Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 8: Monopoly and Duop oly3/24/10This problem set is due 3/31.1. First, we examine a monopolistic …rm. The …rm faces a market describedby the demand function p = A  By, where p is the price the …rm receivesif it sells quantity y of output. The …rm’s cost function is given by C(y) =12y2.(a) Find the pro…t-maximizing quantity of output and the correspondingpro…t for this …rm.(b) Show that if the …rm could sell more output at the (constant) equi-librium price, it would do so. (To do this, take a derivative of the…rm’s payo¤ under the assumption that price is constant rather thana function of output, and then evaluate this derivative at the equi-librium price and quantity from part 1a.)(c) Let p be the equilibrium price and y the equilibrium quantity, sothat p = A By . Now suppose the demand in the market shifts, sothat the new demand curve is p = A0B0y, with A0> A and B0> B,but with it still being the case that p = A0 B0y . Hence, if the…rm does not change its quantity, its price will also not change. Isthe quantity of output y still optimal, given this new demand curve?If not, will the …rm decide to produce more or less? Formulate your…rst-order condition from part 1a in terms of the elasticity of demand,and use this to explain your result in this part.2. Now supp os e you must design a tax on the monopoly. Let the demandfunction be p = A  By.(a) First suppose you consider a sales tax of 10% on this market. Supposethat the …rm has to pay the tax, as is typically the case with salestaxes. Hence, if the …rm produces quantity y of output, the pricepaid by consumers is p = A  By, but the price received by the …rmis :9p = :9(A  By), where the :9 appears because the …rm gets tokeep only ninety pe rcent of the purchase price, paying the remainingten percent to the government in taxes. Find the pro…t maximizingquantity of output for the …rm, the price paid by consumers, theprice received by the …rm, and the …rm’s pro…t. Explain how theseanswers compare to those of part 1a. In particular, do consumerspay more as a result of the tax? Does the price received by the …rmfall? Does the …rm’s pro…t fall?1(b) Instead of a sales tax, the government considers a pro…ts tax. Hence,if the …rm chooses quantity of output y and charges price p = ABy,the government collects t[py 12y2] = t[(A  By)y 12y2] in taxrevenue, leaving (1  t)[py 12y2] = (1  t)[(A  By)y 12y2] as after-tax pro…t for the …rm. Once again, …nd the …rm’s pro…t-maximizingquantity of output and the resulting price. What e¤ect do es th epro…ts tax have on the quantity of output and price? What e¤ectdoes it have on the …rm’s pro…t? In light of your answers, given thata …xed amount of revenue is to be raised, which tax would consumersprefer— a sales tax or pro…ts tax? Why?(c) Now suppose the cost function is given by12x2+F , where F is a …xedcost. For example, F may be the cost of conducting environmentalimpact studies or acquiring the licenses needed to produce. Whate¤ect do es F have on the …rm’s optimal quantity of output, price,and pro…ts? Many communications …rms are monopolies because thegovernment gives them exclusive rights to use certain bands of air-wave lengths. Sometimes the government simply gives the …rms thisexclusive right, while other times it sells the right to the highest bid-der. The latter method has been criticized on the grounds that it willdrive up the …rms’costs and hence the prices charged to consumers.What do you think of this argument?3. Now suppose we have a duopoly market, with two …rms. We have assumedthat …rms 1 and 2 pro d uce perfectly homogeneous products, in the sensethat they sell their output at a price that depends only on the total amountproduced in the market and not on who does the producing. To see howthis assumption might be generalized, let the pro…t functions for the two…rms b e given by (where A > 0, B > 0)1(x1; x2) = (A  Bx1+ Dx2)x1 cx12(x1; x2) = (A  Bx2+ Dx1)x2 cx2:(a) Explain what it means for D to be negative, and what it means for Dto be positive. (Think about substitutes and complements.) If D > 0,we will assume that B > D. What is the ec onomic interpretation ofthis assumption, and what would happen if it did not hold?(b) Find the equilibrium quantities of x1and x2in this market. Todo this, …nd the …rst-order condition for each …rm, and solve theresulting pair of linear equations for x1and x2.Reading Assignment: NS Chapter 14,