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1Unit-1

CONSUMER BEHAVIOUR -I

Unit Structure:

1.0 Objectives

1.1 Introduction

1.2 Revealed Preference

1.2.1 The Feasible Set

1.2.2 The Consumption Decision

1.3 The comparative Statics of Consumer Behaviour

1.4 Income and Substitution Effects

1.5 Summary

1.6 Questions

1.0OBJECTIVES

After going through this unit you will be able to explain the

concepts of -

Revealed Preference

The feasible set

The consumption decision

The comparative statics of consumer Behaviour.

Income and substitution effects.

1.1INTRODUCTION

Samuelson’s revealed preference theory is regarded as

scientific explanation of consumer’s behaviour as against the

psychological explanation provided by Marshallian and Hicks -Allen

theories of demand.

The indifference coordinal theory required less data about the

consumer than the marginal utility (or cardinal) theory. In the

indifference theory one did not have to know the quantities of

utilities of goods. It was enough to know the rankings o f

preferences or consumers. However, to dra w the indifference map

one had to know all the possible combination of goods. This

information had to be supplied by the consumer itthe consumer did

not supply if one could not construct his indifference map.munotes.in

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2To meet this difficulty Prof. Paul Samuelson offered another

theory to explain the consumer’s behaviour in the market.

According to it, the consumer need not supply data on his

preference we could ourselves find out about his preferences by

observing his behaviour by seeing what he buys and at what prices ,

provided his tastes do not change. With this information one may

reconstruct his indifference map. This is known as the Revealed

preference Theory. It may be explained as follows.

When consumer buys one set of goods as against other he

may have reasons for doing so : (a) he likes that particular set more

than the other, (b) that set is cheaper than the other And between

two sets of goods A and B. Suppose the consumer is seen to

buying A but not B, this may not mean that he necessarily prefers A

to B. He have bought A because it is cheaper that B.

Indeed it is possible that even he might have liked B more

than A and may reject that he cannot afford B. However if A and B

cost same amount of money to the consumer and yet he has

bought A and B, the reason could only be that cost he prefers A to

B. Generally, if A is preferred to B, C, D etc. But B, C, D are just as

expensive as A, we may say then that A is revealed preference to

B, C, D or B, C, D are revealed to be inferior to A.

1.2REVEALED PRE FERENCE

We know that utility functions are convenient numerical

representations of preferences and that neither they nor the

consumer’s preferences are directly observable. This subjectivity of

the foundations of consumer theory stimulated interest in t he

development of a theory of demand based solely on observable

and measurable phenomena, namely the bund les actually bought

by a consumer and the prices and money incomes at which they

were bought. The emphasis in this approach is on assumption

about the consumer’s behaviour, which can be observed, rather

than on preferences, which cannot.

As in the utility theory, we assume that the consumer faces a

given price vector, p. and has a fixed money income, M. Our first

behavioral assumption is that the cons umer spends all income.

The second assumption is that only one commodity bundle x

is chosen by the consumer for each price and income situation.

Confronted by a particular p vector and having a particular M, the

consumer will always. Chose the some bund le.

The third assumption is that there exists one and only one

price and income combination at which each bundle is chosen. Formunotes.in

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3a given x there is some P, M situation in which x will be chosen by

the consumer and that situation is unique.

The fourth and crucial as sumption is that the consumer’s

choices are consistent. By this we mean that, if a bundle0xis

chosen and a different bundle1xcould have been chosen, then

when1xis chosen0xmust no longer be a feasible alternative.

To amplify this, let0Pbe the price vector at which0xis

chosen. Then if1xcould have been chos en when0xwas actually

chosen, the cost of1x,01Pxmust be no greater than the cost of0x, which is01Px. This latter is also the con sumer’s money income000MP xwhen0xis chosen.

Similarly, let1Pbe the price vector at which1xis chosen.

Then0xcould not have b een available at price1Potherwise it

would have been chosen. That is, its cost10Pxmust exceed the

cost of1x,1P,1x, which equa l the consumer’s money income1Mwhen1xis chosen. Hence this fourth assumption can be stated

succinctly as00 01 1 1 10P x P x implies P x P x[1.01]

when0xis chosen at01 101,, .P M and x at P MIf0xis chosen

when1xis purchasable0xis said to be revealed preferred to1x.

The statement [1.01] is usually referred to as the we ak around of

revealed preference.

This set of mild behavioural assumptions generates au the

utility based predictions concerning the consumer’s demand

functions. Consider first the sign of the substitution effect. Figure

1.04 shows the consumers initia lbudget line0Bdefined by price

vector0Pand money income0M. The bundle chosen initially on

0

0.Bi s x Bis the budget line after a fall in1Pwith M unchanged, and1xthe new bundle chosen on1B. Our behavioural assumptions do

not place any restriction on the location of1xon1B. As in se ction

2D, it is useful to partition the price effect01xt o xinto a change in

x due solely to relative price change C the substitution effect and a

change due solely to a change in real income. Since we have

forsworn the use of utility functions in this section we cannot use

the indifference curve through0xto define a constant real income.

Instead we adopt the constant purchasing power or slit sky

definition of constant real income Accordingly the consumer’s

money income is lowered until, facing the new prices, the initial

bundle0xcan just be brought. In fig 1.01 the budget line is shiftedmunotes.in

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4inward parallel with1,Buntil at2Bit passes t hrough0x.T h e

consumer confronted with2Bwill buy the bundle2xto the right of0x.T h e r e f o r e0xto2xis the substitution effect and2xto1xthe

income effect of the fall in1P.

X2

X1

X0

X2

B1

B2

B0

OX 1

Fig. 1. 01

We can now prove that if the consumer satisfies assumption

1.01 the substitution effect must always lead to an increase in

consumption of the good whose price has fallen. This is easily done

in the two -good example of fig21.01.xmust lie on2B(by the

assumption that all income is spent) and hence there are three

possibilities :2xcan be to the left or the right of or equal to,02xxcannot be to the left of0xon2Bbecause these bundles are inside

the consumer’s initial feasible set and were rejected in favor of

0,xxcannot equal0xbecause the prices at which2xa n d xare

chosen differ and, by our second assumption different bundles are

chosen in different price income sit uation. Therefore2xmust

contain more1xthan (c.e be to the right of)0x.

This result can be extended to the n -good case, and the proof

is instructive because similar arguments will be used to derive

comparative statics predictions in the theory of the firm. We can

generalize the steps in the analysis of Fig 1.01 as follows.00,pxare the initial price vector and consumption bundle,1p,a n d1xare

the new price vector and consumption bundle. The consumer’s

income in adjusted until at02Mxcan just be purchased at the new

price,1p, so that102px M. Faced with price vector1pand the

compensated money income,2M, the consumer chooses2x,a n dmunotes.in

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5because all income is spent we have that122px M. Hence the

compensating change in M ensures that10 122px M px [1.02]

New2xis chosen when0xis still available (i.e. they are both

on the same budget plane) so that by our consistency assumptio n

1.01 we have00 02px px [1.03]

or :2xwas not purchasable when0xwas bought. Rearranging

1.02 gives10 12 1 0 20px px p x x[1.04]

and similarly 1.03 gives00 02 0 0 20px px p x x[1.05]

subtracting 1.05 from 1.04 gives10 2 00 2 1 0 0 20px x px x p p x xand multiplying by ( -1) we have102 00ppxx[1.06]

This prediction applies irrespective of the number and

direction of price changes, but in the case of a change in the ith

price only,10p and pdiffer only inpiand so 1.06 becomes10 2 0 10 20iiii iiii

ippxx ppxx[1.07]

Hence whenipchanges the substitution effect2iixxis of

opposite sign to the price change. The constant purchasing power

demand curve will therefore slope downwards.

We can also derive the slut sky equation from the behavoura

assumption. Since10 0020M p x and M p xthe compens ating

reduction in M is00 10 0 1 0 1 0 002MM M p x p x p p x p p xmunotes.in

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6and in the case of a change;piniponly we have0iiMp x [1.08]

The price effect of10iii ip on x is x xand this c an be

partitioned into the substitution20iixxand income12iixx

effects:10 20 12ii ii iixx xx xx

Dividing byipgives10 20 12ii i i iiii ixx xx xxppP[1.09]

But from [1.08]0/iipM xand substituting this in the

second term on the right hand side of [1.09] yield1212

.iio ii i

i i

ixxxx xxpp m 0.ii i

ipiimp xxx xxpp M [1.10]

The1mnotation indicates that money income is held constant

in evaluating the rate of change of1xwith respect toipand the

similar notation on the right hand side that purchasing power px

and price vector p are being held constant in evaluating the rate of

chang eo fixwith respect toipand M [1.010] is the descrete

purchasing power version of the slutsky equation of section.

It is possible to show that the utility maximizing theory of the

consumer and the rev ealed preference theory are equivalent .All

the predictions derived from the assumption about preferences in

section 2A can also be derived from the assumption about

behaviour made in this section. A consumer who satisfies the

preference assumptions mill a lso satisfy these behavioral

assumption. Similarly, if the consumer satisfies the behavioural

assumptions, we can construct curves from observed choices

which have all the properties of the indifference curves of section 2 -

1. The consumer acts as if posses sing preferences satisfying the

preference assumptions. (strictly the weak action needs to be

strengthened slightly). Since the two theories are equivalent we will

not consider more of the predictions of the theory of revealed

preference but will instead u se the theory of revealed preferencemunotes.in

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7but will instead use the theory to investigate some properties of

price indices.

1.2.1 The feasible set:

We initially assume that the consumer has a given money

income M, faces constant prices for all of the goods in the utility

function and cannot consume negative quantities of any good.

Then A Consumer’s feasible set defined by these assumptions is

the set of bundles satisfying

11 2 2 ....nn ipx px px p m[1.1]

120, 0, ..... 0n xx xwhereipis the price of good i.

X2

0

22MXPB

O0

11MXPX1

Fig. 1. 02

The feasible set in the two good case is shown in Fig 1. 02as

the triangular area0012 1/Ox x M Pis the maximum amount of1xthat can be bought with income M at a price of012.pxis analogously

defined. The budget constraint is11 2 2px px Min this two -good

case, or:21 1 2/xM p x p [1.2]

which is satisfied by all points on or below the line B from00

12xt o x B,

the upper boundary of the feasible set, is known as the consumer’s

budget line and is defined bymunotes.in

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821 1 2/xM p x P [1.3]

The slope of the budget line is2112tan Mc o n s tdx Pdx P[1.4]

where the notation on the left -hand side is to remind us that thi si s

the rate at which a consumer with fixed income can exchange12x for xon the market. A one unit reduction in purchase of1xreduces expenditure by1P, and so, since 1 unit of2xcosts2P.t h e

consumer can buy12/ppextra units of2x.T h e r e f o r e1u n i to f1xexchanges for12/ppunits of2x.

The consumer’s feasible set has a number of properties

relevant for the analysis of the optimal consumption decision It is.

(a) bounded, from below by the non -negativity constraints on theixand from above by the budget constraint, provided that M is finite

and no price is zero. If, for example,10pthen the budget line

would be a line parallel to the1xaxis through the point022/xM p

and the feasible set would be unbounded to the right : Since1xwould be a free good the consumer would consumes as much of it

as he wished;

(b) closed. Since any bundle on the budget line B or the quantity

does is available;

(c) Co nvers, Since for any two bundles1xand11xin the feasible

set, any bundlexlying on a straight line between them will also be

in the feasible set. Sincexlies between11 1,x and xand they both

satisfy the non-negativity constraints,xwill cost no more than the

consumer’s income lying between1xand11xit must cost no more

than the more expensive of them, say1.xBut since1xlies within

the feasible set, so mustxlies within the feasible set, so mustx.

Hencexis in the feasible set.

(d) non -empty -provided that M > 0 and at least one price is finite

the consumer can buy a positive amount of at last one good.

Consider the effects of changes in M andiPon the fea sible

set, in preparation for section D where we examine their effects on

the consumer’s optimal choice. It money income increases frommunotes.in

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91oMt o M, the consumer’s feasible set expands as the budget line

moves outward parade with its initia l position, as in fig 1. 03(a) with0MMthe intercepts of the budget line B. on the12x and xaxes

respectively are10 2//oM P and M P.Ad o u b l i n go fMf o re x a m p l e ,

will double the value of the intercepts, Since12 02/2 /MP MP when102MM. The slope of the budget line is -12/PPand is unaffected

by changes in M.

Consider next an increase in1P, as shown in fig (1. 03)(b)

since M and2Pare unchanged the budget line will still have the

same2/MPintercept on the2xaxis. An increase in1Pwill cause

the budget line to pivot about02/MPand become more steeply

sloped as12/PPbecomes larger. In fig 1. 03(b) a rise in111Pt oPshifts the1xintercept from101 01/MP t o MPwhere101 1//oMPMP

since111PP.

Equal proportionate increases in all prices cause the budget

line to shift in wards towards the origin as in fig 1. 03(c)suppose12P and Pincrease from12 1 21.P and P to KP and KP where kThen the

slope of the new budget line is unchanged :12 1 2//KP KP P P and the new intercepts are11 2 2// / /M KP M P and M KP M P

Finally, if all prices and M change in the some proportion the

budget lines is unchanged. The intercept on the1st axi sa f t e ra l l

price and M change by the factor K is//iiKM KP M Pso the

intercept is unaffected, as is the slope, which is12 1 2//KP KP P P.

(A) Fig. 1. 03 (B) Fig. 1. 03

X2 X2

B112/MP02/MP02/MPB0

O01/MP11/MPX1O01/MP01/MPX1munotes.in

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10(C) Fig. 1. 03

X22/MP12/MPO101/MP1/MPX1

1.2.2 The consumption decision :

Given the assumption of the previous two sections, the

consumer’s problem of choosing the most preferred bundle from

those available can be formally stated as12 1max , ... . . , 0, 1... , ...ni i i niuxx x s t P X MX i n x x [1.3]

We can derive the conditions which the solution to thi s

problem must satisfy by a diagrammatic analysis of the two good

case. We leave to the latter part of this section a brief confirmation

of our results using more rigorous methods.

From the assumption of section A we can represent the

consumer’s preferen ces by a utility function which has indifference

curves or contours like those of figure 1. 04.

X2 Fig. 1. 04

B

X1

X*I2*2XI1

I0

O*1XX1munotes.in

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11All commodities are assumed to have positive marginal utility

so that bundles on higher indifference curves are preferred to these

on lower indifference curves. This assumption (a consequence of

assumption 4 in section A). Also means that the consumer will

spend all his income since he cannot be maximizing a if he can buy

more of some good with positive marginal utility. The consumer will

therefore choose a bundle on h is budget line B.

In figure 1. 04there is a tangency solution where the optimal

bundle x is such that the highest attainable indifference curve. It is

tangent to the budget line and the consumer consumes some of

both goods the slope of the indifference curve is equal to the slope

of the budget line at the optimum.

22

11tan tancons t M cons tdx dx

dx dx

The negative of the slope of the indifference curve is the

marginal rate of substitution21;MRSand the negative of the slope of

the budget line is the ratio of the price of12x and x.H e n c et h e

consumer’s equilibrium condition can be written as112122PMRSP[1.4]

The consumer is in equilibrium (choosing on optional bundle)

when the rate at which he can substitute one good for another on

the market is equal to the rate at which he is just content to

substitute one good for another.

We can interpret this property of the optimal choice is a some

what different may. If the consumer spent an extra unit of money on1xhe would be able to buy11/Punits of11 1.xxis the gain in utility

from an additional1xunits of1xHence11/Pis the gain in utility

from spending an additional unit of money on12.2 /xPhas an

along us interpretation. The consumer will therefore be maximizing

utility when he allocates his income between1xand2xso that the

marginal utility of expenditure on1xis equal to the marginal utility of

expenditure on2x.1212PP[1.5]

This is exactly the condition o btained by multiplying both sides

of 1.4 by21 ./Pmunotes.in

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12If the consumer’s he would be indifferent between spending it

on12x or x, in either case utility could rise by11 2 2//PP.H e n c e ,

if we call the r ate at which the consumer’s utility increase as income

increases the marginal utility of income, densted bym,w eh a v e

12

12mPP[1.6]

A more plausible optimum when there are many goods would

be a corner point solution, where the optimal bundle x does not

contain positive amounts of all goods, as in Fig 1. 05where no2xis

purchased. In this case the indifference curve at x is steeper than

the budget line i.e has a small slope (re membering that the

indifference curve and the budget line are negatively stopped ).

Hence

22

11tan tancons t M cons tdx dx

dx dx [1.7]

Fig. 1. 05

X201 2

X*

OX 1

and therefore

21 1 2

12

12 2 1tan tancons t M cons tdx P dxMRSdx P dx

Rearranging, this condition ca nb ew r i t t e n1212mPP[1.9]munotes.in

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13The marginal utility of expenditure on the good purchased,1x,

is greater than the marginal utility of expenditure on2x,t h eg o o d

not purchased. Because of the higher marginal utility of expenditure

on1xthan on2xthe consumer would like to move further down the

budget line substituting1xfor2xbut is r estrained by the fact that

consumption of negative amounts of2xis not possible.

A more formal analysis

Since the consumer’s preferences satisfy the assumptions of

section A, the objective function in problem above is continuo us

and strictly quasi -concave From section B the feasible set for the

problem, defined by the budget and non -negativity constraints, is

non-empty, closed, bounded and convex. From the Existence,

Local -Global and uniqueness The orems. the consumer’s

optimiza tion problem has a unique solution and there are no non

global local solutions.

Since there is at least one good with positive marginal utility

the consumer spends all income and hence the budget constraint

can be written as an equality constraint.0ii MP X.I fw e

assume that the solution will be such that some of all goods will be

consumed0 1...ixinwhereixis the optimal level ofix,t h e n

the non -negativity constrai nts are non -binding and we have a

problem to which can be applied the method of lagrange outlined

in Appendix G. The lagrange function derived from is1, ....,ni iLxx MP X [1.10]

1.3THE COMPARATIVE STATICS OF CONSUMER

BEHAVIOUR

The solu tion to the consumer’s optimization problem depends

on the consumer’s preferences, prices and money income. We can

write the solution, which we call the demand for goods, as a

function of prices and money income.12, ,..... , , 1,....,ii n iXD P P P M D P M i n[1.11]

wher e12, ,....nPP PP is the vector of prices, and the form of the

Marshallian demand functioniDdepends on the consumer’s

references.

The properties of feasible sets and the objective function

enable us to place restri ction on the form of the demand functionsmunotes.in

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14first provided that p, M are finale and positive, the optimization

problem must have a solution, since the requirements of the

Existence theorem are satisfied, second, the differentiability of the

indifference curv es and the linearity of the budget constraint imply

that the optimal bundle will vary continuously in response to

changes in prices and income, and that the demand functions are

differentiable. Third, the conditions of the uniqueness theorem are

satisfied and so the demand relationships are functions rather than

correspondences : a unique bundle is chosen at each (P, M)

combination.

We now c onsider the comparative statics properties of the

model. We investigate the effects of changes in the exogenous

variables C prices, money income on the equilibrium value of the

endogenous variables (the consumer’s demand for goods). We

want to predict what happens to the optimal bundle

12** * *

12 , , .... , , ....

nnXX X X D DDas the feasible set varies.

We consider first chang e in the consumer’s money income. In

figure 1. 06,1Bis the initial budget line,*Xthe initial bundle chosen.

An increase in M, with12,PPconstant, will shift the budget line

outward paroled with itself, say to2Bwhere1Xis chosen. A further

increase in M will shift the budget line to3Bwhere11xis chosen

X2 Fig. 1. 06

X0X11ICC4X13X1

X*2X+15B1 B2 B3

OX 1

The income consumption curve is the set of optimal points

trade out as income vaxies in this may, with prices constant. In the

case illustrated both12x and xare normal goods, for which demand

increases as money income rises. However with different

preferences the consumer might have chosen012x or x on B.I f0xhad been chosen (If42Ia n d n o t Ihad been the consumer’smunotes.in

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15indifference curve) then the demand for1xwould have fallen as

money income rose1xwould than be known as an inferiat good. A

rise in M may lead to a rise, a fall, or no change in the demand for a

good. Without knowledge of preferences we cannot predict whether

a particular good will be inferior or normal. The theory of consumer

behavior cannot be tested by considering the effect of changes in M

on the demand for a sin gle good, since any effect is compatible

with the theory.

The theory does predict, however, that all goods cannot be

inferior. If the consumer reduces demand for all goods when

income rises he will be behaving inconsistently. To show this, let*xbe the bundle chosen with an initial money income of11M and Xthe bundle chosen when money income rises to2M.1*..xx i eif

the demand for all goods is reduced, then1Xmust cost less then*xsince prices are held constant1xwas therefore available when*xwas chosen. But when1xwas chosen*xwas still attainable

(Since money income had increased). The consumer there form

preferred*xover x with a money income of1*1M and x over xwith

money income21MM.H ei st h e r e f o r e inconsistent : his behaviour

violates the translate assumption of section A, and our model would

have to be rejected.

We now turn to the effects of changes in price on the

consumer’s demands. Fig 1. 07shows the implications of a fall in

the price of1xwith money income held constant.1Bis the initial

budget line,*xthe initial optimal bundle. A fall in1P,say from11Pt oP, causes th e budget line to shift to12BXis the optimal

bundle on112,BXthe optimal bundle on3,Bwhich results from a

further fall in1Pfrom11 111Pt o P.The price consumption curve (PCC)

is traced out as the set of optimal bundles as goods increases as1Pfalls. However with different preferences the optimal bundle might

have been02x or x on B.I f0xwas the optimal bundle with111PPthen1xwould be a Griffin good, the demand for which falls as its

price falls. We conclude that the demand for a good may fall, rise or

remain unchanged as a result o f change in a price facing the

consumer once again the model yields no definite (refutable)

prediction about the effect on a sin gle endogenous variable (the

demand for a good) of a change in one of the exogenous variables

(in this case a price). If is agai n possible, however, to predict (by

reasoning similar to that employed in the case of a change in M)

that a fall in price will not bad to a reduction in demand for all

goods, and the re ader should supply the argument.munotes.in

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16Fig. 1. 07

X24PCC

X023X11

X*X11X+5B1 B2 B3

OX 1

1.4INCOME AND SUBSTITUTION EFFECTS

The analysis of the effect of price changes on the consume r’s

demands (optimal choices) has suggested that demand for a good

may increase, decrease or remain unchanged, when its price rises;

in other words anything may happen we now examine the effect of

a change in the price of good 1 in more detail in order to see if it is

possible to make more definite (refutable) predictions. We proceed

by making a conceptual experiment. All we can actually following a

price change. However, we can carry out a hypothetical analysis

which decomposes the overall demand change in to two

components. We then use this decomposition to say something

more definite about consumer beha uiour.

In Fig 1. 08. it can be seen that the fall in price of good 1 does two

things :

(a) it reduces the expenditure required to achieve the initial util ity

level1Iallowing the higher utility level2Ito be achieved with the

some expenditure . There has been an increase in the consumers

real income:

(b) it changes the relative prices facing the consumer.munotes.in

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17In Fig 1. 08we accordingly break dawn the change in demand for1xinto :

(a) the income effect, which is the change resulting solely from the

change in real income with relative price held constant; and

(b) the own substituti on effect, which result’s salary from the

change in1Pwith real income held constant.

X2 Fig. 1. 08

X*X1

X011B3 B2

B1

O*1X01X11XX1*1Xa n d Xare the optimal bundles before and after the fall in11 2,P B and Bthe corresponding budget lines. The compensating

variation in money income is that change in M which will make the

consumer just as well off after the price fall as he was before. In

other words, there will be some reduction in M after the price fall

which will ‘cancel out’ the real income gain and return the consumer

to the initial indifference curve.1I.T h eb u d g e tl i ne is shifted

inwards (reducing M) pazallel with the post price fall budget line2Buntil at3Bit is just tangent to the original indifference curve1I.T h e

consumer confronted wit h this budget line would chose bundle0x.

The difference between*xand0xis due to a change in relative

prices with real income (utility) held constant. The difference

between0xand1xis due to the change in money income with

relative prices held constant.*111,Xxand01xare the amounts of1xcontained in the b undles*10,,xxxand

(a)0*11xxis the own substitution effect;munotes.in

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18(b)1011xxis the income effect;

(c)0* 10 1*11 1 1 1 1xx xx xxis the total price effect.

The purpose of carrying out this ex periment in hypothetical

compensation is to demonstrate that the own substitution effect will

always be positive in the case of a price fall and negative for a price

rise. The absolute value of the slope of the indifference curve

declines from left to righ t, i.e. as more12x and less xis consumed the

curve flattens. The fall in1Pflattens the slope of the budget line,

and hence the budget line3Bmust be tangent with1Ito the right of*,xi.e. at a bundle containing more1.xThe income effect is positive in the particular case illustrated in

Fig 1. 08. The income effect reinforces the own substitution effect

since1xcontains more1xthan0x.I f1xhad been inferior the

income effect of the price fall would have been negative and in the

opposite direction to the own substitu tion effect, so that the price

effect would be smaller than the own substitution effect, In fig 1.8

(a) the income effect partially offsets the substitution effect but the

price effect is still positive; a fall in1Pleads to a ris ei nt h ed e m a n d

for1x. In figure 1.8 (b) the negative income effect more than offsets

the positive substitution effect and1xis a giffen good. Hence

inferiority is a necessary, but not sufficient, conditi on for a good to

be a Giffen good.

Fig. 1.8 (a) Fig. 1.8 (b)

This decomposition of the price effect has generated two

further predictions :munotes.in

## Page 19

191) A normal good cannot be a Giffen good. Hence if we o bserve

that a consumer increases demand for a good when money income

rise (other things including prices being held constant), we would

predict that, if its prices should fall, he will want to buy more of it. If

we then observe that he reduced his demand f or the good when its

price falls (and all other prices are constant and his money income

is reasonably close to its original level), then the optimizing model

of consumer behaviors has yielded a false prediction.

2) The own substitution effect is always of opposite sign to the

price change.

The above decomposition of the price effect into an income

and substitution effect is based on the definition, made by Hicks, of

unchanged real income as an unchanged utility level. Stutsky

suggested an alternative definition of a constant real income as the

ability to purchase the bundle of goods bought before the price

change. This constant purchasing power definition has the

advantage that it does not require detailed knowledge of the

consumer’s indifference map.

Figure 1.9 reproduces Figure 1.6 with some additions to show

the relationship between the Hicks and slutsky definitions of a

constant real income. The budget line4Bjust enables the

consumer to buy*x,the initially optimal bundle, at the lower price

of1P. Confronted with this budget line, the consumer actually

chooses1x.T h e price effect has been decomposed into as income

effect113xxand an own substitution effect11xx. The income

effect will again be positive, negative or zero depending on the form

of the indifference map. The substitution effect will, as in the

Hicksian case, always lead to a rise in deman d for a good whose

price has fallerxcannot lie to the left ofxon4Bbecause this

would mean that the consumer is new choosingxwhen*xis still

available, having previously rejectedxin fauour of*x.T h e

transitivity assumption would be violated by such behaviour. The

slutsky definition yields a prediction. (The sign of the substitution

effect) which can be tested without specific knowledge of the

consumer’s indifference curves to cancel out the income effect.

