Download General Equilibrium

MSc Economics
Economic Theory and Applications I
Microeconomics
General Equilibrium
Dr Ken Hori, [email protected]
Birkbeck College, University of London
October 2004
Contents
1 General Equilibrium in a Pure Exchange Economy
1.1 Exchange . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Walrasian Equilibrium . . . . . . . . . . . . . . . . .
1.3 Existence of a Walrasian Equilibrium . . . . . . . . .
1.4 Example 1 . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Example 2 . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Uniqueness of Equilibrium . . . . . . . . . . . . . . .
1.7 Welfare Economics . . . . . . . . . . . . . . . . . . .
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1
1
2
3
6
8
9
9
2 General Equilibrium with Production
2.1 Walrasian Equilibrium with Production
2.2 Example . . . . . . . . . . . . . . . . . .
2.3 Welfare Economics . . . . . . . . . . . .
2.4 Example . . . . . . . . . . . . . . . . . .
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1
Chapter 1
General Equilibrium in a Pure
Exchange Economy
Varian Ch 17.1-7, 21.2-3
1.1
Exchange
In the partial equilibrium set up we assumed that all prices other than the price
of the good being studied are fixed. Now in the general equilibrium model all prices are
variable, and equilibrium requires that all markets clear. Thus GE theory takes account of
all of the interactions between markets, as well as the functioning of the individual markets.
We begin with we consider the case of pure exchange, which is the special case of the GE
model where all of the economic agents are consumers. Later on we will add production in
the economy. In both cases the consumers’ preferences are assumed given exogenously.
So consider an economy with a finite number of consumers and a finite number of
commodities. The consumers are indexed as i = 1, 2, ..., I, and the commodities are indexed
as n = 1, 2, ..., N . A consumption bundle for a consumer is given by an N -dimension nonnegative vector xi = (xi1 , xi2 , ..., xiN ) ∈ <N
+.
Each consumer i has well-behaved preferences ºi on the set of consumption bundles: each commodity is a ‘good’; each consumer is locally non-satiable. Provided ºi satisfy
2
the usual condition, these preferences can be represented by a continuous utility function
Ui (xi ) : <N
+ → <, which is non-decreasing in each argument.
An allocation x is a list of the consumption bundles of all consumers x = (x1 , x2 , ..., xI ),
which is a N × I matrix. The matrix describes what each agent gets of any particular commodity. The space of all possible allocations is denoted by X.
Each consumer has some initial endowment of the various commodities ei =
(ei1 , ei2 , ..., eiN ) ∈ <N
+ . As with allocation, the aggregate endowment is denoted as e =
(e1 , e2 , ..., eI ).
In a pure exchange economy that has no production then, everything that is consumed must come from somebody’s initial endowment. Thus for a particular allocation to
be feasible, the commodities in that allocation must be a redistribution of the aggregate
endowment. Formally, given an endowment e, an allocation is feasible if,
I
X
i=1
xi ≤
I
X
ei
i=1
The simple 2 × 2 case where there are only two consumers and two commodities can be
represented by the Edgeworth Box.
1.2
Walrasian Equilibrium
The first assumption is that with many agents, each agent behave competitively,
i.e. take prices as given, independent of his actions. Thus the market is characterised
by a price vector p = (p1 , p2 , ..., pN ). Secondly the consumers are assumed to be utility
maximisers. Unlike with the Consumer Theory earlier, here the income of consumer i is the
value of his initial endowment p.ei and thus the UMP becomes,
max Ui (xi ) such that p.xi ≤ p.ei
xi
(GE-UMP)
The solution to this problem is the consumer’s demand function xi (p, p.ei ), i.e. demand
for each good given prices p in the economy and his initial endowment ei . However for an
arbitrary price vector p it may not be possible actually to make the desired transaction for
P
the simple reason that the aggregate demand, i xi (p, p.ei ), may not equal the aggregate
P
supply i ei . Thus then the price adjusts to reach the equilibrium, which is defined below,
3
Definition 1.1 (Walrasian Equilibrium) A Walrasian equilibrium for the given pure
exchange consists of a pair (p∗ , x∗ ) such that
1. At prices p∗ , the bundle x∗i solves (GE-UMP) ∀i; and
2. All markets clear:
I
P
i=1
x∗i (p∗ , p∗ .ei ) ≤
positive excess demand.
