MA-I-SEM-I-ECONOMICS-PAPER-I-Microeconomics-I-ENGLISH-munotes

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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,01Pxmust be no greater than the cost of0x, which is01Px. This latter is also the con sumer’s money income000MP xwhen0xis 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 cost10Pxmust 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 MIf0xis 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 Bis 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 effect01xt o xinto 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.xmust 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,02xxcannot be to the left of0xon2Bbecause these bundles are inside
the consumer’s initial feasible set and were rejected in favor of
0,xxcannot equal0xbecause the prices at which2xa n d xare
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,pxare 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 gives10 12 1 0 20px px p x x[1.04]
and similarly 1.03 gives00 02 0 0 20px px p x x[1.05]
subtracting 1.05 from 1.04 gives10 2 00 2 1 0 0 20px x px x p p x xand multiplying by ( -1) we have102 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 pdiffer only inpiand so 1.06 becomes10 2 0 10 20iiii iiii
ippxx ppxx[1.07]
Hence whenipchanges the substitution effect2iixxis 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 xthe compens ating
reduction in M is00 10 0 1 0 1 0 002MM M p x p x p p x p p xmunotes.in

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6and in the case of a change;piniponly we have0iiMp x [1.08]
The price effect of10iii ip on x is x xand this c an be
partitioned into the substitution20iixxand income12iixx
effects:10 20 12ii ii iixx xx xx
Dividing byipgives10 20 12ii i i iiii ixx xx xxppP[1.09]
But from [1.08]0/iipM xand substituting this in the
second term on the right hand side of [1.09] yield1212
.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 xwhereipis the price of good i.
X2
0
22MXPB
O0
11MXPX1
Fig. 1. 02
The feasible set in the two good case is shown in Fig 1. 02as
the triangular area0012 1/Ox x M Pis the maximum amount of1xthat can be bought with income M at a price of012.pxis analogously
defined. The budget constraint is11 2 2px px Min 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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821 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 xon 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,10pthen 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 xand 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) with0MMthe intercepts of the budget line B. on the12x and xaxes
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 oPshifts the1xintercept from101 01/MP t o MPwhere101 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 Pincrease from12 1 21.P and P to KP and KP where kThen 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 Pso 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 as12 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.xxis the gain in utility
from an additional1xunits of1xHence11/Pis the gain in utility
from spending an additional unit of money on12.2 /xPhas 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 ./Pmunotes.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
consumed0 1...ixinwhereixis 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 is1, ....,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 e12, ,....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 DDas 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,PPconstant, will shift the budget line
outward paroled with itself, say to2Bwhere1Xis chosen. A further
increase in M will shift the budget line to3Bwhere11xis chosen
X2 Fig. 1. 06
X0X11ICC4X13X1
X*2X+15B1 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 xare 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 Ihad 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 Xthe bundle chosen when money income rises to2M.1*..xx i eif
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*xover x with a money income of1*1M and x over xwith
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 to12BXis the optimal
bundle on112,BXthe optimal bundle on3,Bwhich 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 with111PPthen1xwould 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
X24PCC
X023X11
X*X11X+5B1 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
X011B3 B2
B1
O*1X01X11XX1*1Xa n d Xare the optimal bundles before and after the fall in11 2,P B and Bthe 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,Xxand01xare the amounts of1xcontained in the b undles*10,,xxxand
(a)0*11xxis the own substitution effect;munotes.in

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18(b)1011xxis the income effect;
(c)0* 10 1*11 1 1 1 1xx xx xxis 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 xis consumed the
curve flattens. The fall in1Pflattens the slope of the budget line,
and hence the budget line3Bmust be tangent with1Ito the right of*,xi.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

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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
effect113xxand an own substitution effect11xx. 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 fallerxcannot lie to the left ofxon4Bbecause this
would mean that the consumer is new choosingxwhen*xis still
available, having previously rejectedxin 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

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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 Mon 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 Pafter 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

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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

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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

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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 ifxexpenditure
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 (withas the lagrange
multiplier) as1.....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 asiipuand dividing the co ndition onixby the condition onixgives.iiiippu………………………………………….. (2.5)munotes.in

