Finding Walrasian equilibria when Walrasian demands are not unique

External question — Economics Stack Exchange Author: Nicolas Torres Original post: https://economics.stackexchange.com/questions/54741 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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.

External answer — Economics Stack Exchange Author: Giskard Original post: https://economics.stackexchange.com/a/54742 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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).

External answer — Economics Stack Exchange Author: mynameparv Original post: https://economics.stackexchange.com/a/54747 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Two Agents: $\{A,B\}$ Two commodities: $\{X,Y\}$ Endowments: $\omega_A=\omega_B=\left(\frac{1}{2},\frac{1}{2}\right)$ Preferences: $u_A(x_A,y_A)=x_A+y_A \quad u_B(x_B,y_B)=2x_A+y_A$ Given preferences of individual $A$ , their demand correspondence is $$(x_A^d,y_A^d)(p_x,p_y,m_A)\in\left\{\begin{matrix} \left(\dfrac{m_A}{p_x},0\right) & \text{if }\dfrac{p_x}{p_y}<1\\ \left(0,\dfrac{m_A}{p_y}\right) & \text{if }\dfrac{p_x}{p_y}>1 \\ \{(x,y)\in \mathbb R^2_+\; \vert \; p_xx+p_yy=m_A\}& \text{if }\dfrac{p_x}{p_y}=1 \end{matrix}\right. $$ where $(p_x,p_y,m_A)$ denotes the prices of commodity $X$ and $Y$ , and the income of agent $A$ respectively. Using similar notation, the demand correspondence for individual $B$ is $$(x_B^d,y_B^d)(p_x,p_y,m_B)\in\left\{\begin{matrix} \left(\dfrac{m_B}{p_x},0\right) & \text{if }\dfrac{p_x}{p_y}<2\\ \left(0,\dfrac{m_B}{p_y}\right) & \text{if } \dfrac{p_x}{p_y}>2 \\ \{(x,y)\in \mathbb R^2_+\; \vert \; p_xx+p_yy=m_B\} & \text{if }\dfrac{p_x}{p_y}=2 \end{matrix}\right.$$ Let commodity $Y$ be the numeraire and normalize its price $p_y\equiv 1$ . Given the value of the endowments, we get that $m_A=m_B=0.5p_x+0.5$ and using this we can express the demand correspondences completely in terms of $p_x$ as follows $$(x_A^d,y_A^d)(p_x)\in\begin{cases} \left(0.5+\frac{0.5}{p_x},0\right) & \text{if }p_x<1\\ (0,0.5p_x+0.5) & \text{if } p_x>1 \\ \{(x,y)\in \mathbb R^2_+\; \vert \; p_xx+y=0.5p_x+0.5\} & \text{if }p_x=1 \end{cases} $$ $$(x_B^d,y_B^d)(p_x)\in\begin{cases} \left(0.5+\frac{0.5}{p_x},0\right) & \text{if }p_x<2\\ \left(0,0.5p_x+0.5\right) & \text{if } p_x>2 \\ \{(x,y)\in \mathbb R^2_+\; \vert \; p_xx+y=0.5p_x+0.5\}& \text{if }p_x=2 \end{cases}$$ We now check for market clearing conditions for commodity $X$ as done in the table below Price Demand for $X$ Supply for $X$ $p_x<1$ $1+\frac{1}{p_x}(>2)$ $1$ $p_x=1$ $[1,2]$ $1$ $p_x \in (1,2)$ $0.5+\frac{0.5}{p_x}(<1)$ $1$ $p_x=2$ $[0,0.75] $ $1$ $p_x>2$ $0$ $1$ Observe that the market for commodity $X$ clears only when $p_x = 1$ . Since $p_y = 1$ , Walras' Law ensures that the market for commodity $Y$ must also clear simultaneously. Hence, the competitive equilibrium allocation is $((0, 1), (1, 0))$ at the price vector $(p_x, p_y) = (1, 1)$ .