# Pure exchange economy: Given an initial endowment are multiple equilibria possible?

Consider a pure exchange economy with two goods ($x_1,x_2$) and two consumers $A,B$. Both users have an initial endowment, $(\omega_1^A,\omega_2^A)$ and $(\omega_1^B,\omega_2^B)$ respectively. A price ratio $p^*$ is an equilibrium price ratio if after both users maximize their utility given their budget, that is $\forall i\in \left\{A,B\right\}$ they solve the problem \begin{align*} \max_{x_1^i,x_2^i} \ & U(x_1^i,x_2^i) \\ \\ \mbox{s.t. } & p \omega_1^i + \omega_2^i = p x_1^i + x_2^i, \end{align*} the maximizing bundles $(x_1^A,x_2^A),(x_1^B,x_2^B)$ are such that the markets are in equilibrium, meaning \begin{align*} x_1^A + x_1^B & = \omega_1^A + \omega_1^B \\ \\ x_2^A + x_2^B & = \omega_2^A + \omega_2^B. \end{align*}

Suppose that $U$ fulfils the usual conditions of convexity, monotonicity (or local non-satiation, pick whichever you like) and continuity. Given an initial endowment is it possible to have two different equilibrium price ratios? The ideal answer would give a simple example, but non-constructive proofs are also okay. I am especially interested in examples where both equilibria are interior points of the Edgeworth-box.

A graphical representation of the problem:

Green is the set of Pareto-optimal points, red and blue are indifference curves, thin lines are budget lines.

Yes. The Debreu version of the Sonnenschein-Mantel-Debreu theorem guarantees that excess demand has to satisfy very little restrictions if there are as many consumers as commodities.

An explicit example of multiple equilibria in a $2\times 2$-exchange economy can be found in

Shapley, L. S., and M. Shubik. “An Example of a Trading Economy with Three Competitive Equilibria.” Journal of Political Economy, vol. 85, no. 4, 1977, pp. 873–875.

In the example, both consumers have even quasi-linear preferences, which are (necessarily for nonuniqueness) linear in different commodities.

• It is a pleasure to have you on the site Michael. In case you are interested, see my follow up question. – Giskard Nov 14 '17 at 14:43
• @denesp Thank you for the kind word, I will take a look at it. – Michael Greinecker Nov 14 '17 at 14:45

Here is another example with two consumers (A and B), two goods (X and Y):

\begin{eqnarray*} u_A(x_A, y_A) & = & \min(x_A, y_A), \ \omega_A = (1, 0) \\ u_B(x_B, y_B) & = & \min(x_B, y_B), \ \omega_B = (0, 1) \end{eqnarray*}

In this case, every feasible allocation $((x_A, y_A), (x_B, y_B))$ satisfying $y_A = x_A$ is a competitive equilibrium, and is supported by price vectors $(p_x, p_y) \in \mathbb{R}^2_+$ such that $p_xx_A +p_yy_A = p_x$ holds. In other words, every $(p_x, p_y) \in \mathbb{R}^2_+ \setminus \{(0,0)\}$ supports some allocation i.e. $((x_A, y_A), (x_B, y_B)) = \left(\left(\dfrac{p_x}{p_x+p_y}, \dfrac{p_x}{p_x+p_y}\right), \left(\dfrac{p_y}{p_x+p_y}, \dfrac{p_y}{p_x+p_y}\right)\right)$ as a competitive equilibrium.