Competitive Equilibrium

Consider a Marketplace consisting of N consumers and L different commodities. Each consumer i has an initial endowment vector w_i of L commodities. Only activities that consumers can perform is to consume and/or trade their commodities (no production is allowed - only pure exchanges). Each consumer has h(er/is) own preferences, described using a utility function u. Let us say that given a vector of quantities of L commodities, the utility function for a given user maps it to a positive real number. Objective of each of the consumers is to maximize their utility by performing pure exchanges of the given commodities. There are natural constraints that the consumers must satisfy. The wealth of a consumer is h(er/is) initial endowment, and the total amount of commodities that a consumer can acquire (or buy) is at most equal to h(er/is) initial wealth (i.e. the commodities that the consumer sells off). The problem we want to address is how to figure out how the exchanges be performed so that each consumer increases the utility? Also is there an equilibrium, an optimum value for each consumer, so that no further exchanges benefit consumers as a whole (only way to increase utility of a consumer is by decreasing the utility of another). To add to complications, the utility functions are private, otherwise the consumers will tend to lie!

To solve this problem, prices are set for each of the commodities. (Call this the price vector p.) Now the consumer i buys the vector of commodities x_i, such that its utility u_i(x_i) is maximized subject to the constraint that amount that the agent pays for acquiring the commodities x_i is at most the amount the agent received for h(er/is) initial endowment (i.e. $p. x_i \le p. w_i$). We solve this maximization problem for each of the consumers. With a given price vector it is possible that the market may not clear. There may be excess demand for certain commodities. In that case the price of that commodity will be raised. The market will clear if all the commodities have been purchased (i.e. sum total of the demand for a commodity equals the sum total of the initial endowment of that commodity, namely Walras Law). Now we need to worry about whether can we find a price vector such that the market clears? We will formalize all this and state and prove the equilibrium theorem.

Let consumer i's endowment vector be w_i= (w_i1, w_i2, ..., w_iL), and u_i be its utility function corresponding to the locally non-satiated preference relation. Denote consumer i's excess demand vector for any given distribution of endowments as z_i(p), where

The vector of aggregate excess demands Z(p), can be described as follows:

The Walrasian equilibrium in this pure exchange economy will be that price vector p^* and allocation x^*, such that all markets clears, i.e. Z(p^*)=0. It turns out that the demand for each consumer x_i will be such that the budget constraint will be binding, i.e., p^*.x_i = p^*.w_i (Walras Law). Summing up these budget constraints for all consumers gives us

p^*.Z(p^*) = 0.

Lemma 1 (From L. Felli's notes)   Let $\{p^*, x^*\}$ be a Walrasian Equilibrium then

1.

if p^*_l > 0, then Z_l(p^*)=0, and

2.

if Z_l(p^*) < 0, then p^*_l=0.

Proof: Walras Law implies that p^*.Z(p^*) = 0, i.e., $\sum_{i=1}^L {p^*}_l.Z_l(p^*) =0$. Note that $Z_l(p^*)\le 0$. We will show that $p^*_l \ge 0$ for all l=1,...,L. Hence the lemma follows.

To show that $p^*_l \ge 0$, for all l=1,..,L, assume that its not true for at least one value of l. Then its easy to see that the utility maximization problem will have no solution (an unbounded value).

subject to

If x_l>0, then increasing x_l unboundedly will not decrease u(x). But then $x_i (i \not= l)$ can be increased unboundedly, and hence the utility can be increased unboundedly, a contradiction to the existence of an optimum. $\Box$

Theorem 2 (Arrow-Debreu Theorem 1954 - Our proof follows the proof from the course notes of Papadimitriou (Lecture 2)   compiled by A. Frome and K. Talwar and the lecture slides of L. Felli (Microeconomics II: Lectures 8/9)). Consider a pure exchange economy. Let Z(p) be the vector of excess demands given by the following equation.

Furthermore, we assume that Z(p) satisfies the following:

1.

Z(p) is a function, continuous and bounded.

2.

Z(p) is homogeneous of degree 0 (scaling of both the price and the initial endowment by the same factor does not alter choices for the consumers).

3.

Z(p) satisfies Walras's law, i.e., p Z(p)=0.

It can be shown that there exists a price vector p^* and an allocation x^* such that Walrasian Equilibrium is achieved, i.e. $Z(p^*) \le 0$.

Proof: The main idea of the proof is the following. Prices will be normalized and restricted to an L-dimensional simplex. If there is an excess demand for a commodity, then its price will be raised and the other prices will be normalized so that they are restricted to simplex. Simplex is compact and convex. It turns out that this process is a continuous mapping of points in simplex to points in simplex, and satisfies all the properties required in Brouwer's fix point theorem. This implies that there is a price vector, which corresponds to the fix point. We will prove that this price vector is a Walrasian Equilibrium Price Vector.

Let $\sum_{i=1}^L p_i = 1$. Define $z'(p)=(z'_1(p), z'_2(p), \cdots, z'_L(p))$, where $z_i'(p)=\max\{0, Z_i(p)\}$. The new price vector $\phi(p)$ is given by

where $\alpha\ge1$ is some normalization constant. It turns out the $\phi(p)$ is a continuous function, and is defined over the simplex, so has a fix point, say p^*. Now

Take the dot product with Z(p*) on both the sides we get

or

Notice that by Walras's law p.Z(p)= 0. Hence z'(p)=0, and therefore $Z(p^*) \le 0$. $\Box$