<p>Given a pure-exchange economy:</p>
<p><span class="math-container">$u_i(x_i,y_i)=\sqrt{x_i}+2\sqrt{y_i}$</span> for <span class="math-container">$i\in\{1,2\}$</span></p>
<p><span class="math-container">$\omega_1=(\omega_1^X,\omega_1^Y)=(6,1)$</span> and <span class="math-container">$\omega_2=(\omega_2^X,\omega_2^Y)=(10,3)$</span></p>
<p>we want to find the competitive equilibrium price <span class="math-container">$p &gt; 0$</span>, where price of <span class="math-container">$X$</span> is <span class="math-container">$1$</span> and price of <span class="math-container">$Y$</span> is <span class="math-container">$p$</span>.</p>
<p>To do so, first we solve the utility maximisation problem of each consumer <span class="math-container">$i$</span>.</p>
<p><span class="math-container">\begin{eqnarray*} \max_{x_i\geq 0, \ y_i\geq 0} &amp; \sqrt{x_i}+2\sqrt{y_i} \\ \text{s.t. } &amp; x_i + py_i \leq \omega_i^X+p\omega_i^Y\end{eqnarray*}</span></p>
<p>Setting up the Lagrangian, we get</p>
<p><span class="math-container">$\mathcal{L} = \sqrt{x_i}+2\sqrt{y_i} - \lambda(x_i + py_i - \omega_i^X-p\omega_i^Y)$</span></p>
<p>First-order conditions are:
<span class="math-container">\begin{eqnarray*} \dfrac{\partial \mathcal{L}}{\partial \mathcal{x_i}} &amp; = &amp; \dfrac{1}{2\sqrt{x_i}}-\lambda= 0\\ \dfrac{\partial\mathcal{L}}{\partial \mathcal{y_i}} &amp; = &amp;  \dfrac{1}{\sqrt{y_i}}-\lambda p= 0 \\ \dfrac{\partial\mathcal{L}}{\partial \lambda } &amp; = &amp;  x_i + py_i - \omega_i^X-p\omega_i^Y = 0\end{eqnarray*}</span></p>
<p>Eliminating <span class="math-container">$\lambda$</span>, we observe that <span class="math-container">$x_i, y_i$</span> satisfy:</p>
<p><span class="math-container">$y_i=\dfrac{4x_i}{p^2}$</span> and</p>
<p><span class="math-container">$x_i + py_i = \omega_i^X+p\omega_i^Y$</span></p>
<p>Solving for <span class="math-container">$x_i$</span>, <span class="math-container">$y_i$</span>, we get
<span class="math-container">$x_i= \dfrac{p(\omega_i^X+p\omega_i^Y)}{p+4}$</span> and <span class="math-container">$y_i= \dfrac{4(\omega_i^X+p\omega_i^Y)}{p(p+4)}$</span>.</p>
<p>To solve for competitive equilibrium price, we can consider market for <span class="math-container">$X$</span> and set total demand for <span class="math-container">$X$</span> equal to total supply of <span class="math-container">$X$</span>, and we get the following condition
<span class="math-container">$\dfrac{p(\omega_1^X+p\omega_1^Y)}{p+4}+ \dfrac{p(\omega_2^X+p\omega_2^Y)}{p+4} =\omega_1^X+\omega_2^X $</span></p>
<p>Equilibrium <span class="math-container">$p$</span> solves the above condition and solving it we get:
<span class="math-container">$p=2\sqrt{\dfrac{\omega_1^X+\omega_2^X}{\omega_1^Y+\omega_2^Y}}$</span></p>
<p>For the specific case when <span class="math-container">$\omega_1=(\omega_1^X,\omega_1^Y)=(6,1)$</span> and <span class="math-container">$\omega_2=(\omega_2^X,\omega_2^Y)=(10,3)$</span>, we get <span class="math-container">$p =4 $</span>. Corresponding competitive equilibrium allocation is <span class="math-container">$(x_1,y_1)= \left(5,\frac{5}{4}\right)$</span> and <span class="math-container">$(x_2,y_2)=\left(11,\frac{11}{4}\right)$</span></p>

CC BY-SA 4.0
2023-09-28T11:12:02.307Z
2023-09-28T11:20:47.583Z
2023-09-28T11:20:47.583Z
Amit
<p>Consider an exchange economy with two consumers (1 and 2) and two goods (X and Y). The preferences of both consumers are represented by a utility function <span class="math-container">$U(X,Y)=\sqrt{X}+2\sqrt{Y}$</span>. Consumer 1 initially has <span class="math-container">$X_1^e$</span> units of good X and <span class="math-container">$Y_1^e$</span> units of good Y, and consumer 2 initially has <span class="math-container">$X_2^e$</span> units of good X and <span class="math-container">$Y_2^e$</span> units of good Y. The two consumers can trade X at the price of 1 and Y at the price of <span class="math-container">$p$</span>.
a) Fully describe the competitive equilibrium in this exchange economy, both as a function of the initial endowments (i.e., <span class="math-container">$X^e$</span> and <span class="math-container">$Y^e$</span>) and for the specific case of <span class="math-container">$(X^e, Y^e) = (6, 1)$</span>, <span class="math-container">$(X_2^e, Y_2^e) = (10, 3)$</span>.</p>
<p>I tried to do <span class="math-container">$L = \sqrt{X} +2\sqrt{Y} + \lambda(X_1^e+ p Y_1^e - X_1^c-p Y_1^c)$</span>. but it definetly did not work.</p>

<exchange>
How do you solve this exchange economy problem using lagrange?
João
# How do you solve this exchange economy problem using lagrange?

