MICRO ECONOMICS FOR SEM-4
EXCHANGE (NOTES)
Introduction
We have studied consumer behaviour in isolation from what was taking place in other parts of the economic system (such
as the factor market, the financial sector, government sector and the rest of the world). Consumer’s decisions were
assumed to be unaffected by what was taking place in other parts of the economy.
It was further assumed that the consumer’s decisions were not influencing the other parts of the economy. The consumer
behaviour analysis was carried out under the CETERIS PARIBUS assumption. -"all other things were assumed to remain
constant". This ceteris paribus assumption DELINKED the subject matter under study from the rest of the economy. Such a
type of analysis which studies one part of the economic system in isolation from rest of the economy is called Partial
Equilibrium Analysis.
Each part/ segment of the economy under study is viewed as independent and self-contained under Partial Equilibrium
Analysis. Under Partial Equilibrium Analysis each part of the economy pursues its own objective/s and strives to achieve
their own equilibrium position without considering the equilibrium position of other constituent parts of the economy.
General Equilibrium Analysis is concerned with the study of economic system as a whole and how ALL the constituent parts
of the economic system work together. General equilibrium is said to be attained when all the constituent parts of the
economic system are simultaneously in equilibrium. Simultaneous equilibrium will be reached in the’ markets when at
positive price, there is neither excess demand nor excess supply and each market is cleared. In addition to the markets
reaching equilibrium, ALL decision making units in the economic system must be able to achieve their individual objectives
simultaneously for a general equilibrium to be established.
General Equilibrium Analysis does not require "other things to be held constant". It DOES AWAY with the ceteris paribus
assumption. It recognizes the fact that the constituent parts of an economy are interdependent and interrelated. Hal
Varian adopts the simplifying assumption of analyzing general equilibrium by considering two consumers (A and B) and two
goods (x1 andx2). Further his analysis assumes that in the economy the two consumers have fixed endowments of goods w 1
and w2. The analysis therefore considers PURE EXCHANGE → wherein the trade among the two consumers is analyzed.
{General equilibrium of production is analyzed in a separate section}
The Edgeworth Box
The Edgeworth Box is used to analyze the exchange of two goods between the two consumers. The Edgeworth Box is
named after the English mathematical economist, F.Y. Edgeworth. Given the above assumptions, A and B are the two
consumers and their relative commodity spaces are represented in figure 1 and 2. The origin OA for the commodity space
pertains to consumer A; the origin OB for the commodity space pertains to consumer B. Both A and B are endowed with an
initial quantity of (x1 and x2). Consumer A is initially, endowed with w1A units of x1 and w2A units of x2. This is shown by point
A1 in figure 1.
Consumer B is Initially endowed with w1B units of x1 and w2B units of x2. This is shown by point B1 in figure 2
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We now combine these two commodity spaces together. This is achieved by rotating consumer B’s commodity axes 180°,
so that B’s origin OB is in the upper–right-hand corner. Figure-3 is the same as figure–1 but figure–2 is turned upside down
i.e. it has been rotated 180° as shown in figure-4. We fit the two commodity spaces together, as shown in figure–5, by
moving them towards each other such that there is an overlap of each axes. The result is the Edgeworth Box.
The student will notice that the length of the box is w1 units of x1 [which is the SUM of initial endowments of A(w1A) and
B(w2B)]. The height of the box is also w2 units of x2 [which is the SUM of initial endowment of A(w2A) and B(w2B)]. Point E
represents, the initial endowments of the two consumers.
It is assumed that the total available supply of x1 is the sum of initial endowment of A and B of x1. The same is true for the
total supply of x2. Consumer A and B’s preferences for x1 and x2 have been shown using indifference curves A’s indifference
curves ICA1, ICA2, ICA3 and ICA4 are normally shaped as shown in figure 6 B’s indifference curves have
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been rotated to conform to the rotation of the axes (Curves ICB1, ICB2, ICB3 and IC84) Indifference curve ICA2 represents A's
level of satisfaction from his initial endowment Indifference curve ICB2 represents B’s level of satisfaction from his initial
endowment.
It will be of interest to note that as one moves from the bottom left to the top right, the satisfaction level of consumer A
increases; and that of B decreases. Consumer A move to higher indifference curves; and B moves to lower indifference
curves.
If A could move to a higher indifference curve, say ICA3 or B could move to ICB3 they could both be better off than they are,
at initial endowment point E.
The initial endowments of good 1 and good 2, given by point E, cannot be a position of equilibrium for the to consumers,
because either both the consumers can gain in welfare (they can become better off) or at least one of then can improve
their satisfaction level without affecting the satisfaction level of the other, if they exchange some amount of goods
(through trade) and move on to the CONTRCT CURVE.
The contract curve is formed by joining the tangential points of consumer A’s and B’s indifference curves, i.e. points Q, R, S
and T. At each point of tangency (Q, R, S and T) the MRSxy for A = MRSxy for B. The contract curve is therefore the locus of
the tangency points of the Indifference curves of the two consumers.
Consumer A and B will, exchange good 1 and good 2, If the trade makes at least One of them better off without making;
the other worse off or makes both of them better off. There will NO EXCHANGE if such exchange makes even ONE
consumer worse off. Let us consider the following possibilities of trade.
Case–I
If through exchange the consumer moved from point E to point S on the contract curve,
(I)
Consumer A will move from ICA2 to ICA3 i.e. he will move to a higher indifference curve representing a higher level
of satisfaction.
(ii)
Consumer B, continues ‘to remain on the same indifference curve ICB2. Although B’s combination of good 1 and
good 2 has changed, he still remains on the same indifference curve.
Case–Il
On the other hand, if the two consumers moved from point E to point R on the contract curve
(i)
Consumer B will move from ICB2 to ICB3 and will enjoy a higher level of satisfaction.
(ii)
Consumer A will continue to remain on ICA2. Consumer A is indifferent between points E and R as they lie on the
same indifference curve ICA2.
Where exactly between the two boundaries, defined by points R and S will the final equilibrium ‘lie shall depend
upon the bargaining power of each consumer.
Case –III
The consumer moved from point E to H on the contract curve as shown in figure-7.
The equilibrium of exchange may be established at point H, where the two consumers gain almost equally as a result of
exchange. ‘Consumer A moves from ICA2 to IC'A and consumer B moves from ICB2 to IC'B. Each of them moves to a higher
indifference curve.
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The lens shaped area (ERFS) in figure 7 is called the Region of Mutual Advantage because both consumers are better off
(than what they were at the original endowment), if they can move to a point in this region All points between R and S are
possible equilibrium points of exchange." We only suppose that the consumers enter into trade to some point in this lens
shaped region. The trade shall continue until there are no more exchanges that are preferred by both consumers. Such a
position is attained at point H.
Suppose A and B are at point H. The region where A can be made better off is given by MHZOB.
The region where B can be made better off is given by JHNOA.
Both regions MHZOB and JHNOA are disjoint. The economic implication is that any movement from point H to any other
point shall make one consumer better off and the other consumer worse off. At allocation H there is no possibility of trade
that makes both consumers better off Allocation H is therefore Pareto efficient.
At a Pareto efficient allocation the IC’s of the two consumers must be tangential to each other inside the ‘box. It is however
possible to have a Pareto efficient allocation in the sides of the Edgeworth box where one consumer has zero consumption
of one good and the other’ consumer has full consumption of the very same good. The student must remember that if the
allocation is at a point where the IC’s of the two consumers intersect then there is a possibility to enter into mutually
advantageous/improving trade.
We are now in, a position to define a Pareto efficient allocation.
A Pareto efficient allocation can be described, as an allocation where:’
1. There is no way to make all the people involved better off; or
2. There is no way to make some individual better off without making someone else worse off; or
3. All the gains from trade have, been exhausted; or
4. There are no mutually advantageous trades to be made, and so on .
Thus a Pareto efficient allocation is one in which there is no feasible reallocation of the goods that would make all
consumers at least as well off and at least one consumer strictly better off,
e A = ( x 1A , x 2A )
= A’s gross demand
e B = ( x 1B , x 2B ) = B’s gross demand ‘
Market Trade
We now explain the trading process that has similarities to the outcome of a perfectly competitive market. There are two
agents A and B. Their endowment is represented by W in’ the Edgeworth Box. The relative prices of good 1 and good 2 is
given by the slope of t 1 t 2 and is equal to − P1 / P2 .
The bundle of goods that a consumer wants to consume at the given price is given by gross demand.
Net demand (also called excess demand) is the difference between gross demand and endowment. Net demand can be
positive (the consumer is a purchaser of that good) or negative (the consumer is a seller of that good) or zero (the
consumer is neither a buyer nor a seller of that good; his endowment is just equal to what he wants to consume for a
particular good).
Net demand = Gross demand – endowment
* *
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In the above figure-8,
For consumer A,
Endowment
W = ( w1A , w 2A ) = (O A R , O AS)
Gross demand
(i)
E A = ( x 1A , x 2A ) = (O A F, O A G )
Net demand for good 1 for consumer A.
e1A = x 1A − w 1A
= OAF − OAR
= −FR
(ii)
Consumer A has an excess supply of FR units of good 1.
Net demand for good 2 for consumer A.
e 2A = x 2A − w 2A
= O A G − O AS
= SG
Consumer A has an excess demand of SG units of good 2.
For Consumer B
Endowment =
w = ( w 1B , w 2B ) = (O B U, O B T)
Gross demand =
(iii)
E B = ( x 1B , x 2B ) = (O B K, O B J )
Net demand for good. 1 for consumer B.
e1B = x 1B − w 1B
= OBK − OBU
= UK = DR
(iv)
Consumer B has an excess demand of UK units of good 1.
Net demand for good 2 for consumer B.
e 2B = x 2B − w 2B
= O B J − O BT
= −JT = −ES
Consumer B has an excess supply of JT units of good 2.
Now consider good 1
Consumer A has an excess supply of FR units of good 1 and consumer B has an excess demand of UK = DR units of good 1.
Now FR  DR  FR > DR
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This means that the amount that A wants to sell is not equal to the amount that B wants to buy. This implies that the total
amount of good 1 that A and B want to consume (their total gross demand, which is OAF + OBK0 is LESS than the total
amount of good 1 available (which is O4V ). Therefore, there is excess supply of FD( = FR – DR) units of good 1.
Now consider good 2
Consumer A has an excess demand of SG units of good 2 and consumer B has an excess supply of JT=ES units of good 2.
Now SG  JT (=ES)  SG > ES
This means that the amount that A wants to buy is not equal to the amount that B wants to sell. This implies that the total
amount of good 2 that A and B want to consume (their total gross demand which is OAG + OBJ) is GREATER than the total
amount of good 2 available (which is QAL). Therefore, there is excess demand of EG(=SG – ES) units of good 2. Thus the
market for good 1 and good 2 is in disequilibrium. Thus given that good 2 has more demand than its total availability,
prices of good 2 shall rise.
* *
Further given that, good 1 has less demand than its total availability, prices of good 1 shall fall. This causes the ‘line t 1 t 2
to became flatter passing through w.
‘Therefore the above adjustment process shall continue until the demand equals supply for each of the goods.
This position is shown In figure 9 where at point M, the total amount that each person wants to buy for each good at the
new prices
P1* and P2* is equal to the total amount of each good available. Therefore there is neither excess supply nor
excess demand of good 1 and good 2.
i.e.
O A x 1A + O B x 1B = O A V
and
O A x 2A + O B x 2B = O A L
Now at M the market is In equilibrium or general equilibrium or Walrasian equilibrium. Please see the definition of general
equilibrium as given at the start of the chapter). This is because all markets (for good 1 and good 2) are cleared and all’
agents (A and B) are maximizing their utility simultaneously.
The calculus conditions describing Pareto efficient allocations: By definition, a Pareto efficient allocation makes each agent
as well-off-as possible given the utility level of the other agent.
The objective is to make agent A as well off as possible given the utility level (u) of agent B. We therefore need to
determine the allocation
(X1A , X 2A , X1B , X 2B ) .
The maximization problem is stated below:
Maximize
U A (X1A , X 2A )
Such that
(1)
U B (X1B , X 2B ) = u
(2)
X1A + X1B = WA1 + WB1 = W 1
(3)
X 2A + X 2B = WA2 + WB2 = W 2
The Lagrangian expression shall therefore be:
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L = U A (X1A , X 2A ) − U B (X1B , X 2B ) − u 
− 1 (X1A + X1B − W 1 )
−  2 (X 2A + X 2B − W 2 )
where
W1 is the total amount of good 1 available.
w2 is the total amount of good 2 available.
 is the Lagrangian multiplier on the utility constraint.
1,2 are the Lagrangian multipliers on the resource constraint.
We differentiate with respect to
L
X1A
L
X 2A
L
X1B
L
X 2B
X1A , X 2A , X1B , X 2B and obtain:
U A
− 1 = 0
X1A
U A
=
− 2 = 0
X 2A
U
= − 1B − 1 = 0
X B
U B
= −
− 2 = 0
X 2B
=
(I)
(II)
(III)
(IV)
We divide I by II and III by IV.
and
U A
= 1
X1A
U A / X1A 1
=
=
= (MRSX1X 2 ) A
U1
U A / X 2A  2
= 2
X 2A
U
 1B = 1
X B
U B / X1B 1
=
=
= (MRSX1X 2 ) B
U B
U B / X 2B  2
 2 = 2
X B
We thus find that at a Pareto efficient allocation,, the marginal rates of substitution for the two agents between the two
goods must be the same,
(MRS x x ) A = (MRS x x ) B =
1 2
1 2
1
2
The above conditions describing Pareto efficiency are similar to Conditions that describe individuals maximization in a
perfectly competitive environment (in absolute terms).
P1
P2
P
)B = 1
P2
(MRS x x ) A =
1 2
(MRS x x
1 2
The Lagrangian multipliers in the Pareto efficiency conditions,  and 2’ are similar, to the prices P1 and P2 under the
consumer choice conditions.
Exchange Equilibrium’ in Algebraic Terms.
Agent A’s demand function for good 1 is
X1A (P1 , P2 )
Agent A’s demand function for good 2 is
X 2A (P1 , P2 )
Agent B’s demand function for good 1 is
X1B (P1 , P2 )
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Agent B’s demand function for good 2 is
X 2B (P1 , P2 ) .
In equilibrium the total demand for each good should be equal to the total supply at a set of prices
X1A (P1* , P2* ) + X1B (P1* , P2* ) = WA1 + WB1
[Demand for good 1 by A plus demand for good 1 by B at prices
(P1* , P2* ) .
(1)
*
1
P
and
*
2
P
is equal to endowment of good 1 by A and
endowment of good 1 by B].
Similarly we can state that for good 2.
X1B (P1* , P2* ) + X 2B (P1* , P2* ) = WA2 + WB2
(2)
We rearrange equation (1) and (2)
X
1
A
(P1* , P2* ) − WA1  + X1B (P1* , P2* ) − WB1  = 0
(3)
The sum of excess demand of agent A for good 1 plus’ the excess demand of agent B for good 1 is equal to zero.
Similarly we can write for good 2.
X
2
A
(P1* , P2* ) − WA2  + X 2B (P1* , P2* ) − WB2  = 0
(4)
Let the net demand ‘function for good ‘1,, b~’ agent A be:
e1A (P1 , P2 ) = X1A (P1 , P2 ) − WA1
(5)
This is also called the excess demand function.
The net demand function for good 1 by agent B.
e1B (P1 , P2 ) = X1B (P1 , P2 ) − WB1
(6)
We now obtain the aggregate excess demand function for good 1 by adding equation (5) and (6).
z1 (P1 , P2 ) = e1A (P1 , P2 ) + e1B (P1 , P2 )
= X1A (P1 , P2 ) + X1B (P1 , P2 ) − WA1 − WB1
The aggregate excess’ demand function for good 2 is defined in a similar manner as
z 2 (P1 , P2 ) = e 2A (P1 , P2 ) + e 2B (P1 , P2 )
= X 2A (P1 , P2 ) + X 2B (P1 , P2 ) − WA2 − WB2
Thus in equilibrium at
(P1* , P2* ) price set aggregate excess demand for each good shall be equal to zero. (from (3) and
(4))
z1 (P1* , P2* ) = 0
z 2 (P1* , P2* ) = 0
We now prove that ‘the value of the aggregate excess demand function is identically equal to zero i.e.
P1 z1 (P1 , P2 ) + P2 z 2 (P1 , P2 ) = 0
Value of aggregate excess demand is’ identically equal to zero FOR ALL possible choice of prices (P 1,P2)and this also
Includes equilibrium prices
(P1* , P2* ) . This is referred as Wairas’ law after the French economist Leon Walras who
contributed substantially to general equilibrium theory.
Proof of Walras Law
We define the budget constraint of the two agents. We know that the value of agents gross" demand is equal to the value
of the agents endowment. In algebraic terms this is stated as (for consumer A).
p1 x 1A (p1 , p 2 ) + p 2 x 2A (p1 , p 2 ) = p1 w 1A + p 2 w 2A
Rewriting the above in terms of net demand.
p1 [ x 1A (p1 , p 2 ) − w 1A ] + p 2 [ x 2A (p1 , p 2 ) − w 2A ] = 0
p1e1A (p1 , p 2 ) + p 2 e 2A (p1 , p 2 ) = 0
This equation says that the value of agent A's net demand is zero.
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This equation shows that the value of a good that A wants to sell plus the value of the other good that A wants to buy
is/MUST be equal ‘to zero i.e. value of purchase + value of sale = 0.
A similar equation is obtained for consumer B
p1 x 1B (p1 , p 2 ) + p 2 x 2B (p1 , p 2 ) = p1 w 1B + p 2 w 2B
p1 [ x 1B (p1 , p 2 ) − w 1B ] + p 2 [p 2 x 2B (p1 , p 2 ) − w 2B ] = 0
p1e1B (p1 , p 2 ) + p 2 e 2B (p1 , p 2 ) = 0
Adding the equations for agent A and agent B together and using the definition of aggregate excess demand,
z 1 ( p1 , p 2 )
and
z 2 (p1 , p 2 ) , we have
p1 [e1A (p1 , p 2 ) + e1B (p1 , p 2 )] + p 2 [e 2A (p1 , p 2 ) − e 2B (p1 , p 2 )] = 0
p1 z 1 ( p1 , p 2 ) + p 2 z 2 ( p1 , p 2 ) = 0
Since the value of each agent’s excess, demand equals zero, the value of the sum of the agent’s excess demands must
equal zero.
It can further be shown that if demand equals supply in one market, demand must also equal supply in the other market.
The Walras’ law must hold for all prices; since each agent must satisfy his or her budget constraint for all prices. Since
Walras’ law holds for all prices, in particular, it holds for a set of prices where the excess demand for good 1 is zero:
z1 (p1* , p *2 ) = 0
According to Walras law it must also be, true that
p1* z1 (p1* , p *2 ) + p *2 z 2 (p1* , p *2 ) = 0
It easily follows from these two equation that if p2 > 0, then we must have
z 2 (p1* , p *2 ) = 0
Thus, as asserted above, if we find a set of prices
(p1* , p *2 ) where the demand for good 1, equals the supply of good 1,
we are guaranteed that the demand for good 2 must equal the supply of good 2. Alternatively, if we find a set of prices
where the demand for good 2 equals the supply of good 2, we are guaranteed that market 1 will be in equilibrium. In
general, if there are markets for k goods, then we only need to find a set of prices where k–1 of the markets are In
equilibrium. Walras’ law then implies that the market for good k will automatically have demand equal to supply.
Relative Prices
As stated above, if there are markets for k-goods, then we only need to find equilibrium in k–1 markets and WALRUS law
would then imply that the market for good k is automatically in equilibrium, This results in k-i independent equations but
as there are markets for k-goods, we need to ‘Find k prices.
To do this; we first need to. understand that if we multiply equilibrium prices
p1* and p *2
by any constant t( > 0), then the.
budget constraint remains unchanged.
p1* x 1 + p *2 x 2 = p1* w 1 + p *2 w 2
(1)
tp x 1 + tp x 2 = tp w 1 + tp w 2
*
1
*
2
*
1
*
2
t (p1* x 1 + p *2 x 2 ) = t (p1* w 1 + p *2 w 2 )
p1* x 1 + p *2 x 2 = p1* w 1 + p *2 w 2
(2)
Equation (1) and (2)are identical.
In the above case, instead of multiplying by a constant t if we multiply by a constant t1 which is in terms of one of the
prices (say p1), then, it is said that all the prices are calculated relative to price p1. The simplest form of this is when we
make price p1 as the numeraire price i.e. peg it equal to one. In this case, we are actually multiplying all the prices by a
constant (in terms of p1)
t' =
1
= 1 . This is explained with the help of the following example:
p1
U A ( x 1A , x 2A ) = ( x 1A )  ( x 2A )1− 
and
U B ( x 1B , x 2B ) = ( x 1B )  ( x 2B )1−
As the above utility functions represent Cobb-Douglas preferences, therefore in equilibrium with prices p1 and p2
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x 1A (p1 , p 2 , m A ) = .
mA
p1
x 2A (p1 , p 2 , m A ) = (1 − )
x 1B (p1 , p 2 , m B ) = .
mA
p2
mB
p1
x 2B (p1 , p 2 , m B ) = (1 − )
Now,
m A = p1 w 1A + p 2 w 2A
and
m B = p1 w 1B + p 2 w 2B
mB
p2
z1 (p1 , p 2 ) = e1A + e1B
= ( x 1A − w 1A ) + ( x 1B − w 1B )
m A  m B
=
+
− w 1A − w 1B
p1
p1
 p w 1 + p 2 w 1A   p1 w 1B + p 2 w 1B 
 + 
 − w 1A − w 1B
=  1 A
p1
p1

 

Similarly,
 p w 2 + p 2 w 2A
z 2 = (1 − ) 1 A
p2


 p w 2 + p 2 w 2B 
 + (1 − ) 1 B
 − w 2A − w 2B
p2



Now we’ need to find p1 and p2 and we know that market for good 1 is in equilibrium i.e. z1(p1,p2) = 0 but there is ‘only one
equation and two variables (p1 and p2). As a result, we peg p = 1, i.e. multiply the prices by constant
t' =
1
= 1.
p1
 p1 w 1A + p 2 w 2A   p1 w 1B + p 2 w 2B 
 + 
 − w 1A − w 1B = 0
z1 (1, p 2 ) = 
p1
p1

 

1
2
1
2
1
1
( w A + 1. p 2 w A ) + ( w B + 1.p 2 w B ) − w A − w B = 0
w 1A +  p 2 w 2A + w 1B + p 2 w 2B − w 1A − w 1B = 0
p 2 (w 2A + w 2B ) + w 1A + w 1B − w 1A − w 1B = 0
p 2 (w 2A + w 2B ) − w 1A (1 − ) − w 1B (1 − ) = 0
p 2 (w 2A + w 2B ) = w 1A (1 − ) + w 1B (1 − )
(1 − q) w 1A + (1 − ) w 1B
(Equilibrium price) p 2 =
w 2A + w 2B
Existence of Equilibrium under Walrus: Law
We are now in a position to generalize the Walras law for markets for k goods. In this case, we need to determine a set of
prices where k–1 markets’ are in equilibrium, then the market for good k shall automatically be in equilibrium where
demand equals supply. The above shall hold true when the following conditions are satisfied.
(I)
The aggregate excess demand function must be continuous. This implies that small changes in prices result in
small changes in quantity demanded.
(2)
Continuity itself requires that no consumer should be able to influence prices through his purchase/sales
behaviour i.e. each consumer’ is very small relative to the size of the market.
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We can therefore conclude that if: the demand for each good varies continuously as. prices vary, then there will. always be
a set of prices where every market is in competitive equilibrium.
First Theorem of welfare Economics
The First Theorem states that a competitive equilibrium is Pareto efficient.
The following conditions need to be satisfied for the First Welfare Theorem to be valid.
(1)
Consumer preferences must be STRICTLY convex.
(2)
Consumers must not have reached the point of satiation with consumption of goods. This implies that the bliss
level of satisfaction must not exist for any consumer.
(3)
Each consumer must be in possession of something of each and every good desired by the other consumers so
that meaningful exchanges can take place.
(4)
The demand function for the good~ must be continuous so that the goods are perfectly divisible.
(5)
There should not be any externalities in consumption.
The following conditions must be present from the production side.
(a)
Increasing returns to scale must not be present. This implies that the production function must exhibit
decreasing returns to scale or constant returns to scale.
(b)
The factors of production must be perfectly divisible.
(c)
There should be no externalities in production.
(d)
The production possibility curve must be strictly concave.
Other Conditions:
I.
The markets (product and factor market) must be perfectly competitive. ‘
II.
There must be existence of perfect knowledge among all the economic agents.
III.
Information must be freely available..
IV.
There should be no transaction costs in resource reallocation.
V.
All types of uncertainties should be ruled out.
Theoretical Proof
When the market is in competitive equilibrium then the set of bundles preferred by A must be above his budget set. The
same shall holds true for B from B’s point of view. Thus the two sets of preferred allocations shall not, intersect. Thus both
consumers shall prefer the equilibrium allocation. We have shown that Pareto efficiency is attained in the Edgeworth Box
when the region where A is better off is disjoint from the region where B is better off. Thus the market equilibrium shall be
Pareto efficient.
Algebraic Proof:
We now algebraically prove that competitive market equilibrium is Pareto efficient using the method of "Reduction Ad
Absurdum". This method first assumes ‘that competitive equilibrium is NOT Pareto efficient and there this’, proposition to
an absurdity.
Let us first assume that the allocation
allocation
( x 1A , x 2A , x 1B , X 2B ) is NOT Pareto efficient. Therefore there shall be some other
( y1A , y 2A , y1B , y 2B ) that is Pareto efficient.
Therefore
y1A + y1B = w 1A + w 1B
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(1)
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y 2A + y 2B = w 2A + w 2B
(2)
( y1A , y 2A )  A( x 1A , x 2A )
(3)
( y1B , y 2B )  A( x 1B , x 2B )
(4)
and
Equations 1 and 2 state that y allocation is feasible and therefore total’ y consumed must be equal to total endowment w
for each of the goods – good 1 and good 2. Equation 3 is drawn from the assumption that
( x 1A , x 2A )
( y1A , y 2A ) is preferred to,
by consumer A and therefore the expenditure on y bundle shall be more than the value of the endowment
for A (shown in equation 5). Equation 4 is drawn from the assumption that
( y1B , y 2B )
is preferred to
( x1B , x 2B )
by
consumer B and therefore the expenditure on the y-bundle shall be more than the value of the endowment for B (shown in
equation 6).
p1 y1A + p 2 y 2A  p1 w 1A + p 2 w 2A
(5)
p1 y1B + p 2 y 2B  p1 w 1B + p 2 w 2B
(6)
We add equation (5) and (6) and obtain equation (7)
p1 ( y1A + y1B ) + p 2 ( y 2A + y 2B )  p1 ( w 1A + w 1B ) + p 2 ( w 2A + w 2B )
(7)
We substitute (1) and (2) in equation (7) to get
p1 ( w 1A + w 1B ) + p 2 ( w 2A + w 2B )  p1 ( w 1A + w 1B ) + p 2 ( w 2A + w 2B )
(8)
Which results in an absurdity as LHS and RHS are Identical,
Therefore the assumption that competitive market equilibrium is not Pareto efficient is highly incorrect.
Thus we have proved the First Theorem of Welfare Economics that all competitive market equilibria are Pareto efficient.
Second Theorem of Welfare Economics
The second theorem states that as long as preferences are convex, every Pareto efficient allocation will be supported as a
competitive equilibrium. Thus for every Pareto efficient allocation there will be a set of prices that ensures. competitive
equilibrium.
Geometric Proof:
Consider figure (7) once again. H is a Pareto efficient allocation that implies that the set of allocations that A prefers
[MHZOB] is disjoint from the set of allocations that B prefers. [JHNOA]. H is a Pareto efficient allocation as the indifference
cuvees of the two consumers
(IC*A , IC*B ) are tangential to each other and it is NOT possible to improve the welfare of
one individual without making the other individual worse off through a reallocation of goods. If the two consumers move
away from H to any other point in the Edgeworth box, than at-least one consumer shall be worse off or both consumers
shall be worse off.
If we draw a straight line through the point of tangency and this straight line does NOT change the preferred bundle of
* *
either of the two consumers, then the slope of this straight line t 1 t 2 given by
p1* / p *2
shall represent the product
price’ ratio which in turn determines the competitive market equilibrium. This is shown in figure 10.
The student must remember that the initial endowment can only be ‘represented by a bundle that lies somewhere on the
*
*
budget line t 1 , t 2 .
Further in some cases it would not be possible to construct a budget line that ensures that the set of allocation that A
prefers is disjoint from the ‘set of allocations that B refers. This has been shown in figure 11. As indifference curves ‘are
having an undulating slope not strictly convex shown as
IC*A1 , IC*A 2
etc B has normal convex shaped indifference
curves. Point e1 is Pareto efficient, but there are no prices at which A and B will want to be at e1. This is because at the
price
p1 / p 2
(given by the slope of the line
t 1 t 2 ), a would like to be at point e2 on IC A 2
and B would like to be at
point e1 on ICB. There will be aggregate excess demand for good 1 and (negative) aggregate excess demand for good 2 at
price ratio
p1 / p 2
. Overall demand shall not be equal to supply at the price ratio
p1 / p 2
.
Thus when preferences are non-convex Pareto efficient allocations cannot achieve competitive market conditions.
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Monopoly in the Edgeworth Box
We stated that a competitive market shall be Pareto’ efficient (First Theorem of Welfare Economics). This implies ‘that if
one or more conditions for existence of perfect competition are violated, then the allocation mechanism shall not be
Pareto Efficient. We now consider a resource allocation mechanism that does not lead to a Pareto Efficient Outcome.
Let there be two consumers A and B constituting the market. Consumer A acts as a monopolist and quotes prices to
consumer B. Given the price set (for good 1 and 2) by A, consumer B shall decide his consumption behaviour (how much to
consume and how much to trade/exchange). We assume that consumer A is aware of the demand of consumer B (how
much consumer B shall purchase when prices vary). Consumer A can be aware of consumer B’s price offer curve.’ [This, is
because the demand curve is derived from the price offer curve]. The price offer curve of B shows all successive positions
of consumer B’s equilibrium at different prices. Now consumer A would like to maximize his ‘utility given the demand
behaviour of B. Therefore A’s equilibrium shall be attained when consumer A’s indifference curve becomes ‘tangential to
consumer B’s price offer curve. This is explained with figure 12. A’s indifference curve ICA shall not be tangential to
consumer B’s indifference curve IC8. Therefore there will be a breakdown of Pareto efficiency condition. The existence of
monopoly results in the allocation defined by point X being Pareto inefficient. In the figure there is a lens-shaped region
(marked as XY). This region is common between the indifference curves ICA and ICB. It shows that both A and B can still be
made better off or atleast one consumer can be made better off without making anyone’ else worse off, Thus the
monopoly allocation defined by X shall be Pareto inefficient, It will however be noticed that consumer B’s indifference
curve is tangential to the budget constraint whose slope is defined by
p1 / p 2
A’s indifference curve is not tangential to the budget constraint with slope
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.
p1 / p 2
.
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Now consumer A starts practicing perfect price discrimination.
There is however one case wherein’ Pareto efficiency can be achieved, under monopoly. This is possible, only when the
monopolist practice first degree (or perfect) price discrimination.
Under perfect price discrimination the monopolist sells each unit of output at the maximum, price that the consumer is
willing to pay. At this maximum price that the consumer is willing to pay, the consumer is JUST INDIFFERENT between
buying or not buying that Unit of the good. In this manner, the monopolist is able to appropriate the entire consumer
surplus.
Suppose in the above discussed. case wherein’ ‘consumer A was the monopolist and was setting price, had practiced
perfect price ‘discrimination. The; initial, endowment is represented by w. A sells one unit of good, 1 to consumer B at the
maximum price that consumer B is willing to pay. B will remain on the same IC8 and consumer A will move to a higher ICA
from ICA1 to ICA2 defined by point F. Then consumer A sells the next unit to consumer B at the maximum price that B is
willing to pay B remains on ICB (as he is just indifferent between buying and not buying) and once again the entire
consumer surplus is appropriated by A. Consumer A moves to ICA3 at point G.
In this manner consumer A would move to his highest possible indifference curve ICA1, which becomes tangential to ICB, at
point E. All the gains in the exchange process are appropriated by A with consumer B remaining on the same utility level
defined by IC8 which passes through w. B is just well off at E as he was at his endowment w. Point E is a Pareto efficient
allocation as consumer A shall be as well off as possible given the indifference curve of consumer B.
Thus existence of perfect price discrimination ensures that a Pareto efficient outcome is attained.
Implications of the First Welfare Theorem
The First ‘Theorem of Welfare Economics states that a perfectly competitive equilibrium is Pareto efficient.
The "invisible hand" (a term used by Adam Smith describing the fair play of market forces of demand and supply) of the
market guides people towards socially desirable. choices. This implies that "an equilibrium produced by competitive
markets will exhaust all possible gains from exchange".
The first theorem of, welfare economics is also called the "Invisible hand theorem":
There are three implicit assumptions of the first theorem of welfare economics. These’ implicit assumptions and their
analysis is discussed below:’’:
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Assumption 1): Each agent is concerned With their own consumption bundle and are not concerned about what other
agents consume.
When one agent is concerned with another agent’s consumption we say that a consumption externality is present. When
consumption externalities ‘are present then the perfectly ‘competitive market equilibrium shall. generally not result in a
Pareto efficient outcome.
Assumption 2): The number of agents is so large that the market power of each agent is zero. The large number of agents
in the market ensures that agents behave competitively. When millions (sometimes billions) of agents are present then the
structure of competitive markets has a desirable outcome that the trading process results in a Pareto efficient allocation.
Assumption 3): The third implicit assumption requires that a competitive equilibrium must actually exist. This can only be
possible only when the number of producers and consumers is extremely large and this ensures that no single producer (or
a single consumer) can influence the market prices by increasing or decreasing his production level (or consumption level).
Each producer (or consumer) has a ‘small share relative to the size of the market. Thus each firm and each consumer is a
price taker.
Usefulness of First Welfare Theorem
Competitive markets ensure that agents need to posses minimum information to make optimal decision. This implies that
a consumer needs to know only the prices of the goods that he is consuming The consumer under perfectly competitive
markets shall not be concerned with the name of the producer, the manner in which the goods have been produced or the
location where the goods have been produced. The only information that a consumer needs is "PRICES". With "price"
information being available, a consumer can determine his demand leading to an. efficient outcome.
Thus the structure of trading defined by perfectly competitive market ensures that information ‘needs are economized.
Implications of the Second Welfare Theorem
"The second welfare theorem tells us that as long as preferences are convex, then every Pareto efficient allocation is’ a
competitive equilibrium for some initial allocation of resources", (Bernheim et. al. page 593). Any Pareto efficient
allocation can be supported by the competitive markets (market mechanism). A’ competitive market determines product
prices. ‘These product prices have two roles to play.
(a)
Allocative Role
(b)
Distributive Role
(a) Allocative Role
The Allocative role of prites is to indicate relative scarcity of the products.
Suppose p1 = 5 and p2 = 4Rs. Per unit then p1/p2 = –5/4.
This implies that 1.25 units of good 2 are equal to 1 unit of good 1 (1x1 = 1.25x2)
Good x1 is relatively scarce and therefore commands’ a high price (relative to price of good x 7).
(b) Distributive Role
Competitive markets not only determine product prices, they also determine factor pries. Factor prices represent the
factor earnings and hence represent factor incomes. Factor incomes, determine how much of different goods different
agents can purchase. Thus, the distributive role of (factor) prices ,is to determine how much different, agents can purchase
and hence consume. What an agent can purchase and consume depends upon his factor income/earnings. Factor incomes
are primarily determined by how much or value of resources (endowment) that an agent initially possess. Therefore we
need to be able to ‘measure the magnitude of’ an agent’s endowment. An agents’ endowment, in most cases, consists of
his own labour power that consists of labour services that an agent can offer for sale (and not actual labour supply). ‘It is
very much possible that the initial endowment of different agents are extremely inequitable. Some agents are naturally
endowed with certain skill sets (artists,’ musicians, ‘sculptors, craftsmen etc.). If the’ initial distribution of endowment is
highly inequitable, the final distribution (competitive allocation) shall also be’ inequitable. Here comes the role of the state.
If the state can redistribute the initial allocation "appropriately" ‘then ,the market mechanism can achieve .a Pareto
efficient allocation’. The state can redistribute the initial allocation of resources through
(i)
Lump-sum transfers
(ii)
Taxes based on the value of endowment
(iii)
Taxes that depend upon the choices that a consumer makes.
(a) Consider the case of lump sum transfers
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Suppose point 3 represents a ‘Pareto efficient allocation for agent A and B. This implies ICA is tangential to ICB and A’s
preferred set of allocations is disjoint from B’s preferred set of allocations.
Case 1: A and B’s initial endowment was given by w, where ICA0 and ICB0 were intersecting. Through the exchange process
the two agents could have moved to point 3, determining the competitive equilibrium. The construction of the straight line
FG (their common tangent) shall help us to determine competitive price ratio |p1/p2|.
Case 2: The initial endowment of agent A and B was given by Z. To reach point, we first undertake a lump sum transfer to
reach point w on FG. WZ units of good 1 are transferred from consumer A to B. The amount of resources received by B or
surrendered by A is fixed. This amount does not depend upon consumer choices. After the lump sum transfer, the agents
are at point w and now we allow the competitive markets to operate. Agent A and B shall enter into the exchange process
and reach point J.
We have shown in case 2 that a Jump sum transfer does not compromise on the issue of "efficiency" because lump sum
transfers do not distort consumer choices.
The state can use lump sum transfers’ and the competitive’ markets (market mechanism) to achieve both efficiency and
equity. In case 2, it is extremely important for the state to know the endowments of each agent in order to determine
whom to tax and whom to subsidize,
b) Taxes based on the value of endowment
"The state could tax one consumer on the basis of value of his endowment and transfer this money to another". There will
be no loss of efficiency as long as the taxes are based on he value of the consumer’s endowment.
c) Taxes based on the choices that consumers make
However when taxes are based on the choices that consumers make, then there shall be inefficiencies.
The above analysis is summarized in the following figure 15
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Hal Varian Workbook Numerical & Detailed Solutions (WORKBOOK)
Q1.
A and B are two consumers consuming two goods x1 and x2. A has an initial endowment of (60,10) with utility
function UA(x1,x2) =
x 1A + x A2 . B has an initial endowment of (20, 30) with utility function
U B ( x 1 , x 2 ) = x 1B + x B2 .
(a)
(b)
(c)
(d)
(e)
(f)
Draw an edgeworth box. Mark the initial endowment E and draw their ICs passing through E.
Which equation needs to be satisfied at any pareto optimum? Show the locus of pareto optimum allocations.
If price of good x1(p1) is the numeraire price = Re.1, what will be the price of good x2(p2)?.
What will be the equilibrium amount of x1 and x2, consumed by A and B?
Can A consume a bundle more than (55, 15)?
If there are 1000 consumers like A and 1000 like B with same endowment and tastes, will the equilibrium prices
for A and B be competitive equilibrium prices? Will demand equal supply for x1 and x2?
Ans. (a)
For A
For B
Endowment = (60, 10)
U A (x + x
A
1
A)
2
Endowment = (20, 30)
U B = x 1B + x B2
Total availability of x1 = 60 + 20 = 80
and
x2 =10 + 30 =40
A’s IC is linear downward sloping passing through E with utility = 60 + 10 = 70 representing perfect substitutes. B’s IC is
convex shaped passing through E with utility = 20  30 = 600 representing Cobb Douglas preferences.
(b)
At pareto optimum,
(MRS x x ) A = (MRS x x ) B
1 2
1 2
(MRS x x ) = −MU / MU A2 = −1
A
1 2
(MRS x x
x B2
) = −MU / MU = − B
x1
B
1 2
A
1
B
1
B
2
x B2
x 1B

1=

x 1B = x B2
(c)
At competitive equilibrium .
and QBF shows all such points.
MRS x x =
1 2
p1
p2
p1
p2
If p1 = Re .1
1=
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9811439887 18

(d)
p 2 = Re .1
For B,
MRSBx x = −
1 2
Budget Line,
x B2
x 1B
x 1B + x B2 = 20  1 + 30  1
x 1B + x B2 = 50
At equilibrium
(MRS x x ) B = −
1 2
p1
p2
x B2
=1
x 1B
x 1B = x B2
Putting this in budget line
x 1B + x B2 = 50
2x 1B = 50
x 1B = 25 = x B2
(e)
(f)
Q2.

x 1A = 80 − 25 = 55

x A2 = 40 − 25 = 15
Cost of bundle (55,15)
= 551 + 151
= 55 + 15
= Rs.70
A’ s income = l60 + l10 = 70
 A cannot a bundle more than (55, 15) because then the cost of that bundle will be more than his income as
at (.55, 15) cost is just equal to his income.
When there are 1000 consumers like A and B each with same taste and endowment, then the equilibrium prices
will be competitive equilibrium prices and demand will be equal to supply for x1 and x2 like for A and B because
their preference and hence MRS are same.
A and B are two consumers consuming two goods x1 and x2. A has an initial endowment of (1,4) with utility
function UA(x1,x2) =
(a)
(b)
(c)
(d)
Ans. (a)
x 1A x A2 ;
B has an initial endowment of (7, 0) with utility function UB(x1,x2) = min
x
B
1
, x B2 
.
Draw an edgeworth box. Mark the initial endowment E and draw their ICs.
Show the locus of pareto optimum allocations.
If price of good x1(p1) = Re. 1 and price of good x2(p2) = Rs. p, write the demand function of x, for consumer
What is the value of A’s initial endowment? Write the demand function of x 2, for A.
For A
For B
Endowment = (1, 4)
UA(x1,x2) =
A
1
x x
A
2
Endowment = (7,0)
UB(x1,x2) = min
x
B
1
, x B2 
Total availability of x1 = 1+ 7 = 8 and x2 = 4 + 0 = 4
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A’s ICs are convex shaped reflecting Cobb Douglas preferences. B’s ICs are
L-shaped reflecting perfect complements. Initial, endowment is marked E.
(b)
Pareto optimality is achieved at the kink of B’s ICs where
(MRS x x ) = (MRS x x )
A
1 2
(c)
x 1B = x B2
as
B
1 2
Therefore, O8F shows the locus of pareto optimum allocations.
p = Re. 1
p2 = Rs. p
For B, Budget Line =
x 1B + px B2 = 7  1 = 7
Equilibrium will be attained when
x 1B = x B2
due to perfect complements.
Putting this in budget line
x 1B + px B2 = 7
x B2 (1 + p) = 7
7
x n2 =
1+ p
(c)
For A, Budget Line
= x 1A + px A2 = 1 1 + p  4
= x 1A + px A2 = 1 + 4p
 Value of his endowment = 1+ 4p
Due to Cobb – Douglas preference, equilibrium level of x2 will be
1
(income)
=
2 (price of x 2 )
1 (1 + 4p)
x A2 =
2 p
1
x A2 =
+2
2p
x A2 =
Q3.
There are 2 consumers A and B consuming 2 goods – x1 and x2. A has an initial endowment of (8, 12) with utility
function
U A ( x 1A , x A2 ) = x 1A + x A2
U B ( x 1B , x B2 ) = x 1B + 4 x B2
B has an initial endowment of (8, 4) with utility function
.
(a)
Draw an endgeworth box with A and B's IC passing through initial, endowment E.
(b)
Which condition must be satisfied at pareto optimum and draw the locus of such allocations?
(c)
Find the competitive equilibrium prices and quantities?
Ans. (a)
For A
For B
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Endowment = (8, 12)
Endowment = (8,4)
U A ( x , x ) = x + 2 x 1A
A
1
A
2
A
1
U B ( x 1B , x B2 ) = x 1B + 4 x 1B
Total availability of x1 = 8 + 8 = 16 and x2 = 12 + 4 = 16
A and B’s ICs are convex shaped passing through E with
U A = 8 + 2 12
and
U B = 8 + 8 = 16
respectively reflecting quasilinear preferences.
(b)
U B = x 1A + 2 x A2
(MRS x x
1 2
MU1A
) =−
= − x A2
A
MU 2
A
U B = x 1B + 4 x B2
(MRS x x
1 2
MU1B
) =−
= − x B2
B
MU 2
B
At pareto optimum .
(MRS x x ) A = (MRS x x ) B
1 2
x = x
A
2
1 2
B
2
x A2 1
=
x B2 4
(1)
Also when MRS are equal there must be neither excess demand nor excess supply i.e. total ‘availability should
equal total demand.

x A2 + x B2 = 16
(2)
Putting (1) in (2)
x A2 + 4x A2 = 16
5x A2 = 16
x A2 = 3.2
Putting this in (1)
x B2 = 4x A2
x B2 = 4  3.2
x B2 = 12.8
As there is a unique pareto optimal allocation therefore, contract ,curve is horizontal , and parallel to x–axis,
represented as MN. This unique allocation is due to the fact that the preferences are quasilinear.
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(d)
According to second welfare theorem when preferences are ‘convex, each pareto optimal allocation is a
competitive equilibrium
Therefore, competitive equilibrium quantity is
At equilibrium,
(MRS x x ) A = −
1 2
x A2 = 3.2 and x B2 = 12.8 .
p1
p2
p1
p2
p
3.2 = 1
p2
p2
1
=
p1
3.2
x A2 =


Q4.
A and B are two consumers consuming two goods – x1 and x2. A has an initial endowment of (3, 2) with utility
function
U A ( x 1A , x A2 ) = x 1A x A2
U B = ( x 1B , x B2 ) = x 1B x B2 .
(a)
A and B together consume twice as much x2 as x1.
Draw an edgeworth box and mark the initial endowment W.
Draw their ICs when both
(b)
(c)
(d)
(e)
and B has an initial endowment of (1, 6) with utility function
U A ( x 1A , x A2 ) = 6
and
U B ( x 1B , x B2 ) = 6 .
Which condition must be satisfied at pareto optimum show the locus of such allocations.
At any pareto efficient allocation, what is the slope of A's IC ?
What is the price ratio and consumption bundles at competitive equilibrium ?
Draw A’s budget line and mark A and B s competitive equilibrium bundle.
Ans. (a)
For A
Endowment (3, 2) Endowment = (1, 6)
For B
U A ( x 1A , x A2 ) = x 1A x A2
U B ( x 1B , x B2 ) = x 1B x B2
Total availability of x1 = 3 + l = 4 and x2 = 2 + 6 = 8
UA = 6
x 1A
1
2
3
x A2
6
3
2
x 1B
1
2
3
x B2
6
3
2
These points form IC,
UB = 6
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These points form IC,,,
A and B’s ICs’ are convex shaped with utility 6 passing through w reflecting. Cobb Douglas preferences.
(b)
U A = x 1A , x A2
(MRS x x
1 2
MU1A
x A2
) =−
=− A
MU A2
x1
A
U B = x 1B , x B2
(MRS x x ) B = −
1 2
MU1B
x B2
=
−
MU B2
x 1B
At pareto optimum
(MRS x x ) A = (MRS x x ) B
1 2
−
(c)
A
2
A
1
x
x
=−
x
x
1 2
B
2
B
1
As they together consume twice as much x2 as x1 therefore, contract curve is formed by, (1, 2), (2, 4); (3, 6’), (4, 8)
represented as OAOB.
At any pareto efficient allocation, they consume twice as much as x2 as x1, therefore,
(MRS x x ) A = −
1 2
x A2
x 1A
x 1A
=− A
x1
(d)
= –2
At competitive equilibrium,
p1
p2
A
x
p
− 2A = − 1
x1
p2
p
2=− 1
p2
MRS = −
(1)
Budget line for A,
p1 x 1A + p 2 x A2 = 3p1 + 2p 2
From (1)

