Market Equilibrium

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Consider a game with following characteristics:

- N players

- m divisible goods

- e_1,e_2,...,e_N \in \Reals^{m}_{+} endowment of each player, i.e. how much the player has of each good

- u_i: \Reals^{m}_{+} \rightarrow \Reals_{+}, ui strictly increasing and concave, the utility function of player i

- prices p\in \Reals^{m}_{+}, i.e. p is the price vector for the m goods. Assume that \sum_{i=1}^{m}p_i = 1

- wealth w_i=p\cdot e_i of each player

- demand for i (given prices p and wealth w) x_i(p,w_i) \in argmax_{x\in \Reals^{m},px\leq w} u_i(x), i.e. demand is a function x_i (p,w_i)\in\Reals^{m}_{+}


Define the aggregate excess demand as Z(p)=\sum_{i=1}^{N}(x_i(p,p\cdot e_i)-e_i), Z(p)\in \Reals^{m}.

We say the market clears if Z(p) = 0. In this case, the aggregate demand for each of the m goods equals the aggregate supply for the good.



[edit]

Theorem:

Given any market, there exist prices p \in \Delta_m such that the market clears.


[edit]

Proof:

We use Brouwer's Fixed Point Theorem:

Given a continuous function f: S\rightarrow S over a convex compact set S\subseteq \Reals^{n}, then \exists x\in S: f(x)=x.

In our case, let f: \Delta_m \rightarrow \Delta_m.

Define f_i(p)=\frac{p_i+max(0,z_i(p))}{1+\sum_{j}max(0,z_j(p))} with
∑fi(p) = 1
i
.

Then according to the Fixed Point Theorem \exists p^*: f(p^*)=p^*, i.e. fi(p * ) = p * for all i. We further assume p_i^*>0 for all i.

Thus we get p_i^*=\frac{p_i^*+max(0,z_i(p^*))}{C} where
C = 1 + ∑max(0,zj(p * ))
j


Consider two cases:

Case 1: C=1

In this case we have z_i(p^*)\leq 0 for i=1,...,m.

Further we have
z(p * )p * = ∑(xi(p * ,p * ei)p * − eip * ) = 0
i
, which yields zi(p * ) = 0 for all i.


Case 2: C>1

In this case we get p_i^*(C-1)=z_i(p^*), which yields zi(p * ) > 0 for all i (due to the assumption p_i^*>0). This, however, is a contradiction to
z(p * )p * = ∑(xi(p * ,p * ei)p * − eip * ) = 0
i
.


Thus, we must be in case 1. Since zi(p * ) = 0 for all i, the market clears at price p * .

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