Quantitative Methods in Maritime Economics [2 ed.] 1870093313

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Quantitative Methods in Maritime

Economics Second Edition J J Evans~& P B Marlow ID ‘II.

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John Evans and Peter Marlow

Second Edition


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Published and distributed by FAIRPLAY PUBLICATIONS LTD. 20 Ullswater Crescent, Coulsdon Business Park, Coulsdon, Surrey CR5 2HR,

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Web: http: //wwwfairplay-publications.co.uk ISBN 1870093 31 3 Copyright © 1990 J. J. Evans and P. B. Marlow First Published 1986 Second Edition April 1990 Reprinted August 1997 All rights reserved. No part of this publication may be reproduced or transmitted, in any fonn or by any means, electronic, mechanical, photocopying, recording or any information storage or retrieval system without permission in writing from the publisher. Printed by ‘Wednesday Press Ltd. Unit 3, Baron Court, Chandlers Way, Temple Farm Industrial Estate, Southend-on-Sea, Essex SS2 SSE.

About the authors John Evans John Evans was born in 1933. He joined the Merchant Navy as an apprentice in I951 and served as a deck oificer with the New Zealand Shipping Company. After attaining the qualification of Extra Master in 1963 he began-a career in teaching at what was then the Welsh College of Advanced Technology. He began to specialise in maritime economics in 1971. He has worked on a consultancy basis for UNCTAD and also for CENSA in connection with which he was one of the main authors of the 1978 UWIST study, Liner Shipping in the US Trades. He is currently a lecturer in the Department of Maritime Studies, University of Wales College of Cardiif. In 1981 he completed his M.Sc. by research in the field of liner freight rates. He was appointed editor of the Journal of Maritime Policy and Management in 1984. He plays golf when he has time. He is married with two sons.

Peter Marlow Peter Marlow is married with three children and was educated at the University College of Wales, Aberystwyth where he obtained a joint honours degree in Economics and Statistics prior to working as a Research Assistant in the Department oi‘ Economics. He subsequently obtained a higher degree by research in the field of quantitative economics. Following a short period as a school teacher he joined the Department of Maritime Studies of the University of Wales Institute of Science and Technology (now the University of Wales College of Cardifl') where he is employed as a lecturer. His main areas of interest include transport economics and the fiscal treatment of shipping and he has undertaken studies for a variety of outside bodies including the Commission of the European Communities, the Organisation for Economic Cooperation and Development, the General Council of British Shipping, and various government departments. He is a Member of the Chartered Institute of Transport and of the Institute of Logistics and Distribution Management. In 1989 he was awarded a doctorate for his work on investment incentives in shipping.

To Ann and Alma


Foreword After spending a number of years teaching university students various aspects of quantitative methods, especially in relation to maritime economics, it became increasingly evident that the literature available for students to consult was widely dispersed and, even then, the content was often not at the level required. Furthermore, many of the text books on quantitative methods tend to be incomprehensible and raise barriers to those with limited mathematical ability: seldom can one find applications in the maritime context. ' We decided therefore to attempt to write a book that would, as far as possible, be selfcontained and cater for the general needs of students of maritime economics and possibly others concemed with shipping andrelated activities. We have not striven to achieve enormous scholastic heights; rather we have tried to provide a self-supporting volume in which many cross-references can be made. We have tried to explain principles, models and methodologies rather than leave the mathematics to explain itself. Since a knowledge of calculus, for example, is desirable in order to follow the mathematical developments in certain chapters we decided to include a chapter containing the rudiments of that subject. Similarly, we have included chapters on analytical geometry, power series and probability theory which contain fundamental information suflicient to enable readers with little or no knowledge of those subjects to be able to follow the main parts of the text; should they wish to acquire a more advanced or detailed understanding, they can consult a number of texts cited in the bibliography. We have not attempted to teach economics as such. We assume that readers have a reasonableqknowledge of this, though one chapter is devoted to a presentation of basic economic principles. A number of important topics are dealt with, e.g., supply and demand; optimum speed, size and distance of ships; correlation —- both simple and multiple; queueing; and decision theory. Examples are given in the maritime context throughout. In this second edition we have taken the opportunity to include some new topics which with hindsight might well have been included in the first. These are: the supply of sea transport; voyage estimates; freight futures; investment appraisal; linear programming; replacement theory; and the efleet of shipping on the balance of payments. The chapter on optimum speed has been enlarged; there has been some restructuring; changes made in the order of chapters; and the opportunity has been taken to make minor changes in the text as necmsary and/or appropriate. We acknowledge with gratitude our appreciation for the information so willingly supplied by the General Council of British Shipping; Worldscale Association (London) Ltd.; and the Far Eastern Freight Conference without whose help the chapters on freight index numbers and CAFs, BAFs & EAIDs would not have been possible; we thank them for giving their permission to reproduce text from some of their oficial publications and other material provided.


We also thank HMSO for permission to reproduce data, extracted from the Pink Book 1988, contained in Table 20.1 (page 230); Taylor 8t Francis Ltd. for permission to reproduce text (ch. 6 (part); ch. 18 (part); and ch. 19), written by one of the authors, that originally appeared in the Journal of Maritime Policy and Management. Our grateful thanks are also due to Mrs. Barbara Fletcher, Director of Education of the Institute of Chartered Shipbrokers; Mr. R. W. Porter, manager of Worldscale (Association) Ltd.; Mr. Mark Lewicki of the Far Eastem Freight Conference; and Mr. David Ellis of Graig Shipping Co., Cardilf for their invaluable assistance in the preparation of certain sections of the book. We are indebted to colleagues in the Department of Maritime Studies of the University of Wales College of Cardiff, especially to Alun Rogers and Andrew Wiltshire of the Cartography Unit who produced the diagrams and figures; also to Sian Poppleton and Tracy Shellard for their unenviable task of typing the manuscript. Any errors or omissions that remain are, of course, the responsibility of the authors. J. J. Evans P. B. Marlow


Contents Chapter

1 Elements of calculus Relevant aspects of analytical geometry Progressions and series Probability theory and distributions Basic economic relationships. The demand and supply ofsea transport Oiflli -al \-I 7 Optimum speed of ships Ships’ costs; voyage estimates; Worldscale E1 Freight futures 10 The optimum size of ships ll Liner freight rates 12, Linear programming and transportation 61:3)Regression and correlation 14 Decision theory 15 Queueing theory 16 The theory and practice of index numbers 17 Currency, bunker and inflation differential factors 18 Investment appraisal in shipping 19 Replacement, obsolescence and modifications of ships 20 Shipping and the balance of payments 21 Calculations in shipping economics Annex I Derivation of formulae used in discounting Annex II Tax relief on capital allowances Annex III To find the PV of interest payments on loan outstanding Annex [V Analysis of an annuity Appendix A Maclaurin’s theorem Appendix B Speed and fuel consumption


Pass 1 l3 24 29 43 61 79 92 116

124 ll!

Qi;€i§> 156 167 I 77 191 197 221 229 235 261 26.4 266 268 270 272

Appendix C Conditions for ship total revenue to remain constant

when one cargo is substituted for another Bibliography References and notes






275 276

Chapter 1 Elements of Calculus The fundamental purpose of difierential calculus is to determine the slope of any given function at a specific point on the curve. 2 Consider the parabola y = % + 6 as shown in Figure 1.1.

Figure 1.1 Y


.o The slope at any point A is defined as the t_a_nge_n_t __Qf_ the angle that the tangent to the curve at A makes with the x axis, i.e. Tan tp in the Figure. Clearly, the task of finding the slope, by drawing, would be tedious and the result quite inaccurate. The angle ¢ is always measured anti-clockwise from the direction of the positive abscissa and can have values in the range 0-180°. A slope of 0° means that the curve is horizontal; from 0-90° the slope is directed upwards to the right; and from 90°-180° the slope is directed downwards to the right.

Figure 1.2 Y

pi I1'

I '\

— t




------I Q

-II\|$iq. _-



.-\._-. _._

1-:,. .


