Exchange economy with two agents, what's the competitive equilibrium?

Question

I'm currently doing this assignment but I'm keep getting stuck by this question. I put the Lagrange function for both agents and get the MRS / Price ratio but what should I do from there?

Answer 1

I’m going to change the notation to one I’m more comfortable with:

Let the goods be $x,y$, the consumers $A,B$; and the respective endowments $(w_{x_A},w_{y_A}), (w_{x_B},w_{y_B})$.

Let’s set as numeraire $p_y = 1$.

Since both utilities are the same, we solve this problem once:

$\max - e^{-x_i} - e^{-y_i}$

s.t. $p_x x_i + y_i = p_x w_{x_i} + w_{y_i}$

The optimality condition is

$MRS_i = p_x$

$\frac{\frac{\partial U_i}{\partial x_i}}{\frac{\partial U_i}{\partial y_i}} = p_x$

$\frac{e^{-x_i}}{e^{-y_i}} = p_x$

$e^{y_i - x_i} = p_x$

$y_i - x_i = \ln(p_x)$

$y_i = x_i + \ln(p_x)$

Plugging into the budget constraint,

$p_x x_i + x_i + \ln(p_x) = p_x w_{x_i} + w_{y_i}$

$(p_x + 1) x_i = p_x w_{x_i} - \ln(p_x) + w_{y_i}$

Solving for $x_i$

${x_i}^\star = \frac{p_x w_{x_i} - \ln(p_x) + w_{y_i}}{p_x + 1}$

Plugging into the expression for $y_i$

${y_i}^\star = \frac{p_x w_{x_i} + p_x \ln(p_x) + w_{y_i}}{p_x + 1}$

Plugging the endowments

${x_A}^\star = \frac{p_x - \ln(p_x) + 5}{p_x + 1}$

${x_B}^\star = \frac{3 p_x - \ln(p_x) + 3}{p_x + 1}$

${y_A}^\star = \frac{p_x + p_x \ln(p_x) + 5}{p_x + 1}$

${y_B}^\star = \frac{3 p_x + p_x \ln(p_x) + 3}{p_x + 1}$

The market for good $x$ clears when

${x_A}^\star + {x_B}^\star = w_{x_A} + w_{x_B}$

$\frac{4 p_x - 2 \ln(p_x) + 8}{p_x + 1} = 4$

$4 p_x - 2 \ln(p_x) + 8 = 4 p_x + 4$

$4 = 2 \ln(p_x)$

$2 = \ln(p_x)$

${p_x}^\star = e^2$

By Walras’s Law, the market for good $y$ also clears.

Plugging the equilibrium price into the demand functions

${x_A}^\star = \frac{e^2 + 3}{e^2 + 1}$

${x_B}^\star = \frac{3 e^2 + 1}{e^2 + 1}$

${y_A}^\star = \frac{3 e^2 + 5}{e^2 + 1}$

${y_B}^\star = \frac{5 e^2 + 3}{e^2 + 1}$

Answer 2

An allocation $((c^*_{a,1},c^*_{a,2}),(c^*_{b,1},c^*_{b,2}))$ is a competitive equilibrium allocation for the given economy supported by the price ratio $\frac{p_1^*}{p_2^*}$ if it satisfies the following:
$1.$ Solution to UMP of $a$ and $b$ :
Given $(p_1^*,p_2^*)$, $(c^*_{a,1},c^*_{a,2})$ is a solution to: $$\begin{align} \max_{c_{a,1},c_{a,2}\geq0} \quad & U_a=-e^{-c_{a,1}}-e^{-c_{a,2}}\\ \textrm{s.t.} \quad & p_1^*c_{a,1}+p_2^*c_{a,2}\leq p_1^*+5p_2^*\end{align}$$ Given $(p_1^*,p_2^*)$, $(c^*_{b,1},c^*_{b,2})$ is a solution to: $$\begin{align} \max_{c_{b,1},c_{b,2}\geq0} \quad & U_b=-e^{-c_{b,1}}-e^{-c_{b,2}}\\ \textrm{s.t.} \quad & p_1^*c_{b,1}+p_2^*c_{b,2}\leq 3p_1^*+3p_2^*\end{align}$$
$2.$ Market Clearing: The optimal demand obtained in $1.$ must be market-clearing i.e.,$$c^*_{a,1}+c^*_{b,1}=4\\ c^*_{a,2}+c^*_{b,2}=8$$

To solve for competitive equilibrium we first need the demand functions of $a$ and $b$.
Since we are interested in relative prices let us normalize price of good 2 to 1. Formally, $p_2\overset{set}{=}1$. This will enable us to find the demand functions in terms of $p_1$ alone.

UMP of agent a: $$\begin{align}\max_{c_{a,1},c_{a,2}\geq0} \quad & -e^{-c_{a,1}}-e^{-c_{a,2}}\\ \textrm{s.t.} \quad & p_1c_{a,1}+c_{a,2}\leq p_1+5\end{align}$$

solving above gives: $(c_{a,1},c_{a,2})^d(p_1)=\left(\frac{5+p_1-\ln p_1}{p_1+1},\frac{p_1(1+\ln p_1)+5}{p_1+1}\right)$

UMP of agent b: $$\begin{align}\max_{c_{b,1},c_{b,2}\geq0} \quad & -e^{-c_{b,1}}-e^{-c_{b,2}}\\ \textrm{s.t.} \quad & p_1c_{b,1}+c_{b,2}\leq 3p_1+3\end{align}$$

solving above gives: $(c_{b,1},c_{b,2})^d(p_1)=\left(\frac{3+3p_1-\ln p_1}{p_1+1},\frac{p_1(3+\ln p_1)+3}{p_1+1}\right)$

Both UMPs are standard ones with concave utility functions so you can use the lagrangian method to solve the above.

Now we can solve for the equilibrium price by using $2.$ and the demand functions.

Market for good 1 clears when: $c_{a,1}(p_1)+c_{b,1}(p_1)=4$ $$\begin{eqnarray} & \frac{5+p_1-\ln p_1}{p_1+1}+\frac{3+3p_1-\ln p_1}{p_1+1}=4 \\ & p_1=e^2\end{eqnarray}$$

Therefore, $((c^*_{a,1},c^*_{a,2}),(c^*_{b,1},c_{b,2}))=\left(\left(\frac{3+e^2}{e^2+1},\frac{3e^2+5}{e^2+1}\right),\left(\frac{1+3e^2}{e^2+1},\frac{5e^2+3}{e^2+1}\right)\right)$ is the competitive equilibrium allocation supported by the equilibrium price ratio $\frac{p_1^*}{p_2^*}=e^2$