I
P
ei , i.e. there is no good for which there is
i=1
The array of final consumption vectors x is known as the final consumption allocation.
The big questions are: does such an equilibrium (necessarily) exist for all economies?
And if it does, is it unique for an economy?
1.3
Existence of a Walrasian Equilibrium
First we will show that non-satiation and desirability of goods lead to demand
equalling supply at equilibrium. For this we use the following function,
Definition 1.2 (Aggregate Excess Demand Function) The aggregate excess demand
function is given by,
z(p) =
I
X
i=1
[xi (p, p.ei ) − ei ]
As each consumer’s demand function is continuous and homogeneous of degree
zero, the aggregate excess demand function also have these properties. Using this we can
state the following law,
Theorem 1.1 (Walras’ Law) For any price vector p the value of the excess demand is
identically zero, i.e.
p.z(p) ≡ 0
Proof. This follows from the individual consumer’s budget constraint and their
non-satiation. Multiply the aggregate excess demand with p,
p.z(p) =
I
X
i=1
[p.xi (p, p.ei ) − p.ei ]
4
But this is zero as xi (p, p.ei ) must satisfy the budget constraint p.xi = p.ei under nonsatiation ∀i = 1, ..., I.
Corollary 1.1 (Market Clearing) If demand equals supply in n − 1 markets and pn > 0,
then demand must equal supply in the nth market.
Here identically zero implies that the value of excess demand is zero for all prices.
What this is stating is that if the workers are non-satiated, then the value of aggregate
demand must equal the value of aggregate consumption from their budget constraints. But
does this lead to the conclusion that the number of goods consumed equals the number of
available supply (i.e. endowment)? After all, the Walrasian equilibrium is defined to be
where the total demand is less than or equal to total supply (i.e. feasible). To investigate
this see that in using Walras’ Law,
Corollary 1.2 If p∗ is a Walrasian equilibrium and zn (p∗ ) < 0, then p∗n = 0, i.e. goods in
excess supply at a Walrasian equilibrium must be a free good.
Proof. At Walrasian equilibrium we have z(p∗ ) ≤ 0. Then since prices are nonnegative, p∗ .z(p∗ ) ≤ 0. But if for zn (p∗ ) < 0 we had p∗n > 0, we would have p∗ .z(p∗ ) < 0
which would violate Walras’ Law.
But we can also assume the following about the individual goods,
Definition 1.3 (Desirability) If pn = 0 then zn (p) > 0 ∀n = 1, ..., N .
Thus we have,
Proposition 1.1 If all goods are desirable, then at a Walrasian equilibrium z(p∗ ) = 0, i.e.
demand will equal supply.
Proof. Corollary 1.2 and the desirability of individual goods imply that p∗n cannot
be zero at a Walrasian equilibrium. Thus p∗n > 0, but Walras’ Law implies then that
z(p∗ ) = 0.
5
We will now use Walras’ law to prove the existence of an equilibrium. First thing
we can do is to normalise prices to the following relative prices (remember the aggregate
excess demand is homogeneous of degree 0 so one can scale the prices with no effect on the
outcome) to reduce the dimension by 1,
Thus
PN
bn
n=1 p
pbn
pn = PN
bm
m=1 p
= 1, which means that the solution p∗ of the Walrasian equilibrium belongs
to the N − 1-dimensional unit simplex,
(
S N−1 =
p ∈ <N
+ :
N
X
n=1
)
pn = 1
We also use the following theorem,
Theorem 1.2 (Brouwer Fixed Point Theorem) If f : S N−1 → S N −1 is a continuous
function from the unit simplex to itself, there is some x∗ ∈ S N −1 such that x∗ = f (x∗ ).
