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.

Dayne · 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+1Bigg)&=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: $x_j^1+x_j^2 = 1$

CES utility function in an Edgeworth box

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