An Algebraic Proof of the Existence of a Competitive Equilibrium in Exchange Economies

The standard excess demand argument for existence of competitive equilibria in exchange economies invokes maximizing the market value of the aggregate excess demand function and thereby adjusting the prices toward equilibrium. By exploiting the Perron-Frobenius theorem on stochastic matrices, we offer an algebraic proof of the existence of a competitive equilibrium without resorting to such a device of excess demand.


Introduction
The standard excess demand argument for existence of competitive equilibriain exchange economies invokes maximizing the value of the excess demand of the economy (the aggregate excess demand) and thereby adjusting the prices toward equilibrium [1]- [3].At least for pedagogical purpose, one may legitimately interpret the adjustment procedure, often referred to as tatonnement, as that in which a price-setting agency, the so-called Walrasian auctioneer, gathers the information as regards to each agent's excess demand and then fine-tunes the prices so as to raise the prices of the commodities that are over-demanded (with a positive aggregate excess demand registered) and to lower the prices of the commodities that are under-demanded (with a negative aggregate excess demand registered).It is true that the excess demand function is a very useful device that renders the fixed point theory nicely applicable to the proof of the existence of a market-clear price vector.Such a technical device, powerful though as it is in remarkably simplifying the formulation, nonetheless suggests a centralized coordination mechanism.Exploiting the Perron-Frobenius theorem on stochastic matrices, a well known result in linear algebra, we provide an alternative proof of the existence of a competitive equilibrium without resorting to such an artificial price-setting mechanism.It is worth pointing out that in so doing our argument does not invoke aggregating the individuals' excess demand, let alone to maximizing the market value of the aggregate excess demand.To the extent that the price-setting agency, as is embodied by the Walrasian auctioneer, personifies the coordination of the decentralized price system, "the invisible hand", and hence rendering it more or less visible, our approach appears to be conceptually more natural.But a cost of awkwardness in algebraic manipulation has to be paid in our undertaking, compared to the rather neat formulation based on the Walrasian tatonnement.However, such a cost may be well justifiable for an economically appealing argument about the important idea of the invisible hand.

The Proof
We first restate the classical result on the existence of a competitive equilibrium.
Theorem.For any pure exchange economy with n commodities and m agents, in which each agent { } such that the endowment of any commodity for the economy as a whole is positive, i.e., 1 0, there exists a competitive equilibrium.Proof.First of all, normalize the endowment of each commodity of the community as one, i.e., , Notice that for any ( ) , , , The sum of each column of F equals ( ) , hence F is a column stochastic matrix.Consequently, the Perron-Frobenius eigenvalue of F is one and there exists at least one corresponding eigenvector n q S ∈ , i.e., Fq 1 One may do so by considering ( ) ( ) for any agent τ and commodity and the prices adjusted to the accordingly changed units used to measure each commodity. 2A careful reader might be concerned with the closedness of the correspondence graph for some or all k , k y is not a limit point but a convex combinations of two vectors each of which is a limit point when one sequence of strictly positive price vectors approaches .such that there exist ( ) ( ) , 1, , , for any k where The correspondence from p to q is closed.
We now prove that n p S ∀ ∈ , {q(p)} is convex.Consider any two elements in {q(p)} say A q and , B q i.e., there exist ( ) ( ) Note the term in the brackets of LHS of Equation ( 1) equals ( ) ( ) Also notice that the RHS of Equation ( 1) equals Substituting the above and ( 2) into (1) yields, ( ) ( ) Thus for an agent  endowed with a positive amount of commodity 1 (existence of such agent is guaranteed by the assumption that Y p τ itself is not a limit point for any sequence of interior price vectors that approaches * p , but a convex combination of two such limit points, the above application applies to each of these two limit points.Thus, * 0.

.
is the percentage share of agent τ 's wealth spent on commodity i. Convexity of preferences implies that for any agent τ , semi-continuous by Berge's maximum theorem.For any hull of the limit points of any possible per-semi-continuous and convex-valued at any n p S ∈ ∂ 2 .Let

∂
. That is, for any n S ∈ , {q(p)} is a convex set.By Kakutani's theorem, there exists , one possibility is that there exist sequences { } ( )

=
for anys such that * 0 s p = , an impossibility in light of the strongly monotone preferences.In the case that ( ) ., all markets clear.
it is not a problem.Consider any τ ∈, is associated.Similar argument yet with more awkward notations applies to another scenario in which