Econ 521 - Week 4:

1. Find all (pure strategy) Nash Equilibria in the following games

(a) Prisoner's Dilemma

(b) Battle of the Sexes

(c) Matching Pennies

2. Basic Cournot game with symmetric firms-

Consider the situation of two firms, with inverse demand function given by

Also, the cost function of each firm i is C_i(q_i) = c_iq_i, where c_i < a. Assume c₁ = c₂ = c. Based on this, answer the following,

(a) Solve for the Nash equilibrium in this game.

(b) Let's assume that the firms decide to collude and each to produce 50% of the total output? Is this collusion sustainable? What if they agreed on a different share of production? Explain.

(c) Compare the situation with a perfect competition case and a monopoly case. Calculate the total equilibrium output, the equilibrium price, the total profit of firms, and consumer surplus in each case.

(d) Now there is a technical change that lowers firm 2's unit cost. That is, now a > c₁ > c₂. How does your answer in (a) and (b) change?

3. Cournot game with differentiated products

Consider the situation of two firms, each producing a different good, where both markets are related with inverse demand function for firm i given by

Also b ∈ [-1, 1] and the cost function of each firm i is C_i(q_i) = cq_i, where c < a, ∀i. Based on this, answer the following,

(a) Interpret b. What happens when b < 0? When b > 0? When b = 0?

(b) Solve for the Nash equilibrium of this game. Plot the best response when b < 0 and when b > 0.

(c) Explain why quantities and prices may differ (if they do) in the situation where b > 0 and b < 0.

4. Cournot's duopoly with a fixed cost f

Consider the game where there are two firms, the inverse demand function is linear and the cost function of each firm i is given by,

where c ≥ 0, f > 0, and c < a. Based on this,

(a) Find the Nash equilibria of this game. Find the range of f such that we have a Nash equilibrium where only one firm produces.

(b) What is the role of f?