X ‘»~+A". :1-—£._42_4._i‘ ;'ti


Now consider the curve shown in Figure 1.2 where y is a function of x (f(x)). A is any point on the curve and B is another point close to A so that AB is a ghord whose slope approximates to the slope of the curve at A. The incremental values of B relative to A are expressed by by and bx in the vertical and horizontal directions respectively. Thus the slope _



of the chord 15 given by Tan BAC =


by :'_-. ._ _- _-. ¢_, =-:._ ‘.j:-'. -_

If for example, the curve represents a part of the function y = x3 + 6 since point A given by (x, y) lies on the curve, so then does point B given by (x + bx), (y + by).




~I I.


and expanding (x + bx)?‘ binomially (see Chapter 3), E \


- 3.2


yj+ by = xi + 3x2bx + Yxbxz + bx3 + 61

or by = 3x2bx + 3xbx2 + bx’ (since y = x3 +


II L4 _fi

by 2 so that 5 = 3x + 3xbx + bx’. I _1a

As bx becomes smaller so will B more closely approach A and the slope of the chord AB will eventually be equal to the slope of the tangent to the cu1've at A.

IL 44.

I. ‘I’

Thus, in the limit, as bx —> 0 (Le. as bx closely approximates to 0 without ever reaching that value): by


I I‘



ax - 3x .

‘*4 -pr"

I "rt

It will be noticed that even though bx tends towards 0, the ratio %I- remains finite. The x



b d limiting value of F: is called the differential coefficient of x being written ax! and it should be understood that this is not a fraction in the ordinary sense. ‘The differential coefficient therefore, gives the precise slope of a curve at any point on it. In this example when x == l the slope is 3 X 13 = 3 and when x = 1/\/3 the slope is 3.(1/'\/3): = 1. In the original equation (y = xi + 6) it may be seen that the effect of the constant is to raise all the values of y by 6 units without affecting the slope at any point.

The General Case d If y = ax“ + b where a and b are constants it may readily be shown that ay = anx"" where n is any real number either positive or negative. It follows that in the case of a straight line of the form ax + by + c = 0, after rearranging ax c y=—?—Eand d the slope, FE = -3 which as is already well known is constant. If the straight line is in the form y = mx + c

d then F3 = m which again is known from fundamental analysis of the straight line.

Difierentiation by Substitution There are some functions that cannot be differentiated by following the general principle shown above. For example, if y = \/(x - 2) another method must be found. A useful method is by substitution: Letu=x-2. Theny=\/u =u1/2 now ix = in"/2


and it may be accepted (without proof) that dy d _ du _=_Y dx


Nov\Isinceu=x-2, %=1

1 I


i—‘%,-Q-. ,

and%;- = iu'1/2 .1


= “X ___ 2)-1/2 or _...L...._.

2\/(x —- 2)

Differentiation of e’


It will be shown (Chapter 3) that xz xi e"=1+x+-2-T+§+... (whereehasavalue2.7l8...)


so that if y = e" -—-.'-I~J1-







--1+X'I-5+.-. =ex

This is the only example where the differential coefficient is equal to the function itself.

-._.§_ ._ _ . ,p_.- lg»--—

d In general: E; (ae‘”‘) = abe""

Differentiation of log,x If y = Log,x then e’ = x, and differentiating both sides of the equation with respect to x, d(e!') Q = 1 dy ' dx

_ ..eY

dlt dx-1

. d(e’) since dy -eY

Hence dx fi-i_1 -ey - X _ d 1 t.e. dx (logex) - X


This is a most important result because of implications for the reverse process known as integration (q.v.) A‘.‘S, -F»-. -»1.an-up»s-rn.¢,=q0h-IMnu.—sv-a4./-Ih—n'~-d¢a?|p'-iI~#¥1J-nmQ—-an— 1

D;'fi'erentiatz'0n of logax i I

where a is any positive number. It can be proved that


log,x = log,x . log,e

0 3: * v


or in words, the difl'erelntial of a product is equal to the first times the

differential of the second plus the second times the differential of the first. d Example: Let p = (3s + 2)3(s2 — 4)2 and it is required to find the differential coefficient —p I

1 . Qt 95‘


In this case v = (s2 — 4)2


1 ?’ S


and u = (3s + 2)3

= §i_">‘*"‘-‘TI ‘I S

d = 3(3t +-2)= . 3 = sot + 2)1 So that F: d

and —v = 2(s2 - 4) -. 2s = 4s(s2 - 4) ds d


Hence FE = (s2 — 4)29(3s + 2)’ + (3s + 2)34s(s2 — 4) which reduces to: g-E = (s2 - 4)(3s + 2)2{9(s2 — 4) + 4s(3s + 2)}


= (52 — 4)(3s + 2)2(21s2 + 8s - 36) 5 1

Differentiation of a Quotient


Let u and v be functions of x so that II

i'=; b meny+by=%-F: u+bu by:-v+bv =

u v


v(u + bu) - u(v + bv) v(v + bv)

I. I I

uv + vbu - uv - ubv vz + vbv


51- 2!

_b_y_vbx “bx "bx-' vz+vbv

In the limit as bx-+0

E- 1‘!

{\ .--~=& ._-l—_,.-—-1- ' "-

__ dl=vdx udx “dx v2

The use of this relationship is generally unnecessary since the quotient can normally be transposed and treated as a product.

:-.a4I I

‘T II I.

I t.

Maximo, Minima and Points of Inflexion A quadratic function of x in the form f(x) = ax’ + bx + c has a maximum or minimum value of x according to the values of a, b and c. A frequently used convention to express the differential coefficient of f(x) is f'(x) ‘_i,--1 _:‘;IA;fi-:.fl_¢.,.


and so f'(x) = 2ax + b


At the turning point the gradient of the curve is equal to 0 so that f '(x) = 2ax + b = O

_ _—b ..X—-2a

. ,_ _ -. -



bz b-b andf(x)-=%a-5+(—2a-l+c b2

"I it


=———+c 4a2a




~- -

1.8.11 Ctllflllllg POI!) IS at


(I-i>6A>(y-6y)6x andas6x—=-0

-t.‘--. -5,,-=--4

T ‘V

ydx=dA l


_ dA t.e. -3-; = y = f(x) Hence A = I f(x)dx + C This gives the total area between the curve and the x axis to the left of a given value of x. However, since C is unknown the solution is indeterminate. Where, however, it is required to find the area between two given values of x it is not necessary to know the value of C since if the two values of x are a and b, C disappears from the expression A=

f(x)dx + CY“ -

f(x)dx + CT“ and a > b

This is normally abbreviated to A_= Ibl f(x)dx. . Example. Find the area under the curve y = —3x2 + 24x — 24. This is a parabola as shown approximately in Figure 1.4.

Figure 1.4 Y

0 5.-1-.5:

Before this can be solved it is necessary to find the x coordinates where the curve cuts this axis. Sincey=0 -3x2+24x — 24=0 i.e.x2-8x+8=0

so that x =

8 1-. \/"86-t_lT'§§ _ _,_,_‘, _,. -._-‘, - H._,

8 : 4\/'2

='—2.'_ = 4 1 2\/2

Y I :-

= 4 + 2.82 = 6.82 or 4 — 2.82 = 1.18

o.s2 Hence area = I (—3x2 + 24x — 24)dx us ...3x3



= [T * T " 24*] = [-6.823 + 1Z(6.8Z)2 - Z4-(5.82)] -' ['-1.183 + l2(|.l8): - 21-l>(1.18)]

= ['/7.25] - [-13.25] = 90.5 units It should be noted that the area of -13.25 units contained between the second pair of brackets represents the shaded area in Figure 1.4. i.e. the area under the x axis between x = 0 and x = 1.18.

.,-s-t.4-. _s_-s.O


_;-._,‘.- .-_q,

-4---__-| .- .q,_- .-


| ‘1. t F t





Chapter 2 Relevant Aspects of Analytical Geometry This subject will be treated in a fairly cursory manner since the subject isvast and much of it is of little concern in economics. This chapter will concentrate on aspects which are

relevant to economists.

The Straight Line Figure 2.1


Y T ,

‘I 1‘


_ 0


i, b


—: ta.





‘L 3 ti B-'7‘


Consider the straight line AD in relation to the x and y axes cutting at origin O. The point P whose coordinates are (x, y) represents any point on the line. PO is equal to x and is ' parallel to OS while RR is equal to y and parallel to TO. Since As TQP and PRS are similar. it follows that: PO



c-- )4


§'ti'=‘1¥""(E>'-Tt)=_';T" where c and b are respectively the intercepts cut off by the straight line on the y and x axes. 13


l Cross-multiplying, xy=cb—cx—by+xy or cx + by - cb = 0 which represents the equation of the straight line.

