<p>I’m going to change the notation to one I’m more comfortable with:</p>
<p>Let the goods be <span class="math-container">$x,y$</span>, the consumers <span class="math-container">$A,B$</span>; and the respective endowments <span class="math-container">$(w_{x_A},w_{y_A}), (w_{x_B},w_{y_B})$</span>.</p>
<p>Let’s set as numeraire <span class="math-container">$p_y = 1$</span>.</p>
<p>Since both utilities are the same, we solve this problem once:</p>
<p><span class="math-container">$\max - e^{-x_i} - e^{-y_i}$</span></p>
<p>s.t. <span class="math-container">$p_x x_i + y_i = p_x w_{x_i} + w_{y_i}$</span></p>
<p>The optimality condition is</p>
<p><span class="math-container">$MRS_i = p_x$</span></p>
<p><span class="math-container">$\frac{\frac{\partial U_i}{\partial x_i}}{\frac{\partial U_i}{\partial y_i}} = p_x$</span></p>
<p><span class="math-container">$\frac{e^{-x_i}}{e^{-y_i}} = p_x$</span></p>
<p><span class="math-container">$e^{y_i - x_i} = p_x$</span></p>
<p><span class="math-container">$y_i - x_i = \ln(p_x)$</span></p>
<p><span class="math-container">$y_i = x_i + \ln(p_x)$</span></p>
<p>Plugging into the budget constraint,</p>
<p><span class="math-container">$p_x x_i + x_i + \ln(p_x) = p_x w_{x_i} + w_{y_i}$</span></p>
<p><span class="math-container">$(p_x + 1) x_i = p_x w_{x_i} - \ln(p_x) + w_{y_i}$</span></p>
<p>Solving for <span class="math-container">$x_i$</span></p>
<p><span class="math-container">${x_i}^\star = \frac{p_x w_{x_i} - \ln(p_x) + w_{y_i}}{p_x + 1}$</span></p>
<p>Plugging into the expression for <span class="math-container">$y_i$</span></p>
<p><span class="math-container">${y_i}^\star = \frac{p_x w_{x_i} + p_x \ln(p_x) + w_{y_i}}{p_x + 1}$</span></p>
<p>Plugging the endowments</p>
<p><span class="math-container">${x_A}^\star = \frac{p_x - \ln(p_x) + 5}{p_x + 1}$</span></p>
<p><span class="math-container">${x_B}^\star = \frac{3 p_x - \ln(p_x) + 3}{p_x + 1}$</span></p>
<p><span class="math-container">${y_A}^\star = \frac{p_x + p_x \ln(p_x) + 5}{p_x + 1}$</span></p>
<p><span class="math-container">${y_B}^\star = \frac{3 p_x + p_x \ln(p_x) + 3}{p_x + 1}$</span></p>
<p>The market for good <span class="math-container">$x$</span> clears when</p>
<p><span class="math-container">${x_A}^\star + {x_B}^\star = w_{x_A} + w_{x_B}$</span></p>
<p><span class="math-container">$\frac{4 p_x - 2 \ln(p_x) + 8}{p_x + 1} = 4$</span></p>
<p><span class="math-container">$4 p_x - 2 \ln(p_x) + 8 = 4 p_x + 4$</span></p>
<p><span class="math-container">$4 = 2 \ln(p_x)$</span></p>
<p><span class="math-container">$2 = \ln(p_x)$</span></p>
<p><span class="math-container">${p_x}^\star = e^2$</span></p>
<p>By Walras’s Law, the market for good <span class="math-container">$y$</span> also clears.</p>
<p>Plugging the equilibrium price into the demand functions</p>
<p><span class="math-container">${x_A}^\star = \frac{e^2 + 3}{e^2 + 1}$</span></p>
<p><span class="math-container">${x_B}^\star = \frac{3 e^2 + 1}{e^2 + 1}$</span></p>
<p><span class="math-container">${y_A}^\star = \frac{3 e^2 + 5}{e^2 + 1}$</span></p>
<p><span class="math-container">${y_B}^\star = \frac{5 e^2 + 3}{e^2 + 1}$</span></p>

