Chapter 8
Equilibrium and Efficiency
1. Markets and Exchange.
2. Equilibrium.
3. Efficiency of the equilibrium outcome.
4. Liquidity Insurance
A textbook account of general equilibrium is available in Hal Varian’s,
Microeconomic Analysis, Third edition, chapter 17, sections 1, 2, 3, and 6.
The last section is based on Frexias and Rochet, Microeconomics of Banking,
pages 20-23 and 192-195.
8.1
Markets and Exchange
Markets involve voluntary exchange among agents. Consider an economy
in which there are n agents and K tradeable commodities. To keep things
simple, consider a pure exchange economy (that is, one in which there is
no production). Each agent has an initial endowment of commodities and
the agents exchange these in the market in order to maximize their utility.
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Let eki be the initial amount of the k-th commodity with individual i.
The vector
 
e1i
e 
 
ei =  2i 
. . .
eKi
represents the i-th individual’s endowment vector. The K × n matrix e =
(e1 , e2 , . . . , en ) denotes the endowment of all individuals.
Similarly, let the vector


x1i
x 
 
xi =  2i 
. . .
xKi
denotes an arbitrary commodity bundle for agent i. The K × n matrix
x ( x1 , x2 , . . . , xn ) denotes a possible allocation in the economy.
In an exchange economy allocations are constrained by the initial endowment. So for a particular allocation to be feasible, the commodities
in that allocation must be a redistribution of the endowment. Formally,
given an endowment e, an allocation x is feasible if
n
n
i =1
i =1
∑ x i ≤ ∑ ei .
8.2
Walrasian equilibrium
Commodities are exchanged in a market, which is characterised by prices.
Let pk denote the price of good k; then p = ( p1 , p2 , . . . , pK ) denotes the
K-dimensional price vector. We assume that consumer has well-behaved
preferences that can be represented by a continuous utility function. For
consumer i we have ui ( xi ), which is non-decreasing in each argument.
Suppose each consumer takes prices as given, and chooses xi to maximise
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her utility subject to the condition that her budget cannot exceed the value
of her endowment.
The i-th consumer’s problem is to
max ui ( xi )
xi
subject to pxi = pei .
(CPi ( p, pei ))
For simplicity, we assume that for the budget constraint holds as an
equality. The solution to the problem, call it xi ( p, pei ) is a demand vector
for agent i.
The economy is given by endowments and preferences of all individuals. At the market equilibrium of this economy, all markets clear.
n
∑ xi ≤
i =1
n
∑ ei .
i =1
Note the weak inequality: the market for some goods may have excess
supply. We can define the exchange equilibrium (sometimes called the
Walrasian equilibrium) for this economy as follows. .
Definition 1 (General Equilibrium). A general equilibrium for the given pureexchange economy consists of a price-allocation pair ( p, x ) such that
1. at prices p, the bundle xi solves CPi ( p, pei ) for all i; and
2. all markets clear:
n
n
i =1
i =1
∑ x i ≤ ∑ ei .
Does such an equilibrium exist? Yes, under suitable restrictions on
preferences. For our purposes, we construct a simple example to illustrate
the idea.
An Example
Consider an economy with two agents and two goods.
3
a
1− a and endowAgent 1Ãhas
! utility function u1 ( x11 , x21 ) = ( x11 ) ( x21 )
1
ment e1 =
.
0
b
1−b and endowAgent 2Ãhas
! utility function u2 ( x12 , x22 ) = ( x12 ) ( x22 )
0
ment e2 =
.
1
Consider an arbitrary price vector ( p1 , p2 ). At these prices, the value of
each agent’s endowment at these prices is p1 e1i + p2 e2i .
For agent 1, we have endowment e11 = 1 and e21 = 0, so the value of
the endowment is p1 . It solves
max ( x11 ) a ( x21 )1−a
x11 ,x21
subject to
p1 x11 + p2 x21 = p1 .
Using the standard Lagrangean method to solve this problem, we find that
the agent 1’s demand at these prices is
x11 ( p1 , p2 ) = a
ap1
x21 ( p1 , p2 ) =
p2
For agent 2, we have endowment e12 = 0 and e22 = 1, so the value of
the endowment is p2 . It solves
max ( x12 )b ( x22 )1−b
x12 ,x22
subject to
p1 x12 + p2 x22 = p2 .
Agent 2’s demand at these prices can be shown to be
bp2
p1
x22 ( p1 , p2 ) = 1 − b.
x12 ( p1 , p2 ) =
4
Market clearing in the market for each good requires
x11 ( p1 , p2 ) + x12 ( p1 , p2 ) = e11 + e12 = 1
x21 ( p1 , p2 ) + x22 ( p1 , p2 ) = e21 + e22 = 1.