2p 2 x 1A + p 2 x A2 = 3  2p 2 + 2p 2
p 2 (2x 1A + x A2 ) = 8p 2
2x 1A + x A2 = 8
As they consume twice as a much as x2 as x1

2x 1A + 2x 1A = 8

4x 1A = 8

x 1A = 2

x A2 = 2x 1A
=22=4
For B,
x 1B = 4 − 2 = 2
VIKAAS WADHWA SIR
9811439887 23
x B2 = 8 − 4 = 4
(c)
A and B’s competitive equilibrium bundle is represented as C (2, 4)
A’s budget line,
2x 1A + x A2 = 8
x 1A = 0, x A2 = 8
When
When
x A2 = 0, x 1A = 4
This is represented as JK.
Q5.
A and B are two consumers consuming two goods – x1 and x2, A has an initial endowment of (0, 12) with utility
function
U A ( x 1A , x A2 ) = x 1A + 2x A2 ,
U B ( x 1B , x B2 ) = minx 1B ,2x B2 
(a)
(b)
(c)
Ans. (a)
B has an initial endowment of
(12, 0) with utility function
Draw an edgeworth box and mark the initial endorment E. Draw their ICs and Locus of. pareto optimal
allocations.
What is the price, ratio of the two goods in equilibrium ?
Write A’s budget line and quantity of x1 and x2 consumed in equilibrium by each of the two consumers.
For A
For B,
Endowment = (0, 12)
U A = x + 2x
A
1
A
2
Endowment= (12, 0)
U B = minx 1B ,2x B2 
Total availability of x1 = 0 + 12 = l2
and
x2 = 12 + 0 = 12
A’s ICs are linear downward sloping with a slope of –1/2 reflecting perfect substitutes. B’s ICs are L–shaped with kink at
x 1B = 2x B2
reflecting perfect complements.
All pareto optimal allocations will lie at the kink of L–shaped ICs because competitive equilibrium always lies at the kink
and each competitive equilibrium is pareto efficient (first Welfare Theorem). The contract curve is represented as MOB.
(b)
U A = x 1A + 2x A2
− MU1A
1
(MRS x x ) A =
=−
A
MU 2
2
1 2
At competitive equilibrium
MRS x x = −
1 2
+
(c)
p1
p2
p
1
=+ 1
2
p2
A’s budget line, .
VIKAAS WADHWA SIR
9811439887 24
p1 x 1A + p 2 x A2 = p1  0 + p 2  12
x 1A + 2x A2 = 24
(1)
Equilibrium will be at the kink of L-shaped IC’s where
x 1B = 2x B2
But as
x 1A + x 1B = 12 ,
therefore,
(12 − x ) = 2(12 − x A2 )
A
1
12 − x 1A = 24 − 2x A2
2x A2 − x 1A = 12
(2)
Adding (1) & (2), we get
4x A2 = 36
x A2 = 9
Putting this in equation (1)
x 1A + 2  9 = 24
x 1A = 24 − 18
x 1A = 6

A’s equilibrium bundle = (6, 9)
B’s equilibrium bundle = (12 – 6, 12 – 9) = (6, 3)
Q6.
As

Consider a pure exchange economy with two-consumers and two goods. At a give a pareto efficient allocation
consumer A has a MRAA = –2. What will be the MRAB of Consumer B?
At any pareto efficient allocation
MRAA = MRABMRA = MRSB
A
MRA = –2
MRA = –2
Q7.
Consumer A loves good y and hates good x. His utility function is
Ans.
(a)
(b)
(c)
(d)
(e)
Ans.
1
U ( x , y) = y − x 2 .
4
Consumer B. likes
both good x and y and his utility function is U(x,y) = y + 2 x. A has an endowment bundle of (8, 0) and B has an
endowment bundle of (8, 16).
Draw A and B’s ICs passing through endowment bundle E.
At pareto optimal allocation, how many units of good x will consumer A choose? Show the locus of pareto
optimal allocations.
At competitive equilibrium how many units of good x will consumer B have? What will be his marginal utility for
good x and y?
If price of good y = Re.1 per unit, then what will be the price of good x when B consumes 16 units of good x ?
Determine the ‘equilibrium allocation of A and B ?
(a) A’s ICs are upward sloping because if consumption of good x increases, consumption of good y has to increase
as x is an economic bad which gives negative utility to the resulting consumption of economic good to rise. The
preference direction of ICs are towards the economic good i.e. North-West. .
VIKAAS WADHWA SIR
9811439887 25
(b)
(c)
B’s ICs are negatively sloped Convex ICs as both the goods are economic goods for him.
The total endowment of good x is 8 + 8 = 16.
The total endowment of good y is also 0 + 16 = 16.
Therefore the Edgeworth box will be a square.
A initially has 8 units of x and 0 units of y and is at point EA.
B initially has 8 units of x and 16 units of y and is at point EB = EA.
At a pareto efficient allocation, A will consume 0 units of x because it is an economic bad for him giving him
negative utility. At all points on the y–axis with 0A as origin, A shall have 0 units of good x and all units of good y.
So it represents the contract curve.
A will give up his initial endowment of good x (8units) which is an economic bad and exchange for units of good y
which Is an economic good for A. When A gives up 8 units of x,’ then all these units of x go to B. B already had B
units of x; therefore total units of good x that B shall have in competitive equilibrium shall be 16.
U ( x , y) = 2 x + y
− MU1 − 1 / x
MRS xy =
=
MU 2
1
1
MU1 =
x
For B, We know
(d)
At 16 units of x, MU1 = 1/4, MU2 = 1
Iff the price good y = 1.
Then equilibrium for B shall be attained when MRSxy =
− px
1
=
py
x
−1 − px
=
py
16
1
Given, py =1;
= px
16
1
Therefore, p x =
4
−
(e)
For A, at equilibrium
good x = 0 as it is an economic bad for him Budget Line,
p x x + p y y = 8p x + 0.p y
1
1
x + 1.y = 8. = 2
4
4
VIKAAS WADHWA SIR
9811439887 26
When x = 0 , y = 2
Therefore equilibrium allocation is (0, 2)
For B, at equilibrium
good x = 16 as pareto optimal allocations, lie on y–axis Budget Line,
1
1
x + y = . .8 + 16
4
4
1
x + y = 18
4
When x = 16, y = 14x = 16, y = 14
Therefore, equilibrium allocation is (16, 14)
Q8.
A and B are two consumers consuming two goods – x1 and x2. They have total B units of x1 and 8 units of x2 to
consume divide between themselves. Both have same utility function U =
(a)
Ans.
ax, x 1 , x 2 m
Draw an edgeworth box. Draw their ICs and show the locus of pareto optimal allocations,
Total availability of x1 = x2 = 8
A and B’s ICs are inverted L– shaped with kink where x1 = x2.
The contract curve will be formed by joining (E1, E2 etc.) all those points where the kinks of both the consumer’s
ICs intersect because then the MRS of A and MRS of B will be equal. This is represented as OAOB.
Q9.
(a)
(b)
(c)
(d)
(e)
Ans.
Consumer A is happiest when he has 8 units of good x1 and 4 units of good x2 per day and his ICs are concetric
around (8,4). His mother believes that perfect bundle would be 2 units of good x 1 and 7 units of good x2 per day
as she feels that the sum of the absolute values of the differences between the amounts of’ each good consumed
and the ideal amounts should be smaller.
Draw the locus of combinations that his mother thinks are exactly as good for A as (6, 6).
Draws the locus of combinations that his mother thinks are e~act!y as good for A as (8, 4).
Draw IC representing the locus of combination that A likes just as well as
(7, 8).
Shade the area where both A and his mother agree that it is better than (7, 8).
Sketch the locus of pareto optimal bundles and to which point does it extend from (8, 4).
(a) Given:
Ideal Actual Differences
x=2
x=6
|4|
y =7
y=6
|1|
Sum of difference = |5|
All bundles actually consumed that should have “sum of the absolute values of the differences between the
amounts of each food consumed and the ideal amounts “equal to 5. are calculated below:
Ideal Actual (J) Differences
x=2
3
|1|
y=7
3
|4|
Sum of difference = |5|
in absolute terms
Ideal
x=2
VIKAAS WADHWA SIR
Actual (K)
4
Differences
|2|
9811439887 27
y=7
4
|3|
Sum of difference = |5|
in absolute terms
Ideal
x=2
y=7
Actual (M) Differences
x=5
|3|
y=5
|2|
Sum of difference = |5|
in absolute terms
Ideal
x=2
y=7
(b)
(c)
Actual (N) Differences
2
|0|
12
|5|
Sum of difference = |5|
in absolute terms
Therefore the combination of bundles that the mother thinks are exactly as good (for A) as (6, 6) shall be:
Combination J (3, 3)
Combination K (4, 4)
Combination M (5, 5)
Combination N (2, 12)
and similar such bundles can be calculated by repeating the exercise explained above. Thus the indifference
curve (for A) as considered by A’s mother shall be Rombus (IC3) shaped as shown in figure below.
This exercise is repeated entirely for bundle (8, 4) also and we determine 1C, for which the sum of the absolute
value of difference is
Ideal Actual Differences
x=2
8
|6|
y=7
4
|3|
Sum of difference = |9|
in absolute terms
(8, 4) bundle shown as point E is consumer A’s bliss point or ideal bundle.
According to the information given in. the question, A’s ICs are concentric circles around his bliss point (8, 4).
Therefore, the locus of combinations that he likes just as well as (7, 8).will be a circle centred at (8, 4) with radius
17
(d)
( (7 − 8)
2
+ (8 − 4) 2
)
. This is shown as ICAB in the following figure.
A will be better off than (7, 8) as he move towards his bliss point (8, 4) and for A’s mother it will be better off
than (7, 8) if he moves towards her ideal bundle (2,7). Therefore, ‘both of them will agree to be better off in the
region shaded with horizontal lines i.e. PQSK.
VIKAAS WADHWA SIR
9811439887 28
(e)
We need to draw all the ICs of A (concentric circles) and his mother (rhombus) and on joining the points where
each concentric circle is tangential to one of the rhombus we get the contract curve as EFE2.
Q10.
There are two kinds of consumers old and young two time periods –1 and 2. They consume one good i.e., x1.
There are N1 old consumers which have, an income in period 1= m1 = 1 and income in period 1= m2 = 0. There are
N2 young consumers which have an income in period 1 = m 1 = 0 and income in period 2 = m2 = F. They consume
c1 in period 1 and c2 in period 2. Their utility function’ is
(a)
U(c1 , c 2 ) = c1 c12−  .
(h)
If p1 = 1 and r → rate of interest, calculate the present value of consumption bundle (c1, c2), income of
consumers, income of young consumers ?
What is the budget line for old and young consumers ?
What will be the demand for x1 in period 1 and 2 for old consumer and young consumer? If r = 0, how much x,
will a young consumer choose’ in period 1 ?
If r = 0, for what value of  will a young consumer choose the same amount in each period? If = 0.55, what
would r be so that young consumers consume the same amount in each period ?
What is the total amount earned by old consumers and young consumers ?
What is the demand of x1 in period 1 by old and young consumers and what is their total demand ?
Write an equation showing that total demand is equal to income. Write an expression for equilibrium value of r
and solve it when N1 = N2, l and
 = 11/21.
Show that the same r equalizes total demand and income in period 2.
Ans.
(a) Present Value of
(b)
(c)
(d)
(e)
(f)
(g)
(c1 , c 2 ) = c1 +
c2
1+ r
Present value of income of old consumers 1+ 0 = 1
1*
1*
=
Present value at income of young consumers = 0 +
1+ r 1+ r
(b)
Budget Line for old consumers,
c1 +
c2
m
= m1 + 2 = 1
1+ r
1+ r
Budget Line for young consumers,
(c)
c2
m12
1*
c1 +
= m1 +
=
1+ r
1+ r 1+ r
 1− 
Utility function, U = c1 c 2
For old consumers, equilibrium consumption


c1 = a.1
(1 − ).1
c2 =
= (1 − ) /(1 + )
1 /(1 + r )
For young consumers, equilibrium consumption
(d)
1*
1+ r

c1 = 

1* /(1 + r )
c 2 = (1 − )
= (1 − )1*
1 /(1 + r )
If r = 0r = 0,
c1 for young consumer = I*
If r = 0 & c1 = c2 for a young consumer, then
c1 = c2
1* = (1– )1*
2 = 1
 = 1/2
Now, If  = 0.55 and c1 = c2 for young consumers, then
cl = c2
VIKAAS WADHWA SIR
9811439887 29
0.55 (1* )
= (1 − 0.55)1*
1+ r
0.55
= 0.45
1+ 4
0.55
r=
−1
0.45
(e)
(f)
r = 0.22
Total earnings of young consumers = N.,X (income of each)
= N2I*
Total earnings of N1X (Income of each)
= N2I
Demand of x1 in period 1 by,
Old consumer = I
Young consumer =
I A
1+ r
 Total demand of consumers =
(c)
aI*
N 1 I + N 2
1+ r
Total demand ~n period .1 = Total earnings in period 1
In period 1.
I *
N 1 I + N 2
= N1I
1+ r
N 2 I *
= N 1 I − N 2 I
1+ r
N 2 I *
1+ r =
−1
N1 I(1 − )
N 2 I *
r=
−1
N1 I(1 − )
It N1 = N2, I = I*,  = 11/21
N 1 I
−1
N1 /(1 − )
11 / 21
=
−1
1 − (11 / 21)
11 / 21
=
−1
1 / 21
1
= = 0.10
10
r=
(h)
Total demand =
in period 2
Total earnings
in period 2
N1 (1 − ) /(1 + r ) + N 2 I *
N1 (1 − ) /(1 + r ) = N 2 I * − N 2 (1 − q)I *
N 2 I*
N 2 (1 − )I *
1+ r =
−
N1 (1 − )I N1 (1 − )I
VIKAAS WADHWA SIR
9811439887 30
N 2 I * − N 2 (1 − )I *
1+ r =
N1 (1 − )I
1+ r =
N 2 I * − N 2 (1 − 1 + )
N1 (1 − )I
N 2 I *
r=
−1
N1 (1 − )I
If N1 = N2, I = I* and  = 11/21
N 1 I
−1
N1 (1 − )I
11 / 21
=
−1
1 − (11 / 21)
11 / 21
1
=
− 1 = = 0.10
10 / 21
10
r=
VIKAAS WADHWA SIR
9811439887 31
EXCHANGE(READINGS)
Up until now we have generally considered the market for a single good in isolation. We have viewed the demand and
supply functions for a good as depending on its price alone, disregarding the prices of other goods. But in general the
price of other goods will affect people’s demands and supplies for a particular good. Certainly the prices of substitutes
and complements for a good will influence the demand for it and, more subtly, the prices of goods that people sell will
affect the amount of income they have and thereby influence how much of other goods they will be able to buy.
Up until now we have been ignoring the effect of these other prices on the market equilibrium. When we discussed the
equilibrium conditions in a particular market, we only looked at part of the problem: how demand and supply were
affected by the price of the particular good we were examining; This is called partial equilibrium analysis.
In this chapter we will begin our study of general equilibrium analysis: how demand and supply conditions interact in
several markets to determine the prices of man goods. As you might suspect this is a complex problem, and we will
have to adopt several simplifications in order to deal with it.
First, we will limit our discussion to time behaviour of competitive markets, so that each consumer or producer will
take prices as given and optimize accordingly. The study of general equilibrium with imperfect competition is very
interesting but too difficult to examine at this point.
Second, we will adopt our usual simplifying assumption of looking at the smallest number of goods and consumers that
we possibly can. In this case, it turns out that many interesting phenomena can be depicted using only two goods and
two consumers. All of the aspects of general equilibrium analysis that we will discuss can be generalized to arbitrary
numbers of consumers and goods, but the exposition is simpler with two of each.
Third we will look at the general equilibrium problem in two stages. We will start with an economy where people have
fixed endowments of goods and examine how they might trade these goods among themselves; no production will be
involved. This case is naturally known as the case of pure exchange. Once we have a clear understanding of pure
exchange markets we will examine production behaviour in the general equilibrium model.
31.1 The Edgeworth Box
There is a convenient graphical tool known as the Edgeworth box that can be used to analyze the exchange of two goods
between two people. The Edgeworth box allows us to depict the endowments and preferences of two individuals in
one convenient diagram, which can be used to study various outcomes of the trading process. In order to understand
the construction of an Edgeworth box it is necessary to examine the indifference, curves and the endowments of the
people involved.
Let us call the two people involved A and B and the two goods involved 1 and 2. We will denote A’s consumption bundle
by
X A = ( x1A , x 2A ) , where x1A ; represents A’s consumption of good 1 and x represents A’s consumption of good
2. Then B’s consumption bundle is denoted by
X B = ( x1B , x 2B ) . A pair of consumption bundles. XA, XB, is called an
allocation. An allocation is a feasible allocation if the total amount of each good consumed is equal to the total amount
available:
x1A + x1B = w1A + w1B
x 2A + X 2B = w 2A + w1B
VIKAAS WADHWA SIR
9811439887 32
Figure 31.1
A particular feasible allocation that is of interest is the initial endowment allocation.
( w1A , w 2A )
and
( w1B , w 2B )
. This is the allocation that the consumers start with. It consists of the amount. of each good that consumers bring to
the market. The will exchange some of these goods with each other in the course of trade to end up at a final allocation.
The Edgeworth box shown in Figure 31.1 can be used to illustrate these concepts graphically. We first use a standard
consumer theory diagram to illustrate the endowment and preferences of consumer A. We can also mark off on these
axes the total amount of each good in the economy the amount that A has plus the amount that B has of each good.
Since we will only be interested in feasible allocations of goods between the two consumers, we can draw a box that
contains the set of possible bundles of the two goods that A can hold.
Note that the bundles in this box also indicate the amount of the goods that B can hold. If there are 10 units of good 1
and 20 units of good 2, then if A holds (7, 12), B must be holding (3.8). We can depict how much A holds of good 1 by
the distance along the horizontal axis from the origin in the lower left–hand corner of the box amid the amount B holds
of good 1 by measuring the distance along time horizontal axis from tine upper right–hand corner. Similarly, distances
along the vertical axes give the amounts of good 2 that A and B hold. Thus the points in this box give us both the bundles
that A can hold and the bundles that B can hold–just measured from different origins. The points in the Edgeworth box
can represent all feasible allocations in this simple economy.
We can depict A’s indifference curves in the usual manner, but B’s indifference curves take a somewhat different form.
To construct them we take a standard diagram for B’s indifference curves, turn it upside down, and “overlay” it on the
Edgeworth box. This gives us B’s indifference curves on the diagram. If we start at A’s origin in tine lower left-hand
corner and move up and to the right, we will be moving to allocations that are more 4 preferred by A. As we move down
and to the left we will he moving to allocations that are more preferred by B. (If you rotate your book and look at the
diagram, this discussion may seem clearer.)
VIKAAS WADHWA SIR
9811439887 33
The Edgeworth box allows us to depict the possible consumption bundles for both consumers the feasible allocations
and the preferences of both consumers. It thereby gives a complete description of the economical relevant
characteristics of the two consumers.
31.2 Trade
Now that we have both sets of preferences and endowments depicted we can begin to analyze the question of how
trade takes place. We start at the original endowment of goods, denoted by the point W in Figure 31.1. Consider the
indifference curves of A and B that pass through this allocation. The region where A is better off than at her endowment
consists of all the bundles above her indifference curve through W. The region where B is better off than at his
endowment consists of all the allocations that are above–from his point of view– -his indifference curve through W.
(This is below his indifference curve from our point of view . . . unless you’ve
still got your book upside down.)
Where is the region of the box where A and B are both made better of f? Clearly it is in the intersection of these’ two
regions. “This is the lens shaped region illustrated in Figure 31.1. Presumably in the course of their negotiations the
two people involved will find some mutually advantageous trade–some trade that will move them to some point inside
the lens-shaped area such as the point M in Figure 31.1.
The particular movement to Al depicted in Figure 31.1 involves person A giving up
acquiring in exchange
up
x 2B − w 2B
x 2A − w 2A
units of good 2. This means that B acquires
x1A − w1A
x1B − w1B
units of good 1 and
units of good land gives
units of good 2.
There is nothing particularly special about the allocation. Any allocation inside the lens–shaped region would be
possible – for every allocation of goods in this region is. an allocation hat makes each consumer better off than he or
she was at time original endowment. We only need to suppose that time consumers trade to Seine point in this region.
Now we can repeat the same analysis at the point Al. We can draw the two indifference curves through M, construct a
new lens-shaped “region of mutual advantage,” and imagine the traders moving to some new point N in this region.
And so it goes  the trade will continue until there are no more trades that are preferred by both parties. What does
such a position look like?
VIKAAS WADHWA SIR
9811439887 34
31.3 Pareto Efficient Allocations
The answer is given in Figure 31.2. At the point M in this diagram the set of points above A’s indifference curve doesn’t
intersect the set of points above B’s indifference curve. The region where A is made better off is disjoint from the region
where B is .made better off. This means that any movement that makes one of the parties better off necessarily makes
thd other party worse off. Thus there are no exchanges that are advantageous for both parties. ‘There are no mutually
improving trades at such an allocation.
An allocation such as this is known as a Pareto efficient allocation. The idea of Pareto efficiency is a very’ important
concept in economics that arises in various guises.
Figure 31.2
A Pareto efficient allocation can be described as an allocation where:
1.
There is no way to make all the people involved better off; or
2.
there is no way to make some individual better off without making someone else worse off; or
3.
all of the gains from trade have been exhausted; or
4.
there are no mutually advantageous trades to be made, and so on.
Indeed we have mentioned the concept of Pareto efficiency several times already in the context of a single market: we
spoke of the Pareto efficient level of output in a single market as being that amount of output where the marginal
willingness to buy equaled the marginal willingness to sell. At any level of output where these two numbers differed,
tbereou1d be a day to make both sides of the market better off by carrying omit a trade. In this chapter we will examine
more deeply the idea of Pareto efficiency involving many goods and many traders.
Note the following simple geometry of Pareto efficient allocations: the indifference curves of tine two agents must be
tangent at any Pareto efficient allocation in the intenio of the box. It is easy to see why. If the two indifference curves
are not tangent at an allocation in the interior of the box, then they must cross. But if they cross, then there must be
some mutually advantageous trades) that point; cannot be Pareto efficient. (It is possible to have Pareto efficient
allocations on tine sides of the box–where one consumer has zero consumption of some good–in which the indifference
curves are not tangent. These boundary cases are not important for tine current discussion.)
VIKAAS WADHWA SIR
9811439887 35
From the tangency condition it is easy to see that there lot of Pareto efficient allocations in the Edgeworth box. In fact,
given any indifference curve for person A, for example, there is an easy way to find a Pareto efficient allocation. Simply
move along. A's indifference curve until you find tine point that is the best point for B. This nil! be a Pareto efficient
point, and thus both indifference curves must be tangent at this point.
The set of all Pareto efficient points in the Edgeworth box is known as the Pareto set, or the contract curve. The latter
name comes from the idea that all “final contracts” for trade must lie on the Pareto set–otherwise they wouldn’t be final
because there would be some improvement that could be made!
In a typical case the contract curve will stretch from A’s origin to B’s origin across time Edgeworth box, as shown in
Figure 31.2. If we start at A’s on origin, A has none of either good and B holds everything. This is Pareto efficient since
the only way A can be made better off is to take something away from B. As we move up the contract curve A is getting
more and more well-off until we final iv get to ifs origin.
The Pareto set describes all the possible outcomes of mutually advantageous trade from starting anywhere in the box.
If we are given the starting point–the initial endowments for each consumer–we can look at the subset of the Pareto
set that each consumer prefers to his initial endowment. This is simply the subset of the Pareto set that lies in the lensshaped region depicted in Figure 31.1. The allocations in this lens-shaped region are the possible outcomes of mutual
trade starting from t-he particular initial endowment depicted in that diagram. But the Pareto set itself doesn’t depend
on the initial endowment, except insofar as the endowment determines the total amounts of both goods that are
available and thereby determines the dimensions of the box.
31.4 Market Trade
The equilibrium of the trading process described above–the set of Pareto efficient allocations is very important, but it
still leaves a lot of ambiguity about where the agents end up. The reason is that the trading process we have described
is very general. Essentially we have only assumed that tine two parties will move to some allocation where they are
both made better off.
If we have a particular trading process, we will have a more precise description of equilibrium. Let’s try to describe a
trading process that; mimics the outcome of a competitive market.
Suppose that we have a third party who is willing to act as an “auctioneer” for the two agents A and B. The auctioneer
chooses a price for good 1 and a price for good 2 and presents these prices to the agents A and B. Each agent then sees
how much his or her endowment is worth at tine prices (p1, p2) and decides how much of each good he or she would
want to buy at those prices.
One warning is in order here. If there are really only two people involved in the transaction, then it doesn’t make much
sense for them to behave in a competitive manner. Instead they would probably attempt to bargain over tine terms of
trade. One way around this difficulty is to think of the Edgeworth box as depicting the average demands in an economy
with only two types of consumers, but with many consumers of each type. Another way to deal with this is to point out
that the behavior is implausible in time two-person case, but it makes perfect sense in tine many-person case, which is
what we are really concerned with.
Either way, we know how to analyze the consumer-choice problem in this framework–it is just time standard consumer
choice problem we described a Chapter 5. In Figure 31.3 we illustrate the two demanded bundles of time two agents.
(Note that the situation depicted in Figure 31.3 is not an equilibrium configuration since the demand by one agent is
not equal to the supply of the other agent.)
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Figure – 31.3
As in Chapter 9 there are two relevant concepts of “demand” in this framework. The gross demand of agent. A for good
1, say, is the total amount of good 1 that he wants at the going prices. The net demand of agent A for good 1 is the
difference between this total demand and the initia1 endownnent of good 1 that agent A holds. In the context of general
equilibrium analysis, net demands are sometimes called excess demands. We will denote tine excess demand of agent
A for good 1 by
e1A .
By definition, if A’s gross demand is and his endowment is u, we have
e1A = x1A − w1A
Time concept of excess demand is probably more natural, but the concept of gross demand is generally more useful.
We will typically use the word “demand” to mean gross demand amid specifically say “net demand or “excess demand”
if that is what we mean.
For arbitrary prices (p1, p2) there is no guarantee that supply will equal demand –in either sense of demand. In terms
of net demand, this means that time amount that A wants to buy (or sell) will not necessarily equal the amount that B
wants to sell (or buy). In terms of gross demand, this means that the total amount that time two agents want hold of
the goods is not equal to the total amount of the goods available. Indeed, this is true in 1 example depicted in Figure
31.3. In this example, the agents will not be able to complete their desired transactions: the markets will not clear.
We say that in this case the market is in disequilibrium. In such a situation, it is natural to suppose that the auctioneer
will change the prices of the goods. If there is excess demand for one of the goods, the auctioneer will raise the price of
that good, and if there is- excess supply for one of the goods, the auctioneer will lower its price.
Suppose that this adjustment process continues until the demand for each of the goods equals the supply What will the
final configuration look like?
The answer is given in Figure 31.4. Here the amount that A wants to buy of good 1 just equals the amount that B wants
to sell of good 1, and similarly for good 2. Said another way, the total amount- that each person wants to buy of each
good at the current prices is equal to the total amount available. We say that the market is in equilibrium. More
precisely, this is called a market equilibrium a competitive equilibrium or a Walrasian equilibrium. Each of these ten
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ms refers to the same thing a set of prices such that each consumer is choosing his or her most-preferred affordable
bundle, and all consumers’ choices are compatible in time sense that demand equals supply in every market.
We know that if each agent is choosing the best bundle that he can afford, then his marginal rate of substitution between
the two goods must be equal to time ratio of the prices. But if all consumers are facing time same prices, then all
consumers will have to have the same marginal rate of substitution between each of the two goods. In terms of Figure
31.4, an equilibrium has the property that each agent’s indifference curve is tangent to his budget line. But since each
agent’s budget line has the slope –p1/p2, this means that two agents’ indifference curves must be tangent to each other.
31.5 The Algebra of Equilibrium If we let
x1A ( p1 , p 2 ) be agent A's demand function for good 1 and x1B ( p1 , p 2 ) be agent B’s demand function for
good 1, and define the analogous expressions for good 2, we can describe this equilibrium as a set of prices
( p1* , p*2 )
such that
x1A ( p1* , p*2 ) + x1B ( p1* , p*2 ) = w1A + w1B
x 2A ( p1* , p*2 ) + x 2B ( p1* , p*2 ) = w 2A + w 2B
These cop at-ions say that in equilibrium the total demand for each good should be equal to time total supply.
Figure 31.4
Another way to describe the equilibrium is to rearrange these two equations to get
[ x1A ( p1* , p*2 ) − w1A ] + [ x1B ( p1* , p*2 ) − w1B ] = 0
[ x 2A ( p1* , p*2 ) − w 2A ] + [ x 2B ( p1* , p*2 ) − w 2B ] = 0
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These equations say that the sum of net demands of each agent for each good should be zero. Or, in other words the
net amount that A chooses to demand (or supply) must be equal to tine net amount that B chooses to supply (or
demand).
Yet another formulation of these equilibrium equations conies from the concept of the aggregate- excess demand
function. Let us denote time net demand function for good 1 by agent A by
e1A ( p1 , p 2 ) = x1A ( p1 , p 2 ) − w1A
and define
e1B ( p1 , p 2 )
The function
in a similar manner.
e1A ( p1 , p 2 )
measures agent A’s net demand or his excess demand the difference between what she
wants to consume of good 1 and what she initially has of good 1. Now let us add together agent A’s net demand for good
I and agent B's net demand for good 1. We get
z1 ( p1 , p 2 ) = e1A ( p1 , p 2 ) + e1B ( p1 , p 2 )
= x1A ( p1 , p 2 ) + x1B ( p1 , p 2 ) − w1A − w1B
which we call the aggregate excess demand for good 1. There is a similar aggregate excess demand for good 2, which
we denote by z2(p1,p2).
Then we can describe an equilibrium
( p1* , p*2 ) by saying that the aggregate excess demand for each good is zero:
z1 ( p1* , p*2 ) = 0
z 2 ( p1* , p*2 ) = 0
Actually, this definition is stronger than necessary. It turns out that if the aggregate excess demand for good 1 is zero,
then the aggregate excess demand for good 2 must necessarily be zero. In order to prove this, it is convenient to first
estabiish a property of the aggregate excess demand function– known as Walras’ law.
31.6 Walras’ Law
Using the notation established above, Walras' law states that
p1z1 ( p1 , p 2 ) + p 2 z 2 ( p1 , p 2 ) = 0
That is, the value of aggregate excess demand is identically zero. To say that the value of aggregate demand is identically
zero means that it is zero for all possible choices of prices, not just equilibrium prices.
The proof of this follows from adding up the two agents’ budget constraints. Consider first agent A. Since her demand
for each good satisfies her budget constraint, we have
p1x1A ( p1 , p 2 ) + p 2 x 2A ( p1 , p 2 ) = p1w1A + p 2 w 2A
or
[ p1x1A ( p1 , p 2 ) − p1w1A ] + [ p 2 x 2A ( p1 , p 2 ) − p 2 w 2A ] = 0
p1e1A ( p1 , p 2 ) + p 2 e 2A ( p1 , p 2 ) = 0
This equation says that the value of agent A ‘s net demand is zero. That is, time value of how munch A wants to buy of
good 1 plus tine value of how much she wants to buy of good 2 must equal zero. (Of course tine amount that she wants
to buy of one of the goods must be negative that is, she intends to sell some of one of time goods to buy more of the
other.)
We have a similar equation for agent B: -
p1x1B ( p1 , p 2 ) + p 2 x 2B ( p1 , p 2 ) = p1w1B + p 2 w 2B
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or
p1 [ x1B ( p1 , p 2 ) − w1B ] + p 2 [ x 2B ( p1 , p 2 ) − w 2B ] = 0
p1e1B ( p1 , p 2 ) + p 2 e 2B ( p1 , p 2 ) = 0
Adding the equations for agent A and agent B together and lising time definition of aggregate excess demand, z i (p1, p2)
and z2(p1 ,p2), we have
p1 [e1A ( p1 , p 2 ) + e1B ( p1 , p 2 )] + p 2 [e 2A ( p1 , p 2 ) + e 2B ( p1 , p 2 )] = 0
p1z1 ( p1 , p 2 ) + p 2 z 2 p1 , p 2 ) = 0
Now we can see where Walras’ law comes from: since the value of each agent’s excess demand equals zero, the value
of the sum of the agents’ excess demands must equal zero.
We can now demonstrate that if demand equals supply in one market, demand must also equal supply in the other
market. Note that Walras’ law must hold for all prices, since each agent must satisfy his or her budget constraint for all
prices. Since Walras’ law holds for all prices, in particular, it holds for a set of prices where the excess demand for good
1 is zero:
z1 ( p1* , p*2 ) = 0
According to Walras’ law it must also be true that
p1*z1 ( p1* , p*2 ) + p*2 z 2 ( p1* , p*2 ) = 0
It easily follow s from these two equations that if p2 > 0 then we must have
z 2 ( p1* , p*2 ) = 0
Thus, as asserted above, if we find a set of prices
( p1* , p*2 ) where the demand for good 1 equals the supply of good 1,
we are guaranteed that the demand for good 2 must equal the supply of good 2. Alternatively, if me find a set of prices
where the demand for good 2 equals the supply of good 2, we are guaranteed that market 1 will he in equilibrium.
In general, if there are markets for k goods, then we only need to finch a set of prices where k – 1 of the market are in
equilibrium. Walras’ law then implies that the market for good k will automatically have demand equal to supply.
31.7 Relative Prices
As we’ve seen above, Walras’ law implies that there are only k – 1 independent equations in a k-good general
equilibrium model: if demand equals supply in k – 1 markets, demand must equal supply in the final market.
But if there are k goods, there will he k prices to be determined. How can you solve for k prices with only k – 1
equations?
The answer is that there are really only k – I independent prices. We saw in Chapter 2 that. if we multiplied all prices
amid income by a positive number 1, then the budget set wouldn’t change, and thus the demanded bundle would not
change either. In the general equilibrium model, each consumer’s income is just time value of his or her endowment at
time market prices. If we multiply all prices by t > 0, we will automatically multiply each consumer’s income by t. Thus,
if we find some equilibrium set of prices
( p1* , p*2 ) then ( tp1* , tp*2 )
are equilibrium prices as well, for any t > 0.
This means that we are free to choose one of the prices and set it equal to a constant. In particular it is often convenient
to set one of the prices equal to 1 so that all of the other prices can be interpreted as being measured relative to it. As
we saw in Chapter 2, such a price is called a numeraire price. If we choose the first price as the numeraire price, then it
is just like multiplying all prices by the constant t = 1/p1.
The requirement that demand equal supply in every market can only be expected to determine the equilibrium relative
prices, since multiplying all prices by a positive number will not change anybody’s demand and supply behaviour.
EXAMPLE An Algebraic Example of Equilibrium
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The Cobb-Douglas utility function described in Chapter 6 has the form
u A ( x1A , x 2A ) = ( x1A )  ( x 2A )1− 
for
person A, and a similar form for person B.
We saw there that this utility function gave rise to the following demand functions:
x1A ( p1 , p 2 , m A ) = a
mA
p1
x 2A ( p1 , p 2 , m A ) = (1 − a )
x1B ( p1 , p 2 , m B ) = b
mA
p2
mB
p1
x 2B ( p1 , p 2 , m B ) = (1 − b)
mB
p1
where a and b-are tine parameters of tine two consumers’ utility functions.
We know that inn equilibrium, the money income of each individual is given by the value of his or her endowment.
m A = p1w1A + p 2 w 2A
m B = p1w1B + p 2 w 2B
Thus the aggregate excess demands for the two goods are
z1 ( p1 , p 2 ) = a
=a
mA
m
+ b B − w1A − w1B
p1
p1
p1w1A + p 2 w 2A
p w1 + p 2 w 2B
+b 1 B
− w1A − w1B
p1
p1
and
z 2 ( p1 , p 2 ) = (1 − a )
= (1 − a )
mA
m
+ (1 − b) B − w 2A − w 2B
p1
p1
p1w1A + p 2 w 2A
p w1 + p 2 w 2B
+ (1 − b) 1 B
− w 2A − w 2B
p2
p2
You should verify that these aggregate demand functions satisfy Walras’ law.
Let us choose p2 as the numeraire price, so that these equations become
p1w1A + p 2 w 2A
p1w1B + p 2 w 2B
z1 ( p1 ,1) = a
+b
− w1A − w1B
p1
p1
z 2 ( p1 ,1) = (1 − a )( p1w1A + p 2 w 2A ) + (1 − b)( p1w1B + p 2 w 2B ) − w 2A − w 2B
All we’ve done here is set p2 = 1.
We now have an equation for the excess demand for good 1, Z1 (p1, 1), and an equation for the excess demand for good
2, z2(p1, 1), with each equation expressed as a function of the relative price of good 1, p 1. In order to find the equilibrium
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price, we set either of these equations equal to zero and solve for p 1 According to Walras’ law, we should get the same
equilibrium price, no matter which equation, we solve
The equilibrium price turns out to be
p1*
=
aw 2A + bw 2B
(1 − a ) w1A + (1 − b) w1B
.
(Skeptics may want to insert this value of p1 into the demand equals supply equations to verify that the equations are
satisfied.)
31.8 The Existence of Equilibrium
In the example given above, we had specific equations for each consumer’s demand function and we could explicitly
solve for tine equilibrium prices. But in general, we don’t have explicit algebraic formulas for each consumer’s
demands. We might well ask how do we know that there is any set of prices such that demand equals supply in every
market? This is known as the question of existence of a competitive equilibrium.
The existence of a competitive equilibrium is important insofar as it serves as a “consistency check” for the various
models that we have examined in previous chapters. What use would it be to build up elaborate theories of the
workings of a competitive equilibrium if such an equilibrium commonly did not exist ?
Early economists noted that in a market with k goods there were k – 1 relative prices to be determined, and theme
were k – 1 equilibrium equations stating that demand should equal supply iii each market. Since time number of
equations equal the number of unknowns, they asserted that t here would be a solution where all of tine equations
were satisfied.
Economists soon discovered that such arguments were fallacious. Merely counting the number of equations and
unknown is not sufficient to prove that an equilibrium solution will exist. However, there are mathematical tools that
can be used to establish the existence of a competitive equilibrium. The crucial assumption turns out to be that the
aggregate excess demand function is a continuous function. This means, roughly speaking, that small changes in prices
should result in only small changes in aggregate demand: a small change in prices should not result in a big jump in the
quantity demanded.
Under what conditions will the aggregate demand functions be continuous? Essentially there are two kinds of
conditions that will guarantee continuity. One is that each individual’s demand function be continuous that small
changes in prices will lead to only small, changes in demand. This turns out to require that each consumer have convex
preferences, which we discussed in Chapter 3. The other condition is more general. Even if consumers themselves have
discontinuous demand behaviour, as long as all consumers are small relative to the size of the market, the aggregate
demand function will be continuous.
This latter condition is quite nice After all the assumption of competitive behavior only makes sense when there are a
lot of consumers who are small relative to the size of time market. This is exactly the condition that we need in order
to get the aggregate demand functions to the continuous.
And continuity is just the ticket to ensure that a competitive equilibrium exists. This the very assumptions that make
the postulated behaviour reasonable will ensure that the equilibrium theory will have content.
31.9 Equilibrium and Efficiency
We have now analyzed market trade in a pure exchange model. This gives us a specific model of trade that we can
compare to the general model of trade that we discussed in the beginning of this chapter. One question that might arise
about the use of a competitive market is whether this mechanism can really exhaust all of the gains from trade. After
we have traded to a competitive equilibrium when demand equals supply in every market, will there be any more
trades that people will desire to carry out ?
This is just another way to ask whether the market equilibrium is Pareto efficient: will the agents desire to make any
more trades after they have traded at the competitive prices?
We can see time answer by inspecting Figure 31.4: it turns out that- time market equilibrium allocation is Pareto
efficient. The proof is this: an allocation iii the Edgeworth box is Pareto efficient if the set of bundles that A prefers
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doesn’t intersect the set of bundles that B prefers. But at the market equilibrium the set of bundles preferred by A must
lie above her budget set and the same thing holds for B, where above” means above from B’s point of view.” Thus tine
two sets of preferred allocations can't
intersect. This means that there are no allocations that both agents
prefer to the equilibrium allocation, so the equilibrium is Pareto efficient.
31.10 The Algebra of Efficiency
We can also show this algebraically. Suppose that we have a market equilibrium that is not Pareto efficient. We will
show that this assumption leads to a logical contradiction.
To say that the market equilibrium is not Pareto efficient means that there is some other feasible allocation
( y1A , y 2A , y1B , y 2B )
such that
y1A + y1B = w1A + w1B
(31.1)
y 2A + y 2B = w 2A + w 2B
(31.2)
( y1A , y 2A )  A ( x1A , x 2A )
(31.3)
( y1B , y 2B )  B ( x1B , x 2B )
(31.3)
and
The first two equations say that the y–allocation is feasible, and the next two equations say that it is preferred by each
agent to the x–allocation. (The symbols
A
and
B
refer to the preferences of agents A amid B.)
But by hypothesis, we have a nnarket equilibrium where each agent is purchasing the best bundle he or she can afford.
If
( y1A , y 2A ) is better than tine bundle that A is choosing, then it must cost more than A can afford, and similarly for
B:
p1y1A + p 2 y 2A  p1w1A + p 2 w 2A
p1y1B + p 2 y 2B  p1w1B + p 2 w 2B
Now add these two equations together to get
p1 ( y1A + y1B ) + p 2 ( y 2A + y 2B )  p1 ( w1A + w1B ) + p 2 ( w 2A + w 2B )
Substitute from equations (31.1) amid (1.2) to get
p1 ( w1A + w1B ) + p 2 ( w 2A + w 2B )  p1 ( w1A + w1B ) + p 2 ( w 2A + w 2B )
which is clearly a contradiction, since the left–hand side anal the right–hand side are time same.
We derived this contradiction by assuming that the market aquarium was not Pareto efficient. Therefore, this
assumption must be wrong. It follows that all market equilibrium are Pareto efficient : result known as the First
Theorem of Welfare Economics.
The First Welfare Theorem guarantees that a competitive market will exhaust all of the gains from trade: an aquarium
allocation achieved by a set of competitive markets will necessarily be Pareto efficient. Such an allocation may not have
any other desirable properties, but it will necessarily be efficient.
In particular, the First Welfare Theorem says nothing about the distribution of economic benefits. The market
equilibrium might not be a “just” allocation – if person A owned everything to begin with, then she would own
everything after trade. That would be efficient, but it would probably not be very fair. But, after all, efficiency does
count for something, and it is reassuring to know that a simple market mechanism like the one we 1-nave described is
capable of achieving an efficient allocation.
EXAMPLE: Monopoly in the Edgeworth Box
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In order to understand the First Welfare Theorem better, it is useful to consider another resource allocation mechanism
that does not lead to efficient outcomes. A nice example of this occurs when one consumer attempts to behave as a
monopolist. Suppose now that there is no auctioneer and that instead, agent A is going to quote prices to agent B, and
agent B will decide how much lie wants to trade at the quoted prices. Suppose further that A knows B’s “demand curve”
and will attempt to choose the set of prices that makes A as well-off as possible, given the demand behavior of B.
In order to examine the equilibrium in this process, it is appropriate to recall the definition of a consumer’s price offer
curve. The price, offer curve, which we discussed in Chapter 6, represents all of time optimal choices of the consumer
at different prices. B’s offer curve represents the bundles that, he will purchase at different prices; that is, it describes
B’s demand behavior. If we draw a budget line for B, then the point where that budget line intersects his offer curve
represents B’s optimal consumption.
Thus, if agent A wants to choose time prices to offer to B that make A as well-off as possible, she should find that ‘point
on B’s offer curve where A has the highest utility. Such a choice is depicted in Figure 31.5.
This optimal choice will be characterized by a tangency condition as usual : A’ indifference curve will be tangent to B’s
offer curve. If B’s offer curve cut A’s indifference curve, there would be sonic point on B’s offer curve that A preferred
– so we couldn’t be at the optimal point for A.
Once we have identified tins point–denoted by X in Figure 31.5–we just draw a budget line to that point from the
endowment. At time prices that generate this budget line, B will choose time bundle A, and A will be as well–off as
possible.
Figure – 31.5
Is this is allocation Pareto efficient ? In general the answer is no. To see this simply note that A’s indifference curve will
not be tangent to the budget line at X and therefore A’s indifference curve will not, be tangent to B indifference curve.
A’s indifference curve is tangent to B’s offer curve.
But it cannot then be tangent to B’s indifference curve. Time monopoly allocation is Pareto inefficient.
In fact. it is Pareto inefficient in exactly the same way as described in the discussion of monopoly in Chapter’ 24. At the
margin A would like to sell more at tine equilibrium prices, but sue can only do so by lowering the price at which she
sells – and this will lower her income received from all her infra marginal sales.
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We saw in Chapter 25 that a perfectly discriminating monopolist would end up producing an efficient level of output.
Recall that a discriminating monopolist was one who was able to sell each unit of a good to the person who was willing
to pay the most for that unit. What does a perfectly discriminating monopolist look like in the Edgeworth box?
Figure 31.6
The answer is depicted in Figure 31.6. Let us start at time initial endowment. W, and imagine A selling each unit of good
1 to B at a different price time price at winch B is just. indifferent between buying or not buying that unit of the good.
Thus, after A sells’ the first unit B will remain on time same indifference curve through W. Then A sells time second
unit of good 1to B for time maximum price he is willing to pay. This means that the allocation in moves further to the
left, but remains on B’s indifference curve through W. Agent A continues to sell units to B in this manner, thereby
moving up B's indifference curve to find her – A's–most preferred point, denoted by an X in Figure 31.6.
It is easy to see that such a point must be Pareto efficient. Agent A will be as well-off as possible given B’s indifference
curve. At such a point, A has managed to extract all of B’s consumer’s surplus: B is no better off than he was at his
endowment.
These two examples provide useful benchmarks with which to think about the First ‘Welfare Theorem. The ordinary
monopolist gives an example of a resource allocation mechanism that results in inefficient equilibria, and the
discri8minating monopolist gives that results in efficient equilibria.
31.11 Efficiency and Equilibrium
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Figure–31.7
The First Welfare Theorem says that the equilibrium in a set of competitive markets is Pareto efficient. What about
time other way around ? Given a Pareto efficient. allocation, can we find prices such that it is a market equilibrium ? It
turns out that the answer is yes, under certain conditions. The argument is illustrated in Figure 31.7.
Let us pick in Pareto efficient allocation. Tim en we know that the set of allocation that A prefers to bier current
assignment is disjoint from the set that B prefers. This implies of course that the two indifference curves are tangent at
the Pareto efficient; allocation. So let us draw in tine straight line that is their common tangent, as in Figure 31.7.
Suppose that the straight line represents the agents’ budget sets. Then if each agent chooses time best bundle on his or
her budget set, the resulting equilibrium will be the marginal Pareto efficient allocation.
Thus the fact that the original allocation is efficient automatically determines the equilibrium prices. Tine endowments
can be any bundles that give rise to the appropriate budget set – that is, bundles that lie somewhere on the constructed
budget line.
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Figure 31.8
Can the construction of such a budget line always he carried out ? Unfortunately, the answer is no. Figure 31.8 gives an
example. Helm time illustrated point X is Pareto efficient, but there are no prices at which A and B will want to consume
at point X. The most obvious candidate is drawn in time diagram, but the optimal demands of a cuts A and B don’t
coincide for that; budget. Agent A wants to demand time bundle Y, but agent B wants the bundle X–demand does rot
equal supply at these prices.
The difference between Figure 31.7 and Figure 31.8 is bat the preferences in Figure 31.7 arc convex while tine ones in
Figure 31.8 are not. If the preferences of both agents are convex, then the common tangent will not intersect either
indifference curve more than once, and everything will work out fine. This observation gives us the Second Theorem
of Welfare.
Economics:
If all agents have convex preferences, then there will always be a set of prices such that each Pareto
efficient allocation is a market p equilibrium for an appropriate assignment; of endowments.
The proof is essentially the geometric argument we gave above. At a Pareto efficient allocation, the bundles preferred
by agent A and by agent B must be disjoint. Thus if both agents have convex preferences we can draw a straight line
between the two sets of preferred bundles that separates one from the other. The slope of this line gives us the relative
prices, and any endowment that puts the two agents on this line will lead to the final market equilibrium being the
original Pareto efficient allocation.
31.12
Implications of the First Welfare Theorem
Time two theorems of welfare economies are among time among the most fundamental results in economics. We have
demonstrated time theorems only in the simple Edgeworth box case, but they are true for much more complex models
with arbitrary numbers of consumers and goods. The welfare theorems have profound implications for the design of
ways to allocate resources.
Let us consider the First Welfare Theorem. This says that any competitive equilibrium, is Parreto efficiently. There are
hardly any explicit assumption in this theorem–it follows almost entirely from the definitions. But there are some
VIKAAS WADHWA SIR
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implicit assumptions. One major assumption is that agents only care about their own consumption of goods, and not
about what other agents consume. If one agent does care about another agent’s consumption, we say that there is a
consumption externality. We shall see that when consumption externalities are present, a competitive equilibrium
need not be Pareto efficient.
To take a simple example, suppose that agent A cares about agent B’s consumption of cigars. Then there is no particular
reason why each agent choosing his or lien own consumption bundle at the market prices will result in a Pareto efficient
allocation. After each person has purchased the best bundle he or she can afford, there may still be ways to make both
of them better off – such as A paying B to smoke fewer cigars. We will discuss externalities in more detail in Chapter
33.
Another important implicit assumption in the First Welfare Theorem is that agents actually behave competitively. If
theme really were two agents as in the Edgeworth box example then it is unlikely that they would each take price as
given. Instead, the agents would probably recognize their market power and would attempt to use their market power
to improve their own positions. The concept of competitive equilibrium only makes sense when there are enough
agents to ensure that each behaves competitively.
Finally, tine First Welfare Theorem is only of interest if a competitive equilibrium actually exists. As we have argued
above, this will be the case if the consumers are sufficiently small relative to the size of the market.
Given these provisos, time First Welfare Theorem is a Pretty strong result: a private market, with each agent seeking
to maximize his or her own utility, will result in an allocation that achieves Pareto efficiency.
The importance of the First Welfare Theorem is that it gives a general mechanism the competitive market that we can
use to ensure Pareto efficient outcomes. If there are only two agents involved, this doesn’t matter very much; it is easy
for two people to get together and examine the possibilities for mutual trades. But if there are thousands, or even
millions, of people involved there must be some kind of structure imposed on the trading process. The First Welfare
Theorem shows that time particular structure of competitive markets has the desirable property of achieving a Pareto
efficient allocation.
If we are dealing with a resource problem involving many people, it is important to note that the use of competitive
markets economizes on the information that any one agent needs to possess. The only timings that a consumer needs
to know to make his consumption decisions are the prices of the goods he is considering consuming’. Consumers don't
need to know anything about how time goods are produced, or who owns what goods, or where the hoods come from
in a competitive market. If each consumer knows only the prices of time goods. he can determine this demands, and if
the market functions well enough to determine the competitive prices, we are guaranteed an efficient outcome. The
fact that competitive markets economize on information in this way is a strong argument in favor of their use as a way
to allocate resources.
31.13 Implications of the Second Welfare Theorem
The Second Theorem of Welfare Economics asserts- that under certain conditions, every Pareto efficient allocation can
be achieved as a competitive equilibrium.
What is the meaning of this result? The Second Welfare Theorem implies that the problems of distribution and
efficiency can be separated. Whatever Pareto efficient allocation you want can be supported by the market mechanism..
The market mechanism is distributionally neutral; whatever your criteiia for a good-or a just distribution of welfare,
you can use competitive markets to achieve it.
Prices play two roles in the market system: an allocative role and a distributive role. The allocative role of prices is to
indicate relative scarcity; the distributive role is to determine how much of different goods different agents can
purchase. The Second Welfare Theorem says that these two roles can be separated: we can redistribute endowments
of goods to determine how much wealth agents have, and then use prices to indicate relative scarcity.
Policy discussions often become confused on this point. One often hears arguments for intervening in pricing decisions
on grounds of distributional equity. However, such intervention is typically misguided. As we have seen above, a
convenient way to achieve efficient allocations is for each agent to face the true social costs of his or her actions and to
make choices that reflect those costs. Thus in a perfectly competitive market the marginal decision of whether to
consume more or less of some good will depend on the price which measures how everyone else values this good on
VIKAAS WADHWA SIR
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the margin. If he considerations of efficiency are inherently marginal decisions–each person should face the correct
marginal tradeoff in making his or her consumption decisions.
The decision about how much different agents should consume is a totally different issue. In a competitive market this
is determined b the value of the resources that a person has to sell. From the viewpoint of the pure theory there is no
reason why the state can't transfer purchasing power endowments–among consumers in any way that is seen fit.
In fact the state doesn’t need to transfer the physical endowments themselves. All that is necessary is to transfer the
purchasing power of the endowment. The state could tax one consumer on the basis of the value of his endowment and
transfer this money to another. As long as the taxes arc based on the value of the consumer's endowment of goods there
will be no loss of efficiency. It is only when taxes depend on the choices that a consumer makes that inefficiencies result,
since in this case, the taxes will affect the consumer’s marginal choices.
It is true that a tax on endowments will generally change people’s behavior. But, according to the First Welfare
Theorem, trade from any initial endowments will result in a Pareto efficient allocation. Thus no matter how one
redistributes endowments, the equilibrium allocation as determined by market forces will still be Pareto efficient.
However, there are practical matters involved. It would be easy to have a lump-sum tax on consumers. We could tax all
consumers with blue eyes, and redistribute the proceeds to consumers with brown eyes. As long as eye color can’t be
changed, there would be no loss in efficiency. Or we could tax consumers with high IQs and redistribute the funds to
consumers with low IQs. Again, as long as IQ can be measured, there is no efficiency loss in this kind of tax.
But there’s the problem. How do we measure people’s endowment of goods? For most people, the bulk of their
endowment consists of their own labor power. People’s endowments of labor consist of the labor that they could
consider selling, not the amount of labor that they actually end up selling. Taxing labor that people decide to sell to the
market is a distortionary tax. If the sale of labor is taxed, the labor supply decision of consumers will be distorted–they
will likely supply less labor than they would have supplied in the absence of a tax. Taxing the potential value of labor
the endowment of labor is not distortionary. The potential value of labor is by definition, something that is not changed
by taxation. Taxing the value of the endowment sounds easy until we realize that it involves identifying and taxing
something that might be sold, rather than taxing something that is sold.
We could imagine a mechanism for levying this kind of tax. Suppose that we considered a society where each consumer
was required to give the money earned in 10 hours of his labor time to the state each week. This kind of tax would be
independent of how much the person actually worked–it would only depend on the endowment of labor, not on how
munch was actually sold. Such a tax is basically transferring some part of each consumer’s endowment of labor time to
the, state. The state could then use these funds to provide various goods, or it could simply transfer these funds to other
agents.
According to the Second Welfare Theorem, this kind of lump-sum taxation would be nondistortionary. Essentially any
Pareto efficient allocation could he achieved by such lump-sum redistribution.
However, no one is advocating such a radical restructuring of the tax system. Most peoples labor supply decisions are
relatively insensitive to variations ill the wage rate. so the efficiency loss from taxing labor may not be too large anyway.
But the message of the Second ‘Welfare Theorem' is important. Price should be used to reflect. scarcity. Lump-sum
transfers of wealth should be used to adjust for distributional goals. To a large degree, these two policy decisions can
be separated.
People’s concern about the distribution of welfare can lead them to advocate various forms of manipulation of prices.
It has been argued, for example, that senior citizens should have access to less expensive telephone service, or that
small users of electricity should pay lower rates than large users. These are basically attempts to redistribute income
through the price system by offering some people lower prices than others.
When you think about it this is a terribly inefficient way to redistribute income. If you want to redistribute income, why
don’t you simply redistribute income’? If you give a person an extra dollar to spend, then he can choose to consume
more of any of the goods that he wants to consume—not necessarily just the good being subsidized.
Summary
1.
General equilibrium refers to the study of how the economy can adjust to have demand equal supply in all
markets at the same time.
2.
The Edgeworth box is a graphical tool to examine such a general equilibrium with 2 consumers and 2 goods.
VIKAAS WADHWA SIR
9811439887 49
3.
A Pareto efficient allocation is one in which there is no feasible reallocation of the goods that would make all
consumers at least as well-off and at least one consumer strictly better off.
4.
Walras’ law states that the value of aggregate excess demand is zero for all prices.
5.
A general equilibrium allocation is one in which each agent chooses a most preferred bundle of goods from
the set of goods that he or aim can afford.
6.
Only relative prices are determined in a general equilibrium system.
7.
If the demand for each good varies continuously as prices vary, then there will always be one set of prices
where demand equals supply in every market: that is, a competitive equilibrium.
8.
The First Theorem of Welfare Economics states that a competitive equilibrium is Parreto efficient.
9.
The Second ‘Theorem of Welfare Economics states that as long as preferences ale convex, then every Pareto
efficient allocation can be supported as a competitive equilibrium.
APPENDIX
Let us examine the calculus conditions describing Pareto efficient allocations. By definition, a Pareto efficient allocation
makes each agent as well-off as possible, given the utility of the other agent. So let us pick as the utility level for agent
B, say, and see how we can make agent A as well-off as possible.
The maximization problem is
max
x1A , x 2A , x1B , x 2B
u A ( x1A , x 2A )
such that
u B ( x1B , x 2B ) = u
x1A + x1B = w1
x 2A + x 2B = w 2
Here
w1 = w1A + w1B is the total amount of good 1 available and w 2 = w 2A + w 2B
2 available. This maximization problem asks us to find the allocation
is the total amount of good
( x1A , x 2A , x1B , x 2B )
that makes person A’s
utility as large as possible, given a fixed level for person B’s utility, and given that the total amount of each good used
is equal to the amount available.
We can write the Lagrangian for this problem as
L = u A ( x1A , x 2A ) − ( u B ( x1B , x 2B ) − u )
− 1 ( x1A + x1B − w1 ) −  2 ( x 2A + x 2B − w 2 )
Here  is the Lagrange multiplier on the utility constraint, and the 's are the Lagrange multipliers on the resource
constraints. When we differentiate with respect to each of the goods, we have four first-order conditions that must hold
at the optimal solution:
L u A
=
− 1 = 0
x1A x1A
L u A
=
− 2 = 0
x 2A x 2A
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u
L
= − 1B − 1 = 0
1
x B
x B
u B
L
=
−

