Suppose two firms, firm 1 and firm 2, operate in a homogeneous good market. The supply of firm i, denoted by q_i, is constrained by installed capacity k_i, i.e., 0 ≤ q_i ≤ ki for i = 1, 2. Firms have zero marginal costs of production for quantities weakly below their capacity. They cannot increase production beyond capacity. There is a group of consumers of size μ̅ who all have unit demand with the same willingness-to-pay r. Demand r is uncertain. The size of demand is determined by the value a random variable μ takes. Suppose that this random variable is uniformly distributed on the unit interval (i.e., on the interval [0, 1]). Due to regulatory intervention there is a price ceiling P ≤ r that firms are allowed to charge (lower prices are admissible). Consider the following two-stage game. At stage 1, before observing the demand realization μ̅ firms make investment decisions simultaneously. The constant cost per unit of capacity is c. After the investment stage, information about capacities become public knowledge. Next, demand is realized and publicly observed. At stage 2, firms compete in prices. Each firm simultaneously and independently sets its price pi. Firms maximize expected profits.

1. Determine the allocation at stage 2 for given capacity choice.

2. Determine the Nash equilibrium at stage 2 for given capacities. Here you have to distinguish between four different parameter regions. Note that in two regions pure-strategy equilibria do not exist and you have to solve for mixed-strategy equilibria. Also note that since all consumer have the same willingness-to-pay you can be agnostic about the rationing rule that is applied. If prices are equal, demand is assumed to be equally split between firms.

3. Analyze the 2-stage game and solve for subgame perfect Nash equilibria .