Our consideration of the comparative static properties of the

model has shown that it does not yield refutable pre dictions about

the overall change in demand for individual goods induced by

ceteris paribus change in a price or money income. In other words,

Fig 1.9munotes.in

## Page 20

202XFig. 1.91XX+

X*

I2

X0I

I2

B1 B3 B4 B2

O*1X01X1X*1X1X0 , 1, 2, .... ,xD iij npi Pi

and0 1, 2, .... ,xD iinMM

for every good and price, only by considering the effect of changes

in1P or Mon goods, or by considering the effect of changes inipand M on a single good or by maki ng more specific assumptions

about the consumer’s preferences can definite predictions be

generated.

Consider, however, the consequences of equal proportionate

change in all prices and M. Suppose M increases to KM (K > 1) and

price to1KPand2KP. The slope of the budget line will be

unaffected. The intercept on the1xaxis is1/MPbefore the

changes in M and prices and11//KM KP M Pafter the ch ange

similarly for the intercept on the2xaxis. Hence the equal

proportionate changes in M and all prices alter neither the slope nor

the intercepts on the budget line and so the feasible set is

unaltered. If the feasible set is u nchanged then so is the optimal

bundle.

The model therefore predicts that the consumer will not suffer

from money illusion ;he will not alter his behaviour if his purchasing

power and relative prices are constant, irrespective of the absolutemunotes.in

## Page 21

21level of p rices and money income more formally, the demand

functioniDfor every commodity is homogeneous of degree zero in

prices and money income, since we have*0,, ,ii ixD K p k M K D i p M D p M [1.12]

1.5SUMMARY

Revealed preference the ory is the scientific explanation of

consumer’s behaviour. The feasible set is the set of bundles

satisfyingPi i M.

Consumer chooses the optimal consumption bundle.

1.6QUESTIONS FOR REVIEW

1. Critically examine the Revealed P reference Theory of

consumer behaviour.

2. Examine the Revealed Preference theory and show how it is

an improvement over the indifference curve analysis.

munotes.in

## Page 22

22Unit-2

CONSUMER BEHAVIOUR -II

Unit Structure :

2.0 Objectives

2.1 Intro duction

2.2 The Expenditure Function

2.3 The Indirect utility function, Roy’s Identity and the slutsky

equation

2.4 Properties of Demand Function

2.5 Choice under uncertainty

2.6 Summary

2.7 Questions for Review

2.0 OBJECTIVES

At the end of this le sson you will be able to explain -

The expenditure function.

The indirect utility function.

The Roy’s Identity

The Slutsky Equation.

Properties of demand function.

Concept of choice under uncertainty.

2.1 INTRODUCTION

In the previous chapter we defi ned the consumer problem as

that of choosing a vectorxto solve the problem max..x s t px M, where p is a price vector and M money income.

From the solution we derived marshallian demand functions, 1, ...iixD p M i n, which express demands as functions of

prices and money income. We observed that we cannot place

restrictions on the signs of the partial derivatives of these functions:

/ 0, / 0 , , ..... ,iiDM pM i j n In particular the demand for

ag o o dd o e sn o tn e cessarily vary inversely with its own price.

However, as a result of a diagrammatic analysis, we were able to

say that this will be trees of normal goods or of inferior goods

whose income effects are weaker than their substitution effects. Wemunotes.in

## Page 23

23know put this analysis on a more rigorous and general basis. We

also consider the problem, central to many applications of

consumer theory, of deriving a money measure of the costs and

benefits incurred by a consumer as a result of price changes. In

doing so, we develo p the methods and concepts of duality theory,

an approach to the analysis of optimization problems which permits

an elegant and concise derivation of comparative static results.

2.2 THE EXPENDITURE FUNCTION

The expenditure function is derived from th ep r o b l e mo f

minimizing the total expenditure necessary for the consumer to

achieve a specified level of utility is :11.....min . . ( ) .....

nii nXXpx st i x x

( ) 0, 1, ....iii x i n……………………. (2.1)

If all prices are positive the first c onstraint in (2.1) will be

satisfied as an equality in the solution, since ifxexpenditure

can be reduced without violating the constraint. If it is further

assumed that all x; are strictly positive in the solution, we can write

the lagrange function for the problem (withas the lagrange

multiplier) as1.....ii nLp x x x ………………. (2.2)

and the necessary conditions for a minimum of L, also the

necessary conditions for a solution of [2.1] are0 1, ....ii

iLpi nx………………... (2.3)

,..... 0inLxx ……………………… (2.4)

The conditions on theixbear a striking resemblance to 1.10

in chapter 1 writing them asiipuand dividing the co ndition onixby the condition onixgives.iiiippu………………………………………….. (2.5)munotes.in

## Page 24

24X2

Fig. 2.102xx*32x1

I0

M1M0M2 M3

OX 1

The ratio o f prices is equated to the marginal rate of

substitution. This is not surprising as examination of the two -good

case in figure 2.1 indicates. The indifference curve I 0shows the

combinations of x 1and x 2which give a utility level of u and the

feasible set for the problem is all points on or above I 0.T h el i n e s

M0,M 1,M 2, are iso expenditure lines similar to the budget lines of

earlier diagrams. M 0, for example, plots all bundles costing m 0,i . e .

satisfying the equation11 2 2 0px px m.T h ep r o blem is to find the

point in the feasible set which is on the lowest isoexpenditure line.

This will, in the tangency solution shown here be where the

indifference curve0is tangent to the isoexpenditure line m 0.T h e

problem confr onting the utility maximizing consumer is to move

along the budget line until the highest indifference curve is reached.

The expenditure minimizing problem is to move along the

indifference curve until the lowest isoexpenditure line is reached.

The optimal*1xin problem (2.1) depend on the prices and the utility

level u :*

1..... , , 1, .......ii n ixH p p uH p u i n……………….. (2.6)

and,iHp uis the Hicksian demand function for x i. Substituting the

optimal values of X iiipxgives.*,,ii i ipx pH p u m p u…………………… (2.7),mp uis the expenditure function, showing the minimum

level of expenditure necessary to achieve a given utility level as a

function of price and the required uti lity level.

The Hicksion demand function is also called the

compensated demand function. In considering the effect of amunotes.in

## Page 25

25change in price on demand with utility held constant (the paralal

derivative/, 1 ,ijHP i jnwe automatically make whatev er

changes in expenditure are required to compensate for the effects

of the price change on real income or utility. This is illustrated in

figure 2.1. Assume P 2remains constant while P 1falls to give a new

family of isoexpenditures lines, with slopes corr esponding to that of

M3in the figure x1is he new expenditure minimizing consumption

bundle, and the change from x*to x1is the effect of making the

relative price change with m varying to keep u constant. The

optimal expenditure line slide round the ind ifference curve from m 0

to m 3as the optimal bundle changes from x*to x1.T h em i n i m i z e d

total expenditure can be read off from the intercepts of m 0and m 3

on the x 2axis. The fall in p 1lowers m from022pxto322px.

Provided the indifference curves are strictly convex to the

origin the optimal x i(and hence the expenditure function) vary

smoothly and continuously with the prices of the goods. Hence the,iHp ufunctions have continuous deriva tives with respect to the

prices. The demand curve we derive from the Hicksian demand

function was represented by curve hh. The slope of the Hicksian or

compensated demand curve,/ 1 , ....iiHp i nis the substitution

effect of the price change. S ince by definition/iiHpis taken with

u held constant.

The expenditure function gives that smallest expenditure, at

a given price vector, that is required to achieve a particular

standard of living or utility level, and describes h ow that expenditure

will change as prices or the required utility level change. The

assumptions made in chapter. Concerning the nature of the

consumer’s preference ordering and indifference sets imply certain

properties of the expenditure function.

(a) The expenditure function is concave in price choose two price

vectors1pand11p, and K such that1ok.D e f i n e1 111.pk p k pWe have to prove that C see the definition of

concaulity in Appendix B) :11 1,, 1 ,MP u k m Pu km P u for given u.

Proof

Let1xand11xsalve the expenditure minimization problem

whine the price vector is respectively1pand11p.B yd e f i n i t i o no f

the expenditure function,11,px m p uand11 11 11,px m p umunotes.in

## Page 26

26likewise, letxsolve the problem when the price vector isp, so

that11,px m p u.S i n c e11xa n d xare solutions to there

respective expenditure minimization problems we must have11 1 1 11 1 1 1P x P x and p x P x[2.8]

Multiplying through the first inequality by k and the second by 1 -k

and su mming gives11 1 1 1 1 1 1 111KP x k p x kp x K p x [2.9]

But by definition ofpthis implies11 1 1 1 1 1 111kp K p p x kp x k p x [2.10]

which is the result we want.

Figure 2.2 illustrates the proof of this important result. It is

obvious that, when the is expenditure lines at which11 1xa n d xare

optimal solutions are shifted so as to pass through point,xthey

must yield higher expenditure, this giving the key inequalities in

(2.8) The rest of the proof then follows by simple algebra.

Figure 2.2

The figure could in one sense be misleading. The inequalities

(which in this case are strict) appear to follow from the convexity of

the indifference curves. Note, however, that the inequalities follo w

simply from the fact that1x(respectively11x)m i n i m i z e spxat price

vector1p(respectively11p)w h i l exmay not 2.8 then follows from

the definition of a minimum. Thus the proof of concavity of themunotes.in

## Page 27

27expenditure function does not depend on convexity of preference.

However the property of uniqueness of solution like11 1xa n d x,a n d

the differe ntiability of Hicksian demands and of the expenditure

function, do. Note that strict converity of preference implies strict

concavity of the expenditure function at an interior solution to

problem [1] Since it implies uniquess of the solution and hence

strict inequalities in [2.8].

Figure 2.3 illustrates the strict concauity of the expenditure

function when the price vectors11 1p and pdiffer only in respect of

one price,1.pThe slope of the expenditure function a tap o i n ti s

equal to the compensated demand for good; at the price1p:

(b) Shephard’s lemma :*

1,/: ,iimpu p X H p uThe proof is just a version of the Envelope Theorem.

Differentiating 2.7 with respect to the 1stprice g ives.

* *

** 1

11nnxiii i i iii ii ix x mxP x xpP p [2.11]

The second equality uses the fact that,iipufrom the first

order condition [2.3] since utility is held constant whenipvaries,

differentiating the constraint [2. 4] with respect toipshows that

**

1/0n

ij i iux pwhich gives the third equality in [2.11]

Thus the partial derivative of the expenditure function with

respect to the 1stprice is the compensated demand for the 1stgood.

In figure 2.3, the slope of the curve at price11 1 1 1

1, ... .... , .ii i inPi s x H p p p uThis can be put intuitively as follows,

suppose a consumer buys 12.5 units of a gas a week at a cost of1Eper unit. The price of gas then rises by p per unit. Shephard’s

lemma says that, to a first approximation, to maintain the same

utility level or standard of living her expenditure must increase by12.5 :iiHPjust enough to maintain consumption at the initial price

level. The qualif ication ‘to a first approximation’ is important. For

finite price changes. Fig 2.3 should thatiiHpoverstates the

required increase in expenditure, since the expenditure function is

strictly concave. As a good’s price goes up, the c onsumer

substitutes away from the good in question, and this reduces the

amount of expenditure otherwise required to keep utility constant.

Shephard’s lemma tells us that for small enough price changes this

distinction can be ignored.

(c)/0i mpwith strict inequality if*0ix.munotes.in

## Page 28

28This follows immediately from shephard’s limma. Since at last

one good must be bought, the expenditure function is non -

decreasing in the price vector p and strictly increasing in at least

one price. Higher prices mean higher expenditure to reach a given

utility.

(d) The expenditure function is homogeneous of degree 1 in prices.

Take a given u value and price vector01pand00,ooLet M m p u p xwhere0xis the expenditure minimizing

bundle at0p,t h a ti s00 0px p xfor all bundle x yielding utility of u

or more. But this imples that00 0 0kp x k p xfor all bundles yielding

at lea st u and so0xis optimal at prices0pand0kp.T h e n00,,mk p u k mp u . Since relative prices do not change, the

optimal bundle is not changed. It has merely become more or less

expensive depending on whether k > 1 or k < 1.

(e) The expenditure function is increasing in u Higher utility at given

prices requires higher expenditure. Rather than use the envelope

theorem again, recall that the lagrange multi plier0in 2.2 is

equal to the derivative/,muis the marginal cost of utility since

it represents threat of change of minimized expenditure with

respect to the required utility level .is the recipr ocal of the

lagrange multipliesin the corresponding utility maximization

problem, i.e.is the inverse of the marginal utility of income note

that, although the assumptions underlying ordinal utility theory allow

the sign ofto be estestablished, we cannot say thatis

necessarily increasing or decreasing, with u, because both are

possible for different, permissible utility functions.,mp u11 1

11 iimP u X H Pu,mp u

OP i11iPPi

Figure 2.3munotes.in

## Page 29

29It is important to be clear about the relation between

expenditure and utility the essential facts about the consumer’s

preferen ce ordering are contained in the structure of her

indifference sets or curves. The minimum expenditure requiredto

reacha given indifference set at given prices is unaffected by any

number we attach to that indifference set to indicate its place in the

orde ring on the other hand, once we have chosen a numerical

representation of the preference ordering a utility function this will

imply a particular relationship between expenditure m and utility u.

But the properties we set out above hold for all permissible utility

functions, and the only general restriction we can place on the

relation between m and u (for a given price vector) is that it is

monotonically increasing.

2.3THE INDIRECT UTILITY FUNCTION, ROY’S

IDENTITY AND THE SLUTSKY EQUATION

The indirect utility function is derived from the consumer

problem of maximizing1....,nxx subject to the budget constraint

iipx Mand non -negativity constraints we saw that theixwhich

are optimal for t his problem will be functions of the*

1: ...., , ,ii i n ip and M x D p p M D p MThe maximized value of**11, .... , ....,nnux x ux xwill therefore also be a function of the

ip and M** *

11... , , .... ( , ,nnux x uD p m D pM u p m [2.12]uis known as the indirect utility function since utility depends

indirectly on prices and money income via the maximization

process, in contrast to the utility function1, ...,nux xwhere u

depends directly on theix.W e can use*uto investigate the effects

of changes in prices and money income on the consumer’s utility.

From the interpretation of the Lagrange multiplier, the effect of

an increase in money income on the maximized utility is

*um[2.13]

The effect of a change inipon*ucan also be found as a

version of the Envelope Theorem, Differentiating*uwith respect toip:munotes.in

## Page 30

30***kkkkii ixxuuPpp p [2.14]

The budget constraint must still be satisfied so that

*0kk

iid dMPxdp dP

and so

*

*0k

ki

ixpxp

or**kikixxpp

Substitution of this in [2.14] gives Ro y’s identity :

****iiiuuxxpm [2.15]

The expression on the right hand side of 2.15 has the

following intuitive explanation An increase inipis a reduction in the

purchasing power of the consumer’s money income M, and by

shephard’s lemmat, to the first orders, her purchasing power falls at

the rate*ixasipvariesis the marginal utility of money income.

The product ofand*ixis the rate at which utility varies with

money income, times the rate at which (the purchasing power of)

money income varies withipand so this product yields the rate of

change of utili ty with respect toip.

Since0, Roy’s identity should that an increase in the price

of good a consumer buys reduces her (maximized) utility or

standard of living to a greater extent, the larger the quant ity of it she

buys.

The indirect utility function tells us that utility depends, via the

maximization process, on the price income situation the consumer

faces. Note that implies 2.13 that the indirect utility function is

monotonically increasing in inc ome, M. thus we can invert the

indirect utility function*,uu p mto obtain the expenditure

function M = m (p, u). A given solution point for a given price vector

can be viewed equivalently as resulting from minimizing

expenditure subjec tt ot h eg i v e nu t i l i t yl e v e lo rm a x i m i z i n gu t i l i t ymunotes.in

## Page 31

31subject to the given expenditure level. We can choose either to

solve the utility maximization problem, obtain the indirect utility

function and invert it to obtain the expenditure function, or to obtain

the expenditure function and then invert it to obtain the indirect

utility function. The two functions are dual to each other, and

contain essentially the same information : the forms of the functions

and their parameters are completely determined by the fo rm of the

original (direct) utility function. But then, since each of these three

functions contains the same information, we can choose any one of

them as the representation of the consumer’s preferences that we

wish to work with.

Duality can be used t og i v ean e a t e rd e r i v a t i o no fr o y ’ s

identity. Setting M -m(p,u), rewrite the indirect utility function as*,,uu p m p u [2.16]

The differentiating through with respect toip,a l l o w i n gmt o

vary in such a may as to hold u constant, gives

**iiuu mopm p[2.17]

which, using shephard’s lemma and 2.13, gives Roy’s identity 2.15

directly.

Since the indirect utility function is ordinal and not cardinal, we

cannot restrict it to be convexs or co ncave (unlike the expenditure

function), bad we can show that is quasi -convex in prices and

income, a property that is useful in many application.

Figure 2.4 illustrates quasi -convexity in prices and income. A

function is quasi convex if, given any poin t in its (convex) domain,

the worse set of the point, i.e. the set of points giving Fig 2.4.

Figure 2.4munotes.in

## Page 32

32the same or lower values of the function, is convex. Take the case

of two goods, where the indirect utility function is*

12,,up p M.In

part (a) of the figure,00 012,pp pis some arbitrary point, and the

indifference curve0uor contour of the indirect utility function,

through that point is convex to the origin if the function is quasi -

conve x. The course set0* 0 * 0 0 0

12 12 1 2,; , , , , wp p p u p p M u p p M use to the north

east of0pchigher prices imply lower utility) and is convex. In (b) of

the figure, the contour*uthrough the point00 * 00

12 12,, ,pp M uppMlies to the north west of the point chigher

price and lower income implies lower utility) and is convex. (Be sure

you can explain the negative and positive slopes of these contours,

respectively) Similarly for any point0

2,pM.

Prove that the indirect utility function is quais convex in prices

and income choose two points in the domain of the function,00,pmand11,pm, such that*00 0 1 1 1,up m u u p m [2.18]

So that11,pmis in the wares set of0,opm.W eh a v et o

show that any convex combination of these two price -income

vectors is also in this worse set of00 * 0,: ,pm u p m u[2.19]

where01 0 11, 1 [ 0 , 1 ]pk p k p k m k m k [2.20 ]

Now take any goods vector x that satisfies the budget

constraintpx mor, given the definitions ofp and m01 0 111kp x k p x km k M[2.21]

For this to hold either00px M[2.22]

or11px M[2.23]

or both. Nowxsatisfying these inequalities cannot yield a higher

utility value than the maximized utility at the corresponding budget

constraint. Hence 2.22 implies*0,up m uand 2.23 impliesmunotes.in

## Page 33

33*1,up m uand since one or both of 2.22 and 2.23 must hold and10uuby assumption, we have establi shed*0,up muas

required.

Figures 2.5 illustrates. Fix the pri ce of good 2 as 1, so that the

intercept of the2xaxis shows total expenditure and the slope of the

budget constraint is -01.pBin the figure corresponds to the budget

constraint00 111 2px x mBcorresponds to1111 2px x mand yieldsa

lower utility value than0.BBcorresponds to11 2,px x Mwhere01 0 1

1 11, 1 .pk p k p m K M k M Balso yields a labour

value of utility than0Bweh a v e .10MM M[2.24]10111ppp[2.25]

ThatBpasses through the intersection point00 0 112,x x of B and Bfollows by nothing that if we sum.00 0 011 2kpx x K M [2.26]

X2

M0M1M0u02X1BB0BO01X1XFigure 2.5

and10 0 111 2 1kp x x M[2.27]munotes.in

## Page 34

34We obtai n

00

11 2px x m[2.28]

Thus11,pmand,pmare both in the worse set of00,pmand,pmis a convex combination of00,pmand11,.pm

2.3.1 The Slutsky Equation : -

The slutsky equation plays a central role in analyzing the

properties of demand functions. It is derived as follows. If we take

as the constraint in the utility maxi mization problem the lev el of

expenditure resulting from solution of the explanditure minimization

problem core equivalently take as the constraint in the latter

problem the level of utility resulting from the solution to the formers)

then the solutions*ixto the two problems, the values of the

Marshallian and Hicksian demand functions, will be identical setting,,mm p uwe can write for the 1stgoods,, ,iiHp u Dp m p u[2.29]

since 2.29 is an identity we can differentia ble through with

respect to the 1stprice, allowing expenditure to change in whatever

way is required to keep utility constant, to obtain

iiiii iHDDmPP M P[2.30]

Using shepherds lemma and rearranging gives the slutsky

equation.ii ij

jiDH DxPP M[2.31[

Taking I = j, so that we consider the effect of a price change

on its own demand, we see from 2.31 that the slope of the

marshalion demand function is the sum of two effect : the

substitution effect,/iiHP, which is the slope of the Hicksian or

compensated demand curve, and the income effect,/iixD m.

Thus the slutsky equation gives a precise statement of the

conclusions of the diagrammatic analysis of chapter 1. We show in

am o m e n tt h a t/0iiHP. Then 2.31 again with i = j establishes

that if the good is normal, so that/0iDm, the slope of its

marshallian demand curve is negative. If the good is inferior, so thatmunotes.in

## Page 35

35/iDMso, the slope is negative, positive or zero depending on

the relative sizes of the absolute values.

ii i iH P and x D M

It is useful to express the slutsky equation in elasticity form.

Again taking i = j, multiplying through 2.31 by/,iipxand the

income term by M / M, givesii ii i iSn [2.32]

whereiiis the marshallian demand elasticity,iiis the Hicksian or

compensated demand elasticity,inis the income elasticity of

demand, and/ii isp x mis the share of good in total expenditure.

Thus the difference between Hicksian and Marshallian elasticity for

a good will be smaller, the smaller its income elasticity and the less

significant it is in the consumer’s budget with,ij2.31 becomessinii iji [2.33]

which emphasizes that cross price marshallian demand elasticities

and on income elasticities weighted by expendi ture shares. Equality

of the Marshalliah cross -price elasticities therefore requires strong

restrictions on preferences.

We define the slutsky matrix as the n x n matrix/iHi pof

Hicksian demand derivatives. It is a straight forwar d extension of

shephard’s lemma and the properties of the expenditure function to

show that this matrix is a symmetric, negative semi definite matrix.

From shephard’s lemma,, 1.....,i

imp uHp u i npwe have2,, 1, ....i

ii imp u Hij npp P[2.34]

Then, from young’s Theorem we have immediately that// ,ii i iHPHpand so the slutsky matrix is symmetric. The

slutsky matrix/iiHPis the matrix of second order partials of the

expenditure function and the concauity of th e expenditure function

implies that matrix is negative semi -definite since

22

1// 0ii mp H pby the definition of negative semimunotes.in

## Page 36

36definiteness, the Hicksian demand curve cannot have a positive

slope. We have seen earlier that strict convexity of pr eference and0ixat the optimum establish the stronger result that

22// 0ii i mp H p.

The Hicksian demand derivative/iiHPis often used to

define complements and substitutes. Two goods i and j are called

Hicksian complement if/0iiHPand Hicksian substitutes if/0iiHP. The advantage of this definition is that symmetry

implies that the nature of the complementarily or substitutability

between the goods canno t change if we take/jiHPrather than/ijHP.T h i sw o u l dn o tb et r u ei fw ed e f i n e dc o m p l e m e n t sa n d

substitutes in terms of the Marshallian demand derivatives.

2.4PROPERTIES OF DEMAND FUNCTIONS

We have seen t hat it is possible to draw definite conclusions

about the effects of price changes on the Hicksian demands. The

Hicksian demand functions are not however, directly observable

since they depend on the consumers, utility level as well as prices

on the other hand, the marshallian demand functions can be

estimated from information on purchases, prices and money

income. The slutsky equation enables us to reformulate the

predictions about the properties and Hicksian demand functions in

terms of the observable Mar shalli an demand functions and thus to

rider the set of testable predictions from consumer theory.

We can summarize the testable implication derived in this and

the previous chapter:

(a) Marshallian demand functions are homeneous of degree zero in

price s and money income:

(b) The Marshallian demand functions satisfy the ‘adding up’

property :*;iiPx M(c) The Hicksian demand derivatives (cross substitution effects)

are symmetric :

//ii iiHPHPor using the slutsky e quation,// / / ;iij i ii i iDP x DM DP x DM(d) The slutsky matrix ///ij ij i iHP DP x DM is

negative semi definite.munotes.in

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37These are all the predictions about the Marshal liand e m a n d

functions which can be made on the basis of the consumer

preference axioms. The converse question of whether a system of

demand functions with these properties implies the existence of a

utility function from which the demand functions could have been

derived is known as the integrability problem. In next section we will

show that this is in fact so by considering the equivalent problem of

retrieving an expenditure function from a set of Marshallian demand

functions which satisfy the above properties.

2.5CHOICE UNDER UNCERTAINTY

Introduction : -

The analysis in the preceding chapters has assumed that all

decisions are taken in conditions of certainty. Any decision would

result in one and only one outcome. When a firm chooses a set of

input quantities, there is only one level of output which will result,

and it knows the profit which it will receive from the sale of each

output, no matter how far in the future production and sale will take

place. Like wise, in planning their purchases of goods and services,

and borrowing or lending decisions, households are assumed to

know with certainty the expenditure and utility associated with each

consumption vector.

But uncertainty is pervasive. There is technological

uncertainty, when the firm is not able to predict for sure the output

level which would result from a given set of input quantities

Machines may break down; crops may be affected by the weather.

There is market uncertainty when a single household or firm is not

able to predict for sure the prices at which it will buy or sell. Market

uncertainty is associated with disequilibr ium and change : If an

economy were permanently in long run static equilibrium, then firms

and house holds would expect to trade at equilibrium prices, which,

by experience, become known. If, however, changes are taking

place through time which change equi librium positions, the

individual agents in the markets cannot know the new equilibrium in

advance, and can only form expectations of prices which they know

may be wrong.

Extension of the theory to take account of uncertainty has two

main aims. It shou ld first tell us something about the usefulness

and validity of the concepts and propositions already derived. What

becomes of the conclusions about the working of a decentralized

price mechanism, for example? Can we still establish existence and

optimalit y of competitive equilibrium? Are the predictions about

household’s and firms responses to changes in parameters

affected qualitatively? The answers are important positively and

normatively. Second, many important aspects of economic activitymunotes.in

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38can not be ad equately analyses without explicit recognition of

uncertainty. For examples, the joint stock limited liability company,

the basic institutional form of the firm in capitalist economies, has

no real rationale in a world of certainty, and neither has the sto ck

market. Insurance furthers markets and speculation cannot be

understood expect in the context of uncertainty Relaxation of the

certainty assumption gives new insights into many other areas, for

example investment decisions.