Proof. We will prove this for N = 2, i.e. to show that for a continuous function
f : [0, 1] → [0, 1], ∃ some x∗ ∈ [0, 1] such that x∗ = f (x∗ ). Consider the function g(x) =
f (x) − x. Then we want to find x∗ where g(x∗ ) = 0. But as 0 ≤ f (x) ≤ 1 ∀x ∈ [0, 1],
g(0) = f (0) − 0 ≥ 0 and g(1) = f (1) − 1 ≤ 0. Thus using the intermediate value theorem,
for continuous f there must be some x∗ ∈ [0, 1] such that g(x∗ ) = f (x∗ ) − x∗ = 0.
Then,
Proposition 1.2 (Existence of Walrasian Equilibria) If z : S N −1 → <N is a continuous function that satisfies Walras’ law p.z(p) ≡ 0, then ∃ some p∗ ∈ S N−1 such that
z(p∗ ) ≤ 0.
Proof. (Varian p.321) Define a map g : S N−1 → S N−1 by1
gn (p) =
pn + max(0, zn (p))
for n = 1, ..., N
P
1+ N
m=1 max(0, zn (p))
1
Intuitively g is a function that increases the price of the good if that good is in excess demand (i.e.
zn (p) > 0).
6
Since
PN
n=1 gn (p)
= 1, g(p) is a point in the simplex S N−1 . It is also a continuous function
as z and the max function are continuous functions. Thus this satisfies the conditions for
Brouwer’s fixed point theorem and hence there is a p∗ such that p∗ = g(p∗ ), i.e.
p∗n =
p∗n + max(0, zn (p∗ ))
P
∗
1+ N
m=1 max(0, zn (p ))
It then suffices to show that p∗ is a Walrasian equilibrium. Rearranging this yields,
p∗n
N
X
m=1
max(0, zn (p∗ )) = max(0, zn (p∗ )) ∀n
Multiplying each of these N equations by zn (p∗ ) and summing up over n,
)( N
)
( N
N
X
X
X
∗
∗
∗
max(0, zn (p ))
pn zn (p ) =
zn (p∗ ) max(0, zn (p∗ ))
m=1
n=1
n=1
But the second {.} is zero by Walras’ law, so we have,
N
X
zn (p∗ ) max(0, zn (p∗ )) = 0
n=1
Now each term of this sum is greater than or equal to zero since each term is either 0 or
(zn (p∗ ))2 . But if any term were strictly greater than zero then the equality would not hold.
Hence every term must be zero, i.e.
zn (p∗ ) ≤ 0 ∀n = 1, ..., N
Note for this proof of existence, desirability is not required. The only requirements
are that the excess demand function be continuous and the Walras’ law, the latter in turn
requiring non-satiation.
1.4
Example 1
¡ A ¢a ¡ A ¢1−a
A
Let consumer A and B have utility functions uA (xA
and
x2
1 , x2 ) = x1
¡
¢
¡
¢
b
1−b
B
B
for consumption of goods 1 and 2. Each agent has an endowxB
uB (xB
1 , x2 ) = x1
2
ment of eA = (1, 0) and eB = (0, 1). The prices of the goods are given by p = (p1 , p2 ).
7
Now we already know the Marshallian demand functions for Cobb-Douglas utility functions
when incomes are mA and mB ; for example for A,
xA
1 (p, m1 ) =
xA
2 (p, m1 ) =
am1
p1
(1 − a)m1
p2
But here mA = p1 × 1 + p2 × 0 = p1 and mB = p1 × 0 + p2 × 1 = p2 . Thus,
xA
1 (p) =
xA
2 (p) =
xB
1 (p) =
xB
2 (p) =
ap1
=a
p1
(1 − a)p1
p2
bp2
p1
(1 − b)p2
=1−b
p2
The aggregate excess demand functions are then,
B
z1 (p) = xA
1 (p) + x1 (p) − eA1 − eB1
bp2
= a+
−1
p1
B
z2 (p) = xA
2 (p) + x2 (p) − eA2 − eB2
(1 − a)p1
=
+ (1 − b) − 1
p2
The first thing to note is that these aggregate excess demand functions are homogeneous of
degree 0 in p as,
btp2
− 1 = z1 (p)
tp1
(1 − a)tp1
− b = z2 (p)
tp2
z1 (tp) = a +
z2 (p) =
Check Walras’ Law,
p.z(p) = p1 z1 (p) + p2 z2 (p)
µ
µ
¶
¶
bp2
(1 − a)p1
= p1 a +
− 1 + p2
−b
p1
p2
= 0
8
as desired. The question is now whether we can find the Walrasian equilibrium price vector
∗
B ∗
p∗ = (p∗1 , p∗2 ) such that the market clears. For good 1 market to clear xA
1 (p ) + x1 (p )
should equal the aggregate endowment 1, or
bp∗2
=1
p∗1
1−a
p∗
⇒ 2∗ =
p1
b
a+
We know by Walras’ law the market for good 2 should also clear. Check this,
(1 − a)p∗1
+ (1 − b) = 1
p∗2
∗
B ∗
xB
1 (p ) + x2 (p ) =
at
p∗2
p∗1
=
1−a
b ,
i.e. the aggregate demand equals the aggregate endowment, as desired. Note
that only relative prices are determined in the equilibrium; a normal practice is to normalise
one of the prices to 1.
