vol. 8, no.2, pp. 1-18, 2005. [8] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall, 2004. [9] G. Fusai and M.C. Recchioni, “Analysis of Quadrature Methods for Pricing Discrete Barrier Options,” Journal of Economic Dynamics and Control, vol. 31, no. 3, pp. 826860, 2007. [10] S. G. Kou and H. Wang, “Option pricing under a double exponential jump diffusion model”, Management. Science, vol. 50, no. 9, pp. 1178-1192, 2004

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[11] R. Lord and C. Kahl, “Optimal Fourier inversion in semi-analytical option pricing”. SSRN working paper, 2006. Available at: http//ssrn.com/abstract=921336. [12] R. Lord, F. Fang, F. Bervoets and C.W. Oosterlee, “A fast and accurate FFT-based method for pricing early-exercise options under Levy processes”, working paper, 2007. [13] T. K. Philips and R. Nelson, “The moment bound is tighter than Chernoff’s bound for positive tail probabilities”. American Statistical Association, vol. 49, no. 2, pp. 175-178, 1995. [14] C. O’Sullivan, “Path Dependent Option Pricing under Levy Processes”, EFA 2005 Moscow Meetings Paper, Febr. 2005. Available at SSRN: http://ssrn.com/abstract=673424. [15] W. Schoutens, Levy Processes in Finance, Wiley, 2003.

L´evy Processes and Option Pricing by Recursive Quadrature

57

[16] M.H. Schultz, Spline analysis, Prentice-Hall, Inc. Englewood Cliffs, N.J., 1973.

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[17] M.A. Sullivan, “Pricing discretely monitored barrier options,” Journal of Computational Finance, vol. 3, no. 4, pp. 35-52, 2000.

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In: Economic Dynamics… Editor: Chester W. Hurlington, pp. 59-76

ISBN: 978-1-60456-911-7 © 2008 Nova Science Publishers, Inc.

Chapter 3

LABOR MARKET IMPERFECTIONS AND ENDOGENOUS BUSINESS CYCLES Alvaro Rodriguez* Department of Economics. Rutgers University. University Heights. 360 Dr Martin Luther King, Jr. Newark, New Jersey 07102

Abstract This paper examines the consequences of introducing labor market imperfections in a rather conventional macro model. In spite of the fact that all agents behave in reasonable way and enjoy perfect foresight, the model allows for a conflict between investment and savings decision that can result in instability. It is found that wage rigidity preserves the stability of economies that were initially stable. Economies that were completely unstable benefit from the introduction of friction: they either become stable or they approach a stable periodic orbit. It is shown that the cycle generated by the model is consistent with procyclical movements in real wages, the rate of capacity utilization and the price level.

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Introduction The two conflicting approaches to the study of business cycles that dominate the current economic literature differ not only in their conception of the cycles but also in their methodology. The real business cycle literature regards fluctuations in economic activity as natural phenomena in the equilibrium path of a frictionless economy. Their models are highly dynamic because they try to explain how random technological shocks affect the rate of capital accumulation and the intertemporal allocation of effort. In contrast the new Keynesian literature regards the fluctuations as the consequence of market failures, but their models tend to be static because the emphasis is on explaining why market equilibrium can be consistent with the existence of involuntary unemployment. Frequently, economic instability is *

E-mail address: [email protected]. Some of the computations in this paper were carried out using Mathematica. Links to the programs used can be found at the site http://mysite.verizon.net/vzeodr6z/endogenous.htm

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60

Alvaro Rodriguez

attributed to the fact that coordination failures lead to the existence of multiple equilibria in an otherwise static model. This paper is a contribution to the literature that looks at how market failures can generate endogenous business cycles through their effect on the rate of capital accumulation. This literature goes back to Kalecki (1935), Hicks (1950), Samuelson (1939) and Kaldor (1940). For more recent discussions on those models see Chang and Smyth (1971), Torre (1977) and Varian (1979). A common problem with their models is that they usually rely on ad-hoc assumptions about the way business firm make decisions about investing in new capital equipment. Investment is assumed to follow a mechanical rule that depends on the current level of output and its rate of change. Because their assumptions leave no room for a rational assessment of the profitability of the new capital equipment, the interest on this type of research has faded. The model presented here, which borrows features from Blanchard (1981) and from the analysis of the classical and the Keynesian model in the first two chapters of Sargent (1987), does not share this limitation. It is a compromise between naive Keynesian models that ignore the role of expectations and general equilibrium models such as those presented in Abel and Blanchard (1983). Here the economy is forward looking because there is a stock market that values the stream of income that will be generated by the existing assets. To avoid reaching conclusions that depend on ad-hoc biases in forecasts it is assumed that there is perfect foresight. Both consumers and producers base their decisions on the stock market values. Consumers do it by consuming according to a variation of the permanent income hypothesis. Producers behave similarly when they invest according to a rule that depends upon Tobin’s q, the ratio of the price of capital to their replacement value. The main difference between our modeling of saving and investment decisions with the one in Abel and Blanchard (1983) is that our model allows for a potential conflict between them. Although all agents behave in a rationally manner, when investing the firms are not passive actors making decisions that reflect the wishes of their owners but instead investment decisions depend on the degree of confidence of the managers. Besides its realism the model considered here has interest because of its closeness to the standard Keynesian view of the economy that lies behind the IS-LM apparatus presented in standard textbooks. First we examine the case of fully flexible prices and wages. It is shown that the model becomes unstable when investors react less rapidly than consumers to changes in the valuation of the assets. The rest of the paper focuses in the main goal which is to analyze the consequences of introducing sticky wages in the economy. The modeling of wages is consistent with the presentation of most textbooks in macroeconomics. The nominal wage rate is a state variable whose rate of change depends on the expected long run rate of inflation and the current conditions in the labor market. The main result is concerning wage sluggishness is that it is in general a stabilizing force. It preserves the stability of the economy when investment is responsive to Tobin’s q, and in the unstable case of timid investors, the unstable economy approaches a business cycle. It is shown that the cycle generated by the model is consistent with some stylized factors such as the procyclical movement of real wages, capacity utilization and the price level.

Labor Market Imperfections and Endogenous Business Cyles

61

The Model The economy produces one homogenous good that can be used for consumption or investment. The price of output is denoted by P. The price of a machine already in operation is allowed to diverge from P. Some of the most common explanations given in the literature for the divergence are: the lack of a market for used capital goods, the presence of costs of adjustment in the production function or the existence of a non-linear transformation curve between capital and consumption goods. The one we would like to emphasize here is that it is due to the existence of risk aversion by the business firms: when confronted with a gap between the valuation of the capital stock in the financial markets and their replacement value the firms respond by investing more but not at a speed that will immediately make the gap disappear. The technology requires that one machine can be operated only by one worker at a time. There is however some flexibility in the level of employment since the hours of operation is a decision variable. If operated continuously a machine produces Φ units of output in one day. Let θ the fraction of the time that is kept working. It is assumed that its daily output will be Φθ units and that its daily depreciation will be cθ2 . In order to analyze the choice of θ we consider first the case of a firm that rents the machine it uses. The firm’s daily profits equal total revenue PΦθ minus the labor costs and the costs of renting the machine. Labor costs are equal to W, the daily wage rate, multiplied by θ. The owner’s compensation can be calculated as follows. First it is necessary to pay for the opportunity cost of the capital plus the depreciation costs. This amount can be obtained by multiplying the value of the machine Q by the sum of the nominal interest rate plus the rate of •

depreciation. To this figure one must deduct the capital gains Q (a dot on top of a variable denotes its derivative with respect to time). Thus the firm’s daily profit is: •

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PΦθ − (rn + cθ 2 )Q + Q − Wθ = 0

(1)

The condition that there are zero profits follows from the assumption of perfect competition. Modern literature on Keynesian models prefers to give firms some monopolistic power so the price chosen is equal to marginal cost multiplied by a mark-up. This assumption is combined with two additional ones concerning the existence of a fixed cost and a market structure that allows free entry. As a consequence, the number of firms is determined by a zero profit condition. The existence of a gap between the price and marginal cost does not make a difference for the issues studied here so we rely on a model very close in spirit to the one found in textbooks presenting the standard IS-LM apparatus. Let q be the ratio of the price of capital relative to its replacement value q = Q÷P. Let r be •

the real interest rate that is the nominal rate rn minus the rate of inflation P÷ P . Let ω stand for the real wage rate W÷P. Using this notation equation (1) can be rewritten in terms of real variables as

62

Alvaro Rodriguez •

Φθ − (r + cθ 2 )q + q − ωθ = 0

(2)

Finally the first order condition for the choice of hours of operation can be obtained differentiating (2)

Φ − 2cθq − ω = 0

(3)

We will assume the existence of an interior solution. The role of expectations in the determination of the equilibrium can be easily illustrated once we simplify the expression for the firm’s profits. According to (3) depreciation costs cθ2 equal half the gross profits (θ times one minus the real wage)1. Substituting this result into (2) one gets a linear differential equation in q which can be solved forwards. ∞

1⎡ W ( s) ⎤ q(t ) = ∫ ⎢Φ − θ ( s) exp[R( s, t )]ds, 2⎣ P( s ) ⎥⎦ t (4). s

R( s, t ) = ∫ − r (v)dv t

The market value of a machine q is the present value of the future revenue that will be obtained by operating the machine in an optimal way. In the calculation, only wage and maintenance costs are subtracted from gross revenue. Since this are the revenues to a firm that owns a machine, the above equation implies that the initial assumption that the firm rents the machines it uses is not binding since the firm is always indifferent between renting and buying. The equation is also quite useful for understanding the relation between the trajectory of real interest rates r(t) and the valuation of capital. Holding everything else constant higher real interest rates imply a lower value of q.

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The Good’s Sector Equilibrium in the goods market requires that gross output must equal consumption plus gross investment. Given the available capital stock K, we have:

KΦθ = ωKθ + bqK + β (q − 1)K + cθ 2 K

(5)

The term in the left hand side is gross output. The first two terms in the right hand side are aggregate consumption. All wage income is consumed and b is the marginal propensity to consume out of wealth. The third term is net investment, β being the sensitivity of investment 1

A more general equation for depreciation like cθ in total costs but it will complicate the algebra.

n

, n > 0 will allow for more a realistic share of depreciation

Labor Market Imperfections and Endogenous Business Cyles

63

to Tobin’s q (the ratio of the price of capital relative to its replacement cost). The last term is the depreciation allowance. According to the permanent income hypothesis consumption should depend on the present value of current and future wages not only on current wages. The present formulation is consistent with the view that market imperfections make more difficult to capitalize labor income than to capitalize capital income. Furthermore our setting incorporates the notion that consumption exhibits excessive volatility regarding fluctuations in current income, a usual characteristic of many Keynesian models. The fact that we do not have to consider future wage income reduces the dimensionality of the model and makes it more tractable. On the other hand introducing a marginal propensity to consume out of wages different from one, makes the notation more cumbersome without adding any substantial change. In the next section we will show that because the model allows b and β to take arbitrary positive values the model can exhibit different dynamic properties. As mentioned before, this is not possible in more conventional models such as the ones in Abel and Blanchard (1983) and Becker (1981), where the firms are risk neutral and are just a veil. Investment and savings decisions are always consistent with the maximization of the consumers’ utility function and in fact the economy behaves as if all decisions were left to a central planner. In their models the presence of a parameter such as β must reflect real characteristics of the economy like costs of adjustment and β will be jointly determined with the parameter “b” by the properties of the consumer’s utility function. Here firms have a mind of their own and the parameter β reflects the degree of confidence by the managers of the firms (who are different from the owners of the firm) on the valuation of the existing capital stock and their willingness to act on the available information. It is important to point out that introducing uncertainty about the forecast of future prices in a conventional model it is unlikely to lead to the results obtained here if the parameters β and b remain connected through their dependence on the risk aversion of the consumers. In other areas the model is quite conventional: All agents behave in a manner consistent with limited rationality and all of them enjoy perfect foresight. The importance given to q in the formulation of aggregate demand is similar to the approach taken in Blanchard (1981) and Phelps (1994). Both models emphasize the importance of the value of the stock market for consumption decisions.

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Labor’s Market When modeling the labor market two major scenarios are considered. In the first one wages are fully flexible. For simplicity we consider a completely inelastic labor supply L1. This feature emphasizes a Keynesian view that most labor fluctuations are involuntary and not the result of adjustments in the labor supply. Then the full equilibrium condition can be written as

θ=

L1 . K

(6)

For very low values of the capital stock, the above equality may imply values of θ higher than what it is feasible, but since the analysis is focused on a neighborhood of the steady state this possibility is ignored.

64

Alvaro Rodriguez

The second case considered here allows for the possibility of unemployment or over employment. A disequilibrium in the labor market results in a rate of increase in the nominal wage rate above or below the expected long run rate of inflation. The equation for the change in nominal wages is •

W = [α (θK − L1 ) + g ]W .

(7)

Where g is the rate of growth of the money supply which will be set equal to zero in the rest of the paper. In this model g is a proxy for the long run rate of inflation. Without such a term the model will imply the existence of a long run rate Phillips curve a result contrary to popular wisdom. Employment is equal to θK, and L1 is the exogenously given labor supply. The difference between the two is a measure of the extent of “overworking”. Thus the parameter α is positive.

Money Market In a model with wage stickiness it is important to have a monetary sector so nominal rigidities do not necessarily imply that the real wage is a predetermined variable. Although prices are affected by wages (recall the zero profit condition (1)) it will be shown that the fact that prices can jump allows expectations of the future to play a decisive role in the economy. We choose a simple, but convenient, specification of the demand for money. It is equal to aY÷rn, where Y stands for real gross output and rn is the nominal interest rate. Given the constant nominal money supply M, the condition that the demand for real money balances must equal its supply can be written as:

M aY = P rn

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Recalling the fact that Y equals equation can be written as

(8)

θΦK and the definitions of both ω and rn the above Mω a Φ θK == • W P r+ P

(9)

Finally the definition of net investment given in (5) yields the following law of motion for K •

K = β (q − 1)K .

(10)

Labor Market Imperfections and Endogenous Business Cyles

65

The Full Employment Case The dynamics of the full employment economy can be analyzed easily. The first step is to combine the first order condition (3) with the condition for equilibrium in the goods market (5) and obtain expressions for the real wage rate ω and the rate of capacity utilization θ as functions of q.

2 c q bq − β + βq W , =ω =Φ− P 2q − 1

θ=

bq − β + qβ c 2q − 1

dω dq

q =1

dθ dq

,

=−

q =1

=

c (b + β ) b

(11)

( β − b) 2 bc

(12)

Since the above two equations do not use either (6) or (7) they hold in the two versions of the model considered here. From (12) one can make q a function of θ. The full employment condition (6) implies that θ is a function of K and thus q itself is a function of K. The equation for the rate of change of K (10) can be written as •

K = β [q( K ) − 1]K

(13)

The stability of the steady state depends upon the sign of the derivative of q with respect to K. At the steady state values (q=1, θ=

b c , K= L1 ) the derivative is given by c b −1

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L 2b b dq dq dθ ⎡ dθ ⎤ = = ⎢ ⎥ − 12 = K dK dθ dK ⎣ dq ⎦ (b − β ) c

(14)

The system is stable if β is greater than b and it is unstable if the inequality is reversed. In some sense, the unstable case is associated with Keynes’s idea of low “animal spirits”. In our setting the problem is basically a mismatch between the animal spirits of those who save and those who invest. It is not that entrepreneurs are making wrong forecasts: they are just too hesitant and move too slowly in response to the available information. Concerning the comovements of the economic variables, it follows from (11) and (12) that in the stable economy as output increases the rate of capacity utilization decreases and the real wages go up. Prices tend to move in opposite direction as output. It is interesting to compare the result concerning the rate of capacity utilization with those of Calvo (1975). He considers a one good neoclassical economy without an independent investment function so q is always equal to one. For such an economy, he shows that the value of θ (the rate of

66

Alvaro Rodriguez

capacity utilization) decreases with K and that the real wage increases with K. The similarity of the results is comforting since it implies that as β increases the economy described here behaves in a way similar to that of a standard neoclassical economy without an investment function.

The Model with Sticky Wages: The Basic Equations When wages are sticky the model becomes more complicated since the interest rate must help in finding the equilibrium for both the goods and the money market. Moreover, the role of expectations on the determination of the equilibrium becomes more apparent. In this section we show that the equations of the model can be summarized as a system of three differential equations in q, the ratio of the price of the capital q to its replacement value, the nominal wage rate W and the capital stock K. The first equation for K is given by (10). The second equation giving the law of motion for W can be obtained combining (12) and (7) • ⎡ ⎛ ⎞⎤ bq − β + qβ − L1 ⎟⎟⎥W W = ⎢α ⎜⎜ K c(2q − 1) ⎢⎣ ⎝ ⎠⎥⎦

(15)

The derivation of the equation for the time derivative of q as a function of K, W and q requires some effort. It is necessary to take the time derivatives of the two equations stating the national account identity and the first order condition for choice of θ and to combine the results with the other equations of the model. The details are presented in Appendix 1. The result is: •

q = φ [W , K , q] =

H− 2 c H− 1 + qL θ − Φ + ωL Hθ Ha K q W Φ + M rw H− k q α + c q θ − Φ + ωLL + M q α ω L1 L

m H− 2 b c q2 + 4 c2 q θ2 + ω H−Φ + ωL − 2 c Hq2 β + q θ Φ − θ ωLL

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(16) where q and w are functions of q defined in () and (). Together (10), (19) and (20) summarize the model as a system of three differential equations in q, W and K. Both W and K are predetermined variables, whose values reflect the past history of the economy. In contrast, q is a jump variable whose value depends on the future trajectory of the variables in the system [see (4)].

The Steady State First we focus our attention on the stationary solution. From (10) it follows that the steady state value of the ratio q must be one. From (11) and (12) it follows that the values of θ and ω

Labor Market Imperfections and Endogenous Business Cyles

are given by

67

b and Φ − 2 bc respectively. We assume that the economy is productive c

enough so it can attain a positive stationary level of real wages. Thus Φ − 2 bc and

Φ − bc are both greater than zero. Regarding K, (20) implies that the stationary value of the capital stock must equal L1 in the expression for

c . Finally, substituting the rest point values of the variables b

dq Mb(Φ − 2 bc) we obtain W = . aL1 dt

Although the interest rate has been eliminated from the system of equations, its steady state value can be obtained without using the complicated expression that makes r a function of q,W,K,θ and ω. Instead, it is sufficient to combine the zero profit condition (2) with the equation for the goods market equilibrium (5). They imply that when q equals one the interest rate must equal b. This equality underscores the fact that the propensity to consume out of wealth b plays here the role assigned to the intertemporal rate of discount in models that maximize an intertemporal utility function. In the long run higher impatience (higher b) leads to a smaller capital stock, to a more intensive rate of capacity utilization and to a lower level of consumption.

The Unstability of the Stationary Solution The behavior of the system in a neighborhood of the steady state can be analyzed using a linear approximation.

⎡ ⎡ ⎤ ⎢ φ1 ⎢ q• ⎥ ⎢ − β )M ( b ωα ⎢W ⎥ = ⎢ ⎢•⎥ ⎢ 2aΦ ⎢K ⎥ ⎢ ⎣⎢ ⎦⎥ ⎢ L1β c ⎢⎣ b

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•

Where

φ2 0 0

⎤ ⎥ φ3 ⎥⎡ q ⎤ αb1.5c −.5 Mω ⎥ ⎢ ⎥ W aL1Φ ⎥ ⎢ ⎥ ⎥ ⎢⎣ K ⎥⎦ ⎥ 0 ⎥⎦

(17)

q P

φ[ ,W , K ] is the function defined in (19) and ω is the real wage rate. The

derivatives are evaluated at the steady state value of the variables. The analysis depends on the eigenvalues of the matrix in (21). They are the solution to

λ3 + Bλ2 + Cλ + D = 0 Where

(18)

68

Alvaro Rodriguez b c M H− 2. b β + 1. b α L1 − 1. α β L1 L − 0.5

− 1. b3ê2 c M + 1.

B=

c M Φ H1. b2 − 1. b β + 1. b α L1 − 1. α β L1 L

b c M β + 1. b

c MΦ

(19) b

c M J- 1.

C= - φ1=

b

c + 0.5 F N H-2. b b + 1. b a L1 + 1. a b L1L

-1. b c M + 1. c M b + 1.

b

c MF

(20)

3 2

D=

− mαβ l1[Φ − 2 bc ]b c m bc [Φ − bc ] + cmβ

(21)

Because one of the variables (q) is a jump variable, the relevant notion of stability is saddle point stability. The steady state is saddle path stable if the matrix in the above equation has two eigenvalues with negative real parts while the third one is real and positive. It is well known that this type of stability can be interpreted as a condition for local uniqueness. Given arbitrary values of the predetermined variables there is a unique initial value of q that puts the economy in a trajectory approaching the steady state2. One of the main results of this paper is that wage stickiness is in general a stabilizing force. The main result in this regard is: Proposition I. The system exhibits saddle path stability if one or two of the following conditions hold: a) α is smaller than

2bβ . L1 ( β + b)

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b) b is greater than b and a is greater than a critical value. A proof of the Proposition is provided in the Appendix. Considering the case b greater than b, the first part of the proposition states that introducing enough price rigidity transforms unstable economies into stable ones. As for the case when b is less than b; the economy is stable if price rigidity is very large or very small. Thus price rigidity tends to preserve stability.3 For the case when β is less than b, numerical simulations of the model showed that for small a as predicted by Proposition I the real part is negative. However we found that increasing the value of α increases the real part of the two stable eigenvalues to the point where two of the eigenvalues become pure imaginary numbers. Afterwards a further increase of α results in a completely unstable system. This change in stability is known as a Hopf bifurcartion.

2

3

This result applies only to trajectories that start and remain close to the steady state. In general it is possible that the stable and unstable manifolds intersect far away from the steady state forming a “homoclinic loop”. In that case an initial value of q in the unstable manifold is consistent with a trajectory that after a long trip eventually approaches the steady state [See Perko (1991) page 114]. The propositions do not rule out the possibility that instability may occur for some intermediate values of α. Numerical experiments indicate that does not occur although I have not been able to obtain a formal proof.

Labor Market Imperfections and Endogenous Business Cyles

69

For in depth discussions of this theorem see Guckenheimer and Holmes (1983) pages 151-152 and Iooss and Joseph (1981) chapters VII and VIII and Marsen and McCraken (1976). The existence of this bifurcation Hopf is established by the following proposition which is proved in Appendix . Proposition II. When β is less than b there exits a value of α such that the system experiences a Hopf bifurcation The remaining issue is what happens when the parameters make the steady state solution unstable. It is well known that the Hopf bifurcation implies the existence of closed orbits. An interesting issue is whether the orbits appear when the steady state is still stable a case known as subcritical bifurcation. The second case is when the orbits coexist with an unstable steady state; such a case is known as a supercritical bifurcation. This is the most interesting case because if the orbits exhibit saddle path stability they are a substitute for the steady state as the final rest point of the system.

Numerical Experiments To answer the questions about the type of bifurcation and the stability of the closed orbits we computed the solution to the model after assigning specific values to the parameters. Applying the formulas for determining the type of the bifurcation and the stability of the orbit we found that in all the cases studied the bifurcation was supercritical and the orbit was stable in the saddle path sense: It has a stable manifold of dimension two4.

g ro s s o utp u t 1.4

1.2

Rea l W ag e

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5

10

15

20

25

P ric e L e ve l

0.8

0.6

Figure 1.

4

For a discussion of the stable and unstable manifold associated with a periodic orbit see Perko (1991) .

70

Alvaro Rodriguez

1.25

Gross Output

1.2 1.15 1.1 1.05

5

10

15

20

25

Gross Investment

0.95

Figure 2.

1.06

Gross Output

1.04 1.02 5

10

15

20

25

0.98

Capacity Utilizatio n

0.96

Capital Stock

0.94

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Figure 3.

1.4

Gross Output

1.2

Tobin’s q 5

10

15

20

25

0.8

0.6

Figure 4.

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Labor Market Imperfections and Endogenous Business Cyles

71

The main results concerning the cyclical behavior of the key economic variables are presented in Figures I, II, III and IV. In those Figures all variables have been divided by their steady state values. For the simulations we choose 2.04 for the steady state value of the capital-output ratio, a value of .7 for the rate of capacity utilization and the steady state value of the interest rate was made equal to .07. The associated values of b and Φ are equal to .07 and .7 respectively. The associated value of c is .14. The chosen values also imply a share of labor in gross output equal to .72. For the figures we choose α equal to 5.1 and set β equal to .068. More about the choice of those values will be said later. Appendix IV gives references to the formulas used in the computation of the orbits. Perhaps the strongest criticism about a theory of unemployment based on wage rigidity is that it may imply countercyclical real wages since employment will be high only when labor is cheap. According to (11), in this model the behavior of the real wage depends on the cyclical movement of q. Figure IV shows that q is countercyclical and thus ω is procyclical although slightly out of phase with gross output.(See Figure I). An interpretation of this result can be obtained considering the zero profit condition (1) that makes the price level P a function of the cost of production. During and expansion nominal wages go up (recall (7)) and the increase in cost results in higher prices. However if q falls with output the price level does not go up as fast as wages. Figure I shows that, consistent with this explanation, the price level is also procyclical. According to (12) when β is smaller than b and q is close to one, the rate of capacity utilization θ is inversely related to q. Thus the model predicts that capacity utilization should move with gross output (See Figure III), a result consistent with the observed facts but at odds with the neoclassical results in Calvo (1975)5. Regarding investment, Figure II shows that gross investment is procyclical and much more volatile that gross output. The key to this result is that both the capital stock and capacity utilization are procyclical so depreciation moves with output. In contrast the model predicts that net investment (not shown in the figures) is countercyclical, a direct consequence of the fact that Tobin’s q is countercyclical. The result seems to be unavoidable in this type of model because if the capital stock is going to return to any of its previous values there must be some countercyclical movement of net investment. Changing the parameters of the model revealed the following links: Lowering the value of β causes the critical of value of α to fall but it increases the length of the business cycle. For example in our simulations we set β at .068 while we use a value of α equal to 5.1, only slightly above the critical value of 4.8. This number implies that an increase in the rate of unemployment of 1% will bring a 5% decrease of nominal wages during a year and resulted in a business cycle that it is approximately 12 years long. The choice of α which it is only slightly above the critical α (4.8) can be made smaller by choosing a smaller β. For example, when β equals .03 the critical α is just .13 but unfortunately the length of the business cycle is approximately 100 years.

5

For a discussion of the procyclical behaviour of the rate of capacity utilization see Burnside et al (1995). Greenwood et al (1988) present a real business cycle model where the rate of capacity utilization moves procyclically. In their model a beneficial supply shock affects only new equipment. The gain in efficiency encourages firms to use the old machines more intensively and depreciate them faster.

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Alvaro Rodriguez

Conclusions It has been customary to blame forecasting errors for the existence of business cycles. Overinvestment during expansions opens the door to excessive pessimism and to underinvestment during contractions. The model presented here offers an alternative interpretation. The cycles occur because price and wage inertia makes possible persistent fluctuations on the profitability of new investment. In such a case investment volatility is just a normal response to those fluctuations. Moreover, the inertia in wages and prices behind the fluctuations is potentially beneficial. Without friction a less than perfect economy may selfdestroy. A likely explanation for the result is that a slow speed of adjustment gives the jump variables (Q and P) more flexibility (and more time) in finding a trajectory that remains close to the steady state. At the heart of the explanation for the cycles is the Keynesian view that investment and savings decisions are made by different people and that their decisions are not always mutually consistent. Future research must look deeper in the way the parameters b and β are determined. A reasonable model must introduce some type of uncertainty that makes forecasting the future difficult. This fact together with some form of “agency problem” may produce the results obtained here. The managers of the firm have as one of their primary objectives not to be removed from their positions. This is more likely to happen if they over invest in times of low economic activity and as a consequence their firm perform worse than the others. Stockholders can be more for tolerant in the opposite case when profits are high and because of underinvestment the firm profits do not raise as fast as the profits other firms. This asymmetry together with the fact that consumption decisions are easier to reverse than investment decisions may explain why consumption tends to react faster than investment to changes in the valuation of assets. Another potential channel by which the same results can be obtained is by considering a model in which consumers have more limitations than firms in obtaining credit. Liquidity constrained consumers can borrow more when asset prices are high and they can offer more collateral for their loans, thus consumption is likely to be more sensitive than investment to a change in the valuation of the assets.

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Appendix 1: Derivation of Equation (16). The analysis needs a law of motion for real wages which can be obtained from (7). Let π be the rate of inflation and assume that the money supply is stationary (that is assume that g equals zero). Then (7) implies that the rate of change of real wages is given by •

ω = [α ( Kθ − L1 ) − π ]ω

(22)

The next additional two equations of the model are obtained by taking the time derivative of their static counterparts. The first one is obtained dividing the national accounts identity (5) by K, differentiating it with respect to time and recalling (15). One gets

Labor Market Imperfections and Endogenous Business Cyles •

73

•

θ [Φ − ω − 2cθ ] − q(b + β ) − θω (α (kθ − L1 ) − π ) = 0

(23)

To obtain the second one it is necessary to differentiate with respect to time of the first order condition for the choice of θ (equation (3)) and to combine the result with (15). •

•

− (α ( Kθ − L1 ) − π )ω − θ 2cq − 2cθ q = 0 .

(24)

The two new equations together with the equation for the demand for money (9), and the zero profit condition (2) are the basis of the analysis. The four equations can be regarded as a

d (θ ). ⎫ ⎧ dq , r,π , ⎬ . One can solve for these four variables as functions of the dt ⎭ ⎩ dt dq other variables in the system. The solution for is the more interesting one. It is given by dt

system in ⎨

(16) in the text.

Appendix 2: Proof of Proposition I It is well known that the polynomial in (18)) can be written as

P (λ ) = (λ − λ1 )(λ − λ2 )(λ − λ3 )

(25)

Where λ1, λ2, λ3 are the solutions to P(λ)=0. The coefficients of the polynomial are related to its roots by the following equalities

C = ∑ λi λ j i≠ j

B = ∑ − λi

(26)

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D = −λ1λ2λ3 Given that the model assumes that the steady state values of real wages is positive (that is

2 bc < Φ ) it follows that the term D in (21) is always negative. This leaves three possibilities a) One root say λ1 is real and positive and the other two are complex numbers. b) The three roots are all real and they are all positive. c) The tree roots are all real, one is positive the other two are negative.

74

Alvaro Rodriguez When α is less than

2bβ the expression for C becomes negative [see (20)] a fact that β +b

rules out possibility b. Case c) implies saddle path stability. If case a) applies C is then equal to the sum of the absolute value of the complex roots plus 2λ1 times the real part of the two remaining complex roots. Since C is negative, the real part of the complex roots must be negative. This proves the first part of the Proposition. In all our numerical calculations the only case that was encountered was case a. The system approaches the steady state in an oscillatory fashion. We proceed to examine the second part of Proposition I concerning the case when β is smaller than b. The fact that the determinant D is negative and the definitions of B and D given in (19) and (21) imply that a sufficient condition for saddle path stability is a positive value of B. That condition rules out the existence of a completely unstable system. According to (19) and our assumptions the denominator of the expression for B is always positive if the steady state value of real wages is positive. Dividing the expression by m c one gets that as

(

)

α becomes arbitrarily large the sign of B depends on the sign of Φ.5 − bc L1 ( β − b) . Given the assumptions it is positive and the proposition is proved.

Appendix III: Proof of Proposition II The Proof uses the following Lemma: Lemma 1. Equation (22) has one real positive solution and two purely imaginary ones if and only if the following conditions are satisfied: D=BC, C > 0, D < 0. The proof uses equations (18)-(21), and the equalities in (26). Suppose that there are 2 pure imaginary roots so

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λ1 is real, λ2 = iγ and λ3 =- iγ. Then C equals γ2 and must be a positive number. Moreover, D equals -λ1γ2 and D is negative when λ1 is positive. Finally B must equal -λ1 and the following condition must be fulfilled P(-B)=-CB+D=0 This proves the necessary part of Proposition IV. The sufficient part of the proposition can be proved using the definition of the discriminant of the polynomial P(λ) [See Dickson (1912)]. It is given by 18BCD-4B3D+B2C2 – 4C3 – 27D2.

Labor Market Imperfections and Endogenous Business Cyles

75

When D equals BC the above expression can be written as -8B2C2-4C3-4B4C.

(27)

Given the assumptions of Proposition IV the discriminant is negative. In such a case the cubic equation (18) has only one real root which must be positive [See Dickson (1912) page 34. Let λ1 be the real root. The fact that G[-B]=G[∑λi] equals zero means that ∑λi =λ1. This is only possible when the real part of the remaining roots is equal to zero. Finally, the positiveness of the real root follows from the fact that the assumptions in Proposition IV require B to be negative. We study proceed to examine the possibility of fulfilling the conditions in Lemma 1 under the assumption that β is smaller than b. According to (21) D is always negative given the restriction that the wage is a positive number. Recalling (20) the denominator of C can be written as

[ bcm(Φ −

]

bc) + cβ m and according to the same assumption it is positive. It

also follows that when β is less than b, the value of C is zero or positive according to whether α is equal or greater than

2bβ . Let G(α) be the function BC-D. It is a quadratic L1 ( β + b) ⎛

2bβ

⎞

⎟⎟ is positive. For large expression in α and from our previous discussion G ⎜⎜ L β b ( + ) ⎠ ⎝ 1

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values of α the sign of G(α) is the same as the sign of the coefficient for α2 in the function G. It can be verified that the sign of that coefficient depends on the sign of β2 – b2. Thus if β is less than b, the function G must be take negative values when α is large. As a consequence there must be a value α0 such that G(α0) equals zero. Since α0 is greater than 2bβπ(b+β)L1 the parameter C is positive and all the conditions of Proposition IV are fulfilled. This in turn implies that condition (a) and condition (b) in Proposition III are also satisfied. Finally, the Hopf bifurcation theorem requires that at the point where the real part of the eigenvalues becomes zero its derivative with respect to a must not vanish. Because the expression for this derivative is quite complex this requirement was confirmed numerically using many values of the parameters.

Computation of the Periodic Orbits The values of the variables are calculated using a power series expansion as a function of an amplitude parameter. We used the formulas provided in pages 125-126 of Iooss and Joseph (1981]. These formulas give a power expansion of order two. Higher order terms can be included using the formulas in pages 144-146 but they are quite cumbersome. The procedure concentrates on the part of the solution that it is spanned by the two eigenvectors associated with the pure imaginary eigenvalues. Iooss and Joseph argue that the contribution of the remaining eigenvectors (one in our case) is small and can be ignored (See pages 140-141]. The result is consistent with the well known fact that for values of α close to the critical value the periodic orbits have a quadratic tangency to the space spanned by the two complex eigenvectors( See Guckeinheimer ande Holmes (1983) page 152). Let Ec be the space

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Alvaro Rodriguez

spanned by the two complex eigenvectors. The part of the solution that it is spanned by the third eigenvector is orthogonal to Ec and therefore plays no role when α is close to the critical value.

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References Abel A and O Blanchard, “An Intertemporal Model of Saving and Investment”, Econometrica. (1983), Vol 51, No 3. 675-692. Becker Robert,”The Duality of a Dynamic Model of Equilibrium and an Optimal Growth Model: The Heterogenous Capital Goods Case”, The Quarterly Journal of Economics, (1981), Vol 96, No 2. 271-300. Blanchard O, “Output, the stock Market and Interest Rates”, American Economic Review, (1981), 71. 114-118. Burnside C, M Eichenbaum and Sergio Rebelo, “Capital Utilization and Returns to Scale’, (1995), in NBER Macroeconomics Annual 1995, Bernanke and J Rotemberg (eds). Cambridge and London: MIT Press. P 67-110. Calvo, Guillermo A, “Efficient and Optimal Utilization of Capital Services”, The American Economic Review, (1975),Vol 65, No 1.181-186. Chang W and D.S. Smyth. “The Existence and Persistence of Cycles in a Nonlinear Model: Kaldor’s 1940 Model Re-examined”. Review of Economic Studies, Vol 38, 37-44. Dickson. L.E. Elementary Theory of Equations. (1912) New York, John Wiley & Sons. Greenwood, J., Hercowitz,Z., Huffman, G. W.: Investment, capacity utilization, and the real business cycle. (1988). American Economic Review, 78, 402-417. Guckenheimer J and P. Holmes., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. (1983). New York, Springer-Verlag. Hicks,J.A. ‘’ Contribution to the Theory of the Business Cycle”, (1950), New York, Oxford University Press. Iooss G and D.D. Joseph. Elementary Stability and Bifurcation Theory. (1980). New York, Springer-Verlag. Kaldor,N.”A Model of the Trade Cycle.” (1940). Economic Journal 50. 78-92. Kalecki, M.”A Macrodynamic Theory of the Business Cycles,” Econometrica, 1935 Marsden J and M. McCraken. The Hopf Bifurcation and its Applications, (1976). New York. Springler-Verlag. Perko, Lawrence. Differential Equations and Dynamical Systems. (1991). New York, Springler-Verlag. Phelps E, Structural Slumps: The Modern Theory of Unemployment, Interest, and Assets. (1994). Cambridge, Harvard University Press. Samuelson, Paul A. ‘Interactions between the Multiplier Analysis and the Principle of Acceleration,” (1939) Review of Economic Statistics Sargent T,”Macroeconomic Theory”, (1987), London, Academic Press. Torre V. “Existence of Limit Cycles and Control in a Complete Keynesian System by Theory of Bifurcations”. (1977), Econometrica . Vol 45. 1457-1466. Varian, H.L. “Catastrophe Theory and the Business Cycle.”(1979) Economic Inquiry, Vol 17. p14-28.

In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 77-97

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 4

A NALYZING E CONOMIC P OLICY U SING H IGH O RDER P ERTURBATIONS Michael Ben-Gad∗ Department of Economics, City University Northampton Square, London EC1V 0HB, UK

Abstract In this chapter I demonstrate the use of high order general perturbations to analyze policy changes in dynamic economic models. The inclusion of high moments in approximating the behavior of dynamic models is particularly necessary for welfare analysis. I apply the method of general perturbations to the analysis of permanent changes to a flat rate tax on the return to capital in the context of the standard Ramsey optimal growth model. Reliance on simple linearizations or quadratic approximations are adequate for generating impulse responses for the variables of interest or the welfare analysis of small policy changes. However when considering the welfare implications of sizable policy changes, the failure to include higher moments can lead not only to quantitatively serious inaccuracies, but even to spurious welfare reversals.

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1. Introduction During the last quarter century macroeconomics has undergone a fundamental transformation. If once macroeconomic models were either static or dynamic but linear in structure, today’s models, based on microeconomic fundamentals are both dynamic and non-linear. Indeed, typically these models yield policy rules or laws of motion that cannot be solved analytically—necessitating the development or adoption from other disciplines of a whole range of new tools to analyze their behavior. These methods often involve some form of linear or quadratic approximation, approximations that are generally adequate for most positive analyses of the effects generated by small shocks or modest and simple changes in policy. However as I will demonstrate, for normative analyses of policy changes, and particularly if the proposed policy changes are large, failure to consider higher order moments can generate results that are often quantitatively, and occasionally qualitatively misleading. ∗

E-mail address: [email protected]

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78

Michael Ben-Gad

In this chapter I consider the welfare effects of large and permanent changes to a flat rate tax on income derived from capital in the context of the standard continuous-time Ramsey optimal growth model. The tax itself is used to fund transfer payments, so changes in the tax rate only affect household welfare indirectly, through the excess burden or deadweight loss the distortionary tax generates. To analyze the change I employ the method of general perturbations first introduced into economics by Kenneth Judd in the 1980’s (Judd 1982, 1985, 1987) and. In those original papers first order perturbations were employed to analyze relatively small changes in fiscal policy. Here because the changes I analyze are quite large, I demonstrate the use of high order perturbations (as outlined in Judd (1999)) and how the inclusion of successively higher moments can change our assessment of the welfare implications of sizable changes in policy. There are three reasons why the method of general perturbations is particularly useful for this type of analysis. First, it yields explicit continuous time formulae that describe the dynamic behavior of the economic variables rather than policy functions defined by grids or collections of points. This is particularly advantageous for analyzing welfare in the context of continuous time models, as such analysis typically requires integration of utility functions that themselves are dependent on the time path of consumption. Second the procedure permits high order approximations that are simple to implement and are analogous to Taylor expansions. Finally, the perturbations procedure is useful for analyzing complicated dynamic changes to policy, beyond the simple permanent changes I analyze here (Ben-Gad (2004, 2006, 2008)). The policy change I consider is a permanent change in a flat rate tax paid on the returns generated by physical capital and used to fund transfers to the representative household. Starting at a baseline rate of 35% I consider the quantitative effect on the welfare of the household of permanently shifting the tax rate from zero to 99%. I demonstrate that for tax cuts that are sufficiently large, failure to include high order moments in the approximation can generate the type of spurious welfare reversals documented by Tesar (1995) and Kim and Kim (2003) when they compare economies with complete and incomplete markets. Furthermore, even though qualitatively, a first order approximation may yield the same predicted welfare outcomes as a fourth order approximation, quantitatively the results may differ substantially. In the model I present, an analysis based on only the first moment of the perturbations method (essentially a linearization) generates modest overestimates of the welfare gains from lowering the tax on income from capital and large underestimates of the welfare losses generated by lowering the tax. This quantitative discrepancy is important, as rarely is such a policy considered in isolation, but only in tandem with considerations involving income distribution and tax incidence. Hence including the non-linearities first order approximations ignore is essential for producing a realistic picture of the trade-offs such policy changes imply. It is important to bear in mind that the optimal growth model considered here is generally perceived to be close to linear, which indeed it is if the changes considered are modest, and only the impulse responses, and not their welfare implications are of interest. If the higher moments in this type of model can prove to be quantitatively important for welfare analysis, or in some cases vital to insuring against spurious welfare reversals, then together the results argue for great caution when analyzing models characterized by far less linear dynamics. Furthermore the simple permanent changes considered here are not par-

Analyzing Economic Policy Using High Order Perturbations

79

ticularly dynamic themselves; they generate one-time jumps in consumption followed by a monotonic convergence process for consumption and capital to the new steady state. More complicated policy changes, such as gradual, temporary, or delayed changes to fiscal policy, even if more modest in scope than those considered here, provide plenty of opportunities for serious welfare miscalculations unless the higher moments are included when attempting to approximate the time path the economy will follow after the policy is announced.

2. The Ramsey Optimal Growth Model with Capital Taxation Consider a representative agent whose income is generated by wagesw (t) from fixed labor supply l, the return r(t) net of the flat rate tax τ (t) on capital holdings k(t) as well as a transfer payment v (t) . The agent maximizes the present value of utility U : R++ → R discounted at the rate ρ, generated by a continuous stream of consumption c (t) : Z ∞ max e−ρt U (c (t)) dt (1) c

0

subject to the continuous time budget constraint: ·

k(t) = w(t)l + (1 − τ (t)) r(t)k(t) − c(t) + v (t) ∀t.

