光华管理学院公共财政作业2

Problem Set2for Public FinanceJuanjuan Meng and Xi WengDepartment of Applied Economics,Guanghua School of ManagementPeking UniversityDue:1p.m.,May12Note:1.There are15questions in total for Question1-4.Each question has6points.And the lastextra question has10points.2.You can write in Chinese or English.3.No late homework will be accepted.Please staple your homework and write your name andID number on the cover.Question1.An island has two lakes x and y,and20fishermen who can choose tofish on either one of the lakes,and keep the average catch on the chosen lake.The total number offish caught on x and y when these arefished by L x and L yfishermen areF x=10L x−12L2xF y=5L y1.What is the total number offish caught?(hint:Find the equilibrium number offishermenon each lake:L x,L y.To do so,suppose you are the20thfisherman,and you know that all 19otherfishermen have already chosen x or y.Then write down under what conditions you would chose lake x and under what conditions lake y.An equilibrium requires you to be indifferent.)2.What allocation offishermen to lakes maximizes the total catch offish on the island?(hint:this is not an equilibrium,but rather the allocation offishermen to x and y by a planner.)3.Identify precisely the externality in this problem.4.If afishing license were required for lake x,what value license(in terms offish)would beneeded to achieve efficiency?5.If a profit maximizing manager of lake x were to hirefishermen at wage rate5fish,what isthe competitive equilibrium?Question2.Consider a group of50residents attending a town meeting.They must choose between three proposals for dealing with town garbage.Proposal1asks the town to provide garbage collection as one of its services.Proposal2calls for the town to hire a private garbage collector to provide collection services,and Proposal3calls for residents to be responsible for getting their own garbage.There are three types of voters.Thefirst type prefers Proposal1to Proposal2 and Proposal2to Proposal3;there are20of these voters.The second type prefers Proposal2 to Proposal3and Proposal3to Proposal1;there are15of these voters.The third type prefers Proposal3to Proposal1and Proposal1to Proposal2;there are15of these.(a)Does there exist a Condorcet winner?(b)Suppose voting proceeds using a Borda count,in which voters list the proposals,in order of preference,on their ballot.The proposal listedfirst(or at the top)on a ballot gets three points,the proposal listed second gets two points and the proposal listed last gets one point.In this situation, with no strategic voting,how many points are gained by each proposal?Which proposal wins?(c)What strategy can the second and third type of voters use to alter the outcome of the Borda count vote in part(b)to one that both types prefer?If they use this strategy,how many points does each proposal get,and which wins?Question3.Romeo loves Juliet and Juliet loves Romeo.Besides love,they consume only one good,spaghetti.Romeo likes spaghetti,but he also likes Juliet to be happy and he knows that spaghetti makes her happy.Juliet likes spaghetti,but she also likes Romeo to be happy and sheknows that spaghetti makes Romeo happy.Romeo’s utility function is U R(S R,S J)=S aR S1−aJandJuliet’s utility function is U J(S R,S J)=S aJ S1−aR,where S J and S R are the amount of spaghetti forRomeo and the amount of spaghetti for Juliet respectively.There is a total of24units of spaghetti to be divided between Romeo and Juliet.a).Suppose that a=2/3.If Romeo got to allocate the24units of spaghetti exactly as he wanted to,how much would he give himself and how much would he give Juliet?If Juliet got toallocate the spaghetti exactly as she wanted to,how much would she take for herself and how much would she give Romeo?b).What are the Pareto optimal allocations?If we treat“spaghetti for Romeo”and“spaghetti for Juliet”as public goods,we would have an economy with two public goods and no private goods.We canfind a Lindahl equilibria byfinding personalized Lindahl prices where p ij is the price that person i pays per unit of j’s consumption.In Lindahl equilibrium the Lindahl prices must be chosen in such a way that given their personalized prices,consumers all agree on the quantity that should be consumed by every consumer and such that for each j,the sum of the prices p ij over all consumers i is one.c).Suppose that Romeo in the previous problem has an intital endowment of18units of spaghetti and Juliet has an initial endowment of6units.Find the Lindahl equilibrium prices and the Lindahl equilibrium quantities of spaghetti for Romeo and Juliet.Question4.Consider a society inhabited byfive citizens.Suppose that the preferences of agent i∈{1,2,3,4,5}over a publicly provided good y and a privately provided good c i are u i=c i+αi log y,whereαi is an intrinsic parameter of agent i withα1=0.2,α2=0.5,α3=0.8,α4=1andα5=2.Assume,in addition,that all individuals have initial resources in private good e i=1for all i.Suppose also that one unit of private good is required to produce one unit of public st,suppose that tofinance the production of the public good,the government raises a tax q on each individual so that agent i’s budget constraint is c i≤1−q.1.In the socially optimal allocation of this economy,what is the amount of y?2.What is the mostly preferred policy q(αi)of agent i?3.Under majority rule,what is the selected policy?Compare this to the social optimum.Howcan we make the social optimum coincide with the equilibrium policy by changing only one of theαi?4.Suppose now that each agent’s preferences are given by u i=c i+|αi−0.6|log y.Redo(1)-(3).Extra Question.Consider the following social choice procedure.If there is a Condorcet winner,it is the social choice.Otherwise,the alternative on top of thefirst person list is the social choice.(That is,if there is no Condorcet winner,then person one acts as a dictator.)Give an example with three people and three alternatives showing that this procedure does not satisfy independence of irrelevant alternatives.。