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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 curve0is 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 iiipxgives.*,,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 jnwe 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 ufunctions 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 nis the substitution
effect of the price change. S ince by definition/iiHpis 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
vectors1pand11p, and K such that1ok.D e f i n e1 111.pk p k pWe 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 uand11 11 11,px m p umunotes.in

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26likewise, letxsolve the problem when the price vector isp, so
that11,px m p u.S i n c e11xa n d xare 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 gives11 1 1 1 1 1 1 111KP x k p x kp x K p x [2.9]
But by definition ofpthis implies11 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 xare
optimal solutions are shifted so as to pass through point,xthey
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

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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 pdiffer 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 uThe 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,iipufrom 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 pwhich 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 price11 1 1 1
1, ... .... , .ii i inPi s x H p p p uThis 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 :iiHPjust 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 thatiiHpoverstates 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 mpwith strict inequality if*0ix.munotes.in

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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 vector01pand00,ooLet M m p u p xwhere0xis the expenditure minimizing
bundle at0p,t h a ti s00 0px p xfor all bundle x yielding utility of u
or more. But this imples that00 0 0kp x k p xfor all bundles yielding
at lea st u and so0xis optimal at prices0pand0kp.T h e n00,,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 plier0in 2.2 is
equal to the derivative/,muis 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 multipliesin 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 ofto be estestablished, we cannot say thatis
necessarily increasing or decreasing, with u, because both are
possible for different, permissible utility functions.,mp u11 1
11 iimP u X H Pu,mp u
OP i11iPPi
Figure 2.3munotes.in

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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 maximizing1....,nxx subject to the budget constraint
iipx Mand 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 MThe maximized value of**11, .... , ....,nnux x ux xwill 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 function1, ...,nux xwhere 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

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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*ixasipvariesis the marginal utility of money income.
The product ofand*ixis 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 mto 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

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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

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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 pis 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 set0* 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 point00 * 00
12 12,, ,pp M uppMlies 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 point0
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,pmand11,pm, such that*00 0 1 1 1,up m u u p m [2.18]
So that11,pmis in the wares set of0,opm.W eh a v et o
show that any convex combination of these two price -income
vectors is also in this worse set of00 * 0,: ,pm u p m u[2.19]
where01 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 mor, given the definitions ofp and m01 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 uand 2.23 impliesmunotes.in

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33*1,up m uand since one or both of 2.22 and 2.23 must hold and10uuby assumption, we have establi shed*0,up muas
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 mBcorresponds to1111 2px x mand yieldsa
lower utility value than0.BBcorresponds to11 2,px x Mwhere01 0 1
1 11, 1 .pk p k p m K M k M Balso yields a labour
value of utility than0Bweh a v e .10MM M[2.24]10111ppp[2.25]
ThatBpasses through the intersection point00 0 112,x x of B and Bfollows by nothing that if we sum.00 0 011 2kpx x K M [2.26]
X2
M0M1M0u02X1BB0BO01X1XFigure 2.5
and10 0 111 2 1kp x x M[2.27]munotes.in

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34We obtai n
00
11 2px x m[2.28]
Thus11,pmand,pmare both in the worse set of00,pmand,pmis a convex combination of00,pmand11,.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 uwe 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

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35/iDMso, 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/,iipxand the
income term by M / M, givesii ii i iSn [2.32]
whereiiis the marshallian demand elasticity,iiis the Hicksian or
compensated demand elasticity,inis the income elasticity of
demand, and/ii isp x mis 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,ij2.31 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 pof
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 npwe have2,, 1, ....i
ii imp u Hij npp P[2.34]
Then, from young’s Theorem we have immediately that// ,ii i iHPHpand so the slutsky matrix is symmetric. The
slutsky matrix/iiHPis 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 pby the definition of negative semimunotes.in

