Finding the competitive equilibrium in an economy with two consumers and two commodities

Question

I am unable to solve the following problem in general equilibrium.

Consider an economy with two consumers and two commodities X and Y. $X_i$ and $Y_i$ are the amounts of commodities present with the i-th consumer where $i=1$ or $i=2$.

The utilities of the consumers are as follows $$U_1(X_1,Y_1,X_2,Y_2)= \min\{X_1,Y_1,X_2,Y_2\} ,\quad U_2(X_1,Y_1,X_2,Y_2)= \min\{X_2,Y_2\} $$ where "min" is the function that takes the minimum value.

Initially consumer 1 is endowed with 10 units of good 1 and 0 units of good 2 whereas consumer 2 is endowed with 0 units of good 1 and 10 units of good 2. Solve for competitive equilibrium.

Progress

I know that consumer 2 would always choose a point on the line $X_2=Y_2$ when in equilibrium. So accordingly, the utility of consumer 1 becomes $\min\{X_1,Y_1,a\}$ where $a=X_2=Y_2$. But I don't know what to do next.

Answer 1

Given a pure exchange economy with externalities $\left((u_1,u_2),\left((\omega^{X}_1, \omega^{Y}_1), (\omega^{X}_2, \omega^{Y}_2)\right)\right)$, competitive equilibrium consists of prices $(p_X^*,p_Y^*)\in\mathbb{R}^2_+$ and an allocation $\left((x_1^*,y_1^*),(x_2^*,y_2^*)\right)\in\mathbb{R}^2_+\times\mathbb{R}^2_+$ satisfying the following:

• Given $(x_2^*,y_2^*)$ and $(p_X^*,p_Y^*)$,

$(x_1^*,y_1^*)$ solves the following problem \begin{eqnarray*} \max_{x_1\geq 0, \ y_1\geq 0} & u_1(x_1,y_1,x_2^*,y_2^*) \\ \text{s.t. } & p_X^*x_1+p_Y^*y_1 \leq p_X^*\omega^{X}_1+p_Y^*\omega_1^Y\end{eqnarray*}

• Given $(x_1^*,y_1^*)$ and $(p_X^*,p_Y^*)$,

$(x_2^*,y_2^*)$ solves the following problem \begin{eqnarray*} \max_{x_2\geq 0, \ y_2\geq 0} & u_2(x_1^*,y_1^*,x_2,y_2) \\ \text{s.t. } & p_X^*x_2+p_Y^*y_2 \leq p_X^*\omega^{X}_2+p_Y^*\omega_2^Y\end{eqnarray*}

• Feasibility: $x_1^*+x_2^* = \omega^{X}_1 + \omega^{X}_2$ and $y_1^*+y_2^* = \omega^{Y}_1 + \omega^{Y}_2$

We are given an economy with $u_1(x_1,y_1,x_2,y_2) = \min(x_1,y_1,x_2,y_2)$ and $u_2(x_1,y_1,x_2,y_2) = \min(x_2,y_2)$ and $\left((\omega^{X}_1, \omega^{Y}_1), (\omega^{X}_2, \omega^{Y}_2)\right)=\left((10, 0), (0, 10)\right)$, here are all the competitive equilibria:

• $(p_X^*,p_Y^*) = (1,0)$ and allocation $\left((x_1^*,y_1^*),(x_2^*,y_2^*)\right)=\left((10, 7), (0, 3)\right)$. In fact, any feasible allocation with $x_1^*=10$ and $y_1^*\geq 0$ is a competitive equilibrium allocation at prices $(p_X^*,p_Y^*) = (1,0)$.

• $(p_X^*,p_Y^*) = (0,1)$ and allocation $\left((x_1^*,y_1^*),(x_2^*,y_2^*)\right)=\left((0, 0), (10, 10)\right)$

• Any $(p_X^*,p_Y^*)$ with $p_X^*>0, p_Y^*=1$, and corresponding allocation $\left((x_1^*,y_1^*),(x_2^*,y_2^*)\right)=\left(\left(\dfrac{10p_X^*}{p_X^*+1}, \dfrac{10p_X^*}{p_X^*+1}\right), \left(\dfrac{10}{p_X^*+1}, \dfrac{10}{p_X^*+1}\right)\right)$