(Asked by: username_0 on 2023-09-27)

Consider an exchange economy with two consumers (1 and 2) and two goods (X and Y). The preferences of both consumers are represented by a utility function $U(X,Y)=\sqrt{X}+2\sqrt{Y}$. Consumer 1 initially has $X\_1^e$ units of good X and $Y\_1^e$ units of good Y, and consumer 2 initially has $X\_2^e$ units of good X and $Y\_2^e$ units of good Y. The two consumers can trade X at the price of 1 and Y at the price of $p$.
a) Fully describe the competitive equilibrium in this exchange economy, both as a function of the initial endowments (i.e., $X^e$ and $Y^e$) and for the specific case of $(X^e, Y^e) = (6, 1)$, $(X\_2^e, Y\_2^e) = (10, 3)$.

I tried to do $L = \sqrt{X} +2\sqrt{Y} + \lambda(X\_1^e+ p Y\_1^e - X\_1^c-p Y\_1^c)$. but it definetly did not work.

**Tags:** exchange


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## 1 answers

### Answer by username_1 on 2023-09-28. Score: 1

Given a pure-exchange economy:

$u\_i(x\_i,y\_i)=\sqrt{x\_i}+2\sqrt{y\_i}$ for $i\in\{1,2\}$

$\omega\_1=(\omega\_1^X,\omega\_1^Y)=(6,1)$ and $\omega\_2=(\omega\_2^X,\omega\_2^Y)=(10,3)$

we want to find the competitive equilibrium price $p > 0$, where price of $X$ is $1$ and price of $Y$ is $p$.

To do so, first we solve the utility maximisation problem of each consumer $i$.

\begin{eqnarray\*} \max\_{x\_i\geq 0, \ y\_i\geq 0} & \sqrt{x\_i}+2\sqrt{y\_i} \\ \text{s.t. } & x\_i + py\_i \leq \omega\_i^X+p\omega\_i^Y\end{eqnarray\*}

Setting up the Lagrangian, we get

$\mathcal{L} = \sqrt{x\_i}+2\sqrt{y\_i} - \lambda(x\_i + py\_i - \omega\_i^X-p\omega\_i^Y)$

First-order conditions are:
\begin{eqnarray\*} \dfrac{\partial \mathcal{L}}{\partial \mathcal{x\_i}} & = & \dfrac{1}{2\sqrt{x\_i}}-\lambda= 0\\ \dfrac{\partial\mathcal{L}}{\partial \mathcal{y\_i}} & = & \dfrac{1}{\sqrt{y\_i}}-\lambda p= 0 \\ \dfrac{\partial\mathcal{L}}{\partial \lambda } & = & x\_i + py\_i - \omega\_i^X-p\omega\_i^Y = 0\end{eqnarray\*}

Eliminating $\lambda$, we observe that $x\_i, y\_i$ satisfy:

$y\_i=\dfrac{4x\_i}{p^2}$ and

$x\_i + py\_i = \omega\_i^X+p\omega\_i^Y$

Solving for $x\_i$, $y\_i$, we get
$x\_i= \dfrac{p(\omega\_i^X+p\omega\_i^Y)}{p+4}$ and $y\_i= \dfrac{4(\omega\_i^X+p\omega\_i^Y)}{p(p+4)}$.

To solve for competitive equilibrium price, we can consider market for $X$ and set total demand for $X$ equal to total supply of $X$, and we get the following condition
$\dfrac{p(\omega\_1^X+p\omega\_1^Y)}{p+4}+ \dfrac{p(\omega\_2^X+p\omega\_2^Y)}{p+4} =\omega\_1^X+\omega\_2^X $

Equilibrium $p$ solves the above condition and solving it we get:
$p=2\sqrt{\dfrac{\omega\_1^X+\omega\_2^X}{\omega\_1^Y+\omega\_2^Y}}$

For the specific case when $\omega\_1=(\omega\_1^X,\omega\_1^Y)=(6,1)$ and $\omega\_2=(\omega\_2^X,\omega\_2^Y)=(10,3)$, we get $p =4 $. Corresponding competitive equilibrium allocation is $(x\_1,y\_1)= \left(5,\frac{5}{4}\right)$ and $(x\_2,y\_2)=\left(11,\frac{11}{4}\right)$


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