Econ 4211A sample solution to the m idterm examSpring 20081. T wo villagers want to hire several Samurai day s to protect their c rops.It costs 4 bags of rice to hire a Samurai for one da y. Assum e (fo rsimplic ity) that the villagers’ preferences are defined o ver bundles ofrice, (consum ed privately) and "safety" (number of Sam u rai days inthe village). The utility of the t wo villagers, a, b are:ua(ra,g)=4lng +lnraub(rb,g)=16lng +lnrbwhere rais the rice (in bag s) consum ed by villager a and rbis the riceconsumed b y villager b, g is the num ber of Sam urai days hired in thevillage. The first village r holds initially Ra=100bags of rice and thesecond one has Rb=120.(a) Write down the resource constrain t of the village (in terms of bagsof rice).ANSWER:ra+ rb+4g =220(b) Formulatetheproblem ofthebenevolentplanner(thewiseman)for this village.ANSWER:(c) Solv e the problem. What is the optimal number of Samurai thatthe villagers should hire together? (Hin t: in this case the wiseman has to solve for both the rice consumption and the num ber1of sam urai — use all the optimality conditions from the wise-manproblem.)ANSW ER: Lagrangian:L (g, ra,rb,λ)=4lng+ln ra+16 lng+ln rb−λ (ra+ rb+4g − 220)F.O.C.:(g :)20g− 4λ =0(ra:)1ra− λ =0(rb:)1rb− λ =0(λ :) ra+ rb+4g = 220The second and the third conditions imply ra= rb=1λ, togetherwith the first, w e get also 5ra= g. Substituting in the last equa-tion, we get2ra+20ra= 220hence ra=10=rb, and g =50.(d) Assume no w the villagers decide on how much protection to hirewithout consulting each other or the wise man (eac h decides ono w n expense on the Sam u rai taking the decision of the other asfixed.)ANSWER:The first villager’s problem:maxga,ra4ln(ga+ gb)+lnras.t.ra+4ga=100Optimality conditions:4raga+ gb=4ra+4ga=1002Villag er a0s reaction function is: ga= −15gb+20.The second (b) villager’s problem:maxgb,rb16 ln (ga+ gb)+lnrbs.t.ra+4ga=120Optimality conditions:16rbga+ gb=4rb+4gb=120Villag er b0s reaction function is: gb= −117ga+48017.Equilib riu m is a solution of the two sim u ltan eo us equation s:ga= −15gb+20gb= −117ga+48017So ga=30521,gb=57521, hence g = ga+ gb=30521+57521=88021=41.905.(e) Dothetwonumbersdiffer? Why? What problem does the WiseMa n help the villagers to solve?AN SWER: The second answer is smaller, protection is a publicgood, when dec idin g independently the two villagers und erp rovidethe public good as compared to the (join t) optim um, the wise manhelps to o v ercom e the free-rider problem.2. Recall the example from class. There are t wo plants in the region thatpollute the air. For each level of pollution the table con ta ins the cost(in, say, thousan ds of dollars per year ) of the optimal operation for eachof the plan ts:AmountofPollution4t3t2t1t 0tPlant A 100 190 600 1200 2000Plan t B 50 80 140 230 3253Assum e also that the marginal damage created b y each additional tonof pollution to the residents of the region is estimated to be 200.(a) How much pollution will be emitted b y the plants if none of themis made liable for the pollution?AN S W E R : Eac h plan t will m inim ize the costs, thus em ittin g 4ton of pollution, total pollution being 8 in the region.(b) Assum e the "Green Watch" group in the community realizes thatthere is a technology a vailable to each plant to eliminat e all thepollution. They appeal to y ou, the ma yor (overseeing the re-gion) to impose a regulation prohibiting any emission from eitherplant. If you are solely in terested in furthering efficiency (findingthe Pareto efficien t allocation) should you agree with the group?W hy? Pro vide calculations to support y ou r answer.AN S WER: Total elimina tion of the pollution is inefficient, as thecost of doing so overweighs the benefits. The benefitfromelimi-nating the pollution is 8 ∗ 200 = 1600, whilethecostincurredbyplant A only is 2000 − 100 = 1900.(c) Suggest a way to restore efficiency in your region. Describe thepolicy y ou will use, the reaction of eac h plant and wh y the result-ing allocation is P ar eto efficient.AN SWER: One way to restore efficiency is to impose a tax on thepolluters at the amount of da m age they create, i.e., t =200perton. In this case the first plant will cut pollution by 1 ton, whichcosts it less than the tax, while further cuts in pollution emissionare more costly than the tax: 600 − 190 > 200. The second plantwill cut the pollution to 0 ton, which costs less than pa yin g thetax according to the table. T h e solution is efficient, as emittingless pollutio n by the first plant costs more than the value of thedam age it creates and emitting mo re pollution by the first plan tis not worth w h ile: it costs less to elimina te the last ton than thedam age it creates.Choose only one of the following two problems (either a or b).3.a Explain, by using your understanding of social insurance, the following4empirica l finding.1Citizen’s belief that (pure) luck determines incomeis positively correlated with the social spending (as a % of GD P).ANS W ER: View ing one’s success as random, less "merit-based" in-creases the idiosyncratic risk in the eyes of individuals who, then, otherthings being equal, are ready to pay more for insurance to protect them-selv e s from " ad verse circumstances". Th is creates grea ter willingnessto pay for social insurance (including unem p loyment insurance and wel-fare).3.b Provide an example of a project (including the stream of benefits/costs)the present value of which is positive under 3% discoun t rate and neg-ative under 7% discount rate. Pro vide one argument in support andone argumen t against imp lementing such a project.ANSWER:Let the project last for 3 periods (years) and generate the followingstream of benefits: b1=0=b2,b3=10, and costs: c1=9;c2= c3=0.Then the presen t value of the project under 3% discount rate is10(1.03)3−91.03=0.413 55The presen t value under 7% rate is10(1.07)3−91.07= −0.248 24Suc h a project has a fla vour of "inv estm e nt in the future" (say, "savethe earth"). A lo w discoun t rate (3%) is favou r ing more future benefitsat the expense of curren t costs (in vestments), so one should view thisproject as acceptable if future benefits are "importan t enough".An argument against such a project (the PV is negative under a highdiscount rate, 7%) is to say that the benefits are too far in the futureand are not justified by curren t costs, in this case future benefits are"less importan t" socially.1See, for example,