CES utility function in an Edgeworth box

Question

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.

Answer 1

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: $x_j^1+x_j^2 = 1$