− 2 = 0
x 2B
x 2B
If we divide the first equation by the second, and the third equation by the fourth, we have
MRSA =
u A / x1A
u A / x 2A
=
1
2
u B / x1B 1
MRSB =
=
u B / x 2B  2
The interpretation of these conditions is given in the text: at a Pareto efficient allocation, the marginal rates of
substitution between the two goods must be the same. Otherwise. there would be some trade that would make each
consumer better off.
Let us recall the conditions that must hold for optimal choice by consumers. If consumer A is maximizing utility subject
to her budget constraint and consumer B is maximizing utility subject to his budget constraint and both consumers
face the same prices for goods 1 and 2, we must have
u A / x1A
u A / x 2A
=
p1
p2
(31.7)
u B / x1B p1
=
u B / x 2B p 2
(31.8)
Note the similarity with the efficiency conditions. The Lagrange multipliers in the efficiency conditions,  and 2 are
just like the prices p1 and p2 in the consumer choice conditions. In fact the Lagrange multipliers in this kind of problem
are sometimes known as shadow prices or efficiency prices.
Every Pareto efficient allocation has to satisfy conditions like those in equations (31.5) and (31.6). Every competitive
equilibrium has to satisfy conditions like those in equations (31.7) and 3 1 8). The conditions describing Pareto
efficiency and the conditions describing individual maximization in a market environment are virtually the same.
WELFARE(NOTES)
Economies have allocated their scarce resources through various mechanisms which include
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a)
Barter
b)
Free trade in open markets
c)
Restricted trade in regulated markets
d)
Central planning
e)
Mixed approach
Therefore there is a need to establish criteria to judge whether an allocation of resources (by a particular social
mechanism) is
(i)
Efficient
(ii)
Equitable
(i)
Efficiency: Economists have developed defined criteria for measuring efficiency. One notion of efficiency was
proposed by the Italian economist, Vilfredo Pareto. An allocation of resources is said to be inefficient i we can
real locate resources so as to make at least one individual better off without making anyone else worse off.
An allocation is said to be (Pareto) efficient if it is impossible to make any individual better off without making
someone else worse off.
(ii)
Equitable: There are various notions of “equity”. This is because equity is a subjective concept and is subject
to different interpretations. There are broadly two notions of equity.
B.
a.
Process oriented
b.
Outcome oriented
Outcome – oriented notion of equity focuses on the processes and mechanisms used to allocate resources that
yield fair outcome/result.
We briefly discuss some outcome–oriented notion of equity that focuses on the distribution of wellbeing.
B1)
Jeremy Bentham, a British philosopher, stated that the society should place equal weight on the well-being of
every individual. This is referred as the principle of utilitarianism. Utilitarians assume that it is possible to
reallocate resources in a manner to increase the well-being of one individual more than it decreases the wellbeing of another.
B2)
John Rawls, an American political theorist, outlined a basis for ranking social outcomes. He stated that social
welfare should be defined in terms of the welfare of the least well off agent in society. This principle is known
as Rawlisianism. Rawlsians assume that there’s a way to identify society’s least happy member.
The concept of Pareto efficiency states nothing about the distribution of welfare across people. In this section, we shall
consider different mechanisms of “ADDING TOGETHER” individual consumer preferences to construct “social
preferences”. In order to do so we first introduce the concept of a welfare function. A welfare function provides a
mechanism to rank different distributions of utility among consumers.
We assume that each consumer is having preferences over the entire allocation of goods among all the consumers. Let
x denote a particular allocation representing what every individual gets of every good. Let y represent a different
allocation. Each individual shall now be in a position to state whether he prefers x to y or y to x or x=y. In this way we
shall have preferences of all agents comprising the society. We now aggregate these preferences to define “social
preference”;
Some mechanisms to aggregate individual preferences to obtain social preferences are discussed below:–
Method – A: Majority Voting
Under this method, allocation "x” is socially preferred “y” when majority of the individuals prefer x to y. A limitation of
this method is that it may not generate a transitive social preference ordering. This limitation is illustrated by
considering the following example from Hal Varian.
There are 3 persons A, B and C who have asked to rank three different allocations x, y and z. Their ranking is given as
follows.
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Persons
A
B
C
Rank 1
x
Y
z
Rank 2
y
Z
x
Rank 3
z
X
y
It will be noticed that persons A and C prefer x to y i.e. a majority (2 out of 3 persons) prefers x to y.
It will be noticed that persons A and B prefer y to .z i.e. a majority (2 out of 3 persons) prefers y to z.
It will be noticed that person B and C prefer z to x i.e. a majority (2 out of 3 persons) prefers z to x.
Thus aggregating individual preferences using the mechanism of majority voting shall not work. This is because the
social preferences resulting from majority voting are not transitive and therefore there is no “best” alternative from
the set of alternatives (x, y and z) from the society’s point of view.
Further the choice of the society shall depend upon the order in which the vote is taken. Consider the following
possibilities:
Possibility One:
The three persons decide to vote first on x Vs. y. The allocation that wins this contest shall then compete against z.
At the first level of contest (x Vs. y); it is the majority that prefers x to y. And at the second level of the contest (x Vs.. z)
it is z that the majority prefers. Therefore, the outcome that the society chooses is z.
Possibility Two:
The three persons decide to vote first on z Vs. x and the second level of the contest is between the winner of (z Vs. x)
and y.
z shall win the first level of the contest. The second level of the contest is between y and z. y shall beat z in the second
level. Therefore, the outcome that the society chooses now is y.
Possibilities one and two show that the overall outcome depends upon the order in which the alternatives are
presented to the agents. This implies that majority voting can be manipulated by changing the order in which things
are voted so as to yield the desired outcome.
Method B: Rank Order Voting
In this method each person ranks the goods according to his preferences and also assigns a number that indicates its
Rank in the ordering. The individual may assign 1 for the best, 2 for the second best and 3 for the third best and so on.
We obtain the results from all persons, Then we add the scores for each alternative across all people. This gives us the
aggregate score for each alternative. The outcome with the lowest aggregate score shall be the socially preferred
outcome. [This is because 1 is for the best score, 2 for the second best etc.] Rank ordering method has a major limitation.
The method can be manipulated when new alternatives are introduced, that can change the final ranking.. This
limitation is illustrated with the following example from Hal Varian. Initially there are 2 alternatives x and y available
to 2’agents A and B.
The preference ordering given by A and .B is given below:
A
B
Rank 1
x
y
Rank 2
y
z
In this case, there would be a tie with each alternative (x & y) getting an aggregate rank of 3.
We now introduce another alternative z for voting/ranking. The ranking given by A and B is shown below:
A
B
Rank 1
x
y
Rank 2
y
z
Rank 3
z
x
Person A has given a score of 1 to x, 2 to y and 3 to z.
Person B has given a score of 1 to y, 2 to z and 3 to x.
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x now has an aggregate score of 4. y has an aggregate score of 3. Thus y shall be preferred to x under the rank – ordering
method.
Thus method A (Majority voting) and method B (Rank ordering) have shown that there is NO PERFECT WAY to
aggregate individual preferences to obtain one social preference outcome.
Arrow Impossibility Theorem:
Kenneth Arrow, the American economist, and the Nobel Prize winner in 1972, has stated that there is NO IDEAL way
to aggregate individual preferences into social preferences. This is called the Arrow’s Impossibility Theorem. There are
3 plausible and desirable criteria to be satisfied to aggregate individual preferences to make one social preference.
These 3 criteria are:
1st Criteria: Given that individual preferences are complete, reflexive and transitive, the social decision mechanism
(that helps to aggregate individual preferences) should result in social preferences that are also complete, reflexive
and transitive.
2nd Criteria: If all individuals prefer alternative x to alternative y, then the social preferences must rank x ahead of y.
3rd Criteria: The preferences between alternative x and y should depend ONLY on how people rank x Vs. y and not on
how they rank other alternatives.
If we aggregate individual preferences to obtain social preferences, then any one of the above criteria described in
Arrow’s impossibility theorem shall be violated. Further, if a social decision mechanism does satisfy criteria 1, 2 and 3
then it will only be under dictatorship (wherein social ranking shall be the ranking of one individual only).
According to Arrow it shall be impossible to device a social. welfare function, that is positively related to individual
choices. In simple words, the society shall not be able to make up its collective mind as to what it wants.
It is impossible to device a mechanism that shall produce a consistent set of preferences for a group from the individual
preferences comprising the group. No social decision mechanism is both rational and egalitarian. A social decision
mechanism that can satisfy. all the three criteria shall be that of dictatorship – wherein one individual determines what
choice to make. Dictatorship lacks the feature of equality. Thus, there is no social welfare function which satisfies the
entire three criterions and which can produce a transitive preference ordering over social states.
Social Welfare Functions
Economists make use of various types of welfare functions to represent distributional judgements about allocations.
Some of these welfare functions are discussed below:
The Classical Utilitarian or Benthamite Welfare function: Jeremy Bentham was the leading philosopher of the
utilitarian, school of thought. According to him, the sole stimulus to human endeavour was self interest and the pursuit
of one’s happiness. This school of thought considered the highest good to be the greatest happiness for the greatest
number.
Suppose person i prefers x to y if and only if ui(x) > ui(y). (i.e. utility from x allocation for person i is greater than utility
from y allocation for the same person i).
“n” is the number of individuals in the society. In order to obtain social. preferences from individual preferences we
add up (summate over i = 1 to n persons) individual utilities and use this result as describing social utility.
In this case, we can say that allocation x is socially preferred to allocation y when
n
n
i =1
i =1
 u i ( x )   u i ( y)
where
n
W(u1(x), u2(x), ………. u(x)) =
 u i (x)
i =1
However one condition that MUST be satisfied in such n aggregation is that the aggregated function must be increasing
in each individual’s utility.
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2. The weighted sum of utility welfare function;
This type of social welfare function is of the form
W (u1 , u 2 , u 3 ,..., u n ) =
n
 a ju i
j=1
where a1, a2, a3 etc. indicates the importance (or weight) that each agents utility has to the overall social welfare.
3. Rawlsian social welfare function:
John Rawls, an American .political theorist at Harvard, stated that social welfare should be defined in terms of welfare
of the least well off agent in the society – a so called minimum approach. The social welfare of an allocation thus
depends upon the welfare of the worst off agent i.e. the person with the minimal utility.
W(u1, u2, …, un) = min{u1, u2, …, un}
The welfare of the rest of the population should only be treated as a tiebreaking rule, for ranking different outcomes
that were irrelevant’ to the worst – off agent.
4. Nietzchean welfare function:
This welfare function states that the value of the allocation’ depends only on the welfare of the best off agent. The
Neitzchean welfare function is the opposite of the Rawisian welfare function.
W(u1, u2, …, un) = max{u1, u2, …, un}
5. Individualistic welfare function (also called the Bergson – Samuelson welfare function):
In this case the individual preferences are being defined NOT over the entire allocations available to all agents but the
individual preferences are being defined over each individual’s own bundle of goods.
xi represent individual is consumption bundle.
uj(xi) represent individual i’s utility.
The social welfare function is of the form:
W = W[u1(x1), u2(x2), … ,un(xn)]
The welfare function is directly a function of the individual’s utility levels and indrectly a function of the individual’s
consumption bundle.
Social Welfare Maximisation
The Bergson–Samuelson social welfare function will take into consideration both the issue of efficiency and (outcome
oriented approach to) equity.
Suppose the economy consists of two agents, A and B. Their preferences are represented by utility functions. Each
allocation of resources leads to a pair of utility levels, one for A and one for, B. Each of these outcomes is plotted as a
point such as J and K. The area OFG shows the utility levels associated with every possible allocation of resources. The
boundary wall of the area OFG, shown by the bold concave line FG is called the utility possibility curve.’ An allocation
of resources that results in a pair of utility for A and B, such as point J, below the boundary FG shall be considered
inefficient. This is because, starting at point J, we can make both A and B better off by moving to a point K. Point J is
inefficient because there is a better way to allocate resources so that atleast one consumer or both consumers are better
off without making anyone else worse off.
Thus all allocations of resources that make consumer A and B to be on the boundary (utility possibility frontier) shall
be efficient.
We have also drawn social indifference curves to represent the judgments associated with a given social welfare
function. Indifference curves that are farther away from the origin (preference direction is northeast) represent higher
level of social welfare.
These indifference curves are also called iso–welfare curves. They represent distributions of those utility that have
constant welfare.
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Maximal welfare point is attained when the utility possibility frontier becomes tangential to the highest possible iso–
welfare curve. This is attained at point e.
This maximal welfare point which is Pareto efficient occurs on the boundary of the utility possibility set.
We can now generalize and state that ANY Pareto efficient allocation (one that lies on the boundary FG of the utility
possibility set) shall be a welfare maximum for some social welfare function. Thus the Bergson Samuelson welfare
function shows a preference for efficiency.
Calculus Conditions for Maximal Welfare
LetX1 and X2 represent the total amount of good 1 and good 2. produced and consumed.
X1 = X1A + X1B
X 2 = X 2A + X 2B
Combination (X1, X2) is on the production possibility frontier if and only if
T(X1, X2) = O
Therefore the maximization problem can be written as
max W (u A ( x1A , x 2A ), u B ( x1B , x 2B )]
Such that T(X1, X2) = O
L = W (u A ( x1A , x 2A ), u B ( x1B , x 2B )] − [T(X1 , X 2 ) − 0]
Differentiating w.r.t each of the choice variable we obtain:
L
w u A ( x1A , x 2A )
T(X1 , X 2 )
=
.
−
=0
x1A u A
x1A
X1
(1)
L
w u A ( x1A , x 2A )
T(X1 , X 2 )
=
.
−
=0
x 2A u A
x 2A
X 2
(2)
L
w u B ( x1B , x 2B )
T(X1 , X 2 )
=
.
−

=0
x1B u B
x1B
X1
(3)
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L
w u B ( x1B , x 2B )
T(X1 , X 2 )
=
.
−

=0
x 2B u B
x 2B
X 2
(4)
Rearranging and’ dividing equation (1) by (2) and.(3) by (4), we obtain the following result.
u A / x1A T /(x1 )
=
u A / x 2A t /(X 2 )
(5)
u B / x1B T /(x1 )
=
u B / x 2B t /(X 2 )
(6)
Equation (5) and (6) can be interpreted, as
MU1A X 2
=
MU 2A X1
MU1B X 2
=
MU 2B X1
a1 , a 2 , a 3
Equation (5) and (6) require that each consumer’s marginal rate of substitution between the goods MUST EQUAL the
marginal rate of transformation.
This implies that the rate at which each consumer is just willing to substitute one good for another must be the same
as the rate at which it is technologically feasible to transform one good into the other. Equations (5) and (6) are welfare
maxima conditions and are identical to the calculus conditions for achieving Pareto efficient allocations. This is because
the allocation resulting from the maximization of the Bergson–Samuelson welfare function is Pareto efficient and every
Pareto efficient allocation maximizes some welfare function.
Fair Allocations:
When an allocation is BOTH equitable and Pareto efficient, it is called a fair allocation.
An allocation is said to be equitable when NO consumer prefers ANY OTHER consumer’s bundle of goods to HIS’ OWN.
If some consumer i DOES PREFER consumer j’s bundle of goods then consumer i ENVIES consume. Consider the
allocation E* in the following, figure2.
OA is the origin for agent A.
OB is the origin for agent B.
OAT1 is the total amount of good x1 available to A & B.
OBT1 is the total amount of good x2 available to A & B.
To test whether an allocation is equitable, we determine the allocations that result when the two consumers SWAP
bundles.
Bundle of A now belongs to B.
Bundle of B now belongs to A.
If the swapped allocation lies below, each consumer’s IC through the original allocation, then the original allocation is
an EQUITABLE allocation.
E* is the original allocation.
Z is the swapped allocation.
Z lies on lCA1 and below lCA2 for A.
Z lies on lCB1 and below lCB2 for B.
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Therefore original allocation at E* is an equitable allocation. E* is NOT ONLY equitable, but is also Pareto efficient. E* is
therefore a fair allocation. .
When the initial allocation is symmetric (allocation is based on equal division), then the market mechanism shall
ensure that it is a fair allocation. Thus fair allocations have provided another alternative to make distributional
judgments.
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Hal Varian Workbook
Numericals & Detailed Solutions
Q1.
Social preference can be determined by Borda Count, Or rank order voting.
There are 10 voters and each voter is asked to rank and give first rank to 1, second rank to 2 and so on For any
two alternatives x and y, if the borda count of x is smaller than or the same as of y, then x is socially “atleast as
good as” y
a)
Is the above condition
i.
Complete
ii.
Reflexive and
iii. Transitive?
b)
If everyone prefers x to y, will the Borda count rank x as socially preferred to y?
c)
If there are 2 voters and 3 alternatives Voter 1 preference is in the order x, z and y and voter 2’s
preference is in the order y, x and z . Calculate the borda count for x, y and z?
d)
If the ranking order changes Voter I preference order becomes x, y and z arid voter 2 preference
order changes to y, z and x. Calculate the Borda Count for x, y and z ?
e)
Will the social preference between x and y only depend on rank of x and y or will it consider rank of z
?
Ans.
(a)
(i)
The above condition is complete because one alternative is preferred to other or the voter s
indifferent.
(ii) Since first alternative is preferred to second and it is atleast as good as itself Thus, it is reflexive.
(in) First alternative is preferred to second, second is preferred to third so first is also preferred to third so
it is transitive.
(b)
Borda count is the summation of the ranks. If everyone ranks x as 1 and y as 2. There are 10 voters
rank of x = 1O
 rank of y = 2O
Thus borda count states x is preferred to y.
(c)
The rank order is
Voters
A
B
Broda count
x
1
2
3
y
3
1
2
z
2
3
5
Borda count for
x=3
y = 4,
z=5
(d)
The new rank order is
Voters
A
B
Broda
x
1
3
4
y
2
1
3
z
3
2
5
Borda Count for
VIKAAS WADHWA SIR
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X=4
Y=3
Z=5
(d)
In the above example after the change, people changed only their ranks of z, but they did not change their
mind about x is preferred to y or y is preferred to x. But when it comes to social preference it is considered
that in case 1, x was preferred to y but in case 2, y was preferred to x.
Q2.
If the utility possibility frontier is give by UA + 2UB = 200. Plot the utility possibility frontier.
(a)
According to Neitzshean social welfare function, what must be the value of UA and UB?
(b)
According to Rawlsian social welfare function, what must be the value of UA and UB ?
(c)
If social welfare is given by w(UA, UB) = UA1/2 UB1/2. What should be the value of UA and UB, if social
welfare is maximized?
(d)
Show the three social maxima on the graph.
Ans.
If utility possibility frontier is
UA + 2UB = 200
This is represented by DA
(a)
Neitzshean social welfare function states
w(UA, UB) = max(UA, UB)
UA + 2UN = 200
If VA = 0
UB = 100
if UB = 0 UA = 200
max of the utility
if UA = 200
(b)
UB = 0.
Rawlsian social welfare function states
W(UA,UB} = min(UA, UB)
Equilibrium will be when
UA = UB
Putting in UA + 2U8 = 200
3UB = 200
UB = 66.67 = UA
(c)
If w(UA, UB) = UA1/2UR1/2
UA = 200.

2
UA = 1OO
UB =
200 1

2 2
UB = 50
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Q3. A parent has two children A and B. She has Rs 1000 to give them. If a is the money given to A and b is the
money given to B. How will she divide the money when
(a)
u (a , b ) = a + b
(b)
1 1
u (a , b ) = − , −
a b
(c)
u(a,b) = loga + logb
(d)
u(a,b) = min{a, b}
(e)
u(a,b) = max{a, b}
(f)
u(a,b) = a2 + b2 and she doesn’t set the MRS equal to 1.
Ans. The budget equation for the parent is a+b = 1000, where a is the amount she gives to A and b is the amount she
gives to B. Slope of budget line =
−
p1
= −1 as she loves
p2
them equally.
(a)
u (a , b ) = a + b
u
1
=
a 2 a
u
1
=
b 2 b
MRS = −
2 b
2 a
+ b
= +1
a
b
=1
a
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b =a
Substituting b = a in the budget equation, we get a = 500, b = 500
(b)
1 1
u (a , b ) = − −
a b
u 1
=
a a 2
u 1
=
a b 2
MRS = −
b2
a2
+ b2
= +1
a2
b
=1
a
a=b
Substituting a = b in budget equation, we get
a = 500, b = 500
(c)
u(a,b) = log a + log b
u 1
=
a a
u 1
=
b b
MRS = −
b
a
+b
= +1
a
Substituting a = b in budget equation, we get
a = 500, b = 500
(d)
u(a, b) = min{a, b}
Solution is at kink, where
a=b
Substituting a = b in budget equation, we get a = 500, b =500
(e)
u(a,b) = max{a, b}
a + b = 1000
if a = 0
if b = 0
b = 1000.
a = 1000
Both give max of the utility

a = 1000, b = 0
or b = 1000, a = 0
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(f)
u(a, b) = a2 + b2
Her indifferences curves are non convex quarter circles and not convex with centre at origin and a, b  0. She will give
money to only one child.
Q4. A parent has 2 children A and B. One unit of good cost Re. 1 for A and Rs. 2 for B. The parent has Rs.1000 to
give
to
her
children.
Her
budget
constraint
becomes
a + 2b = 1000. Who will get more money and consume more when a is the amount of consumption good that A
gets and b is the amount that B gets?
(a)
u(a, b) = a + b
(b)
u(a, b) = ab
(c)
1 1
u (a , b ) = − −
a b
(d)
u(a, b) = max(a, b)
(e)
u(ab) = min{a,b}
Ans. Budget constraint
= a + 2b = 1000
(a) u(a, b) = a + b
Here a and b becomes substitutes and A costs less, so he will get more money and will consume more as b = 0, a = 1000.
(b) u(a,b) = ab
u
=b
a
u
=a
b
MRS = −
b
a
Price ratio =
−
1
2
+ b +1
=
a
2
Substituting the value in budget constraint, we get
2b + 2b = 1000
b = 250, a = 500
They have same amount as
A gets a = 500
B gets 2b = 500
But A consumes more
(c)
1 1
u (a , b ) = − −
a b
+ b2 + 1
=
2
a2
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9811439887 63
b2 1
=
a2 2
2b 2 = a 2  2b = a
Substituting the value in budget constraint, we get
b=
2b + 2b = 1000
1000
= 292.91
2+ 2
a = 2b = 2  292.91 = 414.18
Now,
A gets a = 414.18
and
B gets 2b =2  292.91 = 585.82
So, B wilt get more money but A consumes more.
(d)
u(ab) = max{a, b}
if a = 0,b = 500
if b = 0, a = 1000
 a = 1000, b = 0 will give maximum utility
A will et more money and will consume more.
(e)
u(a, b) = min{a, b}
Equilibrium will be at kink where a = b
Substituting in budget constraint, we get
a + 2b = 1000

a + 2a = 1000
a = b = 333.33
A gets a = 333.33
and B gets 2b = 666.66
B will get more money but both will consume equally.
Q5. A and B have a utility possibility frontier UA + UB2 – 100.
(a)
Plot the utility possibility frontier on the graph
(b)
Calculate the slope of the utility possibility curve
(c)
If A thinks that UA = 75 and UB = 5 is the best distribution of welfare, what must be the A’s weighted
sum of utilities social welfare function?
(d)
If B thinks that UA = 19 and UB = 9 is the best distribution of welfare, what must be B’s weighted sum
of utilities social welfare function?
Ans.
(a)
UA + UA2 = 100
if
UA = 0,
if
UA = 0,
UB = 10
UA = 100
This is represented as JK
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(b)
Slope of the curve
UA + U82 = 100
Let z = UA + UB2 (Level curve)
z
=1
U A
z
= 2U B
U B
MRT =
− MU A
MU B
=
− U A
U B
=
− z U A
U B  z
= −2U B  1
= −2U B
(d)
If (UA,UB) = (75, 5)
MRT = –2U8
|MRS| = |MRT| = 2U8 = 10

w = UA + 2UB

w = UA + 5  2UB
MU B
MU A

|MRS| =

w = UA + 10UB
(e)
If (UA, UB) = (19, 9)
= 10
MRT = –2UB
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9811439887 65
|MRS| = |MRT| = 2UB = 18

w = UA + 2UB

w = UA + 9UB

|MRS| =

w = UA + 18UB
MU B
MU A
= 18
Q6. A and B have identical utility function u(x,x2)= x12 + x22. There are 10 units of x1 and 10 units of x2 in total.
(a)
Draw A and B’s indifference curves in an Edgeworth box and mark the Pareto optimal allocations.
(b)
Find the fair allocations.
Ans.
(a)
u(x1,x2) = x12 + x22
The indifference curves are circles, since x1 and x2 are positive; it would be non convex quarter circles. Pareto efficient
allocations lie on the axes.
(b)
Fair allocation of
u(x, x2) = x12 + x22 is at P and Q because it is equitable and pareto efficient.
Q7. A and B consume x1 and x2. A’s utility function is
U A ( x1 , x 2 ) = 2x1 + x 2 . B’s utility function is
U B ( x1, x 2 ) = x1 + 2x 2
There are 12 units of x1 and x2 each.
(a)
Draw an Edgeworth box showing some of their indifference curves. Mark the Pareto optimal
allocations.
(b)
Write an equation that states
(i)
A likes his bundle and B’s bundle.
(ii) B likes his bundle and A’s bundle.
(c)
x1A + x1B =12
x2A + x2B =12
VIKAAS WADHWA SIR
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Write an equation involving only x1A and x2A such that A prefers his own bundle to B. Write an equation
that involves only x1B and x2B such that B prefers his own bundle to A. Describe the points in the
Edgeworth box.
(d)
Mark the fair allocations.
Ans.
(a)
x1 = 12
x2 = 12
(b)
(i) A likes 2x1 +x2
For A
2x1A + x 2A  2x aB + x B
(ii) B likes x1 + 2x2 .
For B
(c)
x1B + 2x28  x1A + 2x2A
We know that
x1A + x1B = 12
and x2A + x2B = 12
For A
x1B = 12 – x1A
x2B = 12 – x2A
As A prefers his bundle to B, therefore
2x1A + x 2A  2x1B + 2x 2B
2x1A + x 2A  2(12 − x1A ) + 12 − x 2A
2x1A + x 2A  24 − 2x1A + 12 − x 2A
2x1A + x 2A  36 − 2x1A − x 2A
4x1A + 2x 2A  36
2x1A + x 2A  18
Similarly, For B
x1A = 12 − x1B
VIKAAS WADHWA SIR
9811439887 67
x 2A = 12 − x 2B
As B prefers his bundle to A, therefore
x1A + 2x1B  x1A + 2x 2A
x1B + 2x1B  12 − x1B + 2(12 − x 2B )
x1B + 2x1B  12 − x1B + 24 − 2x 2B
x1B + 2x1B  36 − x1B − 2x 2B
2x1B + 4x1B  36
x1B + 2x1B  18
The Area MNOBP describes the points for which A prefers his own bundle to, B and area ROAPS describes the points for
which B prefers, his own bundle to A. This is because in area MNOBP there is no IC of consumer B which is higher than
that of A and in area ROAPS there is no IC of consumer A which is higher than that of B.
(d)
Fair allocation is an allocation which is both equitable and pareto efficient. In the above case, fir allocations
are located from M to P and P to S.
For Example: If we take a point z, then it is pareto efficient and is formed by ICA3 and ICB4, when we swap this bundle,
we get z, which is formed by lower ICAl and ICB1. So neither A nor B envy the swapped bundle z. Therefore is an equitable
bundle & hence it is a fair allocation. Similarly all other bundles can also be proved as fair allocations.
Q8.
A and B are sisters. They would like to consume only one good x1. They love each other, very much.
U A ( x A , x B ) = x A x1−  , U B ( x A , x B ) = x B x1A− 
There are 24 units of x1.
(a)
If  = 2/3, how will A allocate x1 and how will B allocate x1 ?
(b)
What is the Pareto optimal allocation?
(c)
Show A and B’s favourite point in Edgeworth line.,
(d)
If
=
1
, how will A allocate x1 and how will B allocate x1? At the Pareto optimal allocations, what do
3
A and B disagree about?
Ans.
(a)
U A ( x A , x B ) = x A x1B− 
If
=
and
x A + x B = 24
2
3
U A ( x A , x B ) = x 2A/ 3x1B/ 3
As the above utility function represents Cobb douglas preferences, therefore at equilibrium
xA =