As with models of an econo my with certainty, we begin with

the optimization problem of a single decision -taker. The

optimization problem under uncertainty has the some basic

structure as under certainty objects of choice; objective function,

and constraints defining a feasible set of choice objects. The main

interest centers on the first two of these, and, in particular, the

construction of a set of axioms which allows us to define a

preference ordering, represent able by a utility function, over the

objects of choice.

A formualiz ation of ‘uncertainty’

Uncertainty arises because the consequence of a decision is

not a single sure outcome but rather a number of possible

outcomes. our first task in developing a theory of choice under

uncertainty is to set out a precise formulizatio n of the decision

taking situation we can begin by distinguishing three kinds of

variables which play a part in an economic system these are:

(a) The choice variables of the decision taker which are directly

under his control such variables are not only e ndogenous to the

mode of the economic system, but are also endogenous to the

model of the individual economic agent. Example in earlier chapters

include firms output levels and consumers purchases.

(b) Variables whose values are determined by the operat ion of the

economic system, i.e. by the interaction of the choices of individual

economic agents, and which are regarded as parameters by them.

Prices are an example in a competitive economy. Such determined

variables are endogenous to the model of the eco nomic system,

but exogenous to the model of the individual economic agent.

(c) Environmental variables, whose values are determined by

some mechanism outside the economic system and which can be

regarded as parameters of the economic system. They influen ce its

outcome, but are not in turn. affected by it. The weather is an

example, at least for some problems, though, in the light of such

events as global warming, even this could be seen as endogenous

in some models.munotes.in

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39Suppose that the economy operates ove r only two periods,

period 1 (the present) and period 2 (the future). In period 1 the

environmental variables take on specific values which are known to

all economic agents. We assume that the economy produces a

resource allocation and a set of relative pr ices. If there were

complete independence between the decisions made in period 1

and those to be made in period 2, then the state of knowledge at

period 1 about the environmental variables at period 2.5 irrelevant.

In this case, decisions for period 2 can be left until period 2, and do

not affect decision taking at period 1. We assume that this kind of

temporal separability of decision -taking does not exist. At period 1,

economic agents will have to choose values of variables such as

investment (purchase of durable good) and financial assets (bonds

and shares), which effect what they will be able to do in period 2.

Agents plans for the values of variables they will choose at period 2

influenced by their expectation about the value of variable outside

their control at period 2 -determined variables such as prices, and

environmental variables like the wealthier will condition their

choices at period 1. We therefore need a theoretical framework to

analyse the formation of plans and expectation, and their influe nce

on current choices.

We proceed as follows. Suppose there exists a vector of

environmental variables12, ...nee ewhere each environmental

variable is capable of taking on a finite number of values in period

2. LetjEdenote the set of values which can be taken by

environmental variable1. 2, ....jei n.F o re x a m p l e1ecould be

the average temperature over period 2, measured to the nearest

degree centigrade, and1Ecould be the set00

11/ 50 80eC e C,

which has a finite number of elements (since the temperature is

measured in unites of01C) Define a state of the world as a specific

combination of the values of the envir onmental variables, i.e. as a

specific value of the vector12, ....nee esince each element of the

vector can take only a finite number of values the number 5 of

states of the world is also finite, though possibly very large. We

index the sta tes of the world by a number s=1,2 …5 and use the

index to label the value of the choice variable or determined

variables in each state of the world. Thus, for example, we can use

y, to denote the level of income the individual gets in state 5.

There fun damental properties of the set of states of the world

should be clear :

(a) The set is exhaustive, in that it contains all the states of the

world which could possibly obtain at period 2.munotes.in

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40(b) Members of the set are mutually exclusive in that the

occurren ce of any one rules out the occurrence of any other.

(c) The states of the world are outside the control of any decision -

taken, so that the occurrence of any one of them cannot be

influenced by the choice of any economic agent, or indeed by any

coalition of agents.

The definition properties of states of the world are basic to all

subsequent analysis. They can be regarded as an attempt to

eliminate the elements of doubts, apprehension, and muddle which

are part of the every day meaning of the word unce rtainty, and to

give the situation a precise formulization, for purposes of the

theory. Three further assumptions which can be made are:

(a) All decision takers have in their minds the same sets of states

of the world. They classify the possible combinati on of

environmental variables in the same way.

(b) When period 2 arrives, all decision acres will be able to

recognize which state of the world exists, and will all agree on it.

(c) At period 1, each decision taker is able to assign a probability

to the event that a particular state of the world will accurse at period

2. The probabilities may differ for different decision -takers, but all

probability assignments satisfy the basic probability laws. The

probability associated with the 5thstate by decisio nt a k e ri ,d e n o t e’s, lies on the interval110 ,swith'1simplying that I he

regards state 5 as certain not to occur. The probability of one or

another of several states occurring is t he sum of their probabilities

of their simultaneous occurrence being zero, and, in particular one

of the 5 states must occur, i.e.

1

51 51s

Each of these assumptions is quite strong and plays an

important part in what follows. The fi rst is necessary if we are to

portray decision -takers as making agreements in state contingent

terms : in order for one to agree with another that if state 1 occurs I

will do A,A in return for your doing B if state 2 occurs: it is

necessary that they sho uld understand each other’s references to

states.

The second assumption is also required for the formation and

discharge of agreements framed in state -contingent terms. If

parties to an agreement would differ about which state of the world

exists ex pos t, they are unlikely to agree ex ante on some

exchange which is contingent on states of the world. Themunotes.in

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41assumption also rules out problems which might arise from

differences in the information which different decision takers may

possess. Suppose, for exampl e, that individual I cannot tell whether

it is state 1 or state 2 which actually prevails at period 2, while

individual J does know. Then I is unlikely to conclude an agreement

with J under which say, I gains and I loses if state 1 occurs, while J

gains an d I loses if state 2 occurs, because of course. I could be

exploited by J.

Choice under uncertainty.

We now consider the question of optimal choice under

uncertainty. First, we need to define the objects of choice, and then

we can consider the question of the decision taker’s preference

ordering over these choice objects. We present what is usually

called the van -Neumann -Morgenstern theory of Expected utility.

Initially, we assume that there is a single good, which is

measured in units of account, an d which can be thought of a

‘income’. Let5 1, 2, .... 5sydenote an amount of income which

the decision -taker will have if and only if state s occurs C in this

section we shall be concerned only with a single decision -taker and

so do not need t o burden ourselves with a notation which

distinguishes among decision -takers) Assume that the invidual

assigns a probabilitysto state of the world 5, and denote the

vector of probabilities by12 3[ , ... ],while12, ... ,yy yyis

the corresponding vector of state -contingent incomes. Define a

prospect, P, as a given income vector with an associated

probability vector,15 1 5, ..... , ....Py y ychanging the probability vector, or the income vector y (or both)

produces a different prospect. Another term for a prospect would be

a probability distribution of incomes.

The choice objects of our theory are prospects such as P Any

decision has as its only and entire consequenc e some prospect P,

and so choice between alternative actions or decisions is

equivalent to choice between alternative prospects. A preference

ordering over decisions can only be derived from a preference

ordering over their associated prospects.

For exam ple, consider the decision of a market gardener to

insure or not against loss of income through sickness or poor

weather such as sever frost. Decision A is the decision not to

insure, decision B is to insure Associated with A is a prospect,munotes.in

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422AAPywhereAyis on income vector, the components of

which vary across states of the world. In the subset of states in

which he is sick, income will take on one value; in the subset of

states in which there is frost, incom et a k e so na n o t h e rv a l u e ;i nt h e

subset in which he is sick and there is frost, there will be a third

value; and when he is not sick and there is no frost, there will be a

fourth can presumably the highest) value Associated with B is a

certain prospect (as suming that compensation for loss of income

through sickness or frost is complete),Bbpy, where each

element ofByis equal to what income would be in the absence of

sickness and frost, minus the insurance pre mium, which must be

paid in all states of the world. The choice between A and B, i.e. the

decision whether or not to insure, depends on whetherAPis or is

not preferred toBP, To analyses choice under unce rtainty there

fore requires us to construct a theory of the preference ordering

over prospects.

If certain assumption (axioms) concerning a decision -taker’s

preferences are satisfied, then we are able to represent those

preferences the criterion by whi ch he takes his choices in a simple

and appealing way. A test of the appropriateness of the

assumptions would be to show that we can correctly predict

choices not yet observed, on the basis of observation of choice

already made. It should be emphasized tha t our theory is a device

for permitting such predictions, rather than for describing whatever

thought process a decision taker goes through when making

choices. The objects of choice consist of a set of prospects, which

we can denote by12, , ....npp p. The five axioms are described

next.

Axiom 1 : ordering of prospects

Given any two prospects, the decision taker prefers one to the

other, or is indifferent between them, and these relations of

preference and indifference are transitive. In th en o t a t i o no fc h a p e r

for any two prospects,,isppexactly one of the statements.,,,ik jk jkpp pp p pis true, while11ik k ip p and p p p pand similarly for the indifference relation.T h i s oxiom means that

the preference ordering over prospects has the same desirable

properties of completeness and consistency which were attributed

to the preferences ordering over bundles of goods.munotes.in

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43Before stating the second axiom, we need to introduce the

concept of a standard prospect. Given the set of prospects under

consideration we can take all the income values which appear in

them, regardless of the state and the prospect to which they

belong, as defining a set of values of the variable, income. Since

there is a finite number of states and prospects, there is a finite

number of such income values (at most, ns of them) There will be a

greatest and a smallest income value Denote these values byuIy and yrespectively. It follows that al li n c o m ev a l u e sl i eo nt h e

interval,[]Iuyyand we can construct the theory so as to apply to

this entire interval on the real line Define a standard prospect,pas a prospect involleing only the two out comes2uy and ywith

probabilities v and 1 -v respectively, where10v. A specific

standard prospect,10,pcan be written as1101,,upv y y(where for convenience, we do not bother t o write the second

probability11v) we obtain a second standard prospect,110pby

changingiv, the probability of getting the better outcome, to11vso

that11 1101,,upv y ywe can then state the second axiom.

2.6SUMMARY

The expenditure function is derived from the problem of

minimizing the total expenditure necessary for the consumer to

achieve a specified level of utility. Since utility dep ends indirectly on

the prices and money income hence it is refomed to as indirect

utility function.

Slutsky equation plays a central role in analyzing the

properties of demand function.

2.7 QUESTIONS FOR REVIEW

1. Examine the expenditure function.

2. Explain the concept of indirect utility function.

3. Elaborate the concept of Roy’s identity.

4. What are the properties of demand function.

munotes.in

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44Unit-3

TECHNOLOGY OF PRODUCTION AND

PRODUCTION FUNCTION

Unit Structure :

3.0 Objectives

3.1 Introduction

3.2 Technology of Production

3.2.1 Specification of Technology

3.2.1 Input Requirement Set

3.2.2 (i) Isoquant

3.2.2 (ii) Short -run Productio n Possibility Set.

3.3.2 (iii) Production Function

3.2.2 (iv) Transformation Function

3.2.3 Cobb -Douglas Technology

3.2.4 Leontief Technology

3.3 Activity Analysis

3.4 Monotonic Technology

3.5 Convex Technology

3.6 Regular Technology

3.7 The Technical Rate of Substitution

3.8 TRS for Cobb -Douglas Technology

3.9 The Elasticity of Substitution

3.10 Returns to Scale and Efficient Production

3.10.1 The Elasticity of Scale

3.10.2 Returns to Scale and Cobb -Douglas Technology

3.11 Homogeneous and Homothet ic Technology

3.11.1 The CES Production Function

3.12 Summary

3.13 Questions for Review

3.0 OBJECTIVES

After going through this module you will come to know the

concepts, like –

Technology of production,

Specification of technology,munotes.in

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45Input Requirement S et and production function,

Convex Technology

Leontief –Technology

Technical Rate of substitution (TRS)

Elasticity of Substitution

Returns to Scale (Long -Run Production Function)

Efficient Production

Homogeneous Production Function

Homothetic production F unction

The CES Production Function

3.1 INTRODUCTION

The simplest and the most common way to describe the

technology of a firm is the production function, which is generally

studied in intermediate courses. However, there are other ways to

describe firm technologies that are both more general and more

useful. We will discuss several of these ways to represent firm

production possibilities in this unit, along with ways to describe

economicaly relevant aspects of a firm’s technology.

3.2 TECHNOLOGY OF PROD UCTION

A firm produces outputs from various combinations of inputs.

In order to study firm choices we need a convenient way to

summarise the production possibilities of the firm, i.e., which

combinations of inputs and outputs are technologically feasible .

A certain amount of inputs are used to produce certain

amount of outputs per unit time period. We may also want to

distinguish inputs and outputs by the calendar time in which they

are available, the location in which they are available, and even the

circumstances under which they become available. By defining the

inputs and outputs with regard to when and where they are

available, we can capture certain aspects of the temporal or spatial

nature of production.

The level of detail that we will use in spe cifying inputs and

outputs will depend on the problem at hand, but we should remain

aware of the fact that a particular input or output good can be

specified in arbitrarily fine detail.

3.2.1 SPECIFICATION OF TECHNOLOGY:

Suppose the firm has ‘n’ possible goods to serve as inputs

and /or outputs. If a firm usesjjyunits of a good j as an input andmunotes.in

## Page 46

46producesojyof the good as an output, then the net output of good j

is given by

.I ft h en e to u t p u to fg o o dj is positive, then the

firm is producing more of good j than it uses as inputs; if the net

output is negative, then the firm is using more of good j than it

produces.

A production plan is simply a list of net outputs of various

goods. We can represent a p roduction plan by a vector y innRwherejyis negative if thethjgood serve as a net input and

positive if thethjgood serve as a net output. The set of all

technologically feasible production plans is called the firm’s

production -possibilities set and will be denoted by Y, a subset ofnR. The set Y gives us a complete description of the technological

possibilities facing the firm.

When we study the behaviour of a firm in certain economic

environments, we may want to distinguish between production

plans that are “immediately feasible” and those that are “eventually

feasible”. For example, in the short run, some inputs of the firms

are fixed so that only production plans compatible with these fixed

factors are possible. In the log run, such factors may be variable so

that the firm’s technological possibilities may well change.

We will generally assume that such restrictions can be

described by some vector z innR. For example, z could be a list of

maximum amount of the various inputs and outputs that can be

produced in the time period under consideration. The restricted or

short -run production possibilities set will be denoted by Y(z); this

consists of all feasible net output bundles consistent with the

constraint level z.

3.2.2 INPUT REQUIREMENT SET: -

Suppose we are considering a firm that produces only one

output. In this case we write the net output bundle a s( y ,-x) where x

is vector of inputs that can produce y units of output. We can then

define a special case of a restricted production possibilities set, i.e.,

the input requirement set, as -

The input requirement set is the set of all input bundles that

produce at least y nits of outputs.

Here the input requirement set measures inputs as positive

numbers rather than negative.munotes.in

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473.2.2 (i) ISOQUANT

The isoquant gives all input bundles that produce exactly y

units of output. In other words, an isoquant is the combination of all

inputs that produce same level of output i.e., y.

An isoquant can also be defined as:

3.2.2 (ii) SHORT -RUN PRODUCTION POSSIBILITY SET

Suppose a firm produces some output from labour and

capital. Production plans then look like,,ykwhere y is the

level of output,the amounts of labour input, and k the amount of

capital input. We know that the labour can be varied immediately in

the short run but the capital remains fixed at the leve lk.T h e nt h e

short -run production possibility set can be expressed as –

3.2.2 (iii) PRODUCTION FUNCTION

The production function for a firm which has only one output

can be defined as –:fx y i n Ryis the maximu mo u t put associated with x in

y}

3.2.2 (iv) TRANSFORMATION FUNCTION

A production plan y in Y is technologically efficient if there is

noy' in Y such thatyyandyy; in other words, a production

plan is ef ficient if there is no other way to produce more output with

the same inputs or to produce the same output with less inputs.

The set of technologically efficient production plans can be

described by a transformation function:

:nTR R

Where T(y)=0 if and only if y is efficient. The transformation

function gives the maximal vectors of net outputs.

3.2.3 COBB -DOUGLAS TECHNOLOGY

Let ‘a’ be a parameter such that 0 < a < 1. Then the Cobb -

Douglas technology can be defined as –

1. Product ion possibility set -

2. Input requirement set –

3. Isoquantmunotes.in

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48

4. Short -run production possibility set –

5. Transformation function –

6. Production function –

3.2.4 LEONTIEF TECHNOLOGY

Let a >oa n db >o be parameters. Then the Leontief

Technology can be defined as –

The general shape of Cobb -Douglas and Leontief

technology can be depicted diagrammatically as in the figures (a)

and (b) respectively.

Figure 3.1munotes.in

## Page 49

493.3 ACTIVITY ANALYSIS

The most straightforward way of describing produ ction sets

or input requirement sets is simply to list the feasible production

plans. For example, suppose that we can produce an output good

using factor inputs 1 and 2. There are two different activities or

technologies by which this production can take place.

Technique A: One unit of factor 1 and two units of factor 2

produces one unit of output.

Technique B: Two units of factor 1 and one unit of factor 2

produces one unit of output.

Let the output be good 1; and factors be goods 2 and 3.

Then we canrepresent the production possibilities implied by these

twoactivities by the production set –

This input requirement set is depicted in the figure 3.2(A).

Figure 3.2(A)

It may be the case that to produce y units of output we could

just use y times as much of each input for y=1,2, …. In this case

one might think that the set of feasible way to produce y units of

output would be given by

However, this set does not include all the relevant

possibilities. It is true that (y, 2y) will produce y units of output if we

use technique A and that (2y, y) will produce y units of output if we

use technique B -But what if we use a mixture of technique A & B.munotes.in

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50In this case we have to letAybe the amount of output

produced using techniq ue A andABybe the amount produced

using technique B. The V(y) will be given by the set –

So, for example, V(2) ={(2,4),(4,2),(3,3) }.B o t hV ( y )&V ( 2 )

aredepicted in the above figures.

3.4 MONOTONIC TECHNOLOGY

Suppose tha t we had an input vector (3, 2). Is this sufficient

to produce one unit of output? We may argue that since we could

dispose of 2 units of factor 1 and be left with (1,2), it would indeed

be possible to produce 1 unit of output from the inputs (3,2). Thus,

if such free disposal is allowed, it is reasonable to argue that if x is

feasible way to produce y units of output andxis an input vector

with at least as much of each input, thenxshould be a feasible

way to produce y. Thus, the input requirement set should be

monotonic in the following sense.

Monotonicity : x is in V(y) andxis in V(y).

If we assume monotonicity, then the input requirement sets

depicted in figure 4.2 becom e the sets depicted in figure 3.3.

Figure 3.3

This assumption of monotonicity is often an appropriate

assumption for production sets as well. In this context we generally

want to assume that if y is in Y andyythenymust also be in

Y. That is to say that, if y in Y is feasible thenyin Y is also

feasible.munotes.in

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513.5 CONVEX TECHNOLOGY

Let us now consider what the input requirement set looks

like if we want to produce 100 units o f output. As a first step we

might argue that if we multiply the vectors (1,2) and (2,1) by 100,

we should be able just to replicate what we were doing before and

thereby produce 100 times as much. It is clear that not all

production processes will necessa rily allow for this kind of

replication, but it seems to be plausible in many circumstances.

If such replication is possible, then we can conclude that

(100, 200) and (200, 100) are in V(100). Are there any other

possible ways to produce 100 units of outp ut? Well we could

operate 50 processes of technique I and 50 process of activity II.

This would use 150 units of good 1 and 150 units of good 2 to

produce 100 units of output; hence (150 ,150) should be in the

input requirement set. Similarly, we could ope rate 25 process of

activity I and 75 processes of activity II. This implies that

25(100,200) 75(200,100) (175,125) should be in V(100).

More generally, t(100,200) (1 t)(200,100) (100t 200 (1 t) ,2 0 0 t( 1

t)100) Should be V(100) for t = 0, .01, .02 ….

We m ight as well make the obvious approximation here and

let t take on any fractional value between 0 and 1. This leads to a

production set of the form depicted in figure 2.4 A. Thus,

Convexity: If x andxare in V(y), then1tx t xis in V(y), for all01t.That is, V(y) is a Convex set.

We applied the arguments given above to the input

requirement sets, but similar arguments apply to the production

sets. It is common to assume that if y andyare both in Y, then1ty t yis also in Y for01tin other words Y is a convex

set.

Now we will describe a few of the relationships between the

convexity of V(y) and the convexity of Y.

Convex production set implies convex input requirement set.

i.e., if the production set Y is a convex set, then the associated

input requirement set, V(y), is a convex set.

Convex input requirement set is equivalent to quasiconcave

production function . V(y) is a convex set if and only if the production

function f(x) is a quasiconcave function.munotes.in

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523.6 REGULAR TECHNOLOGY

Finally, we will consider a weak regularity condition

concerning V(y)

V(y) is a closed, nonempty set for allyoThe assumption that V(y) is nonempty requires that there is

some conceivable way to produce any given level of output. This is

simply to avoid qualifying statements by phrases like “assuming

that y can be produced” .

The assumption that V(y) is close d is made for technical

reasons and is innocuous in most contexts. Roughly speaking, the

input requirement set must include its own boundary.

3.7 THE TECHNICAL RATE OF SUBSTITUTION

Assume that we have some technology summarized by a smooth

production fun ction and that we are producing at a particular point** *12,yf x x .Suppose that we want to increase the amount of

input 1 and decrease the amount of input 2 so as to maintain a

constant level of output. How can we determine this technical r ate

of substitution between these two factors?

In the two dimensional case, the technical rate of substitution

is just the slope of an isoquant; how one has to adjust2xto keep

output constant when1xchan ges by a small amount, as depicted in

figure 3.4

Figure 3.4

In the ‘n’ –dimensional case, the technical rate of substitution

is the slope of an isoquant surface, measured in a particular

direction.munotes.in

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53Let21xxbe the (implicit) func tion that tells us how much of2xit takes to produce y if we are taking1xunits of the other input.

Then by definition, the function21xxhas to satisfy the following

identity -21 1,fxxx y.

Actually, we require an expression for -*21 1/xx xThen, differentiating the above identity, we get –

This gives us an explicit expression for the technical rate of

substitution.

Here is the another way to derive the technical rate of substitution.

Think of a vector of small changes in the input levels which we write

as12,dx dx dx The associated changes in the output is

approximated by

this expression is known as the

total differential of the function f(x). Consider a particular change in

which only factor 1 and factor 2 changes, and the change is such

that output remains constant. That is1dxand2dxadjust “along an

isoquant”.

Since output remains constant, we have

which can be solved for -

Either the implicit function method or the total differential

method may be used to calculate the technical rate of substitution.

3.8 TRS FOR A COBB -DOUGLAS TECHNOLOGY

Given that112 1 2,aafxx x xwe can take the derivatives to

find-munotes.in

## Page 54

54

It follows that,

3.9 THE ELASTICITY OF SUBSTITUTION

The technical rate of substitution measures the slope of an

isoquant. The elasticity of substitution measures the curvature of an

isoquant. More specifically, the elasticity of substitution measures

the percentage change in the factor ratio divided by the percentage

change in the TRS, with output being held fixed.

If we let21/xx be the change in the factor ratio andTRSbe the change in the technical rate of substitution, then the

elasticity of substitution denoted by''can be given as –

The elasticity of substitution, which is a relatively natural

measure of curvature, ask s how the ratio of factor inputs changes

as the slope of the isoquant changes. If a small change in slope

gives us large change in factor input ratio, then the isoquant is

relatively flat which means that the elasticity of substitution is large.

In practi ce we think of the percentage change as being very

small and take the limit of this expression asgoes to zero. Then,

the expression forbecomes –

It is often convenient to calculateusing the logarithmic

derivative. In general, if y=g(x), the elasticity of y with respect to x

refers to the percentage change in y induced by a small percentage

change in x.munotes.in

## Page 55

55That is,dydy xy

dxdx yx

Provided that x and y are po sitive, this derivative can be written asdInydInx

To prove this, note that by the chain ruledIny d In x dInydInx dx dx

Carrying out the calculations on the left -hand and right -hand

side of the equals sign, we have –

Alternativel y we can use total differential to write –

So that,

Applying this to the elasticity of substitution, we can write –

Here, it should be noted that the absolute value sign in the

denominator is to convert the TRS to a positive number so that the

logarithm makes sense.

The Elasticity of Substitution for the Cobb -Douglas

Production Function:

We have seen above that –munotes.in

## Page 56

56

It follows that,

This in turn implies –

Hence, it is clear from the above expression that the

elasticity of substitutio nf o rt h eC o b b -Douglas production function is

equal to one.

3.10 RETURNS TO SCALE AND EFFICIENT

PRODUCTION

Suppose that we are using some vector of inputs x to

produce some output y and we decide to scale all inputs up or

down by some amount0t.What will happen to the level of

output?

In the case we described earlier, where we wanted only to

scale output up by some amount, we typically assumed that we

could simply replicate what we were doing before and thereby

produce ‘t’ times as much output as before. If this sort of scaling is

always possible, we will say that the technology exhibits constant

returns to scale. More formally, a technology is said to exhibit

constant returns to scale if any of the following are satisfied.

(1)yi nYi m p l i e st yi si nY ,f o ra l l0t;

(2) x in V(y) implies tx is in V(ty), for all0t;

(3) f(tx) = tf(x) for all0t; i.e., the production function f(x) is

homogeneous of degree 1.

The replication argument given above indicates that constant

returns to scale is often a reasonable assumption to make about

technologies. However, there are situations where it is not a

plausible assumption.

One circumstance where constant returns to s cale may be

violated is when we try to “subdivide” a production process. Even ifmunotes.in

## Page 57

57it is always possible to scale operations up by integer amounts, it

may not be possible to scale operations down in the same way.

Another circumstance where the constant retu rns to scale

may be violated is when we want to scale operations up by

noninteger amounts. Certainly, replicating, what we did before is

simply enough, but how do we do one and one half times what we

were doing before.

A third circumstance where constant returns to scale is

inappropriate is when doubling all inputs allows for a more efficient

means of production to be used. Replication says that doubling our

output by doubling our inputs is feasible, but there might be a better

way to produce output. Consi der, for example, a firm that builds an

oil pipeline between two points and uses labour, machines and

steel as inputs to construct the pipeline. He may take the relevant

measure of output for this firm to be the capacity of resulting

pipeline. Then it is c lear that if we double all inputs to the

production process, the output may more than double since

increasing the surface area of a pipe by 2 will increase the volume

by a factor of 4. in this case when output increases by more than

the scale of the input s, we say the technology exhibits increasing

returns to scale.

