Reference no: EM133066336

Consider the following vertical product differentiation duopoly. Suppose that the quality of the product can be described by some number s; E [s; s] C R+. Consumers are identified by 0 E [0; 0] c R+, which measures their preference for quality. Consumers are distributed uniformly on [0; 0]; and are of mass M = 0 -0. A consumer of type 0 receives a utility of vi = 0si - Pi when consuming a unit of good i. Assume this is always sufficiently large that they would buy one unit of the good. Each consumer is, however, always buying only one unit. Two firms compete in the market. We look for the subgame-perfect equilibria of the following two-stage game: firms first choose the quality of their product and then compete in prices. We assume that the marginal cost of production depends on quality. We denote by C(qi; s;) the cost of firm i producing qi units at a quality s; and we assume C(qi; si) = qisi. To guarantee interior solutions in the pricing game, we assume 0 is sufficiently large.

a. Consider the second stage of the game where firms set prices simultaneously, taking the qualities as given. Firm 1 produces quality s1 and firm 2 produces quality $2, with the convention that $1, $2. Derive the Nash equilibrium in prices and express the equilibrium quantities and profits of the two firms at stage 2.

b. Consider now the first stage of the game where firms simulta neously choose the quality of their product. Show that ($1; $2) = (s; 3) or (3; s) are the equilibrium quality choices of the game.