1
xA =
2/3
.24
2 / 3 +1 / 3
=
total unit of x1
2
 24 = 16 units
3
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xB =
1− 
1
xB =
1/ 3
.24
2 / 3 +1 / 3
total unit of x1
1
=  24 = 8 units
3
If A allocates then
x A = 16, x B = 8
Similarly, UB =(xB, xA) = xB2/3 xA1/3
2
x B = .24 = 16
3
1
x A = .24 = 8
3
If B allocates then
x B = 16, x A = 8
(b)
After each gets at least 8 units of x1 , it is a Pareto optimal allocation.
(c)
e1 is A’s favourite point where A gets 16 units and B gets 8 units of x1 and e2 is B’s favourite point where B gets 16 units
and A gets 8 units of x1.
(d)
U A ( x A , x B ) = x A x1B− 
If
=
1
3
U A ( x A , x B ) = x1A/ 3x 2B/ 3
1
x A = .24 = 8
3
2
x B = .24 = 16
3
If A allocates then ‘
x A = 8, x B = 16
Similarly, UB = (xB , xA) = xB1/3xA2/3
1
x B = .24 = 8
3
2
x A = .24 = 16
3
If B allocates then
VIKAAS WADHWA SIR
9811439887 69
x4 = 16, x4 = 8
When
=
1
, A wants to give more of good x1 to B but B does not accept because according to her optimum allocation
3
she wants to give more of good x1 to A but A does not accept either due to her own optimum allocation. As a result, at
Pareto optimal allocations, both A and B disagree with respect to trading with each other as they like each other more
than good x1.
Q9.
A
and
B
hate
each
other
but
love
good
x.
A’s
utility
UA(xA, xB)= xA – xB2 and B’s utility function is UB(xB, xA) = xB – xA2. There are 4 units of x.
function
is
(a)
If A allocates good x, how will he distribute it?
(b)
If B allocates good x, how will he distribute it?
(c)
If each gets 2 units of x. What would be their utility? If total units of x is only 2, then calculate their
utility when both get one unit each. is it Pareto optimal for them to consume 2 units of x in total?
(d)
Is it possible to throw away some units of x and consume equal units of x? If yes, how much should
they throw away?
Ans.
(a) A and B hate each other
UA(xA, xB) = xA – xB2
A hates giving x to B, so he will’ consume all the 4 units of x.
(b) UB(xB, xA) = xB – xA2
B hates giving x to A, so he will consume all the 4 units of x.
(c)
If xA = 2 and XB = 2
UA(xA, xB)
= xA –xB2
= 2 – 22
= –2
UB(xB, xA)
= xB – xA2
= 2 – 22
= –2
If xA = l and xB = 1
UA(xA, xB)
= xA –xB2
= 1 – 12
=0
UB(xA, xA)
= xB –xA2
= 1 – 12
=0
Since both the utilities are 0, it is not Pareto optimal.
(d)
They need to throw some units of x.
If they throw 0 units of x,
Then xA = 2
xB = 2
UA = –2
UB = –2
If they throw 1 unit of x,
UA = 1.5 - (1.5)2
= 1.5 – 2.25
= –0.75
UB = 1.5 - (1.5)2
VIKAAS WADHWA SIR
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= 1.5 – 2.25
= –0.75
If they throw 2 units of x,
XA = 1
XB = l
UA = 1 – 12 = O
UB = 1 – 12 = 0
If they throw 3 units of x,
UA = 0.5 – (0.5)2
= 0.5 – 0.25
= 0.25
UB = 0.5 – (0.5)2
= 0.5 – 0.25
= 0.25
If they throw all units of x,
UA = O – O = O
UB = O – O = O
To maximize both utilities, they must throw 3 Units of x
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WELFARE(READINGS)
Up until now we have focused on considerations of Pareto efficiency in evaluating economic allocations. But there are
other important considerations. It must be remembered that Pareto efficiency has nothing to say about the distribution
of welfare across people; giving everything to one person will typically be Pareto efficient. But the rest of us might not
consider reasonable allocation. in’ this chapter we will investigate some techniques that can be used to formalize ideas
related to time distribution of welfare.
Pareto efficiency is in-n itself a desirable goal if there is some way to make some group of people better off without
hurting other people, why not do it? But there will usually be many Pareto efficient allocations: how can society choose
among them?
The major focus of this chapter will be the idea of a welfare function which provides a way to "add together" different
consumers utilities. More generally, a welfare function provides a way to rank different distributions of utility among
consumer. Before we investigate, the implications of this concept, it is worthwhile considering just how one might go
about "adding together” the individual consumers preferences to construct some kind of "social preferences.”
33.1 Aggregation of Preferences
Let us return to our early discussion of consumer preferences. As usual, we will assume that these preferences are
transitive. Originally, we thought of a consumer’s preferences as being defined over his own bundle-of goods, but now
we want to expand on that concept and think of each consumer as having preferences over the entire allocation of
goods among the consumers. Of course, this includes the possibility that the consumer might not care about what other
people have, just as we had originally assumed. Let us use the symbol x to denote a particular allocation–a description
of what every individual gets of every good. Then given two allocations, x and y, each individual i can say whether or
not he or she prefers x to y. Given the preferences of all the agents, we would like to have a way to “aggregate” them
into one social preference. That is; if e know how all tine individuals rank various allocations we would like to be able
to use this information to develop a social ranking of the various allocations.
This is the problem of social decision making at its most general level. Let's consider a few examples.
One way to aggregate individual preferences is to use some kind of voting. We could agree that x is “socially preferred”
to y if a majority of the individuals prefer x to y. However, there is a problem with this method – it may not generate a
transitive social preference ordering. Consider for example, the case illustrated in Table 33.1.
Table 33.1 Preferences that lead to intransitive voting.
Person A
Person B
Person C
x
y
z
y
z
x
z
x
y
Here we have listed the rankings for three alternatives, x, y, and z. by three people. Note that a majority of time people
prefer x to y, a majority prefer y to z, and a majority prefer z to x. Thus aggregating individual preferences by majority
vote won’t work since, in general, time social preferences measuring from majority voting are not well–behaved
preferences, since they are not transitive. Since the preferences are not transitive, there will be no “best” alternative
from thee set of alternatives (x, y. z). Which outcome society chooses will depend on the order in which time vote is
taken.
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‘To see this suppose that the three people depicted in Table 33.1 decide to vote first on x versus y, and then vote on the
winner of this contest versus z. Since a majority prefer x to y, the second contest will he between x and z, which means
that z will be the outcome.
But what if they decide to vote on z versus x and then pit the winner of this vote against y ? Now z wins the first vote,
but y beats z in the second vote. Which outcome is the overall winner depends crucially on the order in which the
alternatives are presented to the voters.
Another kind of voting mechanism that we might consider is rank-order voting. Here each person ranks the goods
according to his preferences and assigns a number that indicates its rank in his ordering: for example, a 1 for the best
alternative, 2 for the second best, and so on. Then we sum up the scores of each alternative across the people to
determine an aggregate score for each alternative and say that one outcome is socially preferred to another if it has a
lower, score.
In Table 33.2 we have illustrated a possible preference ordering for three allocations x, y, and z by two people. Suppose
first that only alternatives x and y were available. Then in this example x would be given a rank of 1 by person A and 2
by person B. The alternative y would be given just the reverse ranking. Thus the outcome of the voting would be a tie
with each alternative having an aggregate rank of 3.
Table 33.2. The choice between x and y depends on z
Person A
Person B
x
y
y
z
z
x
But now suppose that z is introduced to the ballot, Person A would give x a score of 1, y a score of 2, and z a rank of 3.
Person B would give y a score of 1, z a score of 2, and x a score of 3. This means that x would now have an aggregate
rank of 4. and y would have an aggregate rank of 3. In this case y would be preferred to x by rank–order voting.
The problem with both majority voting and rank–order voting is that’ their outcomes can be manipulated by astute
agents. Majority voting can be manipulated by changing the order on which timings are voted so as to yield the desired
outcome. Rank order voting can be manipulated by introducing new alternatives that change the final ranks of the
relevant alternatives.
The question naturally arises as to whether there are social decision mechanisms–ways of aggregating preferences–
that are immune to this kind of manipulation? Are there ways to “add up” preferences that don’t have the undesirable
properties described above?
Let’s list some things that we would want our social decision mechanism to do:
1.
Given any set of complete, reflexive, and transitive individual preferences, the social decision mechanism
should result in social preferences that satisfy tine same properties.
2.
If everybody prefers alternative x to alternative y, then the social preferences should rank x ahead of y.
3,
The preferences between x and y should depend only on how people rank x versus y and not on how them
rank other alternatives
All three of these requirements seem eminently plausible. Yet it can be quite difficult to find a mechanism that satisfies
all of them. In fact, Kenneth Arrow has proved the following remarkable result.
Arrow’s Impossibility Theorem. If a social decision mechanism satisfies properties 1, 2, and 3, then it must be a
dictatorship: all social rankings are the rankings of one individual.
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Arrow’s Impossibility Theorem is quite surprising. It shows that three very plausible and desirable features of a social
decision mechanism are inconsistent with democracy: there is no “perfect” way to make social decisions. There is no
perfect way to “aggregate” individual preferences to make one social preference. If we want to find a way to aggregate
individual preferences to form social preferences, we will have to give up one of the properties of a social decision
mechanism described in Arrow's theorem.
33.2 Social Welfare Functions
If we were to drop any of the desired features of a social welfare function described above, it would probably he
property 3–that the social preferences between two alternatives only depends on the ranking of those two alternatives.
If we do that, certain kinds of rank-order voting become possibilities.
Given the preferences of each individual i over the allocations, we can construct utility functions, ui(x), that summarize
the individuals’ value judgments: person i prefers x to y if and only if ui(x) > uj(y). Of course, these are just like all utility
functions–they can be scaled in any way that preserves the underlying preference ordering. There is no unique utility
representation.
But let us pick some utility representation and stick with it. Then one way of getting social preferences from individuals’
preferences is to add up the individual utilities and use the resulting number as a kind of social utility. That is, we will
say that allocation x is socially preferred to allocation y if
n
n
i =1
i =1
 u i ( x )   u i ( y)
where -n is the number of individuals in the society.
This works–hut of course it ms totally arbitrary, since our choice of utility representation is totally arbitrary. The
choice of using the sum is also arbitrary. Why not use a weighted sum of utilities? Why not use the product of utilities,
or the sum of time squares of utilities?
One reasonable restriction that we might place on the “aggregating function” is that it be increasing in each individual’s
utility. That way we are assured that if everybody prefers x to y, then the social preferences will prefer x to y.
There is a name for this kind of aggregating function; it is called a social welfare function. A social welfare function is
just some function of the individual utility function: W(u1(x), . . , un(x)). It gives a way to rank different allocations that
depends only on the individual preferences, and it is an increasing function of each individual’s utility.
Let’s look some examples. One special case mentioned above is the sum of the individual utility functions
n
W( u1 ,..., u n ) =  u i
i =1
This is sometimes referred to as a classical utilitarian or Benthamite welfare function. A slight generalization of this
form is the weighted sum of utilities welfare function:
n
W( u1 ,..., u n ) =  a i u i
i =1
Here the weights, a1,an, are supposed to be numbers indicating how important each agent’s utility is to the overall
social welfare. It is natural to take each a as being positive.
Another interesting welfare function is the minimax or Rawlsiàn social welfare function:
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W( u1 ,..., u n ) = min{u1 ,..., u n )
This welfare function says that the social welfare of an allocation depends only on the welfare of the worst off agent–
the person with time minimal utility.
Each of these is a possible way to compare individual utility functions. Each of them represents different ethical
judgments about the comparison between different agents’ welfares. About the only restriction that we will place on
the structure of tine welfare function ,at this point is that it be increasing in each consumer’s utility.
33.3 Welfare Maximization
Once we have a welfare function we can examine the problem of welfare maximization. Let us use tine notation
x ij
to
indicate how much individual has of good j, and suppose that there are n consumers and k goods. Then the allocation
x consists of the list of how much each of the agents bias of each of the good.
If we have a total amount X1, ,Xk bf goods 1,,k to distribute among the consumers, we can pose time welfare
maximization problem:
max W( u1 ( x ),..., u n ( x ))
such that
n
 x1i = X1
i =1
n
 x ik = X k
i =1
Thus we are trying to find the feasible allocation that maximizes social welfare. What properties does such aim
allocation have ?
Time first thing that we should note is that a maximal welfare allocation must be a Pareto efficient allocation. The proof
is easy: suppose that it were not. Then there would be some other feasible allocation that gave everyone at least as
large a utility, and someone strictly greater utility. But the welfare function is an increasing function of each agent’s
utility. Thus this new allocation would have to have higher welfare, which contradicts the assumption that we originally
had a welfare maximum.
We can illustrate this situation in Figure 33.1, where the set U indicates the set of possible utilities in the case of two
individuals. This set is known as the utility possibilities set. The boundary of this set–the utility possibilities frontier is
the set of utility levels associated with Pareto efficient allocations. If an allocation is on the boundary of the utility
possibilities set, then there are no other feasible allocations that yield higher utilities for both agents.
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Figure 33.1
Figure – 33.2
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The "indifference curves” in this diagram are called isowelfare curves since tine depict those distributions of utility
that have constant welfare. As usual the optimal point is characterized by a tangency condition. But tom: our purposes,
the notable thing about this maximal welfare point is that, it is Pareto efficient – it must occur on the boundary of the
uti1ity possibilities set.
The next observation we can make from this diagram is that any Pareto efficient allocation must be a welfare maximum
for some welfare function.
An example is given in Figure 33.2.
In Figure 33.2 we have picked a Pareto efficient allocation and found a set of isowelfare curves for which it yields
maximal ‘welfare. Actually, we can say a bit more than this. If the set of possible utility distributions is a convex set, as
illustrated, then every point a in its frontier is a welfare maximum for a weighted sum of utilities welfare function, as
illustrated in Figure 33.2. The welfare function thus provides a way to single out Pareto efficient allocations: every
welfare maximum is a Pareto efficient allocation, and every Pareto efficient allocation is a welfare maximum.
33.4 Individualistic Social Welfare Functions
Up until now we have been thinking of individual preferences as being defined over entire allocations ma-timer than
over each individuals bundle of goods. But, as we remarked earlier, individuals a might only care about their own
bundles. In this case. we can use xi to denote individual i’s consumption bundle, and let ui(xi) be individual i's utility
level using some fixed representation of utility. Then a social welfare function will have the form
W = W( u1 ( x1 ),..., u n ( x n ))
The welfare function is directly a function of the individuals’ utility levels, but it is indirectly a function of the individual
agents’ consumption bundles. This special form of welfare function is known as an individualistic welfare function or
a Bergson-Samuelson welfare function.
If each agent’s utility depends only on his or her own consumption, then there are no consumption externalities. Thus
the standard results of Chapter 31 apply and we have an intimate relationship between Pareto efficient allocations and
market equilibria : all competitive equilibria are Pareto efficient, and, under appropriate convexity assumptions, all
Pareto efficient allocations are competitive equilibria.
Now we can carry this categorization one step further. Given the relationship between Pareto efficiency arid welfare
maxima described above, we can conclude that all welfare maxima are competitive equilibria and that all competitive
equilibria are welfare maxima for some welfare function
33.5 Fair Allocations
The welfare function approach is a very general way to describe social welfare. But because it is so general it can be
used to summarize the properties of many kinds of moral judgments. On the other hand, it isn’t much use in deciding
what kinds of ethnical judgments might be reasonable ones.
Another approach is to start with some specific moral judgments and then examine their implications for economic
distribution. This is the approach taken in the study of fair allocations. We start with a definition of what might be
considered a fair way to divide a bundle of goods, and then use our understanding of economic analysis to investigate
its implications.
Suppose that you were given some goods to divide fairly among n equally deserving people. How would you do it? It
is probably safe to say that in this problem most people would divide the goods equally among the in agents. Given that
they are by hypothesis equally deserving, what else could you do?
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What is appealing about this idea of equal division ? One appealing feature is that it is symmetric. Each agent has the
same bundle of goods no agent prefers any other agent’s bundle of goods to his or her own, since they all have exactly
the same thing.
Unfortunately, an equal division will not necessarily be Pareto efficient. If agents have different tastes time will
generally desire to trade away from equal division. Let us suppose that this trade takes place and that it moves us to a
Pareto efficient allocation.
The question arises: is this Pareto efficient allocation still fair in any sense? Does trade from equal division inherit any
of the symmetry of the starting point?
The answer is: not necessarily. Consider the following example. We have three people, A, B, and C. A and B have the
same tastes, and C has different tastes. We start from an equal division and suppose that A and C get together and trade.
Then they will typically both be made better off. Now B, who didn’t have the opportunity to trade with C, will envy A
that is, he would prefer A’s bundle to his own. Even though A and B started, with the same allocation, A was luckier in
her trading, and this destroyed the symmetry of the original allocation. This means that arbitrary trading from a-n
equal division will not necessarily preserve the symmetry of time starting point of equal division We might well ask if
there is any allocation that preserves this symmetry ? Is there any way to get an allocation that is both Pareto efficient
and equitable at the same time?
33.6 Envy and Equity
Let us now try to formalize sonic of these ideas. What do we mean by “symmetric” or “equitable” anyway? One possible
set of definitions is as follows.
We say an allocation is equitable if no agent prefers arm other agent’s bundle of goods to his or her own. If some agent
i does prefer some other agent j’s bundle of goods, we say that i envies j. Finally, if an allocation is both equitable and
Pareto efficient, we will say that it is a fair allocation.
These are ways of formalizing the idea of symmetry alluded to above. An equal division allocation has the property that
no agent envies any other agent–but there are many other allocations that have this same property. Consider Figure
33.3. To determine whether any allocation is equitable or riot, just look at the allocation that results if the two agents
swap bundles. If this swapped allocation lies “below” each agent's indifference curve through time original allocation,
then the original allocation is an equitable allocation. (Here “below” means below from the point of view of each agent;
from our point of view time swapped allocation must lie between the
two indifference curves.)
Note also that the allocation in Figure 33.3 is also Pareto efficient. Thus, it is not only equitable, in the sense that we
defined the term , but it is also efficient. By our definition, it is a fair allocation. Is this kind of allocation a fluke, or will
fair allocations typically exist?
It turns out that fair allocations will generally exist and there is an easy want to see that this is so. We start as we did
in the last section, where,
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Figure 33.3
we had an equal division allocation and considered trading to a Pareto efficient allocation. Instead of using just any old
way to trade, let us use the special mechanism of the competitive market. Thus will move us to a new allocation where
each agent is choosing the best bundle of goods he or she can afford at tine equilibrium prices (p1, p2), and we know
from Chapter 31 that such an allocation must be Pareto efficient.
But is it shill equitable? Well, suppose not. Suppose that one of the consumers, say consumer A, envies consumer B.
This means that A prefers what B has to her own bundle. In symbols
( x1A , x 2A )  A ( x1B , x 2B )
But, if A prefers B’s bundle to her own, and if her own bundle is tire best bundle she can afford at the prices (p1, p2) this
means that B’s bundle must cost more than A can afford. in symbols: -
p1w1A + p 2 w 2A  p1x1B + p 2 x 2B .
But this is a contraction ! For by hypothesis. A and B started with exactly the same bundle, since they started from an
equal division. If A can't afford B’s bundle, then B can't afford it either.
Thus we can conclude that it is impossible for A to envy B in these circumstances. A competitive equilbrium from equal
division must be a fair allocation. Thus the market mechanism will preserve certain kinds of equity: if the original
allocation is equally divided, the final allocation must be fair.
Summary
1.
Arrow’s Impossibility Theorem shows that there is no ideal way to aggregate individual preferences into
social preferences.
2.
Nevertheless, economists often use welfare functions of one sort or another to represent distributional
judgments about allocations.
3.
As long as the welfare function is increasing in each individual’s utility, a welfare maximum will be Pareto
efficient. Furthermore, every Pareto efficient allocation can be thought of as maximizing some welfare
function
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4.
The idea of fair allocations provides an alternative way to make distributional judgments. This idea
emphasizes the idea of symmetric treatment.
5.
Even when the initial allocation is symmetric, arbitrary methods of trade will not necessarily produce a fair
allocation. However, it turns out that the market mechanism will provide a fair allocation.
GENERAL EQUILIBRIUM AND WELFARE (READINGS)
The partial equilibrium models of perfect competition that were introduced in Chapter 12 are clearly inadequate for
describing all the effects that occur when changes in one market have repercussions in other markets. Therefore they
are also inadequate for making general welfare statements about how well market economies perform. Instead what
is needed is an economic model that permits us to view many markets simultaneously. In this chapter we will develop
a few simple versions of such models. The Extensions to the chapter, found at the end of the book, show how general
equilibrium models are applied to the real world.
PERFECTLY COMPETITIVE PRICE SYSTEM
The model we will develop in this chapter is primarily an elaboration of the supply– demand mechanism presented in
Chapter 12. Here we will assume that all markets are of the type described in that chapter and refer to such a set of
markets as a perfectly competitive price system. The assumption is that there is- some large number of homogeneous
goods in this simple economy. Included in this list of goods are not only consumption items but also factors of
production. Each of these goods has an equilibrium price, established by the action of supply and demand. At this set
of prices, every market is cleared in the sense that-suppliers are willing to supply the quantity that is demanded and
consumers will demand the quantity that is supplied. We also assume that there are no transaction or transportation
charges and that both individuals and firms have perfect knowledge of prevailing market prices.
The law of one price
Because we assume zero transaction cost and perfect information, each good obeys the law of one price: A
homogeneous good trades at the same price no matter who buys it or which firm sells it. If one good traded at two
different prices, demanders would rush to buy the good where it was cheaper, and firms would try to sell all their
output where the good was more expensive. These actions in themselves would tend to equalize the price of the good.
In the perfectly competitive market each good must have only one price. This is why we may speak unambiguously of
the price of a good.
Behavioural assumptions
The perfectly competitive model assumes that people and firms react to prices in specific ways.
1.
There are assumed to be a large number of people buying: any one good. Each person takes all prices as given
and adjusts his or her behaviour to maximize utility, given the prices and his or her budget constraint. People
may also be suppliers of productive services (e.g., labor), and in such decisions they also regard prices as
given.
2.
There are assumed to be a large number of firms producing each good, and each firm produces only a small
share of the output of any one good. In making input and out-put choices, firms are assumed to operate to
maximize profits. The firms treat all prices as given when making these profit-maximizing decisions.
These various assumptions should be familiar because we have been making them throughout this book. Our purpose
here is to show how an entire economic system operates when all markets work in this way.
AGRAPHICALMODELOFGENERAL EQUILIBRIUM WITH TWO GOODS
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We begin our analysis with a graphical model of general equilibrium involving only two goods, which we will call x
and y. This model will prove useful because it incorporates many of the features of far more complex general
equilibrium representations of the economy.
General equilibrium demand
Ultimately, demand patterns in an economy are determined by individuals’ preferences. For our simple model we will
assume that all individuals have identical preferences, which can be represented by an indifference curve map defined
over quantities of the two goods, x and y. The benefit of this approach for our purposes is that this indifference curve
map (which is identical to the ones used in Chapters 3–6) shows how individuals rank consumption bundles containing
both goods. These rankings are precisely what we mean by demand in a general equilibrium context of course we
cannot illustrate which bundles- of commodities will be chosen until we know the budget constraints that demanders
face. Because incomes are generated as individuals supply labor, capital, and other resources to the production process,
we must delay any detailed illustration until we have examined the forces of production and supply in our model.
General equilibrium supply
Developing a notion of general equilibrium supply in this two-good model is a somewhat more complex process than
describing the demand side of the market because we have not thus far illustrated production and supply of two goods
simultaneously. Our pproach is to use the familiar production possibility curve (see Chapter 1) for this purpose. By
detailing the way in which their curve is constructed, we can illustrate, in a simple con text, the ways in which markets
for outputs and inputs are related.
Edgeworth box diagram for production
Construction of the production possibility curve for two outputs (x and y) begins with the assumption that there are
fixed amounts of capital and labor inputs that must be allocated to the production of the two goods. The possible
allocations of these inputs can be illustrated with an Edgeworth box diagram with dimensions given by the total
amounts of capital and labor available.
Figure 13.1
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In Figure 13.1, the length of the box represents total labor-hours, and the height of the box represents total capitalhours. The lower left corner of the box represents the “origin” for measuring capital and labor devoted to production
of good x. The upper right corner of the box represents the origin for resources devoted to y. Using these conventions,
any point in the box can be regarded as a fully employed allocation of the available resources between goods x and y
Point A, for example represents an allocation in which the indicated number of labor hours are devoted to x production
together with a specified number of hours of capital. Production of goody uses whatever labor and. capital are “left
over:” Point A in Figure 13.1, for example, also shows the exact amount of labor and capital used in the production of
goody. Any other point in the box has a similar interpretation. Thus, the Edgeworth box shows every possible way the
existing capital and labor might be used to produce x and y.
Efficient allocations
Many of the allocations shown in Figure 13.1 are technically inefficient in that it is possible to produce both more x and
more y by shifting capital and labor around a bit. In our model we assume that competitive markets will not exhibit
such inefficient input choices (for reasons we will explore in more detail later in the chapter). Hence we wish to discover the efficient allocations in Figure 13.1 because these illustrate the production outcomes in this model. To do so,
we introduce isoquant maps for good x (using Ox, as the origin) and goad y (using Oy as the origin), as shown in Figure
13.2. In this figure it is clear that the arbitrarily chosen allocation A is inefficient. By reallocating capital and labor, one
can produce both more x than x2 and more y than y2.
Figure 13.2
The efficient allocations in Figure 13.2 are those such as P1. P2. P3, and P4, where the isoquants are tangent to one
another. At any other points in the box diagram the-two goods’ isoquants will intersect, and we can show inefficiency
as we did for point A. At the points of tangency, however, this kind of unambiguous improvement cannot be made. In
going from P2 to P3, for example, more x is being produced, but at the cost of less y being produced therefore P3 is not
more efficient than P2 – both of the points are efficient Tangency of the isoquants for good x ad good y implies that their
slopes are equal. That is, the RTS of capital for labor is equal in x and y production. Later we will show how competitive
input markets will lead firms to make such efficient input choices.
Therefore, the curve joining Ox and Oy that includes all these points of tangency shows all the efficient allocations of
capital and labor. Points off this curve are inefficient in that unambiguous increases in output can be obtained by
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reshuffling inputs between the two goods. Points on the curve OxOy are all efficient allocations, however, because more
x can be produced only by cutting back on y production and vice versa.
Figure 13.3
Production possibility frontier
The efficiency locus in Figure 13.2 shows the maximum output of y that can be produced
for any pre–assigned output of x. We can use this information to construct a production possibility frontier, which
shows the alternative outputs of x and y that can be produced with the fixed capital and labor inputs. In Figure 13.3 the
OxOy locus has been taken from Figure 13.2 and transferred onto a graph with x and y outputs on the axes. At O x, for
example, no resources are devoted to x production; consequently, output is as large as is possible with the existing
resources. Similarly, at O, the output of x is large as possible. The other points on the production possibility frontier
(say, P1, P2, P3, and P4) are derived from the efficiency locus in an identical way. Hence we have derived the following
definition.
Definition
Production possibility frontier. The production possibility frontier shows the alternative combinations of two
outputs that cart be produced with fixed quantities of inputs if those inputs are employed efficiently.
Rate of product transformation
The slope of the production possibility frontier shows how x output can be substituted for y output when total
resources are held constant. For example, for points near Ox on production possibility frontier, the slope is a small
negative number say, –1/4; this implies that, by reducing y output by 1 unit, x output could be increased by 4. Near 0y
on the other hand, the slope is a large negative number (say, –5), implying that y output must be reduced by 5 units to
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permit the production of one more x. The slope of the production possibility frontier clearly shows the possibilities
that exist for trading y for x in production. The negative of this slope is called the rate of product transformation (RPT).
DEFINITION
Rate of product transformation. The rate of product transformation (RPT) between two outputs is the negative of
the slope of the production possibility frontier for those outputs Mathematically,
RPT (of x for y)
= –[slope of production possibility frontier]
=−
dy
(along O x O y )
dx
(13.1)
The RPT records how x can be technically traded for y while continuing to keep the available productive inputs
efficiently employed
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Shape of the production possibility frontier
The production possibility frontier illustrated in Figure 13.3 exhibits an increasing RPT. For output levels near Ox,
relatively
little
y
must
be
sacrificed
for
obtain
one
more
x
(–dy/dx is small). Near 0y, on the other hand, additional x nay be obtained only by substantial reductions in y output
(–dy/dx is large). In this section we will show why this concave shape might be expected to characterize most
production situations.
A first step in that analysis is to recognize that RPT is equal to the ratio of the marginal cost of x (MCx) to the marginal
cost of y (MCy). Intuitively, this result is obvious. Suppose, for example, that x and y are produced only with labor. If it
takes two labor hours to produce one more x, we might say that MCx is equal to2. Similarly, if it takes only one labor
hour to produce an extra y, then MCy is equal to 1. But in this situation it is clear that the RPT is 2: two y must be forgone
to provide enough labor so that x may he increased by one unit. Hence the RPT is equal to the ratio of the marginal
costs of the two goods.
More formally, suppose that the costs (say, in terms of the “disutility” experienced by factor suppliers) of any output
combination are denoted by C(x y). Along the production possibility frontier, C(x, y) will be constant because the inputs
are in fixed supply. If we call this constant level of costs C, we can write C(x, y) this constant level of cost
write
C , we can
C( x, y) − C = 0 . It is this implicit function that underlies the production possibility frontier. Applying the
results from Chapter 2 for such a function yields:
RPT =
dy
C
MC x
=− x =−
dx C ( x ,y )−C =0
Cy
MC y
(13.2)
To demonstrate reasons why the RPT might be expected to increase for clockwise movements along the production
possibility frontier, we can proceed by showing why the ratio of MCx to MCy should increase as x output expands and y
output contracts. We first present two relatively simple arguments that apply only to special cases; then we turn to a
more sophisticated general argument.
Diminishing returns
The most common rationale offered for the concave shape of the production possibility frontier is the assumption that
both goods are produced under conditions of diminishing returns. Hence increasing the output of good x will raise its
marginal cost, whereas decreasing the output of y will reduce it marginal cost. Equation 13.2 then shows that the RPT
will increase for movements along the production possibility frontier from Ox to 0y. A problem with this explanation,
of course, is that it applies only to cases in which both goods exhibit diminishing returns to scale, and that assumption
is at variance with the theoretical reasons for preferring the assumption of constant or even increasing returns to scale
as mentioned elsewhere in this book.
Specialized inputs
If some inputs were “more suited” for x production than for y production (and vice versa), the concave shape of the
production frontier also could be explained. In that case, increases in x output would require drawing progressively
less suitable inputs into the production of that good. Therefore, marginal costs of x would increase. Marginal costs for
y, on the other hand, would decrease because smaller output levels for y would permit the use of only those inputs
most suited for y production. Such an argument might apply, for example, to a farmer with a variety of types of land
under cultivation in different crops. In trying to increase the production of any one crop, the farmer would be forced to
grow it on increasingly unsuitable parcels of land. Although this type of specialized input assumption has considerable
importance in explaining a variety of real-world phenomena, it is nonetheless at variance with our general assumption
of homogeneous factors of production. Hence it cannot serve as a fundamental explanation for concavity.
Differing factor intensities
Even if inputs are homogeneous and production functions exhibit constant returns to scale, the prodoetion possibility
frontier will be concave if goods x and y use inputs in different proportions. In the production box diagram of Figure
13.2, for example, good x is capital intensive relative to goody. That is, at every point along the Ox,0y, contract curve,
the ratio of k to l in x production exceeds the ratio of k to l in y production: The bowed curve Ox0y is always above the
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main diagonal of the Edgeworth box. If, on the other hand, good y had been relatively capital intensive, the contract
curve would have been bowed downward below the diagonal. Although a formal proof that unequal factor intensities
result in a concave production possibility frontier will not be presented here, it is possible to suggest intuitively why
that occurs. Consider any two points on the frontier in Figure 13.3 say, P 1 (with coordinates x1, y4) and P3 (with
coordinates x3, y2). One way of producing an output combination “between” P1 and P3 would be to produce the
combination
x1 + x 3 y 4 + y 2
,
2
2
.
Because of the constant returns-to-scale assumption, that combination would be feasible and would fully use both
factors of production. The combination would lie at the midpoint of a straight-line chord joining points P1 and P3.
Although such a point is feasible, it is not efficient, as can be seen by examining points P1 and P3 in the box diagram of
Figure 13.2. Because of the bowed nature of the contract curve, production at point midway between P1 and P3 would
be off the contract curve: Producing at a point such as P2 would provide more of both goods. Therefore, the production
possibility frontier in Figure 13.3 must “bulge out” beyond the straight line P1P3. Because such a proof could be
constructed for any two points on 0x0y, we have shown that the frontier is concave; that is, the RPT increases as the
output of good X increases. When production is reallocated in a northeast direction along the contract curve (in Figure
13.3), the capital labor ratio decreases in the production of both x and y. Because good x is capital intensive, this change
increases MCx. On the both hand, because good y is labor intensive, MCy decreases.
Hence the relative marginal cost of x (as represented by the RPT) increases.
Opportunity cost and supply
The production possibility curve demonstrates that there are many possible efficient combinations of the two goods
and that producing more of one good necessitates cutting back on the production of some other good. This is precisely
what economists mean by the term opportunity cost. The cost of producing more x can be most readily measured by
the reduction in y output that this entails. Therefore, the cost of one more unit of x is best measured as the RPT (of x
for y) at the prevailing point on the production possibility frontier. The fact that ‘this cost increases as more x is
produced represents the formulation of supply in a general equilibrium context.
EXAMPLE 13.1 Concavity of the Production Possibility Frontier
In this example we look at two characteristics of production functions that may cause the production possibility
frontier to be concave.
Diminishing returns. Suppose that the production of both x and y depends only on labor input and that the production
functions for these goods are
y = f (lx ) − lx0.5
(13.3)
y = f (ly ) = ly0.5
Hence production of each of these goods exhibits diminishing returns to scale. If total labor supply is limited by
lx + ly = 100
(13.4)
then simple substitution shows that the production possibility frontier is given by
x 2 + y 2 = 100 for x, y  0
(13.5)
In this case, the frontier is a quarter-circle and is concave. The RPT can now be computed directly from tie equation for
the production possibility frontier (written in implicit form as
RPT = −
 f  2x x
dy
= − − x  =
=
 f  2y y
dx
y


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f ( x, y) = x 2 + y 2 − 100 = 0 )
(13.5)
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and this slope Increases as x output increases. A numerical illustration of concave starts by noting that the points (10,
0) and (0, 10) both lie on the frontier. A straight line joining these two points would also include the point (5, 5), but
that point lies below the frontier. If equal amounts of labor are devoted to both goods, then production is x = y = which
yields
more of both goods than the midpoint.
Factor intensity. To show how differing factor intensities yield a concave production possibility frontier, suppose that
the two goods are produced under constant returns to scale but with different Cobb–Douglas production functions:
x = f (k, l ) = k 0x.5lx0.5
y = g(k, l ) = k 0y.25ly0.25
(13.7)
Suppose also that total capital and labor are constrained by
k x + k y = 100 ,
lx + ly = 100
(13.8)
It is easy to show that
RTSx =
where,
kx
= Kx ,
lx
K i = k i / li K1.
Being
RTSy =
located
3k y
ly
= Ky
on
the
(13.9)
production
possibility
frontier
requires
RTSx = RTSy, or Kx = 3Ky. That is, no matter how total resources are allocated to production, being on the production
possibility frontier requires that x be the capital-intensive good (because, in some sense, capital is more productive in
x production than in y production). The capital labor ratios in the production of the two goods are also constrained by
the available resources:
kx + ky
lx + l y
where
k
kx
100
+ y = K x + (1 − )K y =
=1
lx + l y lx + l y
100
=
 = I x /(I x + I y )
(13.10)
that is,  is the share of total labor devoted to x production. Using the ion that
K x = 3K y
, we can find the input ratios of the two goods in terms of the overall al1otipn of labor:
Ky =
3
1
, Kx =
1 + 2
1 + 2
(13.11)
Now, we re in a position to phrase the production possibility frontier in terms of the share of or-devoted to x
production:
 3 
x = K l = K (100) = 100

 1 + 2 
0.5
x x
0.5
0.5
x
 1 
y = K 0y.25ly = K 0y.25 (1 − )(100) = 100(1 − )

 1 + 2 
0.25
(13.12)
We could push this algebra even further to eliminate  from these two equations to get an explicit function form for
the production possibility frontier that involves only x and y, but we can show concavity with what we already have.
First, notice that if  = 0 (x production gets no inputs), then x = 0, y = 100. With  = 1, we have x = 100, y = 0. Hence a
linear possibility frontier would include the point (50, 50). But if  = 0.39, say, then
 3 
x = 100

 1 + 2 
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0.5
 3 
= 39

 1.78 
0.5
= 50.6
9811439887 87
 1 
y = 100(1 − )

 1 + 2 
0.25
 1 
= 61

 1.78 
0.25
= 52.8
(13.13)
which shows that the actual frontier is bowed outward beyond a linear frontier. It is worth repeating that both of the
goods in this example are produced under constant returns to scale and that two inputs are fully homogeneous. It is
only the differing input intensities involved in the production of the two goods that yields the concave production
possibility frontier.
QUERY: How would an increase in the total amount of labor available shift the production possibility frontiers in these
examples?
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9811439887 88
Determination of equilibrium prices
Figure 13.4
Given these notions of demand and supply in our simple two-good economy, we can now illustrate how equilibrium
prices are determined. Figure 13.4 shows PP, the production possibility frontier for the economy, and the set of
indifference curves represents individuals’ preferences for these goods. First, consider the price ratio p x/py. At this
price ratio, firms will choose to produce the output combination x1, y1. Profit-maximizing firms will choose the more
profitable point on PP. At x1, y1 the ratio of the two goods’ prices (px/py) is equal to the ratio of the goods marginal costs
(the RPT); thus, profits are maximized there, On the other hand, given this budget constraint (line C), individuals will
demand x'1, y'1. Consequently, with these prices, there is an excess demand for good x (individuals demand more than
is being produced) but an excess supply of good y. The workings of the marketplace will cause px to increase and py to
decrease. The price ratio px/py will increase; the price line will take on a steeper slope. Firms will respond to these
price changes by moving clockwise along-the production possibility frontier; that is, they will increase their production
of good x and decrease their production of good y Similarly, individuals will respond to the changing prices by
substituting y for x in their consumption choices. These actions of both firms and individuals serve to eliminate the
excess demand for x and the excess supply of y as market prices change.
Equilibrium is reached at x*, y* with a price ratio of
p*x / p*y . With this price ratio, supply and demand are equilibrated
for both good x and goody. Given px and py, firms will produce x* and y* in maximizing their profits. Similarly, with a
budget constraint given by C*, individuals will demand x* and y*. The operation of the price system has cleared the
markets for both x and y simultaneously. Therefore, this figure provides a “general equilibrium” view of the supply–
demand process for two markets working together. For this reason we will make considerable use of this figure in our
subsequent analysis.
COMPARATIVE STATICSANALYSIS
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9811439887 89
As in our partial equilibrium analysis, the equilibrium price ratio
p*x / p*y
illustrated in Figure 13.4 will tend to persist
until either preferences or production technologies change. This competitively determined price ratio reflects these
two basic economic forces. If preferences were to shift, say, toward .good x, then
p x / p y , would increase and a new
equilibrium would be established by a clockwise move along the production possibility curve More x and less y would
be produced to meet these changed preferences Similarly, technical progress in the production of good x would shift
the production possibility curve outward, as illustrated in Figure 13.5. This would tend to decrease the relative price
of x and increase the quantity of x consumed (assuming x is a normal good). In the figure the quantity of y
Figure 1.3.5
consumed also increases as a result of the income effect arising from the technical advance; however, a slightly
different drawing of the figure could have reversed that result if the substitution effect had been dominant. Example
13.2 looks at a few such effects.
Example 13.2 Comparative Statics in a General Equilibrium Model
To explore how general equilibrium models work, let’s start with a simple example based on the production possibility
frontier in Example 13.1. In that exmp1e we assumed that production of both goods was characterized by decreasing
returns
x = lx0.5
and
y = ly0.5
and also that total labor available was given by
lx + ly = 100 .
The resulting
production possibility frontier was given by x2 +y2 100, and R.PT = x/y. To complete this model we assume that the
typical individual’s utility function is given by U(x y) = x0.5y0.5 so the demand functions for the two goods are
0.5I
px
0.5I
y = y( p x , p y , l ) =
py
x = x (p x , p y , l ) =
VIKAAS WADHWA SIR
(13.14)
9811439887 90
Base-case equilibrium. Profit maximization by firms requires that
p x / p y = MC x / MC y = RPT = x / y , and
utility-maximizing demand requires that pxPy = y/x. Thus, equilibrium requires that x/y = y/x, or x = y. Inserting this
result into the equation for the production possibility frontier show that
x * = y* = 50 = 7.07
and
px
=1
py
(13.15)
This is the equilibrium for our base case with this model.
The budget constraint. The budget constraint that faces individuals is not especially transparent in this illustration;
therefore, it may be useful to discuss it explicitly. To bring some degree of absolute pricing into the model, let’s consider
all prices in terms of the wage rate, w.
Because total labor supply is 100, it follows that total labor income is
100w. However, because of the diminishing returns assumed for production, each firm also earns profits. For firm x,
say, the total cost function is C(w, x) = wlx = wx2, so px = MCx. = 2wx = 2w
= (px – ACx)x = (px – wx)x =
wx2 =
50 . Therefore, the profits for firm x are x
50w.
A similar computation shows that profits for firm y are also given by 50w. Because general equilibrium models must
obey the national income identity, we assume that consumers are also shareholders in the two firms and treat these
profits also as part of their spendable incomes. Hence total consumer income is
total income = labos income + profits
= 100w + 2(50w) = 200w
(13.16)
This income will just permit consumers to spend 100w on each good by buying units at a price of 2w
50 , so the
model is internally consistent.
A shift in supply. There are only two ways in which this base-case equilibrium can be disturbed: (1) ly changes in
“supply”–that is, by changes in the underlying technology of this economy; or(2) by changes in “demand”–that is, by
changes in preferences. Let’s first consider change in technology. Suppose that there is technical improvement in x
0.5
production so that the production function is = 2 l x . Now the production possibility frontier is given by x2/4 + y2 =
100, and RPT = x/4y. Proceeding as before to find the equilibrium in this model:
px
x
=
p y 4y
px y
=
py x
so
x 2 = 4y
(supply)
(13.17)
(demand)
and the equilibrium is
x * = 2 50 , y* = 50
and
px 1
=
py 2
(13,18)
Technical improvements in x production have caused its relative price to decrease and the consumption of this good
to increase. As in many examples with Cobb–Douglas utility, the income substitution effects of this price decrease on y
demand are precisely offsetting. Technical improvement clearly make consumers better off however. Whereas utility
was previously given
by U(x,y) = x0.5y0.5 =
50
= 7.07, now it has increased to U(x, y) = x0.5y0.5 =
(2 50 ) 0.5 = 2 . 50 = 10 . Technical change has increased consumer welfare substantially.
A shift in demand. If consumer preferences were to switch to favor good y as U(x, y) = x0.1y0.9 , then demand functions
would be given by x = 0.1I/px and y = O.9I/py, and demand equilibrium would require px/py = y/9x. Returning to the
original production possibility frontier to arrive at an overall equilibrium, we have
VIKAAS WADHWA SIR
9811439887 91
px x
=
py y
px
y
=
p y 9x
(supply)
(demand)
(13.19)
so 9x2 = y2 and the equilibrium is given by
x * = 100 , y* = 3 10
and
px 1
=
py 3
(13.20)
Hence, the decrease in demand for x has significantly reduced its relative price. Observe that in this case, however, we
cannot make a welfare comparison to the previous cases because the utility functions has changed.
QUERY: What are the budget constraints in these two alternative scenarios? How is income distributed between wages
and profits in each case? Explain the differences intuitively
GENERAL EQUILIBRIUM MODELING AND FACTOR PRICES
This simple general equilibrium model reinforces Marshall’s observations about the importance of both supply and
demand forces in the price determination process. By providing an explicit connection between the markets for all
goods, the general equilibrium model makes it possible to examine more complex questions about market relationships
than is possible by looking at only one market at a time. General equilibrium modeling also permits an examination of
the connections between goods and factor markets; we can illustrate that with an important historical case.
The Corn Laws debate
High tariffs on grain imports were imposed by the British government following the Napoleonic wars. Debate over the
effects of these Corn Laws dominated the analytical efforts of economists between the years 1829 and 1845. A principal
focus of the debate concerned the effect that elimination of the tariffs would have on factor prices–a question that
continues to have relevance today, as we will see.
VIKAAS WADHWA SIR
9811439887 92
Figure 13.6
The production possibility frontier in Figure 13.6 shows those combinations of grain (x) and manufactured goods (y)
that could be produced by British factors of production. Assuming (somewhat contrary to actuality) that the Corn Laws
completely prevented trade, market equilibrium would be at E with the domestic price ratio, given by
Removal of the tariffs would reduce this price ratio to
p1x / p1y .
p*x / p*y .
Given that new ratio, Britain would produce
combination A and consume combination B. Grain imports would amount to xB – xA, and these would be financed by
export of manufactured goods equal to yA – yB. Overall utility for the typical British consumer would be increased by
the opening of trade. Therefore, use of the production possibility diagram demonstrates the implications that relaxing
the tariffs would have for the production of both goods
Trade and factor prices
By referring to the Edgeworth production box diagram (Figure 13.2) that lies behind the production possibility-frontier
(Figure 13.3), it is also possible to analyze the effect of tariff reductions on factor prices. The movement from point E
to point A in Figure 13.6 is similar to a movement from P3 to P1 in Figure 13.2, where production of x is decreased and
production of y is increased.
This figure also records the reallocation of capital and labor made necessary by such a move. If we assume that grain
production is relatively capital intensive, then the movement from P3 to P1 causes, the ratio of k to I to increase in both
industries. This in turn will cause the relative price of capital to decrease (and the relative price of labor to increase).
Hence we conclude that repeal of the Corn Laws would be harmful to capital owners (i.e., landlords) and helpful to
laborers. It is not surprising that landed interests fought repeal of the laws.
Political support for trade policies
The possibility that trade policies may affect the relative incomes of ‘various factors of production continues to exert
major influence on political debates about-such policies. In the United States for example, exports tend to be intensive
VIKAAS WADHWA SIR
9811439887 93
in their use of skilled labor, whereas imports tend to be intensive m unskilled labor Input By analogy to our discussion
of the Corn Laws, it might thus be expected that further movements toward free trade policies would result in
increasing relative wages for skilled workers and in decreasing relative wages for unskilled workers. Therefore, it is
not surprising that unions representing skilled workers (the machinists or aircraft workers) tend to favor free trade,
whereas unions of unskilled workers (those in textiles, shoes, and related businesses) tend to oppose it.
A MATHEMATICAL MODEL OF EXCHANGE
Although the previous graphical model of general equilibrium with two goods is fairly instructive, it cannot reflect all
the features of general equilibrium modeling with an arbitrary number of goods and productive inputs in the remainder
of this chapter we will illustrate how such a more general model can be constructed, and we will look at some of the
insights that such a model can provide. For most of our presentation we will look only at a model of exchange–
quantities of-various goods already exist and are merely
traded among individuals. In such a model there is
no production. Later in the chapter we will look briefly at how production can be incorporated into the general model
we have constructed.
Vector notation
Most general equilibrium modeling is conducted using vector notation. This provides great flexibility in specifying an
arbitrary number of goods or individuals in the models.
Consequently, this seems to be a good place to offer a brief introduction to such notation.
A vector is simply an ordered array of variables (which each may take on specific values).
Here we will usually adopt the convention that the vectors we use are’ column vectors. Hence we will write an n  1
column vector as
 x1 
x 
x =  2
 
 
x n 
(13.21)
where each xi is a variable that can take on any value. If x and y are two n  1 column vectors, then the (vector) sum of
them is defined as:
 x1   y1   x1 + y1 
x   y   x + y 
2
x + y =  2 +  2 =  2
      
