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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
(c) Suppose that everyone in town pays an equal share of the cost of
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)
(c) If Bob has a total of WB= $100 and Ray has a total of WR= $75
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
(c) Write an equation that can be solved for the Pareto optimal number
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
(c) Suppose that Melvin and Lucy each spend $2,000 on private goods
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
(c) Since aerobics lessons are going to be publicly provided with every-
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