{F —-- qr‘-r.¢-s.s.%, -n-,1. .-t- tn,-as--t.-“:1

Dividing each term by b and rearranging,

11--c+c ht .'-

The gradient of the line is given by Tan ¢ which is equal to -5: if m now replaces — E, the equation can be expressed in the usual form: y = mx + c where c is the intercept on the y axis. t


Equation of straight line joining two given points Given two points (xfyl) and (x;y;), to find the equation of the straight line passing through them.


Q - —*-. .—--1-r .—-s.-1.- '-

Figure 2.2


A X1y1


_ _

Cxv M-U,},-_‘-,uk-.,"__hI___‘ _.,-_.“;-._d,'_,?._.‘_.-



_ a|_ _._=1._ .



‘E xzyz

.__..___'__.. _. _

X In Figure 2.2, (x, y) is any point on the straight line joining (x|y,), (x;y;). Since As ABC and ADE are similar xS¢_--st.‘ -. -or:fh1j.Irni-.4

E_BC AD_AB . 32-31 3'31 lei-=——— Y1"Y2 Y1“? X " X1 _ Y "' Y1

or - --L K2 ' X1

-2sew-" --.1",-—-ZI"—



swhich is a standard "result. Y2 *' Y1

1‘iv l


Second Order Equations A first order equation of the general form ax + by + c = 0 is a straight line and passes through the origin only if c = 0. A second order equation implies that either x or y (or both) is raised to the power of two. Such equations are always curves, although parts may approximate to straight lines. e.g.: y = 4x2 is a parabola passing through the origin. (Fig. 2.3)

Figure 2.3 Y







l The shape of the parabola is changed by varying the coefficient 0f~x2, i.e. the ‘parameter. so that if y = ax: the parabola is of flatter form if a is reduced and vice-versa. Its position with respect to the origin can be changed by introducing constants b and c such that y = a(x — b): + c

it F

A positive value of b moves the parabola bodily to the right while a positive value of c raises it. (Fig. 2.4)

- -

J i

II! III Ii -1(II

['0 -Ii


y=2(x— 10)2+6 = ZXZ — 40X + 206

-r_|-t:|- -

I it


It should be noted that an equation of the general form y = ax: + bx + c is a parabola and always has a maximum or minimum value of y. This value can be positive or negative.

Figure 2.5 Y

-‘,_1___,-Q___‘_ct--‘u,1n_o--H -ugnit¢_-.u,‘_1_-v_ ,'._ _ _ _



'1 ‘v



__-.-‘it a. --


The Rectangular Hyperbola y = 3- where k is constant. (Fig. 2.5) x As a second-order equation, this is of particular interest since the product of x and y is constant. A demand curve represented by such a function would indicate constant total revenue with demand having at all points an elasticity of unity (t'.e. Ed = -1). The curve is

asymptotic to both the x and y axes; tie. it meets them respectively at infinity.

Cubic Functions These include third-order values of x or y and are of the general form:

y = ax’ + bx’ + cx + d where a, b, c, d are constants. (Fig. 2.6(i)) In general a cubic equation unlike a quadratic function has both a maximum and minimum

value of y. An exception occurs in an equation such as y=x3—9x2+27x-27 which factorizes to y = (x — 3):‘ and in this case the maximum and minimum points coincide giving a point of inflexion. (Fig. 2.6(ii))

Adding Functions Two or more functions may be combined to yield a third. Consider the two linear functions y=5x+4andy=2x+7. These may be combined in either of two ways according to the purpose intended. (i) y = 5x + 4 (A) adding vertically y=2x+ 7

gives y = 7x + ll


. . . (l) (Fig. 2.7(i))

(ii) Rearranging.


"'5 5

(A) and adding horizontally



X = 5 -5 -





or 7y = 10x + 43

i.e. y = lg-’-5 + 553-

. . . (2) (Fig. 2.-1(a))


Figure 2.6



i X q~.__-._-q_. _ u t I

i it 1 i

y = (x + 6)(x +'2)(x -1) =x3+7x3+4x-12 li) -fa-» ~I-nu-;.r-q_I-.-. Q

i t. I

4 i

"4-§1"—',_sa.n_- *ntr|-op.-.4. .» l


1: I lI


“*9i 1,“ i.

y = (X ~ 3)’ ' Note point of inflexion

E 7



(ii) l

4-s,_ ,_.nL

Figure 2.7

1 I 1

l .


(i) Y



, tin



(ii) H

Combining Cost Curves Example I A manufacturer produces 100,000 units of goods per annum which he ships regularly to an overseas country. His costs (excluding production) comprise freight, storage prior to shipment and ‘inventory’ or interest on capital tied up in goods during storage. He wishes to minimise these costs per unit. Given the following information, calculate the optimum size of shipment and the corresponding cost per unit:



Value of goods:

$150 per unit

Rate of interest:


Cost of storage per unit per annum: $10 Freight per unit: according to the function (-400.000/x + 10) where x is the size of shipment. The inventory cost/unit if


i. x . 150 . (10/100) o A o where x/2 is the average number of units in

storage at any time. , _ The storage COSI/LIIIII -

x. 10

Inventory plus storage costs/unit __


(75+10 _17.5x

‘10u.000 '

J" 100.000

This is clearly a straight line passing through the origin. Now. adding vertically the two cost functions

c = 400.000/it + 10 c = 17.s>t/'100.000

. . . (1) . . . (ii) >

gives C = 400.0001?‘ + l7.5x/100.000 + 10

. . . (iii)

These functions are plotted in Figure 2.8 below. Graphical methods indicate that the optimum size of shipment is about 50.000 units giving a minimums cost of $24 per unit. This problem, given the function, can however be solved more readily using the differential calculus. for C = 400.000x"‘ + 17.5x/100.000 + 10 dC

., _ _ = ——400.000x'~ + 17.5/100.000 = 0 for a minimum

. X2 e 'i00,000 e 115 “'°“°e 400.000





— , UIIIIS d X-— 2X105—47s09 '

whence C = $26.73 per unit.


r’ - .o@;- i-| s-uiq-


Figure 2.8




50.000 units Example 2 A country Y is the sole producer of a commodity for export to three areas A. B and C. The demand functions are given by: ‘Q1128


Q 500-+-100

= -—-i


‘,2 Z9558

3 A an 0 first 1%


(B) P = “W-F 150


(C) P ' _ 2.000 + 75 where P is the price and Q the quantity that would be bought at that price per unit of time. These functions are plotted in Figure 2.9. In principle these functions must be aggregated or combined horizontally in order to determine the total demand function. However, since a negative demand is not meaningful. only the positive values of these functions can be considered. A lit_tle thought will show that fromoa. pr.i.c.e..o.£.li0. .u_t_1i.t§__d_own__t_9___1,Q.Q_th§ demand will arise from country B only: and from 100 units down to__ 75 the demand will comprise an aggregate of B and A. At prices below 75 units the demand will arise from all three countries. The total demand function will not therefore be a continuous function, being kinked at prices of 100 and 75 units. The top part of thedemand function will therefore be: P = -30/2.000 + 150

l‘ I

Figure 2.9

P 150 l








For the next part it is necessary to combine functions for countries A and B. (i) Q = -500 P + 50.000


-2,000 P

i l

(ii) Q = -—3,——- + 100,000

whence Q =

— 1,5 ( 003+ L000) P + 150,000

P._ —3o+900 °'. '3,s00 7 Figure 2.10 P -|@v__-_,v,.-1_s,'_-"=.-_,u".¢--“|s,_o-,_I\,_tr-° 't

_\ !



--. -Q»-— . _


} .ii




1-l- - I I I.





For prices below 75 units this function must be combined with that of country C. Thus (i) Q = —2,000 P + 150,000

-3,500 P + 150,000 (ii) Q = 13—9,500 P whence Q = -—T- + 300,000

.,.-fi.,m 95

" '9,s00

The aggregate demand will appear as shown in Figure 2.10.