A technology exhibits increasing returns to scale if,

A fourth situation where constant returns to scale may be

violated is by being unable to replicate some inputs.

Consider for example, a 100 acre farm. If we wanted to

produce twice as much output, then we could use twice as much of

each input. But this would imply using twice as much land as well. It

may be that this is impossible to do since more land may not be

available. Even though the technology exhibits constant returns to

scale if we increase all inputs, it may be convenient to think of it as

exhibiting decreasing returns to scale with respect to the inputs

under our control.

More precisely, we have a technology that can be said to

exhibit decreasing returns to scale if,

The most natural case of decreasing returns to scale is the

case where we are unable to replicate some inputs. Thus, we

should expect that the restricted production possibility sets would

typically exhibit dec reasing returns to scale. It turns out that it can

always be assumed that decreasing returns to scale are due to the

presence of some fixed factor input.munotes.in

## Page 58

58Finally, it should be noted that the various kinds of returns to

scale explained above are global in n ature. It may well happen that

a technology exhibits increasing returns to scale for some values of

x and decreasing returns to scale for other values.

3.10.1 THE ELASTICITY OF SCALE

The elasticity of scale measures the percent increase in

output due to a one percent increase in all inputs –that is, due to

an increase in the scale of operations.

Let y =f(x), be the production function. Let t be a positive

scalar, and consider the function y(t)= f (tx). If t =1, we have the

current scale of operations; if t >1, we are scaling all inputs up by t;

and if t <1, we are scaling all inputs down by t.

The elasticity of scale is then given by –

evaluated at t=1

Rearranging this expression, we have -

from the above expression, we may say that the techno logy

exhibits –locally;

(1) Increasing returns to scale, if e(x) >1;

(2) Constant returns to scale, if e(x) =1; and

(3) Decreasing returns to scale, if e(x)<1.

3.10.2 RETURNS TO SCALE AND COBB -DOUGLAS

TECHNOLOGY

Suppose that12abyx x

Then,

Hence,12 1 2,,ft xt x t fxx if and only if a +b=1. It, therefore,

implies that the,

(1) Technology exhibits constant returns to scale, if a+b =1;

(2) Increasing returns to scale, if a+b >1; andmunotes.in

## Page 59

59(3) Decreasing returns to scale if a+b < 1 .

In fact, the elasticity of scale for the Cobb -Douglas technology turns

out to be precisely a+b. To see this consider the definition of

elasticity of substitution –

Evaluating this derivative at t=1 and dividing by

3.11 HOMOGENEOUS AND HOMOTHETIC

TECHNOLOGY

A function f(x) is homogeneous of degree k ifkft x tfx

for allto. The two most important “degrees” in economics are the

zeroth and first degree. A zero degree homogeneous function is

one for which f (tx) = f(x), and first degree homogeneous function is

one for which f(tx)= t f(x).

Comparing this definition to the definition of constant returns

to scale we see that a technology has constant returns to scale if

and only if its production function is ho mogeneous of degree one.

Af u n c t i o n:gR Ris said to be a positive monotonic

transformation if g is strictly increasing function; that is, a function

for which x > y implies that g(x) > g(y).

Ah o m o t h e t i cf u n c t i o ni sam o n o t o n i ct r a n sformation of a

function that is homogeneous of degree one. In other words, f(x) is

homothetic if and only if it can be written as f(x)= g(h(x)), wherehis monotonic function. Both, homogeneous and homothetic

functions are depict ed in the figure 3.5.munotes.in

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60

Figure 3.5

Panel A of the figure 3.5 depicts the function that is

homogeneous of degree one. That is, if x and1xcan both produce y

units of output, then 2x and 21xcan both produc e2 yu n i t so f

output.

Panel B of the figure 3.5 depicts a homothetic function. That

is, if x and1xproduce the same level of output, y, then 2x and 21xcan produce the same level of output, but not necess arily 2y.

Homogeneous and homothetic functions are of interest due

to the simple ways that their isoquants vary as the level of outputs

varies. In the case of a homogeneous function the isoquants are all

just “blown up” versions of a single isoquant. If f (x) is homogeneous

of degree one, then if x and1xproduce y units of output, it follows

that tx and t1xcan produce ty units of output, as depicted in figure

3.5A.

A homothetic function has almost the sam ep r o p e r t y :i fxa n d1xproduce the same level of output, then tx and t1xcan also

produce the same level of output –but it wont necessarily be t

times as much as the original output. The isoquants for a

homothetic technology look just like the isoquants for

homogeneous technology, only the output levels associated with

the isoquants are different.

Homogeneous and homothetic technologies are of interest

since they put specific restrictions on how the techn ical rate of

substitution changes as the scale of production changes. In

particular, for either of these functions the technical rate of

substitution is independent of the scale of production.

3.11.1 THE CES PRODUCTION FUNCTION

The constant elasticity of substitution or CES production

function has the following form;

munotes.in

## Page 61

61It is quite easy to verify that CES function exhibits constant

returns to scale. The CES function contains several other well -

known production functions as special cases, depending on the

value of the parameter

. These are illustrated in figure 3.6.

In figure 3.6 above, panel A depicts the case where

,

panel B the case where

and the panel C the case where

.

The production function contained in the CES function can

be described as –

1) The linear production function (

).

Simple substitution yields –12yx x2) The Cobb -Douglas production function (

). When

the

CES production function is not defined, due to division by zero.

However, we will show that as

approaches zero, the isoquants of

the CES production function looks very much like the isoquants of

the Cobb -Douglas production function.

This is easiest to see using the technical rate of substitution.

Bydirect calculation –

A

approaches zero, this tends to a limit of21xTRSxWhich is simply the TRS for the Cobb -Douglas production function.

3) The Leontief production function

. We have just seen that

the TRS of CES production function is given by equation (1) above,

A P approaches -, this expression approaches -

If21xxthe TRS is negative infinity; if21xxthe TRS is zero. This

means that Q as

approaches -, a CES isoquant looks l ikean

isoquant associated with the Leontief technology.munotes.in

## Page 62

62The CES production function has a constant elasticity of

substitution. In order to verify this, remember that the technical rate

of substitution is given by –

So that,

Taking logs we see tha t,

3.12 SUMMARY

In short production is the creation of utility by transforming

physical units of inputs into physical units of output. Production

function is the technology of combining physical units of inputs to

produce the given level of output.

3.13 QUESTIONS

Q.1 Explain the concept of technology of production.

Q.2 Elaborate the concept of input requirement set.

Q.3 Define and explain the concepts of Cobb -Douglas and Leontief

Technology.

Q.4 Discuss the concept of monotonic, convex and Regular

technology.

Q.5 What is technical rate of substitution? Explain

Q.6 Explain returns to scale and the concept of efficient production.

Q.7 Explain the concept of CES production function.

munotes.in

## Page 63

63Unit-4

COST FUNCTION

Unit Structure :

4.0 Objectives

4.1 Introduction

4.2 Cost Function

4.2.1 Average and Marginal Costs

4.2.2 The Short -run Cobb -Douglas Cost Function

4.2.3 The Geometry of Costs

4.2.4 Long -Run and Short -Run Cost Curve

4.3 Factor Prices and Cost Functions

4.4 Shephard’s Lemma

4.5 The Env elope Theorem

4.6 Duality

4.7 Sufficient Conditions for Cost Functions.

4.8 Summary

4.9 Questions for Review

4.0 OBJECTIVES

After going through this unit you will able to explain the

concepts, like –

Cost Function,

Average and marginal costs,

Long -run an dS h o r t -run costs,

Properties of the cost function,

Shephard’s Lemma,

The Envelope Theorem for Constrained Optimisation,

Duality of cost and Production function,

Geometry of Duality

4.1 INTRODUCTION : -

People without a background in economics usually ma ke a

mistake between cost and price. Price is the amount paid by the

consumer and received by the producer. Cost is the amount spent

by the producer in manufacturing the commodity or the service.munotes.in

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64Cost can be understood in a variety of ways. The

opportu nity cost is the returns from the next best alternative. There

are implicit costs which may not be seen in the accounts

statements and explicit costs which could be clearly understood.

An important division of costs is between Fixed and Variable

Costs. Fixed costs are those which do not depend on the quantity

of output produced, they include costs like rent, payment of loan

installments, permits, etc. Variable costs depend upon the quantity

of output produced and increase with output (for total variabl e

costs).

Another concept of classifying costs is total, average and

marginal costs. Total cost is divided into total fixed and total

variable costs. The total cost refers to the cost incurred in

producing the given quantity of output. The usual total cost function

is of a cubic form. Average cost is the per unit cost of producing

the commodity which can be obtained by dividing the total cost with

quantity of output. Marginal cost is the rate of change in total cost

with respect to output and so ther ec a nn o tb ea n ym a r g i n a lf i x e d

cost by definition.

4.2 COST FUNCTION

The cost function measures the minimum cost of producing

a given level of output for some fixed factor prices. As such it

summarizes information about the technological choices availa ble

to the firms. The behaviour of the cost function can tell us a lot

about the nature of the firm’s technology.

Just as the production function was our primary means of

describing the technological possibilities of production, the cost

function will b e our primary means of describing the economic

possibilities of a firm. Here we will investigate the behaviour of the

cost function(,)cwywith respect to its price and quantity

arguments.

4.2.1 AVERAGE AND MARGINAL COST

Let us con sider the structure of the cost function. In general,

the function can always be expressed simply as the value of the

conditional factor demands.(,) (,)cwy w xwyThis just says that the minimum cost of producingyunits of

output is the cost of the cheapest way to producey.munotes.in

## Page 65

65In the short run some of the factors of production are fixed at

predetermined levels. Letfxbe the vector of fixed factors,vxbe

the vector of variable factors, and break up''winto(), ww wvf,

the vectors of prices of the variable and fixed factors. The short -run

conditional factor demand functions will generally depend onfx,so

we write them as(,, )vfxw y x.T h e n t h e short -run cost function can

be written as –

(,, (,, ))cwyx wx wyx w xvvff f fThe term(,, )vv fwx wyxis called short -run variable cost (SVC), and

the termffwxis the fixed cost (FC).

From these basic units, we can define various derived cost

concepts, as follows –

Short run total cost ( STC )

(,, )vv f f fSTC w x w y x w x

Short run average cost (SAC)(,, )fcwyxSACy

Short run average variable cost (SAVC)(,, )vv fwx wyxSAVCy

Short run average fixed cost (SAFC)ffwxSAFCy

Short run marginal cost (SMC)(,, )fcwyxSMCy

When all factors are variable, the firm will optimize in the choice offx. Hence, the long -run cost function only depends on the factor

prices and level of output as indicated earlier.

We can express this long -run function in terms of the short -

run cost function in the following way. Let(,)fxw ybe the optimal

choice of the fixed factors, and let(,) (,, (,) )vv fxw y xw y x w y be themunotes.in

## Page 66

66long-run optimal choice of the variable factors. Then the long -run

cost function can be written as –(,) (,) (,) (,, (,) )vv f f fcwy wx wy w x wy cwyx wy

The long -run co st function can be used to define cost concepts

similar to those defined above:

Long run average cost(,)cwyLACy

Long run marginal cost(,)cwyLMCy

It should be noted here, that the “long -run average cost” equals

“long -run average variable cost” since all costs are variable in the

long-run; and the “long -run fixed costs” are zero.

4.2.2 THE SHORT -RUN COBB -DOUGLAS COST

FUNCTION :

Suppose the second factor in a Cobb -Douglas technology is

restricted to operate at a level ‘k’. Then the cost minimizing

problem is –

min11 2wx wk

Such that11aayx k

Solving the constraint for1xas a function of y and k gives,

111aaxy k

Thus,

12 1 21

1(, ),, a acw w yk w y k w kThe following variations can also be calculated –

Short -run average cost121a

awkywky munotes.in

## Page 67

67Short -run average variable cost11aaywkShort -run average fixed cost2wky

Short -run margina l cost11aawy

ak4.2.3 THE GEOMETRY OF COSTS

The cost function is the single most useful tool for studying

the economic behaviour of the firm. In a sense, the cost function

summarizes all economically relevant information about the

technology of the firm.

Since, we have taken factor prices to be fixed, costs depend

only on the level of output of a firm. The total cost curve is always

assumed to be monotonic in output : the more you produce the

more it costs. The average cost curve , however, can increase or

decrease with output, depending on whether total cost rise more

than or less then linearly. It is often thought that the most realistic

case, at least in the short -run, is the case where the average cost

curve first decreases an d then increases. The reason for this is as

follows –

In the short -run the cost function has two components : fixed

costs and variable costs. We can therefore write short -run cost as –(,, ) (,, )ff f v v fcwyx w x wx wyxSACyy y SAFC SAVCIn most applications, the short -run fixed factors will be such

things as machines buildings, and other types of capital equipments

while the variable factors will be labour and raw material. Let us

consider how the costs attributable to these factors wil l change as

output changes.

As we increase output, average variable costs may initially

decrease, if there is some initial region of economies of scale.

However, it seems reasonable to suppose that the variable factors

required will increase more or les s linearly until we approach some

capacity level of output determined by the amounts of the fixedmunotes.in

## Page 68

68factors. When we are near to capacity, we need to use more than

a proportional amount of the variable inputs to increase output.

Thus, the average variable cost function should eventually increase

as output increases, as depicted in figure 4.1A. Average fixed

costs must of course decrease with output, as indicated in figure

4.1B. Adding together the average variable cost curve and the

average fixed cost cur ve gives us the U shaped average cost curve

as is depicted in figure 4.1C.ACAFC

AVC

outputAVCACAFC

AVC

outputAFCACAFC

AVC

outputAC

Fig:4.1AF i g : 4.1BF i g : 4.1C

The initial decrease in the average cost is due to the

decrease in average fixed costs; the eventual increase in the

average cost is due to the increase in average variable costs. The

level of output at which the average cost of production is minimized

is sometimes known as the minimal efficient scale.

In the long -run all costs are variable co sts; in such

circumstances increasing average costs seems unreasonable since

a firm could always replicate its production process. Hence, the

reasonable, long -run possibilities should be either constant or

decreasing average costs. On the other hand, cer tain kinds of

firms may not exhibit a long -run constant -returns -to-scale

technology because of long -run fixed factors. If some factors do

remain fixed even in the long -run, the appropriate long -run average

cost curve should presumably be U -shaped.

Let u s now consider the marginal cost curve. What is its

relationship with the average cost curve? Letydenote the point

of minimum average cost; then to the left ofyaverage costs are

declining so that f oryy()0dc y

dy yTaking the derivatives, it gives,

2'( ) ( )0yc y c y

yforyymunotes.in

## Page 69

69This inequality says that marginal cost is less than average cost to

the left of the minimum ave rage cost point. A similar analysis

shows that,

'()()cycyyforyySince both inequalities must hold aty, we have

'(* )(* ) ;*cycyy

That is marginal cost equal average cost at the point of minimum

average cost.

The Cobb -Douglas Cost Curves

The generalized Cobb -Douglas technology has a cost function of

the firm,1()abcy K y1abWhere, k is a function of factor prices and parameters. Thus,

1()()ababcyAC y Kyy

1

'() ()ababKMC y c y yabIf1,abthe cost curves exhibit increasing average costs; if1,abthe cost curves exhibits constant average costs.

4.2.4 LONG -RUN AND SHORT -RUN COST CURVES

Let us now consider the relationship between long -run cost

curves and the short -run cost curves. It is clear that the long -run

cost curves should never lie above any short -run cost curves, since

the short -run cost minimization problem is just a constrained

version of the long -run cost minimization problem.

Let us write the long -run cost function as() (,() )cy cyzy.

Here we have omitted the factor prices since they are assumed

fixed and we let()zybe th e cost minimizing demand for a single

fixed factor. Let*ybe some given level of output, and let*( )zz ybe the associated long run demand for the fixed factor. The short

run cost,(,* )cyz, must be at least as great as the long run cost,munotes.in

## Page 70

70(,() )cyzy, for all levels of output, and the short -run cost will equal

the long -run cost at output*yso(* ,* ) (* ,(* ) )cy z cy zy. Hence,

the long -run and the short -run cost curves must be tangent at*y.

This is just the geometric restatement of the envelope theorem.

The slope of the long -run cost curve at*yis–(* ,(* ) ) (* ,* ) (* ,* ) (* )dc y z y c y z c y z z ydy y z y

But since*zis the optimal choice of the fixed factors at he output

level*y, we must have –

(* ,* )0cy zz

Thus, long -run marginal costs at*yequal short -run marginal costs

at(* ,* )yz.

Finally, we note that if the long -run and short run cost curves are

tangent then the long -run and short -run average cost curves must

also be tangent. A typical configuration is illustrated in figure 4.2

Figure 4.2

4.3 FACTOR PRICES AND COST FUNCTIONS

We turn now to the study of the price behaviour of cost

functions. Several interesting properties follow directly from the

definition of the functions. These properties of the cost functions

aresummarized as below –

Properties of the Cost Functions –

1) Non -decreasing inw:

If',wwthen1(, ) ( , )cw y cwy2) Homogeneous of degree 1 inw:munotes.in

## Page 71

71(, ) ( , )ct wy t cwyfor0t3) Concave inw:

''(( 1 ) , ( , ) ( 1 ) ( , )ct w tw y t cwy tcw yfor01t4) Continuous inw:(,)cwyis continuous as a function of w, for0wProof :

1) Cost function is non -decreasing inw:

Letxand'xbe cost minimizing bundles associated withwand1w.T h e n'wx wxby minimization and'' 'wx w x. Since,'ww.

Putting these inequalities together gives''wx w xas required.

2) Cost function is homogeneous of degree 1 inw:

We show that ifxis the cost minimizing bundle at pricew,t h e nxalso minimizes costs at pricestw.S u p p ose this is not so, and let'xbe a cost minimizing bundle attwso that 'twx twx.B u t t h i s

inequality implies 'wx wx, which contradicts the definition ofx.

Hence, multiplying factor prices by a positive scalartdoes not

change the composition of a cost minimizing bundle, and thus,

costs must rise by exactly a factor oft:(, ) ( , )ct wy t w x t cwy3) Let(,)wxand(' ,' )wxbe two cost -minimizing price factor

combinations and let"( 1 ) 'wt w t wfor any01 .tNow,(" ,) " " "( 1 )'"cw y w x t w x twxSince"xis not necessarily the cheapest way to produceyat price'worw,w eh a v e"( , )wx c w yand'. '' ( ', ).wx c wyThus,(" ,) (,) ( 1 )(' ,)cw y t cwy t c w y4.4 SHEPHARD’S LEMMA

Let(,)ixw ybe the firms conditional factor demand for input''i. Then if the cost function is differentiable at(,)wy,a n d0,iw for1,............,inthen(,)(,)iicwyxw yw1,........,inmunotes.in

## Page 72

72Proof :

Let*xbe a cost –minimizing bundle that produceyat prices*w.

Then define the function,() (,) *gw cwy w xSince,(,)cwyis the cheapest way to producey, this function is

always non -positive, at*,ww(* ) 0 .gwSince, this is the ma ximum value of() ,gwits derivative must vanish:

(* ) (* ,)0i

iigw cw yxww 1,.........,inHence, the cost minimizing input vector is just given by the vector

of derivatives of the cost function with respect to th e prices.

4.5 THE ENVELOPE THEOREM

Shephard’s Lemma is another example of the envelope

theorem. However, in this case we must apply a version of the

envelope theorem that is appropriate for constrained optimization

problems.

Consider a general parame terized constrained maximization

problem of the form –1, 2,,12() m a x ( )xxMa gxxa

Such that1, 2,() 0hxx aIn the case of the cost function –

1, 2, 1 1 2 2 1, 2,() , ()gxx a w x wx hxx a 1, 2)(,fxx y and''acould be one of the prices.

The Langrangian of this problem is1, 2, 1, 2,() () gxx a hxx aLmunotes.in

## Page 73

73and the first order conditions are –

110gh

xx

220gh

xx------------------------------------ (1)

1, 2,() 0hxx aThese conditions determine the optimal choice functions

12(( ) , ( ) ) ,xax awhich in turn determine the maximum value function

12() ( () , () , )Ma gx a x a a ------------------------ (2)

The envelope theorem gives us the formula for deriv ative of the

value function with respect to a parameter in the maximization

problem. Specifically, the formula is –

() (,)()dM a x a

da axx a1, 2, 1, 2,() ()() () gxx a hxx a

aaxx a xx aii ii

These partial derivatives are the derivatives ofgandhwith

respect to a holding1xand2xfixed at their optimal values.

Application of the Envelope Theorem to the Cost Minimization

Problem :

In this problem the parameter''acan be chosen to be one of the

factor prices,iw. The optimal value function()Mais the cost

function(,)cwy.

The envelope theorem asserts that –

(,)(,)(,) ii

iicwyxx w yxx w yii ww

Envelope Theorem: Marginal Cost Revised:

It is another application of the envelope theorem, consider

the derivative of the cost function with respect toy. According to

the envelope theorem, this is given by t he derivative of theLLmunotes.in

## Page 74

74Langrangian with respect toy. The Lagrangian for the cost

minimization problem is

11 2 2 1 , 2 )[( ]wx w x f x x y

Hence,

1, 2,()cww y

y

In other words, the Lagrange multiplier in the cost minimization

problem is simply marginal cost.

4.6 DUALITY

Suppose, set()VO yis an “outer bound” to the true input

requirement set()Vy.G i ven data(,,)tttwxy()VO yis defined to

be

() {:tt tVO y x w x w xfor all t such that}tyyIt is straightforward to verify that()VO yis a closed,

monotonic and conve x technology. Furthermore, it contains any

technology that could have generated the data(,,)tttwxyfor

t = 1,………,T

If we observe choices for many different factor prices, it

seems that()VO yshould “appr oach” the true input requirement set

in some sense. To make this precise, let the factor prices vary over

all possible price vectors0.wThen the natural generation ofVObecomes –*( ) { : ( , ) ( , )Vy x w x w x w yc w y for all0}wRelationship between*( )Vywill contain()Vyand the true

input requirement set() :VyOf course*( )Vywill contain() .VyIn general,*( )Vywill

strictly contain()Vy.F o r e x a m p l e , i n f i g u r e 4.3A we see that the

shaded area can not be ruled out of*( )Vysince the points in this

area satisfy the condition that(,) .wx c w yThe same is true for figure 4.3B.Lmunotes.in

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75

Fig : 4.3AF i g : 4.3B

The cost function can only contain information about the

economically relevant sections of()Vy, namely, those factor

bundles that could actually be the solution to a cost minimization

problem, i.e. that could actually be conditional factor demands.

However, suppose that our original tec hnology is convex and

monotonic. In this case*( )Vywill equal()Vy. This is because, in

the convex monotonic case, each point on the boundry of()Vyis a

cost minimizing factor demand f or some price vector0w. Thus,

the set of points where(,)wx c w yfor all0wwill precisely

describe the input requirement set more formally –

When()Vyequals*( )Vy.S u p p o s e()Vyis regular, convex,

monotonic technology.

Then*( ) ( )VyV yProof: We already know that*( )Vycontains()Vy, so we only

have to show that ifxis in*( )Vythenxmust be in()Vy.

Suppose thatxis not an element of()Vy. Then since()Vyis a closed convex set satisfying the monotonicity hypothesis, we

can apply a version of separating hyperplane theorem to find a

vector*0wsuch that**wx wzfor all z in()Vy.L e t*zbe a

point in()Vythat minimizes cost at the prices*w.T h e n i n

particular we have** * ( * , ) .wx wz c w yBut thenxcan not be in*( )Vy, according to the definition of*( )Vy.

This proposition shows that if the original technology is

convex and monotonic then the cost function associated with the

technology can be used to completely reconstruc tt h eo r i g i n a lmunotes.in

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76technology. If we know the minimal cost of operation for every

possible price vectorw, then we know the entire set of

technological choices open to the firm.

This is a reasonably satisfactory result in the case o f convex

and monotonic technologies but what about less well -behaved

cases? –Suppose we start with some technology()Vy, possibly

non-convex. We find its cost function(,)cwyand then generate*( )Vy. We know from the above results that*( )Vywill not

necessarily be equal to()Vy, unless()Vyhappens to have the

convexity and monotonicity properties. However, suppose we

define–

*( , ) m incw y w xSuch thatxis in*( )VyWhat is the relationship between*( , )cw yand(,)cwy?

When(,)cwyequals*( , )cw y.I t f o l l o w s f r o m t h e d e f i n i t i o n o f t h e

functions that*( , ) ( , )cw y c w yProof: It is easy to see that*( , ) ( , )cw y c w y; since*( )vyalways

contains() ,vythe minimal cost b undle in*( )vymust be at least as

small as the minimal cost bundle in() .vySuppose that for some

prices'w, the cost minimizing bundle'xin*( )vyhas the property

that'' * (' ,) (' ,) .wx c w y cw y But that can not happen, since by

definition of*( ) ' ' ( ' , )vyw xc w y.

This proposition shows that the cost function for the

technology()vyis the same as the cost function for its

convexification*( )Vy. In this sense, the assumption of convex

input requirement sets is not very restrictive from an economic point

of view.

In short, it can be stated that –

(1) Given a cost function we can defin e an input requirement set*( )Vy(2) If the original technology is convex and monotonic, the

constructed technology will be identical with the original

technology.

(3) If the original technology is non -convex or non -monotonic, the

constructed input requirement will be convexified, monotonized

version of the original set, and most importantly, the constructedmunotes.in

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77technology will have the same cost function as the original

technology.

The above three points can be summarized succinctly wi th the

fundamental principle of duality in production : the cost function of a

firm summarizes all the economically relevant aspects of its

technology.

4.7 SUFFICIENT CONDITIONS FOR COST FUNCTIONS

We know that the cost function summarizes all the

econom ically relevant information about a technology. We also

know that all cost functions are non -decreasing, homogeneous,

concave, continuous functions of prices. The question arises :

suppose that you are given a non -decreasing, homogeneous,

concave continu ous function of prices –is it necessarily the cost

function of some technology?

The answer is yes, and the following proposition shows how

to construct such a technology.

When(,)wyis a cost function. Let(,)wybe a differentiable

function satisfying –

1)(, ) ( , )tw y t w y for all0t;

2)(,) 0wyfor0wand0y;

3)(' ,) ( ,)wy w yfor'ww;

4)(,)wyis concave in w.

