# CES utility function in an Edgeworth box

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.

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

• Thank you for your solution. I can see where I made the mistake. Really appreciate it. – Ana Ellis Dec 4 '20 at 9:23