# Finding Walrasian equilibria when Walrasian demands are not unique

I'm trying to solve the following excercise:

Find the Walrasian equilibria for a pure exchange economy where agents' ($$A$$ and $$B$$) preferences and endowments are given by:

$$u_A = x_A + y_A$$

$$u_B = 2 x_A + y_A$$

$$(\omega_{xA},\omega_{yA}) = (\frac{1}{2},\frac{1}{2})$$

$$(\omega_{xB},\omega_{yB}) = (\frac{1}{2},\frac{1}{2})$$

I think the specific parameters for the linear functions and endowments are needed for my issue regarding non-unique demands, that’s why I included them.

I computed the demands for both utilities as usual for linear functions, having set $$p_x = 1$$.

The demands I got (after replacing the endowments) are:

Agent $$A$$:

• $$p_y > 1$$

$$x_A^\star = \frac{1 + p_y}{2}, y_A^\star = 0$$

• $$p_y < 1$$

$$x_A^\star = 0, y_A^\star = \frac{1 + p_y}{2 p_y}$$

• $$p_y = 1$$

$$x_A^\star \in [0,1], y_A^\star = 1 - x_A^\star$$

Agent $$B$$:

• $$p_y > \frac{1}{2}$$

$$x_B^\star = \frac{1+p_y}{2}, y_B^\star = 0$$

• $$p_y < \frac{1}{2}$$

$$x_B^\star = 0, y_B^\star = \frac{1+p_y}{2 p_y}$$

• $$p_y = \frac{1}{2}$$

$$x_B^\star \in [0,\frac{3}{4}], y_B^\star = \frac{3}{2} - 2 x_B^\star$$

I then plot a $$\mathbb{R}^+$$ ray for $$p_y$$ with the demands by cases, taking into account the two different partitions generated by $$A$$ and $$B$$. In all cases where $$p_y \neq 1$$, I get that there cannot be any Walrasian equilibrium.

However, in the case where $$p_y = 1$$, $$A$$'s demands are not unique and I'd get that both markets would be in equilibrium if I chose $$x_A^\star = 0$$, but the markets wouldn't be in equilibrium if instead I chose any $$x_A^\star \in (0,1]$$.

So would $$p_y^\star = 1$$ count as a Walrasian equilibrium or not?

I know there is a way to graphically show the equilibrium situations but I don't understand it with linear functions where $$MRS$$ arguments don't work.

I would appreciate it very much if someone could support their answer with an Edgeworth box graph.

An equilibrium is usually defined as the combination of a price(s) and quantity(/quantities/allocation).

Here $$p_y = 1$$, $$x_A=0$$, $$y_A = 1$$, $$x_B=1$$, $$y_B=0$$ is an equilibrium, $$p_y = 1$$ is an equilibrium price, the consumptions define an equilibrium allocation.

This is true even though given $$p_y = 1$$ the consumers can plan individually optimal consumptions that would not constitute an equilibrium in the economy (i.e.; $$x_A = x_B = 1$$, which is not feasible).

• Thanks! I was confused as the definition I got for a Walrasian Equilibrium was a price vector (or relative prices). Mar 17 at 0:20

let us first write the demand functions of individual $$A$$ and $$B$$

$$(x_A^d,y_A^d)(p_x,p_y,m_A)\in\left\{\begin{matrix} (\frac{m_A}{p_x},0) & ,\frac{p_x}{p_y}<1\\ (0,\frac{m_A}{p_y}) & , \frac{p_x}{p_y}>1 \\ BL_A & , \frac{p_x}{p_y}=1 \end{matrix}\right. \\ where, \; BL_A: Budget \; Line \; of \; A$$

similarly, $$(x_B^d,y_B^d)(p_x,p_y,m_B)\in\left\{\begin{matrix} (\frac{m_B}{p_x},0) & ,\frac{p_x}{p_y}<2\\ (0,\frac{m_B}{p_y}) &, \frac{p_x}{p_y}>2 \\ BL_B & ,\frac{p_x}{p_y}=2 \end{matrix}\right.$$

In order to simplify our analysis, let us consider $$p_y=1$$ i.e., let good $$Y$$ be the numeraire. Using the value of the endowments we also get that $$m_A=m_B=\frac{p_x}{2}+\frac{1}{2}$$

now both demand functions become functions of $$p_x$$ only, so we need to use these demand functions and total endowments for $$X$$ to find $$p_x^*$$ such that demand for good $$X$$ equals the total endowment of good $$X$$ (supply of good $$X$$).

New demand functions are:

$$(x_A^d,y_A^d)(p_x)\in\left\{\begin{matrix} (0.5+\frac{0.5}{p_x},0) & ,p_x<1\\ (0,0.5p_x+0.5) & , p_x>1 \\ BL_A & , p_x=1 \end{matrix}\right. \; (x_B^d,y_B^d)(p_x)\in\left\{\begin{matrix} (0.5+\frac{0.5}{p_x},0) & ,p_x<2\\ (0,0.5p_x+0.5) &, p_x>2 \\ BL_B & ,p_x=2 \end{matrix}\right.$$

In market for good $$X$$:

Price Demand for $$X$$ Supply for $$X$$
$$p_x<1$$ $$1+\frac{1}{p_x}(>2)$$ $$1$$
$$p_x=1$$ $$[1,2] (\ni 1)$$ $$1$$
$$p_x \in (1,2)$$ $$0.5+\frac{0.5}{p_x}(<1)$$ $$1$$
$$p_x=2$$ $$[0,0.75]$$ $$1$$
$$p_x>2$$ $$0$$ $$1$$

Therefore, $$\boxed{\frac{p_x^*}{p_y^*}=1}$$ is the equilibrium price ratio supporting the allocation $$\boxed{((0,1),(1,0))}$$