Economics 393 Test 2 Thursday 28 th June 2018

Transcription

1 Economics 393 Test 2 Thursday 28 th June 2018 Please turn off all electronic devices computers, cell phones, calculators. Answer all questions. Each question is worth 10 marks. 1. Suppose the citizens of the City of Toronto care about only two things aerobics lessons (A) and bread (B). Everyone has the same utility function U = AB. The price of an aerobics lesson is $2 and the price of a loaf of bread is $1. Two million of Toronto s citizens are rich and have an income of $100 each; one million are poor and have an income of $50 each. (a) Find the utility level of each rich person and each poor person in a private equilibrium with no government intervention. (b) Now suppose the government decides to provide aerobics lessons publicly. Assume that majority vote determines the number of aerobics lessons, which must be the same for every citizen. Further assume that the government must balance its budget what it collects in taxes must cover the total costs of providing aerobics lessons to all citizens. Find the utility level of each poor person and each rich person if the government levies a head tax that is, a tax that is the same number of dollars per person, whether rich or poor. (c) Find the utility level of each poor person and each rich person if everything is as in (b) except the government uses a proportional income tax. (a) The utility function for this question is in the Cobb-Douglas family, u(a, B) = A a B b and we know A(p A, p B, wealth) = B(p A, p B, wealth) = a wealth = 1 wealth a + b p A 2 2 b wealth = 1 wealth. a + b p B 2 1 The private-equilibrium row in the table below follows directly from these equations. 1

2 (b) For the government in this question revenue must equal expenditure. With a head tax this means (3 million)(head tax) = (3 million)p A A or, the head tax = 2A. Since the rich have a majority their budget constraint is the one that matters. B = Income head tax = 100 2A or 2A + B = 100. With this budget constraint the rich choose A = 25, with a head tax of 50, and B = 50. This means the poor have to live with A = 25 and no other income for bread after they pay the head tax B = utility = 0. These results are summarized in the head-tax row of the table. (c) With a proportional income tax, t, we have or t[(2 million)(income of rich) + (1 million)(income of poor)] = (3 million)2a, t = 6A 250 = 3A 125. So then the budget constraint of the rich is ( B = (1 t) income = A ) = 4 (125 3A) or Given this budget constrain the rich vote for 12A + 5B = 500. A = = 125 3A and t = = 1 2. From here the results in the last row of the table follow directly. Structure Poor Rich A B Utility A B Utility Private eq. 25/2 25 (1/2) (2)25 2 Head tax (2)25 2 Income tax 125/6 25 (5/6) /6 50 (5/3)25 2 2

3 2. Lunenburg, Nova Scotia is a fishing village. Each fisherman operates one boat. If x is the number of boats the average revenue in dollars per month to each fisherman is given by AR = a bx, a > 0, b > 0. The cost of operating a boat is c dollars per month (this includes the time cost for each fisherman); assume a > c. (a) What would x be in a perfectly-competitive equilibrium with no government intervention? (b) What is the socially-efficient level of x? (c) Now suppose the government levies a licence fee of f dollars per month per boat. What value of f would induce the socially-efficient allocation? (a) In a perfectly-competitive equilibrium with no government intervention profits will equal zero total revenue will equal total cost and thus average revenue equals c. a bx = c or x = a c. b (b) In a socially-efficient allocation x would be chosen to maximize producers surplus. The optimal level of x is (a c)/2b. Producers surplus = (a bx)x cx = (a c)x bx 2. (c) To get the fisherman to act in the socially optimal way we must set f so that average revenue at the level of x in (b) equals c + f. a b a c 2b = a + c 2 = c + f or f = a c Derive the Samuelson conditions for efficient provision of a pure public good. Assume one public good, one private good, and two people. Denote the total supply of the private good by X and the quantity supplied of the public good by G; let F (G, X) = 0 characterize the efficient production technologies. Denote the utility functions by u j (G, x j ), j = A, B. If G units of a public good are produced, G units of the good can be consumed by each person consumption of the good is said to be non-rival ; one person s consumption of the good does not reduce the amount available for anyone else. In this question X is a private good. To find Pareto-efficient allocations we can solve Max G, x B, X λ, µ The first-order conditions for (G, x B, X) are u B ( G, x B) + λ(u A ( G, X x B) u A 0 ) + µf (G, X) 3

