1. Firm I (with discount factor 5= 1 ) has a planning horizon of two periods. In period I it will be the only firm in the market. In period 2 there will be a potential entrant (firm 2) offering an identical product. If entry occurs competition will be in output levels (Cournot competition). Inverse demand in both periods is P= 1 - Q. Firm I has no fixed costs, while firm 2 has a fixed cost equal to . Both firms have a constant marginal cost. Firm 2's 81 marginal cost is . while firm I's marginal cost in period I is - . Firm I can reduce its Marginal cost in period 2 to ce O. ~ |by spending .(2-c ) dollars in period 1. where (>0. The potential entrant will enter if and only if it expects to make positive profits. [Please note: in answering the following questions what is most important is setting up the relevant equations/inequalities and paying attention to whether the relevant functions are convex or concave. Do not waste too much time in calculations. Get back to the actual calculations only after you have answered the required four questions.] (a) What is the minimum amount firm 1 needs to spend in period 1 in order to make sure that the potential entrant will stay out? (b) Determine the value of a call it of, such that for a of firm I will prefer not to deter entry. (c) If firm I decides not to deter entry, what marginal cost will it choose for period 2 (alternatively, how much will it spend in period I in order to reduce its period 2 marginal cost)? (d) If firm I decides to deter entry, what marginal cost will it choose for period 2? ( Alternatively, how much will it spend in period 1 in order to reduce its period 2 marginal cost?)2. There are N consumers, with the same income, denoted by E, but different preferences concerning quality. Each consumer is characterized by a different value of a parameter . which is uniformly distributed in the interval [0.1]. A firm produces a product of quality .x. where x belongs to the interval [4,10]. Each consumer buys at most one unit. If a consumer identified by the value re [0. 1 ] of the parameter does not buy the good. his utility is equal to E. while if he buys one unit of a product of quality & at price p his utility is equal to: E-p+x1. (a) Consider first the case of a single-product monopoly with zero production costs. What quality level x would the monopolist choose? What price would it charge and how many units would it sell? Prove your claims. (b) Continue to assume that production costs are zero. Would a monopolist prefer to offer only one quality or to offer two qualities? That is, would the firm choose to be a single- product firm or a multi-product firm? Give a verbal argument without trying to prove your claim (we'll come back to this in point (2). From now on we continue to assume that the industry is a monopoly and also that the monopolist is offering only one product, However, we relax the assumption of zero production costs and we assume instead that the firm has the following cost function: C(q-x)=eq where q is output and e is a positive constant with 0
