How do I derive the aggregate demand function given two utilities functions?

Question

Assume that we have two people with the same utility function of $U_i = x^{1/2} + y^{1/2}$ where $i=1,2$ and $I_i$ is the income. Let $P_x$ denote price of good $x$ and $P_y$ denote price of good $y$.

I'm being asked to derive the aggregate demand function. The only thing I got so far was finding the market demand for each good per person, which is

$x^*_1 = {I_1}/2P_x$ , $y^*_1 = {I_1}/2P_y$, for person 1

$x^*_2 = {I_2}/2P_x$, $y^*_2 = {I_2}/2P_y$ for person 2

Am I missing something? Please help.

Thanks.

nrivera · Answer

If you have $J$ consumers therefore $J$ demands for a good $X$. Denoting the individual demand of each consumer with $x_j^*$ as you have it, if $X$ is the aggregate demand, it is just the sum of every individual demand:
$X=sum_{j=1}^{J}x_j^*$
Then for your case it's: $x_1^*+x_2^*=frac{(I_1+I_2)}{2P_X}$, and the same with $Y$.

How do I derive the aggregate demand function given two utilities functions?

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