(2)

I assume the instantaneous utility function is of the constant intertemporal elasticity of 1−σ substitution (constant relative risk aversion) form, U (c) = c1−σ , where σ ≥ 0, is the Arrow Pratt measure of relative risk aversion (the inverse of the intertemporal elasticity of substitution). The present value Hamiltonian of the optimization problem: H (c (t) , k (t) , λ (t)) = e

1−σ −ρt c (t)

1−σ

+ λ(t) (w(t)l + (1 − τ (t)) r(t)k(t) − c(t) + v (t)) (3)

yields the necessary first order conditions:

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eρt ∂H : = λ(t) ∂c c(t)σ · ∂H : (1 − τ (t)) λ(t)r(t) = −λ(t) ∂k and the transversality condition:

lim λ(t)k(t) = 0.

t→∞

(4) (5)

(6)

Differentiating (4) with respect to t and substituting into (4) yields the law of motion for consumption: 1 · (7) c(t) = [(1 − τ (t)) r(t) − ρ] c(t) σ In this economy firms combine capital and labor to produce a single good that is both consumed employed as capital, maximizing:

80

Michael Ben-Gad max {F [k(t), l] − w(t)l − (r(t) − δ) k(t)}

(8)

k,l

where F : R2+ → R+ is the production function and δ is the rate of capital depreciation. From the optimization problem (8), in equilibrium the returns to the factors of production are their marginal products: Fl [k(t), l] = w(t), (9) Fk [k(t), l] = r(t) − δ.

(10)

Assuming the production function takes the Cobb-Douglas form F [k, l] = k(t)α l1−α and setting l=1, the laws of motion governing the dynamic behavior of the economy are the law of motion for consumption:  1 · c(t) = (1 − τ (t)) αk(t)α−1 − ρ c(t) (11) σ and from the market clearing condition, the law of motion for the capital stock: ·

k(t) = k(t)α − c(t) − δk (t) .

(12)

In Figure 1 I plot the loci that characterize the relationship between consumption and ·

·

capital when in (11) c = 0, and when in (12) k = 0. In each panel σ =0.5, 1.5, or 2.5, the initial rate of taxation for income from capital is τ = 0.35, and the other parameter values are α = 0.4, δ = 0.1, ρ = 0.04. The intersection between the two loci corresponds to steady state consumption and capital and the arrows in Figure 1 represent the vector field corresponding to the system (11) and (12). Notice neither the steady state values of consumption or capital, or indeed either of the loci are themselves functions of the curvature parameter σ, but its value does subtly influence the dynamics of the system and the shape of the saddle path along which the economy must converge towards steady state. If the government chooses to permanently double the tax rate on income from capital to

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·

0.7, in order to fund more generous transfers, the locus corresponding to k = 0 remains as · it was, but the locus corresponding to c = 0 in Figure 2 shifts to the right.1 The vector fields in Figure 2 correspond to the economy after the change in policy has been announced and each indicates that consumption initially rises with the announced change in policy, before declining along with the capital stock until the system converges to its new steady state equilibrium. Similarly, a decision to eliminate transfers and the capital tax that funds them produces a qualitatively symmetric response in Figure 3—consumption initially declines but then both consumption and capital increase along a saddle path. There are a number of competing methods for evaluating the evolution ofc(t) and k (t) following such changes in policy or other exogenous shocks. It is important to emphasize that all methods, including numerical shooting are approximations. In the next section we consider one method that is both versatile and particularly suited to welfare analysis—the method of general perturbations. After that I demonstrate how it can be used to calculate the behavior of the system (11) and (12) to approximate the equilibrium values ofc(t) and k (t). 1

·

The locus corresponding to k = 0 would shift downward if the tax was used to finance government expenditure rather than a transfer payment.

Analyzing Economic Policy Using High Order Perturbations Σ=0.5 2.0

×

c=0 ×

k=0

Consumption

1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Σ=1.5 2.0

×

c=0 ×

k=0

Consumption

1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Σ=2.5 2.0

×

c=0 ×

k=0

Consumption

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1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Figure 1. Vector field for the baseline model with τ =0.35.

81

82

Michael Ben-Gad Σ=0.5 2.0

×

c=0 ×

k=0

Consumption

1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Σ=1.5 2.0

×

c=0 ×

k=0

Consumption

1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Σ=2.5 2.0

×

c=0 ×

k=0

Consumption

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1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Figure 2. Vector field following abolition of the tax on capital income.

Analyzing Economic Policy Using High Order Perturbations Σ=0.5 2.0

×

c=0 ×

k=0

Consumption

1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Σ=1.5 2.0

×

c=0 ×

k=0

Consumption

1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Σ=2.5 2.0

×

c=0 ×

k=0

Consumption

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1.5

1.0

0.5

0.0 0

1

2

3

4

5

6

7

Capital

Figure 3. Vector field following the doubling of the tax on capital income.

83

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Michael Ben-Gad

3. The Method of General Perturbations Consider a general dynamic system: .

x(t) = Γ [x(t), θ(t)] ,

(13)

where x(t) ∈ Rn × R, m is the number of control variables, n − m is the number of state variables, θ(t) ∈ Rp ×R is a vector of policy variables. Assume that the system (13) is saddle path stable. Assume the vector θ(t) is a function of a set of dynamic perturbations π(t) ∈ Rp ×R, the magnitude of its effect on the system governed by a scalar so that θ(t) = θ + π(t). I rewrite the system as: . x(t; ) = Γ [x(t; ), θ + π(t)] , (14) and differentiate with respect to  : .

x (t; ) = Γx x (t; ) + Γθ π(t),

(15)

where Γx and Γθ are Jacobian matrices evaluated at steady state values (corresponding to  = 0). A first order approximation of the solution to (13) is: x(t) ≈ x(0; 0) + x (t; ).

(16)

To solve the system (15), apply Laplace transforms to both sides: .  Ls x = Γx Ls [x ] + Γθ Ls [π]

(17) R∞

−st where the Laplace transform of a function f (t) is Ls [f h .] i= 0 f (t) e dt and s is an arbitrary positive scalar. Applying the relationship: Ls f = sLs [f ] − f (0) to (17) and solving for Ls [x ] yields:

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Ls [x ] = [sI − Γx ]−1 (x (0) + Γθ Ls [π]) ,

(18)

which can be interpreted as providing a convenient relationship between the time discounted value of the variables of the model Ls [x ], the discounted value of the shocks Ls [π] , and the initial change in the variables at the very moment a policy change becomes known x (0). Since Ls [x ] must be bounded for any positive value ofs it must be bounded for any of the positive eigenvalues of Γx : µi , i ∈ {1, 2, ..., m}.2 The determinants |µi I − Γx | for each i ∈ {1, 2, ..., m} equal zero by definition. Therefore the only way for the system to be bounded when s=µi , i ∈ {1, 2, ..., m}, is for the numerator of [µi I − Γx ]−1 , the adjoint matrix of µi I − Γx multiplied by the vector (x (0) + Γθ Ls [π]) to be equal to zero. Hence the value of the non-zero elements of x (0), the first order approximation of the changes in the control variables that occur the moment the new policy is announced is the solution to:
adj [µi I − Γx ] x (0) + Γθ Lµi [π] = 0, i ∈ {1, 2, ..., m} . 2

(19)

The number of positive eigenvalues must equal the number of controls if the system is saddle path stable.

Analyzing Economic Policy Using High Order Perturbations

85

Taking the inverse Laplace transform of (18) yields the first moment of the approximation: Z t Γx t x (t; ) = e x (0) + eΓx (t−r) Γθ π(r)dr. (20) 0

A second order approximation is obtained by differentiating (15) with respect to : .

x (t; ) = Γx x (t; ) + ω(t),

(21)

where ω(t) is a quadratic function of π(t) and x (t; ) and the tensors Γxx , Γxθ , Γθθ yielding 1 x(t; ) ≈ x(0; 0) + x (t; ) + 2 x (t; ). (22) 2 The process can then be repeated over to produce ever-closer approximations; an approximation of degree Z is: x(t; ) = x(0; 0) +

Z X i ∂ i x(t; ) + OZ i! ∂i

(23)

i=1

As an example, suppose n = 2 and m = p = 1 so that there is one control variable x1 , one state variable x2 , and only one policy variable changes. Furthermore assume the shocks are constant functions that represent permanent changes,π = 1. Then (20) reduces to: x1, (t)

=

γ x12 γ θ 2 − γ x22 γ θ 1

(24)

|Γx | +

(γ x11 γ θ 2 − γ x21 γ θ 1 ) ((γ x11 − µ1 )(γ x22 − µ2 )γ x22 eµ2 t − (γ x22 + µ1 − µ2 )γ x12 γ x21 ) (µ1 − µ2 )µ1 γ x12 |Γx |

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" #
|Γx | − µ1 2 eµ2 t γ x21 γ θ1 − γ x11 γ θ2 1+ x2, (t) = . |Γx | (µ1 − µ2 )µ1




,

(25)

The values of x1, (t) and x2, (t) from (24) and (25) become inputs in the shock processes used to calculate the second moments x1, (t) and x2, (t), which in turn are used to calculate third moments. The process can continue indefinitely, though each successive moment demands ever more computer resources to calculate.

4. Applying the Method of General Perturbations to a Permanent Change in the Rate of Capital Taxation To implement the method of perturbations to the question of a how change in the tax rate on capital income affect the economy, substitute for the tax rate on capital income in (11) and (12) τ (t) = τ + π(t) where π(t) is any bounded dynamic path and  is a small number that regulates its magnitude:   Ls [c ] = Ls [k ]

86 "

Michael Ben-Gad s

−1

  #−1  (1−α)(ρ+(1−τ )δ) ρ ρ (1 − α) δ + 1−τ −  c (0) + σ(1−τ ) ασ  ρ s − 1−τ



α(1−τ ) ρ+(1−τ )δ



1 1−α

+



α(1−τ ) ρ+(1−τ )δ

0

 α

1−α

!



Ls [π]  . (26)

The initial change in the control variable consumption is: α

1

α 1−α ρ(ρ + (1 − α)δ(1 − τ ))(1 − τ ) 1−α −2

c (0) =

1

σ(ρ + δ(1 − τ )) 1−α where µ1 =

1 2



ρ 1−τ

+

r

ρ2 (1−τ )2

+

4(1−α)(ρ+(1−τ )δ) ασ



(1 − α)δ +

Lµ2 [π]

(27)



. It is impor-

ρ 1−τ

tant to emphasize that these policy changes can represent temporary as well as permanent changes in policy. For example a simple change in the tax rate for a limited period of time can be analyzed using combinations of indicator functions.3 For simplicity’s sake I assume the change in the rate of taxation is permanent,π(t) = 1, so the first order perturbations that correspond to (24) and (25) are: 1

α 1−α ρ (1 − τ )µ1 eµ2 t − ρ

c (t) =




1−2α

2−α

,

(28)

(1 − α) (1 − τ ) 1−α (δ + ρ − τ δ) 1−α k (t) =

where µ2 =

1 2



ρ 1−τ

−

r

ρ2 (1−τ )2


1 α α 1−α ρ eµ2 t − 1 (1 − τ ) 1−α

, 2−α (1 − α) (δ + ρ − τ δ) 1−α  )δ) + 4(1−α)(ρ+(1−τ (1 − α)δ + ασ

(29) ρ 1−τ



.

As stated above, the process can be repeated to attain higher order approximations. The second order shock process in (21) are functions of the values ofc (t) and k (t) calculated in (28) and (29):

ω1 (t)

=

2 σ

(

δc (t) − α

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+ (1 − α)





ρ + (1 − τ ) δ α (1 − τ )

 2−α "  1−α

ρ + (1 − α) (1 − τ ) δ α (1 − τ )



ω 2 (t) = −α(1 − α)

!  1 1−α α (1 − τ ) + (1 − α) (1 − τ ) k (t) c (t)(30) ρ + (1 − τ ) δ ! #)   1 1−α α α (1 − τ ) + (1 − )(1 − τ )k (t) k (t) , ρ + (1 − τ ) δ 2



ρ + (1 − τ ) δ α (1 − τ )

 2−α 1−α

k (t).

(31)

In Figure 4, I plot the values of the first four moments of the approximation of consumption, c (t), c (t), c (t), and c (t), for three different values of the parameter that governs the curvature of the utility function, σ = 0.5, 1.5, 2.5. Note three important results. First, the third and fourth moments are larger than the first and second, though of course the respective weights 3 /6 and 4 /24 that multiply the latter in the approximated time path of consumption c(t; ), are significantly smaller than  and 2 /2, that multiply the former (as long as || < 1). Second, lower curvature of the utility function does not imply a lesser role for higher moments in the approximation despite what we might expect—their magnitudes 3

See Ben-Gad (2006, 2008) for the analysis of temporary changes in rates of immigration.

Analyzing Economic Policy Using High Order Perturbations

87

Σ=0.5 3 2 1 0 20

40

60

80

cΕ 100

-1

cΕΕ

-2

cΕΕΕΕ

-3

cΕΕΕ

t

Σ=1.5 0 20

40

60

80

cΕ

100

-1

cΕΕ

-2

cΕΕΕΕ

-3

cΕΕΕ

t

Σ=2.5 0 20

40

60

80

100

t

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cΕ -1

cΕΕ

-2

cΕΕΕΕ

-3

cΕΕΕ

Figure 4. The first four moments of the approximation of consumption, c (t), c (t), c (t), and c (t) for σ =0.5, 1.5, 2.5.

88

Michael Ben-Gad

relative to the lower moments are not substantially different. Finally, the overall magnitude of the third moment is generally higher than all the others—indicating that including at least this degree of non-linearity is likely to be crucial in discerning the subtle changes in the economy that this relatively straightforward change in policy generates. Further evidence for this last point can be seen in the calculated impulse responses of consumption itself following the doubling of the tax in Figures 5 or elimination of the tax in Figure 6. In the latter case, the elimination of the tax means an immediate increase in capital’s net rate of return, providing a strong incentive for higher savings and initially lower consumption. Consumption drops immediately with the announced change in policy. Over time the higher savings generates faster accumulation of capital. The marginal product of capital declines as does its rate of return and the economy converges to a steady state with higher capital and consumption. If we consider that after a century the economy has largely adjusted to the policy change, for each value ofσ, the first order approximation generates an overestimate of the new steady state level of consumption, the second order approximation an underestimate. The third order approximation produces a small overestimate, the fourth order approximation an overestimate as well, though somewhat smaller still. Doubling the rate of tax lowers the net rate of return, and the representative agent responds by dissaving, immediately raising consumption until the stock of capital declines enough to raise its marginal product. Ultimately both consumption and capital are lower and each of the approximations underestimate the decline, with the inclusion of each successive moment closing some portion of the gap. Comparing the fourth order approximation of consumption at t=100 following the tax cut, to the fourth order approximations of consumption following the tax rise, the latter comes closer to the actual long run steady state. Hence we can conclude that non-linearities beyond the first four moments of the approximation play a far more important role for tax increases than tax cuts.

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5. Welfare Calculations In practice lower order approximations in Figures 5 and 6 are often sufficiently accurate for most positive economic analyses. Not so for more normative analyses—as I will demonstrate here, the subtle non-linearities that can be safely ignored in other contexts have profound implications when analyzing welfare effects of large policy changes. Furthermore the more direct numerically driven methods for approximation such as numerical shooting do not produce explicit formulae, but rather specific numerical values whose accuracy is inversely related to the size of the gaps them. Thus when evaluating welfare in the context of a continuous time model, the use of numerical shooting to analyze the behavior of the model necessitates the use of interpolation between each point as the utility function of the time path of consumption is integrated. This creates a second source of inaccuracy. By contrast perturbations methods produce explicit formulae describing the time path of consumption whose explicit integral can often be calculated algebraically (I include the calculations for c (t), c (t), c (t), and c (t) for the three different values of σ in the Appendix). Indeed in this example, because we are only considering simple permanent shocks to the economy, the approximate time path of consumption can be characterized as a simple sum of exponential functions.

Analyzing Economic Policy Using High Order Perturbations

89

Σ=0.5 1.7

1.6

1.5

20

40

60

80

100

1.3

First

1.2

Second Third Fourth

t

Σ=1.5 1.7

1.6

1.5

1.4 20

40

60

80

100

1.3

First

1.2

Second Third Fourth

t

Σ=2.5 1.7

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1.6

1.5

1.4 20

40

60

80

100

1.3

First

1.2

Second Third Fourth

t

Figure 5. The behavior of consumption following the doubling of the tax on income from capital from 0.35 to 0.7 for different orders of the approximation. The thick solid line is steady state consumption for τ =0.7.

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Michael Ben-Gad

Σ=0.5 1.50 First Third Fourth Second

1.45 1.40 1.35 1.30 1.25 1.20 0

20

40

60

80

100

t

Σ=1.5 1.50 First Third Fourth Second

1.45 1.40 1.35 1.30 20

40

60

80

100

t

1.20 1.15

Σ=2.5 1.50 First Third Fourth Second

1.45

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1.40 1.35 1.30 1.25

20

40

60

80

100

t

1.20 1.15

Figure 6. The behavior of consumption following the reduction of the tax on income from capital from 0.35 to zero for different orders of the approximation. The thick solid line is steady state consumption for τ =0.

Analyzing Economic Policy Using High Order Perturbations

91

Table 1. Welfare effects q generated by changing the rate of taxation on income from capital from the baseline rate of 0.35 for σ=0.5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

First Order Approximation

Second Order Approximation

Third Order Approximation

Fourth Order Approximation

0.0184 0.0134 0.0082 0.0028 -0.0028 -0.0086 -0.0145 -0.0207 -0.0270 -0.0335

-0.0007 0.0036 0.0046 0.0024 -0.0032 -0.0123 -0.0249 -0.0412 -0.0614 -0.0856

0.0099 0.0074 0.0054 0.0024 -0.0033 -0.0131 -0.0289 -0.0523 -0.0854 -0.1304

0.0095 0.0073 0.0054 0.0024 -0.0033 -0.0131 -0.0290 -0.0528 -0.0867 -0.1333

To calculate the change in welfare generated by the changes in fiscal policy I calculate the compensating differential q, the fractional change in the value of initial steady state consumption c¯ necessary to equal the utility generated by the time path of consumption c(t; ), following the change in policy: Z 0

∞

e−ρt

((1 + q) c¯)1−σ dt = 1−σ

Z

∞

e−ρt 0

c(t; )1−σ dt. 1−σ

Solving for q yields: " Z q= ρ

∞

e−ρt

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0



c(t; ) c¯

1−σ

#

dt

1 1−σ

− 1.

In Tables 1, 2 and 3, I present the values of q as the tax rate varies from zero to 0.9 for the values of σ = 0.5, 1.5, 2.5 respectively, each calculated using a number of moments that vary from one to four, where once again α = 0.4, δ = 0.1, ρ = 0.04 and the initial rate of taxation on income from capital is τ = 0.35. For small changes all the first order approximations yield welfare effects that are symmetric. A rise in the rate of taxation from 0.35 to 0.4 generates a loss in welfare equivalent to a permanent drop in consumption of 0.28% if σ = 0.5 to 0.21% if σ = 2.5. Lowering the tax rate to 0.3 generates a welfare increases of nearly identical magnitudes. Again it is important to emphasize that these welfare effects are generated by changes in the excess burden the tax on capital income generates, rather than the direct effects of taxation on net income, as all proceeds from the tax are returned to the same representative agent as a transfer.

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Michael Ben-Gad

Table 2. Welfare effects q generated by changing the rate of taxation on income from capital from the baseline rate of 0.35 for σ=1.5.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

First Order Approximation

Second Order Approximation

Third Order Approximation

Fourth Order Approximation

0.0147 0.0109 0.0068 0.0023 -0.0024 -0.0074 -0.0128 -0.0184 -0.0243 -0.0305

-0.0006 0.0030 0.0039 0.0020 -0.0027 -0.0105 -0.0216 -0.0361 -0.0544 -0.0769

0.0075 0.0059 0.0045 0.0020 -0.0028 -0.0112 -0.0248 -0.0453 -0.0748 -0.1159

0.0071 0.0058 0.0045 0.0020 -0.0028 -0.0112 -0.0250 -0.0459 -0.0763 -0.1196

Adding a second moment to the approximation of the time path of consumption and the improvement in welfare generated by cutting the tax rate to 0.3 declines from the equivalent of a permanent increase in consumption of 0.28% to 0.24% if σ = 0.5, from 0.23% to 0.20% if σ = 1.5, and from 0.21% to 0.18% if σ = 2.5. By contrast the welfare losses implied by raising the tax to 0.4 is larger wherever the second moment is included. For σ = 0.5 the loss in welfare is no longer the equivalent of a permanent drop in consumption of 0.28% but rather 0.33%. For σ = 1.5 the welfare loss rises from 0.24% to 0.27% and for σ = 2.5 the welfare loss rises from 0.21% to 0.24%. The asymmetry between the welfare gains generated by tax cuts and the losses generated by welfare gains are precisely analogous to the Harberger triangle—the property of the excess burden increasing at an approximately quadratic rate in the magnitude of the distortionary tax. This quadratic property is completely missed if the analysis relies solely on first order approximations. Within the range of increasing or decreasing the tax rate by five percent, the second order approximation appears quite adequate for any of the values of σ—the addition of higher moments hardly changes the values ofq in Tables 1 to 3. However for larger changes to the tax rate these additional moments are essential.4 Consider the behavior of the compensating differential q for larger tax cuts calculated 4

The magnitudes of both the modest welfare gains generated by a given tax cut and the relatively larger welfare losses generated by a similar-sized tax increase are inversely related to the value ofσ. Hence the higher the intertemporal elasticity of substitution the more sensitive the economy to changes in the tax rate on capital income. Although this parameter plays no role in determining the steady state values of consumption and capital, it does determine the speed of convergence. The more willing the representative agent is to substitute consumption between periods in response to changes in the net rate of return, the faster the agent accumulates or disaccumulates capital in Figures 5 and 6.

Analyzing Economic Policy Using High Order Perturbations

93

Table 3. Welfare effects q generated by changing the rate of taxation on income from capital from the baseline rate of 0.35 for σ=2.5.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

First Order Approximation

Second Order Approximation

Third Order Approximation

Fourth Order Approximation

0.0125 0.0094 0.0059 0.0021 -0.0021 -0.0067 -0.0116 -0.0169 -0.0225 -0.0285

-0.0002 0.0028 0.0035 0.0018 -0.0024 -0.0094 -0.0194 -0.0327 -0.0497 -0.0709

0.0064 0.0052 0.004 0.0018 -0.0025 -0.0100 -0.0221 -0.0406 -0.0677 -0.1063

0.0059 0.0051 0.004 0.0018 -0.0025 -0.0100 -0.0223 -0.0412 -0.0692 -0.1101

when using only the second order approximation. In each case the welfare gain increases with the drop in the tax rate until a threshold that in Figure 7 is 0.182 forσ = 0.5, 0.18 for σ = 1.5, and 0.181 for σ = 2.5. Below these tax rates the welfare improvements decline as the rate of taxation declines, so that lowering the tax rate on income from capital from 0.35 to 0.1 lowers the excess burden by less than merely lowering it to 0.15. Worse still, below the thresholds of 0.012, 0.013 and 0.004 for σ = 0.5, 1.5, and 2.5 respectively, the welfare effect as measured by the second order approximation is negative. This means that lowering the tax rate below these thresholds from the baseline rate of 0.35, actually increases excess burden, leaving the representative agent worse-off. We know this last result cannot possibly be correct as it implies that by abolishing the tax, the perfectly competitive equilibria that emerge are not Pareto optimal—an obvious violation of the First Welfare Theorem. This is spurious welfare result demonstrates why the addition of higher order moments is essential, and indeed with the addition of the third moments in Tables 1-3 and in Figure 7, the anomalous results disappear. Adding one extra moment, the fourth, to the approximation has the effect of slightly correcting downward the welfare benefits generated by the tax cuts, though quantitatively these corrections are small in size as we would expect, comparing the relative sizes of c (t) and c (t) in Figure 4. Based on the fourth order approximation, removing the tax on capital yields a welfare benefit that ranges from an increase in consumption that varies between just under a one percent permanent increase in consumption if the intertemporal elasticity of substitution is high (σ = 0.5) to 0.65% if the intertemporal elasticity of substitution is low (σ = 2.5). For the U.S., U.K., or Japanese economies this is the equivalent of between two quarters (in the first instance) and one quarter (in the second instance) of per-capita output growth. For Germany, France and Italy these represent between four quarters and two and a half

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quarters of per-capita output growth. By contrast should policy makers rely on only first order approximations they would conclude that the welfare benefits from eliminating the tax are twice as big as they are likely to be. Even more alarming is the possiblility that they might rely on second order approximations and conclude that with the elimination of all distortionary taxation, the perfectly competitive economy that emerges will be one characterized by lower utility for even a representative agent. My welfare calculations for the effect of an increases in the tax rate on capital income do not contain the type of spurious welfare reversals generated by tax cuts when calculated using second order approximations. Nonetheless the exclusion of higher moments in the approximation lowers the calculated welfare loss in each case—each successively high order of the approximation raises the magnitude of the welfare loss (lowers the value ofq), though once again the marginal effect generated by inclusion of the fourth moment is very small when compared to those generated by the first three. Doubling the rate of taxation from 0.35 to 0.7 implies a loss of welfare equivalent to a drop in consumption of 5.28%, 4.29%, and 4.13% for values of σ = 0.5, 1.5, and 2.5 respectively. These are significant welfare losses, representing for any developed economy several years of per-capita output growth. Comparing a doubling of the tax to its elimination, the welfare loss from the tax increase ranges from 5.6 to 7 times the increase in welfare generated by the tax cut. Again this Harberger-like property is completely missed if the approximation of the path of consumption following the change in policy only includes the first two moments.

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6. Conclusion The results derived in this chapter demonstrate not only the method for analyzing high order approximations of dynamic non-linear models, but the pitfalls of failing to account for high order non-linearities when considering the welfare effects of policy changes. It is important to emphasize that the model as well as the policy changes considered here are each about the simplest possible. A more complicated change in the tax rate, for example one that was announced at the outset to be a temporary measure, or one whose implementation is preceded by a long delay will produce far more complicated dynamics, necessitating high order approximations even if the magnitudes of the policy changes are far more modest. Similarly, this model has only a single sector, a single representative agent and a single simple distortion. In any richer model, one with heterogenous agents, multiple sectors, external effects, more activities taxed, or graduated tax rates, linear or even quadratic approximations will yield misleading welfare predictions, possibly even spurious welfare reversals, though the proposed changes in policy might well be relatively simple in nature or far more modest in scale than the changes in the tax on capital income considered here.

Appendix The moments of the approximation for consumption for the Ramsey optimal growth model with α = 0.4, δ = 0.1, ρ = 0.04 and τ = 0.35. Note that these are all the sum of

Analyzing Economic Policy Using High Order Perturbations

95

Σ=0.5 0.00 0.2

0.4

0.6

0.8 First

-0.05 Second -0.10 Third -0.15 Fourth -0.20

Σ=1.5 0.00 0.2

0.4

0.6

0.8 First

-0.05 Second -0.10 Third -0.15 Fourth -0.20

Σ=2.5 0.00 0.2

0.4

0.6

0.8 First

-0.05

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Second -0.10

Third

-0.15

Fourth

-0.20

Figure 7. The compensating differential q (on the vertical axis) representing welfare effects of changing the tax rate (on the horizontal axis) from its baseline rate of 0.35 for the first four moments of the approximation of consumption, c (t), c (t), c (t), and c (t).

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exponential functions. The time path of consumption is the sum of these moments weighted by , 2 /2, 3 /6 and 4 /24. σ = 0.5 :

lc (t)

=

−0.272433 + 1.01307e−0.1673t

c (t)

=

−0.831602 − 0.0927452e−0.5019t − 0.745548e−0.3346t + 2.46658e−0.1673t

c (t)

=

−3.41235 + 0.025472e−0.836499t − 0.320379e−0.669199t − 1.83078e−0.5019t −2.07682e−0.3346t + 8.9003e−0.1673t

c (t)

=

−1.91844 − 0.0116596e−1.1711t + 0.43972e−1.0038t + 4.65392e−0.836499t −5.37026e−0.669199t − 27.3845e−0.5019t + 33.3701e−0.3346t − 2.29892e−0.1673t

σ = 1.5 :

lc (t)

=

−0.272433 + 0.654544e−0.0863132t

c (t)

=

−0.831602 − 0.40075e−0.258939t − 0.479341e−0.172626t + 1.92375e−0.0863132t

c (t)

=

−3.41235 + 0.736088e−0.431566t + 0.560929e−0.345253t − 4.23035e−0.258939t −1.25476e−0.172626t + 7.77697e−0.0863132t

c (t)

=

−1.91844 − 2.25338e−0.604192t + 1.61767e−0.517879t + 15.7705e−0.431566t −14.7007e−0.345253t − 23.0643e−0.258939t + 28.2684e−0.172626t − 3.70031e−0.0863132t

σ = 2.5 :

lc (t)

=

−0.272433 + 0.54685e−0.0619867t

c (t)

=

−0.831602 − 0.420346e−0.18596t − 0.399379e−0.123973t + 1.7607e−0.0619867t

c (t)

=

−3.41235 + 0.969317e−0.309934t + 0.572117e−0.247947t − 4.49541e−0.18596t −1.00783e−0.123973t + 7.43954e−0.0619867t

c (t)

=

−1.91844 − 3.72541e−0.433907t + 2.85654e−0.37192t + 18.4319e−0.309934t −18.4476e−0.247947t − 19.8249e−0.18596t + 26.7359e−0.123973t − 4.12126e−0.0619867t

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References [1] Ben-Gad, Michael, “The impact of immigrant dynasties on wage inequality,”Research in Labor Economics 24, 2006, 77-134. [2] Ben-Gad, Michael “Capital-skill complementarities and the immigration surplus,”Review of Economic Dynamics 11, 2008, 335-365. [3] Ben-Gad, Michael, “The economic effects of immigration—a dynamic analysis,”Journal of Economic Dynamics and Control 28, 2004, 1825-1845. [4] Judd, Kenneth L., “An alternative to steady-state comparisons in perfect foresight models,” Economics Letters 10, 1982, 55-59. [5] Judd, Kenneth L., “The welfare cost of factor taxation in a perfect-foresight model,” Journal of Public Economics 28, 1985, 59-83.

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[6] Judd, Kenneth L., “Redistributive taxation in a simple perfect foresight model,”Journal of Political Economy 95, 1987, 675-709. [7] Judd, Kenneth L., Numerical Methods in Economics, MIT Press, 1998. [8] Kim, Jinill, and Sunghyun Henry Kim, “Spurious welfare reversals in international business cycle models,” Journal of International Economics 60, 2003, 471-500.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

[9] Tesar, Linda, “Evaluating the gains from international risksharing,”Carnegie-Rochester Conference Series on Public Policy, 42, 95-143.

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In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 99-120

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 5

I NVARIANCE IN E CONOMIC DYNAMICS AND THE S USTAINABLE D EVELOPMENT I SSUE Vincent Martinet∗and Gilles Rotillon† INRA - UMR Economie Publique EconomiX - Universit´e Paris X

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Abstract In this chapter, we examine how the Economic Dynamics Theory can be used to address the sustainable development issue. Sustainability is related to economic dynamics and intergenerational equity concerns. We first describe how the environmental concerns have been taken into account in the neoclassical approach of the optimal economic growth. In such a framework, the Hamiltonian of the optimization problem is interpreted as a sustainability indicator. We examine how these results are consistent with the way the sustainability issue is described as a requirement to conserve utility level through time. Sustainability is then achieved when genuine saving is positive, “greening” the Net National Product. After that, we use the invariance approach (Noether’s theorem) to exhibit the conditions for conservation laws in economic dynamics to exist. If something is preserved along an optimal economic path, it can be interpreted as a representation of sustainability. We discuss how the existence conditions are related to the economy’s characteristics (time preference and technology). Moreover, these conservation laws are linked to time and state variables transformations in the invariance theory. We relate these transformations to endogenous changes in the economy (environmental preferences and technological progress) and show that the conserved quantities can be interpreted as modified Hamiltonians, defining new sustainability account systems.

Key-words: Sustainability, Optimal control, Invariance ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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Vincent Martinet and Gilles Rotillon

1. Introduction: Sustainable Development Issue and Economic Dynamics Economics is the science that studies optimal use of scarce resources, and an important issue relies on the study of how these scarce resources are used through time, in a dynamic perspective. Since Ramsey (1928) the economics dynamics is related to optimal growth theory. He proposed an intertemporal social welfare function and then tried to obtain the “optimal” rate of savings as the rate which maximized “social utility” subject to some underlying economic constraints. By deriving the “optimal rate of savings” for a society using the standard Benthamite utilitarian calculus, Ramsey defined the Golden rule of an economy, that makes it possible to reach (and sustain) the highest utility level. The optimal growth theory is based on the use of an economic criterion, usually the discounted sum of intertemporal utilities (called the neo-classical criterion) to define optimal growth paths. It results in the definition of an optimal stream of consumption/utility, along with the associated optimal investment decisions. Such an optimal path defines all the trajectory of the economy from a given initial endowment in capitals (including human-made reproducible capital, labor, human capital and also natural resources). Nowadays, the sustainability issue is an important concern. Some societies are asking themselves if the way they consume, invest, use natural resources, can go on without jeopardizing development opportunities for future generations. An economic way to describe that sustainability concern and the related intergenerational equity issue is to require a development path to exhibit a non-decreasing utility to be sustainable (Asheim et al., 2001). In a more general sense, the underlying idea of sustainability in the economic literature can be summarized by the following quotation by the Nobel Price Robert Solow: “If the sustainability means anything more than a vague emotional commitment, it must require that something be conserved for the very long run. It is very important to understand what that thing is: I think it has to be a generalized capacity to produce economic well-being.” (Solow, 1993, p.167-168)

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As stated by Holland (1994, p.169) “On any account of sustainability . . . something or other is supposed to be kept going, or at any rate not allowed to decline other time”. Sustainability is, from that point of view, a conservation issue. Conservation issue in dynamic systems are related to the existence of conservation laws, i.e. invariant quantities along trajectories. The links between economic dynamics and invariance have been emphasized by Samuelson (1970), Sato (1999), Sato and Kim (2002) and Askenazy (2003). In this chapter, we focus on a particular interpretation of conservation laws in economic dynamics: we interpret invariant quantities along optimal growth paths as sustainability indicators. Martinet and Rotillon (2007) links Invariance in optimal growth models with the sustainability issue. Studying simple models, they show that if the neoclassical framework can provide some invariant which can be interpreted as sustainability indicators, it is always under very restrictives conditions. They also conjecture that more complicated models will provide more restrictive conditions for the existence of a conservation law. In a more general model than Martinet and Rotillon (2007), we confirm their

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101

results through an exhaustive study of the feasible production functions. The chapter is organized as follows. Section 2. survey the main results of the neoclassical approach dealing with the environment. Section 3. presents the link between invariance and sustainability given by the Noether theorem and applies it to the study of a canonical model describing an economy using an exhaustible resource as an input in a general production function. We show that only some production functions are possible and we determine the associated invariance laws. Section 4. gives the interpretation of these results, and a discussion. Section 5. concludes.

2. Neoclassical Growth Theory and the Environment In this first section, we describe the links between optimal growth theory and national accounting. National accounting aims at defining wealth indicators. Now that the sustainability issue is predominant, we can examine how sustainability indicators could be build by extending the theoretical national accounting framework.

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2.1. Optimal Growth Theory and National Accounting Today, evaluation and comparison of economic performance between nations throughout the world rest on a national accounting system initiated by Keynes (1940). From a theoretical point of view, national accounting is an attempt to summarize the economic activities of a country, Gross National Product (GDP) and Net National Product (NNP) being the most important aggregates that comes out from the national accounts. But one of the main question about these aggregates is on their significance. What exactly GDP or NNP are seeking to measure? Is there a link with the sustainability issue? The theoretical framework of optimal growth can help to answer to these questions. The seminal result in this field comes with Weitzman (1976) who shows that in a dynamic economy based on the use of capital stocks, if the social objective is to maximize the integral of discounted consumption, the current-value Hamiltonian of the problem corresponds to net national expenditures, defined as the sum of current consumption plus net current investment evaluated at shadow prices of investment in consumption terms. We recall here Weitzman’s result in a more general framework, where the objective of the economy is not the consumption but the utility of consumption, where the strictly concave utility function is an ordinal representation of the preferences of the society, defined up a positive affine transformation. The problem is to maximize the integral of the discounted utility of consumption U (c(t)), in an economy with n capital goods, where S(t) = [S1 (t), . . . , Sn (t)] is the vector of capital stocks, and c(t) ∈ Rn the consumption vector. Each capital stock evolves through time with respect to the consumption and the production. We consider n production functions gi (S(t)), with i = 1, . . . , n. The stocks dynamics read S˙ i (t) = gi (S(t)) − ci (t),

i = 1, . . . , n

(1)

where gi is assume to be concave for all i. We note λi (t) the shadow value of the capital

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Vincent Martinet and Gilles Rotillon

stock i and the current value Hamiltonian of the problem is H(t) = U (c(t)) +

n X

λi (t) [gi (S(t)) − ci (t)]

(2)

i=1

The value function associated is V (S(t))

=

max

Z

c(τ )

s.t.

∞

U (c(τ ))e−δ(τ −t) dτ

(3)

t

S˙ i (τ ) = gi (S(τ )) − ci (τ )

S(t) given This function gives the maximum present value utility that the economy can obtain at date t from stocks S(t) and represents the true welfare of the economy. By standard result we have ∂V = λi (4) ∂Si It immediately implies that n X dV = λi S˙ i (5) dt i=1

which states that welfare is increasing along an optimal path if and only if the value of the stocks (at the shadow prices) is increasing. But taking the time derivative of eq. (3), and using eq. (5) gives immediately that δV = H

(6)

So, the Hamiltonian can be view at the interest at the discount rate of the value function. Moreover, eq. (6) gives Z ∞ H(t)e−δt dt (7) V =

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0

which shows that the Hamiltonian is an average future utility level associated with an optimal path.1 One important result of this literature is that, if the Hamiltonian is autonomous, it stays constant along the optimal path followed by the economy. It means that the income of the society is kept constant. The links between Value (objective functions), income and sustainability are well-developed in Cairns (2007). 1

What is the link between this welfare and a measure of wealth in a national account system? If we approximate u(c∗ ), c∗ being the optimal consumption, by the linear function< λ.c∗ > where λ are the shadow prices of the different capital stocks, we can define a net national product measured in utility units by the “linearized Hamiltonian” given by N N P =< λ.c∗ > + < λ.S˙ > where < x.y > is the inner product of the two vectors x and y. If, as in Weitzman (1976), we take U (c) = c we obtain that the Hamiltonian is equal to the NNP of the economy and that NNP is related to welfare by eq.6. These results are obtained under very restrictive assumptions, which in essence imply that the Hamiltonian is autonomous. If it is not the case, like with exogenous technical progress, Aronsson et al. (1997) or Usher (1994) show that NNP is not related to welfare in a so simple way. We must add to the linear welfare measure the present value of marginal technological progress along the optimal path.

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2.2. Optimal Growth Theory and the Sustainability Issue Until recent decades, neoclassical growth theory has given no much attention to the relationship between economic growth and the environment. In the last thirty years, extensive research has been undertaken to explore the link between economic growth and the environment, and two main problems have been studied. The first one is about the relationship between economic growth and pollution (see Xepapadeas (2005) for a survey). The second one studies the impacts of natural resources on growth process. In the latter case, the sustainability issue cannot be avoided if we consider non-renewable resources. From that point of view, the theoretical framework used to address the sustainability issue relies on the intertemporal allocation of non-renewable resources. This framework allows us to take into account both the intergenerational equity issue, and the environmental concerns, by defining how much of the non-renewable resource stock must be depleted (and thus how much should be preserved), and how the consumption should be spread through time. In this framework, Dasgupta and Heal (1974) have proposed a seminal model where the economy is represented by a two-stock model, with reproducible capital Kt and natural resources St . A part rt of the exhaustible resource is used in the production function ˙ F (K, r) to produce an aggregated good that can be either consumed (ct ) or invested (K). The optimal program reads Z ∞ max e−δt U (ct )dt (8) c(.),r(.) 0

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s.t. K˙ = F (Kt , rt ) − ct S˙ = −rt

(9) (10)

In this model, they show that if there is no technical progress and if the marginal return of capital are decreasing, there is no growth and the consumption decreases toward zero at the infinity. Nevertheless, this paper was about the optimal depletion of an exhaustible resource and the sustainability issue was not on the agenda. The first attempt to deal with resource preservation is Krautkraemer (1985) who introduces the stock of an exhaustible resource in the utility function to take into account the value that society gives to natural resources. The optimization problem becomes Z ∞ e−δt U (ct , St )dt. (11) max c(.),r(.) 0

In this model, the consumption decreases toward zero after some finite time, and a partS ∗ of the resource stock is preserved forever, under the condition thatUS0 (0, S ∗ ) = δUc0 (0, S ∗ ). Altogether, these models define optimal growth paths with natural resources, and raise the issue of the sustainability of the economy, as no positive level of consumption is sustainable with a neo-classical objective function if there is no technological change.

2.3. Sustainability Objectives and Sustainability Indicators The national accounting theoretical framework presented is section 2.1. can be extended to take into account natural resources depletion and environmental degradations. Its

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leads to a “Green” national accounting in which the linearized Hamiltonian at time t is the sum of consumption at time t plus the value of net investment in capital stocks (investment/depletion) evaluated at shadow prices, where these shadow prices encompasses all the way the stocks are valuated in the utility function. Results of eq. (7) holds true, and the Hamiltonian of problem (11) can be interpreted in the same way as a welfare indicator. In the optimal growth literature, the debate on the sustainability issue (in the weak sustainability approach) is generally addressed by requiring the utility level (economic wellbeing) to be sustained. Asheim et al. (2001) argue that sustainability requires the utility to be non decreasing along time, for equity concerns. Stavins et al. (2003) interpret sustainability as ‘intergenerational equity’ plus ‘economic efficiency’. Intergenerational equity is defined by requiring a non decreasing utility through time, and economic efficiency is defined with respect to the neo-classical discounted utilitarian criterion. In these approaches, the social objective (sustainability of the utility) is not considered in the objective function (in the criterion to optimize) but as an added constraint to an economic criterion. The condition for utility to be constant in an autonomous problem (where the current Hamiltonian is constant) is that the value of stocks is constant, which means that resource depletion must be compensated for by capital accumulation. This result is known as “Hartwick’s investment rule” (Hartwick, 1977; Dixit et al., 1980). In this approach, the sustainability requirement is defined as an additional exogenous constraint. According to Krautkraemer (1998) and Cairns and Long (2006), the sustainability concern must be defined within the objective function, and not as an additional constraint.