Then(,)wyis the cost function for the technology defined by*( ) { 0: ( , ),Vy x w x w yfor all0}wProof: Given0wwe define

1(,) (,)( , ) ,......,

nwy wyxwyww And note that since(,)wyis homogeneous of degree 1 inw,

Euler’s law implies that(,)wycan be written as

(,)(,) (,)1 nwywy w w xwyiw ii

Here it should be noted that the monotonicity of(,)wyimplies(,) 0xwymunotes.in

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78Her we need to show that for any given'0 ,w(' , )xw yactually

minimizes'wxover allxin*( ):Vy(' ,) ' (' ,) 'wy w x wy w x for allxin*( ):VyFirst, we show that(' , )xw yis feasible; that is,(' , )xw yis in*() .VyBy the concavity of(,)wyinwwe have –(' ,) ( ,) ( ,) (' )wy w y D w yw w -for all0wUsing Euler’s law as above it reduces to(' ,) ' ( , )wy w x w yfor all0wIt follows from the definition of*() ,Vythat(' , )xw yis in*() .VyNext we show that(,)xwyactually minimizeswxover allxis in

*() ,Vythen by definition it must satisfy.(,)wx w yBut by Euler’s law,(,) (,)wy w xwyThe above two expressions imply –(,)wx wx w yfor allxin*()Vyas required.

4.8 SUMMARY

Concept of cost plays a vital role in determining the

performance of a firm. One requires to known the cost of

production together with the revenue t of i n dt h et o t a la m o u n to f

profits or losses if any. Per unit cost of production i.e. average cost

and average revenue has a greater role in determining the profits or

losses. Marginal cost of production is necessary in knowing the

equilibrium level of o utput.munotes.in

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794.9 QUESTIONS

1)What is cost function?

2)Discuss the concept of average and marginal costs.

3)What is geometry of costs?

4)Explain long -run and short -run cost curves.

5)Explain the Shephard’s Lemma.

6)Explain the Envelope Theorem.

7)Discuss the duality of costs.

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80(Module 3 )

Unit-5

PRICE AND OUTPUT DETERMINATION

UNDER PERFECT COMPETITION

Unit Structure

5.0Objectives

5.1Introduction

5.2Features of Perfect Competition

5.3Introduction to the process of Equilibration

5.4Short-run Equilibrium

5.5Stability of Equilibrium

5.6The Tat onnment Process (TP)

5.7Marshall’s Process

5.8Long -run Equilibrium

5.9Stability in the Long -run

5.10Summary

5.11Questions

5.0OBJECTIVES

This unit will enable you to understand.

The feat ures of perfect competition.

The short -run and long -run equilibrium of a perfectly

competitive firm.

The stability of equilibrium in the short run and long -run.

The Tatonmement Process and

The Marshall’s Process.

5.1INTRODUCTION

A market in which we find perfect competition between a large

number of buyers and a large number of seller of a homogeneous

product and uniform price is called perfect competition market or

perfectly competitive market. In other words in a perfect competition

market all the potential sellers and buyers are fully aware of the

prices at which transactions take place and all the affers made bymunotes.in

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81them and any buyer can purchase any commodity from any of the

sellers at the prices quested by them.

5.2FEATURES OF PERFECT COMPETITIO N

The main features of a perfect competition market are

discussed below.

1) Al a r ge Number of Bayers and sellers : -There are a large

number of buyers and sellers of the commodity in this market. Each

one of them is too small relative to the market and it cannot exert

any perceptible influence on price.

2) Homogenous Product : -The output of each firm in the

market is homogenous, identical or perfectly standardized. As a

result, the buyer cannot distinguish between the output of one firm

and that of a nother and is therefore, indifferent to the particular firm

from which he buys.

3) Freedom of Entry or Exit. Entry (or exit) of the firms into (or

from) the market is free in the perfect competition market. This

means that any new firm is free to start p roduction if it so whishes,

and that any existing firm is free to cease production and leave the

industry if it so whishes. Existing firms cannot bar the entry of new

firms and there are no legal prohibition on entry or exit.

4) Perfect Mobility : -There is perfect mobility of factors of

production geographically (i.e., from one place to the other) as well

as occupationally (i.e., from one job to the other).

5) Perfect knowledge : -there is perfect and complete

knowledge on the part of all buyers and sel lers about the conditions

in the market. For a market to be perfect it is essential that all

buyers and sellers should be aware of what is happening in any

part of the market.

5.3INTRODUCTION TO THE PROCESS OF

EQUILIBRIUM

In preceding chapters we cons idered models of the optimal

choices of consumers and firms. In these models, prices were

always taken as parameters outside the control of the individual

decision -taker. We now examine how these prices are determined

by the interaction of the decisions of such price, taking individuals.

Since the interaction takes place through markets, we examine

theories of markets whose participants act as price -takers, that is

competitive markets.munotes.in

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82Here we drew a distinction between production and supply in

the short run and in the long run we maintain that distinction in

market analysis, since supply conditions are an important

determinant of the market outcome we again think of demand and

supply as rates of flow per unit time. The short run is the period

over which f irms have fixed capacity. In the long run all inputs are

variable. For example, if it takes a year to plan and implem ent

capacity changes then the short run is this year and the long run is

next year. Since decisions for the long run are necessarily planni ng

decisions, expectations must come into the picture. So should

uncertainty.

The chapter adopts a partial equilibrium approach a single

market is considered in isolation. This is not entirely satisfactory,

since there may be interaction between markets .F o re x a m p l e ,w e

shall see that in aggregating firms’ supply curves to obtain a market

supply curve we may wish to take account of the effect of

expansion of aggregate market output on the prices of inputs used

by the firms. The justification for a partia l equilibrium analysis is that

is is simple and can give useful insights. Moreover, the key issues

concerning. The existence and stability of equilibrium can be

introduced in a particularly simple context.

5.4SHORT -RUN EQUILIBRIUM

Let()iixD pbe the 1stconsumer’s demand for the commodity at

price p and()ii

iiXX D p D P[5.1]

be the market demand function the short run supply function of firm

Ii s,iiyS p w[5.2]

whereiyis the output of firm j and w is the price of the variable

input.

It might appear that we could proceed to abstain a market

supply function by aggregating the firm supply functions as we did

the consumers demand function in 5.1, but this is not i ng e n e r a lt h e

case. In deriving the firms supply function we assumed input prices

constant this was a natural assumption to make, since any one firm

is a competitive industry (defined as the set of all producers of a

given commodity) could be expected to b e faced with perfectly

classic input supply curves. Then, as its output price is raised, the

firm could expand its desired production and input levels withoutmunotes.in

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83raising input price. The assumption may not be appropriate for the

industry as a whole, however: as the price at which they can sell

their outputs rises for all firms, expansion in production and input

demands may raise input prices because the increase in demand

for inputs is no longer insignificant, and input supply functions have

positive slopes to the industry as a whole.

Denote the total amount of the variable input used by the

industry by10zy z yIfww z y [5.3]

with10,wzthere are pecuniary external diseconomies: on

increas e in the total output of firms in the industry increases the

price of an input.

The consequences for the firms actual supply are shown in

fig5.1. In the figure, price is assumed to rise from p to p. The firms

initial supplySMCcurve is in each case s,0,pw.I f

simultaneous expansion by all firms raises input prices from01wt o wthe marginal cost curves and short run supply curves of

each firm must rise. Figure 5.1(a) shows one possible Fig5.1.

Figure5.1.

result of the expansion of firms in response to the higher price. The

short run supply curve has risen to1,isp wand so at prices1pthe

firm will want to supply1iyand not0iy. Hence the points on themunotes.in

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84firms supply curve corresponding to p andpwhen all firms

explain , are a and b respectively andispis the locus of all such

price. Supply pairs. clearly, the firms effective market supply curveispwill be less elastic than its ceteris paribus supply curve,isp w, They would concise if input prices were not bid up by

simultaneous explain sion of output by all firms (and there were no

technological externalities.

In (b) of the figure a more extreme case is shown. The

increase in input prices causes a sufficient shift in the firm’s SMC

curve to make the post. adjustment output1iyactually less thaniyand so its effective market supply curvejshas a negative slope.

Thus, although the law of diminishing returns ensures that each

firms ceteris paribus supply curve has a posi tive slope this is not

sufficient to ensure that the firms effective supply curve has a

positive slope, if input prices increase with the expansion of outputs

of all firms.

Denoting the effective industry supply function byypand

substituting 6.3 in 6.2 gives the effective supply function of firm:

,ij jysp w z y p s p [5.4]

and summing gives the effective industry supply functionjj

jjyy s p s p[5.5]

Differentiating 6.4 with respect to the marke t price gives the

effective supply response of firm j (after allowing for the effect of

the increase in induced by the change in output of all firms) as

11 150i

jp jw jdy dysW z z ys pdp dp [5.6]

Since,/ 0jp jss p w pand/0jw iss wwe see that

the firms effective supply could be increasing or decreasing in P.

The change in industry supply as a result of the increases in

p is the sum of the effective changes in the firms supplied and so

from 6.5 and 6.6.

11 ijp jwjj jdy dy dysw z sdp dp dp munotes.in

## Page 85

85Since0jpsand110, 0, 0,je wzssoluing for/dy dpgives

1101ji p

jj ws dy

dp w z s [5.8]

Thus the effective industry supply curve is positively sloped

despite the fact that some of the firms may have negatively sloped

effective supply curves. The slope of the market supply function

depends on the extent to which increases in input demands

increases input prices and the consequent increases in marginal

costs at all output levels. Not tha t at a market supplyss pi.e. a

point on this supply function, each firms marginal cost is exactly

equal to p, given that all output adjustments have been complete.

We define p as the supply price of the corresponding rates of

outpu tiysince it is the price at which each firm would be content to

supply and to go on supplying -the outputiy. At any greater price

firms would find it profitable to expand production; at any lower

price, they would wish to contract.

Figure 5.2 shows a number of possible situations which

might arise when we put the market supply function together with

the demand function. In (a) we show a well behaved case. The

price*p,w i t hd e mand*xequal to supply*y, is obviously on

equilibrium, since sellers are receiving the price they require for the

output they are producing, and this output is being taken off the

market by buyers at that price. There is no reason either for sellers

to change their output (since each*iiys pmaximizes is profit at

price*p) or for buyers to change the amount they buy.

Figure 5.2

In (C) we show a third possi bility, Suppose that firms do not

all have the some AVC, but instead are evenly distributed over a

range of AVC, with the minimum point of the lowest AVC curve

being equal to11p. If there are many sellers, and each seller is anmunotes.in

## Page 86

86insignificant part of the market we can then take the s(p) curve as

continuous, with intercept at11p. However, at price11 1ppdemand is zero -no one would be prepared to pay1por more fo r

this good. It follows that equilibrium in this market implies a zero

output and a price in the interval11 1,ppthe highest price any

buyer would pay is insufficient to cover the AVC of the firm with the

lowest minimum AVC. We have a non produced good which firms

would supply if the price were high enough, but which nobody

wants to buy at such a price. The reader will find it instructive at this

point to construct the excess demand functionszp Dp sp [6.9]

inthese there cases, and illustrate them in a price -excess demand

graph of the type shown in Figure 5.3

Figure 5.3(b) suggests that a discontinuity in a supply or

demand function and thus in the excess demand function. may

imply that there is no equilibri um. This is a matter of some concern,

since our theory of the market predicts the market outcome to be

the equilibrium outcome, and raises the question what do we have

to assume to ensure that the market has an equilibrium? To take

the case of one market i s to give only a provisional answer to the

question since we ignore the interdependence among markets

Nevertheless, it is instructive to consider the existence question in

the simple context of one market.

Figure 5.3

Figure 5.3shows that discontinuity is a problem. Is it then

enough to assume that z(p) is a continuous function of p? clearly

not An equilibrium is a price*0psuch that00zp.I f0,zp o r zp ofor all,p o then z pmay be continous butmunotes.in

## Page 87

87we will not have an equilibrium. This suggests the following

existence theorem for a single market. If.

a) the excess demand functionzpis continuous for0p.

b) there exists a price00psuch that00,zpand

c) there exists a price10psuch that10,zpthen there exists exists an equilibrium price*0psuch that*0zpThe intuition is clear from fig 5.3. If the excess demand curve

is continues us and passes from a point at which excess demand is

positive to a point at which excess demand is negative, it must

cross the pric e axis, giving an equilibrium price.

The significance of the equilibrium price is that it induces

buyers to demand exactly the output that results from individual

sellers’ profit -maximizing decision at that price. Plans are all

mutually consistent and ca nb er e a l i z e d .W en o wt u r nt ot h ee q u a l l y

important question of the sta bility of a market in the short -run.

5.5STABILITY OF EQUILIBRIUM

Stability is an important characteristic of a market since

predictions of the effects of changes in supply or deman d

conditions typically take the form of comparisons of the equilibrium

before and after the change stability, like the question of existence

considered in section A, is also relevant for analyses of welfare,

which typically focus on properties of equilibri a. Such analyses

would have less point if one could not be sure that the market had

an equilibrium to which it would tend.

A market is stable if, when ever the market price is not an

equilibrium price, the price converges over time to on equilibrium

price. The market is locally stable if it tends to an equilibrium when

it starts off in a small neighborhood of that equilibrium and globally

stable when it tends to some equilibrium price whatever its initial

disequilibrium price.

In general we are more i nterested in global stability and

whether the market will eventually end up in some equilibrium local

stability does not imply global stability but, if there is only one

equilibrium, global stability implies local stability. If a market has

multiple equili bria it may be locally stable in the neighbourhood of

some equilibria and unstable in the neighbourhood of other. Global

stability then implies that at least one of the multiple equilibria is

locally stable, though other may be unstable. Even if all themunotes.in

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88equilibria were locally stable this would not imply that the market

was globally stable.

Formally a market is stable if*im p t ptwhere*pis an

equilibrium price,tois time, p(t) is the time path of price and the

initial price*.po pThe analysis of stability is concerned with a market’s

disequilibrium behaviour and requires a theory of how markets

operate out of equilibrium. Any such theory rests on answers to

three fundamental questions.

1]How do the market price or prices respond to non -zero

excess demand?

2]How do buyers and sellers obtain information on the price or

prices being offered and asked in the market?

3]At what point does trading actually take place, i.e. when do

buyers and sellers enter into binding contracts?

There questions are important because answers to them

may differ and differences in the answer lead to significant

differences in the models of disequilibrium. In questions 1 and 2 we

use the phrase price or prices because at this stage theories may

provide for a single price to prevail through out the market even out

of equilibrium, where as other allow there to be differences in prices

offered by buyers and asked by sellers throughout the market.

Whether or not a unique price will always prevail depends on the

answers to questions 2 and 3.

To begin with we consider two contiruious time models of

market adjustment. The first, known as the tatonnement process

(tatonnement can be interp reted as ‘groping’ ) was proposed by

walras. The second, which it can be argued is better suited to

markets with production, was suggested by marshall.

5.6THE TATONNEMENT PROCESS (TP)

The TP is an idealized model of how a market may operate

out of equ ilibrium, in the sense that it may not describe the may a

market works, but under certain conditions a market may operates

as if its adjustment process were a TP there is a central individual,

who can be called the market ‘umpire’, and who has the rate of a

market coordinator. He announces to all decision takers a single

market price (the answer to question 2), which they take as a

parameter in choosing their planned supplies or demands. They

each inform the umpire of their choices and he aggregates them to

find the excess demand at the announced price. He then revisesmunotes.in

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89the announced price by the following rule (the answer to

question 1).

0dpzp tdt[6, 10]

that is, he changes the price at a rate proportionate to the excess

demand. No trading takes place unless and until equilibrium is

reached (the answer to question 3) at which time sellers deliver

their planned supply and buyers take their planned demand. Notice

that in this process there is no contact between buyers and sel lers

out of equilibrium every thing is mediated through the umpire.

Figure 5.4 shows three possible market excess demand

functions. In (a), the excess demand acure has a negative slope. If,

initially, the umpire announces the price0*,ppexcess demand

will be positive and he will revise the announced price upwards

towards*pif the announced price where above*pit would be

revised down words. Since these movements are always in the

equilibrating direction, from wherever the process sarts, equilibrium

will be global stable.

Figure5.4

In (b), the excess demand curve has a positive slope. If the

announced price is initially at0pthe umpire will now reduce price,

since0,zand hence the TP leads away from equilibrium. A

similar result used occur if the initial price were above*.pHence in

this market the equilibrium is globally uns table.

In(c)we have a somewhat more complex case. The excess

demand curve is backward bending, having a negative slope over

one range of prices and a positive slope over another. In this case,

if the initial price were anywhere in the interval1,oppthe TP

would converge to the equilibrium*p. If, however, the initial price

was11 1ppthe market would move away from equilibeium, since

excess demand is positive for1ppand so price would bemunotes.in

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90increased. Therefore the market is not globally stable, since on

initial point sufficiently far from the equilibrium.*pqould lead away

from market equilibrium. The market has two equilibrium positions ,

one at*pand one at1p; the former is locally but not globally

stable, the latter is locally (and therefore globally) unstable.

From this discussion we can deduce the following stability

conditions, i .e. sufficient conditions for the TP to be stable: (a)

equilibrium is globally stable if excess demand is positive whenever

price is less than its equilibrium value and negative when price is

above its equilibrium volue;

(b) equilibrium is locally stable if the condition holds for prices in

all small neighbouxhood of an equilibrium.

For a more formal analysis of stability we can use a distance

function, which measures the distance between two point. Thus

define

2**,pt p pt p [5.11]

which measures the distance between an equilibrium price*pand

some other price p(t) (note that**,0 .pt p pt p)A

necessary condition for the time path of priceptto converge to*pis that/0 ,dd ti.e. the distance between the price path and*pis falling through time. Differentiating we have*2*d p t p dp p t p z p tdt dt

[5.12]

from 5.10 then clearly/0dd tif any only if*)p t p and z p t

have opposite signs, as in the stability condition Note that this is

true regardless of the value of: the ‘speed of adjustment’

parameter determines only now fast, and no whether, the TP

converges to equilibrium.

Is the condition also sufficient for convergence, however? It

may seem ‘intuitively obvious’ that it is, but consider the example of

the function1/ .ya tHere we have/0 , l i m 0dy dt but t y.

So w e have to provide a further argument to justify the claim that*,pt pis not bounded away from zero under the TP.

We do this by establishing a contradiction suppose, without

loss of generality, that*po pand su ppose thatmunotes.in

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91*im p t p where p p .T h ei n t e r v a l ,po pis non -empty,

closed and bounded and the function/dd tis continous, so at

some t we must heave that/dd ttakes an a maximum, by were

strass. Theorem, call this maximum*sNote that, since for*pt pwe must have/0 ,dd tthen s < 0 also for only arbitraryttintegrate to obtain.

**0,,tddt p t p p o pdt [5.13]

and**0ts dt s t[5.14]

Then by definition of*swe must have** *,,pt p po p st [5.15]

or** *,0 ,pt p st p p [5.16]

By choosingtlarge enough, we can make the right hand

side of 5.16 negative, implying we must have on the left -hand side

a negative value of the distance function, which is impossible. Thus

we have the contradiction.

This proof makes precise the intuition that, ifptis always

moving closes to*pwhenever*,pt pit cannot tend to

anything other than*p.

5.7MARSHALL’S PROCESS

Marshall suggested the following altern ative to Walra’sT P

suppose that when sellers bring their output to market they sell it for

whatever it will fetch. Refer to fig 5.5. If supply is less than the

equilibrium supply*ythen the price buyers will be prepared to pay

ifit is auctioned off to the highest bidders, the demand price,0Dp,

exceeds the supply price,0spconversely, if supply exceeds

equilibrium supply auctioning off the available supply causes

demand price to f all below supply price. Marshall argued that when

demand price0pexceeds supply pricespsellers will expandmunotes.in

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92supply, and conversely whenopis less thensp. This is becausespequals each seller’s marginal cost, and so0sppimplies output

explansion increases profits, while whenosppprofits are

increased by an output contraction. This su ggests the adjustment

rule.

0 sdypy pydt [5.17]

Figure 5.5

00 * * 1() () ,SP abS P bD P y x y y x y

where0pyis inverse demand function, giving demand price as a

function of quantity supplied (= quantity traded at any t) and

similarly,spyis the inverse supply function (derived from the

firms marginal cost functions as before). Note that, at equilibrium

quantity**,Dsyp p pmunotes.in

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93Under what conditions is Marshall’s process stable? If output

expands whenDsppand contracts when0sppthen fig 5.5 (a).

Suggests that, when the supply and demand curves have the usual

slopes, the market is stable Figures 6.5(b) and (c) show that, when

the sup ply curve has a negative slope, the process is stable if the

demand curve cuts the supply curve from above but unstable in the

converse case. This is interesting, not only because back ward

bending supply curves are possible but also because the walrasion

TP has precisely the opposite outcomes in these cases. In figure

6.5 (b), the corresponding excess demand functionzp Dp spincreases with price with price and so the

walrasion TP would be unstable. In figure 5.5 (c) z(p) has a

negative slope and so the Walrasion TP is stable. Thus although

the two adjustment processes have then some outcomes in the

‘standard case’, it matiers which we adopt in a ‘non -standard’ case.

To make the stability conditions for Marshall’s process more

precise , we again adopt a distance function approach. Define the

distance function

2**,yt y yt y [5.18]

Then

**22Dsd dyyt y yt y p t p tdt dt [5.19]

using 5.17. Then, for/0dd t,w er e q u i r e*

Dsy t y and p t p thave oppo site signs, confirming the

diagrammatic analysls. We can establish the sufficiency of this

condition along similar lines to those used in the case of the TP

process.

We have already noted that in non -standard cases the

walrasian TP and Marshall’s proce ss have opposite implication for

market stability -it matters whether we take price as adjusting to a

difference in quantities, or quantity as adjusting to a difference in

demand and supply prices. We can also compare the proceses in

terms of the answers to the three questions at the beginning of this

section:

1) Responsiveness of price to non -zero excess demand. In the

standard case of negatively sloped demand and positively sloped

supply, both processes result in market price rising (falling) when

there is positive (negative) excess demand. In the Walrasian case

this happens directly through the TP; in the Marshallian case, it

happens via the auction mechanism which establishes the demand

price.

2) Infromation on price (s). In the TP, this is transmitted

simultaneously to all buyers and sellers by the umpire; in Marshallsmunotes.in

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94process, at each instant the auction mechanism rations off

available output and the demand price is immediately made known.

Buyers never need to know the supply price -sellers know their o wn

marginal costs and so once the demand price is known an output

change can result.

3) When does trade take place? In the TP, only at equilibrium,

under Marshall’s process, at every instant as available supply is

auctioned off marshals process has tradi ng out of equilibrium, with

an efficient rationing rule, so that available supply is auctioned off to

the highest bidders. Alternatively, think of Marshall’s process as

consisting of a sequence of ‘very short -run’ or instantaneous

equilibria, with a vertic al supply curve at each of these equilibria,

and the analysis then establishes conditions under which this

sequence of instantaneous equilibria converges to a full equilibrium

of supply and demand.

Which model is ‘better’ depends on which process captur es

more closely the way a particular market works. Walras’ TP may

seem unrealistic in its reliance on a central ‘umpire’ collecting

buying and selling intentions and announcing an equilibrium price,

but some markets, for example markets in stocks and share s, and

minerals such as gold and silver, are highly organized with brokers

who may function much as a walrasian umpire.

There are two features of both models which are

unsatisfactory in the light of observations of how many markets

work. First, both pro cess are centralized : some device -the umpire

or the auction mechanism ensures that all buyers and sellers

simultaneously face the some price. However, in many real

markets, price formation is decentralized. Individual buyers meat,

haggle and deal with i ndividual sellers, and pressures of excess

demand or supply exert their influence by causing sellers and

buyers to bid price up or down. If information on all the prices being

offered and asked is fully and costlessly available throughout the

market then t his would be equivalent to a centralized adjustment

process. But this is often not the case. Buyers and sellers have to

seek each other out to find the prices at which they are prepard to

trade, and this search process is costly.

Second, in neither mode l do buyers and sellers form

expectations and act upon them In the TP this possibility is simply

excluded. In Marshalls process, sellers must make some forecast

of further price in order to market decisions which determine their

future supply, but this is not module explicitly, being subsumed in

the adjustment rule 6.17. In the rest of this section therefore we

consider the explicit modeling of expectations in market adjustment

processes.munotes.in

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955.8LONG RUN EQUILIBRIUM

We saw that the firm ’sl o n g -run supply c urve is that part of

its long -run marginal cost curve above its long -run overage cost

curve. There are several reasons why the market supply curve

cannot be obtained simply by summing these supply curies:

(a) External preliminary effects. As all firms var yo u t p u t ,i n p u t

prices may change, causing each firm’s cost caves to shift.

(b) External technological effects. Individual firms’ cost curves

shift as a result of expansion of scale by all firms leading to

congestion or improvement in common facilitie s such as

transport or communications.

(c) Changes in the number of firms in the market. As price rises

firms which previously found it unprofitable to produce the

commodity now find it profitable, and so invest in capacity and add

to output. In a compe titive market there are no barriers such as

patents, legal restrications, ownership of raw material sources.

Which impede the entry of new firms. A firm which at the going

price just breaks even, with total revenue equal to long -run total

cost (induding th e opportunity cost of capital and effort supplied by

its owners (s)) is called a marginal firm at that price one which

makes on ‘excess profit’ (total revenue > total long -run opportunity

costs) is called an intro -marginal firm, and one which would market

a loss, but breaks even at a higher price, is called an extrmarginal

firm. As rpice rises, marginal firms become intro -marginal and some

extra marginal firms enter.

It is therefore by no means assured that the long run market

supply curve will be positi vely sloped (see questions 1,2). However,

in figure 5.6 (c) we assume this to be the case. S(p) shows how the

rate of output varies with price when capacity is adjusted and the

number of sellers may change. It should be noted that underlying

this curve is a possibly complex set of adjustments, and the

transition from one point on the curve to another is not so smooth

and effortless as the curve suggest. It should be interpreted as

showing the aggregate output which will be forthcoming at each

price after al l these adjustments have been made. Or, alternatively,

it shows the price at which a given number of firms would remain in

the industry, maintain their capacity and supply in aggregate a

given rate of output. The P -Coordinate of any point y is then the

long-run supply price of that rate of output.

The long -run equilibrium is shown in Figure 5.6 (c) as the

point**,ypAt this point firms are prepared to maintain the rate of

supply*yand consumers are prepare d to buy this output at pricemunotes.in

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96*p. If therefore, the short run supply curve s(p) was as shown in the

figure the short -run equilibrium we have earlier been examining

would also be a long run equilibrium. It would be maintained

indefi nitely in the absence of any change in demand, input prices or

technology.

P P1S2S1S2SLAC 1

P*P*LAC 2

C*

O01yy1 O*2yy2

P

S(P)

S(P)

P*

D(P)

O**xyx,y

Figure 5.6

The other parts of the figure show the implications of the

long-run equilibrium for two ‘representative firms’ In (a), firm 1 is a

marginal firm. At market price*pit chooses a long run profit

maximizing scale of output*1yand at that output*pis equal to its

minimum long-run average cost. Firm 2, on the other hand, shown

in (b) of the figure, is on intra -marginal firm; at its profit -maximizing

scale of output*2y, its long run average cost**cp, and it makes

an excess p rofit equal to** *2pc y. However, such ‘excess profits’munotes.in

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97which may be earned temporarily, will not persist indefinitely, but

rather should be regarded as true opportunity costs to the firm in

long run.

The argument goes as follows the fact that the intra marginal

firms average costs are lower than these of a marginal firm must

reflect the possession of some particularly efficient input, for

example especially fertile soil or exceptionally skilful management.

Since these generate excess profits, we expect other firms to

compete for them, so that after a period long enough for contracts

to lapse, the firms which currently enjoys the seruices of these

super productive inputs will have to pay them what they ask or lose

them. The maximum thes e inputs can extract is the whole of the

excess profits** *2pc yand so what was a profit during the

period when the contract was in force becomes a free opportunity

cost to the firm after that time. Such excess profits are therefore

called quasi -rents, to emphasize that they are not true long -run

excess prof its, but merely rents accruing to the contractual property

rights in certain efficient input serucies, which become transformed

into costs in the long run once this transformation has taken place,

the ‘intro -marginal’ firms LAC curve will rise until its minimum point

is equal to*p. Hence in the long run all firms in the market will be

marginal firms in the sense that they just break even.

Figure 5.6 indust ries the three conditions which must hld in

long-run equilibrium 1) Each firm in the market equates its long -run

marginal cost to price, so that output maximizes profit.

2) For each firm price must equal long run average cost (if

necessary after quotients have been transformed into opportunity

costs) so that profits are zero and no entry or exit takes place.

3) Demand must equal supply.

Condition (1) and (2) then imply that each firm produces at

the minimum point of its long -run average cost curve, as fig5.6(a)

illustrates. This is a strong result on the efficiency of the competitive

market equilibrium, since it implies that total market output is being

produced at the lowest possible cost.

As with the short run supply curves in fig 5.6(b)

discont inuities in the long -run supply curve may imply that

equilibrium does not exist suppose that (a) all firms, whether

currently in the market or not, have identical . u -shaped LAC curves

as shown in figure 5.6 (a) (b) input prices do not vary with industry

output.munotes.in

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98Then, there could be a discontinuity in the long -run market

supply curve at price*pin figure 5.6. At any price below*p,a l l

firms would have the market, and market supply will fall to zero,

while at price*pplanned market supply is*1ymultiplied by the

number of firms which are capable of producing the good with the

given LAC curve. This discontinuity could be avoided if there is

some mechanism which selects potential suppliers in such a may

as to ensure that any given market demand at price*pis just met

by the appropriate number of firms each producing at minimum

long-run average cost then, the long -run market supply c urve would

be a horizontal line at price*pexpansion of market output is

brought about entirely by new entry rather than through output

expansion by existing firms. Long -run equilibrium price can not

differ from*p, and so is entirely cost determined. The level of

demand determines only aggregate output and the equilibrium

number of firms. Note that for a long -run market supply curve which

is a continuous horizontal line we need the least cost output of a

firm*1yin fig 5.6 (a) to be ‘very small’ relative to market demand,

and the number of firms to be ‘very large’.

More simply, if the technology of production is such that

there is no range of outputs over which there are increasi ng returns

to scale, then there is no discontinuity in market supply. For

example, if all firms experience decreasing returns to scale at all

outputs then long -run average and marginal cost curves will be

every where upward sloping and their horizontal sum (taking into

account any input price effects) will have an intercept on the price

axis.

Alternatively, if we assume all firms have identical production

functions with constant returns to scale, and face identical

(constant) input prices, then the long -run market supply curve is

again a horizontal straight line. Each firm’s long -run marginal cost

curve is a horizontal line and coin cides with its long -run average

cost curve, and these are at the some level for all firms. Then, the

only possible equilibri um price is given by this common maxginal

average cost so that price is again completely cost determined

Demand again determines only the aggregate equilibrium market

output. Note that, in such a market model, the equilibrium output of

each firm, as well a s the equilibrium number of firms producing in

the market, are indeterminate.

5.9STABILITY IN THE LONG -RUN

The analysis of the stability of long -run equilibrium in a

competitive market must take into account the interaction betweenmunotes.in

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99short -and long -rundecisions of firms, the effects of new entry and

the role of price expectations. We carry out the analysis for the

case in which input prices increases with aggregate market output,

and all firms have U -shaped cost curves. As shown in figure 5.7,

the long run market output, and all firms have U -shaped cost

curves. As shown in figure 5.7, the long run market supply curve is

upward sloping. It should be thought of as the locus of price -

quantity points at which the long run equilibrium conditions are

satisfie d at each paint, price -long run marginal cost for each firm in

the market, and no further entry or exit will take place at a given

price because firms are just breaking even at that price given that

the quast rents of intra marginal firms have been transfo rmed into

opportunity costs). Thus corresponding to each paint on the curve

is a particular set of firms, each with a profit maximizing capacity

and output level As price rises, output increases along the curves

as a result of both output expansion by exis ting firms and entry of

new firms. However, the actual timl path of price and output may

not lie one the supply curve. For that to happen, we again need the

assumption of rational expectation, as we shall now see.

Figure 5.7

Suppose at year O the mark et is initially in long -run

equilibrium at the price and output pair00pyin figure 5.7 In year

1d e m a n ds h i f tt o1Dp.I nt h es h o r tr u ny e a r1o u t p u tc a no n l y

expand along the short run supply functions1sp,d e t e r m i n e db y

the shrot run marginal cost functions of the firms already in the

market (together with only effects of increasing input prices as

analysed in section A). Thus price in year 1 is established as1p.

Since0pcorresponded to zero profit of the existing firms,1pmust

imply positive profits. The market is clearly not in long runmunotes.in

## Page 100

100equilibrium. What happens next depends upon the assumption we

make abou t price expectations formation.

Begin, as in the cobweb model of section B, with the

assumption of naïve expectations all firms, whether currently in the

market or contemplating entry, expect price1pto prevail next year,

in ye ar 2. The existing firms expand capacity and new firms enter

and install capacity to the extent that planned market output

expands to2y, since this is the aggregate output corresponding to

jong-run profit maximization at price1p. But of course, when period

2 arrives,112,pyis not an equilibrium price will have to fall to2p,

whether demand equals short -run supply as indicated by the sort -

run supply curves2sp. This is determined by the short run

marginal cost curves of all firms in the market initial incumbents and

new entrants in year 2. If all firms again assume, naively, that2pwill prevail in year 3, then ca pacity will be contracted and some

firms will have the market unitl3ywill be the aggregate market

supply that will be planned for year 3. And so on under naïve

expectations, price fluctuates around the equilibrium value*pand

in the case illustrated in figure 6.7 eventually converges to it (in the

absence of further demand change). The fact that capacity can only

be adjusted in the long run introduces the some kind of supply lag

that we assumed for an agr icultural market. The main difference is

that here the short -run supply curve is positively sloped whereas in

the Gobweb model it was in effect vertical. The role of the long -run

supply curve in the present analysis is to show how future price.

Although th e ultimate effect of the demand shift is to move the

market from one point on the long run supply curve to another, the

actual time path of price and output through the adjustment process

lies along the demand curve and describes a diminishing sequence

ofjumps from one side of equilibrium point to the other.

However, our previous criticisms of the naïve expectations

assumption apply equally here. It is irrational for a profit maximizing

firm to form its expectations in this may because then it is

consis tently sacrificing potential profits suppose instead that all

firms have rational expectations, that is, they know the market

model and use its predication as their price expectation. Then, if the

change in demand between periods 0 and 1 is unanticipated, the

year 1 short -run equilibrium is at11,pyas before, but now firms

can predict the now long -run equilibrium price*p.T h i si st h eo n l y

price with the property that the planned output which maximize

profits at that price can actually be realized, i.e sold, on the market

next period Hence existing firms will expand capacity and new firms

will enter so as to expand market output to*y, and the marketmunotes.in

## Page 101

101moves to its long run equilibrium in years 2. If the change in

demand had been fully anticipated at year 0, then the same

argument leads to the conclusion that the market would move to its

new long -run equilibrium in year 1. In that case, the market adjusts

smoothly along its long -run supply curve to change in demand.

5.10SUMMARY

This unit explains the features of perfect expedition and

elaborates the process of price and output determination of a

perfectly competitive firm in the short -run and long run. It also

explains the concept of Tatonnement process and Marshall’s

process.

5.11QUESTIONS

1. Explain the short -run equilibrium of perfectly competitive

firm.

2. Examine the stability of equilibrium of perfectly completive

firm in the short -run.

3. Describe the concept of the Tato nnement Process.

4. Examine the concept of Marshall’s Process.

5. Explain the equilibrium of a perfectly7 competitive firm in the

long-run.

munotes.in

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102Unit-6

PRICE & OUTPUT DETERMINATION

UNDER PERFECT COMPETITION

UNIT STRUCTURE:

6.0Objectives

6.1Introduction

6.2Existence of General Equilibrium

6.3 Stability of Equilibrium

6.4First & Second Fundamental Theorems of Welfare Economics

6.4.1 First Fundamental Theorem of Welfare Economics

6.4.2 Second Fundamental Theorem of Welfare Economi cs

6.5Welfare Effects of Price Changes

6.6Consumer Surplus

6.7Market Failure

6.8Theory of the Second Best

6.9Summary

6.10Questions

6.11 References

6.0OBJECTIVES

After going through this unit you will be able to explain the

concepts of -

Existence & Stablility of General Equilibrium

First & Second Fundamental Theorems of Welfare Economics

The Market Failure

The consumer surplus

Theory of second best

6.1 INTRODUCTION:

The general equilibrium phenomenon is the interdependent

and interrelat ed. General equilibrium indicated equilibrium of

consumer in the market by two forces these are demand and

supply where price of commodity determines at such a point, this

point is called equilibrium point of market. In other words, general

equilibrium con cept is related to price determination in the marketmunotes.in

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103by various forces. How the general equilibrium concept is

interdependent? Consumer demand for a particular commodity

affects by various factors such as taste of consumer, preferences,

price of substitut e commodity, climate, income and many more and

his demand affected by these factors. Therefore, general

equilibrium of market affects because of these factors. So, the

concept of general equilibrium becomes interdependent.

There are two important views t hese are Marshallian general

equilibrium and Walarasian general equilibrium. In Marshallian

equilibrium analysis, Marshall explains partial equilibrium by taking

only two variables to determine prices. He assumed that other

factors being constant. The seco nd view of Walarasian was first

scientific view, because he considered all variable or relevant

variables which plays important role in price determination in the

market or market equilibrium. That is why Walatasian analysis is

called general equilibrium a nalysis.

6.2 EXISTENCE OF GENERAL EQUILIBRIUM:

General equilibrium is the concept which is complex in the

nature, because it is interdependent on various variables. So,

existence of general equilibrium in the market is difficult task. There

are various p roblems are arisen. Among of them three problems

arised by Koutsoyiannis which are as follows -

1.Existence of equilibrium. Does a general equilibrium solution

exist?

2.Uniqueness of equilibrium. If an equilibrium solution exists, is it

unique?

3.Stability of eq uilibrium. If an equilibrium solution exists, is it

stable?

In short, existence of equilibrium in the market is the

condition or situation where neither excess demand exist nor

excess supply exist. It means whatever supply in market at

particular price i s demanded, there is no issue of excess stock. So,

this scenario called market equilibrium. The existence of equilibrium

is shown by below diagram.munotes.in

## Page 104

104

Figure 6.1

In the above figure, price has been shown on Y axis and

quantity on X axis. S1S1 is supply c urve and D1D1 is demand

curve. D1D1 and S1S1 curves intersect to each other at E point. E

point is the equilibrium point at which price and quantity determines

P1 and Q1 respectively. At E point there is existence of equilibrium.

6.3 STABILITY OF EQUILIBR IUM:

If equilibrium exists in market, is it stable? It means when the

various forces of market disturbs the equilibrium, is the tendency of

market towards the equilibrium? If answer is yes, so we can call,

this equilibrium is stable equilibrium.

The stab ility of equilibrium in the market is depends on

shape and slope of demand and supply curve. These two factors

determine whether the equilibrium is stable? For the stable

equilibrium demand curve should be downward sloping and supply

curve should be upward sloping.

Figure 6.2munotes.in

## Page 105

105Above figure shows stable equilibrium. In this figure, DD

demand curve is downward sloping and SS supply curve is upward

sloping which intersects to each other at E point. And at the E point

there is stable equilibrium. If any situa tion, the market equilibrium

disturbs, the price mechanism will make it again stable.

6.4FIRST & SECOND FUNDAMENTAL THEOREMS OF

WELFARE ECONOMICS:

6.4.1 FIRST FUNDAMENTAL THEOREM OF WELFARE

ECONOMICS:

The first fundamental theorem of welfare economics is

related to the concepts of Pareto optimality and the perfect

competition. What is Pareto optimality? Pareto optimality states,

“without making someone worse off, no one will better off.” It means

there is no further improvement possible in social welfar e.

The social welfare equilibrium happens in only perfectly

competitive market, this is known as first fundamental theorem of

welfare economics where all marginal conditions of Pareto will be

fulfilled. In other words, first fundamental theorem of welfare

economics explains Pareto efficiency in economy or social welfare

equilibrium in the perfect competitive market or first fundamental

theorem of welfare economics assumes that general competitive

equilibrium is the Pareto optimal.

Marginal Conditions of P areto Efficiency or

Conditions of Pareto Optimality: -

1.Efficiency in Exchange: The marginal rate of substitution

(MRS) between any two products or commodity must be the

same for every individual who consumes both.

2.Efficiency in Production: The marginal rate of technical

stubstitution (MRTS) between any two factors must be the

same for any two firms using these factors to produce the

same product.

3.Efficiency in Product Mix: The marginal rate of substitution

between any pair of products for any person consumin gb o t hmust be the same as the marginal rate of transformationbetween them.”

4.Efficiency in Consumption or Exchange

5.Pareto Optimality in Consumption or Exchange and Perfect

Competition

6.Pareto Optimality Conditions when theExternal Effects are

Present

7.Efficiency in the Allocation of Factors among Commodities or

Efficiency in Product -Mix or Composition of Output.munotes.in

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106Critics on First Fundamental Theorem of Welfare

Economics:

1)This theorem ensures only about Pareto efficiency not the social

justice.

2)Externaliti es found in consumption and production.

3)Second order condition of the equilibrium must be fulfilled.

4)Economic efficiencies are quite restrictive.

5)The concept of perfect competition is hypothetical in the

practical life imperfect completion prevails.

6.4.2 SECOND FUNDAMENTAL THEOREM OF WELFARE

ECONOMICS:

There is also second fundamental theorem of welfare

economics. This theorem of welfare economics states that “Every

Pareto optimal situation, there is competitive equilibrium.” Given the

initial income dist ribution in the economy or factor endowment in

the economy. In the other word, Pareto efficiency or Pareto

optimality situation found in competitive market equilibrium.

6.5 WELFARE EFFECTS OF PRICE CHANGES:

Welfare of the society is depends on various va riable factors

like as level of production in an economy, government fiscal

policies, variation in tax rate and changes in prices of commodity

etc.

Welfare of a society or individual have main constraints and

these are income and price of commodity and se rvices, because

these two factors have ability to enhance social choices as said by

Amartya Sen. Therefore, change in the prices of commodities and

services effect consumer’s welfare.

In the mid nineteen century, Engineer Jules Dupuit who first

propounde d the concept of economic surplus. Then Alfred Marshall

gave fame to this concept and developed the two major concepts

as consumer surplus and producer surplus with the help of demand

curve and cost curve (supply curve).

6.6 CONSUMER SURPLUS:

The concept of consumer surplus is very useful to

understand that what the welfare effects of price change are ?

“Consumer surplus is the difference between the price which

consumers are willing and able to pay for a good or service and

actually do pay.” In other wor ds, consumer surplus is the difference

between potential price and actual price.munotes.in

## Page 107

107Consumer surplus is the area under the demand curve and market

price.

Consumer Surplus =Potential Price –Actual Price

Consumer Welfare/Consumer Surplus at Initial Level:

Figure 6.3 .1

Consumer Welfare/Consume r Surplus when Price increases:

Figure 6. 3.2

Consumer Welfare/Consumer Surplus when P rice decreases:

Figure 6. 3.3munotes.in

## Page 108

108Above figures 1, 2 and 3 explored the effect of price change

(increase/decrease) on consumer wel fare/ social welfare/ social

advantage.

Figure no. 1 indicates that DD and SS curve which are

demand curve and supply curve intersect to each other at E point

which is the equilibrium point where consumer is ready f orwilling to

pay at OP1 price for OQ q uantity, but actually he pays at OP price.

So, EPP1 is the area of consumer surplus.

Figure no. 2 indicates when the price of commodity rise in

the economy, consumer surplus or welfare of a consumer will

decrease. In this diagram, when SS and DD curves ex ist in the

economy, the area EPD is the area of consumer Surplus. If the

market price increases, supply curve shifts to S1S1 which intersect

to DD demand curve at E1 point and OP1 market price determines

in the economy. Because of price increase, consumer surplus

decrease from EPD to E1P1D. It means inflation in economy

adversely affects the social welfare of society. Therefore,

government of the country and central monetary authority always

tries to control inflation in the economy to save public welfare a nd

public interest.

In the case of price decreases (in Figure 3), consumer

surplus is increased. In the short, when price decreases in the

economy, real income of people and society will increase due to

this able to expand their social choices which leads to increase in

social welfare and in the case of producer surplus adverse situation

will be find.

6.7 MARKET FAILURE:

Market failure is the concept related to Pareto Optimality

criteria or perfect competition. In the condition of perfect

competition, eq uilibrium of market is Pareto optimum. It means ,

there is no further improvement possible or it means market is

successful to attain the Pareto optimum equilibrium. So, this is the

case of market success. Then question is that what is market

failure? Befor e understanding the concept of market failure, there is

need to study the function of market.

Function of market:

The main function of market is that price determination of

commodity in the market where two factors of market determined it

and these are supply of commodity and demand of a commodity.

According to the condition of supply and demand for a commodity,

price of a commodity will determine.munotes.in

## Page 109

109Therefore, if market is working very well at the level of

Pareto to decide price of commodity in the market , it is called

market success and if market is unable to decide prices of

commodities, it is called market failure. It happens due to various

reasons like as externalities, public good, imperfect competition,

asymmetric information etc. In the case of exte rnalities there is very

difficult to decide the prices, because demand is no explicitly given.

So, one factor of the market is partially missing and price of

commodity will not be decided at Pareto level. Samething happens

with public goods where people wa nts various commodity and

services from the government, but they are directly not ready to pay

for it. It means that wants ar e not converting in the demand, that is

why it is known as demand is missing in market. So, if demand is

missing in the market, pri ces of commodities will not decide in the

market. It is the concept of market failure.

6.8THEORY OF THE SECOND BEST:

The Pareto criteria and marginal conditions of Pareto

optimality lead to maximum social welfare or economic efficiency. If

the condition s of Pareto optimality fulfilled, maximum social welfare

will achieve, but if the Pareto optimality not achieved, then what is

the solution for it? Or what is second best solution ?

Prof. Lipsey and Lancaster raised the same question and

developed the the ory of second best.According to them, if the

condition of Pareto efficiency not possible to achieve means

maximum social welfare or maximum social advantage is

unattainable, whether or not efforts should be made to achieve the

second best position by satis fying the remaining marginal

conditions of Pareto optimum.

In the theory of second best, they assert that the theory of

second best or second best solution will not to lead in increase the

social welfare. According to second best theory, social welfare wi ll

not be increased, if any condition of Pareto optimality is not fulfilled.

That is theory of second best solution which is not desirable.

6.9 SUMMARY:

In this unit we have studied theexistence and stability of

equilibrium, thefirst and second theorem of welfare economics, the

concept of market failure, the functions of market, consumer’ s

surplus (at initial level, when price increases and prices decreases ),

the theory of second best etc.munotes.in

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1106.10 QUESTIONS:

Q1. Explain the first & second fundamental theorems of welfare

economics .

Q2. Write notes on following.

Functions of market

The theory of Second Best

Q3. Explain the concept of consumer surplus with the help of

diagram.

Q4. Explain the existence and stability of equilibrium.

6.11 REFERENCES:

Gravelle H. and Rees R. (2004) : Microeconomics, 3rdEdition,

Pearson Edition Ltd, New Delhi.

Varian H (2000) : Intermediate Microeconomics : A Modern

Approach, 8thEdition, W. W. Norton and Company.

Ahuja H. L. (2018), Advanced Microeconomics, 21stEdition, S.

Chand Publication.

Koutsoyiannis A. (1979), Modern Microeconomics, Second

Edition, St. Martin's Press, Inc.

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111Unit-7

MONOPOLY

Unit Structure :

7.0 Objectives

7.1 Introduction

7.2 Market Power and Monopoly Market

7.3 Social Cost of Monopoly Power

7.4 Measurement of Monopoly Power

7.5 Monopoly and Back ward Integration

7.6 Question

7.0OBJECTIVES

Market power may exist in both the buyer’s and seller’s market.

Major objective of this unit is to understand the concept of

market power.

We will also analyse the benefits and social cost of monopoly

power in the seller’s market.

Monopsony or the market power in the buyer’s market will also

be analysed in this unit.

7.1 INTRODUCTION

The neo -classical economics criticized monopoly market and

monopoly power on the ground that misutilises the resources and

leads to inefficiencies in the market. As a result, optimization of

social welfare is not possible. The neo -classical ideology regarding

monopoly power is reflected in many anti -monopolistic legislations

passed by the governments of the countries, like UK, USA etc. The

major objective of such a nti-monopoly laws is to identify the

presence of monopoly power, and then regulate or eliminate any

such monopoly power. The problem in this respect, is that the

monopoly power is not easily identifiable. There is an absence of

unanimity on the factors tha tl e a dt om o n o p o l yp o w e ra n dh e n c ei t

is difficult to quantify monopoly power so that an appropriate action

can be taken to regulate it.

In this unit, we will try to understand the factors that lead to

monopoly power. We will also understand welfare impac to f

monopoly power. We will also discuss how the monopoly powermunotes.in

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112can be measured to enable the government authorities to take

appropriate policy measures.

7.2 MARKET POWER AND MONOPOLY MARKET

7.2.1 Meaning of Monoploy Power :

According to neo -classical e conomic thoughts, monopoly

power of a seller is determined by two factors.

1.The degree of freedom the seller has in deciding his price.

2.The extent to which price exceeds marginal cost or the seller

enjoys long -run abnormal profit.

Since the seller under perfect competition does not have

freedom either to determine price or to charge price higher than

marginal cost, there is no monopoly power present under such

market. Greater or lower degrees of monopoly power may be

present with monopoly or oligop oly markets. Under the monopoly

market, even if there is a single seller, (who does not have any

competitor at the moment) the monopolist may not be able to enjoy

excess profit out of the fear of potential new competitors. Hence,

existence of monopoly with single seller or oligopoly with a few

sellers does not necessarily imply an existence of monopoly power.

7.2.2 Benefits of Monopoly Power

It is important, at this stage, to understand whether the cost

conditions are likely to remain the same when a numbe ro ff i r m s

are combined to become a monopoly. In other words, it is

necessary to analyse whether a monopoly firm (which produces on

a large scale) faces similar cost conditions as compared to a

number of competitive firms producing the same product (on a

smaller scale). Two views need to be considered in this case: -

1. The monopolist can enjoy various economies of scale such as

grater specialization, larger markets, cheaper finance, buying raw

materials in bulk, spending more money on research and

developm ent, applying modern techniques of production and

management, etc. All these will result in the fall in cost of

production. If these economics of scale (leading to fall in cost

production) are large enough leading to substantial fall in cost, the

monopoly price may be smaller than that under perfect competition.

2. The monopolists can charge different price, different buyers, as

he is the sole producer in the market. In other words, there is a

possibility of price discrimination under monopoly and not so under

perfect competition. The monopolist can maximize his profit by

charging higher price from the market where his product faces

inelastic demand and less price from the market where the demand

for his product is highly elastic. The possibility of chargin gd i f f e r e n tmunotes.in

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113price also may promote social welfare as explained in the unit

“Price Discrimination”.

7.3 SOCIAL COST OF MONOPOLY POWER

(WELFARE EFFECTS OF MONOPOLY POWER)

7.3.1 MONOPOLIES WITH THE COSTS HIGHER THAN

COMPETITIVE MARKET.

As you are aware, un der monopoly, the consumers will have

to pay the price which is higher than the marginal cost and the

monopolist enjoys supernormal profit at the expense of consumers.

This result in two things.

a.Consumers face welfare loss as their consumer surplus

declines. (because they have to pay higher price)

b.Producer gains (as he enjoys super normal profit)

In order to understand the welfare loss or gain under

monopoly, the concept of consumer surplus can be sued. (It may

be recalled that the consumer surplu s is the difference between

price the consumer is willing to pay and the price which he actually

pays). Following diagram explains welfare gains or losses arising

out of existence of monopoly power (When the costs under

monopoly are higher than the costs u nder perfect competition).

Figure 7.1Welfare gains / losses from monopoly power

In the diagram 7.1

MD–demand curve for the product.

MR–Marginal Revenue Curve

AC–Average Cost

MC–Marginal Cost

Pc–Price under perfect competition. (The students may recall that

the price under perfect competition is equal to marginal cost.

QCc -Output under perfect competitionmunotes.in

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114MGPc -Consumer surplus (which is equal to the area under

demand curve a006Ed above the price) under perfect competition.

The consumer su rplus perfect competition. The consumer

surplus arises because the consumers are willing to pay higher

price (maximum OM in the case) but they are actually paying less

(OPc)

PM-Price under monopoly.

OP M–Output produced by the monopolist (The monopolis tequates

marginal revenue and marginal cost to determine equilibrium price

&q u a n t i t y . )

It may be noted that for the monopoly market, the equilibrium

price is higher and the equilibrium quantity is lower than that under

perfect competition.

MKP M–Consu mer surplus under monopoly (which is less than that

under perfect competition)

PMKEP C(Area 2) –excess profit earned by the monopolist.

At this point it may be noted that there is a fall in consumer

surplus and a redistribution of income from consumers to producers

(in the form of excess profit which was not so under the perfect

competition).

PMKEP C–Gains to the producer (Area 2)

An important point to be noted here is that the loss to the

consumer in the form of reduction in consumer surplus is mo re than

the gains to the producers in the form of excess profit. Only a part

of the loss to the consumers, is redistributed to the producer. Out of

total loss to the consumer (Area 2 + 3), only a part (area 2) is the

gain to the monopolist. The rest, shown by triangle KGEC (area 3)

is called as the dead –weight loss. It arises due to inefficiency of

resource allocation under monopoly. It is considered as the social

cost of monopoly. In short, monopoly leads to misallocation of

resources and hence there is a social cost of monopoly in the form

of dead weight loss. The extent of welfare reduction depends upon

the price -elasticity of demand for the product and the difference

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115Table 7.1 summaries the welfare impl ications of monopoly power.

