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 \(p >> 0\),

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

Definition 7.1 Given \(A \subset \mathbb{R}^{N}\) and a closed set \(Y \subset \mathbb{R}^{K}\), the correspondence \(f:A \rightarrow Y\) is upper hemicontinuous if it has a closed graph and maps compact sets to bounded sets: for every compact set \(B \subset A\), the image \(f(B)\) is bounded.

Proposition 7.2 (Kakutani) Suppose that \(A \subset \mathbb{R}^{N}\) is a non-empty, compact, convex set and that \(f:A \rightarrow A\) is an upper hemicontinuous correspondence from \(A\) into itself such that the set \(f(x) \subset A\) is non-empty and convex 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 the properties of Proposition 7.1. 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. Throughout, let \[\Delta = \Big\{ p \in \mathbb{R}^{L}_{+} : \textstyle\sum_{l=1}^{L} p_{l} = 1 \Big\}\] be the unit price simplex. Because \(z\) is homogeneous of degree zero (Proposition 7.1), normalizing prices to lie in \(\Delta\) is without loss of generality, and a solution of \(z(p)=0\) with \(p \gg 0\) is a Walrasian equilibrium. The proof constructs a correspondence on \(\Delta\) whose fixed points are exactly the equilibrium price vectors, and then applies Kakutani’s theorem.

Step 1 (the correspondence on the interior). For \(p \gg 0\), where \(z(p)\) is defined, let \[f(p) = \Big\{ q \in \Delta : q \cdot z(p) \ge q' \cdot z(p) \text{ for all } q' \in \Delta \Big\}\] be the set of normalized price vectors that maximize the value of excess demand. Since \(q \cdot z(p) = \sum_{l} q_{l} z_{l}(p)\) is linear in \(q\), the maximum over \(\Delta\) is attained by placing all weight on the goods with the largest excess demand; that is, \(f(p)\) is the convex hull of \(\{ e_{l} : z_{l}(p) = \max_{k} z_{k}(p) \}\), where \(e_{l}\) is the \(l\)-th unit vector. Informally, \(f\) raises the relative price of whatever is in greatest excess demand.

Step 2 (the correspondence on the boundary). For \(p \in \partial\Delta\) — that is, \(p_{l} = 0\) for some \(l\), where \(z(p)\) need not be defined — let \[f(p) = \{ q \in \Delta : q \cdot p = 0 \} = \{ q \in \Delta : q_{l} = 0 \text{ whenever } p_{l} > 0 \}.\] This set is non-empty: if \(p_{l} = 0\) then \(e_{l} \in \Delta\) and \(e_{l} \cdot p = 0\).

Step 3 (\(f\) satisfies the hypotheses of Kakutani’s theorem). The simplex \(\Delta\) is non-empty, compact, and convex, and \(f\) maps \(\Delta\) into itself. Each value \(f(p)\) is non-empty (Steps 1–2) and convex: on the interior it is the set of maximizers of a linear function over the convex set \(\Delta\), and on the boundary it is the intersection of \(\Delta\) with the hyperplane \(\{ q : q \cdot p = 0 \}\). By Definition 7.1 it remains only to show that \(f\) has a closed graph. So take \(p^{n} \to p\) and \(q^{n} \to q\) with \(q^{n} \in f(p^{n})\); we must show \(q \in f(p)\).

First record a uniform upper bound on any single excess demand. For \(p \gg 0\), Walras’ law \(p \cdot z(p) = 0\) gives, for each good \(l\), \[p_{l} z_{l}(p) = -\sum_{k \ne l} p_{k} z_{k}(p) < s \sum_{k \ne l} p_{k} \le s,\] using the lower bound \(-z_{k}(p) < s\) and \(\sum_{k} p_{k} = 1\). Hence \[z_{l}(p) < \frac{s}{p_{l}} \qquad \text{for every good } l \text{ and every } p \gg 0. \tag{7.1}\]

Interior limit. If \(p \gg 0\), then \(p^{n} \gg 0\) for all large \(n\) and \(z\) is continuous at \(p\), so \(z(p^{n}) \to z(p)\). Passing to the limit in \(q^{n} \cdot z(p^{n}) \ge q' \cdot z(p^{n})\) (which holds for every \(q' \in \Delta\)) gives \(q \cdot z(p) \ge q' \cdot z(p)\) for all \(q' \in \Delta\), i.e. \(q \in f(p)\).

Boundary limit. If \(p \in \partial\Delta\), fix any good \(l\) with \(p_{l} > 0\); we show \(q_{l} = 0\). Take \(n\) large enough that \(p^{n}_{l} > 0\) (possible since \(p^{n}_{l} \to p_{l} > 0\)). If \(p^{n} \in \partial\Delta\), then \(q^{n} \in f(p^{n})\) forces \(q^{n}_{l} = 0\) directly, because \(p^{n}_{l} > 0\). If instead \(p^{n} \gg 0\), then by Equation 7.1 the excess demand \(z_{l}(p^{n}) < s/p^{n}_{l}\) stays bounded, whereas the boundary condition of Proposition 7.1 gives \(\max_{k} z_{k}(p^{n}) \to \infty\); so good \(l\) is eventually not among the maximizers, and since \(q^{n}\) places weight only on maximizing goods, again \(q^{n}_{l} = 0\). In either case \(q^{n}_{l} = 0\) for all large \(n\), so \(q_{l} = \lim_{n} q^{n}_{l} = 0\). As this holds for every \(l\) with \(p_{l} > 0\), we have \(q \cdot p = 0\), i.e. \(q \in f(p)\).

In both cases \(q \in f(p)\), so \(f\) has a closed graph and is therefore upper hemicontinuous.

Step 4 (a fixed point exists). By Steps 1–3, \(f : \Delta \rightarrow \Delta\) satisfies every hypothesis of Kakutani’s theorem (Proposition 7.2). Hence there is a price vector \(p^{*} \in \Delta\) with \(p^{*} \in f(p^{*})\).

Step 5 (the fixed point is a Walrasian equilibrium). We first show \(p^{*} \gg 0\). Suppose instead \(p^{*} \in \partial\Delta\). Then \(f(p^{*}) = \{ q : q \cdot p^{*} = 0 \}\), and \(p^{*} \in f(p^{*})\) would force \(p^{*} \cdot p^{*} = \sum_{l} (p^{*}_{l})^{2} = 0\), hence \(p^{*} = 0\) — impossible, since \(\sum_{l} p^{*}_{l} = 1\). So \(p^{*}\) is interior and \(z(p^{*})\) is defined.

Because \(p^{*} \gg 0\), the fixed-point condition \(p^{*} \in f(p^{*})\) says that \(p^{*}\) maximizes \(q \cdot z(p^{*})\) over \(q \in \Delta\). By Walras’ law the maximized value equals \(p^{*} \cdot z(p^{*}) = 0\), so \[q \cdot z(p^{*}) \le 0 \qquad \text{for every } q \in \Delta.\] Taking \(q = e_{l}\) gives \(z_{l}(p^{*}) \le 0\) for every good \(l\): no market is in excess demand. Applying Walras’ law once more, \[0 = p^{*} \cdot z(p^{*}) = \sum_{l} p^{*}_{l}\, z_{l}(p^{*})\] is a sum in which every \(p^{*}_{l} > 0\) and every \(z_{l}(p^{*}) \le 0\). Such a sum can vanish only if each term is zero, so \(z_{l}(p^{*}) = 0\) for all \(l\). Hence \(z(p^{*}) = 0\): every market clears, and \(p^{*}\) is a Walrasian equilibrium. \(\qquad\blacksquare\)