    

x n   y n  x n + y n 
(13.22)
Notice that this sum only is defined if the two vectors are of equal length in fact checking the length of vectors is one
good way of deciding whether one has written a meaningful vector equation.
The (dot) product of two vectors is defined as the sum of the component-by-component product of the elements in the
two vectors. That is: -
xy = x1y1 + x 2 y 2 + ... + x n y n
(13.23)
Notice again that this operation is only defined if the vectors are of the same length. With these few concepts we are
now ready to illustrate the general equilibrium model of exchange.
Utility, initial endowments, and budget constraints
In our model of exchange there are assumed to be n goods and m individuals. Each individual gains utility from the
vector of goods he or she consumes ui(xi) where i = 1. . . m. individuals also possess initial endowments of the goods
i
given by x . Individuals are free to exchange the initial endowments with other individuals or to keep some or all the
endowment for themselves. In their trading individuals are assumed to be price-takers that is, they face a price vector
(p) that specifies the market price for each of the n goods. Each individual seeks to maximize utility and is bound by a
VIKAAS WADHWA SIR
9811439887 94
budget constraint that requires that the total amount spent on consumption equals the total value of his or her
endowment:
px i = px i
(13.24)
Although this budget constraint has a simple form, it may be worth contemplating it for a minute. The right side of
Equation 13.24 is the market value of this individual’s endowment (sometimes referred to as his or her full income).
He or she could “afford” to consume this endowment (and only this endowment) if he or she wished to be self-sufficient.
But the endowment can also be spent on some other consumption bundle (which, presumeably, provides more utility).
Because consuming items in one’s own endowment has an opportunity cost, the terms on the left of Equation 13.24
consider the costs of all items that enter into the final consumption bundle, including endowment goods that are
retained.
Demand functions and homogeneity
The utility maximization problem outlined in the previous section is identical to the one we studied in detail in Part 2
of this book. As we showed in Chapter 4, one outcome of this process is a set of n individual demand functions (one for
each good) in which quantities demanded depend on all prices and income. Here we can denote these in vector form
as
x i (p, x i ) . These demand function are continuous, and, as we showed in Chapter 4, they are homogeneous of degree
0 in all prices and income. This latter property can be indicated in-vector notation by
x i ( tp, tpx i ) = x i (p, px i )
(13.25)
for any t > 0. This property will be useful, because it will permit us to adopt a convenient normalization scheme for
prices, which, because it does not alter relative prices, leaves quantities demanded unchanged.
Equilibrium and Walras’ law
Equilibrium in this simple model of exchange requires that the total quantities of each good demanded be equal to the
total endowment of each good available (remember; there is no production in this model) Because the model used is
similar to the one originally developed by Leon Walras, this equilibrium concept is customarily attributed to him.
Definition.
Walrasian equilibrium. Walrasian equilibrium is an allocation of resources and an associated price vector p*, such
that
m
m
i =1
i =1
 x i ( p* , p* x i ) =  x i
(13.26)
where the summation is taken over the in individuals in this exchange economy.
The n equations in Equation 13.26 state that in equilibrium demand equals supply in each market. This is the
multimarket analog of the single market equilibria examined in the previous chapter. Because there are n prices to be
determined, a simple counting of equations and unknown might suggest that the existence of such a set of prices is
guaranteed by the simultaneous equation solution procedures studied in elementary algebra. Such a supposition would
be incorrect for two reasons. First, the algebraic theorem about simultaneous equation system applies only to linear
equations. Nothing suggests that the demand equations in this problem will be linear in fact, most examp1e of demand
equations we encountered in Part 2 were definitely nonlinear.
A second problem with Equation 13.26 is that the equations are not independent of one another they are related by
what is known as Walras’ law. Because each individual in this exchange economy is bound by a budget constraint of
the form given in Equation 13.24, we can sum over all individuals to obtain
m
m
i =1
i =1
 px i =  px i
VIKAAS WADHWA SIR
9811439887 95
m
 p( x
or
i
− xi ) = 0
(13.27)
i =1
In words, Walras’ law states that the value of all quantities demanded must equal the value of all endowments. This
result holds for any set of prices, not just for equilibrium prices. The general lesson is that the logic of individual budget
constraints necessarily creates a relationship among the prices in any economy. It is this connection that helps to
ensure that a demand–supply equilibrium exists, as we now show.
Existence of equilibrium in the exchange model
The question of whether all markets can reach equilibrium together has fascinated economists for nearly 200 years.
Although intuitive evidence from the real world suggests that this must indeed be possible (market prices do not tend
to fluctuate wildly from one day to the next), proving the result mathematically proved to be rather difficult. Walras
himself thought he had a good proof that relied on evidence from the market to adjust prices toward equilibrium. The
price would increase for any good for which demand exceeded supply and decrease when supply exceeded demand.
Walras believed that if this process continued long enough, a full set of equilibrium prices would eventually be found.
Unfortunately, the pure mathematics of Walra's, solution were difficult to state, and ultimately there was no guarantee
that a solution would be found But Walras idea of adjusting prices toward equilibrium using market forces provided a
starting point for the modern proofs which were largely developed during the 1950s.
A key aspect of the modern proofs of the existence of equilibrium prices is the choice of a good normalization rule.
Homogeneity-of demand functions makes it possible to use any absolute scale for prices, providing that relative prices
are unaffected by this choice.
Such an especially convenient scale is to normalize prices so that they sum to one. Consider an arbitrary set of n nonnegative prices p1, p2, …, pn. We can normalize these to form a new set of prices
p'1 =
pi
(13.28)
n
p
k =1
k
n
These new prices will have the properties that
 p'
k =1
p'i
=
p'i
pi /  p k
p j /  pk
=
pi
pk
k
=1
and that relative price ratios are maintained:
(13.29)
Because this sort of mathematical process can always be done, we will assume, without loss of generality, that the price
vector we use (p) have been normalized in this way.
Therefore, proving the existence of equilibrium prices in our model of exchange amounts to showing that there will
always exist a price vector p* that achieves equilibrium in all markets. That is,
m
m
i =1
i =1
m
m
i =1
i =1
 x i ( p* , p* x i ) =  x i
or
or
 x i ( p* , p* x i ) −  x i = 0
z ( p* ) = 0
(13.30)
where we use z(p) as a shorthand way of recording the “excess demands” for goods at a particular set of prices. In
equilibrium, excess demand is zero in all markets.
VIKAAS WADHWA SIR
9811439887 96
Figure 13.7
Now consider the following way of implementing Walras’ idea that goods in excess demand should have their prices
increased, whereas those in excess supply should have their prices reduced. Starting from any arbitrary set of prices,
Pu’ we define a new set, P1, as
p1 = f (p 0 ) = p 0 + kz(p 0 )
(13.31)
where k is a positive constant. This function will be continuous (because demand functions are continuous), and it will
map one set of normalized prices into another (because of our assumption that all prices are normalized). Hence it will
meet the conditions of the Brouwer’s fixed point theorem, which states that any continuous function from a closed
compact set onto itself (in the present case, from the “unit simplex” onto itself) will have a “fixed point” such that x =
f(x). The theorems illustrated for a single dimension in Figure 13.7. There, no matter what shape the function f(x) takes,
as long as it is continuous it must somewhere cross the 450 line and at that point x = f(x).
If we let p* represent the fixed point identified by Brouwer’s theorem for Equation 13.31, we have:
p* = f (p* ) = p* + kz(p* )
(13.32)
Hence at this point z(p*) = 0; thus, p* is an equilibrium price vector. The proof that Walra's sought is easily accomplished
using an important mathematical result developed a few years after his death. The elegance of the proof may obscure
the fact that it uses a number of assumptions about economic behavior such as: (1) price-taking by all parties; (2)
homogeneity of demand functions; (3) continuity of demand functions; and (4) presence of budget constraints and
Walras’ law. All these play important roles in showing that a system of simple markets can indeed achieve a
multimarket equilibrium.
First theorem of welfare economics
Given that the forces of supply and demand can establish equilibrium prices in the genera! equilibrium model of
exchange we have developed, it is natural to ask what are the welfare consequences of this finding. Adam Smith
hypothesized that market forces provide an “invisible hand” that, leads each market participant to “promote an’ end
[social welfare] which was no part of his intention. Modem welfare economics seeks to understand the extent to ‘which
Smith was correct.
Perhaps the most important welfare result that can be derived from the exchange model is that the resulting Walrasian
equilibrium is “efficient” in the sense that it is not possible to devise some alternative allocation of resources in which
at least some people are better off and no one is worse off. This definition of efficiency was originally developed by
Italian economist Vilfredo Pareto in the early 1900s. Understanding the definition is easiest if we consider what an
“inefficient” allocation might be. The total quantities of goods included in initial endowments would be allocated
inefficiently if it were possible, by shifting goods around among individuals, to make at least one person better off (i.e.,
VIKAAS WADHWA SIR
9811439887 97
receive a higher utility) and no one worse off. Clearly, if individuals’ preferences are to count, such a situation would
be undesirable. Hence we have a formal definition.
Definition.
Pareto efficient allocation. An allocation of the available goods in an exchange economy is efficient if it, is not possible
to devise an alternative allocation in which at least one person is better off and no one is worse off.
A proof that all Walrasian equilibria are Pareto efficient proceeds indirectly. Suppose that p * generates a Walrasian
equilibrium in which the quantity of goods consumed by each person, is denoted by
x k (k = 1,2,..., m) . Now assume
that there is some alternative allocation of the available goods 'x k(k = 1, ..., n) such that, for at least one person, say,
person i, it is that case that ‘x is preferred to *xi. For this person, it must be the case that
p* ' x i  p* * x i
(13.33)
because otherwise this person would have bought the preferred bundle in the first place. If all other individuals are to
be equally well off under this new proposed allocation, it must be the case for them that
p* ' x k = p* * x k ,
k = 1,…,m, k  i
(13.34)
If the new bundle were ‘less expensive, such individuals could not have been minimizing expenditures at p *. Finally, to
be feasible, the new allocation must obey the quantity constraints
m
m
i =1
i =1
 ' xi =  xi
(13.35)
Multiplying Equation 13.35 by p* yields
m
m
i =1
i =1
 p* ' x i =  p' x i
(13.36)
but Equations 13.33 and 13.34 together with Walras’ law applied to the original equilibrium imply that
m
m
m
i =1
i \1
i =1
 p* ' x i   p* * x i =  p* x i
(13.37)
Hence we have a contradiction and must conclude that no such alternative allocation can
exist. Therefore, we can summarize our analysis with the following definition.
Definition.
First theorem of welfare economics. Every WaIrasian equilibrium is Pareto efficient.
The significance of this “theorem” should not be overstated. The theorem ‘does-not say that every Walrasian
equilibrium is in some sense socially desirable. Walrasian equilibria can, for example, exhibit vast inequalities among
individuals arising in part from in equalities in their initial endowments (see the discussion in the next section). The
theorem also assumes price-taking behavior and full information about prices–assumptions that need not hold in other
models. Finally, the theorem does not consider possible effects of one individual’s consumption on another. In the
presence of such externalities, even a perfect competitive price system may not yield Pareto optimal results (see.
Chapter 19).
Still, the theorem does show that Smith’s “invisible hand” conjecture has so the validity. The simple markets in this
exchange world can find equilibrium prices, and at those equilibrium prices the resulting allocation of resources will
be efficient in the Pareto sense. Developing this proof is one of the key achievements of welfare economics:
A graphic illustration of the first theorem
In Figure 13.8 we again use the Edgeworth box diagram, this time to illustrate an exchange economy. In this economy
there are only two goods (x and y) and two individuals (A and B). The total dimensions of the Edgeworth box are
determined by the total
VIKAAS WADHWA SIR
quantities of the two goods available ( x and
y ). Goods allocated to’ individual A are
9811439887 98
recorded using 0A as an origin. Individual B gets those quantities of the two goods that are “left over” and can be
measured using 0B as an origin. Individual A’s indifference curve map is drawn in the usual way, whereas individual
B’s map is drawn from the perspective of 0B. Point B in the Edgeworth box represents the initial endowments of two
individuals. Individual A starts with
xA
and
yA .
Individual B starts with
xB = x − xA
and
yB = y − yA .
Figure -13.8
The initial endowments provide a utility level of
U 2A
for person A and
U 2B
for person B. These levels are clearly
inefficient in the Pareto sense. For example, we could, by reallocating the available goods, increase person B’s utility to
U 3B
while holding person A’s utility constant at
keeping person B on the
U
2
B
U 2A
(point B). Or we could increase person A’s utility to
U 3A
while
indifference curve (point A). Allocations *A and B are Pareto efficient, however, because
at these allocations it is not possible to make either person better off without making the other worse off. There are
many other efficient allocations in the Edgeworth box diagram. These are identified by the tangencies of the two
individuals’ indifference curves. The set of all such efficient points is shown by the line joining 0A to 0B. This line is
sometimes called the “contract curve” because it represents all the Pareto-efficient contracts that might be reached by
these two individuals. Notice, however, that (assuming that no individual would voluntarily opt for a contract that
made him or her worse off) only contracts between points B and A are viable with initial endowments given by point
E.
The line PP in Figure 13.8 shows the competitively established price ratio that is guaranteed by our earlier existence
proof. The line passes through the initial endowments (E) and shows the terms at which these two individuals can
trade away from these initial positions. Notice that such trading is beneficial to both parties that is, it allows them to
get a higher utility level than is provided by their initial endowments. Such trading will continue until all such mutual
beneficial trades have been completed. That ‘will occur at allocation E on the contract curve Because the individuals
indifference curves are tan gent at this point no further trading would yield gains to both parties Therefore the
competitive allocation E* meets the Pareto criterion for efficiency, as we showed mathematically earlier.
Second theorem of welfare economics
The first theorem of welfare economics shows that a Walrasian equilibrium is Pareto efficient, but the social welfare
consequences of this result are limited because of the role played by initial endowments in the demonstration. The
location of the Walrasian equilibrium at E in Figure 13.8 was significantly influenced by the designation of B as the
starting point for trading. Points on the contract curve outside the range of AB are not attainable through voluntary
VIKAAS WADHWA SIR
9811439887 99
transactions, even though these may in fact be more socially desirable than B’ (perhaps because utilities are more
equal). The second theorem of Welfare economics addresses this issue. It states that for any Pareto optimal allocation
of resources there exists a set’ of initial endowments and a related price vector such that this allocation is also a
Walrasian equilibrium. Phrased another way, any Pareto optimal allocation of resources can also be a Walrasian
equilibrium, providing that initial endowments are adjusted accordingly.
Figure 13.9
A graphical proof of the second theorem should suffice. Figure 13.9 repeats the key aspects of the exchange economy
pictures in Figure 13.8. Given the initial endowments at point B, all voluntary Walrasian equilibrium must lie between
points A and B on the contract curve. Suppose, however, that these allocations were thought to be undesirable perhaps
because they involve too much inequality of utility. Assume that the Pareto optimal allocation Q * is believed to be
socially preferable, but it is not attainable from the initial endowments at point B. The second theorem states that one
an draw a price line through Q that is tangent to both individuals’ respective indifference curves. This line is denoted
by P'P' in Figure 13.9. Because the slope of this line shows potential trades these individuals are willing to make, any
point on the line can serve as an initial endowment from which trades lead to Q* One such point is denoted by Q. If a
benevolent government wished to ensure that Q* would emerge as a Walrasian equilibrium, it would have to transfer
initial endowments of the goods from E to
Q
(making person A better off and person B worse off in the process).
Example 13.3 : A Two–Person Exchange Economy
To illustrate these various principles, consider a simple two-person, two-good exchange economy. Suppose that total
quantities of the goods are fixed at
x = y = 1000 . Person A’s utility takes the Cobb–Douglas form:
U A ( x A , y A ) = x 2A/ 3 y 2A/ 3
(13.38)
and person B’s preferences are given by:
U B ( x B , y B ) = x1B/ 3 y 2B/ 3
(13.39)
Notice that person A has a relative preference for good x and person B has a relative preference for good y. Hence, you
might expect that the Pareto-efficient allocations in this model would have the property that person A would consume
relatively more x and person B would consume relatively more y. To find these allocations explicitly, we need to find a
way of dividing the available goods in such a way that the utility of person A is maximized for any preassigned utility
level for person B. Setting up the Lagrangian expression for this problem, we have:
L( x A , y A ) = U A ( x A , y A ) + ( U B (1000 − x A ,1000 − y A ) − U B )
VIKAAS WADHWA SIR
(13.40)
9811439887 100
Substituting for the explicit utility Sanctions assumed here yields
L( x A , y A ) = x 2A/ 3 y 2A/ 3 + [(1000 − x A )1 / 3 (1000 − y A ) 2 / 3 − U B ]
(13.41)
and the first order conditions for a maximum are
1/ 3
L 2  y A 
=  
x A 3  x A 
L 1  x A 
=  
y A 3  y A 
2/3
  1000 − y A 

− 
3  1000 − x A 
2/3
=0
1/ 3
2  1000 − x A 

− 
3  1000 − 6 y A 
=0
(13.42)
Moving the terms in  to the right and dividing the top equation by the bottom gives
 y  1  1000 − y A 

2 A  = 
 x A  2  1000 − x A 
or
xA
4yA
=
1000 − x A 1000 − y A
(13.43)
This equation allows us to identity all the Pareto optimal allocations in this exchange economy. For example, if we were
to arbitrarily choose xA = xB = 500, Equation 13.43 would become
4yA
=1
1000 − y A
so
y A = 200, y B = 800
This allocation is relatively favorable to person B. At this point on the contract curve
(13.44)
U A = 5002 / 32001 / 3 = 369 ,
U B = 5001 / 38002 / 3 = 683 . Notice that although the available quantity of x is divided evenly (by assumption),
most of goody goes to person B as efficiency requires.
Equilibrium price ratio. To calculate the equilibrium price ratio at this point on the contract curve, we need to know
the two individuals’ marginal rates of substitution. For person A,
MRS =
U A / x A
y
200
=2 A =2
= 0.8
U A / y A
xA
500
(13.45)
U B / x B
y
800
= 0.5 A = 0.5
= 0.8
U B / y B
xA
500
(13A6)
and for person B
MRS =
Hence the marginal rates of substitution are indeed equal (as they should be), and they imply a price ratio of
p x / p y = 0.8 .
Initial endowments. Because this equilibrium price ratio will permit these individuals to trade 8 units of y for each
10 units of x, it is a simple matter to devise initial endowments consistent with this Pareto optimum. Consider, for
example, the endowment
x A = 350 , y A = 320 ; x 0 = 650 , y 6 = 680 . If px = 0.8, py = 1, the value of person
A’s initial endowment is 600. If he or she spends two thirds of this amount on good x, it is possible to purchase 500
units of good x and 200 units of good y. This would increase utility from UA = 3502/33201/3 = 340 to 309. Similarly, the
value, of person B’s endowment is 1,200. If he or she spends one third of this on good x, 500 units cm be bought. With
the remaining two thirds of the value of the endowment being spent on goody 800 units can be bought In the process,
B's utility increases from 670 to 683. Thus, trading from the proposed initial endowment to the contract curve is indeed
mutually beneficial (as shown in Figure 13.8)
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QUERRY : Why did starting with the assumption that good x would be divided equally on the contract curve result in
a situation favoring person B throughout this problem? What point on the contract curve would provide equal utility
to persons A and B? What would the price ratio of the two goods be at this point ?
Social welfare functions
Figure 13.9 shows that there are many Pareto-efficient allocations of the available goods in an exchange economy. We
are assured by the second theorem of welfare economics that any of these can be supported by a Walrasian system of
competitively determined prices, providing that initial endowments are adjusted accordingly. A major question for
welfare economics is how (if at all) to develop criteria for choosing among all these allocations. In this section we look
briefly at one strand of this large topic the study of social welfare functions. Simply put, a social welfare function is a
hypothetical scheme for ranking potential allocations of resources based on the utility they provide to individuals. In
mathematical terms:
Social Welfare = SW[U1(x1), U2(x2), …, Um(xm)
(13.47)
The “social planner’s” goal then is to choose allocations of goods among the m individuals in the economy in a way that
maximizes SW. Of course, this exercise is a purely conceptual one in reality there are no clearly articulated social
welfare functions in any economy, and there are serious doubts about whether such a function could ever arise from
some type of democratic process. Still, assuming the existence of such a function can help to illuminate many of the
thorniest problems in welfare economics. A first observation that might be made about the social welfare function in
Equation
13.47 is that any welfare maximum must also be Pareto efficient. If we assume that every individual’s
utility is to “count,” it seems clear that any allocation that permits further Pareto improvements (that make one person
better off and no one else worse oft) cannot be a welfare maximum. Hence achieving a welfare maximum is a problem
in choosing among Pareto efficient allocations and their related Walrasian price systems.
We can make further progress in examining the idea of social welfare maximization by considering the precise
functional form that SW might take. Specifically, if we assume utility is measurable, using the CES form can be
particularly instructive:
U1R U R2
U Rm
SW ( U1 , U 2 ,..., U m ) =
+
+ ... +
R
R
R
,R1
(13.48)
Because we have used this functional form many times before in this book, its properties should by now be familiar.
Specifically, if R = 1, the function becomes:
SW ( U1 , U 2 ,..., U m ) = U1 + U 2 + ... + U m
(13.49)
Thus, utility is a simple sum of the utility of every person in the economy. Such a social welfare function is sometimes
called a utilitarian function. With such a function, social welfare is judged by the aggregate sum of utility (or perhaps
even income) with no regard for how ut9ity (income) is distributed among the members of society.
At the other extreme, consider the case R = –. In this case, social welfare has a “fixed proportions” character and (as
we have seen in many other applications);
SW(U1, U2, …, Um) = Min [U1, U2, ..., Um).
(13.50)
Therefore; this function focuses on the worse-off person in any allocation and chooses that allocation for which this
person has the highest utility. Such a social welfare function is called a maximum function. It was made popular by the
philosopher John Rawls, who argued that if individuals did not know which position they would- ultimately have in
society (i.e., they operate under a “veil of ignorance”), they would opt for this sort of social - welfare function to guard
against being the worse-off person.’7 . However, Rawls’ focus on the bottom of the utility distribution is probably a good
antidote to thinking about social welfare in purely utilitarian terms. It is possible to explore many other potential
functional forms for a hypothetical welfare function. Problem 13.14 looks at some connections between social welfare
functions and the income distribution, for example. But such illustrations largely miss a crucial point if they focus only
on an exchange economy. Because the quantities of goods in such an economy are fixed, issues related to production
incentives do not arise when evaluating social welfare alternatives. In actuality, however, any attempt to redistribute
income (or utility) through taxes and transfers will necessarily affect production incentives and therefore affect -the
size of the Edgeworth - box. Therefore, assessing social welfare will involve studying the trade-off between achieving
VIKAAS WADHWA SIR
9811439887 102
distributional goals and maintaining 1evels of production. To examine such possibilities we must introduce production
into our general equilibrium framework.
A MATHEMATICAL MODEL OF PRODUCTION AND EXCHANGE
Adding production to the model of exchange developed in the previous section is a relatively simple process First the
notion of a good needs to be expanded to include fax
lots of production. Therefore we will assume that our list of n
goods now includes inputs whose prices also will be determined within the general equilibrium model some Inputs
for one firm in a general equilibrium model are produced by other firms. Some of these goods may also be consumed
by individuals (cars are used by both firms and final consumers), and some of these may be used only as intermediate
goods (steel sheets are used only to make cars and are not bought by consumers). Other inputs may be part of individuals’ initial endowments. Most importantly, this is the way labor supply is treated in general equilibrium models
Individuals are endowed with a certain number of potential labor hours. They may sell these to firms by taking jobs at
competitively determined wages, or they may choose to consume the hours themselves in the form of “leisure,” in
making such choices we continue to assume that individuals maximize utility.
We will assume that there are r firms involved in production. Each of these firms is bound by a production function
that describes the physical constraints on the ways the firm can turn inputs into outputs. By convention, outputs of the
firm take a positive sign, whereas inputs take a negative sign. Using this convention, each firm’s production plan can
be described by an n  1 column vector, yj(j = 1 ... r), which contains both positive and negative entries. The only vectors
that. the firm may consider are those that are feasible given the current rate of technology. Sometimes it is
convenient to assume each firm produces only one output. But that is not necessary for a more general treatment of
production.
Firms are assumed to maximize profits. Production functions are assumed to be sufficiently convex to ensure a unique
profit maximum for any set of output and input prices. This rules out both increasing returns to scale technologies and
constant returns because neither yields a unique maxima. Many general equilibrium modes can handle such possibilities, but there is no need to introduce such complexities here. Given these assumptions, the profits for any firm can
be written as
and
 j (p) = py j
if  j (p)  0
yj = 0
if
 j ( p)  0
(1351)
Hence this model has a “long run” orientation n which firms that lose money (at a particular price configuration) hire
no inputs and produce no output. Notice how the convention that outputs have a positive sign and inputs a negative
sign makes impossible to phrase profits in a compact way
Budget constraints and Walras’ law
In an exchange model, individual? purchasing power is determined by the values of their initial endowments. Once
firms are introduced, we must also consider the income stream that may flow from ownership of these firms. To do so,
m
we adopt the simplifying assumption that each individual owns a predefined share, si (where
s
i =1
i
= 1 ) of the profits
of all firms. That is, each person owns an index fund that can claim a proportionate share of all firms’ profits. We can
now rewrite each individual’s budget constraint (from Equation 13.24) as
r
px i = s i  py j + px i
,
i = 1,2,…,m
(l13.5)
j=1
Of course, if all firms were in long-run equilibrium in perfectly competitive industries, all profits would be zero and the
budget constraint in Equation 13.52 would revert to that in Equation 13.24. But allowing for long-term profits does not
greatly complicate our model; therefore, we might as well consider the possibility. As in the exchange model, the
existence of these m budget constraints implies a constraint of the prices that are possible a generalization of Walras’
law. Summing the budget constraints in Equation 13.52 over all individuals yields:
VIKAAS WADHWA SIR
9811439887 103
m
r
m
p  x ( p) = p  y ( p) + p  x i
i
i =1
and letting
j
j=1
(13.53)
i =1
x ( p) =  x i ( p) . y( p) =  y j ( p) , x =  x i
provides a simple statement of Walras’ law:
px (p) = py(p) + px
(13.54)
Notice again that Walras’ law holds for any set of prices because it is based on individuals’ budget constraint.
Walrasian equilibrium
As before, we define a Walrasian equilibrium price vector (p *) as a set of prices at which demand equals supply in all
markets simultaneously. In mathematical terms this means that:
x ( p* ) = y( p* ) + x
(13.55)
Initial endowments continue to play an important role in this equilibrium. For example, it is individuals’ endowments
of potential labor time that provide the most important input for firms’ production processes. Therefore, determination
of equilibrium wage rates is a major output of general equilibrium models operating under Walrasian conditions;
Examining changes in wage rates that result from changes in exogenous influences is perhaps the most important
practical use of such models.
As m the study of an exchange economy, it is possible to use some form of fixed point theorem to show that there exists
a set of equilibrium prices that statics the n equations in Equation 13.55. Because of the constraint of Walras’ law, such
an equilibrium price vector will be unique only up to a scalar multiple that is, any absolute price level that preserves
relative prices can also achieve equilibrium in all markets. Technically, excess demand functions
z(p) = x(p) – y(p) –
x
(13.56)
are homogeneous of degree 0 in prices; therefore, any price vector for which z(p*) = 0 will also have the-property that
z(tp*) = 0 and t > 0. Frequently it is convenient to normalize prices so that they sum to one. But many other
normalization rules can also be used in macroeconomic versions of general equilibrium models it is usually the case
that the absolute level of prices is determined by monetary factors.
Welfare economics in the Walrasian model with production
Adding production to the- model of an exchange economy greatly expands the number of feasible allocations of
resources One way to visualize this is shown in Figure 1310 There PP represents that production possibility frontier
for a two good economy with a fixed endowment of primary factors of production Any point on this frontier is feasible
Consider one such allocation, say allocation A. If this economy were to produce xA and yA, we could use these amounts
for the dimensions of the Edgeworth exchange box shown inside the frontier Any point within this box would also be
a feasible allocation of the available goods between the two people whose preferences are shown. Clearly a similar
argument could be made for any other point on the production possibility frontier.
Despite these complications the first theorem of welfare economics continues to hold in a general equilibrium model
with production At a Walrasian equilibrium there are no further market opportunities (either by producing something
else or by reallocating the available goods among individuals) that would make some one individual (or group of
individuals) better off without making other individuals worse off Adam Smiths invisible hand continues to exert its
logic to ensure that all such mutually beneficial opportunities are exploited (in part because transaction costs are
assumed to be zero).
VIKAAS WADHWA SIR
9811439887 104
Figure 13.10
Again, the general social welfare implications of the first theorem welfare economics are far from clear. There is, of
course, a second theorem, which shows that practically any Walrasian equilibrium can be supported by suitable
changes in initial endowments. One also could hypothesize a social welfare function to choose among these. But most
such exercises are rather uninformative about actual policy issues.
More interesting is the use of the Walrasian mechanism to judge the hypothetical impact of various tax and transfer
policies that seek to achieve specific social welfare criteria. In this case (as we shall see) the fact that Walrasian models
stress interconnections among markets, especially among product and input markets, can yield important and often
surprising results. In the next section we look at a few of these.
COMPUTABLE GENERAL EQUILIBRIUM MODELS
Two advances have spurred the rapid development of general equilibrium models in recent years. First, the theory of
general equilibrium itself has been expanded to include many features of real-world markets such as imperfect
competition, environmental externalities, and complex tax systems. Models that involve uncertainty and that have a
dynamic structure also have been devised, most importantly in the field of macroeconomics. A second related trend
has been the rapid development of computer power and the associated software for solving general equilibrium
models. This has made it possible to study models with virtually any number of goods and types of households. In this
section we will briefly explore some conceptual aspects of these models. The Extensions to the chapter, found at the
end of the book, describe a few important applications.
Structure of general equilibrium models
Specification of any general equilibrium model begins by defining the number of goods to be included in the model.
These “goods” include not only consumption goods but also intermediate goods that are used in the production of other
goods (e.g., capital equipment), productive inputs such as labor or natural resources, and- goods that are- to be produced by the government (public goods). The goal of the model is then to solve for equilibrium prices for all these goods
and to study how thee prices change when conditions change.
Some of the goods in a general equilibrium model are produced by firms. The technology of this production must be
specified by production functions. The most common such specification is to use the types of CES production functions
that we studied in chapters 9 and 10 because these can yield some important insights about the ways in which inputs
are substituted in the face of changing prices. In general, firms are assumed to maximize their profits given their
production functions and given the input and output prices they face.
VIKAAS WADHWA SIR
9811439887 105
Demand is specified in general equilibrium models by defining utility functions for various types of households. Utility
is treated as a function both of goods that are consumed and of inputs that are not supplied to the marketplace (e.g.,
available labor that is not supplied to the market is consumed as leisure). Households are assumed to maximize utility.
Their incomes are determined by the amounts of input they “sell” in the market and by the net result of any taxes they
pay or transfers they receive.
Finally, a full general equilibrium model must specify how the government operates.
If there are taxes in the model, how those taxes are to be spent on transfers or on public goods (which provide utility
to consumers) must be modeled If government borrowing is allowed, the bond market must be explicitly modeled. In
short, the model must fully specify the flow of both source and uses of income that characterize the economy being
modeled.
Solving general equilibrium models
Once technology (supply) and preferences (demand) have been specified, a general equilibrium model must be solved
for equilibrium prices and quantities. The proof earlier in this chapter shows that such a model will generally have such
a solution, but actually finding that solution can sometimes be difficult especially when the number of goods and
households is large. General equilibrium models are usually solved on computer via modifications of an algorithm
originally developed by Herbert Scarf in the 1970s. This algorithm (or more modern versions of it) searches for market
equilibria by mimicking the way markets work. That is, an initial solution is specified and then prices are raised in
markets with excess demand and lowered in markets with excess supply until an equilibrium is found in which all
excess demands are zero. Sometimes multiple equilibria will occur, but usually economic models have sufficient
curvature in the underlying production and utility functions that the equilibrium found by -the Scarf algorithm will be
unique.
Economic insights from general equilibrium models
General equilibrium models provide a number of insights about how economies operate that cannot be obtained from
the types of partial equilibrium models studied in Chapter 12. Some of the most important of these are:
•
All prices are endogenous in economic models. The exogenous elements of models are preferences and
productive technologies.
•
All firms and productive inputs are owned by households. All income ultimately accrues to households.
•
Any model with a government sector is incomplete if it does not specify how tax receipts are used.
•
The “bottom line” in any policy evaluation is the utility of households. Firms and governments are only
intermediaries in getting to this final accounting
•
All taxes distort economic decisions along some dimension. The welfare casts of such distortions must always
be weighed against the benefits of such taxes (in terms of public good production or equity-enhancing
transfers).-
Some of these insights are illustrated in the next two examples. In later chapters we will return to general equilibrium
modeling whenever such a perspective seems necessary to gain a more complete understanding of the topic being
covered.
Example 13.4 A simple general Equilibrium model
Let's look at a simple general equilibrium model with only two households, two consumer goods (x and y) and two
inputs (capital k and labor 1). Each household has an “endowment” of capital and labor that it can choose to retain or
sell in the market. These endowments are denoted by
k1 , l1
and
k 2 , l2
respectively. Households obtain utility from
the amounts of the consumer goods they purchase and from the amount of labor they do not sell into the market (i.e.,
leisure =
li − l j ). The households have simple Cobb–Douglas utility functions:
U1 = x10.5 y10.5 (l1 − l1 ) 0.2 ,
U 2 = x 02.4 y 02.4 (l2 − l2 ) 0.2
VIKAAS WADHWA SIR
(13.57)
9811439887 106
Hence, household 1 has a relatively greater preference for good x than does household 2. Notice that capital does not
enter into these utility functions directly. Consequently, each household will provide its entire endowment of capital
to the marketplace. Households will retain some labor, however, because leisare provides utility directly.
Production of goods x and y is characterized by simple Cobb–Douglas technologies.
x = k 0x.2lx0.8 , y = k 0y.8ly0.8
(13.58)
Thus, in this example, production of x is relatively labor intensive, whereas production of y is capital intensive.
To complete this model we must specify initial endowments of capital and labor. Here we assume that
k1 = 40, l1 = 24 and k 2 = 10, l2 = 24
(13.59)
Although the households have equal labor endowments (i.e., 24 “hour “), household I has significantly more capital
than does household 2.
Base–case simulation. Equations 13.57–13.59 specify our complete general equilibrium model in the absence of a
government, A solution to this model will consist of four equilibrium price (for x, y, k and l) at which households
maximize utility and firms maximize profits.
Because any general equilibrium model can compute only relative prices, we are free to impose a price normalization
scheme. Here we assume that the prices will always sum to unity That is,
p x + p y + p k + pl = 1
(13.60)
Solving for these prices yields
px = 0.363, py = 0.253, pk = 0.136, pl = 0.248
(13.61)
At these prices, total production of x is 23.7 and production of y is 25.3. The utility-maximizing
for household 2, these choices
x1 = 15.7, y1 = 8.1,
l1 − l1 = 24 − 14.8 = 9.2 U1 = 13.5
(13.62)
l2 − l2 = 24 − 18.1 = 5.9 , U 2 = 8.75
(13.63)
for household 2, these choices are
x2 = 8.1, y2 = 11.6,
Observe that household 1 consumes quite a bit of good x but provides less in labor supply than does household 2. This
reflects the greater capital endowment of household I in this base-case simulation. We will return to this base case in
several later simulations.
EXAMPLE 13.5 : The Excess Burden of a Tax
In chapter 12 we showed that taxation may impose an excess burden in addition to the tax revenues collected because
of the incentive effects of the tax. With a general equilibrium model we w much more about this effect. Specifically
assume that the government in the economy of Example 13.4 imposes an ad valorem lax of 0.4 on good x. This
introduces a wedge between what demanders pay for this good x (p x) and what suppliers receive for the good (p'x = (1
– t)px = 0.6px). To complete the model we must specify what happens to the revenues generated by this tax. For
simplicity, we assume that these revenues are rebated to the households in a 50–50 split. In all other respects the
economy remains as described it Example 13.4.
Solving for the new equilibrium prices in this model yields
p x = 0.472, p y = 0.218, p k = 0.121, pl = 0.188
(13.64)
At these prices, total production of x is 17.9, and total production of y is 28.8. Hence the allocation of resources has
shifted significantly toward y production. Even though the relative price of x experienced by consumers (= px/py =
0.472/0.218 = 2.17) has increased significantly from its value (of 1.43) in Example 13.4, the price ratio experienced by
firms (0.6px/py = 1.30) has decreased somewhat from this prior value. Therefore, one might expect, based on a partial
equilibrium analysis, that consumers would demand less of good x and likewise that firm would similarly produce less
of that good. Partial equilibrium analysis would not, however, allows us to predict the increased production of y (which
comes about because the relative price of y has decreased consumers but has increased for firms) nor the reduction in
VIKAAS WADHWA SIR
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relative in price because there is less bgeing produced overall). A more complete picture of all these effects can be
obtained by looking at the final equilibrium positions of the two households. The post tax allocation for household 1 is
x1 = 11.6, y1 = 15.2, l1 − l1 = 11.8, U1 = 12.7
(13.65)
or household 2
x 2 = 6.3, y 2 = 13.2, l2 − l2 = 7.9, U 2 = 8.96
(13.65)
Hence imposition of the tax has made household 1 considerably worse off: utility decreases from 13.5 to 12.7
Household 2 is made slightly better off by this tax and transfer scheme, primarily because it receive a relatively large
share of the tax proceeds that come mainly from household. Although total utility has described (as predicted by the
simple partial equilibrium analysis of excess burden, general equilibrium analysis gives a more complete picture of the
distributional consequences of the tax. Notice also that the total amount of labor supplied decreases as a result of the
tax total leisure increases from 15.1 (hours) to 19.7. Therefore imposition of a tax on good x has a relatively substantial
labor supply effect that is completely invisible in a partial equilibrium model.
VIKAAS WADHWA SIR
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MONOPOLY (NOTES)
Q1.
What do you mean by monopoly? What are the features of monopoly?
The word monopoly comes from two Greek words- “monos” and “polein”. The meaning of mono is single (or atones) and
polein means Selling. Monopoly therefore prevails in the market when there is a single seller. This is the common meaning
of monopoly. In economic theory, monopoly is defined, as a single firm producing commodities which there are no close
substitutes. Following are some characteristics of a Monopoly”.
(a)
In monopoly, the output of the entire industry is controlled by a single seller. The single seller could be organized
as a sole proprietor, or a partnership firm or a joint Stock company. A combination of producers or even the State
could act as the single seller.
(b)
In monopoly, there is no distinction between the firm and the industry – since the firm constitutes the whole
industry.
(c)
The monopoly firm produces a commodity for which there are no closely competing substitutes thus, the
monopoly firms face any direct competition.
(d)
A monopoly will survive in the long run only if entry of new firms does not occur. Impediments that prevent entry
of new firms are caused by Barriers to entry – such barriers to entry may be natural or artificial/created (e.g.,
patent Laws).
(e)
Another distinct feature of monopoly is that it enjoys freedom and independence in fixing the price of the
commodity or the output. It can fix either the price or output, not both.
Example: A very common example of Monopoly is that of Indian Railways.
Another very interesting example of monopoly is The Mysore based Mysore Paints and Varnishes Limited (MPVL) is the sole
authorized supplier indelible ink to the election commission.
Q2.
Explain the main types of Barriers to entry.
The reason a monopoly exists is that other firms find it unprofitable or impossible to enter the market. Barriers to entry are
therefore the source of all monopoly power. If other firm could enter a market then the firm would, by definition, no longer
be a monopoly. There are two general types of barriers to entry: technical barriers and legal barriers.
Technical barriers to entry
A primary technical barrier is that the production of the good in question may exhibit decreasing marginal (and average)
costs over a wide range of output levels: The technology of production is such that relatively large-scale firms are low-cost
producers. In this situation (which is sometimes referred to as natural monopoly), one firm may find it profitable to drive
others out of the industry by cutting prices. Similarly, once a monopoly has been established, entry will be difficult because
any new firm must produce at relatively low levels of output and therefore at relatively high average costs. It is important to
stress that the range of declining costs need only be "large" relative to the market in question. Declining costs on some
absolute scale are not necessary.
Another technical basis of monopoly is special knowledge of a low-cost productive technique. But the problem for the
monopoly that fears entry is keeping this technique uniquely to itself. When matters of technology are involved, this may be
extremely difficult, unless the technology can be protected by a patent. Ownership of unique resources – such as mineral
deposits or land locations, or the possession of unique managerial talents – may also be a lasting basis for maintaining a
monopoly.
Legal barriers to entry
Many pure monopolies are created as a matter of law rather than as a matter of economic conditions. One important
example of a government granted monopoly position is in the legal protection of a product by a patent or copyright.
Prescription drugs, computer, chips and Disney animated movies are examples of profitable products that are shielded (for
a time from direct competition by potential imitators. Because the basic technology for these products is uniquely assigned
to one firm, a monopoly position is established. The defence made of such a governmentally granted monopoly is that the
patent and copyright system makes innovation more profitable and therefore acts as an incentive. Whether the benefits of
such innovative behaviour exceed the costs of having monopolies is an open question that has been much debated by
economists.
VIKAAS WADHWA SIR
9811439887 109
A second example of a legally created monopoly is the awarding of an exclusive franchise to serve a market. These franchises
are awarded in cases of public utility (gas and electric) service, communications services, the post office, some television and
radio station markets, and a variety of other situations. The argument usually put forward in favour of creating these
franchised monopolies is that the industry in question is a natural monopoly: Average cost is diminishing over a broad range
of output levels, and minimum average cost can be achieved only by organizing the industry as a monopoly. The public utility
and communications industries often considered good examples, Certainly, that does appear to be the case for local
electricity and telephone service where a given network robab1y exhibits declining average cost up to the point of universal
coverage. But recent deregulation in telephone services and electricity generation show that, even for these industries, the
natural monopoly rationale may not be all-inclusive. In other cases, franchises may be based largely on political rationales.
This seems to be true for the postal service in the United States and for a number of nationalized industries (airlines, radio
and television, banking) in other countries.
Creation of barriers to entry
Although some barriers to entry may be independent of the monopolist’s own activities, other barriers may result directly
from those activities. For example, firms may develop unique products or technologies resources to prevent potential entry.
The Dc Beers cartel, for example, controls a large fraction of the world's diamond mines. Finally, A would be monopolist may
enlist government aid in devising barriers to entry. It may lobby for legislation that restricts new entrants so as to "maintain
an orderly market" or for health and safety regulations that raise potential entrants’ costs. Because the monopolist has both
special knowledge of its business and significant incentives to pursue these goals, it may have considerable success in
creating such barriers to entry.
The attempt by a monopolist to erect barriers to entry may involve real resource costs. Maintaining secrecy, buying unique
resources, an entangling in political lobbying are all costly activities. A full analysis of monopoly should involve not only
questions of cost minimization and output choice (as under perfect competition) but also an analysis of profit-maximizing
creation of entry barriers. However, we will not provide a detailed investigation of such questions here.1 Instead, we will
generally assume that the mono list can do nothing to affect barriers to entry and that he firm's costs are therefore similar
to what a competitive firm’s costs would be.
Q3.
Explain the nature of Average Revenue and Marginal Revenue UNDER MONOPOLY
Average Revenue and Marginal Revenue
The monopolist’s average revenue – the price it receives per unit sold – is precisely the market demand curve. To choose its
profit-maximizing output level, the monopolist also needs to know its marginal revenue: the change in total revenue that
results from a unit change in output. To see the relationship among total, average, and marginal revenue, consider a firm
facing the following demand curve:
P=6–Q
Table shows the behaviour of total, average, and marginal revenue for this demand curve. Note that total revenue is zero
when the price is Rs.6. At that price, nothing is sold. At a price of Rs.5, however one unit is sold, so total (and marginal
revenue is Rs.5.An increase in quantity sold from 1 to 2 increases revenue from Rs.5 to Rs.8; marginal revenue is thus Rs.3.
As quantity sold increases from 2 to 3, marginal revenue falls to Rs.1, and when quantity increases from 3 to 4 marginal
revenue becomes negative when marginal revenue is positive, revenue is increasing with quantity, but when marginal
revenue is negative, revenue is decreasing
Table Total, Marginal and Average Revenue
Price (P)
Quantity (Q)
Total Revenue (R)
Marginal Revenue
(MR)
Average Revenue
(AR)
Rs.6
0
0
–
–
5
1
5
5
5
4
2
8
3
4
3
3
9
1
3
2
4
8
–1
2
1
5
5
–3
1
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When the demand curve is downward sloping, the price (average revenue) is greater than marginal revenue because all units
are not sold at the same price. If sales are to increase by 1 unit, the price must fall. In that case, all units sold, not just the
additional unit, will earn less revenue. Note, for example, what happens in Table is increased from 1 to 2 units and price is
reduced to Rs.4. Marginal revenue is, Rs.3; Rs.4 (the revenue from the sale of the additional unit of output) less Rs.1 (the
loss of revenue from selling the first unit for Rs.4 instead of Rs.5).Thus, marginal revenue (Rs.3) is less than price (Rs.4).
Figure plots average and marginal revenue for the data in Table. Our demand curve is straight line and, in this case, the
marginal revenue curve has twice the slope of the demand curve (and the same intercept).
Q4.
Profit maximization and output choice under monopoly
Profit Maximization and Output Choice
To maximize profits, a monopolist will choose to produce that output level for which marginal revenue is equal to marginal
cost. Because the monopoly, in contrast to a perfectly competitive firm, faces a negatively sloped market demand curve,
marginal revenue will be less than the market price. To sell an additional unit, the monopoly must lower its price on all units
to be sold if it is to generate the extra demand necessary to absorb this marginal unit. The profit- maximizing output level
for a firm is then the level Q* in Figure. At that level, marginal revenue is equal to marginal costs, and profits are
maximized.Given the monopoly’s decision to produce Q*, the demand curve D indicates that a market price of P* will prevail.
This is the price that demanders as a group are willing to pay for the output of the monopoly. In the market, an equilibrium
price-quantity combination of P*, Q* will be observed. Assuming P* > AC, this output level will be profitable, and the
monopolist will have no incentive to alter output levels unless demand or cost conditions change. Hence we have the
following principle.A profit-maximizing monopolist produces that quantity for which marginal revenue is equal to marginal
cost. In the diagram this quantity is given by Q*, which will yield a price of P* in the market. Monopoly profits can be read as
the rectangle of P*EA C.
Total profits earned by the monopolist can be read directly from Figure. These are shown by the rectangle P *EAG and again
represent the profit per unit (price minus average cost) times the number of units sold. These profits will be positive if market
price exceeds average total cost. If P* <AC, however, then the monopolist can operate only at a long-term loss and will decline
to serve the market.
Because (by assumption) no entry is possible into a monopoly market, the monopolist’s positive profits can exist even in the
long run. For this reason, some authors refer to the profits that a monopoly earns in the long run as monopoly rents. These
VIKAAS WADHWA SIR
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profits can be regarded as a return to that factor that forms the basis of the monopoly (a patent, a favourable location, or a
dynamic entrepreneur, for example); hence another possible owner might be willing to pay that amount in rent for the right
to the monopoly. The potential for profits is the reason why some firms pay other firms for the right to use a patent and why
concessioners at sporting events (and on some highways) are willing to pay for the right to the concession. To the extent
that monopoly rights are given away at less than their true market value (as in radio and television licensing), the wealth of
the recipients of those rights is increased.Although a monopoly may earn positive profits in the long run the size of such
profits will depend on the relationship between the monopolist’s average costs and the demand for its product.
Q5.
Monopoly has no supply curve. Explain this statement.
Absence of Supply Curve: Monopoly
A monopoly does not have a unique supply curve. As stated under the Law of supply, a supply curve tells us the quantity of
a product that a firm is willing and able to produce and sell at each possible price. In case of perfect competition, there was
a unique relationship between price and quantity supplied. The perfectly competitive firm equated MR to MC
(MR = MC) to maximise profits and produced an output level where price was equal to marginal cost (P = MC).
Figure: No unique price-quantity supply relationship: Some quantity (OX) supplied at two prices, (OP1 and OP2) given
different demand conditions (D1 and D2).
This is not the case with a monopoly firm. There is no unique relationship between price and quantity produced and sold.
The monopoly firm equates marginal cost with marginal revenue to maximise profits, but for the monopoly firm marginal
revenue
is
not
equal
to
price
(MR  P). Therefore, the monopoly does not equate marginal cost to price (MC  P). It is therefore possible for different
demand curves, the firm to produce sell the same output but at different prices. It is equally possible for different demand
curves the firm to produce and supply different quantities as the same price. This is explained with the help of figure.
Case One: Monopoly firm produces and supplies the same output at two different prices, under different demand conditions
Same quantity (OX) is supplied at two prices, (OP1 and OP2) given different demand conditions (D1 and D2).
Suppose the monopoly firm faces initially market demand shown by D 1 (in figure.). MR1 is the corresponding marginal
revenue curie. Given cost conditions, the firm reaches equilibrium at E, where MR = SMC (SMC = short run marginal Cost).
The firm produces an output of OX at price OP1. The market demand curve shifts to D2. The corresponding marginal revenue
curve is MR2. Given the cost conditions MR2 interests the SMC also at E, thus producing the same level of output of OX units.
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However this output is sold at different price OP2. Thus, under different demand conditions, a monopoly firm can produce
and supply the same output at two different prices. Therefore, there exists no unique price-quantity supplied relationship.
Case 2. The monopoly firm produces and different quantities at the same price, given demand conditions.
Figure: No unique price-quantity supply’ relationship: Different quantities (OX1 and OX2) supplied at same price (OP) given
different demand conditions (D1 and D2).
It is also possible ‘for a monopoly firm to produce and supply different quantities at the same price, given different demand
conditions. This is explained with the help of figure. D 1 gives the initial demand curve. The corresponding marginal revenue
curve is MR1. Given the cost conditions, the monopoly firm reaches equilibrium at E1, where MR1 = SMC. The firm produces
an output of OX1 and sells it at a price of OP. Now, the demand curve shifts to D2. The corresponding marginal curve is MR2.
The MR2 intersects the SMC at E2, the firm producing an output of OX2 at the same price OP. Thus, given different demand,
conditions, the monopoly firm produces and sells difference outputs at the same price.
Q6.
Differentiate between Perfect Competition and monopoly.
Differences between Perfect Competition and Monopoly
(1) Number of sellers:
In perfect competition, there are a large number of sellers. The numbers of setters are, so large that no single seller is in a
position to influence the market price of the commodity. Monopoly, by definition, consists of a single setter. The total supply
of the commodity is within the control of the single monopoly firm
(2) Nature of the Product:
The product offered by all the firms in the industry under perfect competition, is homogeneous. The product offered by a
monopoly firm is the one which has no close substitute in the market
(3) Entry and Exit conditions:
Although both models assumed that entry and exit of firms is possible in the Long run, entry is blocked under monopoly. The
barriers to entry are so strong that no new firm tries to enter and compete with the monopoly firm. A monopoly firm, if
faced with tosses in. the tong run can go out of business (in the long run).
(4) Demand Curve
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Given the conditions of a large number of buyers and sellers and homogenous product, the demand curve facing a perfectly
competitive firm is perfectly competitive elastic as shown in figure. The firm can sell any amount of the product at the
prevailing market price.
The firm under perfect competition is therefore said to be a “price taker”. The demand curve for the monopoly firm is
downward sloping as shown in figure. This is because the monopoly firm itself constitutes the industry and the demand cure
for an industry is usually downward sloping.
(5) Decision Variables:
The firm under perfect competition has to determine its profit maximising output only. The price at which the film would
sell this output is determined by the forces of market demand and market supply. Given homogeneity of the product and
perfect knowledge about the market conditions among buyers and sellers leaves no scope for promotional activities
(advertising, sales promotion personal selling and publicity) under perfect competition.
Under monopoly with the monopoly firm facing a downward sloping demand curve, the firm can fix either the output or the
price but not both. It faces a trade off between output and price. Once the monopoly firm determines the price, the quantity
that it can sell at that price follows from the demand curve. In this case, the monopoly firm is called a price-maker. If on the
other hand the monopoly firm decides how much to produce and sell, the price per unit, which the monopolist receives,
follows from the demand curve.
Further the monopoly firm can undertake product modification and promotional activities, if is so desires or under
circumstances of potential entry of new firms or government regulations. Thus, the monopoly firm has more decision
variables to be decided and implemented.
Q7.
Explain the social costs of Monopoly Power
OR
Why dead weight loss occur in a Monopoly Power.
The Social Costs of Monopoly Power
In a competitive market, price equals marginal cost. Monopoly power, on the other hand, implies that price exceeds marginal
cost. Because monopoly power results in higher prices and lower quantities produced, we would expect it to makeconsumers worse off-and the firm better off. But suppose we value the welfare of consumers the same as that of producers.
In the aggregate, does monopoly power make consumers and producers better or worse off?
We can answer, this question by comparing the consumer and producer surplus that results when a competitive industry
produces a good with the surplus that results when a monopolist supplies the entire market. (We assume that the
competitive market and the monopolist have the same cost curves.) Figure shows the average and marginal revenue curves
and marginal cost curve for the monopolist. To maximize profit, the firm produces at the point where marginal revenue
equals marginal cost, so that the price and quantity are Pm and Qm. In a competitive market, price must equal marginal cost,
so the competitive price and quantity, PC and QC are found at the intersection of the average revenue (demand) curve and
the marginal cost curve. Now let’s examine how surplus changes if we move from the competitive price and quantity, PC and
QC to the monopoly price and quantity, Pm and Qm.
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Under monopoly, the price is higher and consumers buy less. Because of the higher prices those consumers who buy the
good lose surplus of an amount given by rectangle A. Those consumers who do not buy the good at price P m but who would
buy at price PC also lose surplus – namely an amount given by triangle B. The total loss of consumer surplus is therefore A +
B. The producer1 however, gains rectangle A by selling at the higher price but loses triangle C, the additional profit it would
have earned by selling QCQm at price PC. The total gain in producer surplus is therefore A – C. Subtracting the loss of consumer
surplus from the gain in producer surplus, we see a net loss of surplus give by B + C. This is the deadweight loss from
monopoly power even if the monopolist’s profits were taxed away and redistributed to the consumers of its products there
would be an inefficiency because output would be lower than under conditions of competition. The deadweight toss is the
social cost of this inefficiency.
Rent Seeking
In practice the social cost of monopoly power is likely to exceed the deadweight loss in triangles B and C of Figure. The reason
is that the firm may engage in rent seeking: spending large amounts of money in socially unproductive efforts to acquire,
maintain, or exercise its monopoly power. Rent seeking might involve lobbying activities (and perhaps campaign
contributions) to obtain government regulations that make entry by potential competitors more difficult. Rent-seeking
activity could also involve advertising and legal efforts to avoid antitrust scrutiny. It might also mean installing but not utilizing
extra production capacity to convince potential competitors that they cannot sell enough to make entry worthwhile. We
would expect the economic incentive to incur rent-seeking costs to bear a direct relation to the gains from monopoly power
(i.e., rectangle A minus triangle C.) Therefore, the larger the transfer from consumers to the firm (rectangle A), the larger the
social cost of monopoly.9
Here’s an example. In 1996, the Archer Daniels Midland Company (ADM) successfully lobbied the Clinton administration for
regulations requiring that the ethanol (ethyl alcohol) used in motor vehicle fuel be produced from corn. (The government
had already planned to add ethanol to gasoline in order to reduce the country’s dependence on imported oil.) Ethanol is
chemically the same whether it is produced from corn, potatoes, grain, or anything else. Then why require that it be produced
only from corn? Because ADM had a near monopoly on corn-based ethanol production so the regulation would increase its
gains from monopoly power.
Q8. Thumb Rule for Pricing by a Monopoly Firm
So far we have stated that a monopoly firm would set the price having defined the equilibrium point, where MC =MR.
However in the real world it is extremely difficult for a firm to determine its demand position (AR curve) and therefore it
would be difficult to correctly determine its MR. Further the firm might just know its MC over a limited range of output. With
MR and MC being unknown, (or mere estimates), the firm would not be in a position to apply the marginalist principle of MC
= MR.
Therefore the marginalist principle has been TRANSLATED into a rule of thumb that can be applied in practice by the firm.
According to Rubinfeld and .Pindyck, MR is the first derivative of the Total revenue function.
Therefore,
MR =
TR (PQ)
=
Q
Q
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MR = P + Q
P
Q
The second term of the RHS of above equation is multiplied and divided by P. this is shown below:
MR = P + Q
P
 Q  P 
= P + P 

