Questions

UNIvsRsrrv AT ALa-mv, STATE UNIvERer'v ca New Year: DEPARTMENT oF Econcmcs PhD. l'-'re1imina.r1_,.r Examination in Microeconomics, August 20, 201E! Instructions: Answer any three of the [our mlmhered problems below. Justify your answers whenever possible. 1|Write your answer to each problem in a separate set of attached answer sheets. Write the mnnher ofthe problem you are answering AND Nil-THESE ELSE on the cover of the attached sheets You may not use any electronic devices. The exam lasts 4 hours. 1. A monopolist faces two types of potential customers: A high type with demand curve DILp-j ={l p] anda lowtype withdemand curved{p} = l pwhere I] p1.[t is public knowledge that the proportion of the high type in the population is o:, where [l n: o: c 1. But the monopolist cannot identify an individual's type. The mass of the total population is normalized to 1. Fixed cost and marginal costs are zero. A two part tari [T,p} consists of an 'entry [ee' T E I] and a [linear] price per unit p. 1|When a high type consumer buys Dp] at price p, its consumer surplus S is the area under its demandcurve and above the price [fa high type consumer pays an entryr fee T as well, his net surplus is S T. Similarly, a low type Eacing {T,p] has consumer surplus s and a net surplus sT if he buys. A consumer participates {buys the commodity} if and only if his net surplus is non-negative. a. [f the monopolist uses linear pricing [T = D}, what is its optimal price and prot'E| b. Suppose the monopolist uses a single two-part tari [T,p] where both types partici pate, what is the optimal value of [T,p} and corresponding prot [or the monopolist? c. Suppose the monopolist uses a single tsp-part tariff (T, p} where only the high type participates1 what is the optimal [T,p} and corresponding prot for the monopolist? in your answer, show that the low type will not participate at {Tm}. d. Comparing your answers to parts b and c, give all values of o where serving only the high type is more protable for the monopolist. In particular, check whether giving up low types is agood policy for the monopolist when a: = 115,; or . e Now, suppose the monopolist can oes a menu oftwo two-part tariffs {[Th,ph},['1'i,p;j} where both types 'participate' and a high type selfselects [Tmp and a low type self aelects [T,p. Set up the problem that solves [or the optimal menu that the monopolist can offer. 1ll'll'hy is requiring that both types participate not a problem? i. Solve for the optimal mernr of twopart tariffs {{Tmph}, [T,p} when o: = US. (Hint: the participation constraint for the high type is satised when all other constraints hold. Solve the problem ignoring the incentive coretraint of the low type, then show the incentive constraint of the low type holds at the solution]. g. lCompare the Irionepolist's profits [without computing them} as its offer switches from linear pricing to single two-part tariffs and to a menu of tsp-part taris. \f3. Consider an economy with two types of agents, both of whom have preferences dined over food {to} and prescription medicine [tr]. Type A agents each have thedierentiable utility function adorn, ya] and are endowed with at, units of 5:. Similarly, type B agents each have differentiable utility uzmyhj and are endowed with at, writs of c. Ame this is public information. There are equal numbers of both types. Agenis can give up someoralloftheir rand buy-yat pricep [in unitsofc}. a. First,writedewnthe decision problem each agent Eacesin decidinghow muchrand tr to consume and depict a solution. In order to cover administrative costs, the government imposes [linnp sum] taxes 9,, and pi. {in units of 1:}, respectively, on types A and E. [AIume the gm'e-nrnent can distinguish between the types say, A agents are rich and B are poorand can tell which type a given consumer is.} b. Assuming the agents take the taxes as given, how does this aect their choice prob lems? c. Assume the government must collect atotal oil? from each AB pair [i.e., _1,'.I,,+g.I = G}. Argue that, from an efciency point of view, it does net matter how the tax burden is distributed. d. It is implicitly assmned in part c that there is maeltreetmcntof-equols, that is, that all agents of the same type pay the same tax. [s this necessary for elciency? Explain. Now suppose that instead of covering administrative costs, tax collections were used solely to provide y at the price if a: :1, since by consolidating demand the government is able to negotiate a lower price based on the volume of purchases. In this case, the govemment would tax type i anamountgg, usethisto |:|rurchase'g,.I atthe reducedprice p', and distribute the amount purchased to i; that is, i would receive to, = gifp' {instead of pin'chasing its own tr}. e. First, suppose the amount of the tax, oi, is determined by observing the quantity of 1:.- that agent i would nennally consurneatpricep, asin part a, say pf, but onlychargingi the amount g.- 2 y}, thus allowingi' to consume [,pp'jyf additional units of r. Argue that this would typically be inecient. f. In contrast, suppose g.- is calculated by determining the amount of 93- i would consume at price 1!, say if}, and levying the tax g.- = #1:}. Would the outcome be efcient? Why? g. If u.-, and hence if}, were private information and the scheme in part freo_u.ired t' to \"reveal"I its demand, 1.5, would it be possible for i to do better by am1ouncing other than its true demand? Explain. h. Finally, assume the process of consolidating purchases and negotiating price reduc tions itself entails some administrative cost. Thus, agent 1' will be taxed 11, +3\" where 11,- is the portion devoted to administrative costs (hence, 'g, +31. 2 G} and i is determined as in part f after paying9;. Amuming the agents:I preferences are who, it} = rim, deter mine when it would be better to incur the administrative cost and [ace the lower price 12' rather than avoid the administrative cast and [ace the higher price p