The analysis in this section deals with the situation where

the costs under monopoly are higher than those under the prefect

competition. But this may not always be the case. Next section

deals with the situation where the mo nopolist produces at the cost

lower than that under perfect competition.

7.3.2 MONOPOLIES WITH COST LOWER THAN THE

COMPETITIVE MARKETS

Sometimes, the monopoly firms may be able to produce at

the cost lower than that under competitive market. This may be s o

because of the economic of scale that may be enjoyed by the big

size monopoly firm or due to an easier access to superior

technology as compared to the competitive firm, etc. Under such a

situation, it is possible that the welfare gains associated with m ore

efficiency (Production at a lower cost) may compensate for the

dead -weight loss as shown in the following figure.

Figure 7.2 :

Gains and losses for a monopoly firm with lower costs.

Figure 7.2 depicts a situation where costs are lower (MCM)

under monopoly as compared to the costs under perfect

competition (MCPC). Price under perfect competition (OPc) is lower

than that under monopoly (OPM). it implies that the monopoly firms

enjoy super -normal profits. As explained in the earlier diagrammunotes.in

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116(7.1), mon opoly market faces a deadweight loss equal to the

shaded triangle in the figure. The monopolist makes abnormal

profits, but in this case, the profits are due to lower costs than the

higher price. The cost reduction arised out of various factors

mentioned e arlier, economies on the use of resources which can be

allocated to some other lines of production. These production gains

shown by the shaded area, more than compensate the dead -weight

loss and hence lead to overall improvement in welfare. Thus, in

spite of existence of monopoly power, in spite of existence of

monopoly power, in spite of market concentration, the welfare

improvement will take place. To conclude, the monopolies may or

may not reduce welfare. It would depend on whether and to the

extent to w hich their costs are higher or lower than that in the

competitive industry.

Check Your Progress :

1.Define following terms

a)Monopoly power

b)Economies of scale

c)Price discrimination

d)Social cost of monopoly power

e)Consumer surplus

f)Dead -weight loss

7.4 MEASUREMENT OF MONOPOLY POWER

The degree of monopoly power is measured by taking

perfect competition as a base, professor A. P. Learner has

regarded perfect competition as the market providing socially

optimum (maximum) welfare. Any deviation from perfect

competition implies an existence of monopoly power, according to

him.

Under perfect competition, price is equal to marginal cost at

the equilibrium level. The level of output associated with equilibrium

price implies o ptimum allocation of resources. When the degree of

competition is less than perfect, i.e. under the imperfect market, the

demand curve is downward sloping and price is not equal to

marginal cost. The divergence between price and marginal cost is

an indicat or of the existence of monopoly power, according to Prof.

Lerner. Greater. The divergence between the price and marginalmunotes.in

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117cost, higher is the monopoly power enjoyed by the seller,

symbolically.

Degree of monopolyPM CP

Where P –is eq uilibrium price.

MC-Marginal cost at the equilibrium level of output.

Under perfect competition, difference between marginal cost

and price is zero so

Degree of monopolyPM C OOPP There is an absence of monopoly power under perfect

competition. Greater the value of the indexPM CP,t h eg r e a t e r

isthe degree of monopoly power possessed by the seller.

Lerner’s Measure of monopoly power is criticized on the

following grounds -

1. This measure is not useful in the mar ket where there is non -price

competition or product differentiation. Such as under the

monopolistic competition. In other words, when the products

compete with each other, not in terms of price, but in terms of

product variation, advertising, or any other sales promotion

practices, the above -mentioned formula can not be used to

measure the degree of monopoly power.

2. Another important point of criticism against Lerner’s measure of

monopoly power is that, this measure is based on only one aspect

of monopol y and that is the control over prices. The degree of

control over prices depends on the availability of existing

substitutes. But the monopoly power may also be threatened by

potential substitute which is not considered by this measure.

7.5 MONOPSONY AND BACKWARD INTEGRATION

The term monopsony is opposite of the term monopoly.

Whereas, monopoly refers to a condition or activity in the seller’s

market, monopsony is a seller’s market, monopsony is a condition

or activity in the buyer’s market. In the recent times, the issues

arising out of monopsony are gaining prominence in the developed

markets the buying power of the supermarkets and other retail

chains has been increasing. Hence it is necessary to examine

consequences of growing monopsony power through c onsolidation

mergers and the buyer’s groups. The monopsony may also bring

about wholesale price changer.munotes.in

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118According to K. Lancater, monopsony is the economic term,

used to describe a market involving a buyer with sufficient market

power to exclude competito rs and affect the price paid for its

products. Monopsony in the buyer’s market is the counter part of

monopoly in the seller’s market. Monopsony will generally exist

when there is a corresponsing monopoly in the seller’s market

since. All the firms in the market generally need to purchase similar

products. Thus, if monopoly is held in the output market, the

monopolist will generally hold maonopsony power in the input

market. Vertical integration normally involves a producer’s

integration into next level of production. That means, a producer

may himself, take over distribution of his product. This is forward

integration. Backward integration on the other hand, occurs when

the producer seeks to integrate into his supply market that means,

the producer may hims elf, take over supply of inputs for his firm. A

firm generally uses monopoly power to have forward linkages and

monopsony power to have the backward linkages.

7.6 QUESTIONS

1.Explain in detail the social cost incurred due to existence of

monopoly power.

2.What are the benefits of monopoly power?

3.Explain the concept of dead -weight loss under monopoly with

the help of a diagram.

4.How is the market power measured?

5.What is monopsony? Why does it come into existence?

6.Explain the concept of backward and forward link ages and

existence of market power.

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119Unit-8

PRICE DISCRIMINATION UNDER

MONOPOLY

Unit Structure :

8.0Objectives

8.1Introduction

8.2Price and Outp ut Determination under Monopoly

8.3Price Discrimination

8.4Third Degree Price Discrimi nation : Market Segmentation

8.5First degree discrimination

8.6Second degree Price discrimination

8.7Monopsony

8.8The effect of monophony and output monopoly on the input

market.

8.9Unions as monopoly input suppliers

8.10Bilateral Monopoly

8.11Questions

8.0OBJECTIVES

After going through this unit you will be able to explain the

concepts of -

Monopoly .

Price Discrimination.

Monopsony.

Unions as monopoly input suppliers.

Bilateral Monopoly.

First, Second and Third degree Price discrimination.

Price and output determination under monopoly, price

discriminating monopoly and monop sony.

8.1INTRODUCTION

This unit is designed to explain the concept of monopoly, price

discriminating monopoly, monopsony and bilateral monopoly. This

unit especially deals with the most important decision of price andmunotes.in

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120output determination under monopoly, monopsony and bilateral

monopoly.

8.2PRICE AND OUTPUT DETERMINATION UNDER

MONOPOLY

The monopoly firm is assumed to maximize profit in a stable,

known environment, with given technology and market conditions

We assume diminishing marginal productivity and so, in the

presence of fixed inputs, the firms average and marginal costs will

at some point begin to rise with the rate of output per unit time.

However, we no lo nger assume that diminishing returns to scale set

in at some point : We leave the question open, and permit any one

of increasing constant, or diminishing return to scale to exist over

the range of outputs we are concerned with. The essential

difference fr om the competitive model is the assumption that the

firm faces a downward sloping demand curve. We curie its demand

function in the inverse form./ p D q dp dq[8.1]

where p is price, q is output per unit time and D is the demand

functi on. We do not place restrictions on the second derivative the

function, but do require its first derivative to be negative.

The firms total cost function is0cc q c q[8.2]

where c is total cost per unit time. Maxginal cost is a lways positive,

but we do not place restriction on the second derivative, the slope

of the marginal cost curve. The profit function of the firmq pq c q[8.3]

where TI is profit per unit time we assume that the profit maximizing

output*qis positive Hence*qsatisfies the conditions.11/0q p qdp dq c q[8.4]11 2 2 112/ / 0q dp dq qd p dq c q[8.5]

where [ 8.4] is the first order and [ 8.5] the second order condition.

The term/p qdp dqis the derivative of total revenue pq with

respect to q (taking account 01 [ 8.1], and is marginal revencue.

Thus, [ 8.4] expresses the condition of equality of marginal cost with

marginal revenue. The term222/ /dp dq qd p dqis the derivativemunotes.in

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121of marginal revenue with respect to output and so [ 8.5] is the

condition that the slope of the marginal cost curve must exceed that

of the marginal revenue urve at the optimal point. If marginal costs

are increasing with output whi le, by assumption, marginal revenue

is diminishing with output [7.5] will necessarily be satisfied since in

that case11 2 202 / /c q dp dq qd p dq[8.6]

However, unlike the competitive case, the second -order

condition may also be satisfied if110cq.

More insight into this solution can be gained if we write

marginal revenues, MR, as1/ /MR p q p dp dq[8.7]

Given the definition of the elasticity of demand// 0e p dq dp q[8.8]

We can write a s the relationship between demand elasticity

and marginal revenue:(1 1 / )MR P e[8.9]

Clearly,10eM Rwhile10 ,e Mrand10eM Rcombining [ 8.9] with [ 8.4], we can write the

condition for optimal output as111 /pe c q M C [8.10]

This equation then establishes immediately the two

proposition:

(a) the monopolist’s chosen price always exceeds marginal cost

since its price elasticity is finite;

(b) optimal output is always at a point on the demand curve at

which e < -1( g i v e nt h a t10)cqUnder competition each firm equates marginal cost to price.

Hence the extent of the divergence of price from marginal cost

under monopoly is often reg arded as a measure of the degree of

monopoly power enjoyed by the seller. From [ 8.10]munotes.in

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12211PM CePe[8.11]

The left -hand side, the price marginal cost difference

expressed as a proportion of the price, is the Lerner index of

monopoly power. Thus, ase(the competitive case)

monopoly power tends to zero.

The equilibrium position of the firm implied by its choice of

output*qsatisfying the above conditions is illustrated in fig [ 8.1].I n

(a) of the figure, the demand curve is D(9) and the corresponding

marginal revenue curve is MR. Given the marginal and average

cost curves1cqand AC, profit maximizing output is at*q. Since

this must be sold at a market clearing price. choice of*qrequires

the prices**pD q*1p

Fig8.1

We could therefore regard the equilibrium position as a

choice either of profit maximizing p rice*por of output*q, since

each implies the other. At output*q, profit is the difference between

total revenue**pqand total cost AC*qand is shown by the areas*pabove in Fig [7.1] (a). In (b) of the figure, the some equilibrium

position is shown in terms of total revenue curve is denoted pa, and

its slope at any point measures marginal revenue at that output. Its

concave shape reflects the assumption of diminishing marginal

revenue. The total cost curve is denoted c (q), and its convex

shape reflect the assumption of increasing marginal cost. The total

profit function is the vertical difference between these two curves,

and is shown as the curve TI (q) in the figure. The maximum of this

curve occur at the output*q, which is also the point at which themunotes.in

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123tangents to the total revenue and total costs curves respectively are

parallel , i.e marginal revenue is equal to marginal cast.

The supernormal profit, i.e. profit in excess of all opportunity

costs (including a market determined rate of return on capital which

enters into determination of the average and marginal cost avrves),

isg i v e nb yt h ea r e a**qp c. It can be imputed as a rent to

whatever property right confers the monopoly power and prevents

the new entry which would compete the profits away. It may be that

this right is owned by on individual who leas es it to the firm. If the

supplied is rational and well informed, she will bid up the price of

the base so as just to absorb the supernormal profit, and so the

rent is transformed into an opportunity cost of the monopolist. This

would be true, for example, if the monopolist rented a party clearly

Favorable location. It the monopolist owns the property rights, then

he can impute the profits as the return on this property right. Note

that the identity of the owners of the right does not affect the price

and o utput which will be set by the monopolist. (Since this is

determined by the desire to maximize profit) but simply determines

the division of the spoils note also that the term ‘property right’ is

used here in its widest possible sense: it is meant to indud et h e

ownership not only of land but also of such things as brand names,

public reputations, manual rights, franchises and patents

8.3PRICE DISCRIMINATION

Price discrimination exists when different buyers are charged

different prices for the same good. It is a practice which could not

prevail in a competitive market because of arbitrage. Those buying

at lower prices would resell to those offered higher prices and so a

seller would not gain from discrimination. Its presence therefore

suggests imperfectio n of competition. The producers charge

different prices for the same product, from different units of the

same product at different prices is called price discrimination. By

doing this, the producer tries to capture more & more consumer

surplus, from the b uyer to maximize his profit. Price discriminating

is followed by grouping the consumers. In some cases, forming

groups of consumers is easier as the heterogeneity (differences) of

consumers is directly observable. In some cases, however the

differences amo ng consumers are not visible. In such cases, the

producers have to offer different menus or packages of products at

different prices and allow consumers to choose from among

alternative choice. Thus, price discrimination refers to the act of

manufacturer o f selling the same product at different prices to

different buyers.munotes.in

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1248.3.1 Examples of Price Discrimination:

Monopolist firm many times charges different prices from different

consumers, for the same product and without much cost

differentials. Following examples would clarify the point.

1) A doctor or a lawyer may charge different fees from different

patients / clients for the similar services.

2) A producer may charge different price for the same product at

different parts of a country.

3) Same product / services may be sold to the same buyer at

different price for varying quantity. For example price for 10 kg of

rice may be higher than the price for 100 kg of rice of the same

quality.

4) Consumers may be classified into different categories and by

chan ging the quality of services, different rates may be charged.

For example first class and second class fares in the train, ordinary

or business class in the plane.

All these examples make it clear that price discrimination is quite

common while aiming at maximization of profit. Prof. Stigler’s

definition of price discrimination brings about one more aspect of

the concept. According to Prof. Stigler “Price discrimination is

defined as the sale of technically similar products at prices which

are not proporti onal to marginal costs.”

The concept of price discrimination as indicated in Prof. Stigler’s

definition may be well understood with the help of following

example.

Suppose a hard -bound and color edition of Microeconomics book

by Mankiv costs Rs.500 and a soft-bound block and white edition of

the same book costs Rs.400 for the publisher. Also suppose that he

sells the colored edition for Rs.750 and the black and white edition

for Rs.500. in this example, the manufacturer is said to be

practicing price discr imination as the price differences between two

types of books (750 -500) are more than the cost differences (500 -

400).

8.3.2 POSSIBILITY OF PRICE DISCRIMINATION:

A monopolist can follow price discrimination only under two

fundamental conditions.

1) There should be no possibility of transforming any unit of product

from one market to the other transferability of commodities.

2) There should be no possibility of buyers transferring themselves

from the expensive market to the cheaper market transferability of

demand. It is understand from the above points that the monopolist

can practice price discrimination only if the units of goods or the

units of demand (i.e. the buyers) cannot be transferred from onemunotes.in

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125market to another. It is possible to discriminate among the buyers

only if two types of arbitrage or transaction costs are present.

These are as follows.

1) Arbitrage associated with the transferability of a commodity.

If it is possible to transfer a commodity from one person to another

with very less transac tion cost, price discrimination is not possible.

In other words, price discrimination becomes possible only when it

is costlier to resell a product to another consumer may buy goods

to resell them to the high price consumer. A low price consumer is

the one who gets a commodity at a lower price due to quantity

discounts. A high price consumer the one who does such

discounts, in case the transaction cost is low, the former (low -price)

consumer will buy in bulk and sell it to the latter (high -price)

consumer. This does not allow the discriminating monopolist to

charge different prices from different consumers. Thus, it is

possible to undertake price discrimination in case of services which

have a very low or no transferability. For example, doctor can

charge di scriminating prices to one patient to another. Otherwise, in

case of most of the retail products, price discrimination may be

difficult. Wherever it is difficult, the monopolist may practice partial

discrimination. He may sell his products to retailer at l ower price

and ensure that his product is sold to the final consumer.

2) Arbitrage associated with the transferability of demand.

In case of such arbitrage, the products physically may not be

transferable between the consumers, but the demand for product is

transferred between different packages. For example, the

consumer may be charged different prices based on price -quantity

package or price -quality package makes. The price -quantity

package makes consumer choose between say buying two units of

a product at certain price or buying one unit of a product at some

price. A shirt may cost Rs.250 but if consumer buys two shirts he

may be charged Rs.450 (instead of Rs.500). The price -quality

package discriminates between the consumers on the basis of

quality of a product / service. First class and second class on the

train, ordinary or deluxe room accommodation in a hotel are some

of the examples of price discrimination which help the monopolist to

maximize his profit by charging different price from different

consumer.

The two types of arbitrage discussed above are different in terms of

their impact on price discrimination. As stated earlier, if there is a

possibility of transferring products from one consumer to another,

without much arbitrage or transaction cos t, one consumer may buy

more goods and resell them to the others. Monopolist will not be

able to gain from price discrimination in such a case. The

transferability of demand, on the other hand, includes the

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126different prices from them. In the sections to follow we will try to

analyze the welfare effects of price discrimination.

8.3.3Concluding Remarks:

From the discussion about the possibility and practicality of price

discrimination, it is clear th at there should not be any seepage or

communication between two markets. Thus, price discrimination

depends upon the ability of the monopolist to keep two markets

quite separate. To conclude, price discrimination is possible under

following circumstances: -

1) The nature of product sold is such that there is no possibility of

transferring product / service from one market to another.

2) The geographical distance between two markets is very large or

the markets are separated by the tariff barriers.

3) Legal section is given to charge different prices from different

consumers like electricity for domestic use and for industrial use.

4) Consumer snobbish attitude that higher priced goals are superior

to lower priced ones.

5) Monopolistic or oligopolistic market structure.

Check Your Progress:

1) Define the concept price discrimination

2) State the conditions to be followed by a monopoly for price

discrimination.

3) Explain the concept of arbitrage and state how it is important

factor determining price discrimin ation.

4) Find out more examples of price quantity and price quality

packages offered by the monopolist to discriminate against different

consumers.

5) What is the difference between arbitrage associated with

commodity transfer and the arbitrage associated with demand

transfer?

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8.4FIRST DEGREE DISCRIMINATION

Under third -degree price discrimination the monopolist had

some information he could use to partition buyers into sub markets

and prevent arbitrage between the sub -markets. This, as the name

suggests, is in contrast to.munotes.in

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127(a) first degree price discriminati on, where the monopolist is able to

identify the demand of each individual buyer and prevent arbitrage

among all buyers;

(b) Second -degree price discrimination, where the monopolist

knows the demand characteristics of buyers in genera l,b u td o e s

not know which buyer has which characteristics.

In first degree price discrimination the monopolist can extract

all the consumer surplus of each buyer. An interesting aspect of

this case is that total output of the good is at the level at which each

buyers pays a price equal to marginal cost and monopoly does not

distort the allocation of resources. We have a pareto efficient

outcome, although the monopolist expropriates all the gains from

trade. Any objection to monopoly in this case therefore would have

to be o n grounds of equity fairness of the income distribution rather

than efficiency.

In the second case, the obstacle to price discrimination is that,

if one type of buyer is offered a more fausurable price. quantity

deal than other types, and the monopoly is not able to identity a

buyer’s type, then all buyers will take the most fauourable deal. The

solution for the monopolist is to offer alternative deals which satisfy

a self -selection constraint: a given deal will be preferred to all other

by, and only by , the type for which it is designed.

In the rest of this section we explore first -and second -degree

price discrimination with a simple model. We assume:

(a) two types of buyer in the market, with1nbuyers of the first type

and2nbuyers of the second.

(b) a buyers type is determined by her preferences which for each

type of buyer can be represented by the quasi linear form.1, 2ii i iux y i[8.14]

whereixisthe monopolized good andiyis a composite

commodity representing all other goods;

(c) type 2 buyers have a stronger preference for the good in the

sense that for any x21 1 1

22 2 1 1 1// 0xy x y x y xy MRS u x MRS [8.15]0ioand110:ixbuyers have diminishing marginal utility;munotes.in

## Page 128

128(e) the buyers have identical incomes M, and the price of the

composite commodity is the some for all consumers and is set at

unity. So if12xxo,t h e n12;yyMRecall, that a quasi -linear utility function implies that a

consumers indifference curves in the x, y plane are vertically

parcallel, and there is a zero income effect for good x. The

consumer’s choice problem is

11max

ii i i ix y St px y M Fxy [8.16]

p is the price the monopolist charges, and0Fis fixed

charge that the monopolist may set for the right to buy the good at

price p (examples of such fixed charges are telephone rentals,

entrance charges to a musement parks, subscription fees to a book

or wine club).

First-order conditions include.

10ip[8.17]10[8.18]

Hence1,ixp pyielding demand functions11

ii ixp x p[8.19]iiyMF P x p[8.20]

The indirect utility function is,ii i ivP F xp M F P xP[8.21]

of particular interest are the derivatives

11 1,1ii

ii i i ivVx x Px xPF [8.22]

where the result for/ivpis simply Roy’s identity. In fig u.3, we

show the reservation indifference curvesiufor each of the two

typesof consumers. Since they have the same income M, they are

at the same point when consuming no x, but assumpti on (c) implies

that a type 2 indifference curve is steeper than that of a type 1 at

every x (since1/xy i i iMRS dy dx. The budget line market c in

the figure corresponds to p=c, so thatcixare the respective

consumer’s.munotes.in

## Page 129

129

Fig8.2

demands at that price. In (b) of the figure we show the demand

curves derived from these reservation indifference were. Because

of the quasi -linearity assumption, these are both Hicksian and

Marshallian demand curves, and the area under each betw een

pricesoipand p = c gives the type’s compensating variation, or

maximum uillingness to pay for the right to buy x at price C. These

consumer surpluses are denoted byis, and correspond to the

distances on the y axis shown in (a) of the figure.

We now show that under first degree price discrimination the

monopalist’s optimal policy is to set a price for each type equal to c,

and to set a fixed charge1, 2 .iiFS iThe monopolist sells at

marginal cost and sets separate fixed charges equal to the total

willingness of each type to pay. This requires first that he knows the

type of each buyers, and so can prevent a type 2 buyer taking

advantage of the lower type 1 fixed charge second he must be able

to prevent arbitrage and stop a type 1 buyer reselling to a type 2munotes.in

## Page 130

130buyer at some price between c and22/cFx c, which is the average

price per unit a type 2 buyer pays in this solution.

The idea underlying this policy can be se en in fig 7.3(b). If the

monopolist sets p=c to both types and extracts the total surplus his

profit is12ss. If he sets a higher price, say1PC,a l t h o u g hh e

makes a profit on each unit he sells, the sum of th ese profits and

the remaining consumer surpluses is less than12ssby the sum of

the two shaded triangles. It pays him to expand output and lower

price as long as p > c because his own profit increases precisely by

the difference p -c, which he can recover through the fixed charge.

He will not set a price such as11pc, because the extra surplus he

can recover falls short of the extra cost he incures. And clearly it

would never be worth while to set o fixed char ge1iFSfor only p,

because then he sells nothing to type i.

We can derive this result more formally. The monopolist’s total

profit is11 1 1 1 2 2 2 2 2 1 11 2 22np xp F n p xp F C n xp n xp [8.2

3]

He must not offer a deal which is curse for each consumer

than not buying the good at all. We can express this by the

reservation constraints,1 , 2iii iVP F u i [8.24]

where, recall,iuis the utility is obtains by buying non of good x.

withiasthe lagrange multiplier on these constraints, optimaliip and Fare defined by (see Appendix H)11/0 1 , 2ii i i i i i inx p x x v p i[8.25]/1 , 2ii inv F i o i[8.26]

,, iiii i i i ivp F u v u [8.27]

From [ 8.26] we see that n on-zeroinand/iivFimplyiand so [ 8.27] impliesiivu. Both types of consumers receive only

their reservation utilities. Then using [ 8.22] and [ 8.26] we haveiinandmunotes.in

## Page 131

13111() 0ii i i i i inC x Px x nx[8.28]

implying

ipc[8.29]

The value of1Fthen satisfies,iiivc F uand so must be

equal to consumer surplusisat price c.

We could interpret third degree price discrimination ( analyses

in the first part of this section) as the case in which the monopolist

canidentify each buyer’s type and prevent arbitrage between types,

but for some rea son cannot set fixed charges. He must set a

constant price per unit to all buyers of a given type. Then, profit

maximization implies a price to each type which is above marginal

cost; as we saw earlier. clearly, the monopolist’s profits are leuer

than unde r first degree price discrimination. Buyers are better off

under third degree price discrimination since, although they face a

higher price and so consumer less, they retain some consumer

surplus and are on an indifference curve that must be higher than

there reservation indifference curve. (use fig 8.3.)

8.5SECOND DEGREE PRICE DISCRIMINATION

In the case of second degree price discrimination, the

monopolist is unable to determine the type of the buyer before she

has purchased the good. In that case if he offered only buyer the

option of either12,,C S or C Severy type 2 buyer (a) will as every

type 1 buyer) would chose1,CScan the monopolist do better

than this by offering options chosen so that only o buyer of type I

would want to choose the option designed for her type? In other

words, can the monopolist do better by inducing buyers to reveal

their type by self -selecting the appropriate deal?

Assume that the monopolist knows the number of buyers of

each type,inand can specify in a contract both the quantity of

output he will supply to a buyer and the total charge for that output.

That is, a contract is a pair,iixF. This implies a price per unit/ii ipF xand the contract could be equivalently expressed as

some combination of a fixed charge and constant price per unit, as

in a two -part tariff. The point is that the consumer is offered a

quantity and a fixed charge and not a price and a fixed char ge. We

shall set the reason for this at the end of the following analysis.

The monopolist’s profit is

2

1ii iinF x [8.30]munotes.in

## Page 132

132We again have the reservation constraint, since buyers always

have the option of refusing a contract. These a re now written in

terms of direct utilities, to reflect the fact that quantities are being

specified.iii ixM F u [8.31]

where we use the fact thatiiyMFThere are also self -

selection constraint which ensure th at each type chooses the

appropriate deal we write these as11 1 1 2 2xF xF [8.32]22 2 21xF x F [8.33]

(M cancels out in these expressions)

If,iixFsatisfies these constraints, it will only b e accepted by

type i. (We assume to be able to have a closed feasible set, that if a

buyer is indifferent between the two deals she takes the one

appropriate to her type.)

In principle we now solve for,iixFby maximizingsubject to

[8.31] -[8.33]. However the first order condition for this would not

be instructive. Instead, we first show that, in any optimal solution,

(a) the reservation constraint for a type 2 buyer, a nd (b) the self -

selection constrain tf o rat y p e1b u y e ra r en o n -binding. They can be

dropped from the problem thus simplifying the derivation of the

optimal contract.

We show this in Fig ( 8.4), which reproduces the reservation

indifference curves from Fig ( 8.3) (a)

Fig8.3

(a) Type 2 buyers must be offered21 2xFsuch that22uu. To see

that, note that type 1 buyers must be offered a contract11,xFthat

puts them on or above1u. But since1ulies above2u, such a dealmunotes.in

## Page 133

133must always be better for type 2 bayers than any contract22xFthat puts them on2u.S oo n l yap o i n ta b o v e2ucan satisfy their.

