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
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:
Define the mapping g: S → S by
We want to show that this p* is a Walrasian equilibrium. From the fixed-point property of p*, we have, for i=1, ..., n,
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:
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
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:
The fifth and sixth equations imply MRTSX(K,L) = νK/νL. The last two equations imply MRTSY(K,L) = νK/νL. Therefore, we must have
Equations (5) and (7) imply FYK/ FXK = μX/μY. 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 μX/μY equals MRS(X,Y), Pareto efficiency requires