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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.

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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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