Self-selection constraint.

(b) Type 1 buyers will always strictly prefer their deal to that offered

to type 2 quyers, in an optimal solution. Suppose the optimal deal

offered to type 1 buyers is at a in fig 8.3(it is not relevant to th e

present argument that a is on1u, but me show below that this must

be so). Then the deal offered to type 2 buyers must lie on the type

2 indifference curve passing through a, labeled*2u.I fi tw e r eb e l o w

this, type 2 buyers would prefer a; if above, the monopolist is being

need generous to type 2 buyers because, at any given2x, he could

increase2Fmove verticallur derunuared in the figure) without

violati ng either the reservation or self -selection constraints. (This

incidentally established that the self -selection constraint for type 2

buyers is strictly binding, as we verify later.) Now if the deal offered

to type 2 buyers were on*2uat a point to the left of a, it would be

preferred to a by type 1 buyers and this violates the self -selection

constraint on type 1. It is easy to show that point a it self could not

be offered to both types of buyers in equilibrium.

This laves only p oints on*2uto the right of a as possible deals

to be offered to type 2 buyers, and since these must be strictly

below1uthe type. 1 self selection constraint is non -binding. This

argument also establishes that at an optimum21xx.

As a result of these arguments, the monopolist’s problem is to

find11 2 2,,,xF xFto maximizein 7.30 subject only to 7.31 unit

I-1, and 7.33 using1and2ufor the Lagrange multipliers on 7.31

and 7.33, the first order conditions are,**

11 1 1 2 2 10nc x x [8.34]1*

22 2 20nC x [8.35]

11 20n[8.36]

220n[8.37]** *

11 1 1 1 1 1 10, 0, 0xM F p M F [8.38]** ** * *

22 2 2 1 1 2 2 2 2 2 10, 0, 0xF xF F F [8.39]munotes.in

## Page 134

134From [ 8.37] and [ 8.38] we see that the type 2 self -selection

constraint must bind, and from [ 8.36] and [ 8.38] that the type 1

reservation constraint must bind. Substituting for2in [8.35] gives.1*

22xC [8.40]

implying*22,cxxot h a tt y p e2 consummation is exactly that

under first degree price discrimination. Then, substituti ng for12andin [8.34] gives

1* 1 *1211 2112 1 2nc nxxnn nn [8.41]

Recall that we established in fig 8.4 that we must have**21xx,

so that*1 *

21 22xx c , given diminishing marginal utility. Thus,

writing1*

21xc where0we have

* 2

1112i nxcnn [8.42]

implying that*

11,cxxso that type buyers consume less than under

first degree price discrimination. the optimal va lues*1Fand*2Fthen

follow from soluing the constraints as equalities with the optimal*ixinserted. We know that*1Fwill have type 1 buyers with their

reser vation utilities, while*2Fis such that type 2 buyer retain some

consumer surplus. It follows that, compared with first degree price

discrimination, type 1 buyers are neither better nor curse off, type 2

buyers are better off, and the monopoly makes less profit.

The optimal second -degree price discrimination equilibrium is

illustrated in fig 8.4. The contracts are**11,xFand*22,cxFThe

two most interesting aspects of the solution are firs tt h a t*11,cxxand second that*22cxx. These can be rationalized as follows. At

any1x, the total net surplus can be expropriated from type 1 buyers

since they can be held to their reservati on constraint. Suppose1xwhere set at1cx. The contract for type 2 buyers would have to be a

point on the indifference were2, as show in fig 7.5. Now consider

a small reduction i n11cx from x. Since at1cxin a change in net

surplus from type 1 buyers of just about zero on the other hand, it

permits a downward shift in the indifference curve on which type 2munotes.in

## Page 135

135buyers can be placed, and at any2xthis results in a strictly positive

gain in net surplus to the monopolist. Thus it pays to reduce1xbelow1cxof course, for further reductions in1xthemonopolist will

lose some net sueplus from type 1 buyers, but this must be traded

off against the gain in surplus from type 2 buiyers, and the optimum*1x, just balance these at the margin.

Figure 8.4

To see why*22,cxxnote that it pays the monopolist to

maximize the net surplus of type 2 buyers with respect to outputs

since this then maximizes the value of2Fthat can be set, subject to

the constraint that type 2 buyers would not prefer the type 1

contract.

There is a qualification to the condition in 8.42. note that as1nfalls, given2n,*2xmust also fall. It is then possible, for suitably

small1n,t h a t 8.42 satisfied cannot be satisfied for any10,xin

which case1Fis set sufficiently high that no type 1 buyers enter the

market. The monopolist then knows that the only buyers in the

market are of type 2, and so he can extract all their consumers

surplus, with*22Fs.I nt e r m so ff i g 8.5,*2becomes2.T h e

intuitive explanation is that, when the proportion of t ype 1 buyers is

sufficiently small, the loss in total profit from reducing1xand the

corresponding extracted surplus, is small relative to the gain from

being able to extract more surplus from type 2 buyers. The

equilibrium positi on in Fig 8.5 depends on the proportions of buyers

of the two types as well as on the shapes of the indifference

accrues and the value of c.munotes.in

## Page 136

136The importance of the specification of quantities in the contract

can be seen if we consider the two part tariff si m p l i e db yt h ei m p l i e d

by the equilibrium in fig 8.5. If type 1 buyers took a contract in

which they paid a fixed charge*1cand then a price per unit of**

11 1,ipxthen they would chose consumption*1xand pay

precisely** * *11 1 1cp xFlikewise, if type 2 buyers were set a fixed

charge*2cand paid a price per unit*1

22 2cpx c then they would

chose to consumer2cxand pay in total.**22 2cFC C x.I ft h e

monopolist made these contracts available to all buyers and did not

restrict the quantity that could be bought, fig 8.5should that the

self selection constraint would be violated. Type 2 buyers would

clearly choose a type 1 contract, which would dominate the contract*22,cxFalthough type 1 buyers still prefer their own contract. On

the other land, if the monopolist specified contracts of the form a

fixed charge*1cand a price per unit*p,u pt oam a x i m u mo f*1xunits of consumption, or a fixed charge*2cand a price of c for any

amount of consumption, then the self’s election constraints would

continue to hold.

Figure 8.5

In fact, the tariffs or price schedules that firms with market

power offer often do specify maximum consumption quantities as

well as fixed and variable charge.

A note on terminology, Linear pricing refers to t he case in

which a buyer is charged a fixe dprice p per unit bought, so that her

total expenditure is E=px a linear function. A two -part tariff consists

of a fixed charge. C and a fixed price p per unit bought, so that total

expenditure is the affine funct ion. In this case, the average price

per unit,/,pc xis a non -linear, decreasing function of themunotes.in

## Page 137

137quantity bought. In figure 8.5, the implied unit price**/iiFxto each

type of buyer will not be the same, implying a kind of non -linearity in

the way in which unit price varies with quantity bought. Thus this

kind of price discrimination, as well as two -part tariffs, falls under

the general heading of ‘non -linear pricing’.

To summarize : if a seller can identify each buyer’s type (her

demand function), and revent arbitrage between types, then he

maximizes profit by offering a two part tariff consisting of a unit

price equal to marginal cost c, and a fixed charge which

expropriates all the consumer surplus of the given type. If a seller

cannot identify a buyers type, he must offer optional contract a

higher demand type will choose a contract which offers a unit price

equal to marginal cost and a fixed charges which leaves her with

some consumer surplus, o law demand typ ew i l l choose a contract

which offers a higher price up to a quantity maximum*ixand a

lower fixed charge which never the less appropriates all her

consumer surplus. After natively the contracts may simply specify a

quantity suppli ed and a total charge for that quantity. The aim is to

prevent high demand buyers pretending to be low demand buyers,

and taking the contract the later would be offered under first degree

price discrimination, by making the low demand buyers’ contract

less attractive to the high demand buyers. Finally, if a buyer’s type

can be identified and arbitrage between types can be prevented,

but the seller is constrained to use linear pricing, we have third

degree price discrimination.

8.6THIRD -DEGREE PRICE DISCR IMINATION :

MARKET SEGMENTATION

Suppose that the monopolist can divide the market for his

output into two subgroups between which arbitrage can be

prevented at zero cost. To concentrate on essentials assume that

the costs of supplying the two sub -markets are identical, so that

any price difference between the sub markets will arise from

dissemination, not differences in say, transport or distribution costs.

The monopolist knows the demand, and therefore marginal

revenue, curves, for each graep Let1qand2qbe the quantities

sold to the first and second groups respectively, so that total output12qq q. Take some fixed total output level,0q, and consider the

division of this between the two sub -markets in such a may as to

maximize profit since the total production cost of0qis given, profit

from the division of this between the two markets is maximized if

revenue is maximized. But reve nue is maximized only if1qand2qare chosen such that the marginal revenues in each sub -market aremunotes.in

## Page 138

138equal. To see this let1MRbe the marginal revenue in sub -market 1,

and2MRthat in 2. Suppose12MR MR. Then it would be possible

to take one unit of output from market 2, and sell it in market 1, with

an e tg a i ni nr e v e n u eo f120.MR MRAs long as the marginal

revenues were unequal such possibilities for increasing revenue,

and therefore profit, would exist. Hence a necessary condition for a

profit maximizing allocation of any given total output between the

two markets is that marginal revenues in the markets be equal.

Indetermining the optimal total output level, weare on familiar

ground. If12,MR MRdiffered from marginal cost, it would be

possible to vary output in such a way as to increase total profit by

increasing output when1MR MC, and reducing it in the converse

case. Hence a necessary condition for maximum profit is that12MC MR MR.

Now let12e and ebe the price elasticities of demand in the

respective sub -markets. Then the basic relatio ng i v e ni n 8.9 applies

in this case, so that11 2 211 / 11 /Mc P e p e [8.12]

From the second equality in [7.12] we have12211111pepe[8.13]

If12ee,then clearly12/1ppand there i s no discrimination

There will be price discrimination as long as the clasticities are

unequal at the profit maximizing point Moreover, if12ee,t h e nf r o m

[8.13]12pp, and conversely.

(Remember0.ie) In maximizing profit the monopolist will set

a higher price in the market with the less elastic demand.

The analysis is illustrated in fig 8.6In (a) of the figure are the

demand and marginal revenue curves for such market 1 and in (b)

those for 2. The curve MR in (c) is the horizontal sum of the1MRand2MRcurves. MR has the property that at any total output,0qthe output levels01qand02qwhich have the some marginal

revenues in the sub markets as that at0qsum exactly tomunotes.in

## Page 139

13900 0 012 ,..qi e q q q. The horizontal summation therefore refects the

first condition derived above, that any tota l out put must be divided

between the sub markets in such a may as to equalize their

marginal revenues. The profit maximizing level of total output is

shown at*q,w h e r e*MR MR qis optimally divided between the

submarkets at*1qand*2qwhere the sub -market outputs have

marginal revenues equal to MC and by construction must sum to*q.D e m a n df o r2qis less elastic tha nt h a tf o r1qso that**21pp.

Fig8.6

8.7 PRICE DISCRIMINATION AND SOCIAL WELFARE:

Whether price discrimination promotes social welfare or not is

difficult to say. In other words, there is ambiguity as f ar as the

welfare effect of price discrimination is concerned. It is important to

consider whether social welfare is defined in terms of total output or

distribution of given output. This is because, the total output effects

of price discrimination may hav e positive welfare effects, whereas

distribution may be adversely affected after practicing price

discrimination.

As per the Pareto optimality, one of the maximum condition for

maximization of social welfare is that the marginal rate of

substitution betwe en two goods for different consumers, should be

the same. But when a monopolist follows price discrimination, the

above -mentioned marginal condition is violated. But another aspect

of welfare effects of price -discrimination may be understood by

acknowledgi ng the total output effects of price discrimination.

According to Joan Robinson, price discrimination sometimes may

lead to increase in output. That means, the total output may be

more when the monopolist practices discriminating prices rather

than a unifo rm price. Thus, from the point of view of total output,

when a society prefers more output to less output, price

discrimination may promote social welfare.munotes.in

## Page 140

140To conclude the discussion on welfare effects of

pricediscrimination, following points need to be co nsidered.

1) The losses incurred by the consumers in low electricity market in

the form of reduction in the consumer surplus as the monopolist

charges higher price for them.

2) The gains enjoyed by the consumers in high elasticity market in

the form of in creased consumer surplus, as the monopolist charges

lower price from them.

3) If price discrimination is not exercised by the monopolist and

uniform price policy is followed, there is a possibility that some

markets may be closed for the monopolist (partic ularly the high

elasticity markets as they have to accept monopoly price).

4) Price discrimination leads to redistribution of income from the

consumers in low elasticity markets to the consumers in high

elasticity markets and the monopolist. Since the cons umers in low

elasticity market are generally richer, redistribution may increase

social welfare.

8.7MONOPSONY

Monopsony is defined as a market in which there is a single

buyer of a commodity who confronts many sellers. Each of the

sellers treats the m arket price of the good as a parameter and so

there is a market supply curve for the good which is derived in the

usual way from the supply curves of the individual suppliers. The

single buyer of the good faces a market supply function relating

total suppl y to the price he pays. This can be expressed (in the

inverse form) as1

11 1 10PP Z P[B.1]

where [B.1] shows the price of the commodity which must be paid

to generate a particular supply. Note that the buyer is assumed to

face on upwar d-sloping supply curve; the price required is an

increasing function of the amount supplied.

The market price of the mono psonized input is determined,

given the supply function [B.1], by the buyer’s demand for1z.W e

assume that the monopsonist is a profit maximizing firm, in which

case the demand for1z, and hence its price, is determined by the

firms profit maximizing decision. In the two input, single output case

the firms’s problem is

12 1 1 2 212max,,RFZ Z PZ P ZZZ [B.2]munotes.in

## Page 141

141This is very similar to problem [6.1] except that1pdepends on1zbecause of [B.1] Input 2 is assumed to be bought on a market in

which the firm treats2pasap a r a m e t e r .T h ef i r m so u t p u tm a yb e

sold in a competitive or monopolized market. monopoly need not

imply monopoly. The firm may, for example, be the only employer

of labour in a particular area but be selling its output in a market

where it competes wit h many other firms, and labour may be

relatively immobile.

Necessary conditions for a maximum of [B.2] are (when both12Z and Zare positive at the optimum)1

11 1 10lRF P P Z[B.3]

1

220RF P[B.4]

P1 MBC

S1*1p*12MRP ZO*1zz1

Figure : 8.7

This equilibrium is illustrated for the monopsonized input in

Figure 8.71sis the supply curve of1Zand1MBCplots the marginal

buyer cost111 1PP Zof the single buyer.*12MRP Zis the

marginal revenue products curve for the input given the optimal

level of2ZThe firm maximizes profit with respect to1Zby equating*11 1MRP to MBC at Zto generate this supply of1Zthe firm will set the

monopsony price**11 1PP Z.

The analysis of the single buyer confronting many competitive

sellers is rather similar to the analysis of the single seller

confronting many competing buyers. In each case the firm realizes

that it faces a curve relating price to quantity which summarizes the

response of the competitive side of the market and the firm sets the

quantity or price in the light of this interdependence of price andmunotes.in

## Page 142

142quantity. In each case the market price overstates the marginal

profit contribution of the quantity and in each case this

overstatement depends on the responsiveness of quantity to

changes in price under monopoly the firm equates[1 (1 / )]MR P eto the marginal cost of output, and the less clastic

is demand the greater is the di fference between price and marginal

cost. [B.5] can be written in a similar may. Defining the clasticity of

supply of1Zwith respect to price as11111sdz Pedp Z[B.7]

We see that.

1

11 11

111,1sdpMBC P z Pdz x[B.8]

and so [B -5] becomes

115111 MRP Pe[B-9]

The less elastic is supply with respect to price the greater will

be the difference between1MRPand the price of the input. In other

words, the less responsive to price the input supply is, the greater

the excess of the value of the marginal unit of the input over the

price it receives. This could be regarded as a measure of the

degree of ‘monopsonistic exploitation’.

8.8THE EFFECT OF MONOPSONY AND OUTPUT

MONOP OLY ON THE INPUT MARKET.

When the output is produced from two or more inputs the

analysis of the effect of both monopsony and output monopoly on

the price of one of the inputs is complicated, because the use of the

other input is likely to change as wel l, thus shifting the1MRPcurve.

If the output is produced by a single input this complication does

not arise, and it is possible to show the implication of monopsony

and output monopoly in a single simple diagram such as figures

8.8. Since there is a single output1Zis marginal product depends

only on1Zand so the marginal revenue product1MRPand and the

value of the marginal product1VMPcurves in figure 7.8 are fixed,1Sand1MBCare supply and marginal buyer cost curves. There are

four possible equilibria in this input market, where suppliers treatmunotes.in

## Page 143

143the price of1Zas a parameter. If the firm also treats1Pas given,

i.e. if it acts as if it has no monopsony power and if it also treats out

put price as a parameter then1VMPis its demand curve for1Zand

the market price is01P. If the firm uses its monopsony power but

continues to treat output price as a parameter it will equate11VMP to MBCand set the price11P.I ft h ef i r mm o n o polizes its output

market but regards1Pas a parameta its demand curve for1Zis1MRPand the price of1Zis21P.F i n a l l y ,i ft h ef i r m exercises both

monopoly and monopsony power it equates1MRPand1MBCand

sets a price31P. We see therefore that the price in an input market

is reduced below the competitive level01Pby both monopsony and

monopoly power. The less elastic are the demand for output and

the supply of input functions, the lower will be the price paid to

suppliers of the input.

Fig.8.8

8.9UNION AS MONOPOLY INPUT SUPPLIERS

We define a union as any association of the suppliers of a

partiallar type of labour which is formed with the aim of raising

wages or improving working conditions. A union need not, of

course, be described as such by its members : Many professional

associat ions such as the British Medical Association and the law

society act as unions. Not all unions may be successful in raising

the wages of their members above the competitive level. The union,

like any would be mnonopolist, must be able to control the supply of

labour offered to firms one method of doing this is to ensure that

only union members can sell their labour in that particular market, a

device known as the closed sh op. The closed shop many by itself,

reduce the supply of labour to the market if some potential workers

dislike being union members as such. In general, however, the

closed shop must be coupled with restrictions on the number ofmunotes.in

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144union members if all members are to be employed, since higher

mages will increase the members of workers wishing to join the

union, i.e. become employed at the higher wage.

If the union can act as a monopolist its behaviour will depend

on the objective it pursuers. It may be useful to distinguish between

the objectives of the officials who run the union and those of the

members. In the case of the firm, where conflicts of interest may

exist between shareholders and managers, the extent to which the

managers rursue the interests of the shareholders depends on the

incentive system which relates managerial pay to prof its and on the

threat of products or capital market competition similar mechanisms

may be at work in the case of the union officials salaries can be

related to the pay of members of the union unions which do not

attend sufficiently closely to their members interests may start to

lose member to rival unions officials may be controlled directly

through elections, but here the control machinism may be much

weaker than in a firm. Each union member has only one vote and

so may members must cooperate to change th e officials.

Shareholders vote in proportion to the numbers of shares held and

so a relatively small group of individual shareholders may exercise

effective control.

It is by no means obvious that the political structure of a union

will generate any wel ld e f i n e dp r e f e r e n c eo r d e r i n g ,l e ta l o n eo n e

which reflects the interest of its members. (See the discussion of

the Arrow Impossibility. Theorem in section 13F). However, we will

assume that such a preference ordering exists and can be

represented by a uti lity function,wzwhere w is the wage paid

to union members and z is the number of union members employed

(we assume that hours of work are fixed). We illustrate the

implications of different assumptions about union preferences by

specifying there different forms for.The demand side of the labour market monopolized by the

union is assumed to be competitive and the union is constrained to

choose a wage and employment combination on the labour market

deman d curve D in fig 8.9. MSR is the corresponding ‘marginal

revenue to the sellex’ curve which shows the rate at which the total

wage bill wz varies with z 5 is the supply curve ??? the minimum

wages necessary to attract different numbers of workers into the

industry s plots the reservation wage or supply price of workers.

The competitive equilibrium in the absence of an effective union

monopoly would be at with a wage rate of Wc.

The economic rent earned by a worker is the difference

between the wage paid and the wage necessary to induce that

worker to take a job in the industry. The totalmunotes.in

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1450,wz w wz w z [C.6]

(The union indifference curves are now rectangular hyperbolas

with a horizontal axis atw). SincewandZare constrants, [c.5] is

maximized by maximizingwwZa n dt h eu n i o n ’ so p t i m i z a t i o n

problem is now analogous to that of a monopolist with a constant

‘marginal cost’ ofw.

It is possible to construct many models of the above kinds,

each of which may be appropriate to a particular union or industry.

A model of the way in which the union’s objectives are determined

is necessary in order to be able to p redict. What objectives will be

dominant in what circumstances. This will require a detailed

specification of the political constitution of the union, including the

frequency and type of auction, whether officials are elected or are

appointed and controlle d by elected representatives and so on. In

addition, the theory could be extended to take account of inter

union conflict or cooperation will unions compete for new

members? In what circumstances will unions merge or collude? It

would be interesting to app roach these questions using the

concepts of oligopoly theory developed in chapter.

8.10 BILATERAL MONOPOLY

Bilateral monopoly is a market situation in which a single seller

confronts a single buyer. For definiteness and continuity, we

consider a labour market in which supply is monopolized by a union

and there is a single buyer of labour. Z is the sole input in the

production of an output y = F (Z). The revenue from sale of y is R

(F(Z)) and the MRP curve in fig 7.9 plots the marginal benefit to the

buyer of11:.Z R F MR MP, using the notation of section A. The

average revenue product curve ARP plots //R Z Py Z PAPwhere AP is the average product of z : y / z.

The objective function of the firm is its profit function.,RFz w z w z [0.1]

and its indifference curves in (w,z) space have slope11

0 dRF zFz Wdw z MRP Wdz w Z Z [D.2]

Where11RF M R Pis the firms marginal revenue products for

W < MRP its indifference curves are positively sloped and for w >

MRP they are negatively sloped. Thus its indifference curves aremunotes.in

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146shaped about the MRP curve. If the firm acted as a monoponist

facing competitive labour suppliers, it would announce a wage rate

at which it is willing to hire workers and employ ment would be

determined by the supply curves of the workers.

Suppose that the union has the simple objective function.

0,WZ W W Z W Z [D.3]

Figure 8.9

examined in section C, where0Zis the number of union members

andWis the income or wage of those who are unemployed. The

unions indifference curves are hyperbolas, rectangular to theWWline, with slop e.0 ddw W Wdz Z[D.4]

If the union acted as a monopolist with respect to the labour

supply of its members it would announce a wage rate at which its

members would be willing to supply labour and employment would

be determined by the demand curve for labour.

When a single buyer and a single seller of labour confront

each other it seems implausible that either party will treat a wage

rate announced by the other as parametric and passively adjust

either their supply of demand. Both wi ll realize that they possess

market power in the sense that, by refusing to demand or supplymunotes.in

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147labour at a wage announced by the other, they can prevent any

gains from trade being achieved and thus impose cost on the other.

The two parties must therefore agr ee on a wage and on

employment level before production can occur.

We assume in this section that the agreement between the

union and the firm is the outcome of a cooperative game. In such a

game all the actions of the parties are controlled by a binding

agreement between them specifying what each will do. The

cooperative game approach to bargaining is concerned solely with

the content of the agreement. It ignores the process of bargaining

and negotiation by which agreements are reached (We examine

the al ternative non -cooperative game approach, which does pay

more attention to the bargaining process we attempt to predict the

agreement by requiring that it satsfy certain ‘reasonable’ conditions.

Two obvious conditions to impose are:

(a)Individual rational ity : any agreement should level both parties at

least as well off as they would be if there was no agreement;

(b)Efficiency : there should be no other agreement which would

make on of them better off and the other no worse off.

If an agreement satisf ies these requirements it is an efficient

bargaining solution to the cooperative bargaining game.

Applying these conditions provides a partial answer to the

question of what agreement will be reached by the union and the

firm. If there is no agreement a nd therefore no employment, the

firm will have zero profit. Any agreement which yield a (w,z)

combination on or below its average revenue product curve PAP

will satisfy the individual rationality constraint for the firm. If the

union achievers zero utility if there is no agreement, it will be no

worse off with an agreement at any point on or above the lineWW.

Thus the set of individually rational agreements which make both

parties no worse off is the tria ngle bounded by the vertical axis,WWand pAP in fig 8.9.

Imposing the efficiency requirement further reduces the set of

possible pargains. If the parties’ indifference curves interest at a

point such as1xit is always possible to find another point or

bargain which makes at last on of them better off and the other no

curse off. Thus moving from the agreement1xwhere the

indifference curves11uJ and Iinterest to the agreement2xwill make

the union better off since2xis on the higher in difference curve2Iu. The firm is no increase off at2xsince both point1xto3xwould make both union and firm better off.munotes.in

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148A necessary condition for efficiency is that the parties

indifference curves are tangent.11

00 dddw z R F W dw W Wdz w Z dz Z [D.5]

which implies11RF W[D.6]

All agreements satisfying [D.6.] are efficient. Notice that [D.6]

depends only on the level of employment Z (which enters into11RR)

and not on w. the locus of points where [D.6] is satisfied and the

agreement is efficient is a vertical line atZagreement MRP cutsWWThe set of agreements satisfying individual rationality and

efficiency is the contract curve. In the current model to contract

curve has a particularly simple form. It is the line cc. in fig 8.9

between the PAP andWWcurves where indifference curves are

tangent and the parties no worse off than if they do not agree.

The efficient bargain model predicts the level of employment*Zthe parties will agree on bt it is unable to predict the wage rate

at which the workers will employed. This is perhaps unsurprising

the parti es can agree to choose on employment level which will

maximize their potential gains from agreement the difference

between the firm’s revenue R(F(Z)) and the ‘cost’ of labourWZas

perceived by the union A change in z which increas esRW Zcan

make both parties better off and they can therefore agree to it.

However, for fixed Z , changes in the wage rate have precisely

opposite effects on their utilities.

,wz w zwith z held constant cha nger in w merely make one party better off

at the expense of the other. In figure 10.7 the firm will always reefer

a bargain lower down cc and the union a bargain higher up CC.

One way to remove the indeterminacy of the bilateral

monopoly model is to imp ose additional requirements on the

agreement or solution of the cooperative bargaining game.munotes.in

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1498.11QUESTIONS

1. How the price and output is determined under monopoly?

2. Explain the price and output determination under price

discriminating monopoly.

3. Examine the concept of monoposony.

4. Explain the effect of monopsony and output monopoly on the

input market.

5. What is bilateral monopoly?

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