Chapter 3


Progressions and Series i

An arithmetic progression in the general form. may be expressed as a.a+d.a+2d.a+3d.....a+(n—l)d where a is the first term and d is the common difference between successive terms of the series. The rth term is denoted by a + (r - l)d and the sum (S) of the first n terms is given bv --s--—uIe-'—

S=a+(a+d)+(a+2d)+(a+3d)+...+(a+(n—l)d) which can be re-written as S=(a+(n- l)d)+(a+(n—2)d)+._..+(a+d)+a Adding these two expressions gives 2S=(2a+(n — l)d)+(2a+(n — l)d)+ . . . +(2a+(n- l)d)

= n(.?.a + (n — l)d)

HenceS=%(2a+(n— l)d)=%(a+L)

-tq-v.wI—n.1t'--qv r-HI-.¢~ i‘ t l

where L is the last term. A geometric progression has the general form.

a. ar. ari. ar-‘. . . . ar"" where a is the first term and r the common ratio between successive terms of the series. The pth term is denoted by arl’ " ' and the sum (S) of the first n terms is givenby S=a+ar+ari+ar-‘+...+ar"'""...(i) whence. multiplying throughout by r. rS=ar+ar3+ar-‘+ar‘...+ar"...(ii)

1 i

I----\i-s-1'-t4r-C »iam'.§-w’cIQ'=u-Q;Zi-inI‘,¢-_:qr1__.;1q--vtF'r

and subtracting (ii) from (i) S — rS = a — ar“ il

.'.S(l —r)=a(l —r")


and S=—-——-an _rn) l—r If r < 1. as n becomes larger r" becomes smaller: so that in the limit as n—> ==. r"—> 0. Hence. the sum of an infinite geometric progression is given by S=iwhenr 5C3 SIHCB 5P3 = 60 afld 5C3 =

In fact ,,P, = r!,,C, Consider again the set of letters w. x. y and z. How many combinations of these 4 letters can be made taking 2 at a time? . _ _. _.- _-—

4: ‘C’ I 2:2! Z 2.15.1 I 6 and these can be written down as _‘_.__-.-

wx. wy. wz. xy. xz. yz It may be noted that ,,C, = ,,C,, _,. This identity is sometimes useful in facilitating computations of these combinations.


‘Avs -nu-—.—-1-'I-17‘

l‘ I

it l

The Binomial series (l + x)“ expands to give

".i-—14~*s-. =gr p-I I I


where the (k + l)"‘ term is given by n(n—l)(n-2)(n—3)...(n—k—l)x" 7 7' '1 _ _;_





It! 1'.-_,:fi;:»,-

Adopting the notation used for combinations from the previous section. the binomial expansion becomes. if n is a positive integer. l+nC|nX+nC1cX:+nC3sX3+...+xn

which may be rewritten as 2 ,,C, . x’ r=tI 11

t‘;.--. - ~r-:1 -I“ |I

In general (a + -b)" = 2 ,,C, . a""b'







b+—(-iT)a"'=b3 +...+nab""' +b“



I I i

e.g. (a + b)5 = as + 5a‘b + 10a3b2 + 10a2b3 + Sab‘ + b5 The following properties of the expansion (a + b)“ should be noted: (i) there are n + 1 terms (ii) the exponents of a and b in each term sum to n (iii) the coefficient of any term is "Ck where k is the exponent of either a or b. The coefficients of the terms of successive powers of a and b can be arranged in a triangle known as Pascal’s triangle, from which the coefficients of the terms of the next power's expansion can be derived. Pascal’s triangle is usually written as (a+b)°=








(a + b)3 =

a3 + 3a3b + 3ab3 + bi

(a +‘b)“ =

a" + 4a3b + 6a2b3 + 4ab3 + b‘

(a + b)5 =

as + 5a"b + 10a3b3 + 10a3b3 + 5ab" + bi








. “ .,@-.»@ 1




The coefficients of the terms in the expansion of (a + b)° can easily be derived from the next line of Pascal‘s triangle

1 @ ® 10 5 1 1.

\ 1

1 o ® 20 15 6 1 so that (a + b)“ = a“ + 6a-‘b + 15a"b2 + 20a~‘b-‘ + 15a3b‘ + 6ab5 + b“ A useful expansion for probability theory is that of (q + p)“ which is

(q + P)“ = ..Cs1>°q" + ..CtPq"" + ..¢=1>’q"" + ..CtP’q“" + - - The Exponential Series .

The series (1 +

1 "




expanded binomially gives







2 2n1+16 2n1+1+ ant

and. in the limit as n—> ==


In (1%-H) -1-I-1+5-I-6+iz+...

= 2.7182 . . . which is denoted by the letter e. It is the base for Napierian or natural logarithms (ln).



An exponential series may be written as








writing cx for x gives cx B








and letting e° = a (so that c = loge a) and substituting for c gives :1 2 3] :1 a"=1+x1og,a+x (igfa) +x(c;g!°a) +...


Appendix A contains a note on Maclaurin's Series which is particularly useful in deriving the

expansions of a number of the more common functions.

The Logarithmic Series It is not possible to obtain directly a series for loge x but it can be shown that:

I og,,( 1+)~ x x

x:+x3 2 3

x4+ 4 ...or f 1< x~.. 78) The data are continuous so there is no need to make any adjustment to the numbers (see example 7).



x — |.l 0

5 ~55 7


78 — 70 4.6338


_ 2



Figure 4.6 ..-¢I'"\'Ih-\p—uq,i . I

1"} -=—-.-vg .

i Dry



. ..

Wet "-

’( _., _,_ ,._

I 3)

“' 6



,_ .



'__ I




From Normal tables the probability corresponding to Z = 1.7262 is 0.9579 but this is the area to the left of 78 cm hence Pr(wet) = Pr(rainfall > 78 cm) = 1 - Pr(rainfall < 78 cm)

= 1 — 0.9579 = 0.0421 (ii) Pr (1985 was not dry) = Pr(rainfall > 60 cm in 1985) _x—|i*60—70 UseZ* O -4.6338e—


Find the area to the left of Z = +2.16 in the Normal tables and subtract from one to

get the value corresponding to Z = -2.16 Z = +2.16 corresponds to 0.9846

Z = -2.16 corresponds to 0.0154 but this is the area to the left of rainfall = 60cm. The required area is the one to the right i. e. the Pr(rainfall > 60 cm) which is

1 — 0.0154 = 0.9846. theprobability that 1985 was not a dry year = 0.9846. (iii) Pr(1985 was dry)

= 1 - Pr(1985 not dry) = 0.0154

Pr(1986 was not wet) = 1 — Pr(1986 was wet) = 0.9579

Assuming statistical independence Pr(1985 was dry and 1986 was not wet) = 0.0154 X 0.9579

= 0.01475 (b) The Binomial (or Bernouilli) Distribution

The Binomial distribution is a discrete probability distribution that arises when there are n independent repeated trials each with a constant probability of success (p) and a constant

probability of failure (q) where q = 1 — p. The probability of there being exactly r successes out of n such trials is given by 4

Pr(x = r) = ,,C,p'q“" for r = 0. 1, 2 . . .n and is obtained from the Binomial expansion of (p + q)“ as explained in Chapter 3. The mean of the Binomial distribution is given by np and the variance by npq where

q = 1 L p. It is possible to relate the Binomial and Normal distributions. If n is large and if neither p nor q is close to zero the Binomial distribution closely approximates a Normal distribution with standardised variable given by Z =


This approximation improves as

n increases in magnitude and, in practice, the approximation is good if both np and nq are greater than 5. Allowance must be made for mixing discrete and continuous variables; see the second example which follows.

)-e__ _. I I 1 | I

Example 6 In the selection of graduates for a management training scheme in a large shipping company it is expected that on average only 70% will complete their period of training. What is the probability that out of a group of 5 new recruits 3 or more will fail to finish their training?




Pr( failure to complete training) = 0.3 = p .'.q = 0.7 and n = 5

Pr(3 or more will not complete) = Pr(3) + Pr(4) + Pr(5) Pr(3) = 5C3(0.3)3(0.7)2 = 0.1323 Pr(4) = 5C4(0.3)“(0.7)‘ = 0.0284 Pr(5) = 5C5(0.3)5(0.7)° = 0.0024 .'.Pr(3 or more fail to complete) = 0.1323 + 0.0284 + 0.0024 = 0 1631 Example 7 A major ship chandler plans his orders on the basis that his expected share of sales will average 21% of the total market sales per month. Assuming normality with a

standard deviation of 1% determine (i) the probabilities that his market share will exceed 22.5% at least twice in four successive months (ii) the probability that his market share will lie between 20 6% and 21 8% not less than 20 nor more than 30 times in 50 successive months.

