978-0393123982 Chapter 37 Solution Manual Part 1

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Chapter 37 NAME

Public Goods

Introduction. In previous chapters we studied selfish consumers con-

suming private goods. A unit of private goods consumed by one person

cannot be simultaneously consumed by another. If you eat a ham sand-

wich, I cannot eat the same ham sandwich. (Of course we can both eat

ham sandwiches, but we must eat different ones.) Public goods are a dif-

ferent matter. They can be jointly consumed. You and I can both enjoy

looking at a beautiful garden or watching fireworks at the same time. The

conditions for efficient allocation of public goods are different from those

for private goods. With private goods, efficiency demands that if you and

I both consume ham sandwiches and bananas, then our marginal rates of

substitution must be equal. If our tastes differ, however, we may consume

different amounts of the two private goods.

If you and I live in the same town, then when the local fireworks

show is held, there will be the same amount of fireworks for each of us.

Efficiency does not require that my marginal rate of substitution between

fireworks and ham sandwiches equal yours. Instead, efficiency requires

that the sum of the amount that viewers are willing to pay for a marginal

increase in the amount of fireworks equal the marginal cost of fireworks.

This means that the sum of the absolute values of viewers’ marginal rates

of substitution between fireworks and private goods must equal the mar-

ginal cost of public goods in terms of private goods.

Example: A quiet midwestern town has 5,000 people, all of whom are in-

terested only in private consumption and in the quality of the city streets.

The utility function of person iis U(Xi,G)=Xi+AiG−BiG2,whereXi

is the amount of money that person ihas to spend on private goods and

Gis the amount of money that the town spends on fixing its streets. To

find the Pareto optimal amount of money for this town to spend on fixing

its streets, we must set the sum of the absolute values of marginal rates of

substitution between public and private goods equal to the relative prices

of public and private goods. In this example we measure both goods in

dollar values, so the price ratio is 1. The absolute value of person i’s

marginal rate of substitution between public goods and private goods is

the ratio of the marginal utility of public goods to the marginal utility of

private goods. The marginal utility of private goods is 1 and the marginal

utility of public goods for person iis Ai−BiG. Therefore the absolute

value of person i’s MRS is Ai−BiGand the sum of absolute values

of marginal rates of substitution is i(Ai−BiG)=iAi−(Bi)G.

Therefore Pareto efficiency requires that iAi−(iBi)G= 1. Solving

this for G,wehaveG=(

iAi−1)/iBi.

37.1 (0) Muskrat, Ontario, has 1,000 people. Citizens of Muskrat con-

sume only one private good, Labatt’s ale. There is one public good, the

town skating rink. Although they may differ in other respects, inhabitants

456 PUBLIC GOODS (Ch. 37)

have the same utility function. This function is U(Xi,G)=Xi−100/G,

where Xiis the number of bottles of Labatt’s consumed by citizen iand

Gis the size of the town skating rink, measured in square meters. The

price of Labatt’s ale is $1 per bottle and the price of the skating rink is

$10 per square meter. Everyone who lives in Muskrat has an income of

$1,000 per year.

(a) Write down an expression for the absolute value of the marginal rate

of substitution between skating rink and Labatt’s ale for a typical citizen.

100/G2What is the marginal cost of an extra square meter of skating

(b) Since there are 1,000 people in town, all with the same marginal

rate of substitution, you should now be able to write an equation that

states the condition that the sum of absolute values of marginal rates of

substitution equals marginal cost. Write this equation and solve it for the

the skating rink. Total expenditure by the town on its skating rink will

be $10G. Then the tax bill paid by an individual citizen to pay for the

skating rink is $10G/1,000 = $G/100. Every year the citizens of Muskrat

vote on how big the skating rink should be. Citizens realize that they will

have to pay their share of the cost of the skating rink. Knowing this, a

citizen realizes that if the size of the skating rink is G, then the amount

(d) Therefore we can write a voter’s budget constraint as Xi+G/100 =

1,000. In order to decide how big a skating rink to vote for, a voter simply

solves for the combination of Xiand Gthat maximizes his utility subject

to his budget constraint and votes for that amount of G.HowmuchGis

(e) If the town supplies a skating rink that is the size demanded by the

voters will it be larger than, smaller than, or the same size as the Pareto

(f) Suppose that the Ontario cultural commission decides to promote

Canadian culture by subsidizing local skating rinks. The provincial gov-

ernment will pay 50% of the cost of skating rinks in all towns. The costs

of this subsidy will be shared by all citizens of the province of Ontario.