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If sustainability is about the preservation of something along time, another approach consists in defining a “sustainability criterion”, and examine what that criterion preserves. This approach leads to a debate on the criterion to be used to determine optimal sustainable growth path. Heal (1998) discusses the various criteria proposed to interpret the sustainable development concept. The debate however focuses around two issues: economic efficiency (maximization of a criterion) and intergenerational equity (preservation of “something” in the long-run). One can thus discuss various objective function (sustainability criterion) with respect to the things they preserve for sustainability. Several definitions of sustainability have been proposed (see Dobson, 1996, for a “taxonomy of sustainabilities”). Roughly speaking, the strong sustainability point of view is based on the idea that natural resources (or environmental quality) and human-made capital are complements in the production of well-being rather than substitutes. It leads to require the preservation of some critical natural resources in order to allow future generations to meet their needs. On the contrary, from the weak sustainability point of view, the thing that matters is the economic well-being, whatever are the components of the utility function, and the requirement is on the preservation of utility level, the degradation of natural resources being allowed to be compensated for by an increase in the consumption of other goods. Anyway the issue is to determine what can be sustained, or what has to be sustained. The sustainability debate in the economic literature emerges as a debate on the criterion to optimize for a sustainable development. A criterion defines optimal consumption, re-

Invariance in Economic Dynamics and the Sustainable Development Issue

105

source use, investment... and in general the optimal path is unique and leads to an intertemporal allocation of welfare among present and future generations. Heal (1998) examines the various criteria proposed to cope with the sustainability issue. Each criterion characterizes the optimal (sustainable) path and defines what is preserved for sustainability (if anything is preserved). To describe the criteria, he uses the framework of the intertemporal allocation of a non renewable resource2 , allowing a discussion on both the equity issue and the natural resource preservation issue. The simplest model to address these issues is the “cake-eating economy” model in which the only economic good is the natural non renewable resource St , that is consumed by all generations, over an infinite horizon. The dynamic model reads S˙

=

−ct Z ∞

s.t.

ct dt ≤ S0

0

where ct is the consumption at time t. This model is inspired by the Hotelling (1931) paper on optimal intertemporal use of exhaustible resources.3 The most commonly used criterion is the intertemporal sum of discounted utilities Z ∞ max ∆(t)Ut dt. (12) 0

This criterion, which in general has a solution only if the discount factor ∆(t) decreases toward zero at the infinite time4 , is criticized because it does not take long term utility into account. According to Chichilnisky (1996) this criterion is a dictatorship of the present. Another proposed criterion (Chichilnisky et al., 1995) is inspired by the economic Golden Rule, and defines the highest indefinitely maintainable level of instantaneous utility. The criterion reads max lim Ut (13)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

t→∞

This approach, called the green golden rule, does not take into account the present, and is called a “dictatorship of the future” by Chichilnisky (1996). Solow (1974) and Cairns and Long (2006) address the sustainability issue by using the maximin criterion. This criterion defines the maximal sustainable level of utility, in the sense that it maximizes the utility of the poorest generation:   (14) max min Ut t

This criterion has been criticized as it may lead to maintain the initial poverty by restricting the investment if the first generation is the poorest. As mentioned by Chichilnisky et al. (1995, p.179), “an important task . . . lies in the analysis of criteria that combine maximization of discounted utility with elements related 2

For non renewable resources, the sustainability issue can not be avoided. As mentioned by Heal (1998), although its relevance to a policy debate is seriously limited, this model helps understanding the implication of various criteria. 4 Usually, the discount factor is decreasing at a constant rate, i.e. ∆(t) = e−δt . Other alternatives have been proposed, including hyperbolic discounting which leads to a decreasing discount rate. A general form of hyperbolic discounting is ∆(t) = (1 + αt)−γ/α . Nevertheless, the discount factor is still decreasing and the utility of the far future is not taken into account, which raises the equity issue. 3

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to the long run”. In other words, sustainability issue relies on the definition of criteria that take into account both short and long run. Chichilnisky (1996) develops such a study and provides the general form that the criterion must have:  Z max θ

∞

∆(t)Ut dt + (1 − θ) lim Ut 0



t→∞

(15)

where ∆(t) is the discount factor. Nevertheless, the solution of criteria of this form is not easy to compute (See Heal, 1998). Moreover, the criterion is not unique and depends on the choice of the parameter θ and of the discount factor ∆(t).5 In a recent contribution, Long (2006) proposes, in a complementary way of Chichilnisky (1996), to mix a maximin criterion and the utilitarian discounted criterion. The criterion he proposes, called a “Mixed Bentham-Rawls criterion” reads6  Z max θ 0

∞

 ∆(t)Ut dt + (1 − θ) min Ut . t

(16)

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This criterion is sensitive to the present, to the future and to the least advantaged generation. As mentioned in the beginning of the introduction, sustainability can be interpreted as the requirement to have something preserved in the long-run, in an intergenerational equity perspective. Criteria are more or less preservative with respect to the resource use and the level of consumption. The maximin criterion is the most explicit in the definition of what is preserved, as it targets the preservation of the utility along time. In particular, the equity issue is mainly addressed with respect to the consumption of each generation, and the utility function depends on the instantaneous consumption ct , namely U (ct ). Each criterion leads to a infinite stream of consumption. The emergence of environmental issue has lead to take into account natural resource amenities, which emphasizes the preservation issue. In that perspective, the utilitarian approach can be completed: Natural resources St can be a component of the utility function, U (ct , St ). Heal (1998) examines the implications of such an approach in the sustainability debate. Depending on the criteria, there is more or less consumption and thus “less or more” preservation of resource stocks. In the cake-eating economy, consumption can not been sustained. In the present paper, we analyze what is preserved in optimal growth models, when the economic criterion is the discounted sum of intertemporal utilities. We thus wonder if the Neoclassical paradigm is able to deal with the sustainability issue (i.e. to preserve something for future generations). 5 The term lim can also be replaced by any function that depends only on the limiting behavior of the utility over time, such as long-run average for example. 6 Long (2006) develops the criterion in a discreet time framework and over a finite time horizon. He also considers a constant discount rate. To be consistent within the presentation of the various criteria, we present the criterion in a continuous time framework and over an infinite horizon.

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3. Invariance and Sustainability

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3.1. Interpreting Invariance as Sustainability In mathematics, the existence of symmetries makes it possible to reduce the dimension of analytical problems. This method has been largely used in physics or mechanics to solve systems of partial differential equations, but also in economics for solving optimal growth models with technical change (Samuelson, 1970; Sato, 1999). The Noether theorem (Noether, 1918) is the main tool to exploit symmetries of dynamical optimal programs. This theorem gives conservation laws, i.e. invariant quantities along optimal paths. Such conservation laws can help to solve optimization problems by reducing their dimension. But conservation laws can have direct interpretations. A well-known interpretation in physics is the conservation of energy. In economics, some conservation laws have been interpreted as income or wealth conservation (Samuelson, 1970; Sato, 1999). In the present chapter, we propose to interpret the conservation laws of a dynamical optimal growth problem as sustainability. If sustainability requires to conserve something along time, and if the optimal growth is defined so to maximize an intertemporal social objective function, one can interpret the conservation laws of the optimization problem as the quantities that are preserved for sustainable development by the studied criterion. The criterion that is maximized in an optimal growth problem represents the social preferences, and should encompasses preferences for sustainability. An optimal growth path is the path that maximizes the value function / the criterion. If the criterion represents the sustainable development concerns, the optimal growth path will maximize the sustainability objective function. We wonder if the neoclassical discounted utility criterion is a good “candidate” to be a sustainability criterion. To analyze this question, we will use the Noether theorem to exhibit conservation laws in economic dynamics, along optimal growth paths defined by the neoclassical criterion. If sustainability requires something to be preserved along time, a easy way to discuss the consequences (or the appropriateness) of a criterion for addressing the sustainability issue is to examine what that criterion conserves. Moreover, the general methodology we propose must not depend on a specific representation of the preferences, and thus must be valid for any utility function. We here do the same hypothesis that in Sato (1999). For our issue, it means that the fact that we interpret sustainability as the requirement to conserve something along time (whatever the conserved “thing”) must not depend on specific preferences. Note however that the conservation laws (and thus the “thing” that is preserved along the optimal path) could depend on the preferences (the utility function), and thus could actually change from one preference set to another.

3.2. Noether’s Theorem and Economic Dynamics The Noether theorem (Noether, 1918) makes it possible to exhibit conservation laws, i.e. invariant quantities, along optimal paths defined by a criterion Z ∞ L(t, X, u)dt (17) max u(.)

0

where X stands for the state variables of the dynamical system andu for the controls.

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Such conservation laws are usually not easy to find, but can be exhibited by some state transformations. It is linked to symmetry properties of the problem. We define the transformations on the variables t¯ = φ(t, X, ε), ¯ i = ψ i (t, X, ε) X

(i = 1, . . . , n)

(18)

and their infinitesimal generators7 τ are ξ i respectively. If there are such transformations that satisfy the followingfundamental invariance identity   n  X ∂L ∂L i ∂L dξ i dτ i dτ ˙ τ+ ξ + −X +L =0 (19) i i ∂t ∂X dt dt dt ∂ X˙ i=1

then, the Noether theorem leads to the conservation laws, that is given by ! n n X X ∂L ∂L i x˙ i i τ + ξ = cte, Ω≡ L− ∂ x˙ ∂ x˙ i i=1

(20)

i=1

We thus need to find the conditions for a group of transformation to satisfy thefundamental invariance identity in order to get conservation laws. The problems we study are characterized by a neo-classical optimality criterion Z ∞ e−δt U (ct , St )dt (21) max 0

where the utility function U (c, S) is a representation of an ordinal utility preorder.

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3.3. Invariance in Growth Models with Non-renewable Resources We examine what is conserved along the optimal path defined by the neo-classical criterion in dynamical economic models with non-renewable resources. In the first model we consider, the only economic good is a non-renewable resource that is used alone to produce a consumption good. In the second model, we consider a two goods economy in which the natural resource is used along with a reproducible human-made capital to production a composite good that can be either consumed or invested. Both the models are very usual models in natural resource economics (Dasgupta and Heal, 1979) and in the sustainable development issue literature (Heal, 1998). 3.3.1. The Cake-Eating Economy We consider the following cake-eating economic problem Z ∞ e−δt U (ct , St )dt max c(.)

(22)

0

under the dynamics S˙ = −Q(t)ct . 7

i.e. the first order coefficients of the Taylor series of φ and ψ i about ε = 0.

(23)

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˙ where P (t) = 1 is the technologThis last equation can be rewritten like ct = −P (t)S, Q(t) ical progress enhancing the resource use. The Lagrangian of the problem is thus ˙ = e−δt U (−P (t)S, ˙ St ) L(t, S, S)

(24)

We consider the following one-parameter transformations t¯ = t + τ (t, K, S)ε S¯ = S + ξ(t, K, S)ε

(25) (26)

and examine what are the conditions for invariants to be, i.e. for the application of Noether theorem. If transformations (25) and (26) lead to an invariant, the following fundamental invariance identity must hold8     dτ dτ dξ −δt −δt −δt −δt ˙ ˙ ˙ τ −δe U − e U1 P S + ξe U2 − P e U1 −S + e−δt U = 0 (27) dt dt dt As it must be satisfied for any utility function, we can vanish the factors of the utility function and its derivatives, and get the system dτ = 0 −τ δe−δt U + e−δt U dt   dξ dτ −τ e−δt U1 P˙ S˙ − P e−δt U1 − S˙ = 0 dt dt ξe−δt U2 = 0

(28) (29) (30)

˙

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which leads to τ = w1 eδt , ξ = 0 and PP = δ, where w1 is a constant. By transforming the problem using τ = w1 eδt , we exhibit an invariant if the technological progress has a constant rate of growth, equals to the utility discount rate. Using relation (20), we deduce the conservation law Ω ≡ U − cU1

(31)

Although this canonical model is too simple to address all aspects of the sustainability issue, it is useful for two reasons. Firstly, it allows us to describe the way the Noether theorem can be applied to optimal growth models with natural resources. Secondly, it allows us to compare our results with the existing literature (Hotelling, 1931; Heal, 1998). ˙ 1 is equal to the The preserved quantity, which can be formulated as Ω = U + P (t)SU sum of utility plus the stock depreciation valuated at the consumption price (marginal utility of consumption). 8

We apply the fundamental invariance identity given by eq.(19) to the Lagrangian given by eq. (24).

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3.3.2. A Production-Consumption Economy We now turn to a production-consumption economy, described by the dynamics S˙ = −rt K˙ = F (K, r) − ct − nK

(32) (33)

where F (K, r) is the production function that depends both on the accumulated humanmade capital K and the extracted natural resource r, ct is the consumption, and n the capital depreciation rate. The problem is to maximize the discounted sum of utility Z ∞ max e−δt U (ct , St )dt c(.),r(.) 0

(34)

We consider the transformations t¯ = t + τ (t, S, K)ε S¯ = S + ξ(t, S, K)ε ¯ = K + µ(t, S, K)ε K

(35) (36) (37)

and examine what are the condition on the economy (especially on the form of the production function) for conservation laws to be along optimal growth paths. The Lagrangian associated to this optimization problem is   ˙ − K˙ − nK, S ˙ K)) ˙ = e−δt U F (K, −S) (38) L(t, (S, K), (S,

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A necessary condition for the existence of conservation law is that there exist some set of transformations (τ, ξ, µ) such that the fundamental invariance identity defined by eq. (19) holds true, that is for our specific problem dτ ∂L τ +L dt  ∂t  ∂L dξ ∂L dτ ˙ ξ+ −S + ∂S dt ∂ S˙ dt    ∂L dµ dτ ∂L ˙ µ+ −K =0 + ∂K dt dt ∂ K˙

(39)

Using the Lagrangian expression (38), the previous equation becomes −τ δe−δt U + e−δt U

dτ dt

 dτ dξ ˙ −S −e + ξe dt dt   dτ dµ 0 =0 − n) − e−δt Uc0 − K˙ + µe−δt Uc (FK dt dt −δt

US0

−δt

Uc0 Fr0

(40)

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which is equivalent to 

 dτ U − δτ dt      dτ dµ dτ 0 dξ 0 +Uc − Fr − S˙ + µ(FK − n) − − K˙ dt dt dt dt +US ξ = 0 By vanishing the utility’s factors in order to get a result valid for any utility function, we get the following three equations system ξ = 0 dτ − δτ dt dτ dµ dτ 0 + µ(FK − n) − + K˙ Fr0 S˙ dt dt dt

(41)

= 0

(42)

= 0

(43)

From eq. (42), we get ττ˙ = δ, which leads to τ = w1 eδt , where w1 is a integration constant. We now must define µ from eq. (43). We know that µ is a function of the time and ∂µ ˙ ∂µ ˙ ∂µ state variables: µ(t, S, K). We can write the time derivative of µ as dµ dt = ∂t + S ∂S + K ∂K . Substituting this expression in eq. (43), we get ˙ 1 δeδt = 0 ˙ 1 eδt + µ(F 0 − n) − ∂µ − S˙ ∂µ − K˙ ∂µ + Kw Fr0 Sδw K ∂t ∂S ∂K

(44)

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As the variable transformations depends on time and state variables only,µ is a function of ˙ the previous expression must not t, K and S only (and not of the time derivatives K˙ and S), ˙ As F (K, r) depends on S, ˙ we can not derivate any general statement depend on K˙ and S. ˙ from the previous equation. However, as F (K, r) and its derivatives does not depend on K, ˙ we can already vanish the factors of K in the previous expression and state that a necessary condition for the fundamental invariance identity to holds true is ∂µ ˙ 1 δeδt = 0 ⇒ ∂µ = w1 δeδt + Kw −K˙ ∂K ∂K ⇒ µ(t, K, S) = w1 δeδt K + φ(t, S)

(45)

All is now about the solution of eq. (44), which becomes: 0 ˙ 1 eδt + (w1 δeδt K + φ(t, S))(FK ˙ 0S = 0 − n) − w1 δ 2 eδt K − φ0t − Sφ Fr0 Sδw

(46)

The solution of this equation will depend on the form of the production function, and 0 depends on r or not, i.e. on the separability of the production especially on the fact that FK factor in the production function. We can distinguish the following cases. It appears that all cases can be sorted out in the following six production function classes, depending on the form of the production function, and the fact that its derivatives depends on S˙ or not, and how. Of particular interest is the distinction between separative production functions and non-separative ones.

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A - Separative Production Function Case : F (K, r) = A(K) + B(r) If the production 00 = 0, there are only three interesting cases, depending on function is separative, i.e. if FK,r the form of the function B(r). Sub-case A-1: Linear Production Function F (K, r) = A(K) + br If the production function is separable and linear with respect to the extracted natural resources, eq. (44) becomes ˙ 0 =0 ˙ 1 eδt + (w1 δeδt K + φ(t, S))(A0 − n) − w1 δ 2 eδt K − φ0 − Sφ bSδw K t S

(47)

This equation holds true if the following system is satisfied (w1 δeδt K + φ(t, S))(A0K − n) − w1 δ 2 eδt K − φ0t = 0 δt

bδw1 e −

φ0S

=0

⇒

(48)

δt

φ(t, S) = bδw1 e S + ψ(t)

(49)

Eq. (48) becomes (w1 δeδt K + bδw1 eδt S + ψ(t))(A0K − n) − (w1 δ 2 eδt K + bδ 2 w1 eδt S + ψt0 ) = 0. (50) This expression is in fact (A0K − n)µ(t, K, S) = ∂µ ∂t . Its means that µ(t, K, S) = 0 −n)t (A K h(K, S)e . Given the previously form founded for µ, we can deduce that µ(t, K, S) = eδt (w1 δK + bw1 δS + w2 ), under the necessary condition that δ = A0K − n. It means that the discount rate must be equal to the marginal productivity of capital. This necessary condition is possible (with a constant discount rate as we assume it) only ifA0K does not depend on K which means that the production function is linear with respect to both inputs (F (k, r) = aK + br). For this sub-case A-1 (linear separable production function), we have the following group of transformations τ (t, K, S) = w1 eδt and µ(t, K, S) = eδt (w1 δK + bw1 δS + w2 ), where wi are constants. The  Lagrangian of our problem  under the case A-1 hypothesis is −δt ˙ ˙ ˙ ˙ L(t, K, S, K, S) = e U aK − bS − K − nK; S . According to the Noether theorem, we can define the conservation law of the dynamic optimal path as follows:   ˙ −δt Uc0 + Sbe ˙ −δt Uc0 − w1 eδt δ(K + bS + w3 )e−δt Uc0 (51) Ω = w1 eδt e−δt U + Ke

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which is equivalent to

   Ω = w1 U + Uc0 K˙ + bS˙ − δ(K + bS + w3 )

(52)

Sub-case A-2: Log Production Function F (K, r) = A(K) + b.ln(r) In that case, eq. (44) becomes ˙ 0 =0 −bδw1 eδt + (w1 δeδt K + φ(t, S))(A0K − n) − w1 δ 2 eδt K − φ0t − Sφ S

(53)

This equation holds true if the following system is satisfied φ0S = 0 δt

⇒

φ(t, S) = φ(t) δt

−bδw1 e + (w1 δe K +

φ(t))(A0K

⇒

µ(t, K, S) = µ(t, K) 2 δt

− n) − w1 δ e K −

φ0t

=0

(54) (55)

∂µ This last expression is (A0K − n)µ(t, K) − ( ∂µ ∂t + b ∂K ) = 0. We can here again distinguish two sub-cases, depending on the form of A(K). Either A(K) = aK or A(K) is different.

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• From eq. (55), a production function of the form F (K, r) = aK + b.ln(r) leads to the following equation −bδw1 eδt + (w1 δeδt K + φ(t))(a − n) − w1 δ 2 eδt K − φ0t = 0

(56)

For this relationship to be satisfied, the following conditions must old w1 δeδt (a − n)K − w1 δ 2 eδt K = 0

⇒

δ =a−n

(57)

1 φ(t) − φ0t = bw1 eδt δ

(58)

and δt

−bδw1 e + φ(t)(a − n) −

φ0t

=0

⇔

The solution of this differential equation is φ(t) = −w1 bδteδt . We thus have µ(t, K) = w1 δeδt (K − bt) • If the production function is of any other form, the necessary conditions for eq. (55) imply that both δ = 0 and φ(t) = 0. It would mean that both µ(t, K, S) = 0 and τ = w1 , which means that there are no transformations such that the fundamental invariance identity is satisfied, and thus no conservation laws along the optimal path. For this sub-case A-2 (separable production function with linear capital and “log” resource use), we have the following group of transformations τ (t, K, S) = w1 eδt and µ(t, K, S) = w1 δeδt (K − bt),  where w1 is a constant. In that  case, the Lagrangian reads ˙ S) ˙ = e−δt U aK + b.ln(−S) ˙ − K˙ − nK; S . According to the Noether L(t, K, S, K, theorem, we can define the conservation law of the dynamic optimal path as follows:   b −δt 0 δt −δt −δt 0 ˙ ˙ (59) e U + Ke Uc − S e Uc − eδt w1 δ(K − bt)e−δt Uc0 Ω = w1 e S˙ which is equivalent to

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   Ω = w1 U + Uc0 K˙ − b − δ(K − bt)

(60)

Sub-case A-3: Any Other Production Function F (K, r) = A(K) + B(r) We now consider a production function with a resource production B(r) of any other form. Eq. (44) becomes ˙ 1 eδt + (w1 δeδt K + φ(t, S))(A0 − n) − w1 δ 2 eδt K − φ0 − Sφ ˙ 0 =0 Br0 Sδw t S K

(61)

which is satisfied if the following conditions hold φ0S = 0 ⇒ φ(t, S) = φ(t) δt

(w1 δe K +

Br0 δw1 eδt = 0 φ(t, S))(A0K − n)

⇒ δ=0 −

2 δt

or

w1 δ e K −

(62)

w1 = 0

(63)

φ0t

(64)

=0

Given δ = 0 or w1 = 0, the last equation leads to φ(t)(A0K − n) − φ0t = 0, which means 0 that φ(t) = w2 e(AK −n)t , with the necessary condition that A0K is a constant, and thus that the production function is linear with respect to the capital (F (K, r) = aK + B(r)).

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We thus get the expression of µ(t) = w2 e(a−n)t . For this sub-case A-3 (separable production function, linear with respect to the capital), we have the transformations µ(t, K, S) = w2 e(a−n)t and τ (t, K, S) = w1 (time translation) if the discount rate is nil, or τ = 0 if the discount rate is positive, where wi are constants. The condition on the discount rate allows us to say that the problem is invariant along time translation if there is no discount rate, i.e. if all generations get the same weight in the objective function. Otherwise, there are no time transformation with invariant properties. ˙ S) ˙ In  this case, the  Lagrangian reads L(t, K, S, K, = −δt e U aK + B(r) − K˙ − nK; S According to the Noether theorem, we can define the conservation law of the dynamic optimal path as follows: If δ 6= 0(τ = 0), Ω = −w2 e(a−n−δ)t Uc0    If δ = 0, Ω = w1 U + Uc0 K˙ + Br0 S˙ − w2 e(a−n)t

(65) (66)

Note that in any case, if the production function is additively separable, a necessary condition for the existence of conservation laws is that the production function is linear with respect to the capital stock. B - Non-separative Production Function Case We now turn toward non separative production function. It appears that here again, we can distinguish three cases, depending on the way the resource is used in production. Sub-case B-1: F (K, r) = A(K)r In that first case (which is a particular case of the following case B-2, but that has to be treated on its own from a mathematical point of view), there are conservation laws if eq. (44) is satisfied, which reads for F (K, r) = A(K)r ˙ 0K − n) − w1 δ 2 eδt K − φ0t − Sφ ˙ 0S = 0 (67) ˙ 1 eδt + (w1 δeδt K + φ(t, S))(−SA A(K)Sδw We thus get the following two-equations system A(K)δw1 eδt − A0K (w1 δeδt K + φ(t, S)) − φ0S = 0 δt

2 δt

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−n(w1 δe K + φ(t, S)) − w1 δ e K −

φ0t

=0

(68) (69)

The second equation can be expressed as −nµ(t, K, S) − ∂µ ∂t = 0, which means that µ(t, K, S) = ψ(K, S)e−nt . On the other hand, we know that µ(t, K, S) = w1 δeδt K + φ(t, S). It means that δ = −n and that µ(t, K, S) = (w1 δK + ρ(S))eδt . We thus have to solve eq. (68), which can be denoted A(K)δw1 − A0K w1 δK − A0K ρ(S) − ρ0S = 0

(70)

As ρ only depends on S and not on K, the necessary conditions for this equation to be true are A0K = a (where a is a constant parameter) inducing a linear production function F (k, r) = aKr, and ρ(S) = w2 e−aS . We thus get the expression of µ(t, K, S) = (w1 δK + w2 e−aS )eδt . For this sub-case B-1, we have the following group of transformations τ (t, K, S) = w1 eδt and µ(t, K, S) = (w1 δK + w2 e−aS )eδt , where wi are constants.

Invariance in Economic Dynamics and the Sustainable Development Issue 115   ˙ S) ˙ = e−δt U aKr − K˙ − nK; S . AcIn this case, the Lagrangian is L(t, K, S, K, cording to the Noether theorem, we can define the conservation law of the dynamic optimal path as follows:  
˙ −δt U 0 + Se ˙ −δt aKU 0 − w1 δK + w2 e−aS eδt e−δt U 0 (71) Ω = w1 eδt U e−δt + Ke c c c which is equivalent to    w2 −aS 0 ˙ ˙ Ω = w1 U + Uc K + SaK − δK + e w1

(72)

Sub-case B-2: F (K, r) = A(K)r β A more general production function form is F (K, r) = A(K)r β , with β 6= 1. Note that a Cobb-Douglas production function would be of this form, withA(K) = K α . In that case, eq. (44) becomes ˙ β δw1 eδt +(w1 δeδt K +φ(t, S))(A0 (−S) ˙ β −n)−w1 δ 2 eδt K −φ0 − Sφ ˙ 0 =0 −βA(K)(−S) K t S (73) It leads to the following conditions φ0S = 0

⇒

δt

φ(t, S) = φ(t) ⇒ µ(t, K, S) = µ(t, K)

δt

−βA(K)δw1 e + (w1 δe δt

−n(w1 δe K + φ(t, S))

K + φ(t, S))A0K − w1 δ 2 eδt K − φ0t

(74)

=

0

(75)

=

0

(76)

The last equation is equivalent to −nµ(t, K) − ∂µ ∂t = 0, which means that µ(t, K) = ρ(K)e−nt . As we also know that µ(t, K) = w1 δeδt K + φ(t), we must have δ = −n and we deduce that µ(t, K) = (w1 δK + w2 )eδt . Form eq. (75), we get conditional forms for the production function. There are conservation laws only if the production function is solution of the differential equation

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−βA(K)δw1 + (w1 δK + w2 )A0K = 0

(77)

The “capital part” of the production function must be of the formA(K) = (w1 δK + w2 )β , which results in F (K, r) = (w1 δK + w2 )β r β . For this sub-case B-2, we have the following group of transformations τ (t, K, S) = w1 eδt and µ(t, K, S) = (w1 δK + w2 )eδt , where wi are constants. In this sub-case, the Lagrangian reads   ˙ β − K˙ − nK; S ˙ S) ˙ = e−δt U (w1 K + w2 )β (−S) L(t, K, S, K, According to the Noether theorem, we can define the conservation law of the dynamic optimal path as follows:   ˙ −δt U 0 + Se ˙ −δt U 0 (w1 δK + w2 )β r β−1 −(w1 δK+w2 )eδt e−δt U 0 Ω = w1 eδt e−δt U + Ke c

c

c

(78) which is equivalent to    w2 0 β β−1 ˙ ˙ − (δK + ) Ω = w1 U + Uc K + S(w1 δK + w2 ) r w1

(79)

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Sub-case B-3: any other production function For any other production function F (K, r), eq. (44) is satisfied if µ = 0 and δw1 = 0 (either the discount rate is nil and the system admits invariants under time translations, or there are no invariance properties). There are thus no invariant quantities along the optimal path for such economic systems.

4. Discussion The results can be summarized in a table (for simple transformations set, with particular values for the constant parameters wi ) case:

Production function

A1

aK + br

A2

aK + b.ln(r)

A3

aK + B(r)

B1

aKr

B2

r β (δK)β

B3

F (K, r)

Transformations

Conditions

τ = eδt , ξ = 0 µ = eδt δ(K + bS) τ = eδt , ξ = 0 µ = eδt δ(K − bt) τ = 0, ξ = 0, µ = e(a−n)t τ = 1, ξ = 0, µ = e(a−n)t τ = eδt , ξ = 0 µ = eδt (δK + e−aS ) τ = eδt , ξ = 0 µ = δKeδt τ = 0, ξ = 0, µ = 0

δ =a−n

Ω=U+

Conservation law   ˙ + bS˙ − δ(K + bS) K

Uc0

  ˙ − b − δ(K − bt) Ω = U + Uc0 K

δ =a−n

δ = −n

(a−n−δ)t 0 Ω = −e   Uc ˙ + B 0 S˙ − e(a−n)t Ω = U + Uc0 K r   ˙ + SaK ˙ − δK + e−aS Ω = U + U0 K

δ = −n

  β β−1 ˙ + SδK ˙ Ω = U + Uc0 K r − δK

/

/

/ δ =0

c

From a general point of view, the problem we consider is of the form Z ∞ ˙ L(X(t), X(t), t)dt max

(80)

0

where X(t) are the capital stocks of the economy. One can write the Euler-Lagrange equations −L +

n X

∂L X˙ k =H ∂ X˙ k k=1

(81)

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H is the Hamiltonian of the problem. If we consider a set of transformations t¯ = t + ¯ k = X k + ξ(t, X)ε + o(ε) that satisfy the Fundamental Invariance τ (t, X)ε + o(ε) and X Identity, the Noether theorem tells us that the quantity n X ∂L k ξ Ω = −Hτ + ∂ X˙ k k=1

(82)

is constant along an optimal path. The conservation laws can thus be interpreted as       shadow n inf initesimal inf initesimal X  price of     + Ω = (Hamiltonian)  time  investment  transf ormation k k=1 of X transf ormation in X k (83) The conserved quantity is the Hamiltonian plus the infinitesimal effects of the state variable transformations. Any of the conservation laws Ω of our model can be written in

Invariance in Economic Dynamics and the Sustainable Development Issue

117

that form. In all the cases, using the definition of µ and the form of the production function (and especially taking the marginal productivity of the resource Fr0 ), the conservation law can always be written as   Ω = U + Uc0 K˙ + Fr0 S˙ − µe−δt (84) The conservation law can thus been interpreted as the sum of utility and the stock depletion valuated at consumption price. This result is the same as obtained by Martinet and Rotillon (2007). By writing down   ˙ (85) Ω = U + Uc0 K˙ − µe−δt + Uc0 (Fr0 S) and interpreting K˙ − µe−δt as investment in the “new” capital stock resulting from the variable state K transformation, the conservation law is exactly the Hamiltonian of the transformed problem (with a new definition of the variable states). Our result generalizes the results presented in Martinet and Rotillon (2007), as any of their conservation laws can be written in the previous form, making the “discounted µ” explicit (see proposition 3, 4 and 5 of Martinet and Rotillon (2007)). From a general point of view, conservation laws exist in neo-classical optimal growth programs only under very restrictive conditions: for restrictive classes of production function and for special values of discount rates (see the tabular of results).

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5. Conclusion In this chapter, we examined whether conservation laws exist in neo-classical optimal growth models. We aimed at defining such conserved quantities in order to interpret them as sustainability indicators, trying to embody the Solow’s statement that the sustainability requires to conserve something through time. Our results show that such conservation laws exist only under very restrictive condition. We thus argue that the neo-classical framework is not able to deal with the sustainability issue as it does not conserve anything for future generations, even in quite simple models. When they exist, the conservations are the sum of the utility from the consumption and resource stock level plus a term that can be interpreted as the depreciation of the stocks valuated at a price corresponding to the marginal utility of consumption. This quantity can be interpreted as the true income of the economy (consumption + investment). From the intergenerational equity point of view, one can say nothing about the actual stream of utility. There is no reason that utility stays constant (or not decreasing) through time, as often required to represent intergenerational equity requirement. Note that if the investment in capital were such that it equals the depletion of natural resources stocks, the “genuine saving” of the economy would be nil, and the conserved quantity would be the utility level. It means that if the Hartwick’s investment rule resulted from the optimal decisions, the utility level would be constant. However, such an investment rule is not a characteristic of the optimal path. All these results are consistent with the neo-classical growth theory with environment, and again raises the issue of the definition of a sustainability criterion.

118

Vincent Martinet and Gilles Rotillon

References Aronsson, T., Johansson P. O., & Lofgren, K.-G. (1997), Welfare Measurement, Sustainability and Green Accounting, Edward Elgar. Asheim, G.B., Weitzman, M. L. (2001), Does NNP growth indicate welfare improvement?, Economics Letters, 73, 233-239. Asheim, G., Buchholz, W., & Tungodden, D. (2001). Justifying sustainability. Journal of Environmental Economics and Management, 41, 252-268. Askenazy, P. (2003). ‘Symmetry and optimal control in economics’. Journal of Mathematical Analysis and applications, 282, 603-613. Cairns, R. (2007). ‘Value and Income’. doi:10.1016/j.ecolecon.2007.10.004.

Ecological

Economics,

in

press,

Cairns, R., & Van Long, N. (2006). Maximin: a direct approach to sustainability. Environmental and Development Economics, 11, 275-300. Chichilnisky, G. (1996). An axiomatic approach to sustainable development. Social Choice and Welfare, 13, 219-248. Chichilnisky, G., Heal, G., & Beltratti, A. (1995). The Green Golden Rule. Economics Letters, 49, 175-179. Dasgupta, P., & Heal, G. (1974). The Optimal Depletion of Exhaustible Resources. Review of Economic Studies. Symposium on the Economics of Exhaustible Resources, 41, 1-28. Dasgupta, P., & Heal, G. (1979). Economic Theory and Exhaustible Resources. Cambridge University Press.

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Dasgupta, P.S., Maler, K.-G. (2000) Net national product, wealth and social well-being, Environment and Development Economics, 5, 69-93. Dixit, A., Hammond, P., Hoel, M. (1980) On Hartwick’s rule for regular maximin path of capital accumulation and resource depletion, Review of Economic Studies, 47, 551-556. Dobson, A. (1996). Environment Sustainabilities: an Analysis and a Typology. Environmental Politics, 5(3), 401-428. Hartwick, J. (1977). Intergenerational Equity and the Investing of Rents from Exhaustible Resources. American Economic Review, 67, 972-974. Heal, G. (1998). Valuing the Future: Economic theory and sustainability. Columbia University Press, New York. Heal, G., Kristrom, B. (2005) National Income and the Environment, in Maler K.G. and Vincent J.(eds.), Handbook of Environmental Economics, vol. 3, Elsevier.

Invariance in Economic Dynamics and the Sustainable Development Issue

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Holland, A. (1994). Natural Capital. In Attfield, R. & Belsey, A. (eds.). Philosophy and the Natural Environment. Cambridge University Press, Cambridge. Hotelling, H. (1931). The Economics of Exhaustible Resources. Journal of Political Economy, 39, 137-175. Krautkraemer, J. (1985). Optimal Growth Resource Amenities and the Preservation of Natural Environments. Review of Economic Studies, 52, 153-170. Krautkraemer, J. (1998). Nonrenewable resource scarcity. Journal of Economic Literature, XXXVI, 2065-2107. Keynes, J.M. (1940), How to Pay for the War: A Radical Plan for the Chancellor of the Exchequer, Macmillan. Long, N.V. (2006). A mixed Bentham-Rawls criterion for intergenerational equity. Mimeo, McGill University. Martinet, V., Rotillon, G. (2007). ‘Invariance in growth theory and sustainable development’. Journal of Economic Dynamics and Control, 31, 2827-2846. Noether, E., 1918. English translation: 1971, Invariant Variation Problems.Transport Theory and Statistical Physics 1, 186-207. Ramsey, F. (1928). A Mathematical Theory of Saving, Economic Journal 38, 543-559. Rawls, J. (1971). A theory of Justice. England: Oxford University Press. Samuelson P.A. (1970). Law of conservation of the capital-output ratio, Proceedings of the National Academy of Sciences, Applied Mathematical Science, 67, 1477-1479. Sato, R., 1999. Theory of Technical Change and Economic Invariance. Edward Elgar. Sato, R., Kim, Y., 2002. Hartwick’s rule and economic conservation laws. Journal of Economic Dynamics and Control 26, 437-449.

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Solow, R. (1974). Intergenerational Equity and Exhaustible Resources.Review of Economic Studies: Symposium on the Economics of Exhaustible Resources, 41, 29-45. Solow, R. (1993). An Almost Practical Step Towards Sustainability. Resources Policy, 19, 162-172. Stavins, R., Wagner, A., & Wagner, G. (2003). Interpreting sustainability in economic terms: dynamic efficiency plus intergenerational equity.Economics Letters, 79, 339-343. Stiglitz, J. (1974). Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths. Review of Economic Studies: Symposium on the Economics of Exhaustible Resources, 41, 123-137. Usher, D. (1994), Income and the Hamiltonian, Review of Income and Wealth, Series 40 (2), 123-141.

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WCDE (1987). Our common future (Brundtland report). World Commission on Environment and Development, Oxford University Press. Weitzman M. L. (1976), On the Welfare Significance of National Product in a Dynamic Economy, Quarterly Journal of economics 90, 156-162.

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Xepapadeas, A. (2005), Economic Growth and the Environment, in Maler K.G. and Vincent J.(eds.), Handbook of Environmental Economics, vol. 3, Elsevier.

In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 121-134

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 6

E NDOGENOUS C APITAL U TILIZATION AND D EPRECIATION UNDER C APITAL M AINTENANCE IN THE AK G ROWTH M ODEL ∗ J. Aznar-M´arquez1† and J.R. Ruiz-Tamarit2‡ 1 Universitat Miguel Hern´andez d’Elx (Spain). Av. de la Universidad s/n, Edifici Galia, E-03202 Elx, Spain 2 Department of Economic Analysis, Universitat de Val`encia (Spain), and Department of Economics, Universit´e Catholique de Louvain (Belgium). Facultat d’Economia, Av. dels Tarongers s/n, E-46022 Val`encia, Spain

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Abstract In the traditional exogenous growth model, the decision about how much to save is based on the comparison between costs and benefits of a higher consumption today rather than tomorrow. This endogenously decided savings are automatically channelled to investment, which plays a passive role. The basic model assumes full utilization of the installed capital, and capital depreciation determined exogenously as a constant fraction of the capital stock. These assumptions, however, do not conform to available data. Observed facts show that firms do not always decide to use all the installed capital, and also that they choose the value of the depreciation rate. Actually, the depreciation rate depends on the internal resources devoted to repair and maintenance of the capital stock, which deteriorates either through the use in the production process or simply through the natural process of ageing. Maintenance reduces the depreciation of capital, investment is subject to adjustment costs, and the degree of capital utilization affects itself the maintenance activity. Despite of the empirical evidence, the depreciation rate has been usually regarded as an exogenous and constant parameter. According to the neoclassical growth theory, this one affects negatively the long-run level of the variables and the short-run rates ∗

We thank R. Boucekkine and O. Licandro for their helpful comments. Authors acknowledge the support of the Belgian research programme ARC 03/08-302 and the financial support from the Spanish CICYT, Projects SEC99-0820, SEC2000-0260 and SEJ2004-04579/ECON. Ruiz-Tamarit also acknowledges the grants PR2003-0107 and PR2006-0294 from the Secretar´ıa de Estado de Educaci´on y Universidades, Spanish MECD. † E-mail address: [email protected] ‡ E-mail address: [email protected]. Phone: +34 963828250; Fax: +34 963828249.

122

J. Aznar-M´arquez and J.R. Ruiz-Tamarit of growth. In the new theory of endogenous growth, the constant depreciation rate also affects negatively the long-run rate of growth. But we are interested in the depreciation rate as an endogenous variable, and this leads us to explore simultaneously the determinants of both depreciation and growth. Consequently, in this chapter we study an extended version of the one-sector AK growth model introducing adjustment and maintenance costs. Agents are allowed to under-use the installed capital and to vary the depreciation rate. The model is analyzed using particular functional forms and it is solved in closed-form. We find that adjustment and maintenance costs (efficiency) reduce (increases) investment, depreciation, capital utilization as well as the rate of growth. Impatience reduces the rate of growth but increases depreciation and utilization. Depreciation and utilization depend negatively on the rate of population growth.

Keywords: Maintenance, Depreciation, Capital Utilization, Growth. JEL classification: O40, E22, D90.

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1. Introduction In economic growth theory, two main assumptions associated with the capital stock frequently appear. The former is the hypothesis that physical capital depreciates at a constant exogenous rate. The latter is the hypothesis that physical capital is fully used in the production process. In this study, we overcome the two previous assumptions. In the standard neoclassic growth model, the decision about how much to save is based on the comparison, in welfare terms, between the costs and the benefits of a higher consumption today rather than tomorrow. Once the amount of savings has been decided, they are automatically channelled into investment. Therefore, investment has an entirely passive role. In such a model it is assumed full utilization of the installed capital as well as that depreciation experienced by capital equipment is an exogenously determined constant fraction of capital stock. These assumptions, however, do not conform to observed facts because available data show quite a different reality. Firms do not always decide to use all the installed capital and they are able to change the depreciation rate of capital stock. The way the firms may act on the depreciation rate is by devoting resources to the preservation, that is repair and maintenance, of capital stock which has deteriorated either through the use in the production process or simply through the natural process of ageing. McGrattan and Schmitz (1999) highlights the quantitative relevance of repair and maintenance activities showing that in Canada, for the period 1961-1993, up to 6% of gross national product was devoted to capital repair and maintenance, which is approximately half the expenditure made on the acquisition of new capital goods. In addition, Gylfason and Zoega (2001), using data currently published by the World Bank, studies the relationship between depreciation and growth. Among its results, we would like to emphasize the following ones: i) increased population growth accelerates depreciation, ii) increased efficiency increases depreciation, and iii) increased long-run growth also accelerates depreciation. Despite the above empirical evidence, in growth theory the depreciation rate has been regarded as an exogenous parameter which, in the case of neoclassical models, negatively affects the long-run level of variables and the short-run rates of growth. Moreover, in the

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Endogenous Capital Utilization and Depreciation under Capital Maintenance ... 123 case of endogenous growth models it also affects negatively the long-run rate of growth. In this chapter, we explore the determinants of both depreciation and growth, in the context of the one sector model of growth when a linear production technology is combined with adjustment costs and a technology for capital maintenance. Agents are allowed to underuse the installed capital as well as to vary the depreciation rate. Therefore, in this economy agents decide endogenously the amount of resources devoted to the accumulation of new capital and also the amount devoted to repair and maintenance activities. This latter decision is connected with the matter of the endogeneity of both the capital utilization rate and the depreciation rate. The issue of capital stock maintenance, which allows to break down the strong hypothesis of a constant exogenous depreciation rate even in the absence of obsolescence, has been left to one side for many years after the seminal contributions of the seventies. Nevertheless, in this period various attempts to reintroduce the variability of the depreciation rate have been implemented by means of the hypothesis Depreciation-in-Use. Namely, the line of causality which connects biunivocally high (low) rates of capital utilization, usually associated with high (low) levels of economic activity, with higher (lower) depreciation rates. This hypothesis has been incorporated to microeconomic studies at the firm level [Epstein and Denny (1980), Bischoff and Kokkelenberg (1987), Motahar (1992), Johnson (1994)] as well as to macroeconomic studies related to both the neoclassical growth theory [Rumbos and Auernheimer (2001)] and the real business cycle theory [Burnside and Eichenbaum (1996)]. Although the depreciation rate is transformed into an endogenous variable, this approach is not satisfactory because of the residual role assigned to capital depreciation. More recently, the above hypothesis has been enlarged to include the maintenance activity, which allows the depreciation rate to be a decision variable analogous to the capital utilization rate. There are examples of this at the firm level [Boucekkine and Ruiz-Tamarit (2003)] as well as at an aggregate level in the neoclassical growth theory [Licandro, Puch and Ruiz-Tamarit (2001)] and the real business cycle theory [Licandro and Puch (2000), Collard and Kollintzas (2000)]. As we argued, the basic general equilibrium growth models do not allow for the separation of household saving decisions from investment decisions of firms. However, by introducing adjustment costs connected with gross investment expenditures it is possible to overcome the essentially passive role of investment.1 In this chapter, we adopt the canonical model of Rebelo (1991) and we introduce both an adjustment and a maintenance cost function, which modify the objective functional in a substantial way. In one-sector growth models, the linear technology constitutes a useful referent for easily modeling, directly or asymptotically, the endogenous growth phenomenon. Here, we will check whether the introduction of the above cost functions into the model breaks down the previous linkage. In this framework, depreciation is no longer a residual variable. Together with investment and the rate of capital utilization, it becomes one of the instruments used by economic agents in setting their optimal plans. Briefly, our query is to assess whether incorporating the maintenance and repair expenditures into the aggregate model of economic activity, will substantially change what is known about the short-run convergence hypothesis as well as about the determinants of the long-run rate of growth and other endogenous variables. Our 1

The active role of investment has been studied by Abel and Blanchard (1983) in a neoclassical Ramsey-like model, and also by Barro and Sala-i-Mart´ın (1992) in an endogenous growth model of the AK type.