Using the optimal solutions above, market clearing for the first good amounts
to
bp2
= 1.
a+
p1
Solving this for prices, we can write
p2∗
1−a
=
.
∗
p1
b
Market clearing in the other market requires
(1 − a ) p1
+ 1 − b = 1,
p2
which implies
p2∗
1−a
=
.
∗
p1
b
Two things to note here. One, we obtain the market-clearing price ratio,
not the absolute value of prices. This is not surprising. Only relative prices
matter here. Without loss of generality, we could set p1∗ = 1, so that p2∗ =
1− a
b . Two, the market clearing in both markets leads to the same price ratio.
There is more general result, sometimes called Walras’ Law, behind this:
with K markets, equilibrium in K − 1 markets implies equilibrium in the
remaining market.
The equilibrium price ratio allows us to solve for the equilibrium allocation.
x11 ( p1∗ , p2∗ ) = a
(1 − a ) p ∗1
x21 ( p1∗ , p2∗ ) =
=b
p2∗
bp2
= 1−a
x12 ( p1∗ , p2∗ ) =
p1
x22 ( p1 , p2 ) = 1 − b.
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Hence, the equilibrium of this economy is given by the price-allocation
pair
"µ
!#
¶ Ã
a 1−a
1−a
1,
,
b
b 1−b
8.3
Efficiency of the equilibrium allocation
Definition 2 (Pareto Efficiency). A feasible allocation x is said to be Pareto
efficient if there exists no other feasible allocation x 0 such that all agents i weakly
prefer xi0 to xi , and some agent strictly prefers xi0 to xi .
To demonstrate the efficiency of the equilibrium, we begin with a slightly
modified definition of the equilibrium.
Definition 3. An equilibrium for a given exchange economy consists of a allocationprice pair ( x, p) such that
1. if agent i prefers xi0 to xi , it must be that pxi0 > pxi , and
2. all markets clear:
n
n
i =1
i =1
∑ x i ≤ ∑ ei .
This is equivalent to the earlier definition under our assumption that
all agents exhaust their budget.
Theorem 1 (The First Theorem of Welfare Economics). If ( x, p) is a Walrasian equilibrium, then x is Pareto efficient.
Proof: We prove this result by obtaining a contradiction. Suppose ( x, p)
is a Walrasian equilibrium, and x is not Pareto efficient. That is, suppose
there is some other feasible allocation x 0 which is weakly preferred to x by
all agents, and strictly by some.
1. Since x 0 is feasible,
n
implies
∑
i =1
pxi0
n
≤
n
∑ xi0 ≤
i =1
n
∑ ei .
i =1
∑ pei .
i =1
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Given non-negative prices, this
2. Since x 0 Pareto dominates x, all consumers must like xi0 as much as
xi , and some consumer must strictly prefer xi0 to xi . Since, x is an
equilibrium allocation, it follows from the definition of equilibrium
that pxi0 ≥ pxi for all i, and pxi0 > pxi for some i.
Summing up over all consumers, we get
n
∑
n
pxi <
i =1
∑ pxi0 .
i =1
3. As budget constraints must hold as equalities at the equilibrium:
summing them,
n
n
i =1
i =1
∑ pxi = ∑ pei .
4. Conclusions 1, 2 and 3 contain a contradiction. ¤
8.4
A Model of Liquidity Insurance
We consider an economy with one good and a continuum of individuals
that are ex-ante identical. We analyze the economy over three periods,
t = 0, 1, 2. Each individual is endowed with one unit of the good in period
0. They use this to finance their consumption (C1 , C2 ) in periods 1 and 2.
Individual receive ‘liquidity shocks’. At t = 1, they find out
• with probability π1 that she is of type-1. Individuals of this type are
impatient and care only for consumption in period 1. Their utility
function is of the form u(C1 ),
• with probability π2 = 1 − π1 that she of type 2. Individuals of this
type are patient and care only for consumption in period 2. Their
utility function is of the form ρu(C2 ), where ρ < 1 is a discount factor.
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In ex-ante terms, the consumer’s expected utility can be written as
U (.) = π1 u(C1 ) + π2 ρu(C2 ).
We assume u(.) is increasing and concave.
At time t = 0, each individual faces a choice of two investment technologies.
• The first one is a one-period ‘storage’ technology: you get back one
unit for every unit you store.
• The second technology is an investment technology that returns R >
1 after two periods or L < 1 after one period.
Thus, relative to the storage technology, investment is more productive
over two period but there is a penalty for premature liquidation.
Autarky
Suppose first that there is no trade between agents. This is called autarky.
Each consumer must choose, independently of others, her level of investment in the illiquid technology. Recall that the investment choice is made
at time t = 0, before the individual finds out her type. Suppose the individual invests I and stores the rest 1 − I. Then, if she discovers herself to
be impatient, she consumes what she had stored 1 − I and the liquidated
value LI of her investment.
