Managerial Economics

Indian School of Business

Term 1, 2015-16

HOMEWORK 1

Due: Friday, May 22 at 5pm

Problem 1

Arnab is a risk-averse decision maker whose utility function is given by IIU )( , where I

denotes Arnabs monetary payoff from an investment. Arnab 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 Arnab invest in this factory?

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 (we denote these independent farmers) versus the

farmers cooperating and jointly working on the two pieces of land (collective farmers).

Assume that each farmer (whether independent or collective) can decide on his own how much

time to spend on farming. Let farmer is weekly time spent on farming be denoted hi (in hours

per week; so h1 for farmer 1 and h2 for farmer 2). The farmers 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

collective are more productive since they can specialize: a collective 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 is 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 Stokers course at MIT Sloan

(b) How many hours will an independent farmer work (assuming that farmers choose hi to

maximize their utility)? What is his utility?

(c) How many hours will a cooperative farmer work? What is his utility?

(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?

Problem 3

An annuity provides insurance against out-living ones 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.

(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?

(c) If life expectancy is uniformly distributed in the population (up to a maximum of 100 years)

and potential customers have a sense of their life expectancy, what is the deposit that LEICO

will ultimately end up charging? Who will finally buy this annuity?

Problem 4

The business district of Bombay2, Harimans 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 Tycoons Tunnel (T). The times it takes to commute across the bridge and through

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.

2 Any resemblance to the actual city of Mumbai is purely coincidental. 3 This problem is adapted from Kreps Microeconomics for Managers. 4 The local press grandly calls this a sealink.

(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?

(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?

(c) Suppose Bombays 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?

(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.

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?

(b) In the absence of bargaining, how many weeks would Susan dry her laundry if Carol has a

right to protect her property value?

(c) What is the socially optimal number of weeks the clothesline should be used?

(d) With bargaining, how many weeks would Susan dry her laundry if she has a right to do so?

How much would Carol pay Susan?

(e) With bargaining, how many weeks would Susan dry her laundry if Carol has a right to

protect her property value? How much would Susan pay Carol?