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36definiteness, the Hicksian demand curve cannot have a positive
slope. We have seen earlier that strict convexity of pr eference and0ixat the optimum establish the stronger result that
22// 0ii i mp H p.
The Hicksian demand derivative/iiHPis often used to
define complements and substitutes. Two goods i and j are called
Hicksian complement if/0iiHPand 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/jiHPrather 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 iiHPHPor 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 variables12, ...nee ewhere 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 variable1. 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 set00
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 vector12, ....nee esince 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 ,swith'1simplying 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’. Let5 1, 2, .... 5sydenote 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 probabilitysto state of the world 5, and denote the
vector of probabilities by12 3[ , ... ],while12, ... ,yy yyis
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 ychanging 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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422AAPywhereAyis 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 by12, , ....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,,isppexactly one of the statements.,,,ik jk jkpp pp p pis true, while11ik k ip p and p p p pand 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
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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 yrespectively. 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 ywith
probabilities v and 1 -v respectively, where10v. A specific
standard prospect,10,pcan be written as1101,,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
that11 1101,,upv y ywe 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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45Input 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

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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
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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,,ykwhere 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 Ryis 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 thatyyandyy; 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

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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 andxis an input vector
with at least as much of each input, thenxshould 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) andxis 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 andyythenymust also be in
Y. That is to say that, if y in Y is feasible thenyin 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 andxare in V(y), then1tx t xis 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 andyare both in Y, then1ty t yis also in Y for01tin 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
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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 allyoThe 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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53Let21xxbe 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 function21xxhas to satisfy the following
identity -21 1,fxxx y.
Actually, we require an expression for -*21 1/xx xThen, 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
as12,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 that112 1 2,aafxx x xwe can take the derivatives to
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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 let21/xx be the change in the factor ratio andTRSbe 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 asgoes to zero. Then,
the expression forbecomes –
It is often convenient to calculateusing 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

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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

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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

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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
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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

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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 ifkft 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 Ris 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)), wherehis monotonic function. Both, homogeneous and homothetic
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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;
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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 x2) 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 -
If21xxthe TRS is negative infinity; if21xxthe TRS is zero. This
means that Q as
approaches -, a CES isoquant looks l ikean
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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.
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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
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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

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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

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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,
111aaxy 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

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67Short -run average variable cost11aaywkShort -run average fixed cost2wky
Short -run margina l cost11aawy
ak4.2.3 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 SAVCIn 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
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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? Letydenote the point
of minimum average cost; then to the left ofyaverage costs are
declining so that f oryy()0dc y
dy yTaking the derivatives, it gives,
2'( ) ( )0yc y c y
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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,
'()()cycyyforyySince both inequalities must hold aty, 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 y1abWhere, k is a function of factor prices and parameters. Thus,
1()()ababcyAC y Kyy
1
'() ()ababKMC y c y yabIf1,abthe cost curves exhibit increasing average costs; if1,abthe 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 ybe 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

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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',wwthen1(, ) ( , )cw y cwy2) Homogeneous of degree 1 inw:munotes.in

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71(, ) ( , )ct wy t cwyfor0t3) Concave inw:
''(( 1 ) , ( , ) ( 1 ) ( , )ct w tw y t cwy tcw yfor01t4) Continuous inw:(,)cwyis continuous as a function of w, for0wProof :
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 wfor any01 .tNow,(" ,) " " "( 1 )'"cw y w x t w x twxSince"xis not necessarily the cheapest way to produceyat price'worw,w eh a v e"( , )wx c w yand'. '' ( ', ).wx c wyThus,(" ,) (,) ( 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,............,inthen(,)(,)iicwyxw yw1,........,inmunotes.in

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72Proof :
Let*xbe a cost –minimizing bundle that produceyat prices*w.
Then define the function,() (,) *gw cwy w xSince,(,)cwyis the cheapest way to producey, this function is
always non -positive, at*,ww(* ) 0 .gwSince, this is the ma ximum value of() ,gwits derivative must vanish:
(* ) (* ,)0i
iigw cw yxww  1,.........,inHence, 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 aIn 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 aLmunotes.in

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73and the first order conditions are –
110gh
xx
220gh
xx------------------------------------ (1)
1, 2,() 0hxx aThese 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