_ .-__. ‘.__. -.2.

Figure 4.7


I = £112






gx-it 22.5-21.0 41.5 6 if 10

Pr(Z < 1.5) = 0.9332 from Normal Tables

Pr(x Q 22.5) = 0.9332


and Pr(x > 22.5) = 1 — 0.9332 = 0.0668 Hence p = 0.0668 and q =1 - p = 0. 9332 .~-. _.-_ - 1-.



For (i) use the Binomial distribution “C, p’ q“" with n = 4, r = 0 1, and 2 and

p = 0.0668.


Pr(sales > 22.5% at least twice) = 1 : [Pr(sales > 22.5% never) + Pr(sales > 22.5% once)] Pr(sales > 22.5% never) = 4C0(0.0668)_° (0.9332)‘ = 0.7584 Pr(sales > 22.5% once) = 4C1(0.0668)‘ (0.9332)3 = 0.2172 0.9756

Pr(sales > 22.5% at least twice) = 1 — 0.9756 = 0.0244

q_ ,5 2



nap.. -

lp-. -1 - 1-v n-vnwn

Figure 4.8




20.6 -— 21.0 21.8 - 21.0 Z; = -—T-0-—and Z2 = T .'.Z| =

and Z2 =

From Normal tables the areas are

area to -left of Z2 = 0.7881 area to left of Z1 = 0.3446

.-_._- ._-.

by subtraction the required area between the two Z values = 0.4435.


Figure 4.9



i! = >(—np ‘JFIDQ

i 1


-26- 22 19.5


,_._ __._ .



[___ _ HI



Use p = 0.44, q = 0.56, n = 50 and the Normal approximation where np = 22

.,- .0 -_. - . V

npq = 12.32 \/npq -'= 3.51 Q.

NB The discrete x > 20 is equivalent to x > 19.5 on the continuous scale.


Z; = (30.5 - 22)/3.51 = +2.42 Z, = (19.5 — 22)/3.51'= -0.71 Area to left of Z, = 0.99224. Area to left of Z, = 0.2389. Hence the required probability is


Pr(20 < x < 30) = 0.9922 — 0.2389 = 0.7533

Figure 4.10

-_-'- 1.-_._.-_.



{‘~"»-._1F2-4.J_2»._--42 -3:61“

1 1 1

4‘) 1 1

X .4,‘-.


(c) The Poisson Distribution The negative exponential probability distribution depicted in Figure 4.10 is a continuous distribution which can represent the probability pattern of the time between consecutive

occurrences of a particular event. The total area under the curve is one and the area under the curve between t = 0 and t= T represents the probability of there being a time less than T between consecutive events.

The related discrete probability function giving the number of occurrences during a time interval t has the form e"“ lit ’ Pr(r events in time t) = --£4 for r = 0,1,2... and is known as the Poisson distribution where lit is the average number of occurrences in time t.

The Poisson distribution is sometimes explained as the one which deals with rare events. i. e. with events whose probability of occurrence is very low. The classic example concerns the number of cavalrymen being killed by a horsekick in the course of a year. The important

feature is that the number of times a particular event occurs is very low compared with the number of times the same event does not occur. It is impossible to state how many times something does not happen unless a limit on the total is given. and this demonstrates the essential difference between the Poisson and Binomial distributions. The latter has a clearly

defined number of trials (n) and the event will occur r times and not occur (n - r) times: no such precision concerning non-occurrence of an event is possible under the Poisson distribution.

The mean of the Poisson distribution is denoted by lit which is also equal to the variance. It is possible to relate the Poisson to the Binomial and the Normal distributions with the . . . . - 71. . . . . . . standardised variable being given by Z =

This approximation is valid provided p (the

probability of success) is small and n is large. Under these conditions q will tend to unity and hence the mean (np) will approximate to npq. Conventionally, small samples are those where n < 30, so for n to be large it must exceed 30. Similarly a small value of p would be when p < 0.1 implying a value of q > 0.9. Example 8 A ship is discharging bulk phosphate directly into a fleet of lorries which arrive at the dock. on average. once every twelve minutes. Assuming a Poisson arrival pattern find

(i) the probability that there will be no arrivals in the next hour (ii) the probability that more than one lorry will arrive in the next hour (iii) the probability that exactly three lorries will arrive in the next two-hour period Pr(r. t)

= 65*‘ H. (/11)’

1 arrival every 12 minutes is the equivalent of 5 arrivals every hour so it = 5 For(i)r=0,t=1,andltt=5 e“5 . 5" Pr(0, 1) = —-5-_-— = 6 _ 5 = 0.0067 —=.=- nr——


e“5 .5‘


For (11) Pr(1, 1) = —i,—- = 5 . 6 5 = 0.0335 Pr(0. 1) + Pr(1, 1) = 0.0067 + 0.0335 = 0.0402 Pr(morethan 1 in the hour) = 1 — [Pr(0, 1) + Pr(1, 1)] = 1 - 0.0402 = 0.9598 For(iii)r=3,t=2,A=5

Pr(3,2)e -111.103 3;. 1° 510.1 6000-- 0.00757


Chapter 5 Basic Economic Relationships I

I l "U .

Demand and Elasticity


It will be assumed that readers are familiar with the basic principles of demand and supply and with the derivation of the general shape of such curves as well as with exceptions to the general case. Demand (average revenue) curves normally slope downwards from left to right and reflect an inverse relationship between price and quantity demanded. Supply curves slope upwards from left to right and reflect a direct relationship between price and quantity supplied. In other words. if the price of goods or services rises. less will be bought and more will be supplied.

Figure 5.1




-—4-_{. -_’.q4-1_4- _-v.,_1-—

P2‘ 4-» Pr'cG F






H -

i ’+

— 4


l I8


~ l

OD: O01 Quflfllflv I901-Ighl





8 061 O52 2*“ Quantity supplied


F __ 1

l i;H


1‘ I)

Clearly the demand for and the supply of particular goods or services are not divorced from each other: they simply represent different -sides of the same theory. The two components must be taken together to arrive at an equilibrium price and quantity which satisfy both the

consumers and the producers of the goods or service. (See Figure 5.2) The equilibrium position can be disturbed by a change on either the demand or the supply side (or both). and the “state of the market” or the “market conditions” are determined by the relative strengths of demand and supply. Demand conditions may change due to a change in income while supply conditions could be altered through a change in technology, as for example. with the introduction of containerisation. Such changes will alter the physical position of the demand and/or the supply curves within the axes.

Figure 5.2 l



J '1

I1 i



Pe D



Qe Quantity bought and supplied


In Figure 5.3 there has been a shift in the demand curve from D, to D; (possibly caused by an increase in consumers’ income) and also a shift in the supply curve from S, to S2. The latter represents an increase in supply (i.e. the producers are prepared to supply more at any given price) which could have been caused by, for example, a reduction in the costs of

production. The combined result of these two changes considered together is to give a much larger output at a slightly higher price. It is a simple matter to consider the changes separately as shown in Figure 5.4. This type of analysis leads in two directions. First it leads to a discussion of the revenue


implications of different levels of demand and supply and. second. to a consideration of the rates of change of price and quantity in response to market forces. The revenue aspect is straightforward and revolves around three definitions, namely those of Total Revenue (TR).



Figure 5.3




S1 S2



kg P1








._ .


.';4. u



D2 D1

1 '1.


1 4




Quantity demanded and supplied i 1 ii

Figure 5.4 l,

_..-._-. _i-.__..,fl’







8 P1

\ ,













D 01


02 Ouamitv

Marginal Reinue (MR) and Average Revenée (AR). Total revenue is defined as the

revenue derived from all sales of particular goods or services and is the product of the u-vg-.-_.gnu.-—-—-

number of units sold and the price per unit. Marginal revenue is the change in total revenue brought about by selling one unit more or one giiitzglmegss -— this unit is known as the marginal

unit. Average revenue is El€fiiiéTl“§§ tneaxii Pévenué resulting from all sales of goods or services divided by the number of units sold. In other words

TR = P X Q where P = price and Q = quantity


MR = -

__4,. +_ _

= TR“ 1, — TR“ where n signifies the nth unit of the good.


