Introduction and Perfect Competition Problem Set
May 28, 2021
Question 1
In this question we will consider the short-run equilibrium in a market for some good y for the case of perfect
competition. Suppose each producer has a production function given by y = Lα M β K γ , where L is the amount of
labour, M denotes the amount of intermediate inputs and K = K is the fixed level of capital. We assume that
α + β < 1. Moreover, a firm can avoid paying for capital if it chooses to shut down.
1. Let ω and q be the wage rate and price of intermediate inputs, respectively. Set up the firm’s cost minimization
problem and obtain the levels of L(ω, q, y, K) and M (ω, q, y, K) that minimize cost.
2. Analyze how the optimal amount of intermediate inputs (M (ω, q, y, K)) changes with the wage rate (ω). Explain
the intuition behind your result.
3. Obtain a firm’s cost function and analyze how cost changes with the desired level of output
4. Let p be the price level of y. Write down expressions for firm’s average and marginal costs. What is firm’s
supply curve? Plot marginal cost, average cost and supply curve on the same graph.
5. Suppose that there are N > 1 suppliers in the market. Find the industry supply function.
6. Now consider the demand side of this market. Suppose there are C > 1 consumers, each with utility function
U (y, x) = y µ x1−µ , where x is a numeraire good (i.e. price of x is one). Moreover, suppose consumers are
heterogeneous in their income levels, which each of them having income level denoted by Ic , c = 1, . · · · , C.
Solve a consumer’s utility maximization problem and find the market demand function for y.
7. Solve for equilibrium market price and quantity of y.
8. Suppose that income of some consumer c increases exogenously. What is the impact of this change on the
equilibrium market price and quantity in the market for y?
9. Let C = 10 and suppose that the government is worried about income inequality among consumers and
decides to transfer some amount of income from certain consumers to others. More precisely, suppose that
the government taxes each consumer c = 6, 7, · · · , 10 by a fixed amount F < Ic and pays each consumer
c = 1, 2 · · · , 5 exactly F . What is the impact of this redistribution on equilibrium market price and quantity in
the market for y?
Question 2
Consider a perfectly competitive market for some good q. Each firm’s long-run total cost is given by LRT C(q) =
1 3 2 d
6 q − 3q + 20q. In addition, the market demand function is given by q (p) = 1100 − 50p.
1. Find each firm’s long run supply function.
2. What is the equilibrium quantity that each firm produces in the long-run?
3. Find the long-run equilibrium price and number of firms in the market.
4. Now suppose that the market demand changes to q d (p) = 2200 − 50p. How does the long-run equilibrium
market price change? Is there any change in the long-run equilibrium number of firms?
1
Question 3
Consider a firm, called WXM, which operates in a perfectly competitive market. The short-run total cost is given
by SRT C(q) = 40 + 10q + 0.1q 2 , where q is the amount of output produced. We will assume that all fixed costs are
sunk.
1. Find WXM’s short-run supply curve. Plot the supply curve alongside with short-run average cost and marginal
cost curves.
2. Suppose that the market price is 20 per unit. How many units does WXM produce to maximize its profits?
Calculate WXM’s profits and show it on the graph that you drew in previous part.