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J. Aznar-M´arquez and J.R. Ruiz-Tamarit

technological assumptions allow to expand the basic model in such a way that well-defined investment, depreciation, and utilization functions may be derived. To our knowledge, there are no theoretical contributions focusing on the study of these topics taken together. Consequently, the chapter is devoted to this goal. The remainder of the chapter is organized as follows. Section 2 describes the economy and introduces the assumptions featuring the different parts of the general equilibrium model. In section 3, we solve the intertemporal optimization problem. In section 4 we study the resulting dynamic system and its closed-form solution. In section 5 we get and interpret economic results, connecting with the empirical literature which parallels the present work. Finally, section 6 concludes.

2. The Economy Let us consider an economy populated with many identical infinitely-lived individuals,Nt . Population is assumed to grow at a constant and exogenously given raten ≥ 0. We normalize the initial population to unity and then we get Nt = ent . Moreover, it is assumed that people facing to an infinite planning horizon will discount the future at a positive constant rate ρ > n. Individual preferences are represented by the instantaneous utility function

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U (ct ) =

c1−Φ −1 t , 1−Φ

(1)

where ct is consumption per capita and Φ−1 > 0 is the constant elasticity of intertemporal substitution. Moreover, there are many identical firms producing a single good. We assume that each firm uses a linear technology of the AK type, being the capital stock Kt > 0 the only relevant factor.2 We interpret capital in a broad sense, so that it includes physical capital as well as human capital, which usually comes embodied in workers. In this sense, human capital is considered a rival and excludable factor as physical capital is. Labour, measured as the number of workers and independently of the index of human capital, is considered a perfect substitute for physical capital and it is not necessary for production. Therefore, total current output Yt is a function of the effectively used capital Kt ut , where ut ∈ [0, 1] is the variable proportion of installed capital that firm decides to use, and of the efficiency parameter A which represents a constant technological level. This latter parameter may also be read as the marginal productivity as well as the average productivity of the effectively used capital. Thus, given the constant returns inherent to a linear production function, we write in per capita terms (2) yt = Akt ut . The produced single good may be allocated to consumption, to the accumulation of new capital or to preserving the inherited capital. While current consumption contributes directly to increase welfare, the other uses of output are connected with the increase of the capital stock, which allows for a greater consumption in the future. In this context, 2

Similar results could be derived under a more general production function with constant returns to scale if we introduce, following Romer (1986), the learning-by-investing device together with the knowledge spillovers assumption.

Endogenous Capital Utilization and Depreciation under Capital Maintenance ... 125 accumulation of new capital has not only to do with investment purchases but also with adjustment or installation activities. Moreover, the preservation of old capital depends on maintenance and repair activities. Consequently, we have to introduce in our framework the two corresponding cost functions. First, let us assume that adjustment costs, which are internal to the firm, are represented by a linearly homogeneous function Ψ(It , Kt ), increasing in gross investment, It > 0, and decreasing in the total installed capital stock. Then Ψ(It , Kt ) = Ψ(It /Kt , 1)Kt = φ(it )Kt , where bi2 φ(it ) = t , (3) 2 being it the rate of gross investment over capital and b a positive constant. Second, in order to preserve the inherited capital stock, we assume that period by period it is possible to reduce the depreciation associated with deterioration, which arises from equipment ageing and use3 , by means of the corresponding maintenance and repair activities. These activities entail specific maintenance costs which are internal to the firm and, by assumption, will be represented by a linearly homogeneous functionM (Dt , Kt ut ), decreasing in total depreciation, Dt > 0, and increasing in effectively used capital. Redefining variables we get M (Dt , Kt ut ) = M (Dt /Kt , Kt ut /Kt )Kt = m(δ t , ut )Kt , where 1+ε , m (δ t , ut ) = dδ −ε t ut

(4)

being δ t > 0 the endogenous rate of depreciation over capital stock. In this functionε > 0 roughly represents the elasticity of the average maintenance cost with respect to the depreciation and utilization rates and d is a positive constant. The aggregate resource constraint is Yt = Ct + It + Ψ (It , Kt ) + M (Dt , Kt ut ) where •

It =Kt +δ t Kt . In per capita terms the resource constraint is determined by these two equations   bi2t −ε 1+ε ct + it + + dδ t ut kt = Akt ut , (5) 2 •

kt = (it − δ t − n) kt ,

(6)

•

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where k denotes the time derivative of per capita capital considered in its broad sense.

3. The Optimization Problem In this economy, because of the absence of externalities and other market failures such as imperfections or incompleteness, the competitive equilibrium solution to the intertemporal resources allocation problem is equivalent to the central planner solution. Consequently, every optimal solution may be decentralized as a competitive equilibrium. The planner’s optimization problem is to choose at each moment in time the three controls: the rate of 3 Here we are refering strictly to physical wear and tear but, contrary to the standard procedure, we take this depreciation as an economic phenomenon because firms are assumed to optimally decide how much resources are allocated to maintenance. In this paper we ignore obsolescence as a source of depreciation. Factors usually causing obsolescence are left to one side because of the assumed perfect malleability of capital.

126

J. Aznar-M´arquez and J.R. Ruiz-Tamarit

capital utilization, the rate of investment and the rate of depreciation, which solve the problem Z ∞ 1−Φ ct − 1 −(ρ−n)t max W = e dt s. t. (5), (6) and k0 . (P) 1−Φ {ut ,it ,δ t } 0 The current value Hamiltonian associated with this problem, after dropping time subscripts, may be written as

Hc =

h  Aku − i +

bi2 2

 i1−Φ + dδ −ε u1+ε k −1 + µ (i − δ − n) k,

1−Φ

where µ is a co-state variable. According to the Maximum principle, an interior optimal solution to problem (P) must satisfy the first-order conditions A = (1 + ε) dδ −ε uε ,

(7)

µ = c−Φ (1 + bi) ,

(8)

µ = c−Φ εdδ −1−ε u1+ε ,

(9)

the Euler equation •

−Φ

µ= −c



bi2 − dδ −ε u1+ε Au − i − 2



+ µ (ρ + δ − i) ,

(10)

the constraints (5) and (6), as well as the initial condition k0 > 0 and the transversality condition lim e−(ρ−n)t µt kt = 0. (11)

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t→∞

The multiplier µ defines the shadow price, measured in units of utility, of an additional unit of installed capital. The term (1 + ε) dδ −ε uε ≡ mu (δ, u) is the marginal maintenance cost associated with an increase in the utilization rate. Equation (7) states that this marginal cost must be equal to the marginal productivity of such an increase in the utilization rate, A. The term 1 + bi is the marginal opportunity cost of gross investment. Then, equation (8) states that this marginal cost measured in units of utility must be equal to the shadow price of capital. The term εdδ −1−ε u1+ε ≡ −mδ (δ, u) is the marginal saving in maintenance costs associated with an increase in the depreciation rate. An increase inδ reduces capital stock and, consequently, diminishes maintenance expenditures. Thus, equation (9) states that this marginal saving measured in units of utility must be equal to the shadow price of the lost capital. Moreover, given that the planner has two alternative ways for increasing capital: investment and maintenance, at the optimum the marginal cost of investing has to be equal to the marginal cost of reducing depreciation through maintenance,1 + bi = εdδ −1−ε u1+ε . Now, solving the first order conditions we get the optimal control functions " #  1+ε  ε A 1 ε −1 , i (Θ) = b d 1ε 1 + ε

(12)

Endogenous Capital Utilization and Depreciation under Capital Maintenance ... 127    1+ε 2 ε 1 ε A 1 −1 − 2b 2b d 1ε 1+ε µ Φ k −1 δ (k, µ, Θ) = + , (13)   1+ε   1+ε 1− Φ1 ε ε A 1

1 2b



u (k, µ, Θ) =

ε 1 dε

ε 1 dε

dε

1+ε



A 1+ε

 1+ε 2





ε

ε

A 1+ε

−



1 2b

+ d

1 ε



ε

A 1+ε

A 1+ε

ε 1 dε

1



ε

−1

µ Φ k −1 ,  1+ε 1− Φ1

A 1+ε

(14)

ε

where Θ represents a vector of structural parameters. As we can see, investment only depends on structural parameters and, hence, it takes a constant value as in equation (12). Equations (13) and (14) show that δ and u depend negatively on the state and co-state variables. Moreover, these two variables are linearly and positively related to each other. The accompanying expressions c(k, µ, Θ) and y(k, µ, Θ), with ck = 0, cµ < 0, yk > 0 and yµ < 0, may be derived substituting the previous ones in the resources constraint (5) and the production function (2), respectively. These results show some interesting features of the model. First, given that capital stock and its shadow price evolve in opposite directions, it is difficult at first sight to conclude about the evolution of variables like the capital utilization rate and the depreciation rate. Second, the constancy of the investment rate implies that the gross investment share will move in parallel to the capital-output ratio. Third, per capita consumption evolves inversely to the movement in the shadow price of capital stock, as it happens with production per capita which, in addition, moves directly with capital stock. Finally, consider equation (10) and, after some substitutions and rearrangements, solve it forward subject to the transversality condition (11) which avoids explosive solutions. Hence, we find that µ is determined as the present discounted value of the total marginal product of capital measured in units of utility, µt =

Z

∞

c−Φ s

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t

  R s bi2s −ε 1+ε − dδ s us Aus + e− t (ρ+δ z )dz ds. 2

(15)

In this expression, the discount term takes into account the fact that the depreciation rate is variable.

4. The Dynamic System and the Closed-Form Solution When we substitute the control functions in (6) and (10), the dynamic system which describes the evolution of state and co-state variables becomes 

  = k   •

ε 1 dε



A 1+ε

2bε 1 dε



 1+ε

A 1+ε

ε

2 −1

 1+ε ε



"  1+ε # Φ1 −1   ε −1 ε A  − n k − µΦ, 1  dε 1 + ε

(16)

128

J. Aznar-M´arquez and J.R. Ruiz-Tamarit     2  1+ε   µ= −  •

ε 1 dε

ε

A 1+ε

2bε 1 dε



A 1+ε

−1

  + ρ µ. 

 1+ε ε

(17)

This system is non-linear and it does not admit a linearization because of the lack of a well defined steady state. However, its structure allows for a complete closed-form solution.4 The unique non-explosive particular solution trajectories for the variables of the system are      2  1+ε   ε   ε A  −1     1  1ε 1+ε  d  , (18) t k(t, Θ) = k0 exp − ρ     1+ε    ε  Φ  2bε A     1 1+ε ε d

   2    1+ε   ε  ε A   −1   1   1+ε dε   µ (t, Θ) = µ (0) exp ρ − t ,    1+ε     ε   2bε A     1

(19)

1+ε

dε

with k0 known and  µ (0) =  



(1− Φ1 )

ε 1 dε

ε 1 dε

2bε 1 dε



A 1+ε

A ( 1+ε )

A ( 1+ε )

 1+ε 1−Φ ε

1+ε ε

−1

1 Φ k Φ .

2

+

1+ε ε

(20)

0

ρ Φ

− n

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−1

Then, using (18) and (19) we compute the term µ Φ k −1 which is needed in order to −1 determine the complete particular trajectories for control variables. We find µ Φ k −1 = −1 µ(0) Φ k0−1 , and substituting in equations (12)-(14) we get the explicit closed-form trajectories for i, δ and u, " #   1+ε ε 1 ε A i(Θ) = −1 , (21) b d 1ε 1 + ε  δ(Θ) =

ε 1 dε



A 1+ε

 1+ε 2 ε

−1+ 1− 2bε 1 dε

 u(Θ) =

ε 1 dε



A 1+ε

 1+ε 2 ε

1 Φ



A 1+ε

1 Φ



ε 1 dε



A 1+ε

 1+ε ε

2 −1 +

 1+ε




−1+ 1−   A 2bε 1+ε


ε



ε 1 dε



A 1+ε

 1+ε ε

2 −1

 +

ρ − n, Φ

A 1+ε

1

ρ Φ

ε 1

(22)

−n


.

dε (23)

4

In the Appendix the reader will find the details of this solution.

Endogenous Capital Utilization and Depreciation under Capital Maintenance ... 129 The investment rate, the depreciation rate and the capital utilization rate are constant along the particular solution trajectory. The result concerning the investment rate was known from (12) because of the independence of i with respect to the state and co-state variables. However, in the case of the depreciation and utilization rates, the aforesaid result is due to the compensating effects exerted by capital stock and its shadow price on each of these variables along the optimal solution trajectory. Given that the rate of gross investment over capital is always non-negative, parameter constraints derived in the Appendix from the transversality condition imply non-negative depreciation and utilization rates. The additional constraint on utilization to be lower (or equal) than unity imposes an upper-bound
1 1 ε to the depreciation rate, which must be found in the intervalδ ∈ [0, d ε 1+ε ]. A Moreover, from (2), (5) and the previous trajectories for controls we derive the particular solution trajectories for output and consumption per capita

y(t, Θ) = Au(Θ)k0 exp

c(t, Θ) = Γ(Θ)k0 exp

    



1    Φ   

    

A 1+ε

2bε 1 dε



1    Φ   



ε 1 dε



ε 1 dε



A 1+ε

2bε 1 dε



2 −1

 1+ε ε

 1+ε ε

A 1+ε

2 −1

 1+ε ε

A 1+ε

 1+ε ε

        − ρ t ,     

(24)

        − ρ t .     

(25)

Here Γ(Θ) = Au(Θ) − i(Θ) − φ(i(Θ)) − m(δ(Θ), u(Θ)) > 0 is time independent and c(t,Θ) represents the ratio k(t,Θ) , which is constant along the optimal solution trajectory,

Γ(Θ) =

"

ε 1

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dε



A 1+ε

 1+ε # ε



 1−   

1 Φ






ε 1 dε

2bε 1 dε





A 1+ε

A 1+ε

 1+ε ε



2 −1 +

 1+ε ε

 ρ  − n . Φ 

(26)

5. Balanced Growth Path and Comparative Statics The previous results characterize a balanced growth path. Namely, the model does not show transitional dynamics. Given the complete closed-form solution for each of the involved variables it is easy to conclude about growth rates, finding γ i = γ δ = γ u = 0, 

1  γk = γy = γc = γ =  Φ

ε 1 dε



A 1+ε

2bε 1 dε



(27)

 1+ε

A 1+ε

ε

2 −1

 1+ε ε



  − ρ . 

(28)

130

J. Aznar-M´arquez and J.R. Ruiz-Tamarit

  2(1+ε)   1+ε 2 ε ε A A The latter is positive for ε2 1+ε > (1 + bρ) 2ε1 1+ε − 1. The growth dε dε rate γ does not depend on the initial capital stock k0 and is not related to the initial income level or to any other per capita income level. c(t,Θ) The saving rate, defined as s(t, Θ) = 1 − y(t,Θ) , takes the constant value s(Θ) =

Au(Θ) − Γ(Θ) i(Θ) + φ(i(Θ)) + m(δ(Θ), u(Θ)) = . Au(Θ) Au(Θ)

(29)

Household saving just finances the two kind of expenditures related to the capital accumulation process: gross investment expenditures, including adjustment costs, and capital maintenance expenditures. Moreover, given the absence of transitional dynamics, along the optimal (balanced) growth path welfare depends on the initial consumption level and the constant rate of growth. If we substitute for the known values we get W (Θ) =

Z 0



∞

c(t, Θ)1−Φ − 1 −(ρ−n)t e dt 1−Φ 

     1+ε 1−Φ   ε 1−Φ ε A   k0 1 1+ε  1  1  dε  =  2 Φ − ρ − n  .  1−Φ  (1− 1 ) ε ( A ) 1+ε  ε −1 1 1+ε   Φ ρ ε d   + Φ − n 1+ε 2bε A ε 1 ( 1+ε )

(30)

dε

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Although it is not easy to compute the individual impacts, this expression shows the optimal welfare as a function depending only on the structural parameters of the model. Now, we are able to show some new results arising from a comparative statics exercise for variables i, δ, u and γ. Most of them cannot be found in the theoretical literature because of the insufficiencies of the canonical endogenous growth model, in which the depreciation rate is assumed constant and the capital stock is used at full capacity. From (21), (22), (23) and (28) we find that 1. The greater the productivity of effectively used capital,A, the higher the investment rate as well as the depreciation and capital utilization rates. 2. The higher the weight of installation and maintenance costs in gross product, b and d respectively, the lower the investment rate as well as the depreciation and capital utilization rates. 3. Both the capital utilization rate and the depreciation rate are positively related to impatience characterizing economic agents, which is represented by a low intertemporal elasticity of substitution in consumption, Φ−1 , or a high rate of discount, ρ. 4. The bigger the population growth rate the lower the depreciation and capital utilization rates.

Endogenous Capital Utilization and Depreciation under Capital Maintenance ... 131 5. The investment rate does not depend on preference parameters or on the rate of population growth. 6. The higher the weight of installation and maintenance costs in gross product, b and d respectively, the lower the rate of growth. The new results about depreciation and growth that we have derived along the chapter are mostly consistent with empirical facts. However, the results concerning capital utilization are still to be empirically checked.

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6.

Conclusion

In this chapter, we have introduced the possibility that firms decide endogenously the rate of depreciation and the use of the installed capital stock. This is allowed by the incorporation of an adjustment cost function and a maintenance cost function into the canonical onesector model of endogenous growth due to Sergio Rebelo. These two functions modify the basic cost structure of firms because now the capital accumulation activity implies either purchases of new investment goods, adjustment or installation costs, and maintenance and repair expenditures. Actually, maintenance and repair allow for preserving the installed capital and, hence, for changing the depreciation rate of physical capital. Moreover, given that depreciation is mostly due to the use of the capital stock in the production process, we find a strong connection between the rates of depreciation and capital utilization. As in standard AK models, our model does not provide transitional dynamics. Variables like capital, production, and consumption, all in per capita terms, conform a unique balanced growth path from the beginning, given their initial conditions. In turn, optimal rates of depreciation, capital utilization, and investment are constant despite of being decided endogenously. In this framework depreciation is no longer a residual variable. Moreover, we get a constant saving rate and a constant consumption-capital ratio. This last couple of results is a direct consequence of the linearity of the adjustment and maintenance cost functions with respect to the per capita capital stock. Of course, by dropping this assumption, results could have been different. Among the various original results obtained in this chapter, it is worthy to emphasize some parameter dependences corresponding to the involved endogenous variables. We find that the smaller the installation and maintenance costs and the greater the economy’s efficiency level, the higher the rates of investment, utilization, and depreciation, as well as the higher the rate of growth. Moreover, the higher the rate of population growth, the lower the rates of depreciation and capital utilization, even if this parameter does not affect the investment rate or the rate of growth. Finally, the greater the patience level of economic agents, the higher the rate of growth but the lower the depreciation and utilization rates. Furthermore, because of the direct influence of the utilization rate, a lower level of capital but more intensively used may produce more output. Then, it could be reasonable to find an economy with a lower capital stock that produces and consumes more than another economy with a higher capital stock. However, the absence of convergence also implies that any initial difference in the level of per capita income will be always magnified. As a consequence, our model may explain the great disparity between rich and poor countries

132

J. Aznar-M´arquez and J.R. Ruiz-Tamarit

and its persistence over time. Unfortunately, we are not able to explain the demonstrated ability of some countries to change their positions within the per capita income distribution. Namely, the experiences of growth which are known as miracles and disasters and are associated with overtaking processes.

7. Appendix The dynamic system (16)-(17) may be solved using sequentially the two differential equations together with the boundary conditions. However, to be exhaustive in our search of a closed-form solution and study the issues of existence, uniqueness and positivity, we will take as reference the modified Hamiltonian dynamic system studied in Ruiz-Tamarit and Ventura-Marco (2000), which is of the form •

a a k (t) = ∆k k(t) − Ωk k(t) 11 µ(t) 22 ,

(A.1)

µ (t) = ∆µ µ(t) + Ωµ k(t)a11 −1 µ(t)1+a22 ,

(A.2)

k(t0 ) = k0 ,

(A.3)

lim µ(t)k(t) exp {− (ρ − n) (t − t0 )} = 0.

(A.4)

•

t→∞

The elements ∆k > 0, ∆µ > −∆k , Ωk > 0, Ωµ > 0, a11 T 0, a22 < 0, k0 , t0 and ρ are constant parameters, while k, µ and t are the variables. Authors assume that Ωk > Ωµ , 1 − a11 > 0 and 1 + a22 T 0. It is easy to show that the above dynamic system simpli= fies to (16)-(17) under the specific parameter values ∆k   2  1+ε ε ε A = ρ − H(Θ), H(Θ) = −1 H(Θ) − n, ∆µ 1 1+ε / 2bε 1

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dε



A 1+ε

 1+ε ε

,

Ωk

=



ε 1 dε



A 1+ε

 1+ε  Φ1 −1 ε

dε

> 0,

Ωµ = 0,

a11

= 0 and

a22 = −1 Φ < 0. From these values we get ∆k + ∆µ = ρ − n > 0, where the right hand term represents the effective intertemporal rate of discount. Moreover, ∆k > ρ − n provided that H(Θ) − ρ > 0, which also implies ∆µ < 0 and ∆k > 0. In this context we proceed in three steps. First, define the instrumental variableX(t) = 1 k(t)µ(t) Φ . By totally differentiating and substituting from equations (A.1) and (A.2) we get •

X (t) = ax X(t) − bx .

(A.5)

This is an autonomous non-homogeneous linear differential equation with constant co
∆µ 1 efficients ax = ∆k + Φ = 1 − Φ H(Θ) + Φρ − n ≷ 0 and bx = Ωk > 0. Given the initial condition k0 and a certain, for the moment unknown, initial value µ (t0 ) which allow 1 us to determine the initial condition X(t0 ) = k0 µ(t0 ) Φ , any particular solution to (A.5) must be of the form   bx bx + X(t0 ) − (A.6) exp {ax (t − t0 )} . X(t) = ax ax

Endogenous Capital Utilization and Depreciation under Capital Maintenance ... 133 Once parameters and the initial value of the variables are known, the above expression determines the value of X(t) at any moment in time. In a second step we transform the initial non-linear system and get the two separated, non-autonomous but homogeneous, linear differential equations for the primary variables   • Ωk k(t), (A.7) k (t) = ∆k − X(t) •

µ (t) = ∆µ µ(t).

(A.8)

The expressions for the particular solution are, respectively,        1+ε  Φ1 −1     ε   ε A     Z 1  t 1+ε   dε   k(t) = k0 exp  ds , H(Θ) − n −  X(s) t0           µ(t) = µ(t0 ) exp {− (H(Θ) − ρ) (t − t0 )} .

(A.9)

(A.10)

The third step consists in determining the initial value of the co-state variableµ(t) for which trajectories are non-explosive. Given k0 known, this may be done by determining X(t0 ). All what is needed in this step can be deduced from the transversality condition. This necessary condition, given the signs of the parameters, may be simplified to bx exp {−ax (t − t0 )} b x = 0. +1− (A.11) lim t→∞ ax X(t0 ) ax X(t0 ) In particular, given that bx > 0, this condition holds if, and only if, both ax =


H(Θ) +

ρ Φ

− n > 0 and X(t0 ) =

bx ax

=

ε 1 dε

A ( 1+ε )

1+ε ε

 1 −1 Φ

hold. Coming (1− Φ1 )H(Θ)+ Φρ −n back to (A.6) we find that X(t) will remain constant and equal to its initial stationary value X(t0 ) > 0, ∀t ≥ t0 . Consequently, the non-explosive solution trajectories for variables involved in the modified Hamiltonian dynamic system (A.1)-(A.4) are unique, positive and may be written as in equations (18), (19) and (20). Finally, the constraint Ωµ Ωk Ωk Ωk bx −∆µ = 0 < ∆k = H(Θ)−n < ax = (1− 1 )H(Θ)+ ρ −n also holds and, therefore, we Φ Φ conclude that H(Θ) > ρ. 1−

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1 Φ



References [1] Abel, A. B. and O. J. Blanchard (1983). An Intertemporal Model of Saving and Investment. Econometrica, 51, 675-692. [2] Barro, R. J. and X. Sala-i-Martin (1992). Public Finance in Models of Economic Growth. Review of Economic Studies, 59, 645-661. [3] Bischoff, C. W. and E. C. Kokkelenberg (1987). Capacity Utilization and Depreciation in Use. Applied Economics, 19, 995-1007.

134

J. Aznar-M´arquez and J.R. Ruiz-Tamarit

[4] Boucekkine, R. and J. R. Ruiz-Tamarit (2003). Capital Maintenance and Investment: Complements or Substitutes? Journal of Economics, 78 (1), 1-28. [5] Burnside, C. and M. Eichenbaum (1996). Factor-Hoarding and the Propagation of Business-Cycle Shocks. American Economic Review, 86, 1154-1174. [6] Collard, F. and T. Kollintzas (2000). Maintenance, Utilization, and Depreciation along the Business Cycle. CEPR, DP 2477, UK. [7] Epstein, L. and M. Denny (1980). Endogenous Capital Utilization in a Short-Run Production Model. Journal of Econometrics, 12, 189-207. [8] Gylfason, T. and G. Zoega (2001). Obsolescence. CEPR, DP 2833, UK. [9] Johnson, P. A. (1994). Capital Utilization and Investment when Capital Depreciates in Use: Some Implications and Tests. Journal of Macroeconomics, 16 (2), 243-259. [10] Licandro, O. and L. A. Puch (2000). Capital Utilization, Maintenance Costs and the Business Cycle. Annales d’Economie et de Statistique, 58, 143-164. [11] Licandro, O., L. A. Puch and J. R. Ruiz-Tamarit (2001). Optimal Growth under Endogenous Depreciation, Capital Utilization and Maintenance Costs. Investigaciones Econ´omicas, 25, 543-559. [12] McGrattan, E. and J. A. Schmitz, Jr. (1999). Maintenance and Repair: Too Big to Ignore. Quarterly Review, 23 (4), Federal Reserve Bank of Minneapolis. [13] Motahar, E. (1992). Endogenous Capital Utilization and the q Theory of Investment. Economic Letters, 40, 71-75. [14] Rebelo, S., 1991. Long-Run Policy Analysis and Long-Run Growth.Journal of Political Economy, 99, 500–521.

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[15] Romer, P. (1986). Increasing Returns and Long-Run Growth. Journal of Political Economy, 94, 1002–1037. [16] Ruiz-Tamarit, J. R. and M. Ventura-Marco (2000). Solution to Non-linear MHDS arising from Optimal Growth Problems. FEDEA, DT 2000-16, Spain. [17] Rumbos, B. and L. Auernheimer (2001). Endogenous Capital Utilization in a Neoclassical Growth Model. Atlantic Economic Journal, 29 (2), 121-134.

In: Economic Dynamics… Editor: Chester W. Hurlington, pp. 135-156

ISBN: 978-1-60456-911-7 © 2008 Nova Science Publishers, Inc.

Chapter 7

SOCIAL STATUS, THE SPIRIT OF CAPITALISM, AND THE TERM STRUCTURE OF INTEREST RATES 1 IN STOCHASTIC PRODUCTION ECONOMIES Liutang Gong1,a, Yulei Luo2,b and Heng-fu Zou3,c 1

Guanghua School of Management, Peking University, Beijing, 100871, China Institute for Advanced Study, Wuhan University, Wuhan, 430072, China 2 School of Economics and Finance, University of Hong Kong, Hong Kong 3 Institute for Advanced Study, Wuhan University, Wuhan, 430072, China Development Research Group, The World Bank, Washington, DC 20433, USA

Abstract

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This paper studies capital accumulation and equilibrium interest rates in stochastic production economies with the concern of social status. Given a specific utility function and production function, explicit solutions for capital accumulation and equilibrium interest rates have been derived. With the aid of steady-state distributions for capital stock, the effects of fiscal policies, social-status concern, and stochastic shocks on capital accumulation and equilibrium interest rates have been investigated. A significant finding of this paper is the demonstration of multiple stationary distributions for capital stocks and interest rates with the concern of social status.

Key Words: Stochastic growth; Social status; Fiscal policies; Interest rates. JEL Classification: E0, G1, H0, O0.

1

Project 70271063 supported by the National Natural Science Foundation of China. E-mail address: [email protected] b E-mail address: [email protected] c E-mail address: [email protected]. Mailing address: Heng-fu Zou, Development Research Group, the World Bank, 1818 H St. NW, Washington, DC 20433, USA. a

136

Liutang Gong, Yulei Luo and Heng-fu Zou

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1. Introduction Capital accumulation, interest rates, fiscal policies, and asset pricing under uncertainty have been studied extensively since the 1960s, e.g., Phelps (1962), Levhari and Srinivisan (1969), Brock and Mirman (1972), Mirrlees (1965). Merton (1975) studied the asymptotic theory of growth under uncertainty, and Foldes (1978) explored optimal saving with risk in continuous time. As for the term structure of interest rates, Cox, Ingersoll, and Ross (1981, 1985) considered the equilibrium theory of the term structure of interest rates, and presented the general theory for interest rates in a production economy. Sunderason (1983) provided a plausible equilibrium model, in which the assumption of a constant interest rate is valid. Bhattacharya (1981), Constantinides (1980), and Stapleton and Subrahmanyam (1978) also studied these topics and presented the conditions for a constant interest rate. Constantinides (1980) showed that the term structure of interest rate evolves deterministically over time under the assumptions of perfect capital markets, homogeneous expectations, and the state independent utility. Sunderason (1984) also derived the conclusion of a constant interest rate under Constantinides (1980)'s assumptions on capital markets, expectations, and utility. For the effects of fiscal policies on capital accumulation, interest rates, and asset pricing in stochastic economies, Eaton (1981), Turnovsky (1993, 1995), Grinols and Turnovsky (1993, 1994), and Obstfeld (1994) introduced taxations and government expenditure into the stochastic continuous-time growth and asset-pricing models. Under a linear production technology and other specified assumptions on preferences and stochastic shocks, they have derived explicit solutions of growth rates of consumption, savings, and equilibrium returns on assets. In all these neoclassical models of capital accumulation, interest rates and asset pricing models, wealth accumulation is often taken to be solely driven by one's desire to increase consumption rewards. The representative agent chooses his consumption path to maximize his discounted utility, which is defined only on consumption. This motive is important for wealth accumulation. It is, however, not the only motive. Because man is a social animal, he also accumulates wealth to gain prestige, social status, and power in the society; see Frank (1985), Cole, Mailath and Postlewaite (1992, 1995), Fershtman and Weiss (1993), Zou (1994, 1995), Bakshi and Chen (1996), and Fershtman, Murphy and Weiss (1996). In these wealthis-status models, the representative agent accumulates wealth not only for consumption but also for wealth-induced status. Mathematically, in light of the new perspective, the utility function can be defined on both consumption, c , and wealth, w : u (ct , wt ) . Another interpretation of these models is in line of the spirit of capitalism in the sense of Weber 2 (1958): capitalists accumulate wealth for the sake of wealth . With the wealth-is-status and the-spirit-of-capitalism models, many authors mentioned above have tried to explore diverse implications for growth, savings, interest rates, and asset pricing. Cole, Mailath, and Postlewaite (1992) have demonstrated how the presence of socialstatus concern leads to multiple equilibria in long-run growth. Zou (1994, 1995) has studied the spirit of capitalism and long-run growth and showed that a strong capitalistic spirit can lead to unbounded growth of consumption and capital even under the neoclassical assumption of production technology, Gong and Zou (2002) have studied fiscal policies, asset pricing, 2

See Cole, Mailath, and Postlewaite (1992), Zou (1994); and Bakshi and Chen (1996) for details.

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

137

and capital accumulation in a stochastic model with the spirit of capitalism. Bakshi and Chen (1996) have explored empirically the relationship between the spirit of capitalism and stock market pricing and offered an attempt towards the resolution of the equity premium puzzle in Mehra and Prescott (1985). Smith (2001) has studied the effects of the spirit of capitalism on asset pricing and has shown that when investors care about status they will be more conservative in risk taking and more frugal in consumption spending. Furthermore, stock prices tend to be more volatile with the presence of the spirit of capitalism. This paper explores capital accumulation and equilibrium interest rates in a stochastic model with the spirit of capitalism and with diminishing return to scale technology. Under a CES utility function defined on both consumption and wealth accumulation and a CobbDouglas production function, explicit solutions for capital accumulation and equilibrium interest rates have been derived. These multiple optimal paths and stationary distributions of capital stock and interest rates are quite significantly different from many existing neoclassical models. With the aid of the steady-state distributions for capital stock, the effects of fiscal policies on the long-run economy and the equilibrium interest rates have been investigated. In particular, the equilibrium interest rates are constant when the technology is linear and when the utility function is extended to include the wealth-is-status concern. Moreover, the equilibrium interest rates are a mean reserve process with these special assumptions. This paper is organized as follows: in section 2, we set up a stochastic growth model in a production economy with the social-status concern. Allowing some special utility function and production function with selected parameters, explicit solutions for the optimal paths and stationary distributions of consumption, capital accumulation and interest rates have been derived in section 3. With the aid of the steady-state distribution of the endogenous variables, the effects of fiscal policies, production shocks, and the spirit of capitalism on the long-run economy have been examined in section 4. In section 5, we present the equilibrium interest rates under both a nonlinear technology and a linear technology and analyze the dynamic behavior of equilibrium interest rates and discuss the effects of fiscal policies and stochastic shocks on the interest rates. In section 6, we present some examples to show the existence of multiple stationary distributions of optimal capital accumulation and equilibrium interest rates. We conclude our paper in section 7.

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2. The Model Following Eaton (1981) and Smith (2001), we assume that output y is given by

dy = f (k )dt + ε kdz ,

(1)

where z is the standard Brown motion, ε is the stochastic shocks of production. Equation (1) asserts that the accumulated flow of output over the period (t , t + dt ) , given by the right-hand side of this equation, consists of two components. The deterministic component is described as the first term on the right-hand side, which is the firm’s production technology and has been specified as a neoclassical production function, f ( k ) . The second

138

Liutang Gong, Yulei Luo and Heng-fu Zou

part is the stochastic component, ε kdz , which can be viewed as the shocks to the production and assumed to be temporally independent, normally distributed. Suppose the government levy an income tax and a consumption tax. Then, the agent's 3 budget constraint can be written as

dk = ((1 − τ ) f (k ) − (1 + τ c )c)dt + (1 − τ ′)ε kdz

(2)

τ and τ ' are the tax rates on the deterministic component of capital income and stochastic capital income, respectively, and τ c is the consumption tax rate.

where

With the social status concern, the utility function can be written as u (c, k ) . Suppose the marginal utilities of consumption and capital stock are positive, but diminishing, i.e.

u1 (c, k ) > 0, u2 (c, k ) > 0, u11 (c, k ) < 0, u22 (c, k ) < 0

(3)

The representative agent is to choose his consumption level and capital accumulation path to maximize his expected discounted utility, namely, ∞

max E0 ∫ u (c, k )e − ρt dt 0

subject to a given initial capital stock k (0) and the budget constraint (2). Where 0 < ρ < 1 is the discount rate. Associated with the above optimization problem, the value function J ( k , t ) is defined as ∞

J (k , t ) = max Et ∫ u (c, k )e − ρt dt t

subject to the given initial capital stock k (t ) and the budget constraint (2). Define the current

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value function X ( k ) as

3

Merton (1975) assumed that output is produced by a strictly concave production function,

K (t ) denotes capital stock, L(t ) Y = AF ( K , L)

where

the labor force, and

A(t )

Y = AF ( K , L) ,

is technology progress. Production is

and the labor force follows

dL = aLdt + ε Ldz , z is the standard Brown motion. Defining the capital-labor ratio k = K / L , where

from the Itô's Lemma, we can derive the capital accumulation equation similar to equation (2). Or we assume the technology progress follows

dA / A = adt + ε dz

Defining the efficiency capital (2).

k = K /( AL) , we can also derive the capital accumulation similar to equation

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

139

X ( k ) = J ( k , t )e ρ t

(4)

The recursive equation associated with the above optimization problem is

max{u (c, k ) − ρ X (k ) + X ′(k )((1 − τ ) f (k ) − (1 + τ c )c) + c

1 X ′′(k )(1 − τ ′) 2 ε 2 k 2 } = 0 2

Therefore, we get the first-order condition

uc (c, k ) = (1 + τ c ) X ′(k )

(5)

and the Bellman equation

u (c, k ) − ρ X (k ) + X ′(k )((1 − τ ) f (k ) − (1 + τ c )c) + (1 − τ ′) 2

1 X ′′(k )ε 2 k 2 = 0 2

(6)

Equation (5) states that the marginal utility of consumption equals the after-tax marginal utility of capital stock. Equation (6) determines the value function X ( k ) . In the next section, we will specify the utility function and the production function to present an explicit solution for the value function.

3. An Explicit Solution 3.1. The Explicit Solution under the Separable Utility Function 4

In order to derive an explicit solution, we specify the utility function as

u (c, k ) =

c1−σ k 1−σ +ξ , 1−σ 1−σ

(7)

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where σ > 0 is the constant relative risk aversion, and it also represents the elasticity of intertemporal substitution. ξ ≥ 0 measures the investor's concern with his social status or measures his spirit of capitalism. The larger the parameter

ξ , the stronger the agent's spirit of

capitalism or concern for social status. The production function is specified as

f (k ) = Ak α

(8)

where A > 0 and 0 < α < 1 are positive constants. For the special utility function and production function in equations (7) and (8), we conjecture that the value function takes the following form 4

In Appendix B, we present a similar analysis when the utility function is non-separable.

140

Liutang Gong, Yulei Luo and Heng-fu Zou

X (k ) = a +

b −σ k 1−σ 1−σ

(9)

where a and b are constants, and they are to be determined. Under the specified value function in equation (9), we rewrite the first-order condition (5) as

c −σ = (1 + τ c ) X k = (1 + τ c )b −σ k −σ namely, −1

c = (1 + τ c ) σ bk

(10)

Upon the relationship (10), the Bellman equation (6) is reduced to 1− 1

(1 + τ c ) σ b1−σ k 1−σ k 1−σ b −σ k 1−σ 1− 1 +ξ − ρ (a + ) + b −σ k −σ ((1 − τ ) Ak α − (1 + τ c ) σ bk ) 1−σ 1−σ 1−σ (11) 1 2 2 2 + X kk ε (1 − τ ′) k = 0 2 If α = 1 , from equation (11), we have a = 0 and b is determined by the following equation 1− σ1

0 = σ (1 + τ c )

1 b + ξ bσ − ( ρ − (1 − τ ) A(1 − σ ) + (1 − σ )σ (1 − τ ′) 2 ε 2 ) 2

In general, for the case of α ≠ 1 , we cannot determine the constants a and b . Following Xie (1994), we specified the parameters as a = σ , then from equation (11), we have

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a=

(1 − τ ) A ρ bσ

and b is determined by 1− σ1

0 = (1 + τ c )

1 2

σ b + ξ bσ − ( ρ + (1 − σ )σ (1 − τ ′) 2 ε 2 ).

Summarizing the discussions above, we have Proposition 1. Under the special utility function and production function in equations (7) and (8), if α = 1 , then the explicit solutions for the economy system are

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

141

c −1 = (1 + τ c ) σ b k

(12)

1− σ1

dk = ((1 − τ ) Ak − (1 + τ c )

bk )dt + (1 − τ ′)ε kdz

(13)

and the TVC

lim E ( X ′(k )e − ρt ) = 0

(14)

t →∞

where b is determined by 1− σ1

0 = (1 + τ c ) If

1 2

σ b + ξ bσ − ( ρ − (1 − τ ) A(1 − σ ) + (1 − σ )σ (1 − τ ′) 2 ε 2 ).

(15)

α ≠ 1 and α = σ , then the explicit solutions for the economy are c −1 = (1 + τ c ) σ b , k 1− σ1

dk = ((1 − τ ) Ak α − (1 + τ c )

(12’)

bk )dt + (1 − τ ′)ε kdz

(13’)

and the TVC (14) holds, whereas b is determined by 1− σ1

0 = (1 + τ c )

1 2

σ b + ξ bσ − ( ρ + (1 − σ )σ (1 − τ ′) 2 ε 2 ).

(15’)

When α = 1 , the capital stock follows the stochastic growth path (13), and we get the mean growth rate for the capital stock

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E(

dk 1− 1 ) = ((1 − τ ) A − (1 + τ c ) σ b)dt k

It is easy to show from equation (12’) and the production function that the mean growth rates for consumption level, output, and capital stock are equal. Let us denote the common mean growth rate as φ , which is given by 1− σ1

φ = (1 − τ ) A − (1 + τ c )

b.

From the expression above, it is clear that capital income taxation, consumption taxation, stochastic shocks, and various preference and production parameters jointly determine the growth rate of the economy. Please also note that when the parameters satisfy the condition

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Liutang Gong, Yulei Luo and Heng-fu Zou

α = σ , the deterministic income tax rate has no effects on the equilibrium consumptioncapital stock ratio.

3.2. Steady-State Distributions for Endogenous Variables Similar to the certainty model, we will examine the existence and the properties of the steady state economy. As in Merton (1975), we are seeking the conditions under which there is a unique stationary distribution for the capital stock k , which is time and initial condition independent. From equation (13’), the capital stock follows the following stochastic process, 1− σ1

dk = ((1 − τ ) Ak α − (1 + τ c )

bk )dt + (1 − τ ′)ε kdz

(16)

 b(k )dt + (a(k )) dz , 1/ 2

where we denote a (k ) = (1 − τ ′) Let (1975),

2

1− σ1

ε 2 k 2 and b(k ) = (1 − τ ) Ak α − (1 + τ c )

bk .

π k (k ) be the steady-state density function for the capital stock. As in Merton π k (k ) exists and it can be shown to be π k (k ) =

k 2b( x ) m exp ∫ dx , 0 a( x) a(k )

∞

where m is a constant chosen so that ∫ 0 π k ( x ) dx = 1 . Substituting the expressions for a ( k ) and b( k ) , we have 1− σ1

α k 2((1 − τ ) Ak − (1 + τ ) m c π k (k ) = exp ∫ 2 2 2 2 2 2 0 (1 − τ ′) ε k (1 − τ ′) ε k

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= mk

−2

1− 1 (1+τ c ) σ b+(1−τ ′ )2 ε 2 (1−τ ′ )2 ε 2

Defining variable R = k

α −1

1− 1

(12’).

