Q1. There are two firms (firm 1 and 2) 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.
2. Derive the pure strategy Nash equilibrium and the equilibrium profits when b > 0. Note that two pure strategy Nash equilibria may exist.
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.
Q2. Consider a population of consumers uniformly distributed along the interval from left-hand (x = 0) to right-hand (x=1). The mass of consumers is 1. There are two firms (F1 and F2) that supply homogrnouse goods. The objective of each firm is to maximize its profit. The price is regulated at p. Each of the frims simultaneously chooses its location (i.e. a point on the line between x=0 and x=1). The consumers observe the firm’s choice, and then each consumer buys from the firm whose location is closest to the consumer’s position on the line. If the two firms locate at a same point, they equally split the consumer demand.
For example, if Fi locates as in the following figure, the left-hand consumers buy from F1 and the right-hand consumers buy from F2.
1. In this case, what is a pure-strategy Nash equilibrium? You must explain how to derive it.
2. When there are three firms, no pure-strategy Nash equilibrium exists. Prove it.
3. When there are four firms, what is a pure-strategy Nash equilibrium? You must explain how to derive it.