1 1

l l

The second direction leads to a discussion of the concept of elasticity which may refer to



either demand or supply. Three types of elasticity of demand are considered in the literature and are known respectively as price. income and cross elasticity. of which the mostcommon is plice-elasticity of demand. The usual definitions are


(1) Price elasticity of demand (Ed) is the responsiveness of demand to a change in price


l s


I. tr’

I I»


Price Ed

proportionate change in quantity demanded proportionate change in price

50 Q


Hence Price Ed = § = -6 .


P 4‘_":§>"

r 1.

l ?'


= 6'5 (2) Income elasticity of demand is the responsiveness of demand to a change in income and




is measured in the same way as price elasticity of demand but substituting income for

price in the formula. It will be positive for normal goods. ""--_._...



(3) Cross elasticity of demand (Ex!) is defined as the proportionate change in the quantity demanded of a particular commodity x in response to a proportionate change in the price of a different commodity y. '







E _ Qx __6Qx l’_y_

*>"@'ox‘6Py Pr

_@»Qx 21. *61>y'ox


The cross elasticity of demand will be negative if x and y are complementary goods. and

positive if they are substitutes for each other. Reverting now to price elasticity. which is usually what is meant by the phrase "elasticity of demand”. several points need to be made. First. there is a distinction between arc- and point-elasticity. By definition. the price elasticity of demand is a measure of the

responsiveness of demand to changes in the commodity‘s own price. If the price changes are very small. so that the points are very close together on the demand curve. the point-elasticity of demand is being measured. If the changes of price are not small then

elasticity is being measured along an arc of the demand curve. and problems may arise in calculating a value for .the price-elasticity of demand. For example. consider. in Figure 5.5. a

Figure 5.5 Price

P1 =12 P2=1O

12 ~


(11:20 Q2=25


110 4



I l


20 25 _""'

Quantity demanded

fall in price from £12 to £10 per unit which results in an increase in quantity demanded from 20 to 25 units.


1 l 1 I 1

1 U 1 1

6Q +5 Q = -:5 20 The formula states that Ed = -5? i 1

? T2’ 5






60 664 : 4


u1-i,-4.-».="- Q

1 '5


Hence the elasticity of demand measured along the arc from A to B has a value of -1.5. l Will this be the same value as that of the elasticity of demand measured along the arc from J -4

d .

B to A? Unfortunately not; the arc-elasticity of demand from B to A has a value of -1. The reason for this ambiguity is that A and B are not sufficiently close together to measure point-elasticity of demand and the proportionate changes in price and the quantity are significantly different when taken as proportions of the original and final figures. In other words the direction of change has become important. To measure arc-elasticity in an unambiguous manner the proportionate changes in price and quantity must be expressed as proportions of the average price and average quantity respectively. Hence the correct formula for measuring the value of arc-elasticity is

Ed = o_ ‘(rt + P2) 6P

5P (Q1 + Q2)

(P1 + P2)/2 Secondly. it is conventional to multiply formulae for price elasticity of demand by the number (-1). This has the effect of nullifying the inverse relationship between price and quantity and means that the higher thedgnurnverical !;1lue_ of the price elasticity of demand the more elastic is the demand. ' ' i ' Demand is usually ciassified according to whether it is elastic, inelastic or has an elasticity

which equals unity. It is the value of unity which is critical in this respect and the appropriate terminology may be summarised as:

; l



Value of elasticity

Perfectly elastic

l -

Ed = 1

Elastic . l i

1 < Ed < ac





l __ _ 7_7_




Unity 1"’





1 _


Inelastic __ __






0 < Ed < 1


Perfectly inelastic

Ed = 1

. ‘ l



*'— 7



l 1

_____ *1

Ed = 0

With three exceptions the elasticity of demand varies at every point on the demand curve. The exceptions are shown below in Figure 5.6.

Figure 5.6




.§_ 0.







in at pgflgcfly umu; dgmgnd mm; 5., = Q

{ii} A Dcfioetly inelastic demand curve fie =0


um A rectangular hyperbole flu =1

To obtain an indication of the elasticity of a demand curve at a particular point it is sometimes useful to consider the effect" on. total revenue of varying price. If a small reduction in price leads to (i) a larger proportionate increase in quantity demanded so as to increase total revenue then the demand will be elastic (Ed > I), (ii) a smaller proportionate increase in quantity demanded so as to reduce total revenue then the demand will be inelastic (Ed < 1), (iii) a corresponding increase in quantity demanded so as to leave total revenue unchanged then the elasticity of demand will be unity (Ed == 1). Consider again the previous example:

Figure 5.7 Price

P1 :12




+El;) Q . Ed . _ _l_ where the value of E is conventionally Le‘ MR _ P0 + Ed) expressed as a negatidve number. A demand or other function is often expressed in the form: P=Q" sothat LogP=xLogQ dP _ dQ and P -x Q .r I



°' 5:15"

Now the elasticity of P with respect to Q

_ £91 ._ Q -. iii P d P






Q1115 Q6"

Hence, if P represents price and Q is the quantity of goods demanded, the price elasticity of demand is given by: 1

Ed = I

This relationship is often useful for empirical analysis using multiple regression techniques.

Derived Demand It may be useful to comment here on the idea of a derived demand. A derived demand

exists when goods or service are demanded not for themselves but for their usefulness in producing other goods or services. The demand for any factor of production is a derived

demand as is the demand for sea transport. A fuller discussion of this topic is contained in a subsequent chapter but briefly it can be stated that the elasticity of demand for a factor of

production. or any goods or service the demand for which is derived. will depend on the elasticity of demand for the goods being produced or transported. on the ease with which a substitute can be used. and on the proportion of total costs accounted for by payments to

that factor.

Supply, Costs and Elasticity This chapter has so far dealt with the theory of demand and the revenue implications of changes in price but it is equally important to consider the supply side and the theory of

costs. The basic definitions of cost involve the ideas of total. marginal and average cost. Total cost (TC) is defined as the total cost of all factors of production involved in the production of a certain level of output. It is divided into two components. viz. total fixed

cost (TFC) and total variable cost (TVC).


i'.e. TC = TFC + TVC. Fixed (or indirect) costs are those which do not vary with output while variable (or direct)

costs are those which do vary as output changes. A distinction must be made between the long and short-run time periods which refer, not to any length of calendar time. but to the ability to alter the quantities of factors of production being used. In the long run all factors of production and hence all costs are variable. The short run is defined as the period of time

over which the inputs of some factors cannot be varied. Marginal Cost (MC) is the change in total cost brought about by producing one extra unit and it is usual to distinguish between short-run (SRMC) and long-run marginal cost (LRMC). Average cost is the total cost of production divided by the number of units

produced and will have two components. viz. average fixed cost (AFC) and average variable cost (AVC).

These definitions can be summarised as: TC = TFC + TVC ATC = -F-lg where ATC is Average Total Cost

TFC+TVC =-—-Q——=Ai=c+Avc _ d(TC) MC ' dQ or MC“ = TC“ — TCd__, where n is the nth unit. The supply curve will reflect the costs of production and as these costs change. due perhaps to technological progress. so will the position of the supply curve between the axes.

Figure 5.9


52 -

B sp-



Quantity supplied

Figure 5.9 illustrates an increase in supply in response to changes in the conditions of supply. In this case a reduction of costs permits more to be supplied at the same price and the supply curve moves to the right. A great deal of what was said above concerning demand theory can also be applied to the theory of supply; hence ‘movements along’ must

be distinguished from ‘shifts in’ supply curves and so the responsiveness of supply to changes in price can be measured by the price elasticity of supply. Again the delineating value of elasticity is one: if the elasticity of supply exceeds one then supply is elastic; if the elasticity of supply is less than one then the supply is inelastic; and if the elasticity of supply equals one then the supply curve is said to have unit elasticity. An easy way to determine whether supply is elastic or inelastic at a particular point on the supply curve is to draw a tangent to the supply curve-at that point. If the tangent when extended cuts the price axis the supply will be elastic (E, > 1); if it cuts the quantity axis the supply-twill be inelastic (E, < 1); -and if it passes through the origin the elasticity of supply will be unity.

