Consider an economy that consists of 1000 consumers and two goods. Let p1 be the price of good 1 and p2 that of good 2, where p1 ≥ 0 and p2 ≥ 0.Each individual consumer takes the prices as given a

Consider an economy that consists of 1000 consumers and two goods. Let p1 be the price of good 1 and p2 that of good 2, where p1 ≥ 0 and p2 ≥ 0.

Each individual consumer takes the prices as given and chooses her consumption bundle, (x1, x2) ∈ R 2 +, by maximizing the utility function U(x1, x2) = ln(x1 · x2), subject to the budget constraint p1 · x1 + p2 · x2 = 1000.

a)Solve the system of first-order conditions to find the optimal values of x1 and x2. Your answer might depend on p1 and p2.

b)Suppose the price of good 1 is equals 2, i.e., p1 = 2 and that of good 2 is 1, i.e., p2 = 1. Use your results from (a) to answer the following question: How many units of good 1 and good 2 does an individual consumer demand?

c) Still assume that p1 = 2 and p2 = 1. What is the aggregate demand of good 1? What is the aggregate demand of good 2? That is, how many units of good 1 (good 2, respectively) do all 1000 consumers demand together? (Just sum up all the individual demands from (b)