′

dx

2 A(1 − τ ) k α −1 ) (1 − α )(1 − τ ′) 2 ε 2

, we have

π R ( R) = π k (k ) / |

where

exp(−

bk )

dR m γ −1 |= R exp(− β R) , dk 1 − α

A (1−τ ) γ = 2 (1+τ(1−) α )(1b−+τ(1′)−τε ) ε > 0 , β = (1−α2 )(1 > 0 , and b is determined by equation −τ ′ ) ε c

σ

2

2

2

2

2

2

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

143

Therefore, we have

(1 − α ) β γ m= , Γ(γ ) where Γ(.) is the gamma function . 5

Thus, the steady-state distribution for the capital stock is

k ≤0

0,

π (k ) =

{

(1−α ) β γ Γ (γ )

−

k

1− 1 2((1+τ c ) σ b+(1−τ ′ )2 ε 2 ) (1−τ ′ )2 ε 2

exp(− β k − (1−α ) ), k > 0

(17)

and the moment-generating function

Φ k (θ ) = E{k θ } = The steady-state distribution for R = k

π ( R) =

Γ(γ − θ /(1 − α )) θ /(1−α ) β Γ(γ ) α −1

is

R≤0

0,

{

β

γ

Γ (γ )

(18)

Rγ −1 exp(− β R), R > 0

(19)

and the moment-generating function

Φ R (θ ) = E{Rθ } =

Γ(γ + θ ) −θ β Γ(γ )

(20)

α

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The steady-state distribution for y ( = Ak ) is

π ( y) =

y≤0

0,

{

ηβ

γ

AΓ ( γ )

and the moment-generating function

5

The Gamma function ∞

Γ(α )

Γ(α ) = ∫ xα −1e − x dx . 0

is defined as

( Ay ) − (ηγ +1) exp(− β ( Ay ) −η ),

y>0

(21)

144

Liutang Gong, Yulei Luo and Heng-fu Zou

Φ y (θ ) = E{ yθ } =

Aθ Γ(γ − θ / η ) θ /η β Γ(γ )

(22)

With the aid of these steady-state distributions and moment-generating functions, we can derive quite a few long-run properties of our endogenous variables in the following section of comparative static analysis.

4. Comparative Static Analysis With the given steady-state distributions of the capital stock, output, and the interest rate, we can derive the long-run expected values of capital stock, consumption level, and output ∞

E (k ) = ∫ kπ (k )dk = 0

Γ[γ − 1/(1 − α )] 1/(1−α ) β , Γ(γ ) −1

E (c) = (1 + τ c ) σ bE (k ), E ( y) = where b , β , η , and

AΓ(γ − 1/ η ) 1/η β , Γ(γ )

(23a)

(23b)

(23c)

γ are presented in section 3 above, and Γ(.) is the Gamma function.

4.1. Effects of Uncertainty on Expected Capital, Output, and Consumption As in Zou (1994), the modified golden rule for the long-run capital stock in a deterministic model with the spirit of capitalism can be derived as

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(1 − τ ) f ′(k ) = ρ − (1 + τ c )

uk (c, k ) uc ( c, k )

With our special utility function and production function and with the parameter condition of α = σ , we have

1 −τ (1 − τ ) Aα k α −1 + (1 + τ c )ξ ( Ak α −1 )σ = ρ 1+τc and the associated consumption level and output are

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

c∗ =

145

1−τ A(k ∗ )α , y ∗ = A(k ∗ )α 1+τc

Comparing with the uncertainty case, we have

E ( k ) < k ∗ , E (c ) < c ∗ , E ( y ) < y ∗ Thus, the long-run expected capital stock, expected consumption level, and expected output are smaller than the deterministic steady-state ones, respectively. This is because that the output is a strictly concave function of the capital stock, and Jensen’s inequality implies that an increase in capital risk must reduce the expected capital stock and expected output. The fall in the expected output results in a fall in the expected consumption.

4.2. Effects of the Spirit of Capitalism In our special utility function, we know that the parameter

ξ measures the representative

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agent’s concern with his social status or his spirit of capitalism. Because we have specified the parameters as α = σ , we have σ ∈ [0, 1] .

Figure 1. The effects of the spirit of capitalism on the long-run economy.

146

Liutang Gong, Yulei Luo and Heng-fu Zou

Figure 1 presents the effects of the spirit of capitalism on the economy under the cases of α = 1 (the solid line) and α ≠ 1 and α = σ (the star line). It is easy to see that with a stronger spirit of capitalism, the long-run expected capital stock, consumption level, and output will be higher.

4.3. Effects of Production Shocks

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Figure 2 shows that with increasing production shocks, the long-run expected capital stock, output, and consumption will be decreasing. Therefore, uncertainty in production reduces investment, output and consumption. This result is rather clear-cut because other related studies have indicated an ambiguous result of production shocks on investment and output, see Turnovsky (1993, 2000), Obstfeld (1994), and Gong and Zou (2002).

Figure 2. The effects of production shocks on the long-run economy.

4.4. Effects of Fiscal Policies The solid line in figure 3 shows the effects of income tax rate on the long-run economy. From which, we find the with a rise in the deterministic income tax rate, the long-run capital stock, output, and consumption will be decreasing (solid lines in figure 3). The effects of stochastic income tax rate (starred lines in figure 3) on the economy are just opposite to the effects of

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

147

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deterministic income tax rate: A rising stochastic income tax rate raises expected capital stock, output and consumption. As for the effects of consumption tax rate on the economy, from the circled line in figure 3, we find that with an increasing consumption tax rate, the long-run expected capital stock and output will be rising, whereas the long-run expected consumption will be decreasing. This is true because a rising consumption tax raises the cost of consumption, which leads to a reduction in consumption and an increase in investment, capital stock and output. Please note that this positive effect of a consumption tax rate on capital accumulation and output is a significant feature of stochastic growth model. In the traditional, deterministic literature such as Rebelo (1990), a consumption tax has no effect on the long-run capital accumulation.

Figure 3. The effects of the deterministic income tax rate, the stochastic income tax rate, and the consumption tax rate on the long-run economy.

5. Equilibrium Interest Rates From Cox, Ingersoll, and Ross (1985), we know that the equilibrium interest rates can be written as

r=ρ−

L( X k ) , Xk

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Liutang Gong, Yulei Luo and Heng-fu Zou

where L(.) is the differential operator. 6

Thus we have

Proposition 2. With the utility function and technology in equations (7) and (8), the equilibrium interest rate is given by 1− σ1

r = ρ + σ ((1 − τ ) Ak α −1 − (1 + τ c )

1 b) − ε 2 (1 − τ ′) 2 σ (σ + 1) 2

(24)

when α ≠ 1 and α = σ , where b is determined by equation (15’). Furthermore, the equilibrium interest rate is given by 1− σ1

r = ρ + σ ((1 − τ ) A − (1 + τ c )

1 b) − ε 2 (1 − τ ′) 2 σ (σ + 1) 2

(25)

when α = 1 , where b is determined by equation (15). From proposition 2, the equilibrium interest rate is a constant when the technology is linear. Among many existing studies, Sundareson (1983, 1984) has also presented a constant interest rate for a constant absolute risk aversion utility function in an infinite horizon dynamic portfolio and consumption choice problem. Our model obtains the same result while allowing the utility function to be dependent on both consumption and wealth. Comparative static analysis shows that, with a rise of technology shocks, the equilibrium interest rate will be decreasing; with a rise in the deterministic income tax rate, the equilibrium interest rate will be increasing; but the equilibrium interest rate will be decreasing with a rise of the stochastic income tax rate. Also, we find that with the increase of the consumption tax rate, the equilibrium interest rate will be increasing; please see figure 4. When α ≠ 1 , the equilibrium interest rate is stochastic, not a constant anymore. Using the expression for the equilibrium interest rate, the dynamics for the capital stock can be rewritten as 6

With the utility function and technology in equations (B1) and (8), the equilibrium interest rate is given by

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1− σ1

r = ρ + (σ + λ )((1 − τ ) Ak α −1 − (1 + τ c )

1 b) + (1 − τ ′)2 ε 2 (σ + λ )(−σ − λ − 1) 2

α ≠ 1 and α = σ + λ , where b is determined by ρ 1−σ + 1 (1 − σ )(σ + λ )(1 − τ ′) 2 ε 2 b = 1−σ −λ 2 . 1− (1 + τ c ) σ σ

when

1

On the other hand, the equilibrium interest rate is given by 1− σ1

r = ρ + (σ + λ )((1 − τ ) A − (1 + τ c )

1 b) + (1 − τ ′) 2 ε 2 (σ + λ )(−σ − λ − 1) 2

α = 1 , where b is determined by ρ 1−σ − (1 − τ ) A + 12 (1 − σ )(σ + λ )(1 − τ ′) 2 ε 2 b = 1−σ −λ 1− (1 + τ c ) σ σ

when

1

.

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

149

dk 1 1 = (r − ρ + ε 2 (1 − τ ′) 2 σ (σ + 1))dt + ε dz k σ 2

(26)

.

Thus, the dynamics of the interest rate is

1 1 1− 1 dr = [ (r − ρ + ε 2 (1 − τ ′) 2 σ (σ + 1)) + (1 + τ c ) σ b] σ 2 1 ×{(r − ρ + ε 2 (1 − τ ′) 2 σ (σ + α − 1))dt + (1 − τ ′)ε dz} 2 If

(27)

ε = 0 and ξ = 0 , we have dk 1 = (r − ρ )dt , k σ

dr 1 = (r − ρ )dt r σ

These are dynamic accumulation paths for the capital stock and the interest rate without production shocks and the spirit of capitalism. It is obviously that the equilibrium interest rate will convergent to ρ . Equations (26) and (27) can be used to study the behavior of the interest rate in this economy. For example, when the initial interest rate is very high, say it is larger than

ρ − 12 ε 2 (1 − τ ′)2 σ (σ + 1) , then the capital stock will be growing. And from the expression for the interest rate, it will go down. If the initial interest rate is lower enough, say lower than

ρ − 12 ε 2 (1 − τ ′)2 σ (σ + 1) , then the capital stock will be increasing, thus the interest rate will be go up. Thus, the equilibrium interest rate will fluctuate around a value depending on

ρ , ε 2 , and σ . Similarly, we can find the stationary distribution for the interest rate. For simplicity, we define r = α (1 − τ ) Ak

α −1

, the steady-state distribution and the moment-generating function

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for variable rˆ can be found as

π (r ) =

r ≤0

0,

{

( α (1−βτ ) A )γ Γ (γ )

r γ −1 exp(− α (1−βτ ) A r ), r > 0

θ Γ(γ + θ ) β ( ) −θ Φ r (θ ) = E{r } = Γ(γ ) α (1 − τ ) A

(28)

(29)

Thus, we get the long-run behavior of the equilibrium interest rate

E (r ) =

Γ(γ + 1) β 1 1− 1 ( ) −1 + ρ − σ (1 + τ c ) σ b − ε 2 (1 − τ ′) 2 σ (σ + 1) , (30) 2 Γ(γ ) α (1 − τ ) A

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Liutang Gong, Yulei Luo and Heng-fu Zou

where b is determined by equation (15).

Figure 4. (a) Effects of the spirit of capitalism on the interest rate; (b) Effects of production shocks on the interest rate; (c) Effects of the deterministic income tax rate and the stochastic income tax rate on the interest rate; (d) Effects of the consumption tax rate on the interest rate.

From figure 4, we know that the long-run expected interest rate will be decreasing with a rise in technology shocks and the deterministic income tax rate (the star line in figure 4c). At the same time, the stochastic income tax rate, the consumption tax rate, and the spirit of capitalism all have positive effects on the long-run expected interest rate.

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6. Multiple Optimal Paths and Stationary Distributions From equation (15’), we cannot determine the unique solution for variable b . In this section, we examine the existence of multiple solutions for the consumption-capital ratio for a few selected parameters. Because there exists a unique path for the capital accumulation associated with the consumption-capital ratio, there will exist a unique steady-state 7 distribution associated with each path . Below, we will present examples to show the existence of multiple optimal paths or stationary distributions and their associated long-run expected capital stocks, consumption levels, equilibrium interest rates, and output.

7

For the non-separable utility function in (B1) in Appendix B, we can determine the unique steady state.

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates... If we select the parameters as A = 0.5 ,

151

α = σ = 0.6 , τ = 0.3 , τ ′ = 0.3 , ξ = 0.1 ,

ρ = 0.1 , τ c = 0 , and let ε 2 vary from 0.5, 1, and 1.1, and we get the following results Table 1. Multiple optimal paths when

ε 2 = 0.5 c/k E (k ) E (r ) E (c ) E ( y)

α = σ = 0.6

ε 2 =1

ε 2 = 1.1

path 1 0.0210

path 2 0.3519

path 3 0.1601

path 1 0.4217

path 2 0.2010

path 3 0.0430

path 1 0.2093

path 2 0.4355

path 3 0.0476

150.1

0.8

4.4

0.4

2.0

25.8

1.7

0.4

20.1

0.0559

0.0559

0.0559

0.0118

0.0118

0.0118

0.003

0.003

0.003

3.1568

0.2732

0.6984

0.1810

0.3967

1.1067

0.3619

0.1688

0.9566

4.5098

0.3902

0.9977

0.2585

0.5667

1.5811

0.5171

0.2411

1.3665

For the case of a linear technology , i.e., α = 1 , we have derived the mean growth rate of the economy and the equilibrium interest rate as follows 1− σ1

φ = (1 − τ ) A − (1 + τ c ) 1− σ1

r = ρ + σ ((1 − τ ) A − (1 + τ c )

b

1 b) − ε 2 (1 − τ ′) 2 σ (σ + 1) 2

where b is determined by equation (15). In this case, we select the parameters as:

α = 1 , A = 0.43 , σ = 0.6 , τ = 0.3 , ′ τ = 0.3 , ρ = 0.21 , and τ c = 0 . When ξ = 0 , we have a unique path or stationary

distribution for consumption-capital ratio, the growth rate, and the equilibrium interest rate. When ξ = 0.025 , we have three stationary distributions for these variables. See Table 2 for details.

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Table 2. Multiple optimal paths when

ξ =0

α =1

ξ = 0.025

path 1

path 1

path 2

path3

c/k

0.2473

0.2245

0.2578

0.2301

mean growth rate

0.0537

0.0765

0.0432

0.0709

interest rate

0.007

0.0207

0.0007

0.0174

Finally, we select the parameters as:

α = 1 , A = 0.46 , σ = 0.6 , τ = 0.3 , τ ′ = 0.3 ,

ρ = 0.25 , and τ c = 0 . That is to say, we only change the values of A and the discount rate of

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Liutang Gong, Yulei Luo and Heng-fu Zou

ρ slightly. Again, we have multiple expected values or multiple stationary distributions in the economy. See details in Table 3: Table 3. Multiple optimal paths when

ξ =0 c/k mean growth rate interest rate

path 1 0.3000 0.0220 0.0280

α =1

ξ = 0.025 path 1 0.2744 0.0476 0.0434

path 2 0.3119 0.0101 0.0209

path 3 0.2806 0.0414 0.0397

This existence of multiple stationary distributions for asset accumulation and interest rates is significant different from the unique stationary distribution in Brock and Mirman (1972), Merton (1973), Lucas (1978), Brock (1982), Cox, Ingersoll and Ross (1985), and many other classical models on stochastic capital theory and the term structure of interest rates. In fact, multiple stationary distributions in asset markets and returns may provide a more realistic picture of the real world because it admits the rationality and plausibility of different expectations and heterogeneity, though our model is still in line with the representative agent framework.

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7. Conclusion This paper has studied capital accumulation and the equilibrium interest rates in stochastic production economies with the spirit of capitalism. Under the specified utility function, production function, and selected parameters, we have presented the explicit solutions for consumption and capital accumulation. With the aid of steady-state distributions for the capital stock, we presented the effects of fiscal policies, the spirit of capitalism, and stochastic shocks on the long-run economy. We find that the long-run capital stock, output, and consumption level in the uncertainty case are less than those ones in the deterministic case; the long-run interest rate in the uncertainty case is larger than the deterministic case. These conclusions are different from the one presented by Merton (1975), similar to the one in Smith (1998). As for the effects of the spirit of capitalism on the long-run economy, we find that with the increase of the spirit of capitalism, the long-run interest rate, consumption level, and output will be decreasing. The effect of the spirit of capitalism on the long-run capital stock will be negative when the production shocks are small; its effect will be positive when the production shocks are larger. These findings are different from the ones in Gong and Zou (2001, 2002), and Zou (1994, 1995). The effects of the income tax rate on the long-run economy have also been investigated in this paper, and we have found the similar effects of a deterministic income tax and a stochastic income tax rate presented by Gong and Zou (2002), Turnovsky (1993, 2000). With the rise in the deterministic income tax rate, the long-run capital stock, output, and consumption level will be decreasing, but the interest rate will be increasing. The effects of the stochastic income tax rate on the long-run economy are just opposite to the effects of the

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

153

deterministic income tax rate on the long-run economy. We have also shown that the consumption tax rate will affect the long-run expected capital stock, output, and consumption, which are different from the ones in the traditional, deterministic models. The equilibrium interest rate has been shown under a linear technology and a nonlinear technology, respectively. When the production technology is linear, we can still obtain a constant interest rate for this stochastic model with the spirit of capitalism and social status. This result is similar to the one in Sunderason (1983), who presented the conclusion of constant interest rate under the assumption of CES utility function and a linear technology. Of course, his utility function is independent of the state variable of capital stock. But, with a nonlinear technology, we find that the interest rate follows the mean reserve process and fluctuates around a value depending on the parameters of ρ and σ . Finally, the existence of multiple stochastic optimal paths or multiple stationary distributions for capital accumulation is presented in this paper. This is a main feature of a model with the spirit of capitalism or social-status concern. Associated with the multiple stationary distributions for capital accumulation, there exist multiple expected interest rates. This line of investigation enriches our understanding of the complexity of asset markets and the term structure of interest rates. This paper considers an economy with one consumption good and production technology. A first extension of this paper is to follow Sunderason (1983) to study the equilibrium interest rate in an economy with many consumption goods and production technologies. Secondly, this paper has not considered monetary policy, and we should follow Grinols and Turnovsky (1998) and extend this model to a monetary one with the spirit of capitalism. Thirdly, we can extend this model to consider habit formation, catching up with the Joneses, and the nonexpected utility.

Appendix A: The Steady-State Distribution for a Diffusion Process We follow Merton (1975) and consider the steady-state distribution for a diffusion process. Let X (t ) be the solution to the Itô equation

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dx = b( x)dt + (a ( x))1/ 2 dz , where a (.) and b(.) are twice-differentiable function on [0, ∞) and independent of t with

a( x) > 0 and a(0) = b(0) = 0 . The steady-state distribution will always exist, and it can be expressed as

π ( x) = ∞

x 2b( y ) m exp ∫ dy , 0 a( y ) a( x)

where m is chosen such that ∫ 0 π ( x) dx = 1 .

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Liutang Gong, Yulei Luo and Heng-fu Zou

Appendix B: The Case of Non-separable Utility Function If we specified the utility function as in Bakshi and Chen (1996), Gong and Zou (2002)

c1−σ k − λ , 1−σ

u (c, k ) =

(B1)

where σ is the constant absolute risk aversion, and it is assumed σ > 0 , and λ ≥ 0 when σ ≥ 1 , and λ < 0 otherwise; | λ | measures the investor's concern with his social status or measures his spirit of capitalism. The larger the parameter | λ | , the stronger the agent’s concern for social status. For the specified utility function and production function, we conjecture that the value function takes the following form

X (k ) = a +

b −σ k 1−σ −λ , 1−σ − λ

where a and b are constant, and they are to be determined as follows. From the first-order condition, we have −1

c = (1 + τ c ) σ bk and the Bellman equation (6) is reduced to 1− σ1

(1 + τ c )

b1−σ k 1−σ −λ b −σ k 1−σ −λ 1− 1 ) + b −σ k −σ −λ ((1 − τ ) Ak α − (1 + τ c ) σ bk ) − ρ (a + 1−σ 1−σ − λ 1 + (1 − τ ′) 2 X kk ε 2 k 2 = 0. 2

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If

α = 1 , from the above equation, we have a = 0,

ρ 1−1σ−σ−λ − (1 − τ ) A + 12 (1 − σ )(σ + λ )(1 − τ ′) 2 ε 2 b= 1− (1 + τ c ) σ σ 1

In generally, for the case of α ≠ 1 , we cannot determine the constants a and b . Following Xie (1994), we specified the parameters as α = σ + λ , then, we have

Social Status, the Spirit of Capitalism, and the Tem Structure of Interest Rates...

a=

155

(1 − τ ) A , ρ bσ

ρ 1−1σ−σ−λ + 12 (1 − σ )(σ + λ )(1 − τ ′)2 ε 2 . b= 1− (1 + τ c ) σ σ 1

The remaining discussions are similar to ones in the main text.

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References Bakshi, G. S.and Z. Chen (1996). The spirit of capitalism and stock-market prices. The American Economic Review 86: 133-157. Black, F. (1990). Mean reversion and consumption smoothing. The Review of Finance Studies 3:107-114. Bhattacharya, S. (1981). Notes on multi-period valuation and the pricing of options. The Journal of Finance 36(March):163-180. Brock, W. (1982). Asset prices in a production economy. In The Economics of Information and Uncertainty, edited by J. J. McCall, pp. 1-43. University of Chicago Press. Brock, W. and L. Mirman (1972). Optimal economic growth and uncertainty: The discounted case. Journal of Economic Theory 4:479-513. Cole, H., G. Mailath, and A. Postlewaite (1992). Social norms, saving behavior, and growth. Journal of Political Economy 100: 1092-1125. Cole, H., G. Mailath, and A. Postlewaite (1995). Incorporating concern for relative wealth into economic models. Quarterly Review. Federal Reserve Bank of Minneapolis, vol.19: 12-21. Constantinides, G. M. (1980). Admissible uncertainty in the intertemporal asset pricing model. Journal of Finance Economics 8(March)71-78. Cox, J., J. Ingersoll, and S. Ross (1985) A theory of the term structure of interest rates. Econometrica 53: 385-408. Cox, J., J. Ingersoll, and S. Ross (1981) A reexamination of traditional hypotheses about the term structure of interest rates. The Journal of Finance 36(September):769-799. Eaton, J. (1981). Fiscal policy, inflation, and the accumulation of risk capital. Review of Economic Studies 48: 435-445. Fershtman, C. and Y. Weiss (1993). Social status, culture and economic performance. Economic Journal 103: 946-59. Fershtman, C., Murphy, K., and Y. Weiss (1996). Social status, education, and growth. Journal of Political Economy 106: 108-132. Foldes, L. (1978). Optimal saving and risk in continuous time. Review of Economic Studies 45:39-65. Gong, Liutang and Heng-fu Zou (2001). Money, social status, and capital accumulation in a cash-in-advance model. Journal of Money, Credit, and Banking 33(2): 284-293. Gong, Liutang and Heng-fu Zou (2002). Direct preferences for wealth, the risk premium puzzle, growth, and policy effectiveness. Journal of Economic Dynamics and Control 26: 243-270.

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156

Liutang Gong, Yulei Luo and Heng-fu Zou

Grinols, E. and S. Turnovsky (1993). Risk, the financial market, and macroeconomic equilibrium. Journal of Economic Dynamics and Control 17:1-36 Kurz, M. (1968). Optimal economic growth and wealth effects. International Economic Review 9:348-357. Levhari, D. and T. Srinivisan (1969). Optimal saving under uncertainty. Review of Economic Studies 36:153-163. Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica 46: 1426-1445. Mehra, R. and E. C. Prescott (1985). The Equity premium: A puzzle. Journal of Monetary Economy 15: 145-161. Merton, R.C.(1971). Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3: 373-413. Merton, R.C. (1973). An intertemporal capital asset pricing model. Econometrica 41: 867887. Merton, R. C. (1975). An asymptotic theory of growth under uncertainty. Review of Economic Studies 42: 375-393. Mirlees, J. A. (1965). Optimum accumulation under uncertainty. Working Paper. Obstfeld, M. (1994). Risk-taking, global diversification, and growth. The American Economic Review 84: 1310-29. Phelps, E. S. (1962). The accumulation of risk capital: A sequential utility analysis. Econometrica 30:729-743. Ramsey, F. (1928). A mathematical theory of saving. Economic Journal vol 38:543-559. Rebelo, S. (1991). Long-run policy analysis and long-run growth. Journal of Political Economy 99: 500-521. Smith, W.T. (2001). How does the spirit of capitalism affect stock market prices? Review of Financial Studies 14(4):1215-1232. Stapleton, R. and M. G. Subrahmanyam (1978). A multiperiod equilibrium asset pricing model. Econometrica 46(September):1077-1096. Sunderason, M. (1983). Constant absolute risk aversion preferences and constant equilibrium interest rates. Journal of Finance 38(1):205-212. Sunderason, M. (1984). Consumption and equilibrium interest rates in stochastic production economies. Journal of Finance 39(1):77-92. Turnovsky, S. (1993). Macroeconomic policies, growth, and welfare in a stochastic economy. International Economic Review 35: 953-981. Turnovsky, S. (2000). Methods of Macroeconomic Dynamics. 2nd ed. MIT Press. Xie, D. (1994). Divergence in economic performance: Transitional dynamics with multiple equilibria. Journal of Economic Theory 63: 97-112. Zou, H. (1994). The spirit of capitalism and long-run growth. European Journal of Political Economy 10: 279-93. Zou, H. (1995). The spirit of capitalism and savings behavior. Journal of Economic Behavior and Organization 28: 131-143.

In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 157-169

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 8

O N THE I NVESTMENT-U NCERTAINTY R ELATIONSHIP Kit Pong Wong∗ School of Economics and Finance, University of Hong Kong, Pokfulam Road, Hong Kong

Abstract This paper examines the investment-uncertainty relationship in a canonical real options model. We show that the critical lump-sum payoff of a project that triggers the exercise of the investment option exhibits a U-shaped pattern against the volatility of the project. This is driven by two opposing effects of an increase in the volatility of the project: (i) the usual positive effect on option value, and (ii) a negative effect on option value due to the upward adjustment in the discount rate. We further show that such a U-shaped pattern is inherited by the expected time to exercise the investment option. Thus, for relatively safe projects, greater uncertainty may in fact shorten the expected exercise time and thereby enhance investment. This is in sharp contrast to the negative investment-uncertainty relationship as commonly suggested in the extant literature.

JEL classification: D21; D81; G13 Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Keywords: Real options; Investment timing; Uncertainty

1.

Introduction

The investment-uncertainty relationship has been extensively studied in the literature, which by and large dictates a negative sign to such a relationship (see Caballero, 1991; Leahy and Whited, 1996). The purpose of this paper is to shed more light on this relationship in a canonical real options model of McDonald and Siegel (1986) and Dixit and Pindyck (1994). Viewing investment opportunities as perpetual American call options, firms endogenously devise their investment timing so as to maximize the option values. The optimal decision ∗

E-mail address: [email protected]. Tel.: (852) 2859-1044, fax: (852) 2548-1152 (Corresponding author)

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Kit Pong Wong

rule is that a project should be undertaken at the first instant when the value of the project reaches a critical level (the investment trigger). To gauge the effect of uncertainty on investment, we follow Wong (2007) to use the the expected time to exercise the investment option. If this measure decreases (increases) with the volatility of the project, one can reasonably infer a positive (negative) sign of the investment-uncertainty relationship. Following Sarkar (2000) and Wong (2007, 2008), we employ the single-factor intertemporal capital asset pricing model (CAPM) of Merton (1973a) to determine the risk-adjusted rate of return on the project. We show that the behavior of the investment trigger with respect to the volatility of the project is non-monotonic. When the volatility of the project goes up, the usual positive effect on option value (Merton, 1973b) makes waiting more beneficial. This lifts up the investment trigger. On the other hand, there is a negative effect on option value due to the upward adjustment of the discount rate in accord with the CAPM. This makes waiting more costly and thus pushes down the investment trigger. We show that the negative effect dominates (is dominated by) the positive effect for low (high) levels of uncertainty, thereby rendering a U-shaped pattern of the investment trigger against the volatility of the project. We further show that the expected time of investment inherits the U-shaped pattern of the investment trigger against the volatility of the project. Specifically, the positive effect on option value that calls for shortening the investment time dominates for relatively safe projects, while the negative effect on option value that calls for lengthening the investment time dominates for sufficiently risky projects. Thus, it is quite possible that greater uncertainty may in fact lure firms into making more investment through shortening the expected time to exercise the investment option, especially when projects are relatively safe. This is in sharp contrast to the negative investment-uncertainty relationship commonly found in the extant literature. The rest of this paper is organized as follows. The next section delineates the real options model. Section 3 derives the investment trigger and the value of the investment option. Sections 4 and 5 examine how the investment trigger and the expected time to exercise the investment option respond to an increase in the volatility of the project, respectively. The final section concludes.

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2.

The Model

Consider a canonical real options model a` la McDonald and Siegel (1986) and Dixit and Pindyck (1994). Time, indexed by t ≥ 0, is continuous and the horizon is infinite. Uncertainty is modeled by a complete probability space, (Ω, F, P ). At time 0, a firm owns the right to invest in a project that can be irreversibly undertaken at some endogenously chosen time, τ ≥ 0. The investment time, τ , is uncertain ex ante. Investing in the project costs the firm I > 0, which is instantly paid at time τ . In return, the firm receives a lump-sum payoff, Vτ , so that the net payoff of the project is Vτ − I at time τ . We assume that the lump-sum payoff process, {Vt : t ≥ 0}, is governed by the following geometric Brownian motion: dVt = αVt dt + σVt dZt ,

(1)

On the Investment-Uncertainty Relationship

159

where {Zt : t ≥ 0} is a standard Wiener process defined on (Ω, F, P ), and α > 0 and σ > 0 are the positive drift rate (expected growth rate) and volatility (standard derivation) per unit of time, respectively. The initial value of the lump-sum payoff process, V0 > 0, is known at time 0. Investing in the project is analogous to exercising a perpetual American call option in that the firm has the right, but not the obligation, to invest at some future time to be optimally chosen by the firm. Let F (Vt ) be the value of the investment option at time t ≥ 0. The firm optimally exercises the investment option at the investment time, τ , when the lump-sum payoff, Vτ , reaches a threshold level, V ∗ , from below at the first instant. We refer to V ∗ as the investment trigger. Following Sarkar (2000) and Wong (2007, 2008), we assume that the underlying asset, i.e., the lump-sum payoff, for the investment option can be completely spanned by financial assets traded in the market, where risk-adjusted rates of return on financial assets are determined by the single-factor intertemporal capital asset pricing model (CAPM) of Merton (1973a). Let Yt be the price of an asset or a portfolio of assets that is perfectly correlated with Vt . Denote by ρ > 0 as the correlation of Yt with the market portfolio.1 Then, the price process, {Yt : t ≥ 0}, evolves over time according to the following geometric Brownian motion: dYt = (r + λρσ)Yt dt + σYt dZt , (2) where r > 0 is the constant instantaneous riskless rate of interest, λ > 0 is the constant market price of risk per unit of time, and r + λρσ is the risk-adjusted rate of return on Yt according to the CAPM. Let δ = r + λρσ − α > 0 be the convenience yield or return shortfall on Vt .2

3.

Solution to the Model

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For all Vt ≥ V ∗ , the investment option is immediately exercised so that F (Vt ) = Vt − I. On the other hand, for all Vt < V ∗ , the firm keeps the investment option alive. To derive F (Vt ) in this case, we construct the following dynamic portfolio: (i) Hold the investment option that is worth F (Vt ), and (ii) go short n units of the asset or portfolio of assets that completely spans Vt . The value of this dynamic portfolio is F (Vt ) − nYt . The total return from holding the portfolio over a short time interval, dt, is given by dF (Vt ) − ndYt 1 = F ′ (Vt )dVt + F ′′ (Vt )(dVt )2 − n[(r + λρσ)Yt dt + σYt dZt ] 2   1 2 2 ′′ ′ = αVt F (Vt ) + σ Vt F (Vt ) − n(r + λρσ)Yt dt + [Vt F ′ (Vt ) − nYt ]σdZt , 2

(3)

where the first equality follows from Ito’s Lemma and Eq. (2), and the second equality follows from Eq. (1) and (dVt )2 = σ 2 Vt2 dt. Substituting n = Vt F ′ (Vt )/Yt into Eq. (3) 1

Since most assets would have values that are positively correlated with the market portfolio, we do not consider the case that ρ ≤ 0. 2 The assumption that δ > 0 is called for to ensure a finite investment trigger at the optimum. See Eq. (14).

160 yields

Kit Pong Wong   1 2 2 ′′ Vt F ′ (Vt ) ′ dYt = σ Vt F (Vt ) − δVt F (Vt ) dt, dF (Vt ) − Yt 2

(4)

where δ = r + λρσ − α. Inspection of Eq. (4) reveals that the portfolio with n = Vt F ′ (Vt )/Yt is riskless and thus we must have   1 2 2 ′′ ′ σ Vt F (Vt ) − δVt F (Vt ) dt = r[F (Vt ) − Vt F ′ (Vt )]dt, (5) 2 to rule out arbitrage opportunities. Eliminating dt on both sides of Eq. (5) and rearranging terms yields 1 2 2 ′′ σ Vt F (Vt ) + (r − δ)Vt F ′ (Vt ) − rF (Vt ) = 0. (6) 2 Thus, F (Vt ) must satisfy Eq. (6) for all Vt ∈ (0, V ∗ ), where V ∗ is a free boundary to be optimally chosen by the firm. Eq. (6) is a second-order linear homogeneous ordinary differential equation. The general solution to Eq. (6) is the sum of two powers: F (V ) = A1 V β1 + A2 V β2 ,

(7)

where A1 and A2 are constants to be determined, and β1 and β2 are the positive and negative roots, respectively, for the following fundamental quadratic equation:3 1 2 σ β(β − 1) + (r − δ)β − r = 0. 2

(8)

The two constants, A1 and A2 , and the investment trigger, V ∗ , are determined using considerations that apply at the boundaries of the region, (0, V ∗ ). There are three boundary conditions: F (0) = 0, (9) F (V ∗ ) = V ∗ − I,

(10)

F ′ (V ∗ ) = 1.

(11)

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and The first boundary condition, Eq. (9), simply reflects the fact that zero is an absorbing barrier for the geometric Brownian motion defined in Eq. (1). The second boundary condition, Eq. (10), is the value-matching condition such that the value of the investment option is equal to the net payoff of the project at the investment time, τ . The third boundary condition, Eq. (11), is the smooth-pasting condition, or high-contact condition, such that the investment trigger, V ∗ , is the one that maximizes the value of the investment option.4 For Eq. (9) to hold, Eq. (7) implies that A2 = 0. Thus, we can ignore the negative solution for β in Eq. (8) so that we can simply write β1 = β, where s  1 r−δ 1 r − δ 2 2r β= − + 2. + − (12) 2 σ2 2 σ2 σ 3

Substituting Eq. (7) into Eq. (6) and simplifying the resulting equation yields Eq. (8). Shackleton and Sødal (2005) show that smooth pasting implies rate of return equalization between the option and the levered position that results from exercise. 4

On the Investment-Uncertainty Relationship

161

Note that Eq. (8) can be written as   1 (β − 1) σ 2 β + r = δβ. 2

(13)

Since β > 0 and δ > 0, it is evident from Eq. (13) that β > 1. Using the remaining two boundary conditions, Eqs. (10) and (11), we solve the investment trigger, V ∗ , and the value of the investment option at time 0, F (V0 ), in the following proposition. Proposition 1. The investment trigger, V ∗ , is given by   β ∗ V = I, β−1 and the value of the investment option at time 0 is given by   (V ∗ − I)(V0 /V ∗ )β if V0 < V ∗ , F (V0 ) =  V0 − I if V0 ≥ V ∗ ,

(14)

(15)

where β is defined in Eq. (12). Furthermore, for all V0 < V ∗ , F (V0 ) is strictly convex and greater than V0 − I. Proof. Using Eq. (7) with A2 = 0 and β1 = β defined in Eq. (12), we can write Eqs. (10) and (11) as A1 V ∗β = V ∗ − I, (16) and A1 βV ∗β−1 = 1,

(17)

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respectively. Multiplying β to Eq. (16) and V ∗ to Eq. (17) and subtracting the resulting equations yields Eq. (14). Substituting A2 = 0, β1 = β, and Eq. (16) into Eq. (7) yields Eq. (15). Note that   d V −I βI − (β − 1)V (β − 1)(V ∗ − V ) = = , (18) dV Vβ V β+1 V β+1 where we have used Eq. (14). Thus, if V0 < V ∗ , Eq. (18) implies that V0 − I V∗−I > . V ∗β V0β

(19)

 β V0 > V0 − I. F (V0 ) = (V − I) V∗

(20)

Rewriting inequality (19) yields ∗

Furthermore, for all V0 < V ∗ , we have F ′′ (V0 ) =

β(β − 1)(V ∗ − I)V0β−2 > 0, V ∗β

(21)

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Kit Pong Wong

since β > 1. Hence, Eqs. (20) and (21) imply that F (V0 ) is strictly convex and greater than V0 − I for all V0 < V ∗ .  Since β > 1, Eq. (14) implies that V ∗ > I. That is, the firm finds it optimal to exercise the investment option only when the net payoff of the project is sufficiently positive. The term, (V0 /V ∗ )β , in Eq. (15) can be interpreted as the stochastic discount factor that accounts for both the timing and the probability of one dollar received at the first instant when the investment trigger, V ∗ , is reached from below. If V0 ≥ V ∗ , the investment option is immediately exercised at time 0 so that F (V0 ) = V0 − I. Otherwise, the firm keeps the investment option alive until the investment trigger, V ∗ , is reached from below at the first instant. In this case, F (V0 ) > V0 − I. Figure 1 depicts the value of the investment option at time 0, F (V0 ), as a function of the initial value of the lump-sum payoff, V0 . F (V0) 6

V∗−I

-

0

I

V0

V∗

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Figure 1.

4.

The Trigger-Uncertainty Relationship

In this section, we examine the effect of uncertainty on the investment trigger, V ∗ . To this end, we follow Sarkar (2000) and Wong (2007, 2008) to refer to an increase in uncertainty as an increase in σ, taking all other parameters, r, λ, ρ, and α, as constants. In this case, the increase in σ has a systematic risk component that affects the convenience yield, δ. In accord with the CAPM, we have dδ/dσ = λρ > 0. In contrast, McDonald and Siegel (1986) and Dixit and Pindyck (1994) consider another type of increased uncertainty in which the convenience yield, δ, is held fixed when σ varies, i.e., dδ/dσ = 0. In this regard, the increase in σ has only an idiosyncratic risk component. While we follow the approach of Sarkar (2000) and Wong (2007, 2008), our analysis is readily extended to the case of McDonald and Siegel (1986) and Dixit and Pindyck (1994) by setting dδ/dσ = 0.

On the Investment-Uncertainty Relationship

163

Differentiating Eq. (8) with respect to σ yields     dβ β 2β 2 dδ dδ = 2 − σ(β − 1) = 2 2 − σ(β − 1) , dσ σ (β − 1/2) + r − δ dσ σ β + 2r dσ

(22)

where the second equality follows from Eq. (8). Differentiating Eq. (14) with respect to σ yields   dβ I 2βV ∗ dδ dV ∗ =− = σ(β − 1) − , (23) dσ (β − 1)2 dσ (β − 1)(σ 2 β 2 + 2r) dσ where the second equality follows from Eqs. (14) and (22). If dδ/dσ = 0, it is evident from Eq. (23) that dV ∗ /dσ > 0 for all σ > 0. However, if dδ/dσ = λρ > 0, the triggeruncertainty relationship is no longer monotonic, as is shown in the following proposition. √ Proposition 2. If dδ/dσ = λρ > 0, there exists a unique point, σ ∗ ∈ (0, 2α), defined by s  2r − 2α + λ2 ρ2 2r − 2α + λ2 ρ2 2 ∗ + 2α − , (24) σ = 2λρ 2λρ such that dV ∗ /dσ < (>) 0 for all σ < (>) σ ∗ . Proof. Eq. (23) implies that dV ∗ /dσ has the same sign as that of σ(β − 1) − λρ. Using Eq. (12) and δ = r + λρσ − α, we have s 2 σ 2 − 2α 2α + σ 2 σ(β − 1) − λρ = + λρ + 2r − . (25) 2σ 2σ Note that s 2 2 s 2 2  2α + σ 2 2α + σ 2 σ − 2α σ − 2α + λρ + 2r − + λρ + 2r + 2σ 2σ 2σ 2σ

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=



σ 2 − 2α + λρ 2σ

2

+ 2r −



2α + σ 2 2σ

2

1 [λρσ 2 + (2r − 2α + λ2 ρ2 )σ − 2αλρ]. (26) σ Inspection of Eqs. (25) and (26) reveals that σ(β − 1) − λρ has the same sign as that of λρσ 2 + (2r − 2α + λ2 ρ2 )σ − 2αλρ, which is negative or positive depending on whether σ is lower or higher than σ ∗ , respectively, where σ ∗ is defined in Eq. (24). It is evident from Eq. (24) that σ ∗ > 0. Note that =



2r − 2α + λ2 ρ2 2λρ

2

+ 2α
0 in accord with the CAPM. The upward adjustment of the discount rate reduces the present value of the investment option. This makes waiting less attractive and thus the firm is induced to lower the investment trigger. The two effects act against each other. When there is relatively little uncertainty, it is evident from Eq. (28) that the positive effect is at best second order. The negative effect, on the other hand, is always first order because the risk-adjusted rate of return on the project is linear in σ in accord with the CAPM. When uncertainty becomes greater, the positive effect dominates the negative effect because the significance of the positive effect grows exponentially with σ while that of the negative effect grows only linearly with σ. This explains why V ∗ has a U-shaped pattern against σ with the unique minimum attained at σ ∗ .

5.

