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

<p>\begin{eqnarray*} u_A(x_A, y_A) &amp; = &amp; \min(x_A, y_A), \ \omega_A = (1, 0) \\ u_B(x_B, y_B) &amp; = &amp; \min(x_B, y_B), \ \omega_B = (0, 1) \end{eqnarray*}</p>

<p>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.</p>

CC BY-SA 4.0
2018-06-13T00:22:46.050Z
2018-06-13T03:11:43.667Z
2018-06-13T03:11:43.667Z
Amit
<p>Yes. The <a href="https://doi.org/10.1016/0304-4068(74)90032-9" rel="noreferrer">Debreu version</a> of the Sonnenschein-Mantel-Debreu theorem guarantees that excess demand has to satisfy very little restrictions if there are as many consumers as commodities.</p>

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

<blockquote>
  <p>Shapley, L. S., and M. Shubik. “<a href="http://www.jstor.org/stable/1828711" rel="noreferrer">An Example of a Trading Economy with
  Three Competitive Equilibria.</a>” Journal of Political Economy, vol.
  85, no. 4, 1977, pp. 873–875.</p>
</blockquote>

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

CC BY-SA 3.0
2017-11-14T14:43:20.090Z
It is a pleasure to have you on the site Michael. In case you are interested, see my [follow up question](https://economics.stackexchange.com/questions/19290/pure-exchange-economy-set-of-multiple-equilibria-endowments).
Giskard
CC BY-SA 3.0
2017-11-14T14:45:06.643Z
@denesp Thank you for the kind word, I will take a look at it.
Michael Greinecker
CC BY-SA 3.0
2017-11-14T14:35:21.673Z
2017-11-14T14:35:21.673Z
Michael Greinecker
<p>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*}</p>

<p>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.</p>

<p>A graphical representation of the problem:</p>

<p><a href="https://i.sstatic.net/C67NF.png" rel="noreferrer"><img src="https://i.sstatic.net/C67NF.png" alt="enter image description here"></a>
Green is the set of Pareto-optimal points, red and blue are indifference curves, thin lines are budget lines.</p>

<microeconomics><competitive-equilibrium><pure-exchange-economy><edgeworth-box>
Pure exchange economy: Given an initial endowment are multiple equilibria possible?
Giskard
# Pure exchange economy: Given an initial endowment are multiple equilibria possible?

(Asked by: username_0 on 2017-11-14)

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:

[![enter image description here](https://i.sstatic.net/C67NF.png)](https://i.sstatic.net/C67NF.png)
Green is the set of Pareto-optimal points, red and blue are indifference curves, thin lines are budget lines.

**Tags:** microeconomics, competitive-equilibrium, pure-exchange-economy, edgeworth-box


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## 2 answers

### Answer by username_2 on 2017-11-14. Score: 7 | Accepted answer

Yes. The [Debreu version](https://doi.org/10.1016/0304-4068(74)90032-9) 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.](http://www.jstor.org/stable/1828711)” 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.

> **Comments:**
> **username_0**: It is a pleasure to have you on the site username_2. In case you are interested, see my [follow up question](https://economics.stackexchange.com/questions/19290/pure-exchange-economy-set-of-multiple-equilibria-endowments).  
> **username_2**: @denesp Thank you for the kind word, I will take a look at it.  

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### Answer by username_1 on 2018-06-13. Score: 5

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.


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