1.5
Example 2
(Mas-Colell, Whinston & Green, p.521) Let the utility functions be this time,
¡ A ¢−8
¡ B ¢−8 B
1
A
A 1
B
and uB (xB
+x2 , with endowment allocation
uA (xA
1 , x2 ) = x1 − 8 x2
1 , x2 ) = − 8 x1
8
1
of eA = (2, r) and eB = (r, 2), where r = 2 9 − 2 9 . Then A’s UMP Lagrangian optimisation
is,
max
x1 ,x2 ,λ
L = xA
1 −
The FOCs are,
¡
¢
1 ¡ A ¢−8
A
− λ p1 xA
x2
1 + p2 x2 − 2p1 − rp2
8
1
p1
µ ¶− 1
9
p2
A
= 0 ⇒ x2 =
p1
µ ¶ µ ¶8
p2
p2 9
A
= 0 ⇒ x1 = 2 + r
−
p1
p1
1 − λp1 = 0 ⇒ λ =
¡
xA
2
¢−9
− λp2
A
p1 xA
1 + p2 x2 − 2p1 − rp2
And by symmetry,
xB
1
xB
2
=
µ
p1
p2
¶− 1
= 2+r
9
µ
p1
p2
¶
−
µ
p1
p2
¶8
9
9
Thus for good 1 market clearance,
¶8
+
µ
By substituting for r, this has three solutions
P2
p1
= 2, 1, 12 .
2+r
1.6
µ
p2
p1
¶
−
µ
p2
p1
9
p1
p2
¶− 1
9
=2+r
Uniqueness of Equilibrium
There are more than one ristrictions on the excess demand z that would ensure
the equilibrium to be unique. Here we mention just one of those, using the idea of gross
substitutes,2
Proposition 1.3 If all goods are gross substitutes at all prices, then if p∗ is an equilibrium
price vector, it is the unique equilibrium price vector.
Proof. Suppose p0 is some other equilibrium price vector. Since p∗ > 0 we can
define m = max
p0i
p∗i
6= 0. Let the good at which m is attained be denoted by k, i.e. m =
0
pk
p∗k .
By homogeneity and the fact that p∗ is an equilibrium, we know that z(p∗ ) = z(mp∗ ) = 0.
Now lower each price mp∗i other than pk successively to p0i . Since the price of each good
other than k goes down in the movement from mp∗ to p0 , we must have the demand for
good k going down. Thus zk (p) < 0 which implies that p0 cannot be an equilibrium.
1.7
Welfare Economics
Before we talk about the normative implications of Walrasian equilibria, first we
define the following concept of efficiency,3
Definition 1.4 (Pareto Efficiency) A feasible allocation x is said to be Pareto efficient
if there exists no other feasible allocation x0 such that all agents weakly prefer x0 to x, and
some agent strictly prefers x0 to x.
2
∂z (p)
j
Two goods i and j are gross substitutes at a price vector p if ∂p
> 0 for i 6= j.
i
3
Strictly speaking this is the definition of strong Pareto efficiency. See Varian p.323 for the definition of
weak Pareto efficiency, and the proof that the two are equivalent when the preferences are continuous and
monotonic. Weak Pareto efficiency implies that at optimal x, you cannot make everyone better off.