The Investment-Uncertainty Relationship

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In this section, we examine the sign of the investment-uncertainty relationship in the context of our real options model. To have an interesting case, we assume that V0 < V ∗ so that the investment option is not immediately exercised at time 0. Let Xt = lnVt . Using Ito’s Lemma, Eq. (1) implies that Xt is governed by the following arithmetic Brownian motion:   σ2 dXt = α − dt + σdZt . (29) 2 Let φ(X, τ ) be the probability density function of the investment time, τ , at which the lump-sum payoff of the project reaches the investment trigger, V ∗ , from the initial value, V0 , at the first instant, and X = ln(V ∗ /V0 ). Define L(X, θ) as the Laplace transform, or characteristic function, of φ(X, τ ): Z ∞ exp(−θτ )φ(X, τ ) dτ. (30) L(X, θ) = 0

We solve φ(X, τ ) and L(X, θ) in the following proposition. Proposition 3. The probability density function of the investment time, τ , at which the lump-sum payoff of the project reaches the investment trigger, V ∗ , from the initial value, V0 , at the first instant is given by     2  X 1 σ2 φ(X, τ ) = √ exp − 2 X − α − τ , (31) 2σ τ 2 σ 2πτ 3

On the Investment-Uncertainty Relationship and the Laplace transform of φ(X, τ ) is given by s     σ2 X σ2 2 2 + 2σ θ − α − L(X, θ) = exp − 2 , α− σ 2 2

165

(32)

where X = ln(V ∗ /V0 ) and V0 < V ∗ . Proof. We know that φ(X, τ ) must satisfy the forward Kolmogorov, or Fokker-Planck, equation of motion:   1 2 ∂ 2 φ(X, τ ) σ 2 ∂φ(X, τ ) ∂φ(X, τ ) − α− σ − = 0. (33) 2 ∂X 2 2 ∂X ∂τ Taking Laplace transforms of Eq. (33) term by term gives Z 1 2 ∞ ∂ 2 φ(X, τ ) σ exp(−θτ ) dτ 2 ∂X 2 0 Z ∞  Z ∞ ∂φ(X, τ ) ∂φ(X, τ ) σ2 exp(−θτ ) exp(−θτ ) dτ − dτ = 0. − α− 2 ∂X ∂τ 0 0

Using Eq. (30), we can write Eq. (34) as   σ 2 ∂L(X, θ) 1 2 ∂ 2 L(X, θ) σ − α− − θL(X, θ) = 0, 2 ∂X 2 2 ∂X

(34)

(35)

where the last term on the left-hand side of Eq. (35) follows from integrating by parts. Eq. (35) is a second-order linear differential equation with constant coefficients. The general solution to Eq. (35) is the sum of two exponentials: L(X, θ) = B1 exp[γ1 (θ)X] + B2 exp[γ2 (θ)X],

(36)

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where B1 and B2 are constants to be determined, and γ1 (θ) and γ2 (θ) are the positive and negative roots, respectively, for the following quadratic equation:5   1 2 2 σ2 σ γ − α− γ − θ = 0. (37) 2 2 The two constants, B1 and B2 , are determined using two appropriate boundary conditions. First, note that Z ∞

φ(X, τ ) dτ ≤ 1,

L(X, θ) ≤ L(X, 0) =

(38)

0

for all X > 0. For Eq. (38) to hold, Eq. (36) implies that B1 = 0 or else L(∞, θ) would be unbounded. Second, at X = 0, i.e., at V0 = V ∗ , investment occurs immediately. In this case, φ(0, τ ) becomes the Dirac delta function so that L(0, θ) = exp(−θ × 0) = 1. It then follows from Eq. (36) with B1 = 0 that B2 = 1. Solving Eq. (37) for γ2 (θ) yields  s   σ2 1 σ2 2 2 + 2σ θ − α − . (39) γ2 (θ) = − 2 α− σ 2 2 5

Substituting Eq. (36) into Eq. (35) and simplifying the resulting equation yields Eq. (37).

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Kit Pong Wong

Substituting B1 = 0, B2 = 1, and Eq. (39) into Eq. (36) yields Eq. (32). Inversion of Eq. (32) yields Eq. (31).  Following Wong (2007) and Guti´errez (2007), we adopt the following two definitions to characterize the sign of the investment-uncertainty relationship. Definition 1. Given that the probability of eventual investment is equal to one, the sign of the investment-uncertainty relationship is said to be positive (negative) if, and only if, the expected time of investment decreases (increases) with an increase in uncertainty. Definition 2. Given that the probability of eventual investment is less than one, the sign of the investment-uncertainty relationship is said to be positive (negative) if, and only if, the probability of eventual investment increases (decreases) with an increase in uncertainty. The probability of eventual investment, Π, is given by Z ∞ Π= φ(X, τ ) dτ = L(X, 0).

(40)

0

where the second√equality follows from √ Eq. (30). Eqs. (32) and (40) imply that Π = 1 if, and only if, σ ≤ 2α. For all σ > 2α, Eqs. (32) and (40) imply that   ∗   1−2α/σ2  V V0 2α , = Π = exp − 1 − 2 ln σ V0 V∗ 

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which is less than one. The expected time of investment, E(τ ), is given by Z ∞ ∂L(X, θ) τ φ(X, τ ) dτ = − E(τ ) = , ∂θ 0 θ=0

(41)

(42)

where the second equality follows from Eq. (30). Using Eqs. (32) and (42), we have   ∗  V σ 2 −1 E(τ ) = α − ln , (43) 2 V0 √ which is well-defined only when σ < 2α, i.e., when Π = 1. √ For all σ < 2α, we differentiate Eq. (43) with respect to σ to yield     ∗    V σ 2 −2 dE(τ ) σ 2 1 dV ∗ = α− σln . (44) + α− dσ 2 V0 2 V ∗ dσ √ For all σ > 2α, we differentiate Eq. (41) with respect to σ to yield   ∗    4α V dΠ 2α 1 dV ∗ = −Π 3 ln + 1− 2 . (45) dσ σ V0 σ V ∗ dσ If dδ/dσ = 0, we know from Eq. (23) that dV ∗ /dσ > 0 for all σ > 0. Eqs. (44) and (45) then imply that dE(τ )/dσ > 0 and dΠ/dσ < 0, thereby rendering a positive investmentuncertainty relationship according to Definitions 1 and 2. However, if dδ/dσ = λρ > 0, the

On the Investment-Uncertainty Relationship

167

investment-uncertainty relationship becomes non-monotonic, as is shown in the following proposition. Proposition 4. If dδ/dσ = λρ > 0, there exists a unique point, σ o ∈ (0, σ ∗ ), implicitly defined by dE(τ ) = 0, (46) dσ σ=σo √ such that dE(τ )/dσ < √0 for all σ < σ o , dE(τ )/dσ > 0 for all σ ∈ (σ o , 2α), and dΠ/dσ < 0 for all σ > 2α. ∗ /dσ > 0 for all σ > σ ∗ . Thus, Eq. (44) Proof. From Proposition 2, we know that dV √ implies that√ dE(τ )/dσ > 0 for all σ ∈ [σ ∗ , 2α) and Eq. (45) implies that dΠ/dσ < 0 for all σ > 2α. Define the expression inside the squared brackets on the right-hand side of Eq. (44) as M :  ∗   V σ 2 1 dV ∗ M = σln . (47) + α− V0 2 V ∗ dσ

Differentiating Eq. (47) with respect to σ yields  ∗      dM σ2 V 1 d2 V ∗ 1 dV ∗ 2 = ln + α− − . dσ V0 2 V ∗ dσ 2 V ∗ dσ

Differentiating Eq. (23) with respect to σ yields  2 dβ d2 β 2I I d2 V ∗ = . − dσ 2 (β − 1)3 dσ (β − 1)2 dσ 2 Substituting Eqs. (14), (23), and (49) into Eq. (48) yields  ∗    2  V 2β − 1 dβ 1 d2 β dM σ2 − = ln . + α− dσ V0 2 β 2 (β − 1)2 dσ β(β − 1) dσ 2

(48)

(49)

(50)

Differentiating Eq. (22) with respect to σ yields

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8β 3 [λρ − σ(β − 1)][λρ − σ(2β − 1)] d2 β = dσ 2 (σ 2 β 2 + 2r)2 −

2β 2 (β − 1) 8β 5 σ 2 [λρ − σ(β − 1)]2 − . σ 2 β 2 + 2r (σ 2 β 2 + 2r)3

(51)

Substituting Eqs. (22) and (51) into Eq. (50) yields  ∗   2 dM V 4β [λρ − σ(β − 1)]2 σ2 = ln + α− dσ V0 2 (β − 1)2 (σ 2 β 2 + 2r)2  2β 8β 4 σ 2 [λρ − σ(β − 1)]2 8β 3 σ[λρ − σ(β − 1)] + 2 2 + , + (β − 1)(σ 2 β 2 + 2r)2 σ β + 2r (β − 1)(σ 2 β 2 + 2r)3

(52)

which is unambiguously positive for all σ ∈ (0, σ ∗ ) since in this case we have σ(β − 1) − λρ < 0. It then follows from Eq. (44) that dE(τ )/dσ is strictly increasing in σ for all

168

Kit Pong Wong

σ ∈ (0, σ ∗ ). Eq. (8) implies that β → r/α as σ → 0. Taking limit on both sides of Eq. (44) as σ → 0 therefore yields lim

σ→0

λρ dE(τ ) =− < 0. dσ α(r − α)

(53)

∗ Since dE(τ √ )/dσ is strictly increasing in σ for all σ ∈ (0, σ ) and dE(τ )/dσ o> 0 for ∗all ∗ σ ∈ [σ , 2α), we conclude from Eq. (53) that there exists a unique point, σ ∈ (0, σ ), implicitly defined √ in Eq. (46), such that dE(τ )/dσ < 0 for all σ ∈ (0, σ o ) and dE(τ )/dσ > o 0 for all σ ∈ (σ , 2α). 

To see the intuition of Proposition 4, we use Eq. (28) to recast Eq. (44) as dE(τ ) = dσ



σ2 α− 2

−2  ∗     V σ 2 −1 2β ln + α− σ V0 2 σ 2 β 2 + 2r

  2β dδ σ 2 −1 − α− . 2 (β − 1)(σ 2 β 2 + 2r) dσ

(54)

Inspection of Eq. (54) reveals two effects that govern the expected time of investment when the volatility of the project goes up. As in the previous section, the first term on the right-hand side of Eq. (54) captures the positive effect while the second term captures the negative effect. Proposition 4 states that greater uncertainty may in fact lure the firm into making more investment through shortening the expected time of investment, especially when the project is relatively safe (i.e., σ < σ o ). When the project is sufficiently risky (i.e., σ > σ o ), the usual negative investment-uncertainty relationship as suggested in the extant literature prevails. This non-monotonic investment-uncertainty relationship is driven by the U-shaped pattern of the investment trigger against the volatility of the project. Specifically, the negative effect that calls for shortening the investment time dominates for relatively safe projects, while the positive effect that calls for lengthening the investment time dominates for sufficiently risky projects.

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6.

Conclusion

This paper has examined the investment-uncertainty relationship in a canonical real options model of McDonald and Siegel (1986) and Dixit and Pindyck (1994) with a caveat: Risk-adjusted rates of return on projects are determined by the single-factor intertemporal capital asset pricing model (CAPM) of Merton (1973a). We have shown that the critical lump-sum payoff of a project, i.e., the investment trigger, that triggers the exercise of the investment option exhibits a U-shaped pattern against the volatility of the project. This Ushaped pattern is driven by two opposing effects. When the volatility of the project goes up, the usual positive effect on option value (Merton, 1973b) makes waiting more beneficial. This lifts up the investment trigger. On the other hand, there is a negative effect on option value due to the upward adjustment of the discount rate in accord with the CAPM. This makes waiting more costly and thus pushes down the investment trigger. We have shown that the negative effect dominates (is dominated by) the positive effect for low (high) levels

On the Investment-Uncertainty Relationship

169

of uncertainty. We have further shown that the U-shaped pattern of the investment trigger against the volatility is inherited by the investment-uncertainty relationship. For relatively safe projects, greater uncertainty may in fact shorten the expected time to exercise the investment option and thereby lure firms into making more investment, which is in sharp contrast to the negative investment-uncertainty relationship as commonly suggested in the extant literature.

References

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Caballero, R. J. (1991). On the sign of the investment-uncertainty relationship. American Economic Review, 81, 279–288. Dixit, A. K., & Pindyck, R. S. (1994). Investment under Uncertainty. Princeton, NJ: Princeton University Press. ´ (2007). Devaluating projects and the investment-uncertainty relationship. Guti´errez, O. Journal of Economic Dynamics and Control, 31, 3881–3888. Leahy, J. V., & Whited, T. M. (1996). The effect of uncertainty on investment: Some stylized facts. Journal of Money, Credit and Banking, 28, 64–83. McDonald, R. L., & Siegel, D. (1986). The value of waiting to invest. Quarterly Journal of Economics, 101, 707–727. Merton, R. C. (1973a). An intertemporal capital asset pricing model. Econometrica, 41, 867–887. Merton, R. C. (1973b). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Sarkar, S. (2000). On the investment-uncertainty relationship in a real options model. Journal of Economic Dynamics and Control, 24, 219–225. Shackleton, M. B., & Sødal, S. (2005). Smooth pasting as rate of return equalization. Economics Letters, 89, 200–206. Wong, K. P. (2007). The effect of uncertainty on investment timing in a real options model. Journal of Economic Dynamics and Control, 31, 2152–2167. Wong, K. P. (2008). Does market demand volatility facilitate collusion? Economic Modelling, 25, in press.

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In: Economic Dynamics... Editor: Chester W. Hurlington, pp. 171-197

ISBN 978-1-60456-911-7 c 2008 Nova Science Publishers, Inc.

Chapter 9

R EALIZED VOLATILITY AND C ORRELATION E STIMATORS UNDER N ON -G AUSSIAN M ICROSTRUCTURE N OISE Amir Safari1 , Wei Sun2∗, Detlef Seese1 , Svetlozar Rachev2,3,4† 1 Institute of Applied Informatics and Formal Description Methods University of Karlsruhe, Germany 2 Institute for Statistics and Mathematical Economics University of Karlsruhe, Germany 3 Department of Statistics and Applied Probability University of California, Santa Barbara, USA 4 Chief-Scientist of FinAnalytica INC.

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Abstract Realized volatility and correlation estimators suffer from microstructure noise, resulting in biased and imprecise estimators. This suggests that estimators do not converge for high-frequency levels, where noise especially exists. To solve the problem of noise, some approaches have been suggested in literature. In particular, the subsampling and averaging approach works well. Moreover, the realized volatility literature usually assumes Gaussian microstructure noise despite the fact that noise in real world financial markets does not follow a Gaussian process. In this article, we suggest what we believe to be more realistic microstructure noise processes. The fractional stable noise that we suggest is the most realistic process compared to other noise processes investigated in our simulations. Therefore, realized volatility and correlation estimators should be unbiased and converge faster under this type of noise process. Empirically, some of the estimators exhibit heavy tails and some exhibit dynamic behaviors. This is especially so in the case of absolute based realized correlations which exhibit negative asymmetry in the dependence structure between minute by minute frequency data of CAC and FTSE stock indices. ∗

W. Sun’s research was supported by grants from the Deutschen Forschungsgemeinschaft (DFG). E-mail address: [email protected]. Corresponding author: Prof. Dr. S. Rachev, Institute for Statistics and Mathematical Economics, University of Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany. S. Rachev’s research was supported by grants from Division of Mathematical, Life and Physical Science, College of Letters and Science, University of California, Santa Barbara, and the Deutschen Forschungsgemeinschaft (DFG). †

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Key Words: Fractional Gaussian noise, Fractional stable noise, Microstructure noise, Realized volatility and correlation, Stable noise

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1.

Introduction

Asset return volatilities are central in finance, in particular in asset pricing, portfolio allocation, and market risk measurement. Therefore the field of financial econometrics devotes considerable attention to time-varying volatility and associated tools for its measurement, modeling and forecasting. On the other hand, correlations are critical inputs for many of the common tasks of financial management. Hedges require estimates of the correlation between the assets in the hedge. If the correlations and volatilities are changing, then the hedge ratios should be adjusted to account for the most recent information. Asset allocation and risk assessment also rely on correlations. The most popular approach to obtain volatility estimates is using statistical models that have been proposed in literature on autoregressive conditional heteroscedasticity (ARCH) and stochastic volatility. Another method of extracting information about volatility is to formulate and apply economic models that link the information contained in options to the volatility of the underlying asset. This approach is based on the market’s assessment of future volatility. All these approaches have the following in common: (1) the resulting volatility measures are only valid under the specific assumptions of the models used and (2) they are generally uncertain which or whether any of these specifications provide a good description of actual volatility. Meanwhile, interest in the field of high-frequency finance is growing. Exploiting high-frequency data, volatility and correlation can be measured more accurately in a model-free approach. The realized volatility approach is claimed to be consistent under general nonparametric conditions. In other words, these types of measures provide more precise ex-post observations of the actual volatility compared to the traditional sample variances based on daily or coarser frequency data. In fact, sampling as often as possible would, in theory, produce exact estimates of the true variance in the limit. However, this is not the practical case. The presence of market microstructure noise in high-frequency financial data complicates the estimation of financial volatility and correlation making the approach unreliable. There is a considerable bias in estimation at the higher frequency due to intervention of noise. While the realized volatility approach suggests sampling at the highest possible frequency to attain the highest precision, market microstructure frictions exist at the highest levels of frequency. For this reason, sparse sampling or lower frequencies have been recommended to reduce market microstructure contamination. For example, optimal sampling schemes have been investigated by Bandi and Russell [Ban05b] and A¨ıt-Sahalia et al. [Ait05]. Meanwhile, researchers have investigated some methods to cope with the problem at the highest frequency in the presence of noise. These methods include a kernel-based correction by Zhou [Zho96], a moving average filter by Maheu and McCurdy [Mah02], an autoregressive filter by Bollen and Inder [Bol02], and a subsampling and averaging approach by Zhang et al. [Zha05] However, in the literature on realized volatility, market microstructure noise is usually assumed to follow an i.i.d process. For example, Barndorff-Nielsen, Hansen, Lunde and Shephard [Bar04a], Zhang et al. [Zha05], Bandi and Russell [2005b], and Hansen and Lunde [2006] assume that noise follows a Gaussian process.

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Realized Volatility and Correlation Estimators...

173

As Rachev and Mittnik [Rac00] explain, although earlier theories build on Bachelier’s original theory on speculation assumed the normal distribution, the excess kurtosis found in the studies by Mandelbrot and Fama in the 1960s led them to reject the normal assumption and propose the stable Paretian distribution as a statistical model for asset returns. In subsequent years, a number of empirical investigations supported this conjecture. See, for example, Mittnik, Rachev, and Paolella [Mit98]. The implication of rejecting the random walk hypothesis is that researchers must accept that returns in financial markets are not independent but instead exhibit trends. Sun, Rachev, and Fabozzi [Sun07] argue that in addition to the empirical evidence, there are theoretical arguments that have been put forth for rejecting both the Gaussian assumption and the random walk assumption. One of the most compelling arguments against the Gaussian random walk assumption is that markets exhibit a fractal structure. Samorodnitsky and Taqqu [Sam94] demonstrate that the properties of some self-similar processes can be used to model financial markets that are characterized as being non-Gaussian and non-random walk. Such financial markets have been stylized by long-range dependence, volatility clustering, and heavy tailedness. A distribution that is rich enough to encompass those stylized facts exhibited in return data is the stable distribution. Long-range dependence, self-similar processes, and stable distribution are very closely related [Sun07]. In an empirical study comparing 27 German stocks included in the DAX, Sun, Rachev, and Fabozzi [Sun07] conclude that an ARMA-GARCH model assuming a fractional stable noise outperforms other ARMA-GARCH models assuming independent and identically distributed (i.i.d.), stable, generalized Pareto, generalized extreme value, and fractional Gaussian noises. These results suggest that a non-Gaussian assumption about microstructure noise seems more realistic. Indeed, it seems that the best way of dealing with market microstructure noise may depend on the real world types and properties of noise. Inspired by Sun, Rachev, and Fabozzi [Sun07], this article examines some volatility and correlation estimators under different assumptions about microstructure noise. In particular, the impacts of different assumptions about microstructure noise including i.i.d. or white noise, stable noise, fractional Gaussian noise, and fractional stable noise on accuracy and especially on the bias in estimation of volatility and correlation are investigated and compared. The rest of the article is structured as follows: In section 2, assumptions about price, return, and microstructure noise processes in financial markets are made and then different volatility and correlation estimators are explained. In section 3, self-similar processes of interest are described and some simulation algorithms for generating noise are identified. Behavior of realized volatility and correlation estimators under different real noise processes are investigated via Monte Carlo simulations in section 4. Using 1 minute frequency data for the FTSE and CAC index series, in section 5 we empirically consider the behavior of the estimators. Section 6 concludes and discusses some issues.

2.

Realized Power Volatility and Correlation

To fix the notations, assume that p denotes a price process. The logarithmic price y =log p can be directly observed so that

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y = y ∗ + u,

(1)

where y ∗ denotes the logarithmic efficient price of an asset, i.e., the price that would exist in the absence of market microstructure noise, and u denotes a microstructure noise in the observed logarithmic price as induced by, for instance, asymmetries in information, transaction costs, price discreteness and bid-ask bounce effects. A certain time period t (i.e. one day) is fixed and availability of n high-frequency compounded prices over t is assumed. Given Equation (1), we can readily define continuously returns over any intraperiod interval of length δ = nt and write yti+1 − yti = yt∗i+1 − yt∗i + uti+1 − uti , or rti = rt∗i + ǫti ,

(2)

where rti is a return on day t at time i, and where t = 1, ..., T and i = 1, ..., n. Here n is the number of intraday observations. The following two assumptions are imposed on the price process and market microstructure effects. Price process assumptions: The logarithmic efficient price process, y ∗ , is assumed to be a continuous stochastic volatility semimartingale. More precisely, 1: the logarithmic efficient price process

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yt∗ = αt + mt ,

(3)

αt (with α0 = 0) is a continuous drift process of finite Rvariation defined as Rwhere t+1 t+1 φ(s)ds and mt is a continuous local martingale defined as t σ(s)dW s, with t {Wt : t ≥ 0} denoting a standard Brownian motion. 2: the spot volatility process, σt , is Rthe c´adl´ag process and bounded away from zero. t+1 3: the integrated variance process t σ p (s)ds is bounded, almost surely, for all t < ∞. The price process assumed in (3) is consistent with the asset pricing theory and suggests that the efficient return process evolves over time as a stochastic volatility martingale difference plus an adapted process of finite variation. The stochastic spot volatility can display jumps, diurnal effects, high-persistence (possibly of the long memory type), and nonstationarities. In addition, leverage effects (i.e., dependence between σ and the Brownian motion W ) are allowed. Microstructure noise assumptions: For the microstructure frictions, it is supposed that: ′ 1: microstructure noise in the price process, uti , follows the fractional stable noise. ′ ′ 2: uti are independent of the yt∗i for all i and all n. In general, the characteristics of the noise returns, ǫ’s, may depend on the sampling frequency.

Realized Volatility and Correlation Estimators...

2.1.

175

Volatility Estimators

For the decomposition of continuous logarithmic process (3), the integral of the instantaneous variances over the day, that is,

IP V =

t+1

Z

σ p (s)ds,

(4)

t

where the power or order p is a positive value, provides an ex post measure of the true, latent or Integrated Power Variance (IPV) process associated with day t. In the literature, the special case where p = 2, such that

IV =

Z

t+1

σ 2 (s)ds,

(5)

t

is usually called Integrated Volatility (IV). By cumulating the intraday squared returns, as shown in Merton [Mer80], Andersen et al. [And98], Andersen et al. [And01a] and Andersen et al. [And01b], one can consistently approximate the integrated volatility in equation (5) to a higher precision as the number of intraday observations increases (n → d , denoted as ∞). In particular, one can obtain an estimate of the Realized Volatility, RV d= RV

Tn X

rt2i ,

(6)

ti

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over a period t, with 0 = t0 ≤ t1 ≤ ...tn = T and where i = 1, ..., n is ith intraday observation with an integer n. The basic concept of accumulative intraday squared returns has been extended by Barndorff-Nielsen and Shephard [Bar03a] to a wider class called realized power variation of order p, that is, the sums of absolute powers of increments of a d, process, RP d= RP

Tn X

|rti |p ,

(7)

ti

where i = 1, ..., n is ith intraday observation with an integer n and p, the power or order, is d , as a proxy, is supposed to a positive value. The quantity of Realized Power variation RP approximate the daily increments of the power variation of the semimartingale that drives the underlying logarithmic price process, i.e., Integrated Power Volatility (IPV) in (4). It is supposed to be a consistent and unbiased estimator for IPV, as the frequency increases or n → ∞. While (6) is simply a special case of (7), another well-known special case of (7) is called Realized Absolute volatility estimator (RA), as follows

d= RA

Tn X ti

|rti | ,

(8)

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Amir Safari, Wei Sun, Detlef Seese et al.

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where p = 1 with a fixed t. This estimator is also considered as a proxy for IPV in (4). Barndorff-Nielsen and Shephard [Bar03a] provided a limiting distribution theory for realized power variation. The theoretical validity of the convergence of the estimators depends considerably on the observability of the efficient or equilibrium price process. However, it is widely accepted that the equilibrium price process and, as a consequence, the equilibrium return data are contaminated by market microstructure effects. However, market microstructure noise, especially at very high-frequencies, puts the convergence and in particular the biasedness of the realized variation estimators into challenge [Zho96], [And02], [Zha05]. In practice, microstructure frictions may include for example bid-ask spreads, discreteness of the price grid, asymmetries in information, transaction costs, and lunch-time effects. For solving the problem of microstructure noise, several approaches have been proposed including for example a kernel-based correction introduced by Zhou [Zho96], an optimal sampling introduced by Bandi and Russell [Ban05a], a moving average filter introduced by Maheu and McCurdy [Mah02], an autoregressive filter introduced by Bollen and Inder [Bol02], and a subsampling and averaging approach introduced by Zhang et al. [Zha05]. It has been experimentally found by Ghysels and Sinko [Ghy06b] that the subsampling and averaging class of the estimators is the best volatility predictor of all microstructure noise correctors. Zhang et al. [Zha05] proposed the Two-Scale Realized Volatility estimator (TSRV), which encompasses the realized squared volatility estimators from two time scales. From d avg is obtained which is constructed based the returns on a slow time scale, an estimator RV d all is computed from the returns on a fast on subsampling and averaging procedure, and RV time scale using the latter as a means for bias-corrector of the estimator. Then the estimator T\ SRV is approximated by  n ¯ −1  d n ¯d  T\ SRV = 1 − RV avg − RV (9) all , n n 2 d all is equivalent to (6) and RV d avg = 1 PK P where RV k=1 ti ,ti+1 ∈g (k) rti where the K K number of samples are allocated to g subgrids. The estimator T\ SRV is then consistent and unbiased for integrated volatility (5). Advocating subsampling and averaging approach, Safari and Seese [Saf07] extended the realized power variation (7) to Two-Scale realized Power Volatility (TSPV) estimator which is a consistent and unbiased estimator for IPV in (4) under microstructure noise and at the same time is somewhat immune against large values or jumps compared to TSRV estimator particularly when p = 1. The estimator T\ SP V , which is a generalized case of T\ SRV , is approximated by  n ¯d  n ¯ −1  d RP avg − RP T\ SP V = 1 − all , n n

(10)

d all is equivalent to (7) and RP d avg is computed by where RP K 1 X d RP avg = K

X

k=1 ti ,ti+1 ∈g (k)

|rti |p .

(11)

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. To understand subsampling procedure used here, In (9) and (10) we have n ¯ = n−K+1 K the full grid of G arrival times, G = {t0 , ..., tn }, is first defined and then it is partitioned into K nonoverlapping subgrids g (k) with k = 1, ..., K. The first subgrid starts from t0 and takes every Kth arrival time, i.e., g (1) = (t0 , t0+K , t0+2K , ..., ), the second subgrid starts from t1 and takes every Kth arrival time, i.e., g (2) = (t1 , t1+K , t1+2K , ..., ) and so on. Given the kth subgrid of arrival times, the corresponding realized variation estimator can r d (k) = P be defined as RP (k) |rt | , where ti and ti+1 denote consecutive elements ti ,ti+1 ∈g

i

g (k) .

in For the estimator, the all scale plays a bias-correcting role while the averaging scale reduces the variance of the estimator. The optimal number of subgrids K provided by Zhang et al. [Zha05] is prescribed as K = cn2/3 , 16σǫ4 T Eη 2

(12) R 3 t+1

)1/3 where η 2 = 4 t σ 4 (s)ds. The term R t+1 σǫ4 is square of the variance of noise, while t σ 4 (s)ds is the integrated quarticity. The c2 = 1 RP d and η 2 = 4 (RP d )2 at some reasonable lower frequency. σǫ2 is estimated by σ ǫ 2n 3 The estimator T\ SP V is an unbiased and consistent estimator for IPV in (4) under the microstructure noise as the frequency increases. Both the two-scale estimators are consistent and unbiased under microstructure noise kind of i.i.d process. where c can be estimated by c = (

2.2.

Correlation Estimators

Realized correlation estimator is conditionally constructed based on realized volatility. According to the theory of realized variation, Andersen et al. [And01a] and Andersen et al. [And01b] developed the concept of realized correlation constructed on realized standard PTn d std = RV d 1/2 , and covariance, RCOV \ xy = deviation, RV rtix .rtiy . The Realized ti

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\ xy , is represented as follows Correlation, RCOR

d std,x .RV d std,y ), \ xy = RCOV \ xy /(RV RCOR

(13)

d std,x .RP d std,y ). \ \ xy /(RP RP CORxy = RCOV

(14)

where x and y are high frequency time series of two asset returns. Barndorff-Nielsen and Shephard [Bar04b] provided an asymptotic distribution theory for realized covariance and correlation estimators. They claim that the estimators converge in probability to the corresponding true covariance and correlation. This is quite true under some general conditions, but the microstructure noise remains still as an influential problem. Hence, Safari and Seese [Saf07] continue to extend realized power volatility and in particular two-scale estimator into correlation estimators in order to deal with microstructure noise which may cause a remarkable bias in estimation. Motivated by robustness of absolute transformation in realized power variation against large values specially when p is set around or equal to 1, d std = RP d 1/2 and thus Realized Power realized power standard deviation is derived as RP Correlation as follows

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For two-scale correlation, first a two-scale covariance (TSCOV) has to be defined. The \ T SCOV xy is defined as   n ¯ −1  \ n ¯ \ T\ SCOV xy = 1 − RCOV xy,avg − RCOV xy,all n n

(15)

\ xy,all is the same as RCOV \xy , built on the full grid and the RCOV \ xy,avg where RCOV estimator can be estimated by K X \ xy,avg = 1 RCOV K

X

k=1 ti ,ti+1

rtix .rtiy .

(16)

∈g (k)

Given the two-scale power standard deviation as T\ SP V std = T\ SP V \ Two-Scale Power based Correlation T SP CORxy estimator is prescribed by

\ T SP CORxy = T \ SCOV xy /(T\ SP V std,x .T\ SP V std,y ),

1/2

, then the

(17)

where all estimators are based on a fixed interval of time and where x and y are two assets or high-frequency time series1 . The power or absolute based correlation estimators are expected to estimate the true or Integrated Power Correlation, IP CORxy , consistently and to converge as the frequency increases. We have

IP CORxy = qR t+1

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t

R t+1

Σxy (s)ds , R t+1 σxr (s)ds t σyr (s)ds t

(18)

R t+1 where t Σxy (s)ds is the true or Integrated Covariance. For more information see \ [Saf07]. While the RP CORxy estimator may face to the bias problem due to market mi\ crostructure noise, the two-scale estimator, T SP CORxy , corrects the bias by its subsampling procedure under microstructure noise of type i.i.d. However, this type of noise maybe not to be the real world case. One of the most important and influential issues may occur when utilizing highfrequency data for studying dependence and comovement by covariance and correlation estimators is non-synchronous observations caused by for example missing observations, different working hours per day, and different holidays between countries. Non-synchronous observations mainly lead to unreliable estimations. To cope with such the problem, the best way found insofar is to synchronize observations. Let ti and τj be the instants at which the returns x and y are being observed. Hayashi and Yoshida [Hay05] and Corsi [Cor07] proposed a covariance estimator 1

For the sake of simplicity, throughout of the paper we consider only the covariance and correlation estimators between two assets. However, the matrices of assets can be simply applied on the estimators.

Realized Volatility and Correlation Estimators...

\ xy = RCOV

Tn X Tm X

rtix .rτj y .I[min(ti , τj ) > max(ti−1 , τj−1 )],

179

(19)

τj

ti

where I[.] is the indicator function which takes the value of one only when the observations of two returns instantaneously overlap. This estimator consistently estimates the covariance of non-synchronous processes. Throughout the paper this synchronization is applied for estimation of covariances. In case of two-scale estimator, the indicator function is applied to the model, i.e., this model can be extended to the two-scale model.

3.

Self-similar Processes

The self-similar processes are the processes that are invariant under suitable translations of time and scale. They are important in probability theory because of their connection to limit theorems and they are of great interest in modeling heavy-tailed and long-memory phenomena. In fact, Lamperti [Lam62] used the term semi-stable in order to underline that the role of self-similar processes among stochastic processes is analogous to the role of stable distributions among all distributions. A process {X(t)}t≥0 is called self-similar [Lam62] if for some H > 0, d

X(at) = aH X(t),

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d

for every a > 0, where = denotes equality of all finite-dimensional distributions of the processes on the left and right. The process X(t) is also called an H-self-similar process and the parameter H is called the self-similarity index or Hurst exponent. Weron et al. [Wer05] argue that if we interpret t as time and X(t) as space then above equation tells us that every change of time scale a > 0 corresponds to a change of space scale aH . The bigger H, the more dramatic is the change of the space coordinate. The equation, indeed, means a scale-invariance of the finite-dimensional distributions of X(t). This property of a self-similar process does not imply the same for the sample paths. Therefore, pictures trying to explain self-similarity by some zooming in or out on one sample path, are, by definition, misleading. In contrast to the deterministic self-similarity, the self-similarity of stochastic processes does not mean that the same picture repeats itself exactly as we go closer. It is rather the general impression that remains the same.

3.1.

Fractional Gaussian Processes

Many of the interesting self-similar processes have stationary increments. A process {X(t)}t≥0 is said to have stationary increments if for any b > 0, d

[X(t + b) − X(b)] = [X(t) − X(0)]. The fractional Brownian motion {BH (t)}t≥0 has the integral representation

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Amir Safari, Wei Sun, Detlef Seese et al.

BH (t) =

Z

∞

−∞

H−1/2

[(t − u)+

H−1/2

− (−u)+

]dB(u),

(20)

where x+ =max(x,0) and B(u) is a Brownian motion. It is H-self-similar stationary increments (H-sssi) and it is the only Gaussian process with such properties for 0 < H < 1 [Sam94]. The classic Brownian motion B(t), used by Einstein and Smoluchowski, is simply a special case of the fractional Brownian motion when H = 1/2. In modeling of long-memory phenomena, the stationary increments of H-self-similar processes are of special interest since any H-self-similar process with stationary increments {X(t)}t∈R induces a stationary sequence {Yj }j∈Z , where Yj = X(j + 1) − X(j) and j = ..., −1, 0, 1, ... . The sequence Yj corresponding to the fractional Brownian motion is called fractional Gaussian noise [Mer03]. It is called a standard fractional Gaussian noise if varYj =1 for every j ∈ Z. The fractional Gaussian noise has some remarkable properties. If H=1/2, then its autocovariance function r(k) = R(0, k) = 0 for k 6= 0 and hence it is the sequence of independent identically distributed (i.i.d.) Gaussian random variables. The situation is quite different when H 6= 1/2, namely the Yj ’s are dependent and the time series has the autocovariance function.

3.2.

Fractional Stable Processes

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

While the fractional Brownian motion can capture the effect of long-range dependence, it has less power to capture heavy tailedness [Sun07]. The existence of abrupt discontinuities in financial data, combined with the empirical observation of sample excess kurtosis and unstable variance, confirms the stable Paretian hypothesis identified by Mandelbrot [Man83]. It is natural to introduce the stable Paretian distribution in self-similar processes in order to capture both long-range dependence and heavy tailedness. There are many different extensions of fractional Brownian motion to the stable distribution. The most commonly used extension of the fractional Brownian motion to the α-stable case is the linear fractional stable motion (also called the Levy stable motion). Samorodnitsky and Taqqu  fractional H [Sam94] define the process Zα (t) t∈R by the following integral representation ZαH (t)

=

Z

∞ −∞

H−1/α

[(t − u)+

H−1/α

− (−u)+

]dZα (u),

(21)

where Zα (u) is a symmetric Levy α-stable motion. The integral is well defined for 0 < H < 1 and 0 < α ≤ 2 as a weighted average of the Levy stable motion Zα (u) over the infinite past with the weight given by the above integral kernel denoted by ft (u). The process ZαH (t) is the H-sssi. Assume that H-self-similarity follows from the above integral representation and the fact that the kernel ft (u) is d-self-similar with d = H −1/α, when the integrator Zα (u) is 1/α-self-similar. This implies [Wer05] the following important relation H =d+

1 . α

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The process ZαH (t) is reduced to the fractional Brownian motion if one sets α=2. When H=1/α, then the Levy α-stable motion is obtained which is an extension of the Brownian motion to the α-stable case. Contrary to the Gaussian case (α=2), the Levy α-stable motion (0 < α < 2) is not the only 1/α-self-similar Levy α-stable process with stationary increments (this is true for 0 < α < 1 only). The increment process corresponding to the fractional Levy stable process is called a Fractional Stable Noise (FSN). By analogy to the case of α=2, fractional stable noise has the long-range dependence when H > 1/α and the negative dependence when H < 1/α. If H = 1/α, the increments of fractional Levy stable motion are i.i.d. symmetric α-stable variables. We note that there is no long-range dependence when 0 < α ≤ 1 because H is constrained to lie in the interval (0, 1).

3.3.

Simulation Algorithms for the Noise Processes

A fast Fourier transform method for synthesizing approximate self-similar sample paths for Fractional Gaussian Noise has been presented by Paxson [Pax97]. The method is fast and appears to generate close approximations to true self-similar sample paths. A simulation procedure based on this method that overcomes some of the practical implementation issues has been prescribed by Bardet et al. [Bar03b]. The procedure follows these steps: 1. Choose an even integer M . Define the vector of the Fourier frequencies Ω = (θ1 , ..., θM/2 ), where θt = 2πt/M and compute the vector F = fH (θ), ..., fH (θM/2 ), where fH (θ) =

X 1 sin(πH)Γ(2H + 1)(1 − cosθ) |2πt + θ|−2H−1 , π

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

i∈N

and f H(θ) is the spectral density of fractional Gaussian noise. 2. Generate M/2 i.i.d. exponential (exp(1)) random variables E1 , ..., EM/2 and M/2 i.i.d. uniform (U [0, 1]) random variables U1 , ..., UM/2 . √ 3. Compute Zt = exp(2iπUt ) Ft Et , for t = 1, ..., M/2. e = (0, Z1 , ..., Z(M/2)−1 , ZM/2 , Z¯(M/2)−1 , ..., Z¯1 ). 4. From the M -vector: Z 5. Compute the inverse FFT of the complex Z to obtain the simulated sample path. Using the Fast Fourier Transform (FFT) algorithm, Stoev and Taqqu [Sto04] provide an efficient method for simulation of a class of processes with symmetric α-stable (SαS) distributions, namely the linear fractional stable motion (LFSM) processes. The paths of the LFSM process are generated by using Riemann-sum approximations of its SαS stochastic integral representation. They introduce parameters n, N ∈ ℵ and express the fractional stable noise Y (t) as  H−(1/α)  H−(1/α) ! nN X j j Yn,N (t) := Lα,n (nt − j), − −1 n + n +

(22)

j=1

Where Lα,n (t) := Mα ((j + 1)/n) − Mα (j/n), and j ∈ ℜ. The parameter n is mesh size and the parameter M is the cut-off of the kernel function. The authors use the Fast Fourier Transformation (FFT) algorithm for approximating Yn,N (t). Consider the moving average process Z(m), m ∈ ℵ,

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Z(m) :=

nM X

gH,n (j)Lα(m − j),

(23)

H−(1/α) !  H−(1/α)  j j n−1/α , − −1 n + n +

(24)

j=1

where

gH,n (j) :=

d

and Lα (j) is the series of i.i.d. standard stable Paretian random variables. Since Lα,n (j) = d

n−1/α Lα (j), where j ∈ ℜ, then the letter equations (23) and (24) imply that Yn,N (t) = ˜ α (j) be the n(N + T )-periodic with L ˜ α (j) := Lα (j), for Z(nt), for t = 1, ..., T . Let L j = 1, ..., n(N + T ) and let g˜H,n (j) := gH,n (j), for j = 1, ..., nN , g˜H,n (j) := 0, for j = nN + 1, ..., n(N + T ). Then

d

{Z(m)}nT m=1 =

 +T ) n(N X 

j=1

nT  ˜ α (n − j) g˜H,n (j)L , 

(25)

m=1

because for all m = 1, ..., nT , the summation in equation (23) involves only Lα (j) with indices j in the range −nN ≤ j ≤ nT − 1. Using a circular convolution of the two ˜ α computed by using their Discrete Fourier Transforms n(N +T )-periodic series g˜H,n and L (DFT), the variables Z(n), m = 1, ..., nT (i.e., the fractional stable noise) can be generated. In the next section, assuming that microstructure noise process follows i.i.d. noise, fractional Gaussian noise, stable noise, and fractional stable noise processes, we examine the behavior of the realized volatility estimators nested in GARCH model and realized correlation estimators as model-free and compare performance of the estimators under different noise assumptions.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

4. 4.1.

Behavior on Finite Samples: Simulation Experiments Volatility Simulation Set-up and Results

Drost and Werker [Dro96] and Nelson [Nel90] propose a continuous-time diffusion model of the GARCH(1,1) as dpt = σt dW1,t ,

(26)

dσt2 = θ(ω − σt2 )dt + (2λθ)1/2 σt2 dW2,t ,

(27)

where W1,t and W2,t represent independent standard Brownian motions. For highfrequency observations, such the model seems suitable. But even high-frequency data are

Realized Volatility and Correlation Estimators...

183

observed discretely. However, Drost and Werker [Dro96] argue that the discretely sampled returns from the continuous-time process defined by Eqs. (26) and (27), satisfy the weak GARCH(1,1) model 2 2 2 σ(n),t = ψn + αn r(n),t−1/n + βn σ(n),t−1/n ,

(28)

2 2 where n is the number of observations per day t, and σ(n),t ≡ P(n),t−1/n (r(n),t ) denotes the 2 best linear predictor of r(n),t . The relations between discrete and continuous time model parameters are highlighted by Drost and Werker [Dro96]. This implies that the models (27) and (28) are compatible. In turn, the h-period linear projection from the weak GARCH(1,1) model of (28) is simply specified by Baillie and Bollerslev [Bai92] as



2 P(n),t (r(1/h),t+h ) = P(n),t 

X

j=1,...,nh

X

=

2 

r(n),t+j/n  

2 P(n),t (r(1/h),t+j/n )

j=1,...,nh

=

X

j=1,...,nh

h

i 2 2 2 σ(n) + (αn + βn )j (σ(n),t − σ(n) )

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h i 2 2 2 = nhσ(n) + (αn − βn ) 1 − αn − βnnh × [1 − αn − βn ]−1 (σ(n),t − σ(n) ),

(29)

2 ≡ ψ (1 − α − β )−1 . The model (29) is what Andersen et al. [And98] and where σ(n) n n n Andersen et al. [And99] established for simulations of realized volatility. In realized volatility literature, market microstructure noise is usually assumed to follow an i.i.d process. For example, Barndorff-Nielsen, Hansen, Lunde and Shephard [Bar04a], Zhang et al. [Zha05], Bandi and Russell [2005b], and Hansen and Lunde [2006] assume it follows a Gaussian process. We evaluate the simulations of the model (29) with different microstructure noise assumptions including the i.i.d noise, fractional Gaussian noise, stable noise, and fractional stable noise. Different realized volatility estimators are cast in the model (29) with different noise assumptions and then the bias and variance of estimations are calculated. The Gaussian noise is set equal to 1% of the value of the variable of interest. Random variables for simulations are generated according to minute-by-minute frequency for 4 years assuming 252 working days a year. For generating the non-Gaussian noises, the described procedures in last section is followed based on minute-by-minute frequency observations of CAC 40 and FTSE 100 which will be described in subsection 5.1. The number of sample paths for all simulations is 15,000 realizations. Regarding to two-scale based estimators we allow the sampling observations to be regularly allocated. For all three alternative estimators, we assume equally-distant sampling interval. Three estimators including TSRV, RA, and TSAV (among them the two latter estimators are the RP and TSPV estimators of power 1) are compared. But what is more important here is the comparison of different microstructure noise assumptions. For evaluation of estimators, we use RMSE and Bias statistics.