C1 ( I ) = 1 − I + LI.
If she discovers herself to be patient, she can consume her stored wealth
(this requires re-storing the good from period 1 to 2) and the maturity
value of her investment.
C2 ( I ) = 1 − I + RI.
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Given these, each consumer chooses I to maximize expected utility.
Note that C1 ( I ) = 1 − I (1 − L) ≤ 1, with equality only when I = 0.
Similarly, C2 ( I ) = 1 + I ( R − 1) ≤ R, with equality when I = 1. In general,
the consumption profile (C1 , C2 ) is bounded above by (1, R).
Market Economy
Now suppose consumers are allowed to trade in period 1. Assume there
is a riskless bond, which involves the following contract: you pay p in
period 1 (the price of the bond), which entitles you to one unit of the good
in period 2. It is easy to see that p ≤ 1: if not, bonds would be dominated
by storage, so no one would ever buy bonds.
The ability to trade alters the consumption possibilities relative to autarky. Consider an individual who has invested I and stored 1 − I. If she
discovers herself to be the impatient type she need not liquidate her investment: she could instead sell RI bonds at price p so that total consumption
is
C1 ( I ) = 1 − I + pRI.
If she discovers herself to be patient, her liquid, stored assets, 1 − I, will
allow her to buy 1−p I bonds. Her consumption is now financed by the
proceeds of these bonds and the return on the illiquid investment.
C2 ( I ) =
1−I
+ RI.
p
How do these consumption possibilities compare with autarky? The answer clearly depends on p, the price of bonds.
We argue that market clearing in the bond market requires that p = R1 .
Why? Suppose not. If p > R1 , consumption in both periods in increasing in
I. All agents will choose to hold the highest possible I, and ex post this will
create excess demand in the market for bonds. On the other hand, if p < R1 ,
consumption in both periods in decreasing in I. This will create excess
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supply in the market for bonds. Hence the only bond price compatible
with market clearing is p = R1 .
Evaluating C1 and C2 at this value, we get the market solution
C1M = 1
and
C2M = R,
and to enable this solution we must have I M = π2 . Note that the market
solution Pareto dominates the autarky allocation: no type is worse off and
at least one type is better off.
Pareto Optimal Allocation
While the market solution Pareto dominates the autarkic solution, it may
not be Pareto optimal (i.e., the economy could do even better). Consider
the following problem:
max π1 u(C1 ) + ρπ2 u(C2 )
subject to
C2
= 1.
R
The last relation describes an aggregate resource constraint. The optimal
allocation (C1∗ , C2∗ ) satisfies the first-order condition,
π1 C1 + π2
u0 (C1∗ ) = ρRu0 (C2∗ ).
Does the market allocation (1, R) satisfy this optimality condition? Only
in the very special condition
u0 (1) = ρRu0 ( R).
Thus, the market allocation is optimal only by accident.
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How does optimal consumption (C1∗ , C2∗ ) compare with market-based
consumption (1, R)? Suppose the expression Cu0 (C ) is decreasing. This
is equivalent to assuming the the coefficient of relative risk aversion is
greater than 1). In that case, since R > 1, we have
ρRu0 ( R) < ρu0 (1) < u0 (1),
so that it would be welfare improving to increase C1M above 1 and decrease
C2M below R. Thus
C1∗ > C1M = 1
C2∗ > C2M = R
Here the market allocation is not optimal because complete contingent
markets do not exist. The problem here is that the market economy does
not provide perfect insurance against liquidity shocks. The uncertain realisation of who needs to consume early cannot be observed by everyone.
Financial Intermediation
It may be possible to achieve the Pareto optimal outcome through a financial intermediary. Suppose everyone deposits their endowment with
a bank that promises to pay C1∗ at time t = 1 to all type-1 individuals and
C2∗ at time t = 2 to all type-2 individuals. To meet these payments, the
bank keeps π1 C1∗ – a fraction of the deposits – in storage and invests the
rest in the long-term illiquid project. This could be interpreted as demand
deposit system based on fractional reserve banking.
Is this fractional reserve system stable? If the bank cannot observe the
consumers’ type, the answer depends on the behaviour and expectations
of individuals, especially the type-2 individuals. We consider two cases.
Consider the case where C1∗ > C2∗ . You can check, from the above
optimality condition, that this obtains whenever ρR < 1). This creates an
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arbitrage opportunity: type-2 individuals will withdraw C1∗ at 1 and store
it themselves for consumption in period 2, as this improves on getting C2∗
from the bank in period 2. In other words, by type-2 individuals can do
even better by mimicking type 1 individuals. In other words, everyone
will want to withdraw their deposits in period 1. As (C1∗ , C2∗ ) were set
on the assumption that only a fraction π1 will withdraw early, the bank
cannot meet these demands. In other words, fractional reserve banking
cannot work for this case.
Next, consider the case where C1∗ ≤ C2∗ . (This obtains if ρR ≥ 1). Here
there are two sub-possibilities.
• One, if every type-2 individual expects the bank to honour its commitments, they will be willing to wait. Only type-1 individuals will
withdraw so the proportion of population withdrawing is π1 . As
long as the bank holds π1 C1∗ in liquid assets, it will meet demand for
withdrawals.
• Two, suppose one type-2 individual worries that other type-2 individuals may withdraw early. If so, the bank will be forced to liquidate its long-term assets to meet this demand. In extreme cases, it
may not be able to meet the entire demand for withdrawals and will
fail. If so, it makes sense for this individual to withdraw too.
Thus there are two Nash equilibria: one in which all type-2 consumers
are willing to wait till period 2, and the second in which all depositors,
type 1 and type 2, withdraw at t = 1. This second case can be described as
a bank run. Note that a bank run is inefficient.
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Exercises
Consider the liquidity insurance model described above.
1. Solve for the autarkic allocation and the optimal allocation specializing the utility function to
u(ci ) = ln ci
2. Next solve for the optimal allocation in this time using the utility
function
c1− λ − 1
u ( ci ) = i
1−λ
3. What is the solution with intermediation? Why does it matter if
ρR > 1?
Solutions
1. We solve for the two allocations
Autarky: Under autarky, if the agent choose to invest I in the illiquid
technology, her consumption path is
C1 ( I ) = 1 − I + LI = 1 + I ( L − 1)
C2 ( I ) = 1 − I + RI = 1 + I ( R − 1)
If her utility function is given as u(ci ) = ln ci , she chooses I to maximise
π1 ln(C1 ( I )) + π2 ρ ln(C2 ( I ))
The first-order condition for an interior maximum1 is
π1
1 The
1
ρ
( L − 1) + π2
( R − 1) = 0
C1 ( I )
C2 ( I )
second-order condition follows from the concavity of the u(.) function.
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or
π1
1
ρ
( L − 1) + π2
( R − 1) = 0
1 + I ( L − 1)
1 + I ( R − 1)
which implies
π1
1
ρ
(1 − L ) = π2
( R − 1).
1 + I ( L − 1)
1 + I ( R − 1)
This can be solved for I ∗ . We have
I∗ =
π1 ( L − 1) + π2 ρ ( R − 1)
.
(π1 + π2 ρ)( R − 1)(1 − L)
Thereofore, optimal autarkic consumption is
autarky
C1
autarky
C2
π1 ( L − 1) + π2 ρ ( R − 1)
(π1 + π2 ρ)( R − 1)
π ( L − 1) + π2 ρ ( R − 1)
= 1 + I ∗ ( R − 1) = 1 + 1
(π1 + π2 ρ)(1 − L)
= 1 + I ∗ ( L − 1) = 1 −
Optimal allocation: the choice of an optimal allocation involves maximisation of
π1 u(C1 ) + π2 ρu(C2 )
subject to the resource constraint
π1 C1 + π2
C2
=1
R
The first-order condition for this is
u0 (C1 ) = ρRu0 (C2 )
When u(C ) = ln C, we have u0 (C ) = 1/C, so that the first-order
condition reduces to
ρR
1
=
C1
C2
so that C2 = ρRC1 . Substituting this in the budget constraint, we get
π1 C1 + π2
ρRC1
= C1 (π1 + π2 ρ) = 1,
R
14
so that
1
,
π1 + π2 ρ
ρR
=
.
π1 + π2 ρ
C1∗ =
C2∗
2. Solve for the optimal allocation for the utility function
c1i −λ − 1
u ( ci ) =
1−λ
Here u0 (C ) = C −λ , so that the first-order condition reduces to
1
ρR
=
λ
(C1 )
(C2 )λ
1
so that C2 = (ρR) λ C1 . Substituting this in the budget constraint, we
get
1
1
1
(ρR) λ C1
π1 C1 + π2
= C1 (π1 + π2 ρ λ R λ −1 ) = 1,
R
so that
1
C1∗ =
1
1
π 1 + π 2 ρ λ R λ −1
1
C2∗
=
(ρR) λ
1
1
π 1 + π 2 ρ λ R λ −1
3. What is the solution with intermediation? Why does it matter if
ρR > 1?
A financial intermediary can implement the optimal allocation as follows. All agents deposit their initial endowment with the financial
intermediary. The financial intermediary stores π1 C1∗ and invests the
rest to finance. Type 1 individuals withdraw early and get C1∗ . Type-2
individuals withdraw in period 2 and get C2∗ .
If ρR < 1, the above solutions have C1∗ > C2∗ . But then, type 2 consumers can consume more than C2∗ by withdrawing early (effectively,
pretending to be type 1) and storing it themselves.
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