There are two firms competing in quantity. Firm 1 and 2 set their quantities supplied, q1 and q2, respectively. The production costs are zero. The market price is given by

where a ∈ (0,1/2) and b ≥0. Note that the inverse demand function is kinked at the point (1 - a, a).

This is a simple one-shot game. The firms simultaneously set their quantities. The objective of each firm is to maximize its profit, that is,

1. Derive the pure strategy Nash equilibrium and the equilibrium profits when b = 0.

Ans. 0.5 each

2. Derive the pure strategy Nash equilibrium and the equilibrium profits when b > 0. Note that two pure strategy Nash equilibria may exist.

Ans. All or nothing

3. Dose an increase in b benefit the two firms? This means that you should explain whether or not at increase in the demand size benefits the firm.