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

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

Thanks.

• Do you know what aggregate means? – Giskard Feb 10 at 9:36
• The total demand for a good. So I'm just supposed to just add $x^*_1$ and $x^*_2$ together? – user25621 Feb 10 at 9:46
• Well, if a total is the sum of its parts then that certainly makes sense. – Giskard Feb 10 at 9:52
• Hint: Aggregate demand is the sum of individual demands. – VARulle Feb 10 at 12:00

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 $$Y$$.