MGEC19 HW4 solns PDF

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Warning: TT: undefined function: 22 Managerial Economics Indian School of BusinessHOMEWORK 1 SOLUTIONSProblem 1Madhu is a risk-averse decision maker whose utility function is given by U I)(  I, where Idenotes Madhu’s monetary payoff from an investment. Madhu is considering an investment in machine ...

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Managerial Economics Indian School of Business HOMEWORK 1 SOLUTIONS Problem 1 Madhu is a risk-averse decision maker whose utility function is given by U (I )  I , where I denotes Madhu’s monetary payoff from an investment. Madhu is considering an investment in machine tools factory with a payoff of Rs. 10,00,000 with probability 0.6, and Rs. 250,000 with probability 0.4. If the cost of the investment is Rs. 6,00,000, should Madhu invest in this factory? Expected utility (investment) =

0.6 * 1000000  0.4 * 250000 600000 0.6 *1000 0.4 * 500 775 25 which is positive. Therefore, invest. [10 points] Problem 2 Consider two farmers and two pieces of land.1 We want to know what the effect is of each farmer working his own piece of land versus the farmers cooperating and jointly working on the two pieces of land. To simplify the exposition, we will denote these as ‘independent farmers’ and ‘cooperative farmers’ respectively. Assume that each farmer (whether independent or cooperative) can decide on his own how much time to spend on farming. Let farmer i’s weekly time spent on farming be denoted h i (in hours per week; so h1 for farmer 1 and h2 for farmer 2). The farmer’s productivity (expressed in bushels of grain) is directly proportional to the time he spends on farming. In particular, an independent farmer produces (80*hi) bushels of grain if he works hi hours per week. Farmers in a cooperative are more productive since they can specialize: a cooperative farmer produces (90* hi) bushels of grain if he works hi hours per week. Collective farmers share the output of their farm equally. Let bi be the bushels of grain that farmer i can take home at the end of the year, then bi = 80hi for an independent farmer, while bi = (90h1 + 90h2)/2 for a cooperative farmer. Farmers dislike working and more so as they work more. In particular, farmer i’s utility is Ui = bi – (hi2/2) We assume that a farmer i will choose hi to maximize his utility Ui. (a) Write out the utility functions of an independent farmer and a cooperative farmer completely in terms of h1 and h2. 1

This question is from Berndt, Chapman, Doyle and Stoker’s course at MIT Sloan

The utility of an independent farmer is Ui = 80hi – (hi2/2)

[2 points]

while that of a collective farmer is Ui = 90(h1+ h2)/2 – (hi2/2)

[2 points]

(b) How many hours will an independent farmer work (assuming that farmers choose hi to maximize their utility)? What is his utility? Taking the derivative of the independent farmer’s utility with respect to hours of work and setting it equal to zero yields: 80 – hi = 0 => hi = 80 [1 point] The utility of an independent farmer is 3200.

[1 point]

(c) How many hours will a collective farmer work? What is his utility? Consider farmer 1, with utility: u1 = 45(h1+h2)-h12/2 Taking the derivative of his utility with respect to his hours of work and setting it equal to zero yields h1 = 45. [1 point] u1 = 45(45+45)-2025/2 = 3037.5 [1 point] By symmetry, the same result holds for farmer 2. (d) What is the problem with a collective farm? What would happen (qualitatively) if 100 farmers worked jointly in a collective farm? How could the farmers solve that problem? The problem of the collective farm is that the output is common property, so farmers get only part of the output they produce. The incentives to work change from MR=MC to a fraction of MR = MC. Compare, in particular, independent and collective farmers who consider the gain from putting in one more hour after they have already worked 45 hours. The cost to each of working the 46th hour is 462/2 – 452/2 = 45.5 The gain for an independent worker is 80, which outweighs the cost of 45.5. For a cooperative worker, the gain is only 45 (since half of his output goes to the other farmer), so working the extra hour is not worth it. Note that farmer 1 putting in more hours does not change how many hours farmer 2 works. This problem is magnified with more farmers, as the share of the marginal bundle kept by the farmer (1/no. of farmers) decreases with the number of farmers.

Of the number of hours that someone works is observable, the farmers could write a contract that exploits the gains from specialization and maximizes utility. (The optimal number of hours for each farmer is 90 per week in the cooperative and farmers would get utility of 4050.) If effort is not observable, we would have to explore incentive schemes or screening mechanisms to align incentives. [1 points]

