Page 593 - Microeconomics, Fourth Edition

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PROBLEMS 567
c) Suppose Firms 1 and 2 sign the following contract. firm will choose a quantity, X (for the leader) and Y (for
Firm 1 agrees to pay Firm 2 an amount equal to T dol- the follower). Imagine that you have determined the
lars for every unit of output it (Firm 1) produces. Stackelberg equilibrium for a particular linear demand
Symmetrically, Firm 2 agrees to pay Firm 1 an amount curve and set of marginal costs. Please indicate how X
T dollars for every unit of output it (Firm 2) produces. and Y would change if we then “perturbed” the initial
The payments are justified to the government as a cross- situation in the following way:
licensing agreement whereby Firm 1 pays a royalty for a) The leader’s marginal cost goes down, but the fol-
the use of a patent developed by Firm 2, and similarly, lower’s marginal cost stays the same.
Firm 2 pays a royalty for the use of a patent developed b) The follower’s marginal cost goes down, but the
by Firm 1. What value of T results in the firms achiev- leader’s marginal cost stays the same.
ing the collusive outcome as a Cournot equilibrium?
d) Draw a picture involving reaction functions that 13.19. Suppose that the market demand for cobalt is
shows what is going on in this situation. given by Q 200 P. Suppose that the industry consists
of 10 firms, each with a marginal cost of $40 per unit.
13.15. Consider an oligopoly in which firms choose What is the Cournot equilibrium quantity for each firm?
quantities. The inverse market demand curve is given by What is the equilibrium market price?
P 280 2(X Y ), where X is the quantity of Firm 1,
and Y is the quantity of Firm 2. Each firm has a marginal 13.20. Consider the same setting as in the previous prob-
cost equal to 40. lem, but now suppose that the industry consists of a domi-
a) What is the Cournot equilibrium outputs for each nant firm, Braeutigam Cobalt (BC), which has a constant
firm? What is the market price at the Cournot equilib- marginal cost equal to $40 per unit. There are nine other
rium? What is the profit of each firm? fringe producers, each of whom has a marginal cost curve
b) What is the Stackelberg equilibrium, when Firm 1 MC 40 10q, where q is the output of a typical fringe
acts as the leader? What is the market price at the producer. Assume there are no fixed costs for any producer.
Stackelberg equilibrium? What is the profit of each firm? a) What is the supply curve of the competitive fringe?
b) What is BC’s residual demand curve?
13.16. The market demand curve in a commodity
chemical industry is given by Q 600 3P, where Q is c) Find BC’s profit-maximizing output and price. At this
the quantity demanded per month and P is the market price, what is BC’s market share?
price in dollars. Firms in this industry supply quantities d) Repeat parts (a) to (c) under the assumption that the
every month, and the resulting market price occurs at the competitive fringe consists of 18 firms.
point at which the quantity demanded equals the total
quantity supplied. Suppose there are two firms in this 13.21. Apple’s iPod has been the portable MP3-player
industry, Firm 1 and Firm 2. Each firm has an identical of choice among many gadget enthusiasts. Suppose that
constant marginal cost of $80 per unit. Apple has a constant marginal cost of 4 and that market
demand is given by Q 200 2P.
a) Find the Cournot equilibrium quantities for each
firm. What is the Cournot equilibrium market price? a) If Apple is a monopolist, find its optimal price and
output. What are its profits?
b) Assuming that Firm 1 is the Stackelberg leader, find
the Stackelberg equilibrium quantities for each firm. b) Now suppose there is a competitive fringe of 12
What is the Stackelberg equilibrium price? price-taking firms, each of which has a total cost function
2
TC(q) 3q 20q with corresponding marginal cost
c) Calculate and compare the profit of each firm under the curve MC 6q 20. Find the supply function of the
Cournot and Stackelberg equilibria. Under which equilib- fringe (Hint: A competitive firm supplies along its mar-
rium is overall industry profit the greatest, and why?
ginal cost curve above its shutdown point).
13.17. Consider a market in which the market demand c) If Apple operates as the dominant firm facing compe-
curve is given by P 18 X Y, where X is Firm 1’s tition from the fringe in this market, now what is its
output, and Y is Firm 2’s output. Firm 1 has a marginal optimal output? How many units will fringe providers
cost of 3, while Firm 2 has a marginal cost of 6. sell? What is the market price, and how much profit does
a) Find the Cournot equilibrium outputs in this market. Apple earn?
How much profit does each firm make? d) Graph your answer to part (c).
b) Find the Stackelberg equilibrium in which Firm 1 acts
as the leader. How much profit does each firm make? 13.22. Britney produces pop music albums with the
total cost function TC(Q) 8Q. Market demand for pop
13.18. Consider a market in which we have two firms, music albums is P 56 Q. Suppose there is a compet-
one of which will act as the Stackelberg leader and the itive fringe of price-taking pop music artists, with total
other as the follower. As we know, this means that each supply function Q fringe 2P y, where y 0 is some