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} \ &amp; U(x_1^i,x_2^i) \\
\\
\mbox{s.t. } &amp; 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 &amp; = \omega_1^A + \omega_1^B \\
\\
x_2^A + x_2^B &amp; = \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.