Consider again the Hotelling model as discussed in class. Here we consider a version with entry decisions. Consumers derive utility from consuming the good (poten- tially) sold by firms and incur a linear cost τα when they travel to the firm to acquire the good, where denotes the consumer's location on the line. Firms' marginal cost of production is constant and equal to zero.
(a) Suppose first that only firm 0 is in the market. Solve its profit maximization prob- lem. [Note that we have made no restrictions on the parameters r and 7.]
(b) Now assume that there are two potential firms in the market, located at the end- points of the line. The two firms first decide whether to enter the market. After entry, they compete in prices (or behave as in part (a) if they end up being a mo- nopolist). From now on, assume that < r < 27. The first inequality ensures that the market is covered in the duopoly case (you do not need to establish this, take it as a fact). Moreover, assume that the entry cost e satisfies (c) Consider a planner who can only control the number of firms in the market, but not their pricing. How does the equilibrium number of firms compare to the num- ber of firms chosen by the planner?