CC BY-SA 4.0
2023-05-25T16:01:44.403Z
2023-05-25T16:17:25.860Z
2023-05-25T16:17:25.860Z
Nicolas Torres
<blockquote>
<p>An allocation <span class="math-container">$((c^*_{a,1},c^*_{a,2}),(c^*_{b,1},c^*_{b,2}))$</span> is a <strong><em>competitive equilibrium</em></strong> allocation for the given economy supported by the price ratio <span class="math-container">$\frac{p_1^*}{p_2^*}$</span> if it satisfies the following:<br />
<span class="math-container">$1.$</span> <strong>Solution to UMP of <span class="math-container">$a$</span> and <span class="math-container">$b$</span> :</strong><br />
Given <span class="math-container">$(p_1^*,p_2^*)$</span>, <span class="math-container">$(c^*_{a,1},c^*_{a,2})$</span> is a solution to:
<span class="math-container">$$\begin{align} \max_{c_{a,1},c_{a,2}\geq0} \quad &amp; U_a=-e^{-c_{a,1}}-e^{-c_{a,2}}\\
\textrm{s.t.} \quad &amp; p_1^*c_{a,1}+p_2^*c_{a,2}\leq p_1^*+5p_2^*\end{align}$$</span>
Given <span class="math-container">$(p_1^*,p_2^*)$</span>, <span class="math-container">$(c^*_{b,1},c^*_{b,2})$</span> is a solution to:
<span class="math-container">$$\begin{align} \max_{c_{b,1},c_{b,2}\geq0} \quad &amp; U_b=-e^{-c_{b,1}}-e^{-c_{b,2}}\\
\textrm{s.t.} \quad &amp; p_1^*c_{b,1}+p_2^*c_{b,2}\leq 3p_1^*+3p_2^*\end{align}$$</span><br />
<span class="math-container">$2.$</span> <strong>Market Clearing</strong>: The optimal demand obtained in <span class="math-container">$1.$</span> must be market-clearing i.e.,<span class="math-container">$$c^*_{a,1}+c^*_{b,1}=4\\ c^*_{a,2}+c^*_{b,2}=8$$</span></p>
</blockquote>
<p>To solve for competitive equilibrium we first need the demand functions of <span class="math-container">$a$</span> and <span class="math-container">$b$</span>.<br />
Since we are interested in relative prices let us normalize price of good 2 to 1. Formally, <span class="math-container">$p_2\overset{set}{=}1$</span>. This will enable us to find the demand functions in terms of <span class="math-container">$p_1$</span> alone.</p>
<p>UMP of agent a: <span class="math-container">$$\begin{align}\max_{c_{a,1},c_{a,2}\geq0} \quad &amp; -e^{-c_{a,1}}-e^{-c_{a,2}}\\
\textrm{s.t.} \quad &amp; p_1c_{a,1}+c_{a,2}\leq p_1+5\end{align}$$</span></p>
<p>solving above gives: <span class="math-container">$(c_{a,1},c_{a,2})^d(p_1)=\left(\frac{5+p_1-\ln p_1}{p_1+1},\frac{p_1(1+\ln p_1)+5}{p_1+1}\right)$</span></p>
<p>UMP of agent b: <span class="math-container">$$\begin{align}\max_{c_{b,1},c_{b,2}\geq0} \quad &amp; -e^{-c_{b,1}}-e^{-c_{b,2}}\\
\textrm{s.t.} \quad &amp; p_1c_{b,1}+c_{b,2}\leq 3p_1+3\end{align}$$</span></p>
<p>solving above gives: <span class="math-container">$(c_{b,1},c_{b,2})^d(p_1)=\left(\frac{3+3p_1-\ln p_1}{p_1+1},\frac{p_1(3+\ln p_1)+3}{p_1+1}\right)$</span></p>
<p>Both UMPs are standard ones with concave utility functions so you can use the lagrangian method to solve the above.</p>
<p>Now we can solve for the equilibrium price by using <span class="math-container">$2.$</span> and the demand functions.</p>
<p>Market for good 1 clears when: <span class="math-container">$c_{a,1}(p_1)+c_{b,1}(p_1)=4$</span>
<span class="math-container">$$\begin{eqnarray} 
&amp; \frac{5+p_1-\ln p_1}{p_1+1}+\frac{3+3p_1-\ln p_1}{p_1+1}=4 \\
&amp; p_1=e^2\end{eqnarray}$$</span></p>
<p>Therefore, <span class="math-container">$((c^*_{a,1},c^*_{a,2}),(c^*_{b,1},c_{b,2}))=\left(\left(\frac{3+e^2}{e^2+1},\frac{3e^2+5}{e^2+1}\right),\left(\frac{1+3e^2}{e^2+1},\frac{5e^2+3}{e^2+1}\right)\right)$</span> is the competitive equilibrium allocation supported by the equilibrium price ratio <span class="math-container">$\frac{p_1^*}{p_2^*}=e^2$</span></p>

CC BY-SA 4.0
2023-05-25T17:46:21.207Z
2023-05-31T09:41:36.517Z
2023-05-31T09:41:36.517Z
mynameparv
<p>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?</p>
<p><a href="https://i.sstatic.net/lAgSm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/lAgSm.png" alt="Question" /></a></p>

<finance><competitive-equilibrium><pure-exchange-economy>
Exchange economy with two agents, what's the competitive equilibrium?
carlos21
# Exchange economy with two agents, what's the competitive equilibrium?

(Asked by: username_0 on 2023-05-25)

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?

[![Question](https://i.sstatic.net/lAgSm.png)](https://i.sstatic.net/lAgSm.png)

**Tags:** finance, competitive-equilibrium, pure-exchange-economy

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## 2 answers

### Answer by username_1 on 2023-05-25. Score: 2

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

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

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

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

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


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### Answer by username_2 on 2023-05-25. Score: 1

> 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 $\frac{p\_1^\*}{p\_2^\*}=e^2$


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