Q
 P  Q 
We know that the elasticity of demand is defined as
Ed =
GDQ / P
P/Q
Therefore 1/Ed is equal to part of the second term in the above equation.
Thus,
MR = P + P.
1
Ed
When the firm wants to maximise profits, it equates MR= MC, therefore it can be stated that
MR = P + P.
1
= MC
Ed
.
Simplifying the above equation we get
P − MC = −P
1
Ed
P − MC 1
=
P
Ed
The left-hand side, (P – MC)/P, is the markup over marginal cost as a percentage of price. The relationship says that this
markup should equal minus the inverse of the elasticity of demand. (This figure will be a positive number because the
elasticity of demand is negative). The above equation is further rearranged to obtain price directly as a markup over marginal
cost:
P=
MC
1 + (1 / Ed)
If the elasticity of demand is 5 and marginal cost
Rs.10 (1 – 1/5) = Rs.10/0.80 = Rs.12.5 per unit. We have thus defined
P=
is
Rs.10
per
unit,
price
should
be
MC
1 + (1 / Ed)
The relationship provides a rule of thumb for any firm with monopoly power.
The student needs to remember that, in the above formula, Ed is the price elasticity of demand for the firm, and not the price
elasticity of market demand.If a firm has an estimate of it’s elasticity of demand, the firm can calculate the proper markup.If
the firm’s elasticity of demand is Large, this markup will be small (and we can say that the firm has very little monopoly
power), This is shown with the help of the following figure:
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If the firms’ elasticity of demand is small, this markup will be large (and the firm will have considerable monopoly power).
This is shown with the help of the following figure:
Q9 A monopolist can produce at constant average and marginal costs of AC = MC = 5. The firm faces a market demand curve
given by Q = 53 – P.
Q10
a.
Calculate the profit-maximizing priccquantit3r combination for the monopolist. Also calculate the
monopolist’s profits.
b.
What output level would be produced by this industry under perfect competition (where price = marginal
cost)?
c.
Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of
the monopolist’s profits and the consumer surplus received in case (a). What is the value of the
“deadweight loss” from monopolization?
A monopolist faces a market demand curve given by
Q = 70 – p.
a.
If the monopolist can produce at constant average and marginal costs of
AC = MC = 6, what output level will the monopolist choose in order to maximize profits? V/hat is the
price at this output level?
What are the monopolist’s profits?
b.
Assume instead that the monopolist has a cost structure where total costs are described by
C(Q) = 0.25Q2 – 5Q + 300.
With the monopolist facing the same market demand and marginal revenue, what price-quantity
combination will be chosen now to maximize profits? What will profits be?
c.
Assume now that a third cost structure explains the monopolist’s position, with total costs given by
C(Q) = 0.0133Q3 – 5Q + 250.
Again, calculate the monopolist’s price-quantity combination that maximizes profits. What will profit be?
Hint: Set MC = MR as usual and use the quadratic formula to solve the second-Order equation for Q.
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d.
Q11
Graph the market demand curve, the MR curve, and the three marginal cost curves from parts (a), (b),
and.(c). Notice that the monopolist’s profit-making ability is constrained by (1) the market demand curve
(along with its associated MR curve) and (2) the cost structure underlying production.
A single firm monopolizes the entire market for widgets and can produce at constant average and marginal cost of
AC = MC = 10
Originally, the firm faces a market demand curve given by
Q = 60 – P
a.
Calculate the profit-maximizing price-quantity combination
for the finn. What are the firm’s profits?
b.
Now assume that the market demand curve shifts outward
(becoming steeper) and is given by
Q = 45 – 0.5P.
What is the firm’s profit-maximizing price-quantity
combination now? What are the firm’s profits?
c.
Instead of the assumptions of part (b), assume that the
market demand curve shifts outward (becoming flatter) and is given by
Q = 100 – 2P.
What is the firm’s profit-maximizing price-quantity combination now? What are the firm’s profits?
d.
Q12
Graph the three different situations of parts (a), (b), and (c). Using your results, explain why there is no
real supply curve for a monopoly.
Suppose the market for Hula Hoops is monopolized by a single firm.
a.
Draw the initial equilibrium for such a market.
b.
Now suppose the demand for Hula Hoops shifts outward slightly. Show that, in general (contrary to the
competitive case), it will not be possible to predict the effect of this shift in demand on the market price
of Hula Hoops.
c.
Consider three possible ways in which the price elasticity of demand might change as the demand curve
shifts: it might increase, it might decrease, or it might stay the same. Consider also that marginal costs
for the monopolist might be rising, falling, or constant in the range where MR = MC.
Consequently, there are nine different combinations of types. of demand shifts and marginal cost slope
configurations. Analyze each of these to determine for which it is possible to make a definite prediction
about the effect of the shift in demand on the price of Hula Hoops.
Q13
Suppose a monopoly market has a demand function in which quantity demanded depends not only on market
price (P but also on the amount of advertising the firm does A, measured in dollars). The specific form of this
function is
Q = (20 – P)(I – 0.1A – 0.01A2)
The monopolistic firm’s cost function is given by
C = 10Q + 15 + A
Q14
a.
Suppose there is no advertising (A = 0). What output will the profit-maximizing firm choose? What market
price will this yield? What will be the monopoly’s profits?
b.
Now let the firm also choose its optimal level of advertising expenditure. In this situation, what output
level will be chosen? What price will this yield? What will the level of advertising be? What are the firm’s
profits in this case? Hint: This can be worked out most easily by assuming the monopoly chooses the
profit-maximizing price rather than quantity.
Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of $5 per
unit. Assume the monopoly sells its goods in two different markets separated by some distance:
The demand curve in the first market is given by
Q1 = 55 – P1
and the demand curve in the second market is given by
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Q2 = 70 – 2P2.
Q15
a.
If the monopolist can maintain the separation between the two markets, what level of output should be
produced in each market, and what price will prevail in each market? What are total profits in this
situation?
b.
How would your answer change flit costs demanders only $5 to transport goods between the two
markets? What would be the monopolist’s new profit level in this situation?
c.
How would your answer change if transportation costs were zero and then the firm was forced to follow
a single-price policy?
d.
Suppose the firm could adopt a linear two-part tariff under which marginal prices must be equal in the
two markets but lump-sum entry fees might vary What pricing policy should the firm follow?
Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of $10 per unit.
Monopolized marginal costs rise to $12 per unit because $2 per unit must be paidto lobbyists to retain the widget
producers’ favored position. Suppose the market demand for widgets is given by
QD = 1,000 – 50P.
Q16
a.
Calculate the perfectly competitive and monopoly outputs and prices.
b.
Calculate the total loss of consumer surplus from monopolization of widget production.
c.
Graph your results and explain how they differ from the usual analysis.
Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of
a subsidy.
a.
Why would a lump-sum subsidy not achieve the government’s goal?
b.
Use a graphical proof to show how a per-unit-of-output subsidy might achieve. The government’s goal.
c.
Suppose the government wants its subsidy to maximize1he-difference between the total value of the
good to consumers and the good’s total cost. Show that, in order to achieve this goal, the government
should set where (is the per-unit subsidy and P is the competitive price. Explain your result intuitively.
t
1
=−
P
e Q ,P
Q17
,
Suppose a monopolist produces alkaline batteries that may have various useful lifetimes (A). Suppose also that
consumers’ (inverse) demand depends on batteries’ lifetimes and quantity (Q) purchased according to the function
P(Q,X) = g(X.Q),
where g' < 0. That is, consumers care only about the pro4uctof quantity times lifetime: They are willing to pay
equally for many short-lived batteries or few long-lived ones. Assume also that battery costs are given by
C(Q,) = C(X)Q
where. C'(X) > d.
Show that, in this case, the monopoly will opt for the same level of X a does a competitive industry even though
level of output and prices-may differ. Explain your result. Hint: Treat XQ as a composite commodity.
SOLUTIONS
9
a.
PQ = 53Q – Q2
P = 53 – Q
MR = 53 – 2Q = MC = 5
Q = 24, P = 29,  = (P – AC).Q = 576
b.
MC = P = 5
c.
Competitive Consumers’ Surplus = 2(48)2 = 1152.
VIKAAS WADHWA SIR
P = 5, Q = 48
9811439887 119
Notice that the sum of consumer surplus, profits, and deadweight loss under monopoly equals
competitive consumer surplus.
10
Market demand Q = 70 – P, MR = 70 – 2Q
a.
AC = MC = 6.
To maximize profits set MC = MR.
b.
6 = 70 – 2Q
Q = 32P = 38
 = (P – AC)
Q = (38 – 6)
C=
.25Q2
32 = 1024
– 59 + 300, MC = .5Q – 5.
Set MC = MR
.5Q – 5 = 70 – 20
c.
Q = 30
P = 40
C = .0133Q3 + 5Q + 250.MC = .04Q2 – 5
MC = MR
Therefore: 04Q2 – 5 = 70 – 2Q
or .04Q2 + 2Q – 75 = 0.
Quadratic formula gives Q = 25.
If Q = 25, P = 45
R = 1125
C = 332.8 (MC = 20)  = 792.2
11
a.
AC = MC = 10,
Q = 60 – P,
MR = 60 – 2Q.
For profit maximum, MC = MR
10 = 60 – 2Q
Q = 25 P = 35
 = TR – TC = (25)(35) – (25)(10) = 625.
b.
AC = MC = 10,
Q = 45 – SP,
MR = 90 – 4Q.
For profit maxinium,
MC = MR 10 = 90 – 4Q
Q = 20
P = 50
 = (20)(50) – (20)(10) = 800.
c.
AC = MC = 10,
Q = 100 – 2P,
MR = 50 – Q.
For profit maximum, MC = MR
10 = 50 – Q
Q = 40
P = 30.
 = (40)(30) – (40)(10) = 800.
Note: Here the inverse elasticity rule is clearly illustrated:
Q P
.
P Q
Problem Part
e=
a.
–1(35/25) = –1.4
.71 = (35 – 10)/35
b.
–5(50/20) =–1.25 .80 = (50 – 10)/50
c.
–2(30/40) = –1.5
VIKAAS WADHWA SIR
− 1 P − MC
=
e Q ,P
P
.67 = (30 – 10)/30
9811439887 120
The supply curve for a monopoly is a single point, namely, that quantity price combination which corresponds to
the quantity for which MC = MR.
Any attempt to connect equilibrium points (price-quantity points) on the market demand curves has - little
meaning and brings about a strange shape. One reason for this is that as the demand curve shifts, its elasticity (and
its MR curve) usually changes bringing about widely varying price and quantity changes.
12
a.
b.
There is no supply curve for monopoly; have to examine MR = MC intersection because any shift in
demand is accompanied by a shift in MR curve. Case (1) and case (2) above show that P may rise or fall
in response to an increase in demand.-
c.
Can use inverse elasticity rule to examine this
−e =
P
P
=
P − MC P − MR
As – e falls toward I (becomes less elastic), P–MR increases.
Case-1. MC constant so profit-maximizing MR is constant
If –e ,
P – MR P.
If –e constant, P – MR constant, P constant
If –e, P – MR , P.
Case-2
MC falling so profit-maximizing MR falls:
If –e, P – MR P may rise or fall
If –e constant, P – MR constant, P
If –et, P – MR, P
Case-3
MC rising so profit-maximizing MR must increase -
VIKAAS WADHWA SIR
9811439887 121
If –e, P – MR P
If –e constant, P – MR constant, MR , P
If –e, P – MR P may rise or fall
Q = (20 – P)(I + .1A – .01A2)
13
dK
= .1 − .02A
dA
Let K = l + .1A + .01A2
 = PQ – C = (20P – P2)K – (200 – 10P)K – 15 – A

= (20 − 2P)K + 10K = 0
P
a.
20 – 2P = –10
P = l5 regardless of K or A
If A = 0, Q = 5, C = 65,  = 10
b.
If P = 15,  = 75K – 50K – 15 – A = 25K – 15 – A = 10 + 1.5A – 0.25A2

= 1.5 − 0.5A = 0
A
so A = 3
Q = 5(1 + .3 – .09) = 6.05
PQ = 90.75
C = 60.5 + 15 + 3 = 78.5
 = 12.25; this represents an increase over the case A = 0.
14
The inverse elasticity rule is P = MC/(1 + 1/e)
When the monopoly is subject to an ad
valorem tax of t, tins becomes
a.
P=
MC 1
.
(1 − t ) 1 + 1
e
With linear demand, e falls (becomes more elastic) as price rises. Hence,
Paftertax =
Ppretax
MC
1
MC
1
.

.
=
(1 − t ) 1 + 1
(1 − t ) 1 + 1
(1 − t )
e aftertax
e pretax
Paftertax =
Ppretax
b.
With constant elasticity demand, the inequality in part a becomes an equality so
c.
If the monopoly operates on a negatively sloped portion of its marginal cost curve we have (in the
constant elasticity case)
Paftertax =
d.
(1 − t )
MCpretax 1
Ppretax
MCaftertax 1
.

.
=
(1 − t ) 1 + 1
(1 − t ) 1 + 1 (1 − t )
e
e
The key part of this question is the requirement of equal tax revenues. That is, tP aQa= Qs where the
subscripts refer to the monopoly’s choices under the two tax regimes. Assuming constant MC, profit
maximization requires
MC = Pa (1 − t ).
1
1
1+
e
= Ps .
1
1
1+
e
−.
Combining this with the revenue neutrality condition shows Ps > Pa.
15
a.
Q1 = 55 – F, R1 = (55– Q1)Q1 = 55Q1 –
VIKAAS WADHWA SIR
Q12
9811439887 122
MR1 = 55 – 2Q1 = 5 Q1 = 25, P1 = 30
Q2 = 70 – 2P2
(70Q 2 − Q 2 )
 70 − Q 2 
R2 = 
.Q 2 =
2
 2 
MR = 35 − Q 2 = 5
Q2 = 30, P2 = 20
2
 = (30 – 5)25 + (20 – 5)30 = 1075
b.
Producer wants to maximize price differential in order to maximize profits but maximum price
differential is $1 So P1 = P2 + 5.
 = (P1 – 5)(55 – P1) + (P2 – 5)(70 – 2P2)
Set up Lagrangian ? =  + 5 – P1 + P2

= 60 − 2P1 −  = 0
P1

= 80 − 2P2 +  = 0
P2

= 5 − P1 − P2 = 0

Hence 60 – 2P1 = 4P2 – 80 and P1 = P2 + 5.
c.
130 = 6P2
P2 = 2l.66
P1 = P2 So
 = P - 3P2 – 625
 = 1058.33
P1 = 26.66,

= 140 − 6P = 0
P
P=
d.
140
= 23.33
6
Q1 = 31.67
Q2 = 23.33
 = 1008.33
If the firm adopts a linear tariff of the form T(Q1) = I + rnQi, it can maximize profit by setting m = 5,
1 = .5(55 – 5)(50) = 1250
2 = .5(35 – 5)(60) = 900
and  = 2150.
Notice that in this problem neither market can be uniquely identified as the least willing” buyer so a
solution similar to Example 13.5 is not possible. If the entry fee were constrained to be equal in the two
markets, the firm could set m = 0, and charge a fee of 1225 (the most buyers in market 2 would pay).
This would yield profits of 2450 – 125(5) = 1825 which is inferior to profits yielded with T(Qi).
16
a.
For perfect competition, MC = $10. For monopoly MC = $12.
QD = 1000 – 50P.
The competitive solution is P = MC = $10. Thus Q = 500.
Monopoly :
P = 20 −
1
Q
50
PQ = 20Q −
Produce where MR = MC.MR =
b.
1 2
Q
50
20 −
1
Q = 12.
25
Q = 200, P = $16
See graph below.
Loss of consumer surplus = Competitive CS – monopoly CS
VIKAAS WADHWA SIR
9811439887 123
= 2500 – 400 = 2100.
c.
Of this 2100 loss, 800 is a transfer into monopoly profit, 400 is a loss from increased costs under
monopoly, and 900 is a “pure” deadweight loss.
17
a.
The government wishes the monopoly to expand output toward P = MC. A lump-sum subsidy will have
no effect on the monopolist’s profit maximizing choice, so this will not achieve the goal.
b.
A subsidy per unit of output will effectively shift the MC curve downward. The figure : illustrates this for
the constant MC case
c.
A subsidy (t) must be chosen so that the monopoly chooses the socially optimal quantity, given t. Since
social optimality requires P = MC and profit maximization requires that
MR = MC – t =
 1
P1 + 
 e
substitution yields
t
1
=−
p
e
as was to be shown.
Intuitively, the monopoly creates a gap between price and marginal cost and the optimal subsidy is
chosen to equal that gap expressed as a ratio to price.
VIKAAS WADHWA SIR
9811439887 124
MONOPOLY (READINGS)
A monopoly is a single firm that serves an entire market. This single firm faces the market
demand curve for its output. Using its knowledge of
the demand curve, it takes a decision on how much
to produce. Unlike the perfectly competitive firm’s
output decision (which has no effect on market
price), the monopoly’s output decision will, in fact,
determine the good’s price. In this sense, monopoly
markets and markets characterized by perfect
competition are polar-opposite cases. A monopoly
is a single supplier to a market. This firm may
choose to produce at any point on the market
demand curve.
At times it is more convenient to treat monopolies as having the power to set prices. Technically, a monopoly can choose that
point on the market demand, curve at which it prefers to operate. It may choose either market price or quantity, but not both.
In this chapter we will usually assume that monopolies choose the quantity of Output that maximizes profits and then settle
for the market price that the chosen output level yields.
BARRIERS TO ENTRY
The reason a monopoly exists is that other firms find it unprofitable or impossible to enter the
market. Therefore, barriers to entry are the source of
all monopoly power. If other firms could enter a
market, then the firm would, by definition, no
longer be a monopoly. There are two general types
of barriers to entry: technical barriers and legal
barriers.
Technical barriers to entry
A primary technical barrier is that the production of the good in question may exhibit decreasing
marginal (and average) costs over a wide range of
output levels. The technology of production is such
that relatively large-scale firms are low-cost
producers. In this situation (which is sometimes
referred to as natural monopoly), one firm: may find
it profitable to drive others out of the industry by
cutting prices. Similarly, once a monopoly has been
established, entry will be difficult because any new
firm must produce at relatively low levels of output
and therefore at relatively high average costs. It-is
important to stress that the range of declining costs
need only be “large” relative to the market in question. Declining costs on some absolute scale are not
necessary. For example, the production and delivery
of concrete does not exhibit declining marginal
costs over a broad range of output when compared
with the total U.S. market. However, in any
Particular small town, declining marginal costs may
permit a monopoly to be established. The high costs
of transportation in this industry tend to isolate one
market from another.
Another technical basis of monopoly is special knowledge of a low-cost productive technique. The monopoly has an
incentive to keep its technology secret; but unless this technology is protected by a patent (see next paragraph), this may. be
extremely difficult Ownership of unique resources–such as minimal deposits or land locations, or the possession of unique
managerial talents may also be a lasting basis for maintaining a monopoly.
Legal barriers to entry
Many pure monopolies are created as a matter of law rather than as a matter of economic -conditions. One important example of
a government-granted monopoly position is the legal protection of a product by a patent or copyright Prescription drugs,
computer chips and Disney animated movies are examples of profitable products that are shielded (for a time) from direct
competition ‘by potential imitators. Because the basic technology for these products is uniquely assigned to one firm, a
monopoly position is established. The defense made of ‘such a governmentally granted monopoly is that the patent and
copyright system makes innovation more profitable and therefore acts as an incentive. Whether the benefits of such innovative
behavior exceed the costs of having monopolies is an open question ‘that has been much debated by economists.
A second example of a legally created monopoly is the awarding of an exclusive franchise to serve a market. These
franchises are awarded in cases. of public utility (gas and electric) service, communications services, the post office, some
television and radio station markets, and ‘a variety of other situations. Usually the restriction of entry is combined with a
VIKAAS WADHWA SIR
9811439887 125
regulatory capon the price the franchised monopolist is allowed to charge. The argument usually put forward in favor of
creating these franchised monopolies is that the industry in question is a natural monopoly: average cost is diminishing over
a broad range of output levels, and minimum average cost can be achieved only by organizing the industry as a ‘monopoly.
The public utility and communications industries are often considered good’ examples. Certainly, that does appear to be ‘the
case for local electricity and telephone service where a given network probably exhibits declining average cost up to the
point of universal coverage. But recent deregulation in telephone services and electricity generation’ ‘show that, even for
these industries, the natural monopoly rationale may not be, all-inclusive. In other cases, franchises may be based largely on
political rationales. This seems to be true for the postal service in the United States and for a number of nationalized
industries (airlines, radio and television, banking) in other countries.
Creation of barriers to entry
Although some barriers to entry may be independent of the monopolist’s own activities, other barriers may result directly from
those activities. For example, firms may develop unique products or technologies and take extraordinary steps to keep these
from being copied by competitors. Or firms may buy up unique resources to prevent potential entry. The Dc Beers cartel for
example, controls a large fraction of the world’s diamond mines. Finally, a would-be monopolist may enlist government aid
in devising barriers to entry. It may lobby for legislation’ that restricts new entrants to “maintain an orderly market” or’ for
health and safety. regulations that raise potential entrants’ costs. Because the monopolist has both special knowledge of its
business and significant incentives to pursue these ‘goals, it may have considerable success in creating such barriers to entry.
The attempt by a monopolist to erect barriers td entry may involve real resource costs. Maintaining secrecy, buying unique
resources, and engaging in political lobbying are all costly activities. A full analysis of monopoly should involve not only
questions of cost minimization and output choice (as under perfect competition) but also an analysis of profit maximizing
creation of entry barriers. However, we will not ‘provide a detailed. investigation of such questions here. Instead, we will
take ‘the presence of a single supplier on the market, and this single firm’s cost function, as given.
OUTPUT CHOICE
To maximize profits, a monopoly will choose to produce that output level for which marginal revenue is equal to marginal cost.
Because the monopoly, in contrast to a perfectly competitive firm, faces a negatively sloped market demand curve, marginal
revenue will be less than the market price. To sell an additional unit, the monopoly must lower its price on all units to be
sold if it is to generate the extra demand necessary to absorb this marginal unit. The profit-maximizing output level for. a
firm is then the level Q* in Figure 14.1. At that level, marginal revenue is equal to marginal costs, and profits are maximized.
Figure 14.1
Given the’ ‘monopoly’s decision to produce Q*, the demand curve D indicates that a market price of P* will prevail. This
is the price that demanders as a group are willing to pay for the output of the monopoly. In the market, an equilibrium
price–quantity combination of P*, Q* will be observed. Assuming P* > AC, this output level will be profitable, and the
monopolist will have no incentive to alter output levels unless demand or cost conditions change, hence we have the
VIKAAS WADHWA SIR
9811439887 126
following principle.
Definition.
‘Monopolist’s output. A monopolist will choose to produce that-output for which marginal revenue equals marginal cost
Because the monopolist faces a downward sloping demand curve, market price will exceed marginal revenue and the
firm s marginal cost at this output level.
Profit maximization implies that the gap between a price of a firm’s output and its marginal cost is inversely’ related to the price
elasticity of the demand curve faced by the firm. Applying Equation 1.14 to the case of monopoly yields.
P − MC
1
=−
P
eQ, p
(14.1)
market (eQ,p) because the monopoly is the sole supplier of the good in question. This observation leads to two general conclusions about monopoly
pricing. First, a monopoly will choose to Operate only in regions in which the market demand curve is elastic (eQ,P < –1). If
demand were inelastic, then marginal revenue would be negative and thus could not be equated to marginal cost (which
presumably is positive). Equation 14.1 also shows that eQ,P > –1 implies an (implausible) negative marginal cost.
A second implication of Equation 14.1 is that the firm’s “markup” over marginal cost (measured as a fraction of price) depends
inversely on the elasticity of market demand. For ‘example,’ if eQ,P = –2, then Equation 14.1 shows that P = 2MG, whereas if
eQ,P
=
–10,
then
P = 1.11 MC. Notice also that if the elasticity of demand were constant along the entire demand curve, the proportional markup
over marginal’ cost would remain unchanged in response to changes in input costs. Therefore, market price moves proportionally to marginal cost: Increases in marginal cost will prompt the monopoly to increase its price proportionally and decreases
in marginal cost will cause the monopoly to reduce its price proportionally. Even if elasticity is not constant along the demand
curve, it seems clear -from Figure 14.1 that increases in marginal cost will increase price (although not necessarily in the same
proportion). As long as the demand curve facing the monopoly is downward sloping, upward shifts in MC will prompt the
monopoly to reduce output and thereby obtain a higher price. We will examine all these relationships mathematically in
Examples 14.1 and 14.2.
Monopoly profits
Total profits earned by the monopolist can be read directly from Figure 14.1. These are shown by the rectangle P *EAC and
again represent the profit per unit (price minus average cost) times the number of units sold. These profits will he positive
if market price exceeds average total cost. If p* < AC, however, then the monopolist can operate only at a long-term loss
and will decline to serve the market.
Because (by assumption) no entry is possible into a monopoly market, the monopolist’s positive profits can exist even in
the long run. For this reason, some authors refer to the profits that a monopoly earns in the long rim as monopoly rents.
These profits can be regarded as a return to that factor that forms the basis of the monopoly (e.g., a patent, a favorable
location, or a dynamic entrepreneur, hence another possible owner ‘might be willing to pay that amount in rent for the
right to the monopoly. The’ potential for profits is the- reason why some firms pay other firms for the right to use a patent
and why concessioners at sporting events (and on some highways) are willing to pay for the right to the concession. To the
extent that monopoly rights are given away at less than their true market value (as in radio and television licensing), the
wealth of the recipients of those rights is increased.
Although a monopoly may earn positive profits in the bug run, the size of such profits will depend on the relationship
between the monopolist’s average costs and the demand for its product. Figure 14.2 illustrates two situations in which the
demand,’ marginal revenue, and marginal cost curves are rather similar. As Equation 14.1 suggests. the price-marginal cost
markup is about the same in these two cases. P0ut average costs in Figure 14.2a are considerably lower than in Figure
14.2b. Although the profit-maximizing decisions are similar in the two cases, the level of profits ends up being different. In
Figure 14.2a the monopolist’s price (P*) exceeds the average cost of producing Q* (labeled AC*) by a large extent, and significant profits are obtained. In Figure 14.2b, however, P* = AC* and the monopoly earns zero economic profits, the largest
amount possible in this case. Hence large profits from a monopoly are not inevitable, and the actual extent of economic
profits- may not always be a good guide to the significance of monopolistic influences in a market.
VIKAAS WADHWA SIR
9811439887 127
Figure 14.2
There is no monopoly supply curve
In the theory of perfectly competitive markets , it was possible to speak of an industry supply curve. We constructed the
long-run supply curve by allowing the market demand curve to shift and observing the supply curve that was traced out by
the series of equilibrium price–quantity combinations. This type of construction is not possible for monopolistic markets.
With a fixed market demand curve, the supply “curve” for a monopoly will be only one point–namely, that price–quantity
combination for which
MR = MC. If the demand curve should shift, then the marginal revenue curve would also shift, and a new profit-maximizing
output would be chosen. However, connecting the resulting series of equilibrium points on the market demand curves
would have little meaning. This locus might have a strange shape, depending on how the market demand curve’s elasticity
(and its associated MR curve) changes as the curve is shifted. In this sense the monopoly firm has no well-defined “supply
curve.” Each demand curve is a unique profit-maximizing opportunity for a monopolist.
Example 14.1. Calculating Monopoly Output
Suppose the market for Olympic quality Frisbees (Q measured in Frisbees bought per year) has a linear demand curve of
the form
Q = 2,000 – 20P
(14.2)
P = 100 −
Q
20
(14.3)
and let the costs of a monopoly Frisbee producer be given by
2
C(Q) = 0.05Q + 10,000
(14.4)
To maximize profits, this producer chooses that output level for which MR = MC. To solve this problem we must phrase
both MR and MC as functions of Q alone. Toward this end, write total revenue as
2
P.Q = 100Q −
Q
20
(14.5)
Consequently
MR = 100 −
Q
= MC = 0.1Q
10
(14.6)
and
Q* = 500, P* = 75 .
(14.7)
At the monopoly’s preferred output level,
C(Q) = 0.05(500)2 + 10,000 = 22,500
VIKAAS WADHWA SIR
9811439887 128
AC =
22,500
= 45
500
(14.8)
Using this information, we can calculate profits as
*
*
 = ( P − AC).Q = (75 − 45).500 = 15,000
(14.9)
Observe that at this equilibrium there is a large markup between price (75) and marginal cost (MC = 0.1Q = 50). Yet as long
as entry barriers prevent a new firm from producing Olympic quality Frisbees, this gap and positive economic profits can
persist indefinitely.
QUERY: How would on increase in fixed costs from 10,000 to 12,500 affect the monopoly’s output plans? How would
profits be affected? Suppose total costs shifted to
C(Q) = 0.075Q2 + 10,000.
How would the equilibrium change ?
Example 14.2. Monopoly with Simple Demand Curves
We can derive a few simple facts about monopoly pricing in cases where the demand curve facing the monopoly takes a
simple algebraic form and the firm has constant marginal costs
(i.e. C(Q) = cQ and MC = ).
Linear Demand. Suppose that the inverse demand function facing the monopoly is of the linear form
p = a − bQ . In
this case, PQ = aQ – bQ2 and MR = dPQ/dQ = a – 2bQ. Hence profit maximization requires that
MR = a – 2bQ = MC = c
or
Q=
a −0
2b
(14.10)
Inserting this solution for the profit-maximizing output level back into the inverse demand functions yields, a direct
relationship between price and marginal cost:
P = a − bQ = a −
a −c a +c
=
2
2
(14.11)
In interesting implication is that, in this linear case, dP/dc = 1/2.
That is, only half of the amount of any increase in marginal cost will show up in the market price of the monopoly product.
Constant elasticity demand. If the demand curve facing the monopoly takes the constant elasticity form Q = aPe (where e
is the price elasticity of demand), then we know
MR = P(l + 1/e); and thus profit maximization requires
or
 1
P 1 +  = c
 e
 e 
P = c

1+ e 
(14.12)
Because it must be the case that e < –1 for profit maximization, price will clearly exceed marginal cost, and this gap will be
larger the closer e is to –1.
Notice also that dP/dc = e/(I + e) and so any given increase in marginal cost will increase price by more than this amount.
Of course, as we ‘have already pointed out, the proportional increase in marginal cost and price will be the same.
That is, ep,e = dP/dc.c/P= 1.
QUERY: The demand function in both cases is shifted by the parameter a. Discuss the effects of such a shift for both linear
and constant elasticity demand. Explain your results intuitively.
MONOPOLY AND RESOURCE ALLOCATION
In Chapter 13 we briefly mentioned why the presence of monopoly distorts the allocation of resources. Because the
monopoly produces a level of output for which MC = MR <P, the market price of its good no longer conveys accurate
information about production costs. Hence consumers’ decisions will no longer reflect true opportunity costs of production, and resources will be misallocated. In this section we explore this misallocation in some detail in a partial equilibrium
context.
Basis of comparison
To evaluate the allocational effect of a monopoly, we need a precisely defined basis of comparison. A particularly useful
comparison is provided by a perfectly competitive industry. It is convenient to think of a monopoly as arising from the
“capture” of such a competitive industry and to treat the individual firms that constituted the competitive industry as now
being single plants in the monopolist’s empire. A prototype case would be John D. Rockefeller’s purchase of most of the
VIKAAS WADHWA SIR
9811439887 129
U.S. petroleum refineries in the late nineteenth century and his decision to operate them as part of the Standard Oil trust
We can then compare the performance of this monopoly with the performance of the previously competitive industry to
arrive at a statement about the welfare consequences of monopoly.
A graphical analysis
Figure 14.3 provides a graphical analysis of the welfare effects of monopoly. If this market were competitive, output would
be Qc that is, production would occur where price is equal to long-run average and marginal cost. Under a simple singleprice monopoly, output would be Qm because this is the level of production for which marginal revenue is equal to
marginal cost. The restriction in output from Qc to Qm represents the misallocation brought about through monopolization.
The total value of resources released by this output restriction is shown in Figure 14.3 as area FEQcQm. Essentially, the
monopoly closes down some of the plants that were operating in the competitive case. These transferred inputs can be
productively used elsewhere; thus, area FEQcQm is not a social loss.
Figure 14.3
The restriction in output from Qc to Qm involves a total loss in consumer surplus of PmBEPc. Part of this loss, PmBCPc, is
transferred to the monopoly as increased profit. Another part of the consumers’ loss, BEC, is not transferred to anyone
but is a pure deadweight loss in the market. A second source of deadweight loss is given by area CEF. This is an area of lost
producer surplus that does not get transferred to another source. The total deadweight loss from both sources is area BEF,
sometimes called the deadweight: loss triangle because of its roughly triangular shape. The gain PmBCPc in monopoly
profit from an increased price more than compensates for its loss of producer surplus CEF from the output reduction so
that, overall, the monopolist finds reducing output from Qc to Qm to be profitable.
To illustrate the nature of this deadweight loss, consider Example 14.1, in which we calculated an equilibrium price of $75
and a ‘marginal cost of $50. This gap between price and marginal cost is an indication of the efficiency-improving trades
that are forgone through ‘monopolization. Undoubtedly, there is a would be buyer who is willing to pay’, say, $60 for an
Olympic Frisbee but not $75. A price of $60 would more than cover all, the resource costs involved in Frisbee production,
but the presence of the monopoly prevents such a mutually beneficial transaction between Frisbee users and the providers
of Frisbee making resources. For this reason, the monopoly equilibrium is not Pareto optimal–an alternative allocation of
resources would make all parties better off. Economists have made many attempts to estimate the overall cost of these
deadweight losses, an actual monopoly situations. Most of these estimates are rather small when viewed in the context of
the whole economy. Allocational losses are larger, however, for some narrowly defined industries.
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EXAMPLE 14.3 Welfare Losses and Elasticity
The allocational effects of monopoly can be characterized fairly completely in the case of constant marginal costs and a
constant price elasticity demand curve. To do so, assume again marginal (arid average) costs for a monopolist are given by
c and that the demand has a constant elasticity form of
Q = Pe
(14.13)
where e is the price elasticity of demand (e < –1). We know the competitive price in this market will be
Pc = c
(14.14)
monopoly price is given by
Pm =
c
1 + 1/ e
(14.15)
The consumer surplus associated with any price (P0) can be computed as

CS =  Q( P)dP
P0

=  PedP
P0

pe +1
=
e +1 P
0
=−
P0e +1
(14.16)
e +1
Hence under perfect competition we have
e +1
CSe = −
c
e +1
(14.17)
and, under monopoly
 e 


1 + 1/ e 

CSm = −
e +1
(14.18)
Taking the ratio of these two surplus measures yields
e +1
CSm  1 
CSc
=

 1 + 1/ e 
(14.19)
If e = –2, for example then this ratio as 1/2 consumer surplus under monopoly is half what it is under perfect competition.
For more elastic cases this figure decreases a bit (because output restriction under monopoly are more significant). For
elasticities closer to –1, the ratio increases.
Profits. The transfer from consumer surplus into monopoly profits can also be computed fairly easily in this case.
Monopoly profits are given by
 c

 m = PmQ m − cQ m = 
− c Q m
 1 + 1/ e

 − c / e  c 
=


 1 + 1 / e  1 + 1 / e 
 c 
= −

 1 + 1/ e 
e+1
.
1
e
e
(14.20)
Dividing this expression by Equation 14.17 yields
e +1
 m  e + `  1 
CSc
=


 e  1 + 1 / e 
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9811439887 131
 e 
=

1+ e 
e
(14.21)
For e = –2 this ratio is 1/4 Hence one fourth of the consumer surplus enjoyed under perfect competition is transferred into
monopoly profits Therefore the deadweight loss from monopoly in this case is also a fourth of the level of consumer
surplus under perfect competition.
QUERY : Suppose e = –1.5. What fraction consumer surplus is lost through monopolization? How much is transferred into
monopoly profits? Why do these results differ from the case
e = –2 ?
MONOPOLY, PRODUCT QUALITY AND DURABILITY
The market power enjoyed by a monopoly may be exercised along dimensions other than the market pricing of its product.
If the monopoly has some, leeway in the type, quality, or diversity of the goods it produces, then it would not be surprising
for the firm’s decisions to differ from those that might prevail under a competitive organization of the market. Whether a
monopoly will produce higher-quality or lower-quality goods than would be produced under competition is unclear,
however. It-all depends on the firm’s costs and the nature of consumer demand.
A formal treatment of quality
Suppose consumers’ willingness to pay for quality (X) is given by the inverse demand function P(Q, X), where
and
P
0
Q
P
 0.
X
If the costs of producing Q and X are given by C(Q, X), the monopoly will choose Q and X to maximize
 = P(Q,X)Q – C(Q,X)
(14.22)
The first-order conditions for a maximum are

P
= P(Q, X ) + Q
− CQ = 0
Q
Q

P
=Q
− CX = 0
X
X
(14.23)
(14.24)
The first of these equations repeats the usual rule that marginal revenue equals marginal cost for output decisions. The
second equation states that, when Q is appropriately set, the monopoly should choose that level of quality for which the
marginal revenue attainable from increasing the quality of its output by one unit is equal to the marginal cost of making
such an increase. As might have been expected, the assumption of profit maximization requires the’ monopolist to
proceed to the margin of profitability along all the dimensions it can. Notice, in particular, that the marginal demander's
valuation of quality per unit is multiplied by the monopolist’s output level when determining the profit-maximizing choice.
The level of product quality chosen under competitive conditions will also be the one that maximizes net social welfare:
SW = 
Q*
P(Q, X )dQ − C(Q, X )
(14.25)
0
where Q* is the output level determined through the competitive process of marginal cost
pricing, given X. Differentiation of Equation 14.25 with respect to X yields the first-order
condition for a maximum:
Q* = 0
SW
=
PX (Q, X )dQ − CX
0
X
(14.26)
The monopolist’s choice of quality in Equation 14.24 targets the marginal consumer. The monopolist cares about the
marginal consumer’s valuation of quality because increasing the attractiveness of the product to the marginal consumer is
how it increases sales. The perfectly competitive market ends up providing a quality level in Equation 14.26, maximizing
total consumer surplus (the total after subtracting the cost of providing that quality level), which is the same as the quality
level that maximizes consumer surplus for the average consumer. Therefore, eyen if a monopoly and a perfectly
competitive industry choose the same output level, thy might opt for differing quality levels because each is concerned
with a different margin in its decision making. Only by knowing the specifics of the problem is it possible to predict the
direction of these differences. For an example, see Problem 14.9; more detail on the theory of product quality and
monopoly is provided in Problem l4.11.
The durability of goods
VIKAAS WADHWA SIR
9811439887 132
Much of the research on the effect of monopolization on quality has focused on durable goods. These are goods such as
automobiles, houses, or refrigerators that provide services to their owners over several periods rather than being
completely consumed soon after they are bought. The element of time that enters into the theory of durable goods leads
to many interesting problems and paradoxes. Initial interest in the topic started with the question of whether monopolies
would produce goods that lasted as long as similar goods produced under perfect competition. The intuitive notion that
monopolies would “underproduce” durability (just as they choose an output below the competitive level) was -soon shown
to be incorrect by the Australian economist Peter Swan in the early 1970s.
Swan’s insight was to view the demand for durable goods as the demand for a flow of services (i.e., automobile
transportation) over several periods. He argued that both a monopoly and a competitive market -would seek to minimize
the cost of providing this flow to consumers. The monopoly would, of course, choose an output level that restricted the
flow of services to maximize profits, but assuming constant returns to scale in production there is no reason that durability
per se would be affected b market structure. This result is sometimes referred to as Swan' s independence assumption.
Output decisions can be treated independently, from decisions about product durability.
Subsequent research on the Swan result has focused on showing how it can be undermined by different assumptions
about the nature of a particular durable good or by relaxing the implicit assumption that all demanders are the same. For
example, the result depends critically on how durable goods deteriorate. The simplest type of deterioration is illustrated by
a durable good, such as a lightbulb, that provides a constant stream’ of services until it becomes worthless. With this type
of good, Equations 14.24 and 14.26 are identical, so Swan’s independence result holds. Even when goods deteriorate
smoothly, the independence result continues to hold if a constant flow, of services can be maintained by simply replacing
what has been used–this requires that’ new goods and old goods be perfect substitutes and infinitely divisible. Outdoor
house paint may, more or less, meet this requirement. On the other hand, most goods clearly do not. It is just not possible
to replace a run down refrigerator with, say half of a new one. Once such more complex forms of deterioration are
considered, Swan's result may not hold because we can no longer fall back on the notion of providing a given flow of
services at minimal cost over time. In these more complex cases, however, it is not always e case t at a monopoly will
produce less durability, than will a competitive market it all depends, on the nature of the demand for durability.
Time inconsistency and heterogeneous demand
Focusing on the service flow from durable goods provides important insights on durability, but it does leave an important
question unanswered–when should the monopoly produce the actual durable goods needed to provide the desired service
flow? Suppose, for example, that a lightbulb monopoly decides that its profit-maximizing output decision is to ‘supply the
services provided by, 1million 60-watt bulbs, If the firm decides to produce 1 million bulbs in the first period, what is it to
do in the second period (say, before any of the original bulbs burn out)? Because the monopoly chooses a point on the
service demand curve where P > MC, it has a clear incentive to produce more bulbs in the second period by cutting price a
bit. Bill consumers can anticipate this, so’ they may reduce their first period demand, waiting for a bargain. Hence the
monopoly’s profit-maximizing plan will unravel. Ronald Coase was the first economist to note this “time inconsistency’0
that arises when a monopoly produces a durable good. Coase argued that its presence would severely undercut potential
monopoly power in the limit, competitive pricing is the only outcome that can prevail in the durable goods ,case. Only if
the monopoly can succeed in making a credible commitment not to produce more in the second period can it succeed in its
plan to achieve monopoly profits on the service flow from durable goods.
Recent modeling of the durable goods question has examined how a monopolist’s choices are affected in situations where
-there are different types of demanders. In such cases, questions about the optimal choice of durability and about credible
commitments become even more complicated. The monopolist must not only settle on an optimal scheme for each
category of buyers, it must also ensure that the scheme intended for (say) type-I demanders is not, also attractive to type-2
demanders. Studying these sorts of models would take us too far a field, but some illustrations of how such “incentive
compatibility constraints” work are provided in the Extensions to this chapter at the end of the book and in Chapter 18.
PRICE DISCRIMINATION
In some circumstances a monopoly may be able to increase profits by departing from a single-price policy for its output.
The possibility of selling identical goods at different prices is called price discrimination.
Price discrimination. A monopoly engages in price discrimination if it is able to sell otherwise identical units of output at
different prices.
Examples of price discrimination include senior citizen discounts for restaurant meals (which could instead be viewed as a
price premium for younger customers), coffee sold at a lower price per ounce when bought in larger ‘cup sizes, ,and
different (net) tuition charged to different college students after subtracting their more or less, generous financial and
awards. A “nonexample” of price discrimination might be higher auto, insurance ‘premiums charged to younger drivers. It
might be clearer to think of the insurance policies sold to younger and older drivers as being different products to the
‘extent that younger drivers are riskier and result in many more claims having to be paid.
Whether a price discrimination strategy is feasible depends crucially on the inability of buyers of the good to practice
arbitrage. in the absence of transactions or information costs, the “law of one price” implies that a homogeneous good
must sell everywhere for the same price. Consequently, price discrimination schemes are doomed to failure because
demanders who can-buy from the monopoly at lower prices will be more attractive sources of the good – for those who
VIKAAS WADHWA SIR
9811439887 133
must pay high prices–than is the monopoly itself. Profit seeking middlemen will destroy any discriminatory pricing scheme.
However, when resale is costly or can be prevented entirely, then price discrimination becomes possible.
First-degree or perfect price discrimination
If each buyer can be separately identified by a monopolist, then it may be possible to charge each the maximum price he
Or she would willingly pay for the good. This strategy of perfect (or first-degree) price discrimination would then extract all
available consumer surplus, leaving demanders as a group indifferent between buying the monopolist’s good or doing
without it. The strategy is illustrated in Figure 14.4. The figure assumes that buyers are arranged. in descending order of
willingness to pay. The first buyer is willing to pay up to P1 for Q1 units of output; therefore, the monopolist charges P1 and
obtains total revenues of P1Q1 as indicated by the lightly shaded rectangle. A second buyer is willing to pay up to P2 for Q2 –
Q1 units of output; therefore, the monopolist obtain total revenue of P2(Q2 – Q1) from this buyer. Notice that this strategy
cannot succeed unless the second buyer is unable to resell the output he or she buys at P 2 to the first buyer (who pays
P1 > P3).
The monopolist will proceed in this-way up to the marginal buyer, the last buyer who is willing to pay at least the good’s
marginal cost (labeled Mc in Figure 14.4). Hence total quantity produced will be Q*, Total revenues collected will be given
by the area DEQ*O. All consumer surplus has been extracted by the monopolist, and there is no deadweight loss in this
situation. (Compare Figures 14.3 and 14.4.) Therefore, the allocation of resources under perfect price discrimination is
efficient, although it does entail a large transfer from consumer surplus into monopoly profits.
Figure 14.4
Example 14.4 : First Degree Price Discrimination
Consider again the Frisbee monopolist in Example 14.1. Because there are relatively few high quality Frisbees sold, the
monopolist may find it possible to discriminate perfectly among a few world class flippers. In this case, it will choose to
produce that quantity for which the marginal exactly the marginal cost of a Frisbee:
P = 100 −
Q
= MC = 0.1Q
20
(14.27)
Hence
Q* = 666
and at the, margin, ‘price and marginal cost are given by
P = MC = 66.6
Now, we can compute total revenues by integration:
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(14.28)
9811439887 134
R=
Q*
0
Q =666