10
This implies that at Pareto efficient x, you cannot make one person better off
without making hurting someone else. Next slightly modify the definition of the Walrasian
equilibrium,
Definition 1.5 (Walrasian Equilibrium) A Walrasian equilibrium for the given pure
exchange consists of a pair (p∗ , x∗ ) such that
1. If agent i prefers x0i to x∗i , it must be that p.x0i > p.ei ; and
2. All markets clear:
I
P
i=1
x∗i =
I
P
ei .
i=1
This definition is equivalent to the original definition, as long as the desirability
assumption is satisfied, i.e. there are no free goods. Now we can state the following,
Theorem 1.3 (First Theorem of Welfare Economics) If (x, p) is a Walrasian Equilibrium, then x is Pareto efficient.
Proof. Prove this by obtaining a contradiction. Suppose (x, p) is a Walrasian
Equilibrium but x is not Pareto efficient. That is, suppose there is some other feasible
allocation x0 which is weakly preferred to x by all agents, and strictly by some. Then,
1. Since x0 is feasible,
I
P
i=1
p.x0i ≤
I
P
p.ei , given non-negative prices.
i=1
2. Since x0 Pareto dominates x, all consumers must like x0i as much as xi , and some
consumer must strictly prefer x0i to xi . Since x is an equilibrium allocation, it follows
from the definition of equilibrium that p.x0i ≥ p.ei for all i, and p.x0i > p.ei for some
i. Summing up over i we have,
I
X
p.x0i >
i=1
I
X
p.ei
i=1
1. and 2. contradict.
Note that a market equilibrium is not ‘optimal’ in any ethical sense, since the
market equilibrium may be very ‘unfair’. Now conversely we can state that every Pareto
efficient allocation is a Walrasian equilibrium,
11
Theorem 1.4 (Second Theorem of Welfare Economics) Assume that preferences are
convex, continuous, non-decreasing and locally non-satiable. Let x∗ be a Pareto efficient allocation which is strictly positive (i.e. x∗in > 0 ∀i, n). Then if we redistribute endowments
among all consumers suitably, x∗ can be obtained as a Walrasian equilibrium allocation.
Proof. (Varian p.326 - slightly different version) Given x∗ , let
©
ª
Zi = zi ∈ <N : zi Âi x∗i
This is the set of all consumption bundles that agent i prefers to x∗i . Then define
(
)
I
I
X
X
∗
N
Zi = z ∈ < : z =
zi with zi ∈ Zi
Z =
i=1
i=1
i.e. the set of bundles that can be allocated among the I consumers in a manner that
strictly Pareto dominates x∗ . Further let,
(
+
Z =
N
z∈<
:z≤
I
X
x∗i
i=1
)
Since each Zi is convex from the fact that preferences are convex, Z ∗ is convex as the sum
of convex sets is convex. Z + is obviously convex. Now Pareto efficiency of x∗ ensures that
Z ∗ and Z + do not intersect. Then by the Separating Hyperplane Theorem,4 there exists a
non-zero, N -dimensional vector (call it p) and a scaler (call it b) such that
p.z ≤ b ∀z ∈ Z + and
p.z ≥ b ∀z ∈ Z ∗
The claim is that this p, combined with x∗ , forms a Walrasian equilibrium. In particular
we must have p.x∗ = b, and for any redistribution e such that p.e∗i = p.x∗i ∀i, (p, x∗ ) must
be a Walrasian equilibrium. In effect we have found a set of prices p which support the
allocation x∗ as an equilibrium. What is required is to ensure that p is a plausible price
vector, i.e. it is non-negative, and if agent i strictly prefers yi to xi , then p.yi > p.xi .
These follow from the monotonicity, continuity and local non-satiation assumptions of the
preferences.
4
The Separating Hyperplane Theorem states that if A and B are two non-empty, disjoint, convex sets in
< , then there exists a linear functional p such that p.x ≥ p.y ∀x ∈ A, y ∈ B.
