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PURE EXCHANGE ECONOMY

• Let x = { x¹, ..., x^L } represent an allocation of goods to L consumers. (Each element of the above list is itself a vector of n goods.)
• Let w = { w¹, ..., w^L } represent the initial endowments of the L consumers.
• Let p = ( p₁, ..., p_n ) represent a vector of n prices.

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

• x^*l is preferred to x^l for all x^l that satisfies the budget constraint p^* x^l ≤ p^* w^l That is, all consumers maximize their utility.
• Demand does not exceed supply for each good. That is, Σ_l ( x^*l - w^l ) ≤ 0

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

• If each consumer has strictly increasing and strictly convex preferences, then z(p) is continuous.
• z(p) is homogeneous of degree 0.
• Walras' law holds: p z(p) = 0.

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 x^l = p w^l Sum over all l, we get p Σ_l x^l = p Σ_l w^l which is precisely Walras' law.

The practical significance of Walras' law is that if z₁ = z₂ = ... = z_n-1 = 0 and if p_n > 0, then z_n 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 : &sum_i=1ⁿ p_i = 1 }

Define the mapping g: S → S by

g_i(p) = [ p_i + max{0,z_i(p)} ] / [ 1 + ∑_j max{0,z_j(p)} ] Note that g is continuous and the range of g is in S because ∑_i g_i = 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,

p_i^* = [ p_i^* + max{0,z_i(p^*)} ] / [ 1 + ∑_j max{0,z_j(p^*)} ] Cross-multiply to get: p_i^* ∑_j max{0,z_j(p^*)} = max{0,z_i(p^*)} Multiply the i-th equation z_i(p^*) and sum over all the n equations: ∑_i p_i^* z_i(p^*) ∑_j max{0,z_j(p^*)} = ∑_i z_i(p^*) max{0,z_i(p^*)} By Walras' law, the left side of the above equation is 0, so ∑_i z_i(p^*) max{0,z_i(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 z_i(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 w^l < p x'^l Sum over all l and using the fact that x' is feasible, we arrive at a contradiction: p ∑_l w^l < p ∑_l x'^l = p ∑_l w^l

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

u^l(x^~l) ≥ u^l(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. u^l(x^~l) = u^l(x^*l) Now, this equation implies that if x^~l solves max u^l(x^l) s.t. p^~ x^l ≤ 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:

• Let y = { y¹, ..., y^K } represent the net output supply of K firms in the economy. (Input demands are treated as negative output. Again, each element of the list is a vector of n goods.)
• Let T^lk represent the share of firm k owned by consumer l.

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

• x^*l maximizes l's utility subject to the budget constraint p^* x^l ≤ p^* w^l + ∑_k T^lk p^* y^k
• p^* y^*k(k) ≥ p^* y^k for all y^k that is feasible given firm k's production technology. (That is, all firms maximize their profits.)
• Demand does not exceed supply for each good. That is, ∑_l ( x^*l - w^l ) ≤ ∑_k y^*k

It can be proved that a competitive equilibrium exists if

• preferences are convex (utility functions are quasi-concave)
• production sets are convex (production functions are concave)
• a number of technical assumptions a satisfied

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 U^A(X^A, Y^A)
subject to U^B(X^B, Y^B) = u
X^A + X^B = F^X(K^X, L^X)
Y^A + Y^B = F^Y(K^Y, L^Y)
K^X + K^Y = K
L^X + L^Y = L
Form the Lagrangian: L = U^A(X^A, Y^A) + λ (U^B (X^B,Y^B) - u) +
μ^X (F^X (K^X,L^X) - X^A - X^B) + μ^Y (F^Y (K^Y,L^Y) - Y^A - Y^B) +
ν^K (K - K^X - K^Y) + ν^L (L - L^X - L^Y) The FOCs are: (1) U^A_X - μ^X = 0
(2) U^A_Y - μ^Y = 0
(3) λ U^B_X - μ^X = 0
(4) λ U^B_Y - μ^Y = 0
(5) μ^X F^X_K - ν^K = 0
(6) μ^X F^X_L - ν^L = 0
(7) μ^Y F^Y_K - ν^K = 0
(8) μ^Y F^Y_L - ν^L = 0
The first two equations imply MRS^A(X,Y) = μ^X/μ^Y. The third and fourth equations imply MRS^B(X,Y) = μ^X/μ^Y. Therefore, we must have MRS^A(X,Y) = MRS^B(X,Y)

The fifth and sixth equations imply MRTS^X(K,L) = ν^K/ν^L. The last two equations imply MRTS^Y(K,L) = ν^K/ν^L. Therefore, we must have

MRTS^X(K,L) = MRTS^Y(K,L)

Equations (5) and (7) imply F^Y_K/ F^X_K = μ^X/μ^Y. 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 F^Y_L/ F^X_L, because equations (6) and (8) imply that this ratio is also equal to F^Y_K/ F^X_K. Since μ^X/μ^Y 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:

• MRS between any two goods must be the same across any pair of consumers--achieved by consumers maximizing utility
• MRTS between any two goods (factors) must be the same across any pair of producers--achieved by producers minimizing costs
• MRS between any two goods must be the same as their MRT--achieved by producers maximizing profits

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Lecture Notes

Competitive Markets Coase theorem