Q 2 
P(Q)dQ = 100Q −

40 

Q =0
(14.29)
= 55,511.
Total costs are
C(Q) = 0.05Q2 + 10,000 = 32,178
(14,30)
total profits are given by
 = R – C = 23,333
(14.31)
which represents a substantial increase over tile single-price policy examined in Example 14.1 (which yie1d 15,000).
Query : What is the maximum price any Frisbee buyer pays in this case? Use this to obtain a geometric definition of profits.
Third-degree price discrimination through market separation
First degree price discrimination pose a considerable information burden for the monopoly it must know the demand
function for each potential buyer. A less strigent requirement would be to assume the monopoly can separate its buyers
into relatively few identifiable markets (such as “rural–urban,” “domestic–foreign” or “prime-time–off prime”) and pursue
a separate monopoly pricing policy in each market. Knowledge of the price elasticities of demand in these markets is
sufficient to pursue such a policy. The monopoly then sets a price in each markets, according to the inverse elastic rule.
Assuming that marginal cost is the same in all markets, the result is a pricing policy in which
or


1
1
P1 1 +  = Pj 1 + 
 ej 
 ei 


(14.32)
Pi (1 + 1 / e j )
=
Pj (1 + 1 / ei )
(14,33)
where Pi and Pj are the prices charged in markets i and j, which have pike elasticities of demand given by ei and ej. An
immediate consequence of this pricing policy is that the
Figure 14.5
profit-maximizing price will be higher in markets in which demand is Less elastic. If, for examp1e, ei = –2 and e1 = –3, then
Equation 14.33 shows that Pi/Pj = 4/3–prices will be one third higher in market i, the less elastic market.
Figure 14.5 illustrates this result for two markets that the monopoly can serve at constant marginal cost (MG). Demand is
less elastic in market 1 than in market 2; thus, the gap between price and marginal. revenue is larger in ‘the former market.
VIKAAS WADHWA SIR
9811439887 135
Profit maximization requires that the firm produce
Q1* in market I and Q*2
in market 2, resulting in a higher price in the
less elastic market. As long as arbitrage between the two markets can be prevented, this price difference can persist. The
two price discriminatory policy is clearly more profitable for the monopoly than a single-price policy would be because the
firm can always opt for the latter policy should market conditions warrant.
The welfare consequences of third degree price discrimination are, in principle, ambiguous. Relative to a single price policy,
the discriminating policy requires raising the price in the less elastic market and reducing it in the more elastic one. Hence
the changes have an offsetting effect on total allocational losses. A more complete ‘analysis suggests the intuitively
plausible conclusion. that the multiple-price policy will be allcatiotially superior to a single-price policy only in situations in
which total output is increased through discrimination.
Example 14.5 illustrates a simple case of linear demand curves in which ? single-price policy does result in greater
allocational Iosses.
Example 14.5 Third–Degree Price Discrimination
Suppose that a monopoly producer of widgets has a constant marginal cost of c = 6 and sells its product in two separated
markets whose inverse demand functions are
P1 = 24 – Qt
and
P2 = 12 – 0.5Q2
(14.34)
Notice that consumers in market 1 are more eager to buy than are consumers in market 2 in the sense that the former are
willing to pay more for any given quantity. Using the results for linear demand curves from Example 14.2 shows that. the
profit-maximizing price-quantity combinations in these two markets are:
24 + 6
= 15
2
12 + 6
P2* =
= 9 , Q*2 = 6
2
P1* =
(14 35)
With this pricing strategy profits are  = (15 – 6)9 + (9 – 6) 6 = 81 + 18 = 99.
We can compute the deadweight losses in the two markets by recognizing that the competitive output (with P = MC = 6) in
market 1 is 18 and in market 2 is 12
DW
= DW1 + DW2
*
*
= 0.5( P1 − 6)(18 − 9) + 0.5( P2 − 6)(12 − 6)
= 40.5 + 9 = 49.5
(14.36)
A single prince policy. In this case constraining the monopoly to charge a single price would duce welfare Under a single
price policy the monopoly would simply cease serving market 2 because it can; maximize profits by charging a price of 15,
and at that price no widgets will be bought in market 2 (because the maximum willingness to pay is 12). Therefore, total
deadweight loss in this situation is increased from its level in Equation 1436 because total potential consumer surplus in
market 2 is now lost:
DW
= DW1 + DW2
= 40.5 + 05(12 – 6)(12 – 0)
= 40.5 + 36
= 76.5
(1437)
This illustrates a situation where third-degree price discrimination is welfare improving over a single price policy–when the
discriminatory policy permits "smaller" markets to be served.
Whether such a situation is common is an important policy question (e.g., consider the case of U.S. pharmaceutical
manufacturers charging higher prices at home than abroad).
QUERY : Suppose these markets were no longer separated. How would you construct the market demand in this situation?
Would the monopolists profit maximizing single price still
SECOND–DEGREE PRICE DISCRIMINATION THROUGH PRICE SCHEDULES
The examples of price discrimination examined in the previous section require the monopoly to separate demanders into a
number of categories and then choose a profit-maximizing price for each such category. An alternative approach would be
for the monopoly to choose a (possibly rather complex) price schedule that provides incentives for demanders to separate
themselves depending on how much they wish to buy. Such schemes include-quantity discounts, minimum purchase
requirements or “cover” charges, and tie-in sales. These plans would he adopted by -a monopoly if they yielded greater
profits than would a single-price policy, after accounting for any possible costs of implementing the price schedule.
Because the schedules will result in demanders paying different prices for identical goods, this form of (second degree)
price discrimination is feasible only when there are no arbitrage possibilities. Here we look at one simple case.
The Extensions to this chapter at the end of the book and portions of Chapter 18 look at other aspects of second-degree
price discrimination.
Two-part tariffs
VIKAAS WADHWA SIR
9811439887 136
One form of pricing schedule that has been extensively studied is a linear two-part tariff, under which demanders must pay
a fixed fee for the right to consume a good and a uniform price for each unit consumed. The prototype case, first studied
by Walter Oi, is an amusement- park (perhaps Disneyland) that sets a basic entry fee coupled with a stated marginal price
for each amusement used. Mathematically, this scheme can be represented by the tariff any demander must pay to
purchase q units of a good:
T(q) = a + pq,
(14.38)
where a is the fixed fee and p is the marginal price to be paid. The monopolist’s goal then is to choose a and p to maximize
profits, given the demand for this product. Because the average price paid by any demander is given by
p=
T a
= +p
q q
(14.39)
this tariff is feasible only when those who pay low average prices (those for whom q is large) cannot resell the good tO
those who must pay high average prices (those for whom q is small).
One approach described by Oi for establishing the parameters of this linear tariff would be for the firm to set the marginal
price, p, equal to MC and then set a to extract the maximum consumer surplus from a given set of buyers. One might
imagine buyers being arrayed according to willingness to pay. The choice of p MC would then maximize consumer surplus
for this group, and a could be set equal to the surplus enjoyed by the least eager buyer. He or she would then be
indifferent about buying the good, but all other buyers would experience net gains from the purchase.
This feasible tariff might not be the most profitable, however. Consider the effects on profits of a small-increase in p above
MC. This would result in no net change in the profits earned from the least willing buyer Quantity demanded would drop
slightly at the margin. where p = MC, and some of what had previously been consumer surplus (and therefore part of the
fixed fee, a) would be converted into variable profits because-now p > MC. For all other demanders, profits would be
increased by the price increase.
Although each will pay a bit -less iii fixed charges, profits per unit bought will increase to a greater extent. In some cases it
is possible to make an explicit calculation of the optimal two part tariff. Example 14.6 provides an illustration. More
generally, however, optimal schedules will depend on a variety of contingencies. Some of the, possibilities are examined in
the Extensions to this chapter at the end of the book.
Example 14.6 : Two Part Tariffs
To illustrate mathematics of two-part tariffs, let’s return to the demand equations introduce in Example 14.5 but now
assume that they apply to two specific demanders:
q1 = 24 – p
(14.40)
q2 = 24 – 2p2
where now the p's refer to the marginal prices faced by these two buyers.
p1 = p 2 = MC = 6 .
Hence in this case, q1 = 18 and q2 = 12. With this marginal price, demander 2 (the less eager of the two) obtains consumer
surplus of
36 (= 0.5(12 – 6) 12).
That is the maximal entry fee that might be charged without causing this person to leave the market. Consequently, the
two-part tariff in this case would be T(q) = 36 + 6q. If the monopolist opted for this pricing scheme, its profits would be
An Oi tariff. Implementing the two-part tariff suggested by Oi would require the monopolist to set
 = R − C = T(q1 ) + T(q 2 ) − AC(q1 + q 2 )
= 72 + 6.30 − 6.30 = 72
(14.41)
These fall short of those obtained in Example 14.5.
The optimal tariff. The optimal two-part tariff in this situation can be computed by noting that total profit with such a tariff
are  = 2a + (p – MC)(q1 + q2).
Here the entry fee, a, must equal ‘the consumer surplus obtained by person 2.
Inserting the specific parameters of this ‘problem yields
 = 0.5.2q 2 (12 − p) + (p − 6)(q1 + q 2 )
= (24 − 2p)(12 − p) + (p − 6)(48 − 3p)
= 18p − p 2
(14.42)
Hence maximum profits are obtained when p = 9 and a = 0.5(24 – 2p)(l2 – p).
Therefore, the optimal tariff is T(q) = 9 + 9q.
With this tariff, q1 = 15 and q2 = 6, and the monopolists profits are
81 [= 2(9) + (9 – 6) . (15 + 6)].
The monopolist might opt for this pricing scheme if it were under political pressure to have a uniform pricing policy and to
agree not to price demander 2 “out of the market.” The two-part tariff permits a degree of differential pricing (p1 = 9.60, p2
= 9.75) but appears “fair” because all buyers face schedule.
QUERRY : Suppose a monopolist could choose a different entry fee for each demander, What pricing policy would be
followed ?
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9811439887 137
REGULATION OF MONOPOLY
The regulation of natural monopolies is an important subject in applied economic analysis, The utility, communications,
and transportation industries are highly’ ‘regulated in most countries, and devising regulatory procedures that induce
these industries to operate in a desirable way is an important practical problem. Here we will examine a few aspects of the
regulation of monopolies that relate to pricing policies.
Marginal cost pricing and the natural monopoly dilemma
Many economists believe it is important for the prices charged by regulated monopolies to reflect marginal costs of
production accurately. In this way the deadweight loss may be
Figure 14.6
minimized. The principal problem raised by an enforced policy of marginal cost pricing is that it will require natural
monopolies to operate at a loss. Natural monopolies, by definition, exhibit decreasing average costs over a broad range of
output levels. The cost curves for such a firm might look like those shown in Figure 14.6. In the absence of regulation, the
monopoly would produce output level QA and receive a price of PA for its product.
Profits in this situation are given by the rectangle PAABC. A regulatory agency might instead set a price of PR for the
monopoly. At this price, QR is demanded, and the marginal cost of producing this output level is also PR. Consequently,
marginal cost pricing has been achieved. Unfortunately, because of the negative slope of the firm’s average cost curve, the
price PR (= marginal cost) decreases below average costs. With this regulated price, the monopoly must operate at a loss of
GFEPR. Because no firm can operate indefinitely at a loss, this poses a dilemma for the regulatory agency: Either it must
abandon its goal of marginal cost pricing, or the government must subsidize the monopoly forever.
Two-tier pricing systems
One way out of the marginal cost pricing dilemma is the implementation of a multiprice system. Under such a system the
monopoly is permitted to charge some users a high price while maintaining a low price for marginal users. In this way the
demanders paying the high price in effect subsidize the losses of the low-price customers. Such a pricing scheme is shown
in Figure 14.7. Here the regulatory commission has decided that some users’ will pay a relatively high price, P1. At this
price, Q1 is demanded.
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9811439887 138
Figure 14.7
Other users (presumably those who would not buy the good at the P, price) are offered a lower price, P2. This lower price
generates additional demand of Q2 – Q1. Consequently, a total output of Q2 is produced at an average cost of A. With this
pricing system, the profits on the sales to high-price demanders (given by the rectangle P1DBA) balance the losses incurred
on the low-priced sales (BFEC). Furthermore, for the “marginal user,” the marginal cost pricing rule is being followed: It is
the “intramarginal” user who subsidizes the firm so it does not operate at a loss. Although in practice it may not be so
simple to establish pricing schemes that maintain marginal cost pricing and cover operating costs, many regulatory
commissions do use price schedules that intentionally discriminate against some users (e.g., businesses) to the advantage
of others (consumers).
Rate of return regulation
Another approach followed in many regulatory situations is to permit the monopoly to charge a price above marginal cost
that is sufficient to earn a “fair” rate of return on investment. Much analytical effort is then devoted to defining the “fair”
rate concept and to developing ways in which it might be measured. From an economic point of view, some of the most
interesting questions about this procedure concern how the regulatory activity affects the firm’s input choices. If, for
example, the rate of return allowed to firms exceeds what owners might obtain on investment under competitive
circumstances, there will be an incentive to use relatively more capital input than would truly minimize costs. And if
regulators delay in making rate decisions, this may give firms cost-minimizing incentives that would not otherwise exist.
We will now briefly examine a formal model of such possibilities:
A formal model
“ Suppose a regulated utility has a production function of the form
q = f(k, l)
This firm’s actual rate of return on capital is then define as
s=
pf (k, l ) − wl
k
(14.43)
(14.44)
where p is the price of the firm’s output (which depends on q) and w is the wage rate for labor input. If s is ‘constrained by
regulation to be equal to (say) s , then the firm’s problem is to maximize profits
 = pf (k, l) - wl − vk
(14.45)
subject to this regulatory constraint. The Lagrangian for this problem is
 = pf (k, l ) − wl − vk + [ wl + s k − pf (k, l )] (14.46)
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Notice that if  = 0, regulation is ineffective and the monopoly behaves like any profit-maximizing firm. If  = 1, Equation
14.46 reduce to
(14.47)
 = ( s − v) k
which, assuming s  v (which it must be if the firm is not to earn less than the prevailing rate of return on capital
elsewhere), means this monopoly will hire infinite amounts of capital an implausible result. Hence 0 <  < 1. The first–order
conditions for a maximum are

= pf l − w + ( w − pf l ) = 0
l

= pf k − w + ( s − pf k ) = 0
k

= wl + s k − pf (k, l ) = 0

(14.48)
The first of these conditions implies that the regulated monopoly will hire additional-labor input up to the point at which
pfl = w – a result that holds for any profit-maximizing firm.
For capital input, however, the second condition implies that
or
Because
(1 − )pf k = v −  s
v − s
 ( s − v)
pf k =
=v−
1− 
1− 
s  v and  < 1, Equation 14.50 implies
pf k  v
(14.49)
(14.50)
(14.51)
The firm will hire more capital (and achieve ‘a lower marginal productivity of capital) than it would under unregulated
conditions. Therefore, “overcapitalization” may be a regulatory induced misallocation of resources for some utilities.
Although we shall not do so here, it is possible to examine other regulatory questions using this general analytical
framework.
DYNAMIC VIEWS OF MONOPOLY
The static view that monopolistic practices distort the allocation of resources provides the principal economic rationale for
favoring antimonopoly policies. Not all economists believe that the static analysis should be definitive, however. Some
authors, most notably J.A. Schumpeter, have stressed the beneficial role that monopoly profits can play in the process of
economic development. These authors place considerable emphasis on innovation and the ability of particular types of
firms to achieve technical advances. In this context the profits that monopolistic firms earn provide funds that can be
invested in research and development. Whereas perfectly competitive firms must be content with a normal return on
invested capital, monopolies have ‘surplus' fund with which to undertake the risky process of research. More important,
perhaps, the possibility of attaining a monopolistic position or the desire to maintain such a position–provides an
important incentive to keep ‘one step ahead of potential competitors. Innovations in new products and cost saving
production techniques may be integrally related to the possibility of monopolization. Without such a monopolistic position,
the full ‘benefits of innovation could alit be obtained by the innovating firm.
Schumpeter stresses the point that the monopolization of a market may make it less costly for a firm to plan its activities.
Being the ‘only source of supply for a product eliminates many of the contingencies that a firm in a competitive market
must face. For example, a monopoly may not have to spend as much on selling ‘expenses (e.g., advertising, brand
identification, and visiting retailers) as would be the case in a more competitive industry. Similarly, a monopoly may know
more about the specific demand curve for its product and may more readily adapt to changing demand conditions; Of
course, whether any of these purported benefits of monopolies outweigh their allocational and distributional
disadvantages is an empirical question. Issues of innovation and cost savings can not be answered by recourse to a priori
arguments; detailed investigation of real-world markets is a necessity.
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IMPERFECT COMPETITION(READINGS)
This chapter discusses oligopoly markets, falling between the extremes of perfect competition and monopoly.
Definition
Oligopoly. A market with relatively few firms but more than one.
Oligopolies raise the possibility of strategic interaction among firms. To analyze this strategic interaction rigorously, we will
apply the concepts from game theory that were introduced in Chapter 8. Our game-theoretic analysis will show that small
changes in details concerning the variables firms choose, the timing of their moves, or their information about market
conditions or rival actions can have a dramatic effect on market outcomes. The first half of the chapter deals with shortterm decisions such as pricing and output, and the second half covers longer-term decisions such as investment,
advertising, and entry.
SHORT-RUN DECISIONS: PRICING AND OUTPUT
It is difficult to predict exactly the possible outcomes for price and output when there are few firms; prices depend on how
aggressively firms compete, which in turn depends on which strategic variables firms choose, how much information firms
have about rivals, and how often firms, interact with, ,each other in the market.
For example, consider the Bertrand game studied in the next section. The game involves two identical firms choosing
prices simultaneously for their identical products in their one meeting in the market. The Bertrand game has a Nash
equilibrium at point C in Figure 15.1. Even though there may be only two firms iii the market, in this equilibrium they
behave as though they were perfectly competitive, setting price equal to marginal cost and earning zero profit. We will
discuss whether the Bertrand game is a realistic depiction of actual firm behavior, but an analysis of the model shows that
it is possible to think up rigorous
game-theoretic models in which one extreme–the competitive outcome–can emerge in concentrated markets with few
firms.
At the other extreme, as indicated by point M in Figure 15.1, firms as a group may act as a cartel, recognizing that they can
affect price and coordinate their decisions. Indeed, they may be able to act as a perfect cartel and achieve the highest
possible profits– namely, the profit a monopoly would earn in the market. One way to maintain a cartel is to bind firms
with explicit pricing rules. Such explicit pricing rules are often prohibited by antitrust law. But firms need not resort to
explicit pricing rules if they interact on the market repeatedly; they can collude tacitly. High collusive prices can be
maintained with Market equilibrium under imperfect competition can occur at many points on the demand curve. In the
figure which assumes that marginal costs are constant over all output ranges, the equilibrium of the Bertrand game occurs
at point C also corresponding to the perfectly competitive outcome. The perfect cartel outcome occurs at point M also
corresponding to the monopoly outcome Many solutions may occur between points M and C, depending on the specific
assumptions made about how firms compete For example the equilibrium of the Cournot game might occur at a point such
as A The deadweight loss given by the shaded triangle increases as one moves from point C to M
FIGURE 15.1
the tacit threat of a price war if any firm undercuts. We will analyze this game formally and discuss the difficulty of
maintaining collusion.
The Bertrand and cartel models determine the outer limits between which actual prices in an imperfectly competitive
market are set (one such intermediate-price is represented by point A in Figure 15.1). This band of outcomes may be wide,
and given the plethora of available models there may be model for nearly every point within the band. For example, in a
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later section we will show how the Cournot model, in which firms set quantities rather–than prices as in the Bertrand
model, leads to an outcome (such as point A) somewhere between C and M in Figure 15.1.
It is important to know where the industry is on the line between points C and M because total welfare (as measured by
the sum of consumer surplus and firms’ profits; see Chapter 12) depends on the location of this point. At point C, total
welfare is as high as possible; at point A, total welfare is lower by the area of the shaded triangle 3. In Chapter 12, this
shortfall in total welfare relative to the highest possible level was called deadweight loss.
At point M, deadweight loss is even greater and is given by the area of shaded regions 1, 2, and 3. The closer the
imperfectly competitive outcome to C and the farther from M, the higher is total welfare and the better off society will be.
BERTRAND MODEL
The Bertrand model is named after the economist who first proposed it. The model is a game involving two identical firms,
labeled 1 and 2, producing identical products at a constant marginal cost (and constant average cost) c. The firms chose
prices p1 and p2 simultaneously in a single period of competition. Because firms products are perfect substitutes, all sales
go to the firm with the lowest price. Sales are split evenly if p1 = p2. Let D(p) be market demand.
We will look for the Nash equilibrium. The game has a continuum of actions, as does Example 8.5 (the Tragedy of the
Commons) in Chapter 8. Unlike Example 8.5, we cannot use calculus to derive best-response functions because the profit
functions are not differentiable here. Starting from equal prices, if one firm lowers its price by the smallest amount, then
its sales and profit would essentially double. We will proceed by first guessing what the Nash equilibrium is and then
spending some time to verify that our guess was in fact correct.
NASH EQUILIBRIUM OF THE BERTRAND GAME
The only pure-strategy Nash equilibrium of the Bertrand game is
p1* = p*2 = c . That is, the Nash equilibrium involves
both firms charging marginal cost. In saying that this is the only Nash equilibrium, we are making two statements that need
to- be verified:
This outcome is a Nash equilibrium, and there is no other Nash equilibrium.
To verify that this outcome is a Nash equilibrium, we need to show that both firms are playing a best response to each
other–or, in other words, that neither firm has an incentive to deviate to some other strategy. In equilibrium, firms charge
a price equal to marginal cost, which in turn is equal to average cost. But a price equal to average cost means firms earn
zero profit in equilibrium. Can a firm earn more than the zero it earns in equilibrium by deviating to some other price? No.
If it deviates to a higher price, then it will make no sales and therefore no profit, not strictly more than in equilibrium. If it
deviates to a lower price, then it will make sales but will be earning a negative margin on each unit sold because price
would be below marginal cost. Thus, the firm would earn negative profit, less than in equilibrium. Because there is no
possible profitable deviation for the firm, we have succeeded in verifying that both firms’ charging marginal cost is a Nash
equilibrium.
It is clear that marginal cost pricing is the only pure-strategy Nash equilibrium. If prices exceeded marginal cost, the highprice firm would gain by undercutting the other slightly and capturing all the market demand. More formally, to verify that
p1* = p*2 = c is the only Nash equilibrium, we will go one by one through an exhaustive list of cases for various values of
p1, p2, and c, verifying that none besides p1 = p2 = c is a Nash equilibrium. To reduce the number of cases, assume firm 1 is
the low-price firm–that is, p1  p2. The same conclusions would be reached taking 2 to be the low-price firm.
There are three exhaustive cases: (1) c > p1 (ii) c < p1, and (iii) c = p1.
Case (i) cannot be a Nash equilibrium. Firm 1 earns a negative margin p1 – c on every unit it sells, and because it makes
positive sales, it must earn negative profit. It could earn higher profit by deviating to a higher price. For example, firm 1
could guarantee itself zero profit by deviating to p1 = c.
Case (ii) cannot be a Nash equilibrium either. At best, firm 2 gets only half of market demand (if p 1 = p2) and at worst gets
no demand (if p1 < p2). Firm 2 could capture all the market demand by undercutting firm l’s price by a tiny amount  . This 
could be chosen small enough that market price and total market profit are hardly affected. If p1 = p2 before the deviation,
the deviation would essentially double firm 2’s profit. If p1 < p2 before the deviation, the deviation would result in firm 2
moving from zero to positive profit. In either case, firm 2’s deviation would be profitable.
Case (iii) includes the sub case of p1 = p2 = c, which we saw is a Nash equilibrium. The only remaining sub case in which p1 
p2 is c = p1 < p2. This sub case cannot be a Nash equilibrium: Firm 1 earns zero profit here but could earn positive profit by
deviating to a price slightly above c but still below p2.
Although the analysis focused on the game with two firms, it is clear that the same outcome would arise for any number of
*
*
*
firms n  2. The Nash equilibrium of the n-firm Bertrand game is p1 = p 2 = .... = p n = c.
Bertrand Paradox
The Nash equilibrium of the Bertrand model is the same as the perfectly competitive outcome. Price it set to marginal cost,
and firms earn zero profit. This result–that the Nash equilibrium in the Bertrand model is the same as in perfect
competition even though there may be only two firms in the market–is called the Bertrand paradox It is paradoxical that
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competition between as few as two firms would be so tough. The Bertrand paradox is a general result in the sense that we
did not specify the marginal cost c or the demand curve; therefore, the result holds for any c and any downward-sloping
demand curve.
In another sense, the Bertrand paradox is not general; it can be undone by changing various of the model’s other
assumptions. Each of the next several sections will present a different model generated by changing a different one of the
Bertrand assumptions. In the next section, for example, we will assume that firms choose quantity rather than price,
leading to what is called the Cournot game. We will see that firms do not end up charging marginal cost and earning zero
profit in the Cournot game. In subsequent sections, we will show that the Bertrand paradox can also be avoided if still
other assumptions, are changed: if firms face capacity constraints rather than being able to produce an unlimited amount:
at cost c, if products are slightly differentiated rather than being perfect substitutes, or if firms engage in repeated
interaction rather than one round of competition.
COURNOT MODEL
The Cournot model, named after the economist who proposed it, is similar to Bertand model except that firms are
assumed to simultaneously choose quantities rather than prices. As we will see, this simple change in strategic variable will
lead to a big change iq implications. Price will be above marginal cost, and firms will earn positive profit in 4ie Nash
equilibrium of the Cournot game. It is somewhat surprising (but nonetheless an important point to keep in mind) that this
simple change in choice variable matters iii the strategic setting of an oligopoly when it did not matter with a monopoly:
The monopolist obtained the same profit-maximizing outcome whether it chose prices or quantities.
We will start with a general version of the Cournot game with n firms indexed by
i = 1, …, n. Each firm chooses its output qi of an identical product simultaneously. The outputs are combined into a total
industry output Q = q1 + q2 + … + qn, resulting in market price P(Q). Observe that P(Q) is the inverse, demand curve corresponding to the market demand curve Q = D(P). Assume market demand is downward sloping and so inverse demand is,
too; that is, P'(Q) < 0. Firm i’s profit equals its total revenue, P(Q)q1, minus its total cost, Ci(qi):
i = P (Q)q i − C i (q i )
(15.1)
Nash equilibrium of the Cournot game
Unlike the Bertrand game, the profit function (15.1) in the Cournot game is differentiable; hence we can proceed to solve
for the Nash equilibrium of this game just as we did in Example 8.5, the Tragedy of the Commons; That is, we find each firm
i's best response by taking the first-order condition of the objective function (15.l)with respect to qi;
i
= P(Q) + P' (Q)q i − Ci (q i ) = 0
q i          
(15.2)
MR
MC
Equation 15.2 must hold for all i = 1,... ,n in the Nash equilibrium.
According to Equation 15.2, the familiar condition for profit maximization from
Chapter 11–marginal revenue (MR) equals marginal cost (MC)–holds for the Cournot firm. As we will see from an analysis
of the particular form that the marginal revenue term takes for the Cournot firm, price is above the perfectly competitive
level (above marginal cost) but below the level in a perfect cartel that maximizes firms’ joint profits.
In order for Equation 15.2 to equal 0, price must exceed marginal cost by the magnitude of the “wedge” term P’(Q)qi. If the
Cournot firm produces another’ unit on top of its existing production of q1 units, then, because demand is downward
sloping, the additional unit causes market price to decrease by P'(Q), leading to a loss of revenue of P’(Q)qi (the wedge
term) from firm i’s existing production.’
To compare the Cournot outcome with the perfect cartel outcome, note that the objective for the cartel is to maximize
joint profit:
n
n
n
(15.3)
 j = P (Q) q j − C j (q j )
j=1
j=1
j=1
Taking the first-order condition of Equation 15.3 with respect to qi gives



n
  n 
  j = P(Q) + P' (Q) q j − C i(qi ) = 0
q i  j=1 
MC
     j=1
(15.4)
MR
This first-order condition is similar to Equation 15.2 except that the wedge term,
n
P ' (Q)  q j = P ' (Q)Q
(15.5)
j=1
is larger in magnitude. with a perfect cartel than with Cournot firms. In maximizing joint profits, the cartel accounts for the
fact that an additional unit of firm i’s output, by reducing market price, reduces the revenue earned on all firms’ existing
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output. Hence P’(Q) is multiplied, by total cartel output Q in Equation 15.5. The Cournot firm accounts for the reduction in
revenue only from its own existing output qj. Hence Cournot firms will end up overproducing relative to the joint profitmaximizing outcome. That is, the extra production’ in the Cournot outcome relative to a perfect cartel will end up in lower
joint profit for the firms. What firms would regard as overproduction is good for society because it means that the Cournot
outcome (point A, referring back to Figure 15.1) will involve more total welfare than the perfect cartel outcome (point M in
Figure 15.1).
Example 15.1 : Natural–Spring Duopoly
As a numerical example of some of these ideas, we will consider a case with just two firms and simple demand and cost
functions, Following Cournot’s nineteenth-century example of two natural springs,’ we assume that each spring owner has
a large supply of (possibly healthful) water and faces the problem of how much to provide the market. A firm’s cost of
pumping and bottling q1 lifers is Ci(qi) = cqi implying that marginal costs are a constant c per liter, Inverse demand for Spring
water is
P (Q) = a − Q
(15.6)
where a is the demand intercept (measuring the strength of spring water demand) and
Q = q1 + q 2
is total spring water output. We will now examine various models of how this market might operate
Bertrand model. In the Nash equilibrium of the Bertrand game, the two firms set price equal to marginal cost. Hence
*
*
*
market price is P = c , total output is Q = a − c , firm profit is i = 0 , and total profit for all firms  = 0 .
For the Bertrand quantity to be positive we must have a > c, which we will assume throughout the problem.
Cournot model. The solution for the Nash equilibrium follows Example 8.6 closely. Profits for the two Cournot firms are
 =P(Q)q1 – cq1 = (a – q1 – q2 – c)q1,
(15.7)
 =P(Q)q2 – cq2 = (a – q1 – q2 – c)q2,
Using the first-order conditions to solve for the best-response functions, we obtain
q1 =
a − q2 − c
,
2
q2 =
a − q1 − c
2
(15.8)
Solving Equations 15.8 simultaneously yields the Nash equilibrium
q1* = q*2 =
a −c
2
(15.9)
Thus, total output is Q* = (2/3)(a – c). Substituting total output into the inverse demand curve implies an equilibrium price
of P* = (a + 2c)/3. Substituting price and outputs into the profit functions (Equations 15.7) implies
2
1
1* = *2 =  (a − c) 2 , so total market profit equals  * = 1* = *2 = (a − c) 2 .
9
9
Perfect cartel. The objective function for a perfect cartel involves joint profits
1 + 2 = (a – q1 – q2 – c)q1 + (a – q1 – q2 – c)q2
(15.10)
The two first-order conditions for maximizing Equation 15.10 with respect to q1 and q2 are the same:


(1 +  2 ) =
(1 +  2 ) = a − 2q1 − 2q 2 − c = 0
q1
q 2
(1511)
The first-order conditions do not pin down market shares for firms in a perfect cartel because they produce identical
1
q1* + q*2 = Q* = (a − c) .
2
1
*
Substituting total output into inverse demand implies that the cartel price is P = (a + c) .
2
1
*
2
Substituting price and ‘quantities into Equation 15.10 implies a total cartel profit of  = (a − c) .
4
products at constant marginal cost. But Equation 15.11 does pin down total output:
Comparison. Moving from the Bertrand model to the Cournot model to a perfect cartel, because a > c we can show that
quantity Q* decreases from a – c to (2/3)(a – c) to
(1/2)(a – c). it can also be shown that price P* industry profit  increase. For example, if
a = 120 and c = 0 (implying that inverse demand is P(Q) = 120 – Q and that production is costless), then market quantity is
120 with Bertrand competition, 80 with Cournot competition, and 60 with a perfect cartel. Price increases from 0 to 40 to
60 across the cases, and industry profit increases from 0 to 3,200 to 3,600.
QUERY : In a perfect cartel, do firms play a best response to each other's quantities? if not, in which direction would they
like to change their outputs? What does this say about the stability of cartels ?
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EXAMPLE 15.2 Cournot Best–Response Diagram
Continuing with the natural-spring duopoly from Example 15.1, it is instructive to solve for the Nash equilibrium using in
graphical methods. We will graph the best response functions given in Equation 15.8, the intersection between the best
responses is the Nash equilibrium. As background you may want to review a similar diagram (Figure 8.8) for the Tragedy of
the Commons.
The linear best–response functions are most easily graphed by plotting their intercepts, as shown in Figure 15.2. The bestresponse functions intersect at the point
q1* = q *2 =
( a − c)
. which was the Nash equilibrium of the Cournot game
3
computed using algebraic methods in Example 15.1.
Figure 15.2
Figure 15.2 displays firms isoprofit curves An isoprofit curve for firm 1 is the locus of quantity pairs providing it with the
same profit level To compute the isoprofit curve associated with a profit level of (say) 100 we start by setting Equation 15.7
equal to 100
1 = (a − q1 − q 2 − c)q1 = 100
(15.12)
Then we solve for q2 to facilitate graphing the isoptofit
q 2 = a − c − q1 −
100
q1
(15.13)
Several example isoproilts for firm 1 are shown in the figure. As profit increases from 100 to 200 to yet higher levels the
associated isoprofits shrink down to the monopoly point which is the highest isoprofit on the diagram To understand why
the individual isoprofits are shaped like frowns refer back to Equation 1513 As q, approaches 0 the last term
(–100/q1) dominates causing the left side of the frown to turn down. As q1 increases the
–q1 term in Equation 15.13 begins, to dominate, causing the tight side of the frown to turn down.
Figure 15 3 shows how to use best response diagrams to quickly tell how changes in such underlying parameters as the
demand intercept a or marginal cost c would affect the equilibrium Figure 15.3a depicts an increase in both firms marginal
cost c The best responses shift inward resulting in a new equilibrium that involves lower out put for both Although firms
have the same marginal cost in this example, one can imagine a model in which firms have different marginal cost
parameters and, so can be varied independently. Figure 15.3b depicts an increase in just firm l’s marginal cost; only firm
1’s best response shifts. The new equilibrium involves lower output for firm 1 and higher output for firm 2. Although firm
2’s best response does not shift, it still increases its output as it anticipates a reduction in firm 1’s output and best responds
to this anticipated output reduction.
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QUERY: Explain why firm l’s individual isoprollts reach a peak on its best-response function in Figure 15.2. What would firm
2’s isoprofits look like in Figure 15.2? How would you represent an increase in demand intercept a in Figure 15.3?
Figure 15.3
Varying the number of Cournot firms
The Cournot model is particularly useful for policy analysis because it can represent the whole range of outcomes from
perfect competition to perfect cartel/monopoly (i.e., the whole range of points between C and M in Figure 15.1) by varying
the number of firms n from n =  to n = 1. For simplicity, consider the case of identical firms, which here means the n firms
sharing the same cost function C(qi). In equilibrium, firms will produce the same share of total output: qi = Q/n.
Substituting qi = Q/n into Equation 15.12, the wedge term becomes P’(Q)Q/n, The wedge term disappears as n grows large;
firms become infinitesimally small. An infinitesimally small firm effectively becomes a price-taker because it produces so
little that any decrease in market price from an increase in output hardly affects its revenue. Price approaches marginal
cost and the market outcome approaches the perfectly competitive one. As n decreases- to 1, the wedge term approaches
that in Equation 15.5, implying the Cournot outcome approaches that of a petfect cartel.
As the Cournot firm’s market share grows, it internalizes the revenue loss from a decrease
ill market price to a greater extent.
Example 15.3 Natural–Spring Oligopoly
Return to the natural spring in Example 15.1, but now consider a variable number n of firms rather than just two. The proflt
of one of them, firm 1, is
i = P(Q)q i − cq i = (a − Q − c)q i = (a − q i − Q −i − c)q i
(15.14)
It is convenient to express total output as Q = qi + Q–i, where Q–i = Q – qi is the output of all firm except for i. Taking the
first-order condition of Equation 15.14 with respect to qi, we recognize that firm i takes Q–i as a given and thus treats it as a
constant in the differentiation,
i
= a − 2q i = Q −i − c = 0
q i
(15.15)
which holds for all i = 1,2,…,n.
They to solving the system of n equations for the n equilibrium quantities is to recognize that the Nash equilibrium involves
equal quantities because firms are symmetric. Symmetry implies that
Q*−1 = Q* − q*i = q*i = (n − 1)q*i
(15.16)
Substituting Equation 15.16 into 15.15 yields
a − 2q*i − (n − 1)q*i − c = 0
(15.17)
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or
q*i = (a − c) /(n + 1) .
Total market output is
 n 
Q* = nq*i = 
( a − c )
 n +1
(15.18)
and market price is
 1   n 
P * = a − Q* = 
a + 
c
 n +1  n +1
*
*
*
Substituting for q i , Q and P into the firm’s profit Equation 15.14, we have that total profit for all firm is
 =
*
n*i
a −c
= n

 n +1 
(15.19)
2
(15.20)
Setting n = 1 in Equations 15.18–15.20 gives the monopoly outcome, which gives the same price total output and profit as
in the perfect cartel case computed in Example 15.1. Letting n grow without bound in Equations 15.18-15.20 gives the
perfectly competitive outcome, the me same output computed in Example 15.1 for the Bertrand case.
QUERY : We used the trick of imposing symmetry after taking the first-order condition for firm is quantity choice. It might
seem simpler to impose symmetry before taking the first-order condition.
Why would this be a mistake? How would the incorrect expressions for quantity, price, and profit compare with the
correct ones here?
Prices or quantities?
Moving from price competition in the Bertrand model to quantity competition in the Cournot model changes the market
outcome dramatically. This change is surprising on first thought. After all, the monopoly outcome from Chapter 14 is the
same whether we assume the monopolist sets price or quantity. Further thought suggests why price and quantity are such
different strategic variables. Starting from equal prices, a small reduction in one firm’s price allows it to steal all the market
demand from its competitors. This sharp benefit from undercutting makes price competition extremely “tough.” Quantity
competition is “softer.” Starting from equal quantities, a small increase in one firm’s quantity has only a marginal effect on
the revenue that other firms receive from their existing output. Firms have less of an incentive to out produce each other
with quantity competition than to undercut each other with price competition.
An advantage of the Cournot model is its realistic implication that the industry grows more competitive as the number n of
firms entering the market increases from monopoly to ‘perfect competition. In the Bertrand model there is a discontinuous
jump from monopoly to perfect competition if just two firm enter, and additional entry beyond two has no additional
effect on the market outcome.
An apparent disadvantage of the Cournot model is that firms in real-world markets tend to set prices rather than
quantities, contrary to the Cournot assumption that firms choose quantities. For example, grocers advertise prices for
orange juice, say, $3.00 a container, in new paper circulars rather than the number of containers it stocks. As we will see in
the next section, the Cournot model applies even to the orange juice market if we reinterpret quantity to be the firm’s
capacity, defined as the most the firm can sell given the capital it has in place and other available inputs in the short run.
CAPACITY CONSTRAINTS
For the Bertrand model to generate the Bertrand paradox (the result that two firms essentially behave as perfect
competitors), firms roust have unlimited capacities. Starting from equal prices, if a firm lowers its price the slightest
amount, then its demand essentially doubles. The firm can satisfy this increased demand because it has no capacity
constraints, giving firms a big incentive to undercut. If the undercutting firm could not serve all the demand at its lower
price because of capacity constraints, that would leave some residual demand for the higher-priced firm and would
decrease the incentive to undercut.
Consider a two-stage game in which firms build capacity in the first stage and firms choose
prices p1 and p2 in the
second stage. Firms cannot sell more in the second stage than the capacity built in the first stage. If the cost of building
capacity is sufficiently high, it turns out that the subgame - perfect equilibrium of this sequential game leads to the same
outcome as the Nash equilibrium of the Cournot model.
To see this result, we will analyze. the game using backward induction. Consider the second-stage pricing game supposing
q1 and q 2 in the first stage. Let
production is at capacity for both firms. A situation in which -,
the firms have already built capacities
p
be the price that would prevail when
p1 = p 2  p
(15.21)
is not a Nash equilibrium. At this price, total quantity demanded exceeds total capacity;
therefore, firm I could increase its profits by raising price slightly and continuing to sell
VIKAAS WADHWA SIR
q1 .
9811439887 147
Similarly,
p1 = p 2  p
(15.22)
is not a Nash equilibrium because now total sales fall short of capacity At least one firm (say, firm 1) is selling less than its
capacity. By cutting price slightly, firm l can increase its profits by selling up to its capacity,
q1 . Hence the Nash
equilibrium of this second stage game is for firms to choose the- price at which quantity demanded exactly equals the total
capacity built in the first stage:
p1 = p 2 = p
(15.23)
Anticipating that the price will be set such that firms sell all their capacity, the first stage capacity choice game is essentially
the same as the Cournot game. Therefore, the equilibrium quantities, price, and profits will be the same as in the Cournot
game. Thus, even in markets (such as orange juice sold in grocery stores) where it looks like firms are setting prices, the
Cournot model may prove more realistic than it first seems.
PRODUCT DIFFERENTIATION
Another way to avoid the Bertrand paradox is to replace the assumption that the firms’ products are identical with the
assumption that firms produce differentiated products. Many (if not most) real-world markets exhibit product
differentiation. For example, toothpaste brands vary some what from supplier to supplier–differing in flavor, fluoride
content, whitening agents, endorsement from the American Dental Association, and so forth. Even if suppliers’ product
attributes are similar, suppliers may still be differentiated in another dimension: physical location. Because demanders will
be closer to some suppliers than to others, they may prefer nearby sellers because buying from them involves
less travel time.
Meaning of “the market”
The possibility of product differentiation introduces some fuzziness into what we mean by the market for a good. With
identical products, demanders were assumed to be indifferent about which firm’s output they bought; hence they shop at
the lowest-price firm, leading to the law of one price. The law of one price no longer holds if demanders strictly prefer one
supplier to another at equal prices. Are green-gel and white-paste toothpastes in the same market or in two different
Ones? Is a pizza parlor at the outskirts of town in the same market as one in the middle of town?
With differentiated products, we will take the market to be a group of closely related products that are more substitutable
among each other (as measured by cross-price elasticities) than with goods outside the group. We will be somewhat loose
with this definition, avoiding precise thresholds for how high the cross-price elasticity must be between goods within the
group (and how-low with outside goods). Arguments about which goods should be included in a product group often
dominate antirust proceedings, and we will try to avoid this contention here.
Bertrand competition with differentiated products
Return to the Bertrand model but now suppose there are n firms that simultaneously choose prices pi (i = 1,…, n) for their
differentiated products. Product i has its own specific attributes ai, possibly reflecting special options, quality, brand
advertising, or location A product may be endowed with the attribute (orange juice is by definition made from oranges and
cranberry juice from cranberries), or the attribute may be the result of the firm’s choice and spending level (the orange
juice supplier can spend more and make its juice from fresh oranges rather than from frozen concentrate). The various
attributes serve to differentiate the products. Firm i’s demand is
q i (pi , P−i , a i , A −i )
(15.24)
where P–i is a list of all other firms’ prices besides i’s, and A–i is a list of all other firms’
attributes besides i’s. Firm i’s total cost is
ci (q i , a i )
(15.25)
and profit is thus
i = p i q i − C i ( q i , a i )
(15.26)
With differentiated products, the profit function (Equation 15.26) is differentiable, so we do not need to solve for the Nash
equilibrium on a case-by-case basis as we did in the Bertrand model with identical products. We can solve for the Nash
equilibrium as in the Cournot model, solving for best-response functions by taking each firm’s first-order condition (here
with respect to price rather than quantity). The first-order condition from
Equation 15.26 with respect to pi is
i
q C q
= q i + pi i − i . i = 0
pi
p q p
    i  i  i
(15.27)
A
B
The first two terms (labeled A)on the right side of Equation 15.27 are a sort of marginal revenue–not the usual marginal
revenue from an increase in quantity, but rather the marginal revenue from an increase in price. The increase in price
VIKAAS WADHWA SIR
9811439887 148
increases revenue on existing sales of qi units, but we must also consider the negative effect of the reduction in sales (
q i / pi multiplied by the price pi) that would have been earned on- these sales. The last term, labeled B, is the cost
savings associated with the reduced sales that accompany an increased price.
The Nash equilibrium can be found by simultaneously solving the system of first-order conditions in Equation 15.27 for all i
= 1, ..., n. If the attributes a, are also choice variables (rather than just endowments), there will be another set of first-order
conditions to consider. For firm i, the first-order condition with respect to ai has the form
i
q C C q
= pi i − i − i . i = 0
a i
a i a i q i a i
(15.28)
The simultaneous solution of these first–order conditions can be complex, and they yield few definitive conclusions about
the nature of market equilibrium. Some insights from particular cases will be developed in the next two examples.
EXAMPLE 15.4 Toothpaste as a Differentiated Product
Suppose that two firms produce toothpaste, one a green gel and the other a white paste. To simplify the calculation,
suppose that production is costless. Demand for product i is
q i = a i − pi +
pj
(15.29)
2
The positive coefficient on pj, the other good’s price, indicates that the goods are gross substitutes. Firm is demand is
increasing in the attribute ai, which we will take to be demanders’ inherent preference for the variety in question; we will
suppose that this is an endowment rather than a choice variable for the firm (and so will abstract from the role of
advertising to promote preference for a variety).
Algebraic solution. Firm i’s profit is
pj 