N
12
Chapter 2
General Equilibrium with
Production
Varian Ch.18.1-18.6
2.1
Walrasian Equilibrium with Production
In this chapter we consider an economy with production. This introduces three
additional features into the model, more goods to distribute, the issue of labour supply, and
profits. Consider an economy with,
1. a finite number of commodities, n = 1, 2, ..., N ,
2. a finite number of firms, j = 1, 2, ..., J,
3. a finite number of consumers, i = 1, 2, ..., I.
As well as the consumers’ preferences, here the production function is also assumed
given exogenously. The firms are assumed to be profit-maximisers operating in competitive
markets. The profits are distributed to an individual i according to his ownership θij > 0
of the firm j. Note then,
I
X
i=1
θij = 1 ∀j = 1, ..., J
13
This ownership of shares, and the initial endowments of goods, are assumed exogenously
given. This ownership now adds to the value of the endowment of the consumer. Thus if
yj (p) denote the production plan (i.e. negative/positive elements are inputs/outputs, and
thus p.yj is the profit) of the j th firm at prices p, the budget constraint for consumer i is
now,
p.xi (p) =
J
X
θij p.yj (p) + p.ei
j=1
The consumers are assumed to be utility maximisers, and thus assuming strict convex
preferences we can derive the Marshallian demand functions xi (p) given prices p. Then the
aggregate excess demand function is given by,
z(p) =
I
X
i=1
xi (p) −
J
X
j=1
yj (p) −
I
X
ei
i=1
We can now repeat the exercise for the pure exchange economy. First,
Definition 2.1 (Walrasian Equilibrium) A Walrasian equilibrium for the given economy consists of a price vecor p, an array of production plans yj , one for each of J firms,
and an array of consumption plans x∗ , such that
1. for each individual i, the bundle xi (p) maximises utility at prices p subject to the
budget constraint
p.xi (p) =
J
X
θij p.yj (p) + p.ei
j=1
2. for each firm j, the production plan yj (p) maximises profits p.yj (p) at prices p; and
3. all markets clear:
I
X
i=1
xi (p) ≤
i.e. this holds element by element.
I
X
ei +
i=1
J
X
yj (p)
j=1
Also,
Theorem 2.1 (Walras’ Law) If z(p) is as defined above, then
p.z(p) = 0 ∀p
14
Proof. Substituting for z(p),
p.z(p) =
I
X
i=1
J
X
p.xi (p) −
j=1
I
X
p.yj (p) −
p.ei
i=1
Now using the budget constraints,
p.z(p) =
J
I X
X
θij p.yj (p) +
i=1 j=1
=
à I
J
X
X
j=1
i=1
θij
!
I
X
i=1
p.yj (p) −
= 0
p.ei −
J
X
J
X
j=1
p.yj (p) −
I
X
p.ei
i=1
p.yj (p)
j=1
Thus Walras’ law holds for the same reason that it holds in the pure exchange
ecnomy: each consumer satisfies his budget constraint, so the economy as a whole has to
satisfy an aggregate budget constraint. Thus this is only driven by the local non-satiation
assumption of individual preferences. As before then, if there is excess supply such that
zn (p) < 0, then pn = 0, i.e. it is a free good. However desirability implies that for any
commodity with pn = 0, zn (p) > 0. Thus desirability and non-satiation implies that all
markets clear exactly in a Walrasian equilibrium, i.e.
I
X
i=1
2.2
xi (p) =
I
X
i=1
ei +
J
X
yj (p)
j=1
Example
An economy with two goods (time, which can be consumed as leisure or supplied
as labour, and a consumption good), one firm, and I identical individuals. All agents (firms
1
and consumers) behave competitively. The firm has a production function f (l) = l 2 where
l is labour input. Each individual has an equal share in the firm, and endowment of L
hours of time, and utility function u(x1 , x2 ) = xa1 x21−a , where x1 is leisure consumed, x2
is the amount of the consumption good consumed, and 0 < a < 1. Find the Walrasian
equilibrium.