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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}tyyIt 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.wThen the natural generation ofVObecomes –*( ) { : ( , ) ( , )Vy x w x w x w yc w y for all0}wRelationship 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 yThe 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 yfor all0wwill 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 yProof: 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*0wsuch that**wx wzfor 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 yBut 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
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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 xSuch 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 yProof: 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(,)wyis a cost function. Let(,)wybe a differentiable
function satisfying –
1)(, ) ( , )tw y t w y for all0t;
2)(,) 0wyfor0wand0y;
3)(' ,) ( ,)wy w yfor'ww;
4)(,)wyis concave in w.
Then(,)wyis the cost function for the technology defined by*( ) { 0: ( , ),Vy x w x w yfor all0}wProof: Given0wwe define
1(,) (,)( , ) ,......,
nwy wyxwyww      And note that since(,)wyis homogeneous of degree 1 inw,
Euler’s law implies that(,)wycan be written as
(,)(,) (,)1  nwywy w w xwyiw ii
Here it should be noted that the monotonicity of(,)wyimplies(,) 0xwymunotes.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(,)wyinwwe have –(' ,) ( ,) ( ,) (' )wy w y D w yw w  -for all0wUsing Euler’s law as above it reduces to(' ,) ' ( , )wy w x w yfor all0wIt 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 yBut by Euler’s law,(,) (,)wy w xwyThe above two expressions imply –(,)wx wx w yfor 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 pbe 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 by10zy z yIfww z y [5.3]
with10,wzthere 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 supplySMCcurve is in each case s,0,pw.I f
simultaneous expansion by all firms raises input prices from01wt o wthe 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 to1,isp wand 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 andispis the locus of all such
price. Supply pairs. clearly, the firms effective market supply curveispwill 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 byypand
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 functionjj
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 pand/0jw iss wwe 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

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85Since0jpsand110, 0, 0,je wzssoluing for/dy dpgives
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 supplyss pi.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 pmaximizes 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

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86insignificant part of the market we can then take the s(p) curve as
continuous, with intercept at11p. However, at price11 1ppdemand 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,ppthe 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 functionszp 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*0psuch that00zp.I f0,zp o r zp ofor all,p o then z pmay be continous butmunotes.in

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87we will not have an equilibrium. This suggests the following
existence theorem for a single market. If.
a) the excess demand functionzpis continuous for0p.
b) there exists a price00psuch that00,zpand
c) there exists a price10psuch that10,zpthen there exists exists an equilibrium price*0psuch that*0zpThe 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 ptwhere*pis an
equilibrium price,tois time, p(t) is the time path of price and the
initial price*.po pThe 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*,ppexcess 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,zand 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,oppthe TP
would converge to the equilibrium*p. If, however, the initial price
was11 1ppthe market would move away from equilibeium, since
excess demand is positive for1ppand 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 priceptto converge to*pis that/0 ,dd ti.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 tif 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 tHere 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 pis not bounded away from zero under the TP.
We do this by establishing a contradiction suppose, without
loss of generality, that*po pand 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 pis non -empty,
closed and bounded and the function/dd tis continous, so at
some t we must heave that/dd ttakes an a maximum, by were
strass. Theorem, call this maximum*sNote that, since for*pt pwe must have/0 ,dd tthen s < 0 also for only arbitraryttintegrate 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, ifptis always
moving closes to*pwhenever*,pt pit 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 so0sppimplies output
explansion increases profits, while whenosppprofits 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
where0pyis 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 pmunotes.in

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93Under what conditions is Marshall’s process stable? If output
expands whenDsppand contracts when0sppthen 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 functionzp Dp spincreases 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 thave 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
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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**,ypAt 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 P1S2S1S2SLAC 1
P*P*LAC 2
C*
O01yy1 O*2yy2
P
S(P)
S(P)
P*
D(P)
O**xyx,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 yand 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 pair00pyin figure 5.7 In year
1d e m a n ds h i f tt o1Dp.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 functions1sp,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

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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,pyis not an equilibrium price will have to fall to2p,
whether demand equals short -run supply as indicated by the sort -
run supply curves2sp. 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 at11,pyas 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

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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

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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

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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
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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

