1. Let the input prices be w = (w1, w2) and output price be p. Derive the cost function c (w, y) and the output supply function y (w, p) for firms with the following production functions:

(a) f(x1, x2) = √x₁ + 2√x₂
(b) f(x1, x2) = min[√x₁, 2√x₂

2. Consider a competitive firm which uses three inputs K (capital), L (labor), and E (electricity) to produce output y. The input prices are w = (w_K = 4, w_t., = 1, w_E = 2) and the output price is p = 1. The firm faces the following production function:

f (K, L, E) = K (√L + √E).

In the short-run, the firm's capital level is fixed at f , but can choose labor and electricity as it wishes. In the long-run, the firm can also vary its capital level.

(a) Derive the short-run cost function c (w, y; K).

(b) Then derive the long-run cost function c (w, y).

3. Consider an economy with 2 goods and 40 consumers, all with the same utility function:

u(x₁, x₂) = √x₁x₂.

The goods prices are p1 = a and p2 = 1. Among the consumers, 20 of them each have an income of $100, 10 of them each have an income of $50, and the remaining 10 consumers each have an income of $200. There are 71, firms operating in the competitive market for good 1. Each firm has the cost function c(q) = q₂.

(a) Solve for the equilibrium price a. [Hint: The aggregate demand for good 1 is the sum of all consumers' Marshallian demands for good 1:

20 x₁ (p₁, p2, y= 100) + 10 x₁ (p₁ + p₂ y = 50 ) + 10 x₁(p₁, p₂, y = 200)

(b) In the long-run, firms are free to enter and leave the market. It costs each firm an amount $F to enter this market if it wishes to do so. Solve for the number of firms in the market of good 1 in the long run. Does this number increase with entry cost F?

(c) Take the same aggregate demand function for good 1 derived in part (a). Consider a firm's cost function as c(q) = 1/2q³ -100q instead. Calculate the equilibrium price of good 1 respectively for two alternative market structures: (i) a competitive industry of 10 firms; (ii) a monopolistic firm. Briefly explain how and why the prices differ under different market structures.

4. Consider the economy in the example on page 24 of Notes 10. There are n firms in a competitive industry. The market demand function is given by p = 50 - 2Q. A firm's cost function is C (q) = q². In class, we have calculated the total surplus is S₁ = 50²n/4(n+1). Now consider how taxation can change the total surplus.

(a) Suppose the government imposes a per-unit tax t > 0 on consumers. That is, for each unit of goods purchased, the consumer pays p + t. As a result, the market demand function becomes p + t = 50 - 2Q. The tax goes to the government and firms only receive p for each unit of output sold. So the profit of a firm is pq - q². Derive the total surplus, denoted by s₂. Is s₂ > s₁?

(b) Instead of the per-unit tax, suppose the government imposes a pro­portional tax rate r > 0 on consumers. That is, the consumer pays a after-tax price of (1 + τ) p for each unit of goods, and the market demand becomes (1 + τ) p = 50 - 2Q. Once again, all tax revenues go to the government. Derive the total surplus, denoted by s₃. Ls S₃ > s₁?