15
The way to solve this is to first solve the optimisation problems of each agents (i.e.
the firm and the consumers) assuming the prices as given, and then find the equilibrium
prices that clears the markets. Thus letting (w, p) be the wage rate and the price of the
consumption good, first consider the firm’s optimisation:
1
max pl 2 − wl
l
The FOC is
1 ∗− 1
pl 2 − w = 0
2
³ p ´2
⇒ l∗ =
2w
The SOC is
3
1 ³ p ´−3
1
< 0 as desired
− pl∗− 2 = − p
4
4 2w
Thus the firm’s labour demand function, the output supply function and the profit function
are, respectively,
³ p ´2
2w
1
p
y(p, w) = l 2 =
2w
l(p, w) =
1
π(p, w) = pl 2 − wl =
p2
4w
Next consider the consumers’ optimisation where the consumption bundle (x1 , x2 ) =
(L − li , ci ),
max xa1 x1−a
subject to wx1 + px2 = wL +
2
x1 ,x2
p2
π(p, w)
= wL +
I
4wI
The Marshallian demand functions are,
´
³
p2
¶
µ
a
wL
+
2
4wI
p
= L − li =
x1 p, wL +
4wI
w
³
´
p2
¶
µ
(1
−
a)
wL
+
2
4wI
p
=
x2 p, wL +
4wI
p
Thus the individual labour supply is,
li = (1 − a)L −
a ³ p ´2
I 2w
16
and hence the aggregate labour supply is,
l(p, w) = (1 − a)IL − a
³ p ´2
2w
The aggregate demand for the consumption goods is simply
¶
µ
p
wIL
+
c(p, w) = (1 − a)
p
4w
Now we can find the market clearing prices. For example for labour market clearance,
³ p ´2 ³ p ´2
=
(1 − a)IL − a
2w
2w
½µ
¶ ¾1
2
1−a
p
=2
IL
⇒
w
1+a
By Walras’ law the consumption goods market should also clear. Check this:
¶
µ
p
p
wIL
+
=
(1 − a)
p
4w
2w
½µ
¶ ¾1
2
1−a
p
=2
⇒
IL
w
1+a
as desired.
2.3
Welfare Economics
We will now modify the Welfare Theorems.
Theorem 2.2 (First Theorem of Welfare Economics) If (x, y, p) is a Walrasian Equilibrium, then x is Pareto efficient.
Proof. (Varian p.345) Suppose not, i.e. let (x, y, p) be a Walrasian Equilibrium
but there exists a Pareto dominating allocation (x0 , y0 ). Then,
1. Since x0 is feasible,
I
P
i=1
p.x0i ≤
I
P
i=1
p.ei +
J
P
j=1
p.yj , given non-negative prices.
17
2. Since consumers are utility maximisers,
p.x0i > p.ei +
J
X
j=1
θij p.yj ∀i = 1, ..., I
Summing over the consumers we have
I
X
p.x0i >
i=1
where we have used
PI
i=1 θ ij
I
X
p.ei +
i=1
J
X
p.yj
j=1
= 1.
1. and 2. contradict.
Theorem 2.3 (Second Theorem of Welfare Economics) Assume that preferences are
convex, continuous, non-decreasing and locally non-satiable, and the production sets are convex. Let (x, y) be a strictly positive Pareto efficient ‘consumption allocation-production plan
pair’. Then (x, y) is the ‘allocation plan’ in a Walrasian equilibrium if we first redistribute
endowments and share-holdings among consumers.
Proof. Uses Seperating Hyperplane Theorem as before.
2.4
Example
An economy consists of two non-satiated individuals (1 and 2) who supply labour
(L) and demand a consumption good (X). The individuals have identical preferences represented by the utility function Ui = Xia (1 − Li )1−a , 0 < a < 1, i = 1, 2. The two individuals
differ in their ability to produce the consumption good from a unit of labour: for each
unit of labour supplied they produce w1 and w2 respectively. The economy’s production
constraint is therefore X1 + X2 = w1 L1 + w2 L2 .
Let us find the economy’s utility possibility frontier. This is derived by maximising
the utility of one individual subject to the production constraint and the constraint that
the other individual obtains a specified value of utility U . We will treat this as an equalityconstrained optimisation problem, and assume interior solutions whose SOCs are satisfied.
