This problem extends that of the multi-product monopolist presented in the lecture. Consider two markets which are each monopolized by the same firm. The demand functions for each are given by Q1 = 100 - pi + pz (1) Q2 = 100 - p2 + P: (2) where Q, is the total quantity demanded in market j and p, is the unit price in that market. Assume that the firm's cost function is given by c(q,) = 20q, in both markets. (To be clear, the subscript i refers to firms and the subscript j refers to markets 1 and 2. qi, refers to firm i's choice of quantity to supply to market j. Because each market has a single firm supplying it by assumption, Q, = q.-) Solve the multi-product monopolist's problem. 1. Write out the profit function for the same monopolist selling into each of the two markets. 2. Write down the first order conditions for the monopolist's problem in general form (i.e., not using the specific parameterizations given for demand and cost functions). 3. Derive the general solution for the the Lerner Index in each of the two markets. How does it differ from the case of a single-product monopolist? 4. Now plug in the parameterizations for demand and cost and solve for equilibrium price and output in each market. There are four total outcomes to solve for: Q1. Q2. Pi. pr- 5. What is the residual demand elasticity facing the monopolist at the equilibrium price and output in each market? 6. How does your solution in (4) and (5) differ from each of these three cases: . Separate monopolists in each market, with no relationship between them (so Q; = 100 - P;). . Separate monopolists in each market, but with the same demand functions (so Q, = 100 - p, + p-, . where the subscript -j refers to the price in the other market.)