Tacit Collusion:
Two firms, which have zero marginal cost and no fixed cost, produce some good, each producing qi ≥ 0, i ∈ {1, 2}. The demand for this good is given by p = 200 - Q, where Q = q1 + q2.
a. First consider the case of Cournot competition, in which each form chooses qi and this game is infinitely repeated with a discount factor δ
b. For which values of δ can you support the firms' equally splitting monopoly profits in each period as a subgame-perfect equilibrium that uses grim-trigger strategies (i.e., after one deviates from the proposed split, they resort to the static Cournot-Nash equilibrium thereafter)? (Note: Be careful in defining the strategies of the firms!)
c. Now assume that the firms compete a la Bertrand, each choosing a ' price pi ≥ 0, where the lowest-price firm gets all the demand and in case of a tie they split the market. Solve for the static stage-game Bertrand-Nash equilibrium.
d. For which values of δ can you support the firms' equally splitting monopoly profits in each period as a subgame-perfect equilibrium that uses grim-trigger strategies (i.e., after one deviates from the proposed split, they resort to the static Bertrand-Nash equilibrium thereafter)? (Note: Be careful in defining the strategies of the firms!)
e. Now instead of using grim-trigger strategies, try to support the firms' equally splitting monopoly profits as a subgame-perfect equilibrium in which after a deviation firms would resort to the static Bertrand competition for only two periods. For which values of δ will this work? Why is this answer different than your answer in (d)?