# ECON 100 The First Order Condition in Microeconomics & Equilibrium Question

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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 1s 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 1s profit as a function of ??2 .
c. Find firm 2s reaction function and give firm 2s 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 is 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 firms 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 1s 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
2s 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
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