There are hundreds of towns like Muskrat in Ontario. It is true that to

pay for this subsidy, taxes paid to the provincial government will have

to be increased. But there are hundreds of towns from which this tax

NAME 457

is collected, so that the effect of an increase in expenditures in Muskrat

on the taxes its citizens have to pay to the state can be safely neglected.

Now, approximately how large a skating rink would citizens of Muskrat

37.2 (0) Ten people have dinner together at an expensive restaurant

and agree that the total bill will be divided equally among them.

(a) What is the additional cost to any one of them of ordering an appetizer

37.3 (0) Cowﬂop, Wisconsin, has 1,000 people. Every year they have

a fireworks show on the Fourth of July. The citizens are interested in

only two things—drinking milk and watching fireworks. Fireworks cost 1

gallon of milk per unit. People in Cowﬂop are all pretty much the same.

In fact, they have identical utility functions. The utility function of each

citizen iis Ui(xi,g)=xi+√g/20, where xiis the number of gallons

of milk per year consumed by citizen iand gis the number of units of

fireworks exploded in the town’s Fourth of July extravaganza. (Private

use of fireworks is outlawed.)

(a) Solve for the absolute value of each citizen’s marginal rate of substi-

37.4 (0) Bob and Ray are two hungry economics majors who are sharing

an apartment for the year. In a ﬂea market they spot a 25-year-old sofa

that would look great in their living room.

Bob’s utility function is uB(S, MB)=(1+S)MB, and Ray’s utility

function is uR(S, MR)=(2+S)MR. In these expressions MBand MRare

the amounts of money that Bob and Ray have to spend on other goods,

S= 1 if they get the sofa, and S= 0 if they don’t get the sofa. Bob has

WBdollars to spend, and Ray has WRdollars.

458 PUBLIC GOODS (Ch. 37)

to spend on sofas and other stuff, they could buy the sofa and have a

Pareto improvement over not buying it so long as the cost of the sofa is

37.5 (0) Bonnie and Clyde are business partners. Whenever they work,

they have to work together. Their only source of income is profit from

their partnership. Their total profit per year is 50H,whereHis the

number of hours that they work per year. Since they must work together,

they both must work the same number of hours, so the variable “hours of

labor” is like a public “bad” for the two person community consisting of

Bonnie and Clyde. Bonnie’s utility function is UB(CB,H)=CB−.02H2

and Clyde’s utility function is UC(CC,H)=CC−.005H2,whereCBand

CCare the annual amounts of money spent on consumption for Bonnie

and for Clyde.

(a) If the number of hours that they both work is H, what is the ratio

of Bonnie’s marginal utility of hours of work to her marginal utility of

(b) If Bonnie and Clyde are both working Hhours, then the total amount

of money that would be needed to compensate them both for having to

work an extra hour is the sum of what is needed to compensate Bonnie

and the amount that is needed to compensate Clyde. This amount is

approximately equal to the sum of the absolute values of their marginal

rates of substitution between work and money. Write an expression for

this model is formally the same as a model with one public good Hand

one private good, income.)

NAME 459

37.6 (0) Lucy and Melvin share an apartment. They spend some of

their income on private goods like food and clothing that they consume

separately and some of their income on public goods like the refrigerator,

the household heating, and the rent, which they share. Lucy’s utility

function is 2XL+Gand Melvin’s utility function is XMG,whereXLand

XMare the amounts of money spent on private goods for Lucy and for

Melvin and where Gis the amount of money that they spend on public

goods. Lucy and Melvin have a total of $8,000 per year between them to

spend on private goods for each of them and on public goods.