18
The Lagrangian for individual 1’s utility maximisation problem is then,
£
¤
£(X1 , X2 , L1 , L2 , λ, µ) = X1a (1 − L1 )1−a − λ U − X2a (1 − L2 )1−a
−µ [X1 + X2 − w1 L1 − w2 L2 ]
The FOCs are,
X1 :
aX1a−1 (1 − L1 )1−a − µ = 0
(2.1)
X2 :
λaX2a−1 (1 − L2 )1−a − µ = 0
(2.2)
L1 :
− (1 − a)X1a (1 − L1 )−a + µw1 = 0
(2.3)
L2 :
− λ(1 − a)X2a (1 − L2 )−a + µw2 = 0
(2.4)
λ :
U − X2a (1 − L2 )1−a = 0
(2.5)
µ :
w1 L1 + w2 L2 − X1 − X2 = 0
(2.6)
Substituting (2.1) into (2.3) and (2.2) into (2.4) to eliminate µ yields,
(1 − a)X1
aw1
(1 − a)X2
= 1−
aw2
L1 = 1 −
(2.7)
L2
(2.8)
Substituting these into (2.6) then yields, after simplification,
X1 = a(w1 + w2 ) − X2
Then we can express U1 in terms of X2 only using this and (2.7),
U1 = X1a (1 − L1 )1−a
¶
µ
1 − a 1−a
=
X1
aw1
µ
¶
1 − a 1−a
=
{a(w1 + w2 ) − X2 }
aw1
But from (2.5) using (2.8),
U=
µ
1−a
aw2
¶1−a
(2.9)
X2
Inverting this for X2 and substituting into (2.8) thus yields an expression for U1 in terms
of w1 , w2 and U ,
U1 =
µ
1−a
aw1
)
¶
¶1−a (
µ
aw2 1−a
U
a(w1 + w2 ) −
1−a
19
Thus for any values of U2 = U ,
U1 =
µ
1−a
aw1
)
¶
¶1−a (
µ
aw2 1−a
U2
a(w1 + w2 ) −
1−a
(2.10)
which is the desired utility possibility frontier. Any point on this frontier is Pareto efficient.
Now consider the competitive equilibrium of this economy where the price of the
consumption good is normalised to 1 and the wages are given by wi . The Lagrangian for
each individual’s UMP is, for i = 1, 2,
£ = Xia (1 − Li )1−a + αi (wi Li − Xi )
It is easily seen that the Marshallian demands are,
Xi = awi
Li = a
which leads to the indirect utility function
Vi = aa (1 − a)1−a wia
The question is whether this competitive equilibrium lies on the utility possibility frontier.
To check this substitute for U2 in (2.10),
(
)
¶
µ
µ
¶
1 − a 1−a
aw2 1−a a
1−a a
U1 =
a (1 − a) w2
a(w1 + w2 ) −
aw1
1−a
= aa (1 − a)1−a w1a
= V1
Thus the equilibrium is Pareto efficient, demonstrating the FTWE.
Now consider the question whether any points on the utility possibility frontier can
be attained as a competitive equilibrium with suitable redistribution of income by means
of this lump-sum transfer. To do this consider a lump-sum transfer Xi = wi Li − Ti with
T1 = −T2 . The Lagrangian for the UMP is now,
£ = Xia (1 − Li )1−a + αi (wi Li − Ti − Xi )
20
which leads to the Marshallian demands
Xi = a(wi − Ti )
Ti (1 − a)
Li = a +
wi
and the indirect utility function
−(1−a)
Vi = aa (1 − a)1−a (wi − Ti ) wi
Thus using T1 = −T2 ,
−(1−a)
V1 = aa (1 − a)1−a (w1 − T1 ) w1
−(1−a)
V2 = aa (1 − a)1−a (w2 + T1 ) w2
Check that this satisfies (2.10) by substituting for V2 ,
(
)
¶
¶
µ
µ
1 − a 1−a
aw2 1−a a
−(1−a)
1−a
U1 =
a (1 − a)
(w2 + T1 ) w2
a(w1 + w2 ) −
aw1
1−a
−(1−a)
= aa (1 − a)1−a (w1 − T1 ) w1
= V1
This demonstrates the STWE.