(15.30)
i = pi q i − Ci (q i ) = pi  a i − pi + 
2

where Ci (q i ) = 0 because i’s production is costless. The first-order condition for profit maximization with respect to pi
is
pj
i
= a i − 2p i + = 0
pi
2
(15.31)
Solving for pi gives the following best response functions for i = 1, 2
1
p 
p1 =  a1 + 2 ,
2
2
1
p 
p2 =  a 2 + 1 
2
2
(15.32)
Solving Equation 15.32 simultaneously gives the Nash equilibrium prices
p*i =
8
2
ai + q j
15
15
(15.33)
The associated profit are
*i
2 
8
=  ai + a j 
15 
 15
2
(15.34)
Firm i's equilibrium price is not only increasing in its own attribute, ai, but also in the other product's attribute, aj. An
increase in aj causes firm j to increase its price, which increases firm is demand and thus the price i charges.
Graphical solution. We could also have solved for equilibrium prices graphically, as in Figure 15.4. The best responses in
Equation 15.32 are upward sloping. They intersect at the’ Nash equilibrium, point E. The isoprofit curves for firm I are
smile-shaped. To see this, take the expression for firm 1’s profit in Equation 15.30, set it equal to a certain profit level (say,
100), and solve for p2 to facilitate graphing it on the best-response diagram. We have
p2 =
100
+ p1 − a
p1
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(15.35)
9811439887 149
Figure 15.4
EXAMPLE 15.5 : Hotelling's Beach
A simple model in which identical products are differentiated because of the location their suppliers (spatial
differentiation) was provided by H. Hotelling in the 1920s. As shown in Figure 15.5, two ice cream stands, labeled A anti B,
are located along a beach of length L. The stands make identical ice cream cones, which for simplicity are assumed to be
costless to produce. Let a and b represent the firms’ locations on the beach. (We will take the locations of the ice cream
stands as given; in a later example we will revisit firms’ equilibrium location choices.) Assume that demanders are located
uniformly along the beach, one at each unit of length. Carrying ice cream a distance d back to one’s beach umbrella costs
td because ice cream melts more the higher the temperature t and the further one must walk. Consistent with the
Bertrand assumption, firms choose prices pA and pB simultaneously.
Determining demands. Let x be the location of the consumer who is indifferent between buying from the two ice cream
stands. The following condition must be satisfied by x:
p A + t ( x − a ) 2 = p B + t (b − x ) 2
(15.36)
Figure 15.5
The left side of Equation 15.36 is the generalized cost of buying from A (including the price paid and the cost of
transporting the ice cream the distance x – a). Similarly, the right side is the generalized cost of buying from B. Solving
Equation 15.36 for x yields
x=
b + a pB − pA
+
2
2t (p − a )
(15.37)
If price are equal, the indifferent consumer is located midway between a and b. If A’s price is less than B's, then x shifts
toward endpoint L. (This is the case shown in Figure 15.5.)
Because all demander between 0 and x buy from A and because there is one consumer per unit distance, it follows that A’s
demand equals x:
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9811439887 150
q A ( p A , p B , a , b) = x =
b + a pB − pA
+
2
2t (b − a )
The remaining L – x consumers constitute B’s demand:
q B ( p B , p A , a , b) = L − x = L −
b + a pA − pB
+
2
2t (b − a )
(15.38)
(15.39)
Solving for Nash equilibrium. The Nash equilibrium is found in the same way as in Example 15.4 except that, for demands,
we use Equations 15.38 and 1539 in place of Equation 15.29. Skipping the details of the calculations, the Nash equilibrium
prices are
t
p*A = (b − a )(2L + a + b)
3
(15.40)
These prices will depend on the precise location of the two stands and will differ from each
other. For example, if we assume that the beach is L = 100 yards long, a = 40 yards, b = 70 yards, and t = $0.O,0l (one tenth
*
*
of a penny), then p A = $3.10 and p B = $2.90 . These price differences arise only from the locational aspects of
this problem–the cones themselves are identical and costless to produce. Because A is somewhat more favorably located
than B, it can charge a higher price for its cones without losing too much business to B. Using Equation 15.38 shows that
x=
110
3.10 − 2.90
+
 52
2 (2)(0.001)(110)
(15.41)
so stand A sells 52 cones whereas B sells only 48 despite its lower price At point x, the consumer is indifferent between
walking the 12 yards to A and paying $3.10 or walking 18 yards to B and paving $2.90 The equilibrium as inefficient in that
a consumer slightly to the tight of x would incur a shorter walk by patronizing A but still chooses B because of A's power to
set higher prices
Equilibrium profits are
t
(b − a )(2L + a + b) 2 ,
18
t
*B = (b − a )(4L − a − b) 2
18
*A =
(15.42)
Somewhat surprisingly, the ice cream stands benefit from faster melting as measured here by the transportation cost t. For
*
*
example if we take L = 100, a = 40, b = 70 and t = $0.001 as in the previous paragraph then  A = $160 and  B = $140
*
(rounding to the nearest dollar) If transportation costs doubled to t = $0.002 then profits would double to  A = $320 and
*B = $280
The transportation/melting cost is the only source of differentiation in the model If t = 0, then we can see from Equation
15.40 that prices equal 0 (which is marginal cost given that production is costless) and from Equation 15.42 that profits
equal 0 in other words, the Bertrand paradox results.
QUERY: What happens to prices and profits if ice cream stands locate in the same spot? If they locate at the opposite ends
of the beach?
Consumer search and price dispersion
Hotelling’s model analyzed in Example 15.5 suggests the possibility that competitors may have some ability to charge
prices above marginal cost and earn positive profits even if the physical characteristics of the goods they sell are identical.
Firms’ various locations– closer to some demanders and farther from others–may lead to spatial differentiation. The
Internet makes, the physical location of stores less relevant to consumers, especially if shipping charges are independent of
distance (or are not assessed). Even in this setting, firms can avoid the Bertrand paradox if we drop the assumption that
demanders know every firm’s price in the market. Instead we will assume that demanders face a small cost s, called a
search cost, to visit the store (or click to its website) to find its price.
Peter Diamond, winner of the Nobel Prize in economics in 2OIO, developed a model in which demanders search by picking
one of the n stores at random and learning its ‘price. Demanders know the equilibrium distribution of prices but not which
store is charging which price. Demanders get their first price search for free but then must pay s for additional searches.
They need at most one unit of the good, and they all have the same gross surplus v for the one unit.
Not only do stores manage to avoid the Bertrand paradox in this model, they obtain the polar opposite outcome: All
charge the monopoly price v, which extracts all consumer surplus! This outcome holds no matter how small the search cost
s is–as long as s is positive (say, a penny). It is easy to see that all stores charging v is an equilibrium. if all charge the same
VIKAAS WADHWA SIR
9811439887 151
price v, then demanders may as well buy from the first Store they search because additional searches are costly and do not
end up revealing a lower price. It can also be seen that this is the only equilibrium. Consider any outcome in which at least
one store charges less than v, and consider the lowest-price store (label it i) in this outcome.
Store i could raise its price Pi by as much as s and still make all the sales it did before. The lowest price a demander could
expect to pay elsewhere is no less than pi, and the demander would have to pay the costs to find this other price.
Less extreme equilibria are found in models where consumers have different search costs. For example, suppose one group
of consumers can, search for, free and another group has to pay s per search. In equilibrium, there will be some price
dispersion across stores. One set of stores serves the low–search-cost demanders (and the lucky high– search-cost
consumers who-happen to stumble on a bargain). These bargain stores sell at marginal cost. The other stores serve the
high-search-cost demanders at a price that makes these demanders indifferent between buying immediately and taking a
chance that the next price search will uncover a bargain store.
TACIT COLLUSION
In Chapter 8, we showed that players may be able to earn higher payoffs in the subgame perfect equilibrium of an
infinitely repeated game than from simply repeating the Nash equilibrium from the single-period game indefinitely. For
example, we saw that, if players are patient enough, they can cooperate on playing silent in the infinitely repeated version
of the Prisoners’ Dilemma rather than finking on each other: each period. From the perspective of oligopoly theory, the
issue is whether firms must endure the Bertrand paradox (marginal cost pricing and zero profits) in each period of a
repeated game or whether they might instead achieve more profitable outcomes through tacit collusion.
A distinction should be drawn between tacit collusion and the formation of an explicit cartel. An explicit cartel involves
legal agreements enforced with external sanctions if the agreements (e.g., to sustain high prices or low outputs) are
violated. Tacit collusion can only be enforced through punishments internal to the market–that is, only those that can be
generated within a subgame-perfect equilibrium of a repeated game. Antitrust laws generally forbid the formation of
explicit cartels, so tacit collusion is usually the only way
for firms to raise prices above the static level.
Finitely repeated game
Taking the Bertrand game to be the stage game, Selten’s theorem from Chapter 8 tells us that repeating the stage game
any finite number of times T does not change the outcome. The only subgame-perfect equilibrium of the finitely repeated
Bertrand game is to repeat the stage-game Nash equilibrium–marginal cost pricing–in each of the pea-kids. The -game
unravels through backward induction. In any subgame staring in period T, the unique Nash equilibrium will be played
regardless of what happened before. Because the outcome in period T – 1 does not affect the outcome in the next period,
it is as though period T – 1 is the last period, and the unique Nash’ equilibrium must be played then, too. Applying
backward induction, the game unravels in this manner all the way back to the first period.
Infinitely repeated game
If the stage game is repeated infinitely many periods, however, the folk theorem applies. The folk theorem indicates that
any feasible and individually rational payoff can be sustained each period in an infinitely repeated game as long as the
discount factor, , is close enough to unity. Recall that the discount factor is the value in the present period of one event
and a mixed-strategy equilibrium giving her the lowest payoff of all three). In the sequential version of the game, if a player
were given the choice between being the first mover and having the ability to commit to attending an event or being the
second mover and having the flexibility to be able to meet up With the first wherever he or she showed up, a player would
always choose the ability to commit. The first mover can guarantee his or her preferred outcome as the unique subgameperfect equilibrium by committing to attend his or her favorite event.
Sunk costs
Expenditures on irreversible investments are called sunk costs.
Sunk cost. A sunk cost is an expenditure on an investment that-cannot be reversed and has no resale value.
Sunk costs include expenditures on unique types- of equipment (e.g., a newsprint-making machine) or job specific training
for workers (developing the skills to use the newsprint machine). There ia sometimes confusion between sunk costs and
what we have called fixed costs. They are similar in -that they do not vary with the firm’s output level in a production
period and are incurred even if no output is produced in that period. But instead of being incurred periodically, as are
many faxed costs (heat for the factory, salaries for secretaries and other administrators), sunk costs are incurred only once
in connection with a single investment. Some fixed costs may be avoided over a sufficiently long run–say, by reselling the
plant and equipment involved–but sunk costs can never be recovered because the investments involved cannot be moved
to a different use. When the firm makes a sunk investment, it has committed itself to that investment, and this may have
important consequences for its strategic behavior.)
First-mover advantage in the Stackelberg model
The simplest setting to illustrate the first-mover advantage is in the Stackelberg model, named after the economist who
first analyzed it. The model is similar to a duopoly version of the Cournat model except that rather than simultaneously
VIKAAS WADHWA SIR
9811439887 152
choosing ,the quantities of their identical outputs–firms move sequentially, with firm 1 (the leader) choosing its output first
and-then firm 2 (the follower) choosing after observing firm l’s output.
We use backward induction to solve for the subgame-perfect equilibrium of this sequential game. Begin with the follower’s
output choice. Firm 2 chooses the output q2 that maximizes its own profit, taking firm 1’s output as -given. In other words;
firm 2 best responds to firm 1’s output. This results in the same best-response function for firm 2 as we computed in the
Cournot game from the first-order condition (Equation-15.2). Label this best-response function BR2(q1).
Turn then to the leader’s output choice. Firm 1 recognizes that it can influence the follower’s action because the follower
best responds to l’s observed output, Substituting BR2(q1) into the profit function for firm 1 given by Equation 15.1, we
have
1 = P(q1 + BR 2 (q1 ))q1 − C1 (q1 )
(15.56)
The first-order condition with respect to q2 is
1
= P(Q) + p(Q)q1 + P(Q)BR 2 (q1 )q1 − Ci (q i ) = 0.
      
q1
(15.57)
S
This is the same first-order condition computed in the Cournot model (see Equation 15.2) except for the addition of the
term S, which accounts for the strategic effect of firm 1’s output on firm 2’s. The strategic effect S will lead firm 1 to
produce more than it would have in a Cournot model. By overproducing, firm 1 leads firm 2 to reduce q2 by the amount
BR 2 (qi); the fall in firm 2’s output increases market price, thus increasing the revenue that farm 1 earns on its existing
sales. We know that q2 decreases with an increase in q1 because best-response functions under quantity competition are
generally downward sloping; see Figure 15.2 for an illustration.
The strategic effect would be absent if the leader’s output choice were unobservable to the follower or if the leader could
reverse its output choice in secret. The leader must be able to commit to an observable output choice or else firms are
back in the Cournot game. It is easy to see that the leader prefers the Stackelberg game to the Cournot game. The leader
could always reproduce -the outcome from the Cournot game by choosing its Cournot output in the Stackelberg game. The
leader can do even better by producing more than its Cournot output, thereby taking advantage of the strategic effect S.
EXAMPLE 15.8 : Staclelberg Springs
Recall the two natural spring owners from Example 15.1. Now rather than having them choose outputs simultaneously as
in the Cournot game assume that they choose outputs sequentially as in the Sktackelberg game, with firm 1 being the
leader and firm 2 the follower.
Firm 2’s output. We will solve for the subgame perfect e1 using backward induction, starting with firm 2's output choice
We already found firm 2 s best response function to Equation 15.8 repeited here:
q2 =
a − q1 − c
2
(15.58)
Firm 1's output. Now fold the game back to solve for firm 1's output choice. Substituting firm 2's best response from
Equation 15.58 into firm 1's profit function from Equation 15.56 yields
1

 a − q1 − c  
i = a − q1 − 
 − c q1 = (a − q1 − c)q1
2
2

 

(15.9)
Taking e first-order condition,
1 1
= (a − 2q1 − c) = 0
q1 2
(a − c)
(a − c)
*
*
and solving gives q1 =
. Substituting q1 back into firm 2's best response function gives q 2 =
2
4
1
*
* 1
2
2
are 1 =  (a − c) and  2 = 
( a − c )
8
 16 
*
*
*
To provide a numerical example suppose a = 120 and c = 0. Then q1 = 60, q 2 = 30 , 1 = $1,800 and
(15.60)
Profits
*2 = $900 . Firm 1 produces twice as much and earns twice as much as firm 2.
Recall from the simultaneous Cournot game in Example 15.1 that for these numerical values, total market output was 80
and total industry profit was 3,200 implying that each of the two firms produced 80/2 = 40 units and earned $3,20012 = $1
600. Therefore when firm 1 is the first mover in a sequential game it produces (60 – 40)140 = 50% more and earns (1800 –
1600)/1600 = 125% more than in the simultaneous game.
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9811439887 153
Graphing the Stackelberg outcome. Figure 15.6 illustrates the Stackelberg equilibrium on a best response function
diagram. The leader realizes that the follower will always best respond so the resulting outcome will always be on the
follower s best response function. The leader effectively picks the point on the followers best response function that
maximizes the leader is profit The highest isoprofit (highest in terms of profit level but recall from Figure 15.2 that higher
profit levels are reached as one moves down toward the horizontal axis) as reached at the point S of tangency between
firm 1's isoprofit and firm best response function. This is the Stackelberg equilibrium. Compared with the Cournot
equilibrium at point C the Stackelberg equilibrium involves higher output and profit for firm 1 Firm is profit is higher
because by committing to the high output level, firm 2 is forced to respond by reducing its output.
Commitment as required for the outcome to stray from firm 1's best response function as happens at point S. If firm 1
could secretly reduce q1 (perhaps because q1 is actual capacity that can be secretly reduced by reselling capital equipment
for close to its purchase price to a manufacturer of another product that uses similar capital equipment) then it would
move back to its best response, firm 2 would best respond to this lower quantity, and so on, following the dotted arrows
from S back to C .
Figure–15.6
QUERY : What would be the Outcome if the identity of the first mover were not given and instead firms had to complete
to be the first? How would firms vie for this position? Do these considerations help explain overinvestment in Internet
firms and telecommunications during the "dot com bubble ?"
Contrast with price leadership
In the Stackelberg game, the leader uses what has been called a “top dog” strategy, aggressively overproducing to force
the follower to scale back its production. The leader earns more than in the associated simultaneous game (Cournot),
whereas the follower earns less. Although it is generally true that the leader prefers the sequential game to the
simultaneous game (the leader can do at least as well, and generally better, by playing its Nash equilibrium strategy from
the simultaneous game), it is not generally true that the leader harms the follower by behaving as a “top dog.” Sometimes
the leader benefits by behaving as a “puppy dog,” as illustrated in Example 15.9.
EXAMPLE 15.9 Price Leadership Game
Return to Example 15.4, in which two firms chose price for differentiated toothpaste brands simultaneously. So that the
following calculations do not become too tedious, we make the simplifying assumptions that a1 = a2 = 1 and c = 0.
Substituting these parameters back into Example 15.4 shows that equilibrium prices are 2/3  0.667 and profits are
4/9  0.444 for each firm.
Now consider the game in which firm 1 chooses price before firm 2. We will solve for the subgame-perfect equilibrium
using backward induction, starting with firm 2’s move. Firm 2’s best response to its rivals choice p1 is the same as
computed in Example 15.4–which on substituting a2 = 1 and c = 0 into Equation 15.32, is
VIKAAS WADHWA SIR
9811439887 154
p2 =
 p1
+
2 4
(15.61)
Fold the game back to firm 1’s move. Substituting firm 2’s best response into firm 1’s profit function from Equation 15.30
gives
1  1 p  p

1 = p1 1 − p1 +  + 1  = 1 (10 − 7 p1 )
2  2 4  8

Taking the first order condition and solving for the equilibrium price we obtain
15.61 gives
(15.62)
p1*  0.714
p*2  0.679 . Equilibrium profits are 1*  0.446 and *2 
Substituting into Equation
0.460. Both firms’ prices and profits are
higher in this sequential game than in the simultaneous ones but now the follower earns even more than the leader.
As illustrated in the best response function diagram in Figure 15.7 firm 1 commits to a high price to induce firm 2 to raise
its price also, essentially ‘softening’ the competition between them.
Figure–15.7
The leader-needs a moderate price increase (from 0.667 to 0.714) to induce the follower to increase its price slightly (from
0.667 to 0679), so the leader’s profits do not increase as much as the follower’s.
QUERY: What choice variable realistically is easier to commit to, prices or quantities? What business strategies do firms use
to increase their commitment to their list prices?
We say that the first mover is playing a “puppy dog” strategy in Example 15.9 because it increases its price relative to the
simultaneous-move game; when translated into outputs, this means that the first mover ends up producing less than in the
simultaneous-move game. It is as though the first mover strikes a less aggressive posture in the market and so leads its
rival to compete less aggressively.
A comparison of- Figures 15.6 and 15.7 suggests the crucial difference between the games that leads the first mover to
play a “top dog” strategy in the quantity game and a “puppy dog” strategy in the price game: The best-response functions
have different slopes. The goal is to induce the follower to compete less-aggressively. The slopes of the -best-response
functions determine whether the leader can best do that by playing aggressively itself or by softening its strategy. The first
mover plays a “top dog” strategy in the sequential quantity game or indeed any game in which best responses slope down.
When best responses slope down, playing more aggressively induces a rival to respond by competing less aggressively.
Conversely, the first mover plays a “puppy dog” strategy in the price game or any game in which best responses slope up.
When best responses slope up, playing less aggressively induces a rival to respond by competing less aggressively.
Therefore, knowing the slope of firibs’ best responses provides considerable insight into the sort of strategies firms will
choose if they have commitment power. The Extensions to this chapter at the end of the book provide further technical
details, including shortcuts for determining the slope of a firm’s best response function just by looking at its profit function.
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9811439887 155
STRATEGIC ENTRY DETERRENCE
We saw that, by committing to an action, a first mover may be able to manipulate the second mover into being a less
aggressive competitor. In this section we will see that the first mover may be able to prevent the entry of the second
mover entirely, leaving the first mover as the sole firm in the market. In this case, the firm may not behave as an unconstrained monopolist because it may have distorted its actions to fend off the rival’s entry.
In deciding whether to deter the second mover’s entry, the first mover must weigh the costs and benefits relative to
accommodating entry–that is, allowing. entry to happen.
Accommodating entry does not mean behaving non–strategically. The first mover would move off its best response
function to manipulate the second mover into being less competitive, as described in the previous section. The cost of
deterring entry is that the first mover would have to- move off its best-response function even further than it would if it
accommodates entry. The benefit is that it operates alone in the market and has market demand to itself. Deterring entry
is relatively easy for the first mover if the second mover must pay a substantial sunk cost to enter the market.
EXAMPLE 15.10 : Deterring Entry of a Natural Spring
Recall Example 15.8, where two natural-spring owners choose outputs sequentially. We now add an entry stage: In
particular, after observing firm 1’s initial quantity choice, firm 2 decides whether to enter the market. Entry requires the
expenditure of sunk cost K2, after which firm 2 lose output. Market demand and cost are as in Example 15.8. To simplify
the calculations, we will take the specific numerical values a = 120 and c = 0 [implying that inverse demand is P(Q) = 120 –
Q, and that production is costless). To further simplify, we will abstract from firm i’s entry decision and assume that it has
already sunk any cost needed to enter before the start of the game. We will look for conditions under which firm 1 prefers
to deter rather than accommodate firm 2’s entry.
acc
Accommodating entry. Start by computing firm l’s profit if it accommodates firm 2’s entry denote 1 . This has already
been done in Example 15.8, in which there was no issue of deterring firm 2's entry. There we found firm 1’s equilibrium
acc
2
acc
output to be (a – c)/2 = q1
and its profit to be (a − c) / 8 = 1 . Substituting the specific numerical values a =
acc
acc
120 and c = 0, we have q1 = 60 and 1 = (120 – 0)2/8 = 1,800.
Deterring entry. Next, compute firm 1’s profit if it deters firm 2’s entry, denoted
produce an amount
q1det
high enough that, even if firm 2 best responds to
1det . To deter entry, firm 1 needs to
q1det ,
it cannot earn enough profit to cover
its sunk cost K2. We know from Equation 15.58 that firm 2's best-response function is
q2 =
120 − q1
2
(15.63)
Substituting for q2 in firm 2’s profit function (Equation 15.7) and simplifying gives
det 2
 120 − q1
 2 = 
2


 −K


(15.64)
Setting firm 2’s profit in Equation 15.64 equal to 0 and solving yields
(15.65)
q1det = 120 − 2 K 2
q1det is the firm 1 output needed to keep firm 2 out of the market. At this output level, firm 1’s profit is
1det = 2 K 2 120 − 2 K 2
(15.66)
det
which we found by substituting q1 , a = 120, and c = 0 into firm 1’s profit function from Equation 15.7. We also set q2 = 0
because, if firm I is successful in deterring entry, it operates alone in the market.
acc
det
Comparison. The final step is to juxtapose 1 and 1
to find the condition under which firm 1 prefers deterring to
det
acc
accommodating entry. To simplify the algebra, let x = 2 K 2 . Then 1 = 1 is
x 2 − 120x + 1800 = 0
(15.67)
Applying the quadratic formula yields
(
x=
120  7200
2
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)
(15.68)
9811439887 156
Taking. the smaller root (because we will be looking for a minimum threshold), we have
x = 17.6 (rounding to the nearest decimal). Substituting x = 17.6 into
2
x = 2 K2
and solving for K2 yields
2
 x   17.6 
K2 =   = 
  77
2  2 
(15.69)
If K2 = 77, then entry is so cheap for firm 2 that firm 1 would have to increase its output all the way to
q1det = 102
in
order to deter entry. This is a significant distortion above what it would produce when accommodating entry:
q1acc = 60 . If K2 < 77, then the output distortion needed to deter entry wastes so much profit that firm 1 prefers to
accommodate entry. If K2 > 77, output need not be distorted as much to deter entry thus, firm 1 prefers to deter entry.
QUERY: Suppose the first mover must pay the same entry cost as the second, K1 = K2 = K. Suppose further that K is high
enough that the first mover prefers to deter rather than accommodate the second mover's entry. Would this sunk cost not
be high enough to keep the first mover out of the market, too? Why or why not?
A real-world example of overproduction (or overcapacity) to deter entry is provided by the 1945 antitrust case against
Alcoa, a U.S. aluminum manufacturer. A U.S. federal court ruled that Alcoa maintained much higher capacity than was
needed to serve the market as a strategy to deter rivals’ entry, and it held that Alcoa was in violation of antitrust laws.
To recap what we have learned in the last two sections: with quantity competition, the first mover plays a “top dog”
strategy regardless of whether it deters or accommodates the second mover’s entry. True, the entry-deterring strategy is
more aggressive than the entry-accommodating one, but this difference is one of degree rather than kind. However, with
price competition (as in Example 15.9), the first mover’s entry-deterring strategy would differ in kind from its entryaccommodating strategy. It would play a “puppy dog” strategy if it wished to accommodate entry because this is how it
manipulates the second mover into playing less aggressively. It plays a “top dog” strategy of lowering its price relative to
the simultaneous game if it wants to deter entry. Two general principles emerge.
•
Entry deterrence is always accomplished by a “top dog” strategy whether competition is in quantities or prices, or
(more generally) whether best–response functions slope down or up. The first mover simply wants to create an
inhospitable environment for the second mover.
•
If firm 1 wants to accommodate entry, whether it should play a “puppy dog” or “top dog” strategy depends on the
nature of competition–in particular, on the slope of the best-response functions.
SIGNALING
The preceding sections have shown that the first mover’s ability to commit may afford it a big strategic advantage In this
section we will analyze another possible first mover advantage: the ability to signal. If the second mover has incomplete
information about market conditions (e.g., costs, demand), then it may try to learn about these conditions by observing
how the first mover behaves. The first mover may try to distort its actions to manipulate what the second learns. The
analysis in this section is closely tied to the material on signaling games in Chapter 8, and the reader may want to review
that material before proceeding with this section.
The ability to signal may be a plausible benefit of being a first mover in some settings in which the benefit we studied
earlier–commitment–is implausible. For example, in industries where the capital equipment is readily adapted to
manufacture other products, costs are not very “sunk”; thus, capacity commitments may not be especially credible. The
first mover can reduce its capacity with little loss. For another example, the price-leadership game involved a commitment
to price. It is hard to see what sunk costs are involved in setting a price and thus what commitment value it has. Yet even in
the absence of commitment value, prices may have strategic, signaling value.
Entry-deterrence model
Consider the incomplete information- game in Figure 15.8 The game involves a first mover (firm 1) and a second mover
(firm 2) that choose prices for their differentiated products. Firm 1 has private information about its marginal cost, which
can take on one of two values: high with probability Pr(H) or low with probability Pr(L) = 1 – Pr(H). In period 1, firm 1 serves
the market alone. At the end of the period, firm 2 observes firm 1’s price and decides whether to enter the market. If it
enters, it sinks an entry cost K2 and learns the true level of firm l’s costs; then firms compete as duopolists, in the second
period, choosing prices for differentiated products as in Example 15.4 or 15.5. (We do not need to be specific about the
exact form of demands.) If firm 2 does not enter, it obtains a payoff of zero, and firm 1 again operates alone in the market.
Assume there is no discounting between periods.
Firm 2 draws- inferences about firm 1’s cost from the price that firm 1 charges in the first period. Firm 2 earns chore if it
competes against the high-cost type because the
VIKAAS WADHWA SIR
9811439887 157
Figure – 15.8
high-cost type’s price will be higher, and as we saw in Examples 15.4 and 15.5, the higher the riva1’s price for a
differentiated product, the higher the firm’s own demand and profit.
t
Let Di be the duopoly profit (not including entry costs) for firm i  {l, 2} if firm 1 is of type t  {L, H}. To make the model
L
H
interesting, we will suppose D 2  K 2  D 2 , so that firm 2 earns more than its entry cost if it faces the high-cost
type but not if it faces the low-cost type. Otherwise, the information in firm l’s signal would be useless because firm 2
would always enter or always-stay out regardless of firm 1’s type.
To simplify the model, we will ‘suppose that the low-cost type only has one relevant action in the first period–namely,
L
setting its monopoly price p1 . The high-cost type can choose one of two prices: can set the monopoly price associated
H
L
with its type, p1 , or it can choose the same price as the low type, p1 . Presumably, the optimal monopoly price is
L
1
H
increasing in marginal cost; thus, p1  p1 . Let M1 be firm 1’s monopoly profit if it is of type t  {L, H} (the profit if it
is alone and charges its optimal monopoly price
p1H
if it is the high type and
p1L
type’s loss relative to the optimal monopoly profit in the first period if it charges
price
p1H . Thus, if the high type charges p1H
it earns
in the first period, then it earns
if it is the low type). Let R be the high
p1L
M1H
rather than its optimal monopoly
in that period, but if it charges
p1L ,
M1H − R .
Separating equilibrium
We will look for two kinds of perfect Bayesian equilibria: separating and pooling, in a separating equilibrium, the different-’
types of the first mover must choose different actions. Here, there is only one such possibility for firm 1: The low-cost type
H
L
chooses p1 and the high-cost type chooses p1 , Firm 2 learns firm 1’s type from these actions perfectly and stays out on
L
H
1
seeing p1 and enters on seeing p1 . It remain ’ to check whether the high-cost type would prefer to deviate to p1 . In
H
H
H
equilibrium the high type earns a total profit of M1 + D1 : M1 in the first period because it charges its optimal
H
monopoly price, and D1 in the second because firm 2 enters and the firms compete as duopolists. If the high type were
L
H
to deviate to p1 , then it would earn M1 − R in the first period, the loss R coming from charging a price other than
VIKAAS WADHWA SIR
9811439887 158
its first-period optimum, but firm 2 would think it is the low type and would not enter. Hence firm 1 would earn
the second period, for a total of
2M1H − R
M1H
in
across periods: For deviation to be unprofitable we must have
M1H + D1H  2M1H − R
(15.70)
or (after rearranging)
R  M1H − D1H
(15.71)
That is, the high-type’s loss from distorting its price from its monopoly optimum in the first period exceeds its gain from
deterring firm 2’s entry in the second period.
If the condition in Equation 15.71 does not hold, there still may be a separating equilibrium in an expanded game in which
L
the low type can charge other prices besides p1 .
The high type could distort its price downward below
p1L ,
increasing the first-period loss the high type would suffer from
pooling with the low type to such an extent that the high type would rather charge
p1H
even if this results in firm 2’s
entry.
Pooling equilibrium
If the condition in Equation 15.71 does not hold, then the high type would prefer to pool with the low type if pooling
deters entry. Pooling deters entry if firm 2’s prior belief that firm 1 is the high type, Pr(H)–which is equal to its posterior
belief in a pooling equilibrium is low enough that firm 2’s expected payoff from entering,
Pr(H)D H2 + [1 − Pr(H)]D L2 − K 2
(15.72)
is less than its payoff of zero-from-staying-out of the market.
Predatory pricing
The incomplete-information model of entry deterrence has been used to explain why a rational firm might want to engage
in predatory pricing, the practice of charging an artificially low price to prevent potential rivals from entering or to force
existing rivals to exit. The predatory firm sacrifices profits in the short run to gain a monopoly position in future periods.
Predatory pricing is prohibited by antitrust laws, in the most famous antitrust case, dating back to 1911, John D.
Rockefeller–owner of the Standard Oil Company that controlled a substantial majority of refined oil in the United States–
was accused of attempting to monopolize the oil market by cutting prices dramatically to drive rivals out and then raising
prices after rivals had exited the market or sold out to Standard Oil. Predatory pricing remains a controversial antitrust
issue because of the difficulty in distinguishing between predatory conduct, which authorities would like to prevent, and
competitive conduct, which authorities would like to promote. In addition, economists initially had trouble developing
game-theoretic models in which predatory pricing is rational and credible.
Suitably ‘interpreted, predatory pricing may emerge as a rational strategy in the incomplete-information model of entry
deterrence. Predatory pricing can show up in a separating equilibrium–in particular, in the expanded model where the lowcost type can separate only by reducing price below its monopoly optimum. Total welfare is actually higher in this
separating equilibrium than it would be in its full-information counterpart. Firm 2’s entry decision is the same in both
outcomes, but the low-cost type’s price may be lower (to signal its type) in the predatory outcome.
Predatory pricing can also show up in a pooling equilibrium. In this case it is the high–cost type that charges an artificially
low price, pricing below its first-period optimum to keep firm 2 out of the market. Whether social welfare is lower in the
pooling equilibrium than in a full information setting is unclear In the first period price is lower (and total welfare
presumably higher) in the pooling equilibrium than in a full information setting. On the other hand, deterring firm 2’s entry
results in higher second-period prices and lower welfare Weighing the first-period gain against the second period loss
would require detailed knowledge of demand curves discount factors and so forth. The incomplete-information model of
entry deterrence is not the only model of predatory pricing that economists have developed. Another model involves
frictions in the market for financial capital that stem perhaps from informational problems (between borrowers and
lenders) of the sort we will discuss in Chapter 18. With limits on borrowing, firms may only have limited resources to
“make a go” in a market. A larger firm may force financially strapped rivals to endure losses until their resources are
exhausted and they are forced to exit the market.
HOW MANY FIRMS ENTER ?
To this point, we have taken the number of firms in the market as given, often assuming that there are at most two firms
(as in Examples 15.1, 15.3, and 15.10). We did allow for a general number of firms, n, in some of our analysis (as in
Examples 15.3 and (5.7) but were silent about how this number n was determined. In this section, we provide a gametheoretic analysis of the number of firms by introducing a first stage in which a large number of potential entrants can each
choose whether to enter: We will abstract from first-mover advantages, entry deterrence, and other strategic
considerations by assuming that firms make their entry decisions simultaneously. Strategic considerations are interesting
VIKAAS WADHWA SIR
9811439887 159
and important, but we have already developed some insights into strategic considerations from the previous sections and–
by abstracting from them–we can simplify the analysis here.
Barriers to entry
For the market to be oligopolistic with a finite number of firms rather than perfectly competitive with an infinite number of
infinitesimal firms, some factors; called barriers to entry, must eventually make entry unattractive or impossible. We
discussed many of these factors at length in the previous chapter on monopoly. If a sunk cost is required to enter the
market, then–even if firms can freely choose whether to enter only a limited number of firms will choose to enter in
equilibrium because competition among more than that number would drive profits below the level needed to recoup the
sunk entry cost. Government intervention in the form of patents or licensing requirements may prevent firms from
entering even if it would be profitable for them to do so.
Some of the new concepts discussed in this chapter may introduce additional barriers to entry. Search costs may prevent
consumers from finding new entrants with lower prices and/or higher quality than existing firms. Product differentiation
may raise entry barriers because of strong brand loyalty. Existing firms may bolster brand loyalty through expensive
advertising campaigns, and softening this brand loyalty may require entrants to conduct similarly expensive advertising
campaigns. Existing firms may take other strategic measures to deter entry, such as committing to a high capacity or
output level, engaging in predatory pricing, or other measures discussed in previous sections.
Long-run equilibrium
Consider the following game-theoretic model of entry in the long run. A large number of symmetric firms are potential
entrants into a market. Firms make their entry decisions simultaneously. Entry requires the expenditure of sunk cost K. Let
n be the number of firms that decide to enter. In the next stage, the n firms engage in some form of competition over a
sequence of periods during which they earn the present discounted value of some constant profit stream. To simplify, we
will usually collapse the sequence of periods of competition into a single period Let g(n) be the profit earned by an
individual firm in this competition subgame [not including the sunk cost, so g(n) is a gross profit). Presumably, the more
firms in the market, the more competitive the market is and the less an individual firm earns, so g'(n) < 0.
We will look for a subgame-perfect equilibrium in pure strategies. This will be the number of firms, n*, satisfying two
conditions. First, the n* entering firms earn enough to cover their entry cost: g (n*)  K. Otherwise, at least one of them
would have preferred to have deviated to not entering. Second, an additional firm cannot earn enough to cover its entry
cost: g(n* + I)  K. Otherwise, a firm that remained out of the market could have profitably deviated by entering. Given that
g'(n) < 0, we can put these two conditions together and say that n* is the greatest integer satisfying g(n*)  K.
This condition is reminiscent of the zero-profit condition for long-run equilibrium under perfect competition. The slight
nuance here is that active firms are allowed to earn positive profits. Especially if K is large relative to the size of the market,
there may only be a few long-run entrants (thus, the market looks like a canonical oligopoly) earning well above what they
need to cover their sunk costs, yet an additional firm does not enter because its entry would depress individual profit
enough that the entrant could not cover its large sunk cost.
Is the long-run equilibrium efficient? Does the oligopoly involve too few or too many firms relative to what a benevolent
social planner would choose for the market? Suppose the social planner can choose the number of firms (restricting entry
through licenses and promoting entry through subsidizing the entry cost) but cannot regulate prices or other competitive
conduct of the firms once in the market. The social planner would choose n to maximize
CS(n) + ng(n) – nK
(15.73)
where CS(n) is equilibrium consumer surplus in an oligopoly with n firms, ng(n) is total equilibrium profit (gross of sunk
entry costs) across all firms, and nK is the total expenditure on sunk entry costs. Let n** be the social planner’s optimum.
In general, the long-run equilibrium number of firms, n*, may be greater or less than the social optimum, n**, depending on
two offsetting effects: the appropriability effect and the business-stealing effect.
•
The social planner takes account of the benefit of increased consumer surplus from lower prices, but firms do not
appropriate consumer surplus and so do not take into account this benefit. This appropriability effect would lead
a social planner to choose more entry than. in the long run equilibrium: n** > n*.
•
Working in the opposite direction is that entry causes the profits of existing firms to decrease, as indicated by the
derivative g'(n) < 0. Entry increases the competitiveness of the market, destroying some of firms’ profits. In
addition, the entrant “steals” some market share from existing firms–hence the term business-stealing effect. The
marginal firm does not take other firms’ loss in profits ‘when making its entry decision, whereas the social planner
would. The business-stealing effect biases long-run equilibrium toward more entry than a social planner would
choose: n** < n*.
Depending on the functional forms for demand and costs, the appropriability effect dominates in some cases, and there is
less entry in long-run equilibrium than is efficient. In other cases, the business-stealing dominates, and there is more entry
in long-equilibrium than is efficient, as in Example 15.11.
EXAMPLE 15.11. Cournot in the Long Run
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9811439887 160
Long-run equilibrium. Return to Example 15.3 of a Gout-not oligopoly. We will determine the long-run equilibrium
number of firms in the market. Let K be the sunk cost a firm must pay to enter the market in an initial entry stage. Suppose
there is one period of Cournot competition after entry. To further simplify the calculations, assume that a = 1 and c = 0.
Substituting these values back into Example 15.3, we have that an individual firm’s gross profit is
2
 1 
g(n ) = 

 n +1
(15.74)
The long-run equilibrium number of firms is the greatest integer n* satisfying g(n*)  K. Ignoring integer problems, n*
satisfies
n* =
1
−1
K
(15.75)
Social planner’s problem. We first compote the individual terms in the social planner’s objective function (Equation 15.73).
Consumer surplus equals the area of the shaded triangle in Figure 15.9, which, using the formula for the area of a triangle,
is
2
1
n
CS(n ) = Q(n )a − P(n ) =
2
2(n + 1) 2
(15.76)
here the last equality comes from substituting for price and quantity from Equations 15.18 and 15.19. Total profits for all
firms (gross of sunk costs) equal the area of the shaded rectangle
ng(n ) = Q(n )P(n ) =
n
(n + 1) 2
(15.11)
Substituting from Equations 15.76 and 15.77 into the social planner’s objective function (Equation 15.73) gives
2
n
n
+
− nK
2
2(n + 1)
(n + 1) 2
(15.78)
After removing positive constants, the first–order condition with respect to n is
3
1 − K (n + 1) = 0
(15.79)
Figure –15.9
Implying that
n ** =
1
−1
K1 / 3
VIKAAS WADHWA SIR
(15.80)
9811439887 161
Ignoring integer problems, this is the optimal number of firms for a social planner.
Comparison. If K < 1 (a condition required for there to be any entry), then n** < n*, and so there is more entry in long-run
equilibrium than a social planner would choose. To take a particular numerical example, let K = 0.1. Then n* = 2.16 and n**
= 1.15, implying that the market would be a duopoly in long-run equilibrium, but a social planner would have preferred a
monopoly.
Feedback effect
We found that certain factors decreased the stringency of competition and increased firms’ profits (e.g., quantity rather
than price corn-petition, product differentiation, search costs, discount factors sufficient to sustain collusion). A feedback
effect is that the more profitable the market is for a given number of firms, the more firms will enter the market, making
the market more competitive and less profitable than it would be if the number of firms were fixed.
To take an extreme example, compare the Bertrand and Cournot games. Taking as given that the market involves two
identical producers, we would say that the Bertrand game is much more competitive and less profitable than the Cournot
game. This conclusion would be reversed if firms facing a sunk entry cost were allowed to make rational entry decision.
Only one firm would choose to enter the Bertrand market. A second firm would drive gross profit to zero, and so its entry
cost would, not be covered. The long-run equilibrium outcome would involve a monopolist and thus the highest prices and
profits possible, exactly the opposite of our conclusions when the number of firms was fixed! On the other hand, the
Cournot market may have space for several entrants driving prices and profits- below their monopoly levels in the Bertrand
market.
The moderating effect of entry should lead economists to be careful when drawing conclusions about oligopoly outcomes.
Product differentiation, search costs, collusion, and other factors may reduce competition and increase profits in the short
run, but they may also lead to increased entry and competition in the long run and thus have ambiguous effects overall on
prices and profits. Perhaps the only truly robust conclusions about prices and profits in the long run involve sunk costs.
Greater sunk costs constrain entry even in the long run, so we can confidently say that prices and profits will tend to be
higher in industries requiring higher sunk costs (as a percentage of sales) to enter.
INNOVATION
At the end of the previous chapter, we asked which market structure–monopoly or perfect competition–leads to more
innovation in new products and cost-reducing processes. If monopoly is more innovative, will the long-run benefits of
innovation offset the short-run deadweight loss of monopoly? The same questions can be asked in the context of oligopoly. Do concentrated market structures, with few firms perhaps charging high prices, provide better incentives for
innovation? Which is more innovative, a large or a small firm? An established firm or an entrant? Answers to these
questions can help inform policy toward mergers, entry regulation, and small-firm subsidies.
As we will see with the aid of some simple models, there is no definite answer as to what level of concentration is best for
long-run total welfare. We will derive some general tradeoffs, but quantifying these trade-offs to determine whether a
particular market would be more innovative if it were concentrated or unconcentrated will depend on the nature of
competition for innovation, the nature of competition for consumers, and the specification of demand and cost functions.
The same can be said for determining what firm size or age is most innovative.
The models we introduce here are of product-in-n-ovations, the invention of a product (e.g., plasma televisions) that did
not exist before. Another class of innovations is that of process innovations, which reduce the cost of producing- existing
products–for example, the use of robot technology in automobile manufacture.
Monopoly on innovation
Begin by supposing that only a single firm, call it firm 1, has the capacity to innovate. For example, a pharmaceutical
manufacturer may have an idea for a malaria vaccine that no other firm is aware of. How much would the first be willing to
complete research and development for the vaccine and to test it with large-scale clinical trials? How does this willingness
to spend (which we will take as a measure of the innovativeness of the firm) depend on concentration of firms in the
market?
Suppose first that there is currently no other vaccine available for malaria. If firm I successfully develops the vaccine, then
it will be a monopolist. Letting M be the monopoly profit, firm 1 would be willing to spend as much as M to develop the
vaccine.
Next, to examine the case of a less concentrated market, suppose that another firm (firm 2) already has a vaccine on the
market for which firm 1’s would be a therapeutic substitute.
If firm 1 also develops its vaccine, the firms compete as duopolists. Let D be the duopoly profit. In a Bertrand model with
identical products, D = 0, but D > 0 in other models for example, models involving quantity competition or collusion. Firm
1 would be willing to spend as much as D to develop the vaccine in this case. Comparing the two cases, because M > D,
it follows that firm 1 would be willing to spend more (and, by this measure, would be more innovative) in a more
concentrated market. The general principle here can be labeled a dissipation effect: Competition dissipates some of the
profit from innovation and thus reduces incentives to innovate. The dissipation effect is part of the rationale behind the
patent system. A patent grants monopoly rights to an inventor, intentionally restricting competition to ensure higher
profits and greater innovation Incentives.
Another comparison that can be made is to see which firm, 1 or 2, has more of an incentive to innovate given that it has a
monopoly on the initial idea. Firm 1 is initially out of the market and must develop the new vaccine to enter. Firm 2 is
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already in the malaria market with its first vaccine but can consider developing a second one as well, which we will
continue to assume is a perfect substitute. As shown in the previous paragraph, firm 1 would be willing to pay up to D for
the innovation. Firm 2 would not be willing to pay anything because it is currently a monopolist in the malaria vaccine market and would continue as a monopolist whether or not it developed the second medicine. (Crucial to this conclusion is
that the firm with the initial idea can decline to develop it but still not worry that the other firm will take the idea; we will
change this assumption in the next subsection.) Therefore, the potential competitor (firm 1) is more innovative by our
measure than the existing monopolist (firth 2). The general principle here has been labeled a replacement effect: Firms
gain less incremental profit and thus have less incentive to innovate if the new product replaces an existing product
already’ making profit than if the firm is a new entrant in the market. The replacement effect can explain turnover in
certain industries where old firms become increasingly conservative and are eventually displaced by innovative and quickly
growing startups, as Microsoft displaced IBM as the dominant company in the computer industry and as Google now
threatens to replace Microsoft.
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