(a) What is the absolute value of Lucy’s marginal rate of substitution

(b) Write an equation that expresses the condition for provision of the

for themselves and they spend the remaining $4,000 on public goods. Is

(d) Give an example of another Pareto optimal outcome in which Melvin

gets more than $2,000 and Lucy gets less than $2,000 worth of private

(e) Give an example of another Pareto optimum in which Lucy gets

(f) Describe the set of Pareto optimal allocations. The allocations

(g) The Pareto optima that treat Lucy better and Melvin worse will have

(more of, less of, the same amount of) public good as the Pareto optimum

460 PUBLIC GOODS (Ch. 37)

37.7 (0) This problem is set in a fanciful location, but it deals with a

very practical issue that concerns residents of this earth. The question

is, “In a Democracy, when can we expect that a majority of citizens will

favor having the government supply pure private goods publicly?” This

problem also deals with the efficiency issues raised by public provision

of private goods. We leave it to you to see whether you can think of

important examples of publicly supplied private goods in modern Western

economies.

On the planet Jumpo there are two goods, aerobics lessons and

bread. The citizens all have Cobb-Douglas utility functions of the form

Ui(Ai,B

i)=A1/2

iB1/2

i,whereAiand Biare i’s consumptions of aerobics

lessons and bread. Although tastes are all the same, there are two differ-

ent income groups, the rich and the poor. Each rich creature on Jumpo

has an income of 100 fondas and every poor creature has an income of

50 fondas (the currency unit on Jumpo). There are two million poor

creatures and one million rich creatures on Jumpo. Bread is sold in the

usual way, but aerobics lessons are provided by the state despite the fact

that they are private goods. The state gives the same amount of aerobics

lessons to every creature on Jumpo. The price of bread is 1 fonda per

loaf. The cost to the state of aerobics lessons is 2 fondas per lesson. This

cost of the state-provided lessons is paid for by taxes collected from the

citizens of Jumpo. The government has no other expenses than providing

aerobics lessons and collects no more or less taxes than the amount needed

to pay for them. Jumpo is a democracy, and the amount of aerobics to

be supplied will be determined by majority vote.

(a) Suppose that the cost of the aerobics lessons provided by the state

is paid for by making every creature on Jumpo pay an equal amount of

taxes. On planets, such as Jumpo, where every creature has exactly one

head, such a tax is known as a “head tax.” If every citizen of Jumpo gets

20 lessons, how much will be total government expenditures on lessons?

(b) More generally, when everybody pays the same amount of taxes, if x

lessons are provided by the government to each creature, the total cost

NAME 461

body getting the same amount and nobody able to get more lessons from

another source, each creature faces a choice problem that is formally the

same as that faced by a consumer, i, who is trying to maximize a Cobb-

Douglas utility function subject to the budget constraint 2A+B=I,

where Iis its income. Explain why this is the case. If Alessons

(d) Suppose that the aerobics lessons are paid for by a head tax and all

lessons are provided by the government in equal amounts to everyone.

How many lessons would the rich people prefer to have supplied? 25.

How many would the poor people prefer to have supplied? 12.5.

(Hint: In each case you just have to solve for the Cobb-Douglas demand

with an appropriate budget.)

(e) If the outcome is determined by majority rule, how many aerobics

(f) Suppose that aerobics lessons are “privatized,” so that no lessons are

supplied publicly and no taxes are collected. Every creature is allowed to

buy as many lessons as it likes and as much bread as it likes. Suppose

that the price of bread stays at 1 fonda per unit and the price of lessons

stays at 2 fondas per unit. How many aerobics lessons will the rich get?

(g) Suppose that aerobics lessons remain publicly supplied but are paid

for by a proportional income tax. The tax rate is set so that tax rev-

enue pays for the lessons. If Aaerobics lessons are offered to each

creature on Jumpo, the tax bill for a rich person will be 3Afondas

and the tax bill for a poor person will be 1.5Afondas. If Alessons

are given to each creature, show that total tax revenue collected will