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Econ 100C, Problem Set 5

1. Exactly two firms are competing by choosing quantity in a market. The first has the cost function

??1 (??1 ) = 3??12. The second has the cost function ??2 (??2 ) = 4??22. Inverse market demand is

equal to P(Q) = 120 Q , where ?? = ??1 + ??2 .

a. Find firm 1s reaction function. The reaction function gives its profit maximizing output

for any fixed ??2 .

b. If ??2 is fixed and firm 1 maximizes profit, calculate the price in this market as a function

of ??2 . Give firm 1s profit as a function of ??2 .

c. Find firm 2s reaction function and give firm 2s profits as a function of ??1 .

d. Calculate the Cournot equilibrium quantities. Find the equilibrium price and the profit

for each firm.

e. Now suppose that the two firms manage to collude successfully. Write the joint profit

function and find the joint profit maximizing quantities. Find the equilibrium price and

the joint profit. Comment on the change in welfare from Cournot to collusion.

2. Exactly n firms are competing by choosing quantity in a market. Each has the cost function of

???? (???? ) = ????2 . Inverse market demand is equal to P(Q) = 120 Q , where ?? = ??1 + ??2 + ? +

???? .

a. Find firm is reaction function, that is its profit maximizing output for any fixed output

for the other n-1 firms.

b. Calculate the Cournot equilibrium quantities.

c. Find the market price, each firms profit and total industry profits. Show that each of

these converges to the competitive outcome as n approaches infinity.

3. Exactly n+1 firms are competing by choosing quantity in a market. One has the cost function

??1 (??1 ) = 5??1 . The other n firms have cost functions ???? (???? ) = 4???? for ?? = 2, 3,
, ?? + 1. Inverse

market demand is equal to P(Q) = 120 2Q , where ?? = ??1 + ??2 + ? + ????+1 .

a. Find firm 1s reaction function.

b. Find other firms reaction functions.

c. Calculate the Cournot equilibrium quantities. Find the equilibrium price and the profit

for each firm.

d. For what values of n will firm 1 stop producing?

4. The demand that duopoly quantity-setting firms face is ?? = 90 ? 2??1 ? 2??2. Firm 1 has no

marginal cost of production, but Firm 2 has a marginal cost of $30. How much does each firm

produce if they move simultaneously? What is the equilibrium price?

5. A duopoly faces a market demand of p =120 Q. Firm 1 has a constant marginal cost of 20. Firm

2s constant marginal cost is 40. Calculate the output of each firm, market output, and price if

there is (a) collusion between the firms (i.e., the Cartel model) or (b) competition in quantity

between the firms (i.e., the Cournot model).

6. To examine the trade-off between efficiency and market power from a merger, consider a

market with two firms that sell identical products. Firm 1 has a constant marginal cost of 1, and

Firm 2 has a constant marginal cost of 2. The market demand is Q = 15 P.

a. Solve for the Cournot equilibrium price, quantities, profits, consumer surplus, and

deadweight loss.

b. If the firms merge and produce at the lower marginal cost, how do equilibrium values

change?

c. Discuss the change in efficiency (average cost of producing the output) and welfare

consumer surplus, producer surplus (or profit), and deadweight loss.

7. Acura and Volvo offer warranties on their automobiles, where wA and wV are the numbers of

years of an Acura and Volvo warranty, respectively. The revenue for Firm i, i = A for Acura and V

for Volvo, is 27000 wi / (wA + wV). Its cost of providing the warranty is Ci = 2000 wi. Acura and

Volvo participate in a warranty-setting game in which they simultaneously set warranties.

Calculate the Nash equilibrium warranties.

8. Duopoly quantity-setting firms face the market demand ?? = 150 ? ??1 ? ??2 . Each firm has a

marginal cost of $60 per unit.

a. What is the Cournot equilibrium?

b. What is the Stackelberg equilibrium when Firm 1 moves first?

9. Determine the Stackelberg equilibrium with one leader firm and two follower firms if the market

inverse demand curve is ??(??) = ?? ? ?? ??, and each firm faces a constant marginal cost, m, and

no fixed cost. (Suppose the leader moves first, and the two followers move simultaneously after

observing the leaders output.)

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