7  The Existence of General Equilibrium

7.1 Introduction

This section states and proves an existence theorem for Walrasian Equilibrium.

7.2 Existence of equilibrium.

Let \(z(p)\) be aggregate excess demand.

Proposition 7.1 Assume that \(\succsim_{i}\) is continuous, strictly convex and strongly monotone. Suppose also that \(\sum_{i}\omega_{i} >> 0\). Then for all \(p >> 0\),

  1. \(z(\cdot)\) is continuous.
  2. \(z(\cdot)\) is homogenous of degree 0.
  3. \(p \cdot z(p) = 0\) for all \(p\). (Walras’ law)
  4. There is an \(s > 0\) such that \(z_{l}(p) > -s\) for every good \(l\) and all \(p\).
  5. If \(p^{n} \rightarrow p\), where \(p \ne 0\), and \(p_{l} = 0\) for some \(l\), then \(\max \{z(p^n)\} \rightarrow \infty\).

Definition 7.1 Given \(A \in R^N\) and the closed set \(Y \subset R^K\), the correspondence \(f:A \rightarrow Y\) is upper hemicontinuous if it has a closed graph and the images of compact sets are bouncded. That is, for every compact set \(B \subset A\), the set \(f(B)\) is bounded.

Proposition 7.2 (Kakutani) Suppose that \(A \subset R^N\) is a non-empty, compact, convex set and that \(f:A \rightarrow A\) is an upper hemicontinuous correspondence form \(A\) into itself with the property that that set \(f(A) \subset A\) is nonempty and onvex for every \(x \in A\). Then \(f()\) has a fixed point; that is, there is an \(x \in A\) such that \(x \in f(x)\).

Proposition 7.3 (Existence) Let \(z(p)\) be a function defined for all \(p \in \mathbb{R}^{L}_{++}\) satisfying 17.B.2. Then the system of equations \(z(p) = 0\) has a solution. Hence a Walrasian equilibrium exists for a pure exchange economy with strictly positive endowments and where consumers have continuous, strictly convex and strongly monotone preferences.

Proof. Step 1. Let \(f(p) = \{q \in \Delta: z(p)\cdot q \ge z(p)\cdot q' for all q' \in \Delta\}\).

Step 2. If \(p \in boundary \Delta\) \(f(p) = \{q \in \Delta: p\cdot q = 0\} = \{q \in \Delta: q_{l} = 0 \mbox{ if } p_{l} > 0\}\)

Step 3. A fixed point of \(f\) is an equilibrium. (DISCUSS BOUNDARY)

Step 4. The fixed point correspondence is convex valued and upper hemicontinuous. Convex valued by the definition. Upper-hemicontinuous: use a sequence and show that its limit is in the graph.

Step 5. A fixed point exists.