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Table 1. Results of volatility simulations assuming different noise (simulated based on CAC data) Assumptions White noise Fractional Gaussian noise Stable noise Fractional Stable noise

T\ SRV RMSE Bias 0.001612 1.686e-005 0.001487 1.675e-005 0.001489 1.676e-005 0.001306 1.502e-005

d

RA RMSE 0.001125 0.001060 0.001072 0.000885

Bias 1.749e-005 1.688e-005 1.692e-005 1.644e-005

T\ SAV RMSE Bias 0.000853 1.383e-005 0.000820 1.347e-005 0.000821 1.349e-005 0.000819 1.318e-005

Table 2. Results of volatility simulations assuming different noise (simulated based on FTSE data) Assumptions White noise Fractional Gaussian noise Stable noise Fractional Stable noise

RMSE 0.002859 0.002853 0.002731 0.002548

Bias 1.113e-004 1.107e-004 9.685e-005 7.071e-005

d

RA

T\ SRV RMSE 0.002125 0.002104 0.001873 0.001546

Bias 1.750e-004 1.750e-004 1.607e-004 1.418e-004

T\ SAV RMSE 0.001847 0.001823 0.001765 0.001508

Bias 1.002e-005 9.993e-006 9.885e-006 8.453e-006

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

In terms of RMSE and Bias of estimation, the results of Monte Carlo simulations are contained in Table 1. A horizontal comparison of different volatility estimators is an indication of different consistency of the estimators. In general, the TSAV estimator yields less variation and bias than others at minute-by-minute simulated frequency. This is in line with the results of [Saf07]. Table 2, which provides the results of simulations using the simulated noise values based on 1 minute frequency real data of FTSE, yields the same results. However, a vertical comparison of the values contained in the tables is more important purpose of the paper. Both the tables report that the GARCH(1,1) model assuming the fractional stable noise outperforms the other models assuming the White noise, Fractional Gaussian noise, and Stable noise. Among the models with different noises, the model assuming White noise has the worst results of fitting. These results are consistent with those of obtained by Sun, Rachev, and Fabozzi [Sun07].

4.2.

Correlation Simulation Set-up and Results

Meddahi [Med02] and Barndorff-Nielsen and Shephard [Bar04b] suggest actual correlation estimator used for simulation study as follows Rt

t−1 Σxy (s)ds , Rt t Σ (s)ds Σ (s)ds t−1 xx t−1 yy

qR

where Σxy represents actual covariance and Σxx shows the variance. The differences between realized correlation estimators (13), (14), and (17) and the actual correlation represent errors of realized correlation estimators in estimating the true correlation. For correlation simulation we follow Meddahi [Med02] and Barndorff-Nielsen and Shephard [Bar04b] and

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Table 3. Results of correlation simulation Assumptions White noise Fractional Gaussian noise Stable noise Fractional Stable noise

\ xy RSCOR RMSE Bias 0.003841 5.478e-005 0.003839 5.478e-005 0.003802 5.463e-005 0.003754 5.418e-005

\ xy RACOR RMSE Bias 0.003341 4.654e-004 0.003317 4.652e-004 0.003321 4.657e-004 0.003206 4.574e-004

\ xy T SACOR RMSE Bias 0.002760 2.783e-005 0.002760 2.744e-005 0.002764 2.346e-005 0.002608 1.837e-005

estimate the errors between realized correlation and actual correlation estimators. Obviously, this set-up for simulation is model free. As for microstructure noise, the size of the standard deviation of the White noise is set again equal to 1% of the generated data. Other types of noise have been previously simulated and used for volatility estimators based on minute-by-minute real CAC and FTSE data that are used here for correlation simulation. Other conditions for volatility simulations are held. According to Table 3, the White noise which is usually assumed when modeling realized volatility, has the worst errors in terms of RMSE and Bias of estimation. Instead, models with Fractional Stable noise have the best performance of estimation. This fact is true for the three correlation estimators. In the following section, distributional and dynamic properties of estimators will be experimentally compared. Since there exists no two-scale realized squared correlation, we compare the results of our measures with realized squared based correlation.

5.

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5.1.

Empirical Behaviors of the Estimators Data Description

The empirical behaviors of the described realized volatility and correlation estimators are studied here utilizing CAC40 and FTSE100 stock index data at every 1 minute frequency. The sample indices cover a period of shorter than 4 calendar years from Oct. 27, 2003 to July 10, 2007. This period totally includes 929 trading days with 436425 observations. A few important basic statistics of returns or increments on index series are appeared in Table 4. The mean of the returns are positive at 1 minute frequency, implying an average positive return over this period at minute by minute frequency for investors who have invested in an assumed portfolio of index. The table reports excess kurtosis much higher than 3 (coefficients equal to 524 for CAC and to 434 for FTSE). The distributions of the time series are highly leptokurtic. Simply, this is an indication of heavy tail in distribution of the time series. This conveys that there is a higher probability for extreme events to appear in our data sample than that in normally distributed data sample. Also, departure from symmetry in distribution of the time series can be observed clearly by the skewness in the coefficient reported in the table equal to -0.807 and -0.375 for CAC and FTSE. They are negative, meaning that negative extreme values which are more common than positive extreme values are generally populated in the left side of the distribution. The popular Jarque-Bera test of normality simply indicates that the investigated time series with p-value equal to 2.2e-16 at

186

Amir Safari, Wei Sun, Detlef Seese et al. Table 4. Some descriptive statistics and test of return of indices Statistic Minimum Maximum Mean Median Sum Variance Skewness Kurtosis Jarque-Bera test of normality Hurst exponent by Whittle Estimator

CAC40 -2.74e-02 3.15e-02 1.33e-06 0.00e+00 5.89e-01 1.37e-07 -8.07e-01 5.24e+02 2.2e-16 5.17e-01

FTSE100 -2.16e-02 1.87e-02 1.03e-06 0.00e+00 4.54e-01 8.20e-08 -3.75e-01 4.34e+02 2.2e-16 5.24e-01

95 percent of confidence do not form a normal distribution. The Chi squared with 2 degrees of freedom in Jarque-Bera test of normality is 5.06e+09 and 3.46e+09 for CAC and FTSE, respectively. Finally, estimated Hurst exponents approximated by the Whittle [Whi63] Estimator2 equal to 0.517 (std=0.00037) and 0.524 (std=0.00029) imply almost a random walk and no memory in returns series of CAC and FTSE. These facts are in line with common sense. As Rachev et al. [Rac05] state empirical evidence does not support the assumption that many important variables in finance follow a normal distribution.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

5.2.

Unconditional and Conditional Distributions

We now turn our attention to the distribution of volatility and correlation time series constructed based on different estimators. It would be interesting if we could observe some of those important facts that have been documented in the literature of financial time series over a long history of research here in high-frequency based time-varying volatilities and correlations, too. Figure 1 depicts the time series of realized volatility estimators. Some periods of market turmoil with higher volatility on the time series are evident. Over these market turmoils, CAC and FTSE almost simultaneously display high and low volatilities. Hence two markets look to be interdependent on volatility. Simply squared based volatility tends to report volatility much smaller than two others on average, as can be seen in Table 5. However, the means of realized volatility approximated by absolute based estimators are close. In comparison, all estimators show a mean of volatility reported in Table 5 much higher than a constant variance of returns reported in Table 4.

2

Throughout the paper the Whittle Estimator is approximated by the FARIMA model.

Realized Volatility and Correlation Estimators... 4 e−04

Two−scale squared based volatility of FTSE over the time

0

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600

0 e+00

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8 e−04

Two−scale squared based volatility of CAC over the time

187

800

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Absolute based volatility of FTSE over the time

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0.05 0.15 0.25

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Absolute based volatility of CAC over the time

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800

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Two−scale absolute based volatility of FTSE over the time

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0.20

0.20

0.35

Two−scale absolute based volatility of CAC over the time

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400

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800

0

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800

Figure 1. Time series of realized volatility measures constructed based on two-scale squared, absolute, and two-scale absolute estimators. Volatility can be observed as time-varying. Two markets display almost simultaneously high and low volatilities. Hence they look to be interdependent on volatility.

Kernel density of two−scale squared based volatility in FTSE

0

0 4000

10000

10000

Kernel density of two−scale squared based volatility in CAC

0 e+00

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8 e−04

0 e+00

30 20 0

10

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0.05

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Kernel density of two−scale absolute based volatility in FTSE

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12

Kernel density of two−scale absolute based volatility in CAC

0

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

4 e−04

Kernel density of absolute based volatility in FTSE

20

Kernel density of absolute based volatility in CAC

2 e−04

0.0

0.1

0.2

0.3

0.4

0.05

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0.15

0.20

0.25

0.30

Figure 2. Kernel density distributions of different realized volatility series look skewed rightward. The shapes are not exactly the same. The tails of squared based volatility of series are heavier than others.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Table 5. Basic statistics and tests of realized volatility measures Statistic

d

CAC

d

FTSE

T\ SRV

RA

T\ SAV

T\ SRV

RA

T\ SAV

Mean

6.49e-05

9.33e-02

1.30e-01

3.89e-05

7.66e-02

8.94e-02

Median

4.38e-05

8.74e-02

1.22e-01

2.69e-05

7.27e-02

8.49e-02

Variance

5.83e-09

7.63e-04

1.49e-03

1.82e-09

3.81e-04

5.19e-04

Skewness

5.66e+00

2.11e+00

2.11e+00

5.53e+00

2.87e+00

2.92e+00

Kurtosis

4.87e+01

6.85e+00

6.82e+00

4.41e+01

1.39e+01

1.48e+01

Jarque-Bera

2.2e-16(97415)

2.2e-16(2527)

2.2e-16(2508)

2.2e-16(80431)

2.2e-16(8874)

2.2e-16(9941)

Hurst via Whittle

0.541(4.8e-09)

0.572(0.00033)

0.576(0.00064)

0.551(1.4e-09)

0.591(0.00017)

0.596(0.00023)

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According to Table 5, the unconditional distributions of volatilities are positively skewed. The kurtosis coefficients are all higher than that of a normal distribution. Meanwhile, this coefficient for squared based volatility is much higher than that of others. Positive extreme values in volatility time series, which are the reason for high leptokurtosis, lead their unconditional distributions to possess a right heavy tail. These heavy right tails can be seen in Figure 2. Exploiting 5 minute exchange rate data, Andersen et al. [And01b] found that the distributions of realized daily variances are skewed to the right and leptokurtic. In line with these findings, using 5 minute stock exchange data, Andersen et al. [And01a] also confirm that the unconditional distributions of realized variances are highly right-skewed. With p-value of 2.2e-16, null hypothesis of normality for all volatility series is significantly rejected with the Jarque-Bera test. The values in parenthesis reported in the table for Jarque-Bera test is Chi squared with 2 degrees of freedom. Plots in Figure 3 are more informative and compare the empirical distribution with that of the normal. The size of discrepancy from bisector represents deviation from normality. In the right tail, the discrepancy is much higher.

0

4

QQ plot of two−scale squared volatility in FTSE

−4

−4

0

4

QQ plot of two−scale squared volatility in CAC

−4

−2

0

2

4

−4

2

4

4 0 −4

0 −4

−4

−2

0

2

4

−4

−2

0

2

4

0

4

QQ plot of two−scale absolute volatility in FTSE

−4

−4

0

4

QQ plot of two−scale absolute volatility in CAC

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

0

QQ plot of absolute volatility in FTSE

4

QQ plot of absolute volatility in CAC

−2

−4

−2

0

2

4

−4

−2

0

2

4

Figure 3. Quantile Quantile-normal plots compare the empirical volatility series to the theoretical distribution. More matching to the straight line means more approaching to normality. The right tails are easily observable by longer distance from the straight line. Regarding to different estimators for multivariate conditional correlations, it is easy to study dependence structure between CAC and FTSE. Now with respect to theoretical developments, we are able to empirically investigate how high-frequency correlations between the markets change over the time. The aforementioned estimators of correlation are applied on the returns series described in Table 3. For the purpose of comparison, we proceed with a squared based, absolute based, and two-scale realized correlation estimators. After estimation of dependence structure between returns of the markets, the descrip-

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Amir Safari, Wei Sun, Detlef Seese et al.

tive statistics of time-varying realized correlation series are given in Table 6. On average, there exists a positive dependence between two returns of the markets, meanwhile the dependency is reported by squared based correlation so stronger than absolute based ones. According to the constant variance of correlation series, squared based correlation highly fluctuates around its mean. Graphically, the fluctuations of correlations are shown in Figure 4, the top plots. With regard to our 1 minute frequency data, we found that there is a slightly asymmetry to the right side of distribution of squared based correlation whereas absolute based correlations include negative asymmetry in their distributions. This finding is consistent with that of Safari and Seese [Saf07] where a relationship between the returns of NASDAQ and DAX at 5 minute frequency was investigated. While the distribution of squared based correlation looks platykurtic, the distribution of others indicates excess kurtosis and therefore existence of fat tail. Since there are some extreme negative values in absolute based correlation series and they dominate positive extreme values, the heavy tails present in left side of the distributions. Kernel density plots in Figure 4 exhibit the heavy tails. Longer negative tail in multivariate absolute based realized correlations can be documented in such a way that the extreme values are usually populated in the left tail of the distributions as can be observed in QQ plots of Figure 4. Jarque-Bera test for null normality reveals statistically none of distributions pose normality at the 5 percent level of significance. Absolute based correlation

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−0.00010 0.00000

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−5 e−05

−0.00020

0.1 −0.1 0

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Two−scale absolute based correlation

0.00005

Squared based correlation

−4

−2

0

2

4

−4

−2

0

2

4

Figure 4. Some characteristics of distribution of realized correlations are graphically depicted. Obviously realized correlations, based on first row plots, oscillate over the time. Negative asymmetry is present in absolute based correlations. Longer left tail in dependence structure of the absolute based correlations is obvious considering density plots.

Presence of negative asymmetry in correlation conveys the meaning that negative shocks in returns have greater impact than positive shocks in CAC and FTSE markets. In the other words, downside comoves are greater than upside comoves between markets. Negative asymmetry in the correlation between stock markets and even between stock market and other markets (i.e. bond market) has been documented by several researchers

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Table 6. Basic statistics and test of realized correlations

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Statistic Mean Median Variance Skewness Kurtosis Jarque-Bera test Hurst via Whittle

\ xy RSCOR 4.47e-04 1.88e-03 1.94e-03 3.06e-02 8.00e-01 2.9e-06(25.5) 0.513(7.00e-05)

\ xy RACOR 3.98e-07 6.83e-07 5.27e-10 -8.86e-01 9.66e+00 2.2e-16(3769.9) 0.558(6.64e-05)

\ xy T SACOR 2.11e-07 3.16e-07 1.21e-10 -3.68e-02 5.07e+00 2.2e-16(1003.4) 0.563(9.01e-05)

in different ways. Kroner and Ng [Kro98] introduce Asymmetric Dynamic Covariance (ADC) model which encompasses several popular multivariate GARCH models within its framework while allowing for asymmetric effects to appear within the conditional variancecovariance process. Cappiello, Engle and Sheppard [Cap06] propose a new generalized autoregressive conditional heteroskedastic process, the asymmetric generalized dynamic conditional correlation (AG-DCC) model, to permit conditional asymmetries in correlation dynamics. Using equity returns, Longin and Solnik [Lon01] derive the distribution of extreme correlation for a wide class of return distributions. Empirically, they reject the null hypothesis of multivariate normality for the negative tail, but not for the positive tail. They suggest that correlation increases in bear markets, but not in bull markets. According to Rachev [Rac05] actually correlation is one particular measure of dependence among many. Another approach is to model dependency using copulas. From copula theory, we find that for continuous multivariate distribution functions, the univariate margins and the multivariate dependence structure can be separated, and the dependence structure can be represented by a copula. Embrechts et al. [Emb03] argue that copulas provide a natural way to study and measure dependence between random variables. Exploiting copula theory, Patton [Pat04] construct models of the time-varying dependence structure that allow for different dependence during bear markets than bull markets. Stock returns appear to be more highly correlated during market downturns than during market upturns. Patton [Pat06] extended the theory of copulas to allow for conditioning variables for evaluating asymmetry in dependence, and employed it to construct flexible models of conditional dependence structure in the joint density of the some exchange rates. Two different copulas were estimated: the copula associated with the bivariate normal distribution and the symmetrized Joe-Clayton copula, which allows for general asymmetric dependence. Time variation in the dependence structure between the two exchange rates was captured by allowing the parameters of the two copulas to vary over the sample period. He found evidence that the mark-dollar and yen-dollar exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating.

5.3.

Dynamics of Volatility and Correlation Series

For this empirical section, it is attempted to understand in details some facts of human activity in the financial markets that are particularly well documented for financial data and

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more recently for high-frequency time series. An important task of related investigations is to unveil several stylized facts of financial markets, which consistently appear on different markets and in different periods of time, and that any candidate model should convincingly explain. With regards to the fact that realized volatility and correlation estimators are in principle model free, in bellow the presence or absence of some of these stylized facts in time-varying volatilities and correlations series is empirically investigated. log−log

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timated based on CAC time series. For autocorrelation functions and long memory autocorrelation function, the number of lags is arbitrarily selected equal to 200. Left plots are autocorrelation functions and right ones are long memory autocorrelation functions. Obviously plots suggest that realized volatility is a long memory process, although absolute based estimators confirm this suggestion stronger.

Either a slow decay in autocorrelation of volatility series can be reported or not, left panels on Figures 5 and 6 exhibit the autocorrelation plots for CAC and FTSE. Autocorrelations for squared based volatility series in both CAC and FTSE last to almost 20 lags, almost one calendar month (regarding to almost 22 working days in a month), in the meantime autocorrelations in absolute based volatility series still remain meaningful until more than 40 lags, almost two calendar months. This important fact conveys that a shock in the volatility process will have a long-lasting impact. As a matter of fact, autocorrelation may be an indication of a long memory process. As such, long memory may be an interesting signature for series dynamics. If the decay in the autocorrelation function is slower than a hyperbolic rate, i.e., the correlation function decreases algebraically with increasing integer lag, then the time series would possess a long memory. Thus, it makes sense to investigate the decay on a double logarithmic scale and to estimate the decay exponent. The right panels on the figures display long memory behavior of the estimators in volatilities constructed on minute-by-minute frequency series. Graphically, if the time series exhibits long memory behavior, it can be seen as a straight line in the plot on the right panels of Figures 5 and 6. Because of longer meaningful autocorrelation in absolute based series, it is visible that

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timated based on FTSE time series. For autocorrelation functions and long memory autocorrelation function, the number of lags is arbitrarily selected equal to 200. Obviously, plots suggest that realized volatility is a long memory process, although absolute based estimators confirm this suggestion stronger.

the corresponding log-log plots of long memory decays slower than that of squared based estimator. Reported in Table 4, the Hurst Exponent estimated through the Whittle estimation for two-scale squared based, absolute based, and two-scale absolute based estimators are equal to 0.541, 0.572, and 0.576 and to 0.551, 0.591, and 0.596 for CAC and FTSE respectively. The values in parenthesis are standard deviation. Consistent with Andersen et al. [And99], there is indeed an evidence to suggest that volatility is a long memory process. Figure 7 seeks for possible regular patterns in dependency structure across equity markets among the correlation estimators. If the dependency structure between the markets obeys squared based model of correlation, then there does not exist autocorrelation and long memory in dependency of the markets, as the top plots of the figure state. Meanwhile, if the dependency patterns are ruled by absolute based models of correlations in real world, although not too long but the correlation series include somewhat autocorrelation and long memory process implying regular behavior of dependency. This can be seen in lower plots of the figure. Reported in Table 5, the Hurst Exponents with the values of 0.513, 0.558, and 0.563 for squared, absolute and two-scale absolute based correlations indicate differences.

6.

Conclusion and Discussion

The consistency of volatility estimators differs given they are constructed differently. This fact is valid for correlation estimators as well. Actually, squared based estimators are more sensitive to the larger values. Moreover, adopting the two-scale approach when the construction of the estimators corrects the bias of the estimation. Therefore, the performance of the two-scale power estimator dominates the performance of the others. The simulation

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correlations between CAC and FTSE. For autocorrelation function and long memory autocorrelation function, the number of lags is arbitrarily selected equal to 200. The top row plots, which belong to squared based correlation, do not indicate autocorrelation and long memory while the others do somewhat.

studies revealed these facts. Different assumptions about the type of microstructure noise have a meaningful impact on the accuracy and bias of the estimators nested in a continuous GARCH process. All estimators are affected by the assumptions and almost all the estimators react identically to the change in assumptions. This implies that all the estimators are almost identically sensitive to the changes in assumptions about the type of noise and they act in the same way when the assumptions change. Meanwhile, all estimators perform the best when the fractional stable noise assumption is adopted and the worst when the i.i.d. assumption about noise is made. This turns out that the underlying microstructure noise process actually follows fractals and a stable distribution. As such, there exists self-similarity in the real world microstructure noise including long range dependence as well as heavy tailedness in minute-by-minute stock index data. Heavy tailedness in volatility estimators is empirically obvious. The volatility estimators unveil some dynamic behavior investigated by the Hurst parameter. In particular, absolute based correlation estimators are consistent with the other studies described here to reveal negative asymmetry in dependence structure. Like copulas, the absolute based correlations indicate a negative asymmetry in dependence between the markets conveying that negative shocks in the returns have greater impact than positive shocks. In the other word, downside comoves are greater than upside comoves between the markets. In our simulations using minute-by-minute frequency data, the most realistic noise assumption was revealed to be the fractional stable noise. But, if market microstructure noise behaves differently at different frequencies, the problem of microstructure noise would be more complicated. For example, tick-by-tick frequency data might show another type of

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noise process and might not include fractals. Even the coarser frequencies might approach to other types of noise. These need further investigations.

References [Ait05] A¨ıt-Sahalia, Y., Mykland, P. A., Zhang, L., 2005,“How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise”, The Review of Financial Studies Vol. 18, No. 2, pp. 351-416. [And98] Andersen, T.G., Bollerslev, T., 1998,“Answering the skeptics: Yes, standard volatility models do provide accurate forecasts”, International Economic Review, 39, pp. 885-905. [And99] Andersen, T.G., Bollerslev, T., Lange, S., 1999,“Forecasting financial market volatility: Sample frequency vis-a-vis forecast horizon”, Journal of Empirical Finance, 6, pp. 457-477. [And01a] Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens H., 2001,“The distribution of realized stock return volatility”, Journal of Financial Economics, 61, pp. 43-76. [And01b] Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P., 2001,“The distribution of realized exchange rate volatility”, Journal of the American Statistical Association, Vol. 96, No. 453, pp. 42-55. [And02] Andreou, E., Ghysels, E., 2002,“Rolling-sample volatility estimators: Some new theoretical, simulation, and empirical results”, Journal of Business and Economic Statistics 20(3), pp. 363-376. [Bai92] Baillie, R. T., Bollerslev, T., 1992,“Prediction in dynamic models with time dependent conditional variances”, Journal of Econometrics, 52, pp. 91-113.

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[Ban05a] Bandi, F., Russell, J., 2005,“Volatility”, Forthcoming in the Handbook of Financial Engineering, Elsevier. Edited by J. R. Birge and V. Linetsky. [Ban05b] Bandi, F., Russell, J., 2005,“Microstructure noise, realized volatility, and optimal sampling”, Working paper, Graduate School of Business, University of Chicago. [Bar03a] Barndorff-Nielsen, O.E., Shephard, N., 2003,“Realized power variation and stochastic volatility models”, Bernoulli, 9(2), pp. 243-265. [Bar03b] Bardet, J., Lang, G., Oppenheim, G., Philippe, A., Taqqu, M., 2003,“Generators of long-range dependent processes: A survey”, In P. Doulkhan, G. Oppenheim, M. Taqqu (Eds.), Theory and applications of long-range dependence, Birkh¨aser: Boston. [Bar04a] Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2004,“Regular and modified kernel-based estimators of integrated variance: The case with independent noise”, CAF: Center for Analytical Finance, University of Aarhus, Working Paper No. 196.

196

Amir Safari, Wei Sun, Detlef Seese et al.

[Bar04b] Barndorff-Nielsen, O.E., Shephard, N., 2004,“Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics”, Econometrica, Vol. 72, No. 3, pp. 885-925. [Cap06] Cappiello, L., Engle, R. F., Sheppard, K.,“Asymmetric dynamics in the correlations of global equity and bond returns”, Journal of Financial Econometrics, Vol. 4, No. 4, pp. 537-572. [Cor07] Corsi, F., 2007,“Realized Correlation Tick-by-Tick”, Discussion Paper 2007-02, Department of Economics, University of St. Gallen. [Dro96] Drost, F.C., Werker, B.J.M., 1996,“Closing the GARCH gap: Continuous time GARCH modeling”, Journal of Econometrics, 74, pp. 31-57. [Emb03] Embrechts, P., Lindskog, F., McNeil, A., 2003,“Modelling dependence with copulas and applications to risk management,” in Rachev, S., T., Handbook of heavy tailed distributions in finance, Elsevier North-Holland. [Hay05] Hayashi, T., Yoshida, N., 2005, “On covariance estimation of non-synchronously observed diffusion processes”, Bernoulli, 11(2), pp. 359-379. [Kro98] Kroner, K. F., Ng, V. K., 1998,“Modeling asymmetric comovements of asset returns”, The review of financial studies, Vol. 11, No. 4, pp. 817-844. [Lam62] Lamperti, J. W., 1962,“Semi-stable stochastic processes”, Transactions of the American Society, 104 (62), pp. 62-78. [Lon01] Longin, F., Solnik, B., 2001,“Extreme Correlation of International Equity Markets”, Journal of Finance, 56, pp. 649-76. [Man83] Mandelbrot, B., B., 1983, The Fractal Geometry of Nature, San Francisko: W.H. Freeman.

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[Med02] Meddahi, N., 2002,“A theoretical comparison between integrated and realized volatility”, Journal of Applied Econometrics, 17, pp. 479-508. [Mer80] Merton, R. C., 1980,“On estimating the expected return on the market: An exploratory investigation”, Journal of Financial Economics, 8, pp. 323-361. [Mer03] Mercik, S., Weron, K., Burnecki, K., Weron, A., 2003,“Enigma of self-similarity of fractional levy stable motions”, Acta Physica Polonica B, 34, 3773. [Mit98] Mittnik, S., Rachev, S., Paolella, M. S., 2003,“ Stable Paretian modeling in finance: Some empirical and theoretical aspects”, Adler, R. et al., editors. A practical guide to heavy Tails: Statistical techniques and applications, Birkh¨aser: Boston, pp. 79-110. [Nel90] Nelson, D. B., 1990,“ARCH models as diffusion approximations”, Journal of Econometrics, 45, pp. 7-38.

Realized Volatility and Correlation Estimators...

197

[Pat04] Patton, A. J., 2004,“On the out-of-sample importance of skewness and asymmetric dependence for asset allocation”, Journal of Financial Econometrics, Vol. 2, No. 1, pp. 130-168. [Pat06] Patton, A. J., 2006,“Modelling asymmetric exchange rate dependence”, International Economic Review, Vol. 47, No. 2, pp. 527-556. [Pax97] Paxson, V., 1997,“Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic”, Computer Communication Review, 27 (5), pp. 5-18. [Rac00] Rachev, S., Mittnik, S., 2000, Stable Paretian models in finance, Wiley: New York. [Rac05] Rachev, S., T., Menn, C., Fabozzi, F. J., Fat-tailed and skewed asset return distributions: Implications for risk management, portfolio selection, and option pricing, Wiley Finance, 2005. [Saf07] Safari, A., Seese, D., 2007,“Distributional and dynamical properties of realized volatility and correlation”, Manuscript. [Sam94] Samorodnitsky, G., Taqqu, M. S., Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, London, 1994. [Skl59] Sklar, A., 1959,“Fonctions de r´epartition a´ n dimensions et leurs marges”, Publi´ cations de lInstitut Statistique de l´ Universit´e de Paris 8, pp. 229-31. [Sol96] Solnik, B., Boucrelle, C., and Le Fur, Y., 1996,“International Market Correlation and Volatility”, Financial Analysts Journal, September-October, pp. 17-34. [Sto04] Stoev, S., Taqqu, M. S., 2004,“Simulation methods for linear fractional stable motion and FARIMA using the fast Fourier transform”, Fractals, Vol. 12, No. 1, pp. 95-121.

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[Sun07] Sun, W., Rachev, S., Fabozzi, F. J., 2007,“Fractals or I.I.D. : Evidence of longrange dependence and heavy tailedness from modeling German market returns”, Journal of Economics and Business, 59, 575-595. [Wer05] Weron, A., Burnecki, K., Mercik, S., Weron, K., 2005,“Complete description of all self-similar models driven by Levy stable noise”, Physical Review E, 71, p. 016113. [Whi63] Whittle, P., 1963,“On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral matrix”, Biometrika, 40, pp. 129-134. [Zha05] Zhang, L., Mykland, P.A., A¨ıt-Sahalia, Y., 2005,“A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High Frequency Data”, Journal of the American Statistical Association,Vol. 100, No. 472, pp. 1394-1411. [Zho96] Zhou, B., 1996,“High-frequency data and volatility in foreign-exchange rates”, Journal of Business and Economic Statistics, 14 (1), pp. 45-52.

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In: Economic Dynamics… Editor: Chester W. Hurlington, pp. 199-208

ISBN: 978-1-60456-911-7 © 2008 Nova Science Publishers, Inc.

Chapter 10

WHY INCREASED KNOWLEDGE DOES NOT NECESSARILY IMPROVE TRADING SUCCESS: A MONTE-CARLO SIMULATION# Jürgen Huber1,* and Michael Kirchler2 Univ. of Innsbruck, Dept. of Banking and Finance, Innsbruck, Austria

Abstract In this paper we investigate the widely held belief that success in the stock market can largely be attributed to the information underlying the trading decisions. We present results from a Monte-Carlo simulation of a financial market with asymmetrically informed traders whose information is based on future dividends. We observe a J-shaped distribution of returns: while the best informed can outperform all others, average informed traders have lower returns than the worst informed.

Keywords: Value of information, asymmetric information, Monte-Carlo simulation

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1. Introduction The statement “the more information I have, the better” is accepted by most people, as it seems intuitively obvious. The most frequently cited reference in this respect in economics is #

Most of the analyses presented in this paper are also covered in the longer paper “When more information can be harmful: Evidence from experimental asset markets” in “Information, Interaction, and (In)Efficiency in Financial Markets”, Festschrift for Klaus Schredelseker, Innsbruck, 2008. Some material in the Introduction and the Conclusion are similar to the respective sections in “J-shaped returns to timing advantage in access to information – Experimental evidence and a tentative explanation” by Jürgen Huber, published in the Journal of Economic Dynamics and Control, Vol. 31, 2007, p. 2536-2572. Copyright permission by Elsevier is gratefully acknowledged. 1 E-mail address: [email protected] * Authors address: University of Innsbruck, Department of Banking and Finance, Universitaetsstrasse 15, A-6020 Innsbruck, Austria, and 2 E-mail [email protected].

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probably Blackwell (1951, 1953). However, the widespread belief that having more information is always better (or at least never worse) in financial markets is surprising, given that researchers in many different disciplines have shown that more public or private information is not always better for those who use it. Especially game theoretical models teach us that more information may harm some or even all agents. Almost 60 years ago von Neumann and Morgenstern (1947) argued that a player may find it advantageous to forego some information. Savage (1954) lists several cases where information can be disadvantageous due to psychological reasons or because it makes bets impossible. Hirshleifer (1971) shows that public information in markets with risk-averse individuals can make them worse off as it may destroy insurance opportunities that would have been available without the public information. Gersbach (2000a, 2000b) shows that the value of public information in social choice situations may be negative for a majority of voters. Even in disciplines like supply chain management, models show that information can be harmful, as e.g. in Iyer et al. (2005). In a good overview Bassan et al. (1997) present several game theoretical settings where information can be advantageous to both players, disadvantageous to both, or advantageous to one and disadvantageous to the other – depending on the specific setting. Their main message is that “in an interactive decision framework with incomplete information, the relevant issue is that of interactive knowledge rather than simply knowledge per se.” (Bassan et al. 1997, 3). For us, the game-theoretical approaches are especially interesting, as we understand the market as a strategic game where investors try to outsmart each other. We think that Gibbons’ (1992, 63) conclusion that in game theory “having more information … can make a player worse off” also holds true for financial markets. Schredelseker (1984) claims, that information may be harmful for traders in a market context. He argues that an uninformed trader can only choose stocks randomly, earning on average the market return. If insiders are able to outperform the market, some traders (the average informed) have to receive returns below the market return. Schredelseker (2001) shows in a binomial setting that information is harmful to the average informed only if all traders actively use their information. However, if traders can learn and switch strategies they will do so until a situation is reached, where information is just useless but not harmful. We found the same result in experiments (Huber et al. 2006, 2008). According to Lakonishok/Lee (2001), Lin/Howe (1990), Krahnen/Rieck/Theissen (1999), and Jeng/Metrick/Zeckhauser (2003) insiders gain above-average returns, whereby stronger results can be found for purchases than for sales. After including transaction costs the evidence is mixed (e.g. Lin/Howe (1990) report insignificant results after transaction costs).1 Another strand of literature on private information deals with the performance of 'professionals' on financial markets. Here, the evidence regarding mutual fund managers speaks a clear language. Jensen (1965), and Malkiel (1995, 2003a, 2003b) find that between 60 and 80% of mutual funds underperform the relevant benchmark index. According to many authors this underperformance is due to transaction costs. After excluding these costs from the analysis mutual funds returns roughly equal the market return. But, it is very difficult to account for the survivorship bias in practice with the consequence that funds that are taken from the market during the examination period due to inferior performance are not included 1

Most of the authors subsume the investment behavior of insiders under a contrarian investment style, as they sell stocks that have risen in the past and they buy stocks that have fallen recently.

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in the sample. After correcting for these shortcomings, the average return of mutual funds before transaction costs will even be lower. Jensen, surprised by these results, wrote, “One must realize that these analysts are extremely well endowed. Moreover, they operate in the securities markets every day and have wide-ranging contacts and associations in both business and financial communities.” If they cannot beat the market, how can a small investor taking advice from his bank or some stock market newsletter expect to? This paper is organized as follows: After the introduction we present our market model in section 2. Section 3 presents results from a Monte-Carlo-simulation and section 4 concludes the paper.

2. Market Model

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2.1. Information Structure and Model Description In the past thirty years several authors (e.g. Grossman/Stiglitz 1980, Hellwig 1982, Figlewski 1982, Kyle 1985, Copeland and Friedman 1992, Ackert et al. 2002) have developed models with asymmetrically informed traders. However, all these models are limited to only two information levels: “uninformed” and “informed”. We present a model with more than two information levels. This is not only a quantitative, but also a qualitative change: with just two levels of information, it is no surprise that the informed will never perform worse than the uninformed. This allows us to analyze differences in trading profits among uninformed, average informed, and insiders. We believe that in real-world markets, relevant fundamental information is first known to insiders. A company’s managers know about major future developments in their enterprise well before word begins to spread through the market. This is supported by insider trading literature which shows that insiders gather information first and so they can generate aboveaverage returns, as for example Lakonishok/Lee (2001), Lin/Howe (1990), Krahnen/Rieck/Theissen (1999), and Jeng/Metrick/Zeckhauser (2003) show. Results of event studies support this reasoning by showing that abnormal returns start to accumulate long before companies’ earnings announcements, this means before the information becomes publicly available (e.g. Campbell/Lo/MacKinlay 1997). This indicates that better informed traders start trading on the information before it is published. We extend the concept of markets with two information levels to any desired number of information levels. We generally speak of Ix with x specifying the number of future dividends known to a trader. A trader with information level I1 knows the dividend for the current period. A trader with I2 knows the dividends for the current and the next period, and so on. Figure 1 illustrates this for five information levels. Dividends in period I1 I2 I3 I4 I5

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This design implies that information trickles down through the market from the best informed to the broad public over time. Even traders I1 get the same dividend information as I5 – only four periods later. For the sake of simplicity we assume that traders know the exact value of future dividends and that they never get wrong information. The underlying dividend process was designed as a random walk process without drift:

Dk = Dk −1 + ε

(1)

Dk represents the dividend in period k and ε is normally distributed with N (0; σ ) with a standard deviation of 0.12. The starting dividend D0 equals 0.8. From the dividends each agent computes the conditional present value of the stock (CPVj,k) given his information. This is calculated using Gordon’s formula, discounting the known dividends and assuming the last one as an infinite stream which is also discounted. CPVj,k stands for the conditional present value of the asset in period k, j represents the index for the information level of the trader, and re is the risk-adjusted interest rate.

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Furthermore, at the end of the period, the current dividend is paid out and the risk-free rate is paid for cash holdings. At the start of the next period, each trader receives new information previously known only to the next-best-informed trader – or completely unknown, in the case of I5. This means that the former dividend for period (k+1) is the dividend for period k one period later. The informational advantage of better informed traders is therefore one of time, as they can buy the risky asset at low prices before dividends rise and sell at high prices if dividends will fall in the future.

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3. Monte-Carlo Simulation Knowing which dynamics in fundamentals dominate seems to be crucial to judge on the usefulness of information in markets. To explore this we developed a simple Monte Carlo simulation with a call market mechanism. To keep things simple, we conduct the simulation with the following assumptions: the CPVs of all information levels are ranked from the highest to the lowest and the median is set as market price as this is the price where a call market clears. All information levels with a lower CPV take a short position of one stock; all with a higher CPV take a long position of one stock. We check on which side of the market the insider is (‘long’ if his CPV is higher than the median CPV and ‘short’ if it is lower) and which other traders are on the same side. The insider’s CPV is assumed to be a perfect estimate of the intrinsic value of the stock. The profit or loss for each trader in each simulation run is therefore determined by comparing the price (=median CPV) with the CPV of the insider (CPVinsider). When the insider’s CPV is below the median (i.e. the insider sold the asset) the loss for buyers equals price minus CPVinsider. The sellers gain the respective amount. Whenever the insider buys the asset

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(CPVinsider is higher than the median) the profit for buyers is CPVinsider minus price, while sellers lose this amount. The trader with the median CPV has no profit or loss in the respective simulation run. If the insider is the median trader no profits or losses occur. With this setting we conduct one million simulation runs to get reliable results. We conducted simulations with several numbers of information levels ranging from 5 to 255 – always finding consistent results. Here we exemplarily present results for 15 information levels as these are enough to get clear patterns and yet not too many to make it impossible to present them. The figure below presents the average net returns per simulation run, clearly showing, that there is no linear or monotone relationship.

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Figure 2. Return distribution in the Monte-Carlo simulation with 15 information levels.

While more information improves the net return starting with information level I6, I1 has higher returns than the better informed I2 to I6.2 I5 has the worst performance and I9 is the first to “beat the market” i.e. the first with a positive net return. For the first eight traders their information did not pay off. The simulation allows us to take a closer look at what causes this result. First we examine how often each of the information levels in on the right side (i.e. on the same side as the best informed), on the wrong side, and the median trader. The respective percentages for one million runs are displayed below. The frequency of an information level being on the right side of the market looks very similar as the return distribution presented above. In addition we see that the average informed are most frequently in the median position, while the most extreme (best and worst informed) have the lowest shares of being the median trader. We also see that I10 is the first

2

We conducted a Wilcoxon signed ranks test with 65,000 simulation runs and found I1 to have significantly higher returns than each of I2 to I6 at the 1 percent level.

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to be on the right side more often than on the wrong side while I9 (which is the first with a positive net return) is still wrong more often than right.

Market positions of different information levels 100%

percentage of cases

80%

60%

median right side wrong side

40%

20%

0% 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

information level

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Figure 3. Percentages the different information levels are on each market side in the Monte-Carlo simulation.

To better understand the results above, we conduct an additional analysis: we expect better informed traders to be able to estimate the intrinsic value (=CPVinsider) with a smaller estimation error than worse informed ones. When we examine the average absolute deviation of each information level’s CPV from CPVinsider this assumption is confirmed, as I1 has the highest average estimation error, which then decreases almost linearly. This is shown in figure 4. With these results it is still not clear why we see a negative slope in the distribution of net returns. However, we have to be aware that in a call market it is not important how far a trader’s estimate deviates from the intrinsic value, but the only relevant question is whether a trader is on the right or wrong side of the market, i.e. on the same side of the median CPV as the best informed. All transactions are settled at this median CPV (=market price), no matter if a trader’s CPV deviates just marginally or significantly from it. Next we calculate for each information level the average absolute deviation of the respective CPV from the median CPV. This is done separately for the cases when an agent ends up on the right vs. the wrong side of the market. The results are also displayed in figure 12 and they can add to our understanding: we see that with increasing information level the differences between a trader’s CPV and the median CPV decrease when the trader ends on the wrong side of the market. This means, that the worse informed sometimes are really far off the median CPV, while this is not true for better informed traders. When they end up on the wrong side this is often a close call. Still, the losses may be very high.

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Differences between traders CPVs and median or insider 16 14

difference

12 10 8 6 4 2 0 1

2

3

4

5

6

7 8 9 10 information level

11

12

13

14

15

difference to insider's estimate (=intrinsic value) difference to median when on the right side difference to median when on the wrong side

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Figure 4. Average differences between an information level’s CPV and the median CPV (=price) when on the right or wrong side of the market and to I15’s CPV.

The following example illustrates the point: suppose the CPV for I1 to I5 is 40.1, for I6 to I12 it is 39.9, for I13 it is 40, for I14 it is 44, and 50 for I15, which is also the intrinsic value. The median is I13’s CPV of 40, which becomes the market price. Seven traders (I6 to I12) are below and go short, while seven traders (I1 to I6, I14, and I15) go long – which in this case is the right decision. The profit for those who are long is quite high (50-40=10), while those who are short lose 10 each, even though their estimation error is just 0.1. The results for the right side of the market look quite different: here the deviations are highest for the best informed, meaning that they are quite sure about their choice, while the difference is much smaller for worse informed. Here we find a non-linear relationship again, as the worst informed have a higher estimation error than average informed. The final question we have to answer is, how much traders earn (loose) per case when they are on the right (wrong) side of the market. The respective numbers for 15 information levels are displayed below. We see that the losses per case on the wrong side are essentially the same up to I7 and then slowly decrease. When on the right side the seven highest information levels earn somewhat more per case on the right side than do worse informed. The results presented here are clearly not a final answer, as we set some unrealistically simplifying assumptions and equated being on the same side as the insider with being successful. However, they may still contribute to an understanding of what happens in markets and how this influences the distribution of returns across different information levels.

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Net return per case on right/wrong side 15 10

net return

5 0 -5 -10

right side wrong side

-15 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

information level

Figure 5. Net return per simulation run when a trader is on the right/wrong side of the market.