Figure 5.10


Pa (Pl

r I l

../ '



Pr'ce _





Quantity (Q)


Any linear supply curve which passes through the origin will have constant elasticity throughout its length (E, = 1) as will perfectly horizontal (E, = ==) and perfectly vertical (E, = 0) supply curves. Apart from these three cases the elasticity of supply (E,) will vary at every point on a supply curve.


E1I ?_

Figure 5.11 Pi-in



Price I


_ _E_ _ _ _.


8 l ‘



T" lil 5, = o



"' liil§s= ii»








;\b 4*-7 eAif

liiil Es = 1 _.T,.e:h_ . _-L4 F'. I

Relationships Between Cost Curves Basic economic theory suggests that cost curves could be ‘U’-shaped in both the short and long run. This shape stems from the law of diminishing returns and from the long-run average cost curve being an envelope curve of the ‘U’-shapedshort-run average cost curves. More advanced economic theory suggests that the cost curves might be ‘L’-shaped or

- _.-.- .-.---,_“'


Figure 5.12



Output saucer-shaped and incorporates the idea of reserve capacity. For the purposes of this book

cost curves will be treated as ‘U’-shaped and certain relationships will then follow. From the diagram above it can be seen that the average cost (AC) continues to fall so long as MC lies below it, and rises when MC is above it; and that the MC curve cuts the AC curve from below at the lowest point of the latter. Proof Let AC = f(Q) Then TC = Q . f(Q) Cl TC

and MC = la?) =5 Qf’(Q) + f(Q) When AC ..= MC.

f(Q) = Qf'(Q) + f(Q) i.e. Qf'(Q) = 0

or f'(Q) = 0 - _ but since AC = f(Q)

it follows that AC is at a minimum when MC = AC The AC curve depicted above is the average total cost curve which as defined earlier contains two components. average fixed cost and average variable cost. Figure 5.13 explains the relationships between these curves.



It was stated above that supply curves reflect the costs of production but in fact the relationship may be stronger than that suggested by this statement. Under conditions of perfect competition the firm's marginal cost curve above AVC has the identical shape of the

firm‘s supply curve. and it follows that the supply curve for a competitive industry is the


Figure 5.13 0.4



QE °~1





ATC :- Average Total Cost

6 Output




AVG :- Average Variable Cost AFC :- Average Fixed Cost

MC :- Marginal Cost

horizontal sum of the marginal cost curves of all the individual firms in the industry. To put

this into a shipping context. simply consider the firm as an individual ship and the industry as a fleet: the supply curve for a fleet is the horizontal sum of the marginal cost curves of all the individual vessels in the fleet. (See also chapter 6)

Relationships Between Revenue Curves ‘l

For any downward sloping demand curve (or average revenue curve) there is a marginal revenue curve which is also downward sloping but is‘ twice as steep as the average revenue curve. Figure 5.14. using a straight line demand curve. depicts this relationship and point R represents the point of maximum revenue.

I l

l l

Figure 5.14 it



l t‘ l






D = AR Quantity

xy._.-_-F-» 1 I


If the demand curve is a straight line with the equation P = mQ + C then total revenue, given by P X Q. will be mQ2 + CQ and marginal revenue will be 2mQ + C which is the derivative of total revenue with respect to output. The MR curve is coterminous with the AR curve on the price axis, falls twice as steeply and consequently bisects the base of the triangle formed by the demand curve and the axes. This relationship is true for all downward sloping demand curves (except under conditions of perfect price discrimination). However. under perfect competition. a market structure often applied to tramp shipping. there ceases to be a distinction between average revenue and marginal revenue. The perfectly horizontal demand curve facing the firm makes it impossible to draw a separate marginal revenue curve. This follows because the market price is assumed to be unaffected by variations in output. and so the marginal revenue arising from the sale of one more unit is constant and equal to price.

Figure 5.15 Price



If profit maximisation is the objective of the shipping company then it should expand output up to -the point where MC = MR. The industry as a whole will be in equilibrium (no tendency for new firms to enter or for existing firms to leave it or for firms to alter their level of output) when AC = AR.

Figure 5.16 Short-run equilibrium of firm lil ' ""°'





AC oi-undaQui~ mi'Irlllprdin





AC orndlou





1.2" ' '

'-:"i#-'- " iii _: F. I.‘ “ifiii;-2-I-:\-.1;




-aw--_- _.-_. _.-:?-_ic~_




Figure 5.17 Long-run equilibrium of firm and industry M

F l


Price ‘





l _l




l l






l l



' ti





ll l. 5




Consumers’ Surplus, Producers’ Surplus, and Price Discrimination

.,._L ._*_-4 l



The interaction of demand and supply as shown in Figure 5.18 determines not only the

equilibrium price and quantity but also the values of consumers’ and producers’ surpluses. Consumers‘ surplus (CS) is represented by the area below the demand curve but above the price and is a measure of indirect benefit to the consumer. It represents the difl'erence between the money that consumers in total are preapred to pay (PP) for goods or services and the money actually paid (PA), i.e. CS = P, — PA; this area of consumers’ surplus is often the focus of price discriminatory actions. Producers’ surplus is the area above the supply curve but below price and represents the diflerence between the amount at which a producer is prepared to supply a certain level of output and the amount at which he actually does supply that level of output; alternatively, it may be regarded as the difference between the money obtained for goods and the direct costs of producing them.

l l .v-v-4




Figure 5.18 Price

l‘ Consumers’ surplus



L 1‘ \-

_ T_, ._ _v

Pe Producers’ surplus wqw_ 4m‘E;m

.l) l





ii l I


Price A

Figure 5.19

19, P,

E1 E2



If price falls the area of consumers‘ surplus will increase. The diagram above clearly shows that when the price falls from P, to P2 the area of consumers’ surplus increases from AP,E, to AP,E,. The net change is given by the area P,P,E,E,. Price discrimination is defined as the practice of selling identical goods or services to different buyers at different prices or the selling of different units of identical goods or services to the same buyer at different prices. Good examples arise in service industries where prices differ as between individuals and firms, and also in passenger transport where peak and off-peak charges differ for the same journey. Liner shipping is often accused of practising price discrimination. Three conditions are necessary for the successful application of price discrimination. First there must be some degree of monopoly power on the part of the supplier so that some control can be exercised over the level of output or frequency of service being offered. Second it must be possible to split the market into separate submarkets where the different prices can be charged without any fear of the lower-priced goods finding their way into the higher-priced market. Finally the price elasticity of demand must be different in each submarket to enable the higher price(s) to be charged. The determination of equilibrium positions under the assumption of profit maximisation depends on equating marginal cost and marginal revenue for the total market. This will determine the total output which can then be divided between the submarkets in accordance with the marginalist rules and sold at prices determined by the demand curve in each market.

Figure 5.20 Prim _

Price l ,

lil Market A

Price iii) Market B

[iiil Totil MOI‘!!!

"' 1 11 x 1 1 x Q

1 x @ : $11



: ¢ — 1 1--‘h-ii-—












. ._ 0



l _._. O

.LL‘_ -II I--"'-—"—





Figure 5.20 shows how the different prices in each market are determined and how the total output OT is divided between the two markets (OA + OB = OT). Under perfect price discrimination the producer aims to appropriate the entire area of consumer surplus by charging different prices for each unit sold. In practice this is extremely difficult and it rarely, if ever, occurs; the producer would usually settle for second or third-degree discrimination with different prices being charged for different blocks of output.

i I S


It is interesting to note that, under perfect price discrimination, price is equal to the marginal revenue as each unit is sold for a different price. The demand and MR curves coincide even though the demand curve is downward-sloping. Laingm provides the following diagram (Figure 5.21) showing how the cargo volume (X) may be determined under a perfectly discriminating price structure. The point of this diagram is that the MR curve becomes the demand curve when perfect price discrimination occurs and no single price prevails in the market. If profit maximisation was the aim then the volume of cargo carried would be OW but this would result in an area of supernormal profits (TUV) which could encourage new entrants to the trade. In order to deter such potential entrants the operator may decide to expand the volume of cargo carried to OX. At this point AC = AR and normal profits are being earned. The quantity WX is carried at rates below the average cost, i.e. these cargoes cost more to carry than they earn in revenue, and the super normal profits from other cargoes are used to compensate for these losses. Under this approach normal profits are being eamed over maximum volumes of cargo.


i. l



Figure 5.21


Freight rate

4. 0

‘|' '>


-' -




.. . .. _ z I



*AC =MC -.-_._.i _.-¢,:n‘?.;


.i.l O



Volume of cargo

In general if P = f(Q) 7"7*1"‘__‘4i1“_"

then under perfect price discrimination


i.u_;-.- , ":;-3


AR=%rf(Q)dQ 0

Hence if P = mQ + C 2 TR=%+CQ


=ll‘l-MC). otherwise profits could be increased by reducing the volume of each cargo until the respective MR5 were equal to their MCs. The vessel would not however be full. Example A container ship has a capacity of 1.500 twenty-foot containers (TEUs).

