MGEC41H3 Industrial Organization Winter 2023
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MGEC41H3
Industrial Organization
Winter 2023
Problem set 2 – Due: February 17, 2023
This problem set is worth a total of 100 points. You should complete the assignment individually and upload it on Quercus. If you work in groups, write the names of your group members on the front page of your answer sheets. If you are late in turning in your work by 24 hours, you lose 15% of your grade. If you are late in turning in your work by 48 hours, you lose 30% of your grade. If you turn in your assignment after 48 hours from due date and time, you get 0 for that assignment.
Question 1 (30 points)
Consider an industry with two firms producing a homogeneous good. Let q1 and q2 be the output quantities for firms 1 and 2, respectively. The inverse demand function in the industry is P = 200 - Q, where Q = q1 + q2 . Firm 1 has a cost function of C(q1 ) = q 1(2) and firm 2 has a cost function of C(q2 ) = 4q2(2) + 10.
Assume in parts (a)-(d) that firm 1 moves first and sets her output and firm 2 responds by setting her output afterwards.
(a) (4 points) Set up firm 2’s problem and calculate firm 2’s best response function q2 = BR(q1 ), i.e. the profit maximization output of firm 2 as a function of firm 1’s output.
(b) (6 points) Set up firm 1’s problem and calculate firm 1’s profit using the best response func- tion you obtained in (a). Then find firm 1’s and 2’s profit maximizing output.
(c) (10 points) Find the market price, the profits for both firms and the consumer surplus for the market.
(d) (5 points) Compare the firms 1’s and 2’s output you obtained in (b). What are the features of the market/firms that accentuate the output differences between the firms?
Suppose now firms 1 and 2 set output simultaneously.
(e) (5 points) Find the Cournot equilibrium price. Do consumers prefer Cournot competition or Stackelberg competition? (Hint: You do not need to solve for consumer surplus to answer this question.)
Question 2 (35 points)
Suppose there are two firms that produce a homogeneous good with market demand P = 100 - Q. Each consumer buys only one unit of a good. Assume both firms face a constant marginal cost c = 28 and no fixed costs.
(a) (2 points) If firms compete in prices in a Bertrand fashion, what are the optimal prices and total demand in this market?
(b) (3 points) Denote the equilibrium prices you found in (a) by (p1(a), p2(a)). Suppose firms face the same cost structure, but now each firm can only produce up to Q = 36 units. If firm 1 sets p1 = p1(a), can firm 2 increase profits by charging higher prices?
(c) (3 points) Given the results in (a) and (b), can you conclude that the strategy profile (p1(a), p2(a)) is a Nash equilibrium with capacity limitations? Explain.
(d) (5 points) Suppose instead both firms charge monopoly prices in this economy and split the market evenly. Given the demand function, how much each firm will produce? What are the equilibrium prices? Calculate each firm’s profits in this case.
(e) (5 points) Suppose firm 1 continue charging the monopoly price you found in (d). Show that firm 2 can earn higher profits than in (d) if she decides to undercut prices. Is the strategy profile with both firms charging the monopoly price a Nash equilibrium with capacity limita- tions? Explain.
(f) (7 points) (7 points) Suppose firm 2 continues undercutting prices as in (e). Show that firm 1 also has incentives to charge even lower prices than firm 2. For which values of p1 will firm 1 obtain higher profits undercutting firm 2, given firm 2’s price? Denote the lower bound of prices by p¯.
(g) (10 points) Finally, show that no firm will have incentives to charge prices below and above marginal costs. What can you conclude about the existence of a Nash equilibrium in this setting?
Question 3 (35 points)
Suppose there are two firms, each with a cost function C(q) = 3q, that play a repeated game facing demand P = 150 - Q. Each period, they choose quantities simultaneously. The firms compete for infinitely many periods. Both firms discount future profits with a discount factor of β where 0 s β < 1, so the discounted profits are π0 + βπ1 + β2 π2 + ...
(a) (5 points) Suppose these firms decide to form a cartel. What is the total discounted profit of each firm from following the cartel agreement in all periods?
(b) (10 points) If firm 1 decides to collude and firm 2 decides to “cheat” on the cartel agreement, what is firm 2’s profit for the period which she cheats? (Hint: derive best response functions for firm 2 when she decides to charge monopoly prices over her residual demand)
(c) (10 points) Suppose that each firm says it will follow the “grim trigger” strategy. What is the total discounted profit for each firm associated with cheating?
(d) (10 points) What is the value of of β such that both firms are indifferent between cheating and following the cartel agreement?
2023-02-18