General Equilibrium


PURE EXCHANGE ECONOMY

A competitive equilibrium (or Walrasian equilibrium) is represented by a list (x*, p*) such that

Define an aggregate excess demand function z(p) = ∑l x*l(p) - wl.

Proof of Walras' law. Since utility functions are strictly increasing, the budget constraint holds as an equality. The budget constraint for consumer l is

p xl = p wl
Sum over all l, we get
p Σl xl = p Σl wl
which is precisely Walras' law.

The practical significance of Walras' law is that if z1 = z2 = ... = zn-1 = 0 and if pn > 0, then zn must be zero. This says that in computing the competitive equilibrium, we just have to make sure that n-1 of the markets clear. Once this is satisfied, the n-th market also clears.

To prove the existence of a Walrasian equilibrium, we need a fixed-point theorem:

Brouwer's fixed-point theorem. If f: S → S is a continuous mapping from a compact and convex set S to itself, then there exists some x in S such that x=f(x).

Proof of existence of Walrasian equilibrium. Since excess demand functions are homogeneous of degree 0, whenever z(p*) ≤ 0, we have z(tp*) ≤ 0. In other words, whenever (x*, p*) is a competitive equilibrium, (x*, tp*) is also a competitive equilibrium. We normalize prices in such a way that they always sum to 1. Hence, we restrict our attention to prices that belong to the n-1 dimensional unit simplex:

S = { p in R+n : &sumi=1n pi = 1 }

Define the mapping g: S → S by

gi(p) = [ pi + max{0,zi(p)} ] / [ 1 + ∑j max{0,zj(p)} ]
Note that g is continuous and the range of g is in S because ∑i gi = 1. So g is a continuous mapping from S to S. By Brouwer's fixed-point theorem, there exists a p* such that p* = g(p*).

We want to show that this p* is a Walrasian equilibrium. From the fixed-point property of p*, we have, for i=1, ..., n,

pi* = [ pi* + max{0,zi(p*)} ] / [ 1 + ∑j max{0,zj(p*)} ]
Cross-multiply to get:
pi*j max{0,zj(p*)} = max{0,zi(p*)}
Multiply the i-th equation zi(p*) and sum over all the n equations:
i pi* zi(p*) ∑j max{0,zj(p*)} = ∑i zi(p*) max{0,zi(p*)}
By Walras' law, the left side of the above equation is 0, so
i zi(p*) max{0,zi(p*)} = 0
But each of the n terms in this sum is non-negative. So for the sum to be equal to 0, we must have zi(p*) ≤ 0 for each i.

Note that a Walrasian equilibrium need not be unique.

First Welfare Theorem. If an allocation (x, p) is a Walrasian equilibrium, then x is a Pareto efficient allocation.

Proof. Let x' be a feasible allocation that everyone prefers to x. Then, for every consumer l, the bundle x'l must be beyond l's budget:

p wl < p x'l
Sum over all l and using the fact that x' is feasible, we arrive at a contradiction:
p ∑l wl < p ∑l x'l = p ∑l wl

Second Welfare Theorem. Suppose x* is a Pareto efficient allocation in an economy with endowment vector w. Assume that preferences are convex. If the endowments are redistributed so that the new endowment vector is x*. Then x* is a competitive equilibrium allocation associated with this economy with endowment vector x*.

Proof. Since preferences are convex, the aggregate excess demand function for the economy with endowment vector x* is continuous, so a Walrasian equilibrium exists. Let (x~, p~) be a Walrasian equilibrium for this economy. We want to show that (x*, p~) is a Walrasian equilibrium.

Since in a Walrasian equilibrium everyone prefers the equilibrium bundle to his endowment bundle, we must have

ul(x~l) ≥ ul(x*l)
But x* is a Pareto efficient allocation. There cannot be any other feasible bundle that makes everyone strictly better off. So the above inequality must hold as an equality.
ul(x~l) = ul(x*l)
Now, this equation implies that if x~l solves
max ul(xl) s.t. p~ xl ≤ p~ x*l
then x*l must also be a solution to the same problem. In other words, x*l maximizes l's utility subject to the budget constraint at prices p~. Furthermore, x* ≤ w by definition of feasibility. So (p~, x*) must be a competitive equilibrium.

It should be noted that the endowments need not be redistributed to x* to make the second welfare theorem work. Any new endowment vector x' that satisfies p~ x'l = p~ x*l for all l will equally does the trick.

ECONOMY WITH PRODUCTION

We modify the description of the pure exchange economy by introducing the following:

A competitive equilibrium is represented by a list (x*, y*, p*) such that

It can be proved that a competitive equilibrium exists if

The convexity assumptions are sufficient to guarantee that the demand and supply functions are continuous in prices. Note that the assumption of convex production sets is not innocuous. For example, the usual U-shaped cost curves used in partial equilibrium analysis do not satisfy convexity. Note also that competitive equilibrium need not be unique. It can also be shown that the first and second welfare theorems continue to hold.

CHARACTERIZATION OF WELFARE THEOREMS USING MARGINAL CONDITIONS

Consider a simple model with two persons (A and B), two final goods (X and Y), and two inputs (K and L). Pareto efficiency is modeled by:

maximize UA(XA, YA)
subject to UB(XB, YB) = u
XA + XB = FX(KX, LX)
YA + YB = FY(KY, LY)
KX + KY = K
LX + LY = L
Form the Lagrangian:
L = UA(XA, YA) + λ (UB (XB,YB) - u) +
μX (FX (KX,LX) - XA - XB) + μY (FY (KY,LY) - YA - YB) +
νK (K - KX - KY) + νL (L - LX - LY)
The FOCs are:
(1) UAX - μX = 0
(2) UAY - μY = 0
(3) λ UBX - μX = 0
(4) λ UBY - μY = 0
(5) μX FXK - νK = 0
(6) μX FXL - νL = 0
(7) μY FYK - νK = 0
(8) μY FYL - νL = 0
The first two equations imply MRSA(X,Y) = μXY. The third and fourth equations imply MRSB(X,Y) = μXY. Therefore, we must have
MRSA(X,Y) = MRSB(X,Y)

The fifth and sixth equations imply MRTSX(K,L) = νKL. The last two equations imply MRTSY(K,L) = νKL. Therefore, we must have

MRTSX(K,L) = MRTSY(K,L)

Equations (5) and (7) imply FYK/ FXK = μXY. The left hand side of this equality is called the marginal rate of transformation between X and Y. That is, it tells us how much additional Y can be produced if the output of X is reduced by one unit. Note that MRT can also be defined as FYL/ FXL, because equations (6) and (8) imply that this ratio is also equal to FYK/ FXK. Since μXY equals MRS(X,Y), Pareto efficiency requires

MRT(X,Y) = MRS(X,Y)
In summary, Pareto efficient requires the following three sets of marginal conditions to be satisfied:


[up] Lecture Notes

[previous] Competitive Markets [next] Coase theorem