Exercises 3 and 4: Supply, The economy from Exercises 3 and 4 also consists of 1000 firms.

Exercise 3 Each firm produces both goods, i.e., good 1 and good 2. Each firm takes the market prices p₁ 0 and p₂ 0 as given and chooses output to maximize profits. If a firm produces x₁ units of good 1 and x₂ units of good 2, with (x₁, x₂) R₊², it has the total costs of

C(x₁, x₁) = (x₁²+ 0.5x₂² )/1200 .

1. For given prices p₁ and p₂, find the revenue, R(x₁, x₂), of a single firm.

2. Find the profit function, (x₁, x₂), of a single firm.

3. Solve the profit-maximization problem, that is, derive the system of firstorder conditions and solve this system. That is, find the critical point of . Your solution might depend on p₁ and on p₂.

4. Check that the critical point is indeed a (local) maximizer.

Exercise 4

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 3, part (3) to answer the following question: How many units of good 1 and good 2 does an individual firm supply?

2. Still assume that p₁ = 2 and p₂ = 1. What is the aggregate supply of good 1? What is the aggregate supply of good 2? That is, how many units of good 1 (good 2, respectively) do all 1000 firms supply together? (Just sum up all the individual supplies from Exercise 4, part (1).)

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