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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,
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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.
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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 is0cc 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 firmq 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 dqis 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 term222/ /dp dq qd p dqis 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 case11 2 202 / /c q dp dq qd p dq[8.6]
However, unlike the competitive case, the second -order
condition may also be satisfied if110cq.
More insight into this solution can be gained if we write
marginal revenues, MR, as1/ /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 Rwhile10 ,e Mrand10eM Rcombining [ 8.9] with [ 8.4], we can write the
condition for optimal output as111 /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 t10)cqUnder 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 curves1cqand 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**pqand 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
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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 x21 1 1
22 2 1 1 1// 0xy x y x y xy MRS u x MRS   [8.15]0ioand110:ixbuyers have diminishing marginal utility;munotes.in

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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;yyMRecall, 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, and0Fis 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]
Hence1,ixp pyielding demand functions11
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/ivpis 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

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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 charge1, 2 .iiFS iThe 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

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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 than12ssby 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 ge1iFSfor only p,
because then he sells nothing to type i.
We can derive this result more formally. The monopolist’s total
profit is11 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.
withiasthe lagrange multiplier on these constraints, optimaliip and Fare 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/iivFimplyiand so [ 8.27] impliesiivu. Both types of consumers receive only
their reservation utilities. Then using [ 8.22] and [ 8.26] we haveiinandmunotes.in

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13111() 0ii i i i i inC x Px x nx[8.28]
implying
ipc[8.29]
The value of1Fthen satisfies,iiivc F uand 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 either12,,C S or C Severy type 2 buyer (a) will as every
type 1 buyer) would chose1,CScan 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 xand 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

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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 thatiiyMFThere are also self -
selection constraint which ensure th at each type chooses the
appropriate deal we write these as11 1 1 2 2xF xF [8.32]22 2 21xF x F [8.33]
(M cancels out in these expressions)
If,iixFsatisfies 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,iixFby maximizingsubject 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 offered21 2xFsuch that22uu. To see
that, note that type 1 buyers must be offered a contract11,xFthat
puts them on or above1u. But since1ulies above2u, such a dealmunotes.in

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133must always be better for type 2 bayers than any contract22xFthat 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
find11 2 2,,,xF xFto maximizein 7.30 subject only to 7.31 unit
I-1, and 7.33 using1and2ufor 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

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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 for2in [8.35] gives.1*
22xC [8.40]
implying*22,cxxot h a tt y p e2 consummation is exactly that
under first degree price discrimination. Then, substituti ng for12andin [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,
writing1*
21xc where0we have
* 2
1112i nxcnn [8.42]
implying that*
11,cxxso 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,xFand*22,cxFThe
two most interesting aspects of the solution are firs tt h a t*11,cxxand 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

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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,cxxnote 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,xin
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,*2becomes2.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

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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,ipxthen they would chose consumption*1xand pay
precisely** * *11 1 1cp xFlikewise, 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,cxFalthough 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 xis a non -linear, decreasing function of themunotes.in

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137quantity bought. In figure 8.5, the implied unit price**/iiFxto 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

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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 MRAs 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. If12,MR MRdiffered 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 ebe 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 that11 2 211 / 11 /Mc P e p e [8.12]
From the second equality in [7.12] we have12211111pepe[8.13]
If12ee,then clearly12/1ppand 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 the1MRand2MRcurves. 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

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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 qis 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 r2qis less elastic tha nt h a tf o r1qso 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

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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) as1
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

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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 Zare 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 of1Zand1MBCplots the marginal
buyer cost111 1PP Zof the single buyer.*12MRP Zis 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 Zto 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

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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 eto 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

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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 MBCand 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,wzwhere 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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1450,wz w wz w z [C.6]
(The union indifference curves are now rectangular hyperbolas
with a horizontal axis atw). SincewandZare constrants, [c.5] is
maximized by maximizingwwZa 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’ ofw.
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 PAPwhere 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 slope11
0 dRF zFz Wdw z MRP Wdz w Z Z    [D.2]
Where11RF M R Pis 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
andWis the income or wage of those who are unemployed. The
unions indifference curves are hyperbolas, rectangular to theWWline, 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 lineWW.
Thus the set of individually rational agreements which make both
parties no worse off is the tria ngle bounded by the vertical axis,WWand 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 Iinterest 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 implies11RF 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 cutsWWThe 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 andWWcurves 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 labourWZas
perceived by the union A change in z which increas esRW Zcan
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 zwith 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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