<p>There seems to be some confusion in the expression for <span class="math-container">$x^*_i$</span> in the question that whether <span class="math-container">$i$</span> is for consumer of for the good. Assuming <span class="math-container">$i$</span> is for consumer:</p>
<p>Let <span class="math-container">$x^*_i = (x_1^i,x_2^i)'$</span> be the equilibrium bundle for consumer <span class="math-container">$i$</span>.</p>
<p>Since utility function is same for both, from MRS we have:</p>
<p><span class="math-container">\begin{align}
\frac{x_1^i}{x_2^i}=\bigg(\frac{p_1}{p_2}\bigg)^{s-1} \tag{$i=1,2$}
\end{align}</span></p>
<p>Budget constraint for <span class="math-container">$i$</span>:</p>
<p><span class="math-container">\begin{align}
p_1x_1^i+p_2x_2^i&amp;= p_iw  \\
x_2^ip_2 \Bigg(\bigg(\frac{p_1}{p_2}\bigg)^s+1\Bigg)&amp;=p_iw \tag{using MRS}\\
x_2^i \bigg(\frac{p_1^s+p_2^s}{p_2^{s-1}}\bigg)^s&amp;=p_iw \\
x_2^i &amp;=\frac{p_2^{s-1}}{p_1^s+p_2^s}p_iw
\end{align}</span></p>
<p>So,:</p>
<p><span class="math-container">$$x^*_i(p,pw) = \Bigg(\frac{p_1^{s-1}}{p_1^s+p_2^s}p_iw,\frac{p_2^{s-1}}{p_1^s+p_2^s}p_iw \Bigg)$$</span></p>
<p>The question can be solved further, for <span class="math-container">$p_1/p_2$</span> using the constraint: <span class="math-container">$x_j^1+x_j^2 = 1$</span></p>

CC BY-SA 4.0
2020-12-04T09:23:17.333Z
Thank you for your solution. I can see where I made the mistake. Really appreciate it.
Ana Ellis
CC BY-SA 4.0
2020-12-04T05:03:11.637Z
2020-12-04T06:17:48.640Z
2020-12-04T06:17:48.640Z
Dayne
<p>Two consumers have the CES utility function <span class="math-container">$x_1^\beta +x_2^\beta$</span>, for <span class="math-container">$0&lt;\beta&lt;1$</span>, their initial endowments are <span class="math-container">$w^1=(1,0)$</span>, <span class="math-container">$w^2=(0,1)$</span> Draw the Core of this economy in an Edgeworth box. Note and verify that the demand of the CES utility function is <span class="math-container">$x_i^*(p,pw)={\dfrac{p_i^{(s-1)}}{(p_1^s+p_2^s)}}$$pw$</span>, where <span class="math-container">$s={\dfrac{\beta}{\beta-1}}$</span></p>
<p>I have drawn the IC of the CES function, that I guess are the similar to this in a sense in order to find the core. <a href="https://dismaldocket.files.wordpress.com/2013/02/pareto-set.jpg" rel="nofollow noreferrer">https://dismaldocket.files.wordpress.com/2013/02/pareto-set.jpg</a></p>
<p>For the finding the demand I was looking at equating their MRS=<span class="math-container">$\dfrac{\beta x_1^{\beta-1}}{\beta x_2^{\beta-1}}$</span> = <span class="math-container">$\dfrac{p_1}{p_2}$</span>
by substituting this to the budget equation I get that <span class="math-container">$x_1^*$</span>=<span class="math-container">$\dfrac{w \cdot p_1}{p_1^2+p_2^{\beta/(\beta-1)}}$</span></p>
<p>However I most probably have done miscalculations or am completely sidetracked :). Any suggestions is more than welcomed.</p>

<microeconomics><demand><ces-function><edgeworth-box>
CES utility function in an Edgeworth box
Ana Ellis
# CES utility function in an Edgeworth box

(Asked by: username_0 on 2020-12-03)

Two consumers have the CES utility function $x\_1^\beta +x\_2^\beta$, for $0<\beta<1$, their initial endowments are $w^1=(1,0)$, $w^2=(0,1)$ Draw the Core of this economy in an Edgeworth box. Note and verify that the demand of the CES utility function is $x\_i^\*(p,pw)={\dfrac{p\_i^{(s-1)}}{(p\_1^s+p\_2^s)}}$$pw$, where $s={\dfrac{\beta}{\beta-1}}$

I have drawn the IC of the CES function, that I guess are the similar to this in a sense in order to find the core. <https://dismaldocket.files.wordpress.com/2013/02/pareto-set.jpg>

For the finding the demand I was looking at equating their MRS=$\dfrac{\beta x\_1^{\beta-1}}{\beta x\_2^{\beta-1}}$ = $\dfrac{p\_1}{p\_2}$
by substituting this to the budget equation I get that $x\_1^\*$=$\dfrac{w \cdot p\_1}{p\_1^2+p\_2^{\beta/(\beta-1)}}$

However I most probably have done miscalculations or am completely sidetracked :). Any suggestions is more than welcomed.

**Tags:** microeconomics, demand, ces-function, edgeworth-box


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## 1 answers

### Answer by username_1 on 2020-12-04. Score: 4 | Accepted answer

There seems to be some confusion in the expression for $x^\*\_i$ in the question that whether $i$ is for consumer of for the good. Assuming $i$ is for consumer:

Let $x^\*\_i = (x\_1^i,x\_2^i)'$ be the equilibrium bundle for consumer $i$.

Since utility function is same for both, from MRS we have:

\begin{align}
\frac{x\_1^i}{x\_2^i}=\bigg(\frac{p\_1}{p\_2}\bigg)^{s-1} \tag{$i=1,2$}
\end{align}

Budget constraint for $i$:

\begin{align}
p\_1x\_1^i+p\_2x\_2^i&= p\_iw \\
x\_2^ip\_2 \Bigg(\bigg(\frac{p\_1}{p\_2}\bigg)^s+1\Bigg)&=p\_iw \tag{using MRS}\\
x\_2^i \bigg(\frac{p\_1^s+p\_2^s}{p\_2^{s-1}}\bigg)^s&=p\_iw \\
x\_2^i &=\frac{p\_2^{s-1}}{p\_1^s+p\_2^s}p\_iw
\end{align}

So,:

$$x^\*\_i(p,pw) = \Bigg(\frac{p\_1^{s-1}}{p\_1^s+p\_2^s}p\_iw,\frac{p\_2^{s-1}}{p\_1^s+p\_2^s}p\_iw \Bigg)$$

The question can be solved further, for $p\_1/p\_2$ using the constraint: $x\_j^1+x\_j^2 = 1$

> **Comments:**
> **username_0**: Thank you for your solution. I can see where I made the mistake. Really appreciate it.  

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