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4. Conclusion In this paper we presented results from a Monte-Carlo simulation of a financial market with asymmetrically informed traders. Our goal was to examine how information influences the distribution of net returns. Fama once wrote, “fundamental analysis is a fairly useless procedure both for the average analyst and the average investor” (in Bernstein 1992, 161). The distribution of net returns in our simulation corroborates this statement: while the best informed can outperform the market, all others can not. More information does not help these underperforming traders, as the average informed (e.g. I5) perform worse than I1. Yet after seven decades of studies consistently showing that gathering information and trading on it is not necessarily a way to succeed in the stock market, professionally managed funds are still a big industry, and newspapers and media providing stock market information make money even though evidence suggests that using this information makes investors worse off than a simple random strategy or index investment would. We think the impressive growth of index funds since their introduction in the early 1970ies can be interpreted as a rational reaction by market participants if they find that they cannot beat the market by trading on information. Probably investors become more experienced, probably also more modest and more willing to accept earning the market return instead of a promise – often unfulfilled – of earning more. William Fouse, who initiated the emission of the first index fund in 1970, warned about the “quicksand premise that increasing knowledge about a company guarantees greater forecasting success” (in Bernstein 1992, 245). In the early 1990ies about one-third of institutional money was already invested in

Why Increased Knowledge Does not Necessarily Improve Trading Success…

207

index funds (Bernstein, 1992). Bogle (1999) reports that in 1995 about 40 percent of all funds were invested in index instruments. In addition Cremers and Petajisto (2006) report a ‘silent indexation’ of actively managed mutual funds: while twenty years ago 99 percent of funds had 60 or more percent of their assets under active management the respective share dropped to less than 60 percent of fund in 2003. One out of eight of these actively managed funds actually had less than 20 percent of his assets under active management while the largest part is invested in the index. Still active information gathering is widespread. We think Cowles showed great understanding of the human psyche when he wrote, “Even if I did my negative surveys every five years, or others continued them when I’m gone, it wouldn’t matter. People are still going to subscribe to these services. They want to believe that someone really knows. A world in which nobody really knows can be frightening.” (Bernstein 1992, 38).

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References Ackert, L., Church, B., Zhang, P., 2002. Market behaviour in the presence of divergent and imperfect private information: experimental evidence from Canada, China and the United States. Journal of Economic Behavior and Organization 47, 435-450. Bassan, B., Scarsini, M., Zamir, S., 1997, “I don’t want to know!”: Can it be rational? Hebrew University Jerusalem. Center for Rationality and Interactive Decision Theory, Discussion paper 158. Bernstein, P.L., 1992. Captial Ideas, The Improbable Origins of Modern Wall Street. The Free Press, New York. Blackwell, D., 1951. Comparison of experiments. In: Neyman, J. (Ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkely. Blackwell, D., 1953. Equivalent comparison of experiments. Annals of Mathematical Statistics 24, 265-272. Bogle, J., 1999. Common Sense on Mutual Funds. Wiley, New York. Copeland, T., Friedman, D., 1992. The Market Value of Information: Experimental Results. Journal of Business 65, 241-265. Cremers, M., Petajisto, A., 2006. Active and passive positions of mutual funds. Working paper. Figlewski, S., 1982. Information diversity and market behavior, Journal of Finance 37, 87-102. Gersbach, H., 2000a. Public information and social choice. Social Choice and Welfare 17, 25-31. Gersbach, H., 2000b. Size and distributional uncertainty, public information and the information paradox. Social Choice and Welfare 17, 241-246. Gibbons, R., 1992. A Primer in Game Theory. Prentice Hall. Gosport. Grossman, S.J., Stiglitz, J.E., 1980. On the impossibility of informationally efficient prices. American Economic Review 70, 393-408. Hellwig, M., 1982. Rational expectation equilibrium with conditioning on past prices: A mean-variance example. Journal of Economic Theory 26, 279-312. Hirshleifer, J., 1971. The Private and Social Value of Information and the Reward to Incentive Activity. American Economic Review 61, 561-574.

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Huber, J., Kirchler, M., Sutter, M., 2008. Is more information always better? Experimental financial markets with cumulative information. Journal of Economic Behavior and Organization 65, 86-104. Huber, J., Kirchler, M., Sutter, M., 2006. Vom Nutzen zusätzlicher Information auf Märkten mit unterschiedlich informierten Händlern - Eine experimentelle Studie. Zeitschrift für Betriebswirtschaftliche Forschung 58, 38-61. Jeng, L.A., Metrick, A., Zeckhauser, R., 2003. Estimating the returns to insider trading: A performance-evaluation perspective. Review of Economics and Statistics 85, 453-471. Jensen, M.C., 1965. The Performance of Mutual Funds in the Period 1945-64. Journal of Finance 20, 587-616. Krahnen, J.P., Rieck, C., Theissen, E., 1999. Insider trading and portfolio structure in experimental asset markets with a long-lived asset. European Journal of Finance 5, 29-50. Kyle, A.S., 1985. Continuous auctions and insider trading. Econometrica 53, 1315-1335. Lakonishok, J., Lee, I., 2001. Are insider trades informative? The Review of Financial Studies 14, 79-111. Lin, J., Howe, J., 1990. Insider trading in the OTC market. Journal of Finance 45, 1273-1284. Malkiel, B.G., 1995. Returns from investing in equity mutual funds 1971 to 1991. Journal of Finance 50, 549-572. Malkiel, B.G., 2003a. Passive Investment Strategies and Efficient Markets. European Financial Management 9, 1-10. Malkiel, B.G., 2003b. The Efficient Market Hypothesis and Its Critics. Journal of Economic Perspectives 17, 59-82. Savage, L.J., 1954. The Foundations of Statistics. Wiley, New York. Schredelseker, K., 1984. Anlagestrategie und Informationsnutzen am Aktienmarkt. Zeitschrift für Betriebswirtschaftliche Forschung 36, 44-59. Schredelseker, K., 2001. Is the usefulness approach useful? Some reflections on the utility of public information. In Contemporary Issues in Accounting Regulation, McLeay, Stuart and Riccaboni, Angelo (Eds.), pp. 135-153.

INDEX

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A Aβ, 117, 118, 119, 120 access, 209 accounting, 105, 107, 108 accuracy, viii, 35, 37, 46, 48, 52, 93, 183, 204 ACF, 202, 203, 204 ADC, 201 adjustment, ix, 65, 67, 76, 127, 128, 129, 131, 136, 138, 165, 166, 173, 179 administrative, 2 ageing, ix, 127, 128, 131 agent, 83, 93, 96, 97, 98, 99, 144, 146, 147, 153, 160, 162, 212, 214 agents, viii, 4, 63, 64, 67, 99, 129, 137, 138, 210 aggregate demand, 67 aggregates, 105 aid, ix, 143, 145, 152, 160 algorithm, 37, 39, 41, 42, 45, 46, 48, 50, 53, 191 alternative, 2, 3, 8, 9, 10, 12, 13, 14, 16, 17, 24, 25, 27, 40, 76, 101, 132, 193 alternatives, vii, 35, 36, 109 amplitude, 79 analysts, 211 annual rate, 9 anomalous, 98 application, 39, 40, 113 arbitrage, 168 ARC, 127 arithmetic, 42, 173 Asian, 36 assessment, 64, 82, 182 assets, vii, 16, 35, 64, 76, 144, 167, 182, 188, 217 associations, 211 assumptions, ix, 46, 64, 78, 79, 106, 127, 128, 130, 144, 145, 182, 183, 184, 192, 193, 204, 212, 215 asymmetric information, 209 asymmetry, x, 76, 97, 181, 200, 201, 204 asymptotic, 144, 164, 187 asymptotically, 129

attention, 3, 36, 50, 70, 107, 182, 196 Australia, 10 Austria, 209 autocorrelation, 202, 203, 204 autonomous, 106, 108, 139 availability, 184 averaging, ix, 181, 182, 186, 187 aversion, 67

B barrier, viii, 36, 37, 38, 39, 42, 43, 46, 48, 49, 50, 52, 61, 168 barriers, 50 behavior, viii, 22, 23, 50, 71, 75, 81, 82, 84, 93, 94, 95, 97, 110, 145, 157, 163, 164, 166, 183, 192, 202, 203, 204, 210, 217 Beijing, 143 Belgium, 127 benchmark, vii, 1, 3, 9, 10, 11, 12, 15, 16, 17, 22, 23, 24, 25, 27, 28, 29, 46, 210 benchmarks, 40 benefits, viii, 98, 99, 127, 128 bias, 182, 183, 187, 188, 193, 194, 203, 204, 210 bifurcation, 73, 79 binding, 66 blame, 76 bond market, 200 Boston, 205, 206 boundary conditions, 139, 169, 174 bounds, 40, 44, 45, 48 Brownian motion, 166, 168, 173, 184, 189, 190, 191, 192 Brundtland report, 126 business, vii, 33, 63, 64, 65, 75, 76, 80, 102, 129, 211 business cycle, vii, 33, 63, 64, 75, 76, 80, 102, 129

C calculus, 104

Atlantic, 142

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210

Index

calibration, vii, 1, 9, 10, 11, 12, 15, 23, 24, 25, 28, 29, 46, 48 California, 181, 217 Canada, 128, 217 capacity, viii, 63, 64, 69, 70, 71, 74, 75, 80, 104, 141 capital, vii, viii, ix, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 29, 32, 33, 34, 63, 64, 65, 66, 67, 70, 71, 75, 81, 82, 83, 84, 86, 87, 89, 93, 94, 95, 96, 97, 98, 99, 104, 105, 106, 107, 108, 112, 114, 117, 118, 119, 120, 121, 123, 124, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 143, 144, 145, 146, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 160, 161, 163, 164, 166, 167, 179, 180 capital accumulation, ix, 21, 63, 64, 108, 124, 136, 138, 143, 144, 145, 146, 155, 158, 160, 161, 163 capital gains, 65 capital goods, 65, 105, 128 capital markets, 144 capitalism, 144, 145, 147, 149, 151, 152, 153, 154, 155, 157, 158, 159, 160, 161, 162, 163, 164 cast, 193 cations, 7, 207 causality, 129 certainty, 150 CES, 145, 161 Chi square, 196, 199 Chicago, 32, 163, 205 China, 143, 217 classes, 115, 123 classical, 64, 160 classification, 128, 165 classified, 2 clustering, 183 collateral, 76 collusion, 180 Columbia, 124 Columbia University, 124 commodity, 5 communities, 211 compensation, 65 competition, 65 complementary, 110 complex numbers, 77 complexity, 161 components, 108, 145 composite, 54, 112 composition, 32 computation, 30, 42, 75 computer, 89 computing, 3, 10, 36 concave, 5, 11, 15, 21, 26, 105, 146, 153 concentrates, 79 conception, 63 Conditional Distribution, 196 conditioning, 201, 217 confidence, 64, 67, 196 conflict, viii, 63, 64 conjecture, 104, 147, 162, 183

consensus, 2 conservation, viii, 103, 104, 111, 112, 113, 114, 117, 118, 119, 120, 121, 122, 123, 125 constant rate, 7, 50, 109, 113, 136 constraints, 104, 132, 135 construction, 42, 203 consumers, 64, 67, 76 consumption, viii, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 29, 65, 66, 67, 71, 76, 82, 83, 84, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 123, 127, 128, 130, 133, 135, 136, 137, 138, 144, 145, 146, 147, 149, 150, 152, 153, 154, 155, 156, 158, 160, 161, 163, 164 contamination, 182 contractions, 76 contracts, 39 control, 11, 31, 88, 89, 90, 103, 124, 132, 133, 134 convergence, vii, 21, 34, 35, 36, 37, 46, 51, 52, 53, 83, 97, 129, 138, 186 convex, 11, 169, 170 coordination, 64 copulas, 201, 204, 206 COR, 187 correlation, ix, x, 39, 167, 181, 182, 183, 187, 188, 192, 194, 195, 196, 199, 200, 201, 202, 203, 204, 206, 207 correlation function, 202 correlations, x, 181, 182, 196, 199, 200, 201, 202, 203, 204, 206 costs, viii, ix, 65, 66, 67, 127, 128, 129, 131, 132, 136, 138, 166, 210 CPU, 46, 47, 50 credit, 76 critical value, 72, 75, 79 criticism, 75 culture, 163 cycles, 63, 76

D damping, 39 debt, 6 decay, 202 decentralized, 131 decisions, x, 16, 64, 67, 76, 104, 123, 129, 209 decomposition, 185 defense, 2 deficiency, 10 deficit, 3, 6 definition, 28, 55, 68, 78, 88, 104, 110, 123, 189 degradation, 108 degree, ix, 64, 67, 89, 93, 127 degrees of freedom, 199 delta, 26 demand, 68, 77, 180 density, 36, 37, 38, 39, 41, 42, 45, 46, 52, 53, 150, 191, 197, 200, 201

Index depreciation, ix, 3, 6, 9, 26, 65, 66, 67, 75, 84, 113, 114, 123, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138 derivatives, 70, 71, 113, 115 desire, 144 deterministic, 145, 146, 150, 152, 153, 154, 155, 156, 158, 160, 161, 189 developed countries, 2, 6 deviation, 187, 199, 214 DFT, 192 dictatorship, 109 differential equations, 139, 140 diffusion, 46, 60, 161, 192, 206 diffusion process, 161, 206 dimensionality, 67 Dirac delta function, 174 discount rate, ix, 106, 109, 110, 113, 117, 119, 121, 123, 146, 159, 165, 166, 173, 179 discounting, 109, 212 discreteness, 184, 186 discretization, 36 disequilibrium, 68 dissaving, 93 distortions, 26 distribution, x, 36, 42, 139, 145, 150, 151, 157, 158, 159, 160, 161, 183, 186, 187, 190, 195, 196, 199, 200, 201, 204, 205, 209, 213, 214, 215, 216 diurnal, 184 divergence, 65 diversification, 164 diversity, 217 dividends, x, 209, 211, 212 downsizing, vii, 1, 2, 3, 12, 16, 17, 21, 22, 23, 27, 29, 33 Duality, 80 dynamic systems, 104 dynamic theory, vii dynamical properties, 207 dynamical system, 111

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E E6, 1 earnings, 211 Ecological Economics, 124 econometrics, 182 economic, viii, 2, 4, 24, 29, 32, 34, 63, 69, 75, 76, 81, 82, 93, 101, 103, 104, 105, 107, 108, 109, 110, 111, 112, 121, 125, 128, 129, 130, 131, 137, 138, 163, 164, 182 economic activity, 63, 76, 129 economic efficiency, 108 economic fundamentals, 24, 29 economic growth, viii, 32, 34, 103, 107, 128, 163, 164 economic performance, 105, 163, 164 economic policy, 2 economic problem, 112 economic systems, 121

211

economics, 33, 82, 104, 111, 112, 124, 126, 206, 209 economies, vii, viii, ix, 1, 2, 6, 9, 10, 27, 33, 63, 72, 82, 98, 143, 144, 160, 164 economy, viii, 2, 3, 4, 9, 10, 11, 12, 16, 17, 23, 27, 28, 31, 33, 34, 63, 64, 65, 67, 68, 69, 70, 71, 72, 76, 83, 84, 89, 93, 97, 99, 103, 104, 105, 106, 107, 109, 110, 112, 114, 121, 123, 129, 130, 131, 138, 144, 145, 148, 149, 150, 153, 154, 155, 157, 159, 160, 161, 163, 164 education, 163 efficiency level, 138 Efficient Market Hypothesis, 218 eigenvalues, 30, 71, 72, 79, 88 eigenvector, 80 Einstein, 190 elasticity, vii, 1, 3, 5, 9, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 29, 83, 97, 98, 130, 131, 137, 147 emission, 216 emotional, 104 employment, 65, 68, 75 endogeneity, 12, 129 endogenous, vii, viii, ix, 1, 2, 10, 27, 31, 33, 34, 63, 64, 103, 128, 129, 131, 136, 138, 145, 152 energy, 111 England, 125 English, 125 enterprise, 211 entrepreneurs, 69 environment, 2, 105, 107, 123 environmental, viii, 103, 107, 108, 110 environmental degradation, 107 equality, 5, 67, 71, 167, 171, 176, 189 equilibrium, vii, ix, 3, 4, 6, 7, 9, 10, 11, 24, 27, 29, 33, 63, 64, 66, 67, 69, 70, 71, 84, 129, 130, 131, 143, 144, 145, 150, 155, 156, 157, 158, 159, 160, 161, 164, 186, 217 equilibrium price, 186 equipment, 9, 64, 75, 128, 131 equity, viii, 34, 103, 104, 107, 108, 109, 110, 123, 125, 145, 201, 203, 206, 218 equity market, 203 estimating, 42, 194, 206 estimator, 185, 186, 187, 188, 189, 194, 203 estimators, ix, x, 181, 183, 186, 187, 188, 192, 193, 194, 195, 196, 197, 199, 202, 203, 204, 205 Euler, 132 Euler-Lagrange equations, 121 European, 32, 42, 47, 48, 51, 164, 218 evidence, ix, 93, 127, 128, 183, 196, 201, 203, 209, 210, 216, 217 evolution, 84, 133 exchange rate, 199, 201, 205, 207 exchange rates, 201 exclusion, 99 execution, 50 exercise, ix, 6, 39, 40, 42, 50, 51, 52, 136, 165, 166, 168, 170, 179, 180 exogenous, viii, ix, 2, 6, 84, 106, 108, 127, 128, 129 exotic, vii, 35, 36

212

Index

expansions, 76 expenditures, 2, 3, 6, 10, 129, 132, 136, 138 explosive, 30, 133 exponential, vii, 35, 36, 37, 46, 60, 93, 101, 191 exponential functions, 93, 101 externalities, 4, 131

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F failure, viii, 81, 82 family, 5 fat, 200 fax, 165 Federal Reserve, 34, 142, 163 Federal Reserve Bank, 142, 163 fertility, vii FFT, 36, 37, 48, 60, 191 finance, 3, 6, 12, 84, 182, 196, 206, 207 financial markets, ix, 65, 181, 183, 201, 202, 210, 218 financial support, 127 financial system, vii financing, 2 firms, ix, 2, 4, 64, 65, 67, 75, 76, 83, 127, 128, 129, 130, 131, 138, 165, 166, 180 first generation, 109 fiscal policy, vii, 1, 2, 4, 7, 10, 27, 30, 32, 82, 83, 96 flexibility, 65, 76 flow, 145 fluctuations, 63, 67, 76, 200 focusing, 12, 130 forecasting, 76, 182, 216 Fourier, 36, 39, 42, 43, 45, 46, 52, 53, 60, 191, 192, 207 fractal structure, 183 fractals, 204, 205 fractional Gaussian noise, 183, 190, 191, 192, 193, 207 France, 98 freedom, 196 friction, viii, 63, 76 full capacity, 136 full employment, 69 fund transfers, 82 funds, 84, 210, 216, 217 Fur, 207

G game theory, 210 games, vii Gamma, 151, 152 gauge, 166 Gaussian, vii, viii, ix, x, 35, 36, 37, 39, 41, 42, 45, 46, 51, 52, 53, 181, 182, 183, 189, 190, 191, 193, 194, 195 Gaussian random variables, 190 GDP, 2, 105

generation, 109, 110 Germany, 98, 181 government, 2, 4, 6, 10, 11, 12, 17, 34, 84, 144, 146 government expenditure, 144 grants, 127, 181 greening, viii, 103 grids, 82 gross investment, 75, 129, 131, 132, 133, 135, 136 growth, vii, viii, ix, 1, 2, 3, 6, 7, 9, 10, 12, 15, 16, 17, 18, 21, 23, 26, 27, 29, 32, 33, 34, 68, 81, 82, 98, 99, 104, 105, 107, 108, 110, 111, 113, 114, 123, 124, 125, 127, 128, 129, 135, 136, 138, 139, 143, 144, 145, 149, 155, 159, 160, 163, 164, 167, 216 growth rate, 7, 10, 135, 144, 149, 159, 160, 167 growth theory, ix, 104, 105, 107, 123, 125, 127, 128, 129

H Hamiltonian, viii, 83, 103, 105, 106, 108, 121, 122, 123, 125, 132, 139, 140 harm, 210 harmful, 26, 209, 210 Harvard, 80 health, 2 heart, 76 Hebrew, 217 heterogeneity, 160 heteroscedasticity, 182 high-frequency, 181, 182, 184, 188, 192, 196, 199, 202 Holland, 104, 125 homogeneous, 65, 131, 140, 144, 168 Hong Kong, 143, 165 horizon, 109, 110, 130, 156, 166, 205 household, 11, 82, 129 households, 4, 5, 7, 16, 26 human, 104, 114, 130, 201, 217 human capital, 104, 130 Hurst parameter, 204 hybrid, vii, 35, 37, 39, 46, 52 hyperbolic, 109, 202 hypothesis, 111, 117, 128, 129, 183, 190, 201

I identity, 70, 76, 112, 113, 114, 115, 118 idiosyncratic, 170 immigration, 91, 101 implementation, 99, 191 incentive, 16, 21, 93 incidence, 2 inclusion, viii, 81, 82, 93, 99 income, vii, 1, 3, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 29, 64, 66, 67, 82, 83, 84, 86, 87, 89, 94, 95, 96, 97, 98, 99, 106, 111, 123, 136, 146, 149, 150, 154, 155, 156, 158, 160, 161

213

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Index income distribution, 82 income tax, vii, 1, 3, 9, 10, 12, 15, 16, 17, 21, 22, 23, 26, 27, 29, 146, 149, 150, 154, 155, 156, 158, 160, 161 increasing returns, 4 independence, vii, 1, 16, 17, 26, 27, 135 indication, 194, 195, 202 indicators, 104, 105, 123 indices, x, 181, 192, 195, 196 induction, 41 industry, 6, 216 inelastic, 15, 67 inequality, 44, 53, 55, 56, 58, 59, 69, 101, 153, 169, 171 inertia, 76 infinite, 11, 38, 40, 109, 110, 130, 156, 166, 190, 212 inflation, 64, 65, 68, 76, 163 infrastructure, 2, 9, 34 inherited, ix, 130, 131, 165, 180 initial state, 7, 10, 11 Inspection, 168, 171, 179 instability, viii, 63, 72 instruments, 129, 217 insurance, 210 integration, 38, 39, 41, 42, 53, 82, 115 Interaction, 209 interest rates, ix, 66, 143, 144, 145, 155, 158, 160, 161, 163, 164 intergenerational, vii, viii, 103, 104, 107, 108, 110, 123, 125 international, 10, 102 interpretation, 75, 76, 104, 105, 111, 144 interval, 37, 38, 39, 40, 41, 44, 53, 135, 167, 184, 188, 191, 193 intervention, 17, 19, 20, 29, 182 intrinsic, 212, 214, 215 intrinsic value, 212, 214, 215 intuition, 173, 179 invariants, 113, 121 inversion, vii, 35, 36, 37, 39, 52, 60 investment, viii, ix, 1, 2, 3, 5, 10, 12, 15, 17, 21, 26, 27, 28, 32, 34, 63, 64, 65, 66, 69, 70, 75, 76, 104, 105, 108, 109, 121, 123, 127, 128, 129, 130, 131, 132, 133, 135, 136, 138, 154, 155, 165, 166, 167, 168, 169, 170, 173, 174, 176, 177, 179, 180, 210, 216 investment rate, 133, 135, 136, 138 investment ratio, 15, 17, 21, 26 investors, 64, 145, 195, 210, 216 involuntary unemployment, 63 IP, 188 IPV, 185, 186, 187 IS-LM, 64, 65 isolation, 82 Italy, 35, 98 iteration, 38, 47

J Jacobian, 88 Japan, 10 Japanese, 98 Jerusalem, 217 judge, 212 justification, 6

K kernel, 39, 41, 42, 43, 58, 190, 191 Keynes, 69, 105, 125 Keynesian, 63, 64, 65, 67, 76, 80 Keynesian model, 64, 65, 67 King, 5, 33, 63 Kolmogorov, 174

L L1, 43, 53, 54, 67, 68, 69, 70, 71, 72, 76, 77, 78, 79 labor, viii, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 29, 34, 63, 64, 65, 67, 68, 75, 83, 104, 146 labor force, 146 Lagrangian, 113, 114, 117, 118, 119, 120 language, 210 Laplace transforms, 60, 88, 174 lattice, 36 lattices, 36 law, 2, 30, 68, 70, 76, 83, 84, 104, 113, 114, 117, 118, 119, 120, 121, 123 laws, viii, 81, 84, 103, 104, 105, 111, 112, 114, 118, 119, 120, 121, 122, 123, 125 laws of motion, 81, 84 lead, viii, 64, 67, 81, 109, 110, 113, 144, 188, 199 leisure, 3, 5, 12, 17, 21, 22, 23, 26 lifetime, 11 limitation, 64 limitations, 76 linear, 21, 26, 66, 71, 81, 82, 99, 106, 117, 118, 119, 129, 130, 139, 140, 144, 145, 156, 159, 161, 168, 173, 174, 190, 191, 193, 207, 213 linear function, 106 linkage, 129 links, 75, 104, 105, 106 literature, vii, viii, ix, 2, 9, 24, 35, 36, 37, 63, 64, 65, 104, 106, 108, 112, 113, 130, 136, 155, 165, 166, 179, 180, 181, 182, 185, 193, 196, 210, 211 loans, 76 locus, 84 London, 32, 34, 80, 81, 207 long-term, 7, 21 losses, 97, 213, 214, 215

214

Index

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M machines, 66, 75 macroeconomic, 3, 17, 19, 20, 22, 81, 129, 164 macroeconomic models, 81 macroeconomics, 64, 81 maintenance, ix, 66, 127, 128, 129, 131, 132, 133, 135, 136, 138, 140, 142 management, 182, 210, 217 manifold, 11, 72, 73 manifolds, 72 manipulation, 42 mapping, 5, 24 marginal product, 4, 84, 93, 117, 123, 130, 132 marginal utility, 2, 113, 123, 147 market, viii, x, 36, 37, 63, 64, 65, 66, 67, 68, 69, 70, 71, 84, 131, 145, 164, 167, 180, 182, 183, 184, 186, 188, 193, 196, 200, 201, 204, 205, 206, 207, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218 market failure, 63, 64, 131 market prices, 164 market value, 64, 66 markets, 7, 82, 160, 161, 183, 196, 197, 199, 200, 201, 202, 203, 204, 209, 210, 211, 212, 215, 218 martingale, 37, 184 Mathematica, 63 mathematical, 119, 164 mathematics, 111 matrix, 41, 42, 47, 50, 71, 72, 88, 207 Mb, 71 measurement, 182 measures, 147, 153, 162, 182, 195, 197, 198 mechanical, 64 mechanics, 111 media, 216 median, 212, 213, 214, 215 memory, 184, 196, 202, 203, 204 microstructure, ix, x, 181, 182, 183, 184, 186, 187, 188, 192, 193, 195, 204 Ministry of Education, 1 misleading, 81, 99, 189 MIT, 80, 102, 164 modeling, 64, 67, 129, 182, 189, 190, 195, 206, 207 models, vii, viii, 2, 10, 29, 35, 36, 46, 48, 50, 63, 64, 67, 71, 81, 82, 99, 101, 102, 104, 107, 110, 111, 112, 113, 123, 128, 129, 138, 144, 145, 160, 161, 163, 182, 183, 193, 194, 195, 201, 203, 205, 206, 207, 210, 211 monetary policy, 161 money, 6, 68, 70, 76, 77, 216 money supply, 68, 76 monotone, 15, 213 Monte-Carlo, x, 36, 183, 194, 209, 212, 213, 214, 216 Monte-Carlo simulation, x, 209, 213, 216 Moscow, 60 motion, 30, 68, 70, 76, 83, 84, 145, 146, 167, 174, 190, 191, 207 movement, 64, 75, 133

multidimensional, 36 multiplier, 132 multivariate, 199, 200, 201, 207 multivariate distribution, 201 mutual funds, 210, 211, 217, 218

N NASDAQ, 200 national, 70, 76, 105, 106, 107, 108, 124, 128, 164 National Academy of Sciences, 125 national expenditure, 105 national product, 124, 128 natural, ix, 41, 57, 63, 104, 107, 108, 109, 110, 112, 113, 114, 117, 123, 127, 128, 190, 201 natural resources, 104, 107, 108, 113, 123 negative relation, 15, 26, 27 net income, 96 net investment, 66, 68, 75, 108 net national product, 106 network, 207 New England, 34 New Jersey, 63 New York, 60, 80, 124, 217, 218 newspapers, 216 Newton, 52, 60 nodes, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54 noise, ix, x, 181, 182, 183, 184, 186, 187, 188, 190, 191, 192, 193, 194, 195, 204, 205, 207 nonlinear, 145, 161 non-linear, 65, 81, 99, 134, 140, 215 non-linearities, 82, 93, 99 non-linearity, 93 nonparametric, 182 non-random, 183 non-renewable, 107, 112 non-renewable resources, 107, 112 normal, 76, 183, 196, 199, 201 normal distribution, 183, 196, 199, 201 norms, 43, 163 null hypothesis, 199

O obligation, 167 observations, 182, 184, 185, 188, 189, 192, 193, 195 OECD, 32 one dimension, 36 operator, 156 optimization, viii, 46, 83, 84, 103, 107, 111, 114, 130, 131, 146, 147 orbit, viii, 63, 73 OTC, 218

Index

private sector, 3, 16, 26, 34 probability, 36, 37, 38, 39, 41, 42, 166, 170, 173, 176, 187, 189, 195 paper, vii, ix, 1, 2, 3, 4, 6, 9, 29, 33, 35, 37, 39, 42, 52, probability density function, 36, 38, 39, 41, 42, 173 60, 63, 64, 68, 72, 107, 109, 110, 131, 143, 145, probability theory, 189 160, 161, 164, 165, 166, 179, 188, 189, 194, 196, procedures, 193 205, 206, 209, 211, 216, 217 producers, 64 paradox, 217 production, ix, 2, 4, 7, 9, 27, 33, 65, 75, 84, 105, 107, parameter, vii, ix, 1, 3, 9, 23, 24, 28, 38, 40, 48, 67, 68, 108, 112, 114, 115, 116, 117, 118, 119, 120, 121, 79, 84, 90, 97, 110, 119, 127, 128, 130, 135, 138, 123, 127, 128, 129, 130, 133, 138, 143, 144, 145, 139, 147, 152, 153, 162, 189, 191 146, 147, 148, 149, 152, 154, 157, 158, 160, 161, Pareto, 98, 183 162, 163, 164 Pareto optimal, 98 production function, ix, 27, 65, 84, 105, 107, 114, 115, Paris, 103, 207 116, 117, 118, 119, 120, 121, 123, 130, 133, 143, partial differential equations, 111 145, 146, 147, 148, 149, 152, 160, 162 partition, 37 production technology, 4, 129, 144, 161 passive, viii, 64, 127, 128, 129, 217 productivity, 2, 24, 32, 33, 130, 136 pay off, 213 productivity growth, 32 per capita, 130, 131, 133, 135, 136, 138, 139 profit, 65, 68, 71, 75, 77, 212, 213, 215 per capita income, 136, 138, 139 profitability, 64, 76 performance, 34, 192, 195, 203, 210, 213 profits, 65, 66, 76, 211, 213 periodic, viii, 63, 73, 79 program, 46, 107 permanent income hypothesis, 64, 67 property, 97, 99, 189 permit, 201 proposition, 72, 73, 78, 123, 156, 169, 171, 173, 178 perturbations, viii, 81, 82, 84, 88, 89, 90, 93 proxy, 4, 68, 185, 186 pessimism, 76 psyche, 217 pH, 191 psychological, 210 Phillips curve, 68 public, vii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, physics, 111 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, pku, 143 32, 33, 34, 210, 212, 217, 218 planning, 130 public capital, vii, 1, 2, 3, 4, 6, 9, 11, 12, 15, 16, 17, 18, plausibility, 160 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 32, 33 play, 68, 93 public debt, 2, 6 policy instruments, 11 public enterprises, 10 policy makers, 99 public expenditures, 2, 6, 8 policy variables, 88 public finance, 6 political, 2 public investment, vii, 1, 2, 3, 4, 10, 11, 12, 13, 14, 15, pollution, 107 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, polynomial, 77, 78 33 poor, 138 public sector, 2, 5, 6, 11 population, ix, 5, 128, 130, 137, 138 population growth, ix, 5, 128, 137, 138 portfolio, 156, 164, 167, 168, 182, 195, 207, 218 Q Portugal, 10 poverty, 109 query, 129 power, 65, 79, 144, 185, 186, 187, 188, 190, 193, 203, 205 R powers, 168, 185 preference, viii, 26, 103, 111, 138, 149 random, 63, 183, 191, 192, 196, 201, 212, 216 premium, 34, 145, 163, 164 random walk, 183, 196, 212 present value, 66, 67, 83, 106, 173, 212 range, 11, 16, 17, 23, 36, 81, 97, 192, 204, 207 preservative, 110 rate of return, 9, 10, 13, 14, 15, 19, 20, 21, 93, 97, 166, prestige, 144 167, 168, 173, 180 prices, 7, 46, 48, 49, 50, 51, 53, 64, 67, 68, 75, 76, 108, rationality, 67, 160 145, 163, 164, 184, 212, 217 real wage, viii, 63, 64, 65, 66, 68, 69, 70, 71, 75, 76, printing, 6 77, 78 private, vii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 19, 20, realism, 64 21, 22, 23, 24, 26, 27, 29, 34, 210, 217 reality, 128 private investment, 3, 5, 16, 22

P

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215

216

Index

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reasoning, 16, 211 recall, 22, 41, 42, 43, 44, 50, 54, 68, 75, 105 recalling, 76 recovery, 22 recursion, 40, 41, 42, 56 reduction, 16, 17, 22, 95, 155 regional, 34 regression, 206 regular, 42, 45, 54, 55, 124, 203 relationship, ix, 3, 12, 15, 18, 21, 23, 24, 26, 27, 29, 84, 88, 107, 118, 128, 145, 148, 165, 166, 171, 173, 176, 177, 178, 179, 180, 200, 213, 215 relative size, 98 relevance, 12, 29, 109, 128 renewable resource, 109 repair, ix, 127, 128, 129, 131, 138 research, vii, 64, 76, 107, 127, 181, 196 researchers, 182, 183, 200, 210 resolution, 145 resources, ix, 2, 5, 16, 26, 89, 109, 110, 117, 127, 128, 129, 131, 133 returns, x, 4, 7, 82, 84, 130, 144, 160, 183, 184, 185, 186, 187, 188, 189, 193, 195, 196, 199, 200, 201, 204, 206, 207, 209, 210, 211, 213, 214, 215, 216, 218 returns to scale, 7, 130 revenue, 6, 66 rewards, 144 rigidity, viii, 63, 72, 75 risk, 9, 36, 42, 65, 67, 83, 144, 145, 147, 153, 156, 162, 163, 164, 167, 170, 182, 206, 207 risk assessment, 182 risk aversion, 9, 65, 83, 147, 156, 162, 164 risk management, 206, 207 robustness, 3, 23, 187

series, 31, 79, 183, 190, 192, 195, 196, 197, 199, 200, 202, 203 services, 3, 217 shadow prices, 105, 106, 108 shape, 84 shares, 213 shock, 89, 90, 202 shocks, ix, 63, 81, 84, 88, 89, 93, 143, 144, 145, 146, 149, 154, 156, 157, 158, 160, 200, 204 short run, 26 short-term, 16 sign, vii, 1, 29, 69, 78, 79, 165, 166, 171, 173, 176, 180 signs, 140 similarity, 70 simulation, 1, 6, 183, 191, 194, 195, 203, 205, 212, 213, 214, 216 simulations, x, 72, 75, 181, 183, 193, 194, 195, 204, 213 singular, 42, 50 skeptics, 205 skewness, 195, 207 smoothing, 163 social, ix, 2, 104, 105, 108, 111, 124, 143, 144, 146, 147, 153, 161, 162, 163, 210, 217 social security, 2 social status, ix, 143, 144, 146, 147, 153, 161, 162, 163 social welfare, 104 society, 104, 105, 106, 107, 144 Solow, 104, 109, 123, 125 solutions, ix, 5, 36, 77, 133, 143, 144, 145, 148, 149, 158, 160 Spain, 1, 127, 142 speculation, 183 speed, 46, 65, 76, 97 spillovers, 130 S St. Louis, 34 stability, viii, 31, 63, 64, 69, 72, 73, 78 sacrifice, 3, 12, 22 standard deviation, 187, 188, 195, 203, 212 sales, 210 standards, 9 sample, 182, 189, 190, 191, 193, 195, 201, 211 state-owned, 6 sample variance, 182 stationary distributions, ix, 143, 145, 158, 159, 160, sampling, 182, 184, 186, 193, 205 161 saving rate, 15, 136, 138 statistics, 193, 195, 196, 198, 200, 201 savings, viii, 12, 13, 14, 15, 16, 23, 63, 67, 76, 93, 104, steady state, 67, 69, 71, 72, 73, 75, 76, 77, 78, 83, 84, 127, 128, 144, 164 88, 93, 94, 95, 96, 97, 134, 158 savings rate, 12, 13, 14 steady-state growth, 15, 17 scalar, 88 stochastic, ix, 34, 36, 60, 143, 144, 145, 146, 149, 150, scarce resources, 104 154, 155, 156, 158, 160, 161, 164, 170, 182, 184, scarcity, 125 189, 191, 205, 206, 207 science, 104 stochastic model, 145, 161 search, vii, 11, 139 stochastic processes, 36, 189, 206 SEC, 127 stock, ix, x, 5, 11, 37, 64, 65, 66, 67, 70, 71, 75, 80, 84, securities, 211 93, 105, 106, 107, 113, 119, 123, 127, 128, 129, self-similarity, 189, 204, 206 130, 131, 132, 133, 135, 136, 138, 143, 145, 146, sensitivity, 3, 4, 9, 23, 29, 66 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, Sensitivity Analysis, 23, 24 160, 161, 164, 181, 195, 199, 200, 204, 205, 209, separation, 129 211, 212, 216

217

Index stock exchange, 199 stock markets, 200 stock price, 37 strategic, 210 strategies, 26, 210 STRUCTURE, 143 substitutes, 108 substitution, vii, 1, 3, 5, 16, 24, 26, 29, 83, 97, 98, 130, 137, 147 substitution effect, 16 Sun, 181, 182, 183, 184, 186, 188, 190, 192, 194, 196, 200, 202, 204, 206, 207 supercritical, 73 supply, 2, 3, 12, 15, 34, 67, 68, 75, 83, 210 supply chain, 210 supply shock, 75 surplus, 3, 101 surprise, 211 sustainability, viii, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 123, 124, 125 sustainable development, viii, 103, 108, 111, 124, 125 symmetry, 112, 195 synchronization, 189 synthesis, 207 systematic, 170 systems, viii, 103, 111

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T targets, 110 tax cuts, 93, 97, 98, 99 tax incidence, 82 tax increase, 93, 97, 99 tax rates, 5, 7, 9, 98, 99, 146 tax system, vii, 1, 2, 3, 6, 9, 10, 11, 12, 16, 17, 18, 23, 26, 27, 29 taxation, 3, 26, 84, 90, 96, 97, 98, 99, 101, 102, 149 taxes, vii, 1, 3, 6, 8, 9, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 29, 33 taxonomy, 108 Taylor expansion, 82 Taylor series, 112 technical change, 111 technological, viii, 4, 63, 103, 106, 107, 113, 130 technological change, 107 technological progress, viii, 103, 106, 113 technology, vii, viii, 1, 7, 65, 103, 129, 130, 144, 145, 146, 156, 158, 159, 161 textbooks, 64, 65 theoretical, 12, 105, 107, 130, 136, 183, 186, 199, 205, 206, 210 theory, vii, viii, ix, 75, 103, 124, 125, 128, 129, 144, 160, 163, 164, 182, 183, 184, 186, 187, 201 third boundary condition, 168 third order, 52, 93 threshold, 40, 98, 167 threshold level, 167 thresholds, 98

time, viii, ix, 2, 3, 4, 5, 9, 11, 23, 26, 31, 36, 38, 40, 41, 42, 46, 47, 48, 50, 65, 70, 76, 77, 82, 83, 88, 90, 93, 96, 97, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 115, 119, 121, 123, 131, 132, 135, 139, 140, 144, 150, 158, 163, 165, 166, 167, 168, 169, 170, 173, 176, 179, 180, 184, 186, 187, 188, 189, 190, 193, 195, 196, 197, 199, 200, 202, 203, 205, 206, 212 time frame, 110 time series, 9, 11, 31, 187, 188, 195, 196, 199, 202, 203 timing, 32, 165, 170, 180, 209 total costs, 66 total revenue, 65 trade-off, 3, 12 trading, x, 195, 209, 211, 216, 218 traffic, 207 trajectory, 66, 70, 72, 76, 104, 135 transaction costs, 184, 186, 210, 211 transactions, 214 transfer, 5, 82, 83, 84, 96 transfer payments, 82 transformation, 38, 65, 81, 105, 112, 119, 121, 123, 187 transformations, viii, 103, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122 transition, vii, 1, 2, 3, 9, 12, 16, 17, 21, 22, 23, 29, 36, 37, 38, 39, 41, 42, 45, 46, 52, 53 translation, 119, 125 triggers, ix, 165, 179

U uncertainty, ix, 67, 76, 144, 153, 154, 160, 163, 164, 165, 166, 170, 171, 173, 176, 177, 179, 180, 217 unemployment, 68, 75 uniform, 37, 44, 53, 191 univariate, 39, 201 uti, 184

V validity, 186 values, vii, 1, 2, 3, 7, 9, 10, 11, 13, 14, 15, 17, 23, 26, 27, 30, 31, 50, 51, 64, 67, 69, 70, 71, 72, 73, 75, 77, 79, 84, 88, 89, 90, 93, 96, 97, 99, 121, 123, 136, 139, 152, 159, 160, 165, 167, 186, 187, 194, 195, 199, 200, 203 variability, 129 variable, ix, 3, 10, 19, 20, 30, 31, 39, 43, 64, 65, 68, 70, 72, 89, 90, 115, 122, 123, 128, 129, 130, 132, 133, 138, 139, 140, 150, 157, 158, 161, 193 variables, viii, ix, 3, 4, 6, 7, 9, 10, 12, 17, 19, 20, 22, 30, 31, 65, 69, 70, 71, 72, 75, 76, 77, 79, 81, 82, 88, 103, 111, 112, 115, 127, 128, 129, 131, 133, 134, 135, 136, 138, 139, 140, 145, 152, 159, 191, 192, 193, 196, 201

218

Index

variance, 182, 184, 187, 190, 193, 194, 196, 200, 201, 205 variation, 64, 184, 185, 186, 187, 194, 201, 205 vector, 30, 84, 88, 105, 133, 191 visible, 202 volatility, ix, x, 36, 48, 67, 76, 165, 166, 167, 179, 180, 181, 182, 183, 184, 185, 186, 187, 192, 193, 194, 195, 196, 197, 198, 199, 202, 203, 204, 205, 206, 207 voters, 210

98, 99, 100, 101, 102, 106, 108, 109, 124, 128, 130, 136, 164 welfare loss, 82, 97, 99 wellbeing, 108 well-being, 104, 108, 124 wisdom, 68 workers, 130 working hours, 16, 188 World Bank, 128, 143 WP, 32, 66 writing, 123

W X

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wage rate, 64, 65, 68, 70 wages, 2, 64, 67, 68, 70, 75, 76, 83 X-axis, 19, 20 Washington, 143 wealth, 66, 71, 105, 106, 111, 124, 144, 145, 156, 163, Y 164 wealth effects, 164 Y-axis, 19, 20 wear, 131 welfare, vii, viii, 1, 2, 3, 4, 7, 10, 11, 12, 16, 17, 18, 21, yield, 37, 48, 81, 82, 96, 99, 167, 170, 173, 176 22, 23, 26, 27, 28, 29, 33, 81, 82, 83, 84, 93, 96, 97,