Commodities in containers are available as indicated by their demand functions:



Number (TEUs)


$300 $400 $500 $600

S00 600 400 300

— — — - u-la-rt-no-

It is required to find the number of containers of each commodity and the rate to charge in order to maximise profits. It is assumed that the marginal cost of handling the containers is negligible. If ql, C12, q3, q4 are the respective numbers of containers of commodities A. B, C. and D

then ¢l1+q2 ‘H-ls +q4=1=500---(1) and the equations corresponding to the marginal revenues of each, assuming linearity. can be deduced:

600 MR1=-%q,+600...(ii) 390

MR2‘ = — W2 + 800 . . . (111)

1.000 MR, = --456-q, + 1.000.. .(iv) 1,200 300

MR4 = _""""iq4+1,200...(V)

Solving equations (i)-(v) for values of q give (to the nearest integer) Number qi= Q2: qs= Q4:

Freight Rate

388 500 346 266

fl = fz =

n=5m f4 = £668

Revenue £142,396 £233 ,500 £196,528 £177,688 -nip-pi



MR1: MR2 = MR3 = MR4 =

Assuming that MC is negligible, the usual way of maximising profits (MC = MR) yields a total revenue of £680,000 and 300 TEUs being shutout. If the ship s capacity is l 800 TEUs, both methods yield the same result.

Incidence of Costs of Sea Transport Figure 6.4 D

Pd P P,



Figure 6.4 shows the supply curve (SS1) of goods from an exporting country and the demand curve (DDI) of an importing country. If there were no transport costs, the price paid and received for the goods would be P; but when the freight element is equal to AB for



example. the importing country suffers an increase in price equal to PP, while the exporting country sustains a fall in price of PP,. Clearly PP, + PP, = f where f is the freight rate per unit of goods transported. The relative magnitudes of PP, and PP, depend upon the slopes of the supply and demand functions in the area under consideration. Now the proportion of freight borne by the importer = a/f and the proportion of freight borne by the exporter = b/f DO








5Q P P vi _ andE,=6.§,-S-=-6 0'0

-41 -.

.E__F!P E,



P, a




ll. is




Figure 6.6


em, _____ _._+__ 6Ps I






50. Now df = dP,, — dP, . . . (i) (negative sign appears because dP,, and dP, move with opposite signs) Pd dQ P, dQ and Ed I b . dmt E, E . dps

(by definition)

.,,,Pd d_ to Ps . l

8 Ed .





. E5

_ EsEd



" at T Q (E,P., 4 P,E,, "'(“) P d but dP,, = E-3 .

d dP E0 = Yd . Ed

and substituting in (ii) gives

dP,, E, E,Ed K jdf ‘P; E,i=7,, -' P,E, .

dP.i _


"8' Jr ' EIP,'- i>,i§,, at

_Q(_.§.a_) .,,(_na._) E,P,, - P,Ed E,Pd - P,Ed (Ea + 1)


= Pd . Q . Es [(EsPd - PsEd)]

Q(Ed + 1)


= —-1-Y— where K = P—-E- which, as shown earlier is a constant for linear supply Cl and demand functions. S


“Ed = "1 IhCflT = O

Similarly it may be shown that the change in the value of exports (VE) with a change in freight is given by: d(VE) : QP,E,,(l + E,) df T P,,E, — P,Ed

= 9§l._+L~l,,,h,,e K = E153 (1/K_


_ d(VE) Qii

The Elasticity of supply is given by:

E, = (dQ/Q)(P/< 200.000 120 3.180..(15.24)-,.21000

= \/__._._-----—-— = 3363

whence speed in knots = 226.8/24 = 9.45. The corresponding gross profits/day for a range of speeds are:

Speed (kts) Gross Profit/Day ($) Speed (kts) Gross Profit/Day ($) 8 8.5 9 9.5 10

3.437 3.641 3.763 3.798 3.743

3.339 2,512 1,225 -562 2

Hence at 9.5 kts the gross profit/day is $3.798 compared with a daily loss of $2 886 at

15 kts. The above table shows that the profit is very sensitive to changes in speed except near the optimum where there is no great difference in the 9-10 kt range of speeds In practice the Cube law does not hold good over an extended range of speeds so that while

the Cube law is operative near design speeds, at lower speeds a square law may be more appropriate.

One sourcel“ suggests that the fuel consumption per day obeys a law of the form Consumption = CD x J where CD is the consumption at design speed and J is dependent on s such that J=As3+Bs1+Cs

If s is given in knots then coefficients A, B and C are of the general order 0.000167 -0.00098 and 0.0363 respectively


i.e. J = 0.0001675?’ — 0.0009852 + 0.03635

Given suficient data relating to speed and fuel consumption the coefficients and hence the curve can be determined by the method of Ordinary Least Squares as shown in Appendix B. However, using fuel consumption/day = CD(As3 + Bsz + Cs) in example 1 and remembering that the unit of s in this expression is the knot, the equation for profit/day becomes RW

GS = T - CR - pCD(As3 + B52 + Cs) 24s

and differentiating with respect to s d GS 24 RW {as ) 1 de e pCD(3As3 + 2Bs + C) = Ofor a maximum

-3A1+2B S S +C—g€-liv—dpCD


which reduces to s =

4 Rw — C — 2B s i-—— Pdcp

It will be noticed that the variable s appears on both sides of the equation, but in the righthand expression it is not of great significance. The equation can be solved quite easily by entering say 15 kts for s in the RHS and after evaluating the expression to yield s on the LHS this value can be substituted in lieu of 15 kts and the expression evaluated once more. This process of iteration is continued until there is no significant difference between the values of s in each side of the equation. Thus, using the values for constants already given, the following values are obtained.

1st Iteration 2nd Iteration 3rd Iteration

(s) LHS 11.97 kts 11.52 kts 11.46 kts

(s) RI-IS 15 kts 11.97 kts 11.52 kts

Hence using the expression for fuel consumption/day the optimum speed is approximately 11i kts compared with 9i kts using the Cube law. The difference between the result depends on the values of A, B and C in the expression for J and upon the extent that the optimum speed differs from the design speed. For the remainder of the exposition it will, for simplicity, be assumed that the Cube law holds good. ' When time in port and disbursements, cargo handling costs, canal dues etc are considered, the daily gross surplus is given by Rw


pks3.§ s




where D are the disbursements in port and t is the time in port in days Differentiating this expression with respect to s gives

d(GS) _ (RW — D)




2 .( 1)

(la) S2


_ (_—_d_) _ (Q

if2( 1)






= 0 for a maximum RW — D pkszd 2pks -___-__i__--_=0 a 2 =


and assuming that r, is greater than r; this function indicates that an increase in S1 will result in a decrease in TR and vice-versa. Differentiating TR partially with respect to S2 gives. in a similar way: 3TR E wi S 1 — S )(1'2 —21'1)_H(ii)


(Si " 52)

which will normally be negative. implying that an increase in S; causes a reduction in total revenue. Differentiating TR partially with respect to r, gives C

3TR at 5T1


(5 - 52) .. ;_eS*2)...(11) (S1

which is always positive. and substituting this in equation (i) gives

are art} (r,~i,) BS]



.(S1'_S2)... DIV

and similarly


6TR (r2 — r1) 61';

(S1 _ S2)

which is the same as (iv).

( )

Examining equation (iv) it will become clear that BTR _ 6TR _ _

-5-5- will be > -E?-— if (r; - r1) is >(S1 - S2) and conversely

1 1 BTR _ 6TR _ _ _ _ _ K will be < 6Tif (r2 —-— rl) is