公共财政学答案.pdf

Public Economics Homework #1 Binzhen Wu Assigned: Sep. 27, 2012 Due in two weeks: 9:50am, Oct. 11/12 (Thursday/Friday) 1. (10 Points) Pareto Optimal Consider a case of pure exchange, with no externalities. Amy has the utility function UA =0.5xA + 0.5yA and Betty has the utility function UB = 0.5xB0.5yB 0.5. The resource constraint is xA + xB = 10 and yA + yB = 8. Consider the following two allocations. One is xA = 0, yA = 0 and xB = 10, yB = 8; the other is xA = 5, yA = 4 and xB = 5, yB = 4. Are these two allocations Pareto efficient? Explain. If the answer is no, find a Pareto improvement. ANS: The first one is Pareto efficient, the latter one is not. For the first allocation: MRSx,yA = MUxA/MUyA = 1/1 = 1 MRSx,yB = MUxB/MUyB = 0.5*0.5xB-0.5yB0.5/ 0.5*0.5xB0.5yB-0.5 = yB/xB =8/10 So MRSx,yA MRSx,yB. However, this allocation is Pareto optimal, because this allocation is a corner allocation. We do not require MRSx,yA = MRSx,yB for corner solutions. For this allocation, there is no way to make A better off, without making B worse off. Therefore it is Pareto optimal. Second allocation (inner solution): MRSx,yA = MUxA/MUyA = 1/1 = 1 MRSx,yB = MUxB/MUyB = 0.5*0.5xB-0.5yB0.5/ 0.5*0.5xB0.5yB-0.5 = yB/xB = 4/5 MRSx,yA MRSx,yB, it is not Pareto efficient. For the second allocation, Amy gets relatively more utility from x, while Betty gets relatively more utility from y. Amys possible Pareto improvement is to give Amy one more unit of X, give Betty one more unit of Y. The new allocation is xA = 6, yA = 6 and xB = 4, yB = 4. (Receive full credit as long as it is Pareto improvement.) Under the new allocation, Amys utility is 0.5*6+0.5*6 = 6; Bettys utility is 0.5*square (4*4) = 2. Under the old allocation, Amys utility is 0.5*5+0.5*7 = 6; Bettys utility is 0.5*square (15) 2. Therefore, Betty is better off while Amy is not worse off. This is a Pareto improvement. There seems to be some errors in this part of answer, a Pareto Improvement which is also the answer for most students could be xA = 5.5, yA = 3.5 and xB = 4.5, yB = 4.5 2. (30 Points) Competitive Equilibrium Suppose Fred is the only consumer in the economy. Fred derives utility from two things: clothes and books. He has an income of $24 and a utility function given by: U(C, B) = C3/4B1/4. There is one firm that produces clothes using labor (L) and capital (K) and a technology given by: fC(L,K) = L1/4K1/4.There is one firm that produces books with a technology given by: fB(L,K) = L1/3K1/3 . The price of labor PL = $1 per unit and the price of capital PK = $4 per unit. Solve for the competitive equilibrium in this economy. a. What is Freds demand for clothes? For books? b. What are the firms supply curves (for clothes? for books?)? c. What are the prices (PC and PB) when markets are clear? d. Suppose the price of capital increases, how will the price of clothes and books change? e. Suppose there is another consumer Joan, she also has preference U(C, B) = C3/4B1/4 and an endowment of $24. What is the aggregate demand curve for clothes now? What are the market-clear price for clothes PC? ANS: a. Max C3/4B1/4 s.t. PCC + PBB = 24 .(1) F.O.C MUC/MUB = PC/PB or 3B/C = PC/PB .(2) Combine (1) and (2) We have C = 18/PC and B = 6/PB b. For producer of clothes Max PCLC1/4KC1/4 wLC rKC F.O.C. w.r.t LC and KC, and we will have LC/KC = r/w .(1) C = LC1/4KC1/4 .(2) Combine (1) and (2) KC = (r/w)-1/2C2 LC = (r/w)1/2C2 The cost function CostC(w,r,C) = wLC + rKC = 2(rw)1/2C2 In a completely competitive market PC = MCC = 4(rw)1/2C With r = 1 and w = 4, the supply function for clothes is C=PC/8 With same procedure, we obtain a general supply function of book B(r,w) = PB2/9rw or B = PB2/36 For clothes, Demand: C = 18/PC Supply: C = PC/8 PC = 12, C = 3/2 For books Demand: B = 6/PB Supply: B = PB2/36 PB = 6, B = 1 c. see (b) above d. PC and PB are all increasing in r and w. So price increases. e. The demand of Joan is identical to that of Fred. So the aggregate demand for clothes will be C = 36/PC The supply curve is still C = PC/8 So the market-clear price for clothes PC = 12 square (2) 3. (10 Points) Social Surplus Consider a free market with demand equal to Q = 1600-10P and supply equal to Q = 30P a. What is the value of consumer surplus? What is the value of producer surplus? b. Now the government imposes a $10 per unit subsidy on the production of the good. What is the consumer surplus now? The producer surplus? Why is there a dead weight loss associated with the subsidy, and what is the size of the loss? ANS: a. Demand Q = 160010P Supply Q = 30P P = 40, Q = 1200 CS = 0.5*(16040)*1200 = 72000 PS = 0.5*40*1200 = 24000 b. Demand Q = 1600-10P Supply Q = 30(P+10) P = 65/2, Q = 1275 CS = 0.5*(16032.5)*1275 = 81281.25 PS = 0.5*(65/2+10)*1275 = 27 093.75 Deadweight loss: 0.5*10*(1275-1200) = 375 When producer is subsidized by s, there is excess supply and production is inefficient. Consumer surplus and producer surplus increase, however, the social gain is negative. There is deadweight loss. 4. (20 Points) Public Goods Suppose Paul and Scott are the only residents of a city (others may pass through and possibly commit crimes). Each has a utility function over cigarettes (x) and total policemen (M), of the form: U = 2*log (x) + log (M). The total policemen hired, M, is the sum of the number hired by each person, so M = MP+MS. Paul and Scott both have income of 100, and