Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 9: Monopoly - Duop oly3/31/10This problem set is due 4/7.1. Third Degree Price Discrimination. Consider a monopolist (say a lo-cal movie theatre in Fort Lauderdale) which has two distinct client groups,adults and seniors. The inverse demand for the adults is given byp (qA) = a  bqA;and the inverse demand of retirees is given byp (qB) =a3b3qR:(a) Describe the demand function in the two markets graphically andthen compute the demand elasticity in each market.(b) Compute the demand function q (p) under the assu mption that themovie theater can only o¤er a single price to both segments of themarket. (Hint: at a given price add the demand of the adults andsenior market. You need to go from the inverse demand function tothe demand function.) Illustrate the aggregate demand function incontrast to the demand functions in each segment. Now compute theoptimal price of the movie theatre when it can only o¤er a single andcommon price to the market segments. Who goes to the movies andwho doesn’t?(c) Next we allow the movie theatre to o¤er di¤erent p rices in each se g-ment and customers cannot misrepresent their identity. What is theoptimal price in each one of the markets?(d) Compare the welfare of the consume rs and the revenue of the con-sumer after the introduction of di¤erent prices across di¤erent seg-ments. Explain.2. The duopoly model we have been working with views …rms as choosingtheir quantities of output, with the market then setting a common pricefor the two …rms. Here is an alternative model. Firms 1 and 2 set prices p1and p2. If p1< p2, …rm 1 sells quantity A  Bp1and …rm 2 sells nothing.Similarly, if p1> p2, …rm 2 sells quantity A Bp2and …rm 1 sells nothing.If p1= p2, each …rm sells half of the quantity ABp1= ABp2. Supposethat each …rm has constant marginal cost c, so that the cost of producingoutput xifor …rm i is C(xi) = cxi.1(a) Find the equilibrium prices p1and p2in this market. This is nota job for calculus , because the …rms’ payo¤s are not di¤erentiablefunctions of p1and p2. Instead, try a few combinations of prices, andfor each one , ask yourself whether each …rm is doing the best it cangiven the other …rm’s price, or whether either …rm has an incentive tochange its price. You have an equilibrium when each …rm is settinga price that maximizes its pro…ts, given the price of the other …rm.(b) How does the outcome you’ve found in [2a] compare to the equilib-rium of the quantity-setting model, or to the competitive outcomein this market? Now suppose you are involved in a case before theJustice Department, in which a merger is to be evaluated that willleave a market with only two …rms. The concern is that this mergerwill lead to higher consumer prices. If you represented one of the…rms that wanted to merge, which model of the resulting duopolymarket would you be inclined to use as the basis for your analysis?Which would you use if you worked for the Justice Department? Asan outsider, which do you think is more appropriate?3. Pure Strategy Nash Equilibrium.(a) Find all pure strategy equilibria in the following games, often referredto as “Battles of Sexes”:BobBallet SoccerAnn Ballet 2; 1 0; 0Soccer 0; 0 1; 2(b) and “Hawk-Dove”:Defend AttackDefend 3; 3 1; 4Attack 4; 1 0; 04. Voluntary Contribution Games. Consider the voluntary contributiongame where for all i 2 I:ui(g1; :::; gI) = ln (w  gi) + ln0@IXj=1gj1A.and each agent has to make a choice as to how much of the public goodgihe wants to contribute and how much to keep for private consumptionw  gi, where the wealth level w is the same for all the agents.(a) De…ne the situation as a game and de…ne the Nash equilibrium.2(b) Derive the socially optimal allocation in which each agent has thesame welfare weight (equal to one)(c) Derive the unique symmetric Nash equilibrium.(d) Compare the socially optimal contribution with the equilibrium con-tribution. What can you say about the relationship as I ! 1.5. Reading Assignment: NS Chapter 8, 14,