Consider a monopolist providing a product over several periods. The monopolist  has constant marginal costs of production of c in each period. Consumers consider to consume one unit of a good in each period. The consumer population is of mass 1. Half of all consumers have a high valuation ɣ_H for the product, which is the same in each period. The other consumers have a low valuation ɣ_L with ɣ_H > ɣ_L > c. While high-type consumers have to buy the product in the period in which they consume it, low-type consumers are assumed to be able to store one unit for one period at no cost. The common discount factor is d .We assume that even a product which is consumed one period later generates a value larger than production costs, dɣ_L > c. 

1. Suppose that there is an infinite time horizon and that the monopolist commits to a price path {p^t}_{t =1, 2…..} before the market opens. Determine the profit-maximizing price path under the constraint that the price has to be the same in all periods. Determine the monopolist’s profits.

2. Consider the same setting as in δ1) without the restriction that the prices have to be constant over time. What is the profit-maximizing price path? Under which conditions does the monopolist prefer non-constant prices over constant prices? Determine the maximal profits of the monopolist? Explain your result.

3. Consider now a two-period setting. Suppose that the monopolist cannot commit to a price path. In particular, the monopolist sets p1 and, after selling in the first period, he sets p² at the beginning of the second period. Are there parameter constellations such that the monopolist sets different prices in the two periods? If your answer is negative provide a proof why non-constant prices cannot be a subgame perfect Nash equilibrium. Other- wise, characterize the set of parameters under which a subgame perfect Nash equilibrium with non-constant prices is supported.