the price of both cigarettes and a policeman is 1. They are limited, for the purposes of this problem, to providing between 0 and 100 policemen. a. How many policemen are hired if the government does not intervene? How many are paid for by Paul? By Scott? b. What is the socially optimal number of policemen? If you answer differs from a), why? c. Suppose the broader county government is not happy with the private equilibrium, and it decides to provide 10 policemen for the town. It taxes Paul and Scott equally to pay for the new hires, but each is free to hire additional policemen if they would like. What is the new total number of policemen? How does your answer compare to a)? Have we achieved the social optimum? Why or why not? The “free” in the problem means each individual could feel free to choose any amount of policemen they would like to consume, not that policemen are provided free (at the price 0). ANS: a. For Paul Private solution considers, Max 2*log (xP) + log (M) s.t. xP+ MP = 100 Remember, M = MP+MS F.O.C. w.r.t. MP 2/ (100-MP) = 1/ (MS+ MP) or 2M = 100- MP or MP = 100/3 - 2MS/3 .(1) Scott, who faces a symmetric problem, will have a same F.O.C. (reaction function) 2/ (100-MS) = 1/ (MP+ MS) or 2M = 100- MS or MS = 100/3 - 2MP/3 .(2) Combining (1) and (2) gives MP = MS = 20, total number of policemen is M = 40 b. Social optimum occurs when the sum of the marginal rate of substitution equals the marginal rate of transformation. MRTM,x = PM/Px = 1 MRSP = MUM/MUx = ! ! ! ! = ! !(!) = MRSP + MRSS = ! ! = MRT = 1 Furthermore, we know the social budget constraint is xP + xS + M = 200 Then we obtain M = 66.67 The level of policemen is greater than that in a). The reason is that when individuals consider their own benefit, ignoring the fact that an additional policeman brings benefits to other people, the public goods are underprovided. They are not altruism, and everyone wants to be a free rider. This is like an positive externality. c. The procedure is the same as in part (a) For Paul, Max 2*log (xP) + log (MP+MS +10) s.t. xP+ MP = 95 The reaction function MP = 25 - 2MS/3 And MS = 25 - 2MP/3 for Scott So solve the equations and we have MP = MS = 15, which implies that the total number of policemen M = 15+15+10 = 40 This level is still same as in part (a), social optimum is not achieved since both of them are taxed equally and the amount is less than the amount of what they originally provide without government intervention. So they just pay taxes as if they buy the policemen. In this case, government provision fully crowds out private provision 5. (20 Points) Public Goods Now, suppose in question 4 above, we still have Paul and Scott paid for cigarettes and policemen. However, since they now live together, the cigarettes become a public good as they can enjoy the smoke in the air. So the utility function changes to U = 2*log (X) + log (M), where X and M are total cigarettes and total policemen. i.e., X = XP+XS, M = MP+MS. Assume their income is still 100, and the price of both cigarettes and a policeman is 1. a. How many cigarettes and policemen will be paid by each of them if they independently make decision? b. Is the level of choice in part (a) social optimal? Why? c. To achieve social optimal, is it necessary for them to consume equally on M or X? ANS: a. For Paul, Max 2*log (X) + log (M) s.t. XP + MP = 100 and XS + MS = 100 F.O.C w.r.t. XP and MP We will have reaction functions 3MP = 100 + XS 2MS 3XP = 200 + 2MS XS Symmetrically, we have reaction function for Scott, 3MS = 100 + XP 2MP 3XS = 200 + 2MP XP And we obtain that XS = XP = 200/3 and MP = MS = 100/3 Some students argued about the unnecessary of XS = XP and MP = MS in this question. As this is a private provision question for two identical individuals, their consumption choice should be the same facing their own maximization problem. b. It is social optimal. Notice that these two individuals have the same interest, that is, when one individual gets better off, the other individual is also better off. Therefore, we can make both individuals the best by maximizing one individuals utility subject to resource constraint. At this allocation, there is no potential pareto improvement. Max U =2*log (X) + log (M) X+M = 200 We have X = 400/3 and M = 200/3 Since the answer in (a) (MP = MS = 100/3, XS = XP = 200/3) satisfies the condition, it is social optimal. c. The consumption bundle is not unique. Any point on the dash line (for example A) would be an optimal allocation for (XS, MS). (XP, MP) = (X*- XS, M*- MS) will reach optimum (point B). Allocations on dash line are all Nash equilibrium. 6. (10 Points) Gruber, Chapter 7, Q12 Andrew, Beth, and Cathy live in Lindhville. Andrews demand for bike paths, a public good, is given by Q = 12 2P. Beths demand is Q = 18 P, and Cathys is Q = 8 P/3. The marginal cost of building a bike path is MC = 21. The town government decides to use the following procedure for deciding how many paths to build. It asks each resident how many paths they want, and it builds the largest number asked for by any resident. To pay for these paths, it then taxes Andrew, Beth, and Cathy the prices a, b, and c per path, respectively, where a + b + c = MC. (The residents know these tax rates before stating how many paths they want.) a. If the taxes are set so that each resident shares the cost evenly (a = b = c), how man