Problem 3 An annuity provides insurance against out-living one’s financial resources. LEICO, a life insurance company, takes a deposit from customers at age 60 years, and returns an annual payment of Rs. 5000 till their death. (a) Calculate the break-even deposit for LEICO if average population-wide life expectancy is 80 years. Assume a 5% interest rate. Rs. 62311 is the deposit. [3 points] (b) If potential customers have a sense of their life expectancy, based on factors such as the longevity of their parents, who will purchase the annuity with the deposit you have calculated above? Customers who expect to live for a long time, in particular, for those whose life expectancy is greater than 80 years. [3 points] (c) If life expectancy is uniformly distributed in the population (up to a maximum of 100 years), what is the deposit that that LEICO will ultimately end up charging? Who will finally buy this annuity? Only those who live for 100 years will end up buying the annuity. [2 points] LEICO will charge then 85795 which is the break even value for those customers. [2 points] Problem 4 The business district of Bombay2, Hariman’s Joint, sits on an island.3 Most of the people who work in this district commute from the mainland. Specifically, 400,000 people make this commute. Bombayites are in love with their cars, so each of the 400,000 people drives to and from work in a private car; there is no carpooling. There are two routes from the mainland into (and out of) the business district, the Borli Bridge (B)4 and the Tycoon’s Tunnel (T). The times it takes to commute across the bridge and through 2 3 4

Any resemblance to an actual city is purely coincidental. This problem is adapted from Kreps’ Microeconomics for Managers. The local press grandly calls this a “sealink”.

the tunnel depend on the number of individuals nB and nT who take the bridge and the tunnel, respectively. Specifically, if nB people come via the bridge, the commute time via the bridge is 30+ nB/20000 minutes, and if nT people come via the tunnel, the commute time via the tunnel is 40+ nT/5000 minutes. (a) Suppose each of the 400,000 people who make this commute takes either the bridge or the tunnel; that is, nB+nT =400,000. People choose whether to take the bridge or the tunnel depending on which takes less time, so in equilibrium, the numbers nB and nT are chosen so that the two commute times are equal. What are nB and nT? Equate commute times, 40 + nT/5000 = 30 + nB/20000 We also know that nB + nT = 400,000 Solve as nT = 40,000 [2 points] and nB =360,000[2 points] (b) We define the total commute time as nB times the commute time via the bridge plus nT times the commute time via the tunnel. In your answer to part a, what is the total commute time? This gives a commute time of 48 minutes for both routes, and hence total commute time of 19.2 million minutes. [2 points] (c) Suppose Bombay’s mayor could control the number of people who come via the bridge and via the tunnel. She chooses these numbers to minimize the total commute time. How should she allocate the 400,000 commuters between the bridge and the tunnel to minimize total commute time? We want to minimize (30 + nB/20000)nB + (40+ nT/5000)nT subject to nB + nT = 400000. One way to solve is to substitute 400000-nB for nT, take the derivative and set it equal to zero. We get nB = 60,000 and nT = 340,000 The optimal commute time is 47 minutes by bridge and 52 minutes by tunnel. [2 points] (d) Except for the congestion on the bridge and tunnel, there is a 0 marginal cost of getting commuters across the bridge and the tunnel. For this reason, transit across the bridge and through the tunnel has been kept free. But the mayor is considering whether to impose a toll

on one or the other. If a toll of tB is imposed on the bridge and tT on the tunnel, consumers will rearrange their commute so that (10tB+commute time through the bridge in minutes) equals (10tT +commute time through the tunnel in minutes). In other words, 10 minutes of commute time is worth Rs. 1 to commuters. Find values for tB and tT, where one is 0, so that, facing these tolls, commuters arrange their commute in the manner that minimizes total commute time. You can read the answer right off of the last line of part c. To get commuters to divide 60000 via tunnel and 340000 via the bridge, the toll has to be set to overcome the 5 minute difference in commute times, in favor of the bridge. So set a toll of Rs. 0.50 for the bridge. [2 points] Problem 5 Awbrey Butte is an exclusive neighborhood of big, modern houses surrounded by native pines in the Oregon mountains (see attached article). Resident Susan Taylor likes to use a clothesline to dry her laundry, and thus saves $80 on average in annual energy costs (apart from the obvious environmental benefits). Neighborhood manager Carol Haworth is concerned that seeing laundry outside might give potential home-buyers the idea that residents are too poor to afford dryers, and that will drive down property values. Suppose the decline in property values is proportional to the number of weeks a year Susan dries her laundry outside. Number of weeks clothesline is used

Total benefits to Susan Taylor

Total annualized loss to Carol Hayworth

0

0

0

10

$16

$10

20

$32

$25

30

$48

$45

40

$64

$70

50

$80

$100

(a) In the absence of bargaining, how many weeks would Susan dry her laundry if she has a right to do so? 50 weeks [2 points] (b) In the absence of bargaining, how many weeks would Susan dry her laundry if Carol has a right to protect her property value?

0 weeks [2 points] (c) What is the socially optimal number of weeks (from the six points in the table above) the clothesline should be used? 20 weeks [2 points] (d) With bargaining, how many weeks would Susan dry her laundry if she has a right to do so? What is the minimum Carol will pay Susan? 20 weeks. [1 point] Carol will pay Susan $48. Susan’s payoff is preserved ($80). Carol will be better off because her cost is $25 + $48 = $73 $0. [1 point]...