4 Rewrite (1), (2) and (3) as G + λ ua G + µ F G = 0 (1) x B λ ua x A = 0 (2) λ ua x + µ F A X = 0. (3) G + λ ua G = µ F G x = F B λ ua = µ xa X Now divide each term in (4) by the corresponding term in (5) to obtain: (4) (5) MRS B GX + MRS A GX = MC of G MC of X. 4. Every weekday morning N commuters travel from North Guelph to South Guelph. They can either go around the town on the Hanlon or they can use Gordon Street which runs through town. The Hanlon is uncongested and takes a minutes; Gordon Street takes c + N/b minutes where b and c are constants and N is the number of vehicles using the Gordon-Street route. Assume a > c and N > 2b(a c). (i) What is total commuting time in the private equilibrium? (ii) What is total commuting time in a socially-efficient allocation? (iii) Assume all commuters are identical and let w be the value of one minute of a commuter s time. What toll on the Gordon-Street route would result in a Pareto-efficient equilibrium? (iv) If Guelph decides to widen Gordon Street and reduces commute time on Gordon Street to c + N/(2b) minutes would the new private equilibrium be Pareto efficient? Justify your answer. (i) IF both routes are used the time taken on each route will be the same in the private equilibrium. As above let N denote the number of people using the Gordon-Street route. a = c + N b or N = b(a c). Since N > 2b(a c) both routes are used and total commute time is an. (ii) In a socially-efficient allocation the social planner would choose N to minimize total commuting time. 4

5 ( Total time = c + N ) N + a(n N). b The N that minimizes total commuting time is b(a c)/2 and the time the Gordon-Street route will take would be c + 1 b(a c) = a + c b 2 2 Comparing the private and socially-efficient allocations we see that in the socially-efficient allocation the time saved per commuter on the Gordon-Street route would be a a + c = a c 2 2. So total time saved in the socially-efficient allocation would be a c b(a c) 2 2 = b(a c)2. 4 Therefore total time taken in the socially-efficient allocation must be an b(a c) 2 /4. (iii) The toll must make commuters indifferent between using the Hanlon (with no toll) and using Gordon Street, paying the toll but saving (a c)/2 minutes in commute time. Since w is the value of a minute saved the toll should be (a c)w/2. (iv) IF both routes are used, after the Gordon-Street widening, the time taken on each route will be the same in the new private equilibrium. a = c + N 2b or N = 2b(a c). Since N > 2b(a c) both routes are used and total commute time is still an. The new private equilibrium is not Pareto efficient and the widening of Gordon Street would be a waste of money. 5. (a) In the two-type car insurance model discussed in class, when the insurance companies cannot distinguish the good from the bad drivers, use a carefully-drawn and clearlylabeled w g -w b diagram, and a clear explanation, to prove that a pooling equilibrium cannot exist in this model. (b) Now suppose: initial wealth is w 0 = 100; the loss in the event of an accident is, L = 50; the probability that a good driver does not have an accident is π g = 2/3; the probability that a bad driver does not have an accident is π b = 1/3; there is an equal number of good and bad drivers; and the Bernoulli utility of wealth function is u(w) = ln(w). Does an equilibrium with car insurance exist? Defend your answer carefully. (a) See the diagram at the end of this pdf. Let the candidate for a pooling equilibrium be at point A on the market line (the m-line). Since the good driver indifference curve through 5

6 A must be steeper than the bad driver indifference curve through A there must be points, say like B, below the bad driver indifference curve and above the good driver indifference curve, southeast of A and below the good-driver zero-profit line. If contract B were offered it would attract only the good drivers and it would therefore make positive profits because it would attract only the good drivers (it is below the g-line). The firms holding the contract at A will now lose money because their pool of customers has disproportionately more bad drivers. Thus A cannot be sustained. (b) We have already proved there can never be a pooling equilibrium so either there is no equilibrium with car insurance or there is a separating equilibrium. In any separating equilibrium all the bad drivers will be on the b line where w b = w g. Call this point A. Label the point where the bad driver indifference curve through A cuts the g line point B. For a separating equilibrium to exist the m line cannot cut the good driver indifference curve through B. We have seen that the slope of any zero profit line in absolute value terms is π/(1 π) and here π m = 1 2 π g π m = = 1 2. Thus the absolute value of the slope of the m line is 1. The MRS for any good driver indifference curve at any point is π g 1 π g MU wg MU wb = 2 w b w g. In this question this expression is 1 at the endowment point (100, 50). This means that there is a good driver indifference curve tangent to the m line at the endowment point. The good driver indiference curve through B has to be higher which proves the m line has to be below the good driver indifference curve through B. This proves that a separating equilibrium exists with the parameters specified in this question. 6

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