Question

Problem 2. Consider a duopoly with identical firms with no fixed cost and marginal cost of c. Let the
inverse demand curve for the industry be p(Y ) = A − bY , where Y is the total industry output. Let y1 and
y2 be the output of each firm. Assume that the firms in the industry each choose quantity, and then let the
market determine the price they will receive.
(a) Compute the total market output, the price, the quantity for each firm, and the profit for each

firm, under Cournot competition. Make sure to start at the beginning, setting up each firm’s profit-
maximization problem and proceeding from there.

(b) Compute the total market output, the price, the quantity for each firm, and the profit for each firm, if
the firms collude and choose their quantities together to maximize total profits. Assume that they split
the total output evenly. Again, make sure to set up the joint profit-maximization problem and proceed
from there. [Hint: you will find that you cannot solve the system of two first-order conditions. Indeed,
you will find that the two first-order conditions are identical, leaving you with only one equation with
two unknowns. To solve this problem, simply take advantage of the assumption that they split the
market evenly, y1 = y2. Now you have two equations with two unknowns.]
(c) Now suppose that one firm sticks to the collusive quanity, while the other one cheats. Compute the
profit maximizing output of the cheater, the market price, and the profit for each firm.

Econ 100A, Fall 2019
Prof: Dan Acland

Problem Set #12
Page 2

(d) Using the results you have computed above, argue that the collusive agreement is not a Nash equi-
librium. In addition, argue that if firms have common knowledge of rationality (each knows that the

other is rational, and knows that the other knows they are rational, etc) the only Nash equilibrium is
for both firms to choose the Cournot equilibrium quantity.
(e) Again, using the results you have computed above, argue that this situation can be thought of as a
prisoner’s dilemma. (Recall that we characterized a prisoner’s dilemma as a situation in which agents
are unable to cooperate, even when to do so would maximize their individual payoffs, because the
private cost of cooperation is greater than the private benefit of cooperation.)

Answer 1

We are supposed to do only four sub-parts to a question. For solution to other parts of the question please post as separate question.