Reference no: EM132541796

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

Exercise 1 : Each individual consumer takes the prices as given and chooses her consumption bundle, (x1, x2) ∈ R2+, by maximizing the utility function U(x1, x2) = ln(x12 x2), subject to the budget constraint p1 · x1 + p2 · x2 = 900.

1. Write out the Lagrangian function for the consumer's problem.

2. Write out the system of first-order conditions for the consumer's problem.

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

4. Suppose the price of good 1 is 2, i.e., p1 = 2 and that of good 2 is 1, i.e., p2 = 1. Approximate the change in a consumer's utility that results from 10 units decrease of her budget from 900 to 890.

5. Check that the critical point satisfies the second-order condition.

Exercise 2

1. Suppose the price of good 1 is 2, i.e., p1 = 2 and that of good 2 is 1, i.e., p2 = 1. Use your results from Exercise 1, part (3) to answer the following question: How many units of good 1 and good 2 does an individual consumer demand?

2. 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? (Sum up all the individual demands from Exercise 2, part (1).)

3. The aggregate demand function of a good takes the market prices as argument and maps them into the total units that are demanded. That is, for given prices the aggregate demand function of a good tells us how many units of that good are demanded by all 1000 consumers together. Use your results from exercise 1, part (3) to derive that aggregate demand function for each good.