PUBLIC GOODS
Snyder and Nicholson, Copyright ©2008 by Thomson South-Western. All rights reserved.
Public Goods
• Public goods are nonrival
– the use of the good does not prevent
others from using it e.g. knowledge
• Pure Public goods are nonexclusive
– once they are produced, they provide
benefits to an entire group
– it is impossible to restrict these benefits to
the specific groups of individuals who pay
for them
Attributes of Public Goods
• A good is nonrival if consumption of
additional units of the good involves
zero social marginal costs of production
Attributes of Public Goods
• A good is exclusive if it is relatively easy
to exclude individuals from benefiting
from the good once it is produced
• A good is nonexclusive if it is
impossible, or very costly, to exclude
individuals from benefiting from the
good
Attributes of Public Goods
• Some examples of these types of goods
include:
Exclusive
Yes
Rival
No
Yes
Hot dogs,
cars,
houses
Bridges,
swimming
pools
No
Fishing
grounds,
clean air
National
defense,
mosquito
control
Public Good
• A good is a pure public good if, once
produced, no one can be excluded from
benefiting from its availability and if the
good is nonrival -- the marginal cost of
an additional consumer is zero
Public Goods and
Resource Allocation
• We will use a simple general equilibrium
model with two individuals (A and B)
• There are only two goods
– good y is an ordinary private good
• each person begins with an allocation
(yA and yB)
– good x is a public good that is produced
using y
x = f(ysA + ysB)
Public Goods and
Resource Allocation
• Resulting utilities for these individuals are
UA[x,(yA - ysA)]
UB[x,(yB - ysB)]
• The level of x enters identically into each
person’s utility curve
– it is nonexclusive and nonrival
• each person’s consumption is unrelated to what
he contributes to production
• each consumes the total amount produced
Public Goods and
Resource Allocation
• The necessary conditions for efficient
resource allocation consist of choosing
the levels of ysA and ysB that maximize
one person’s (A’s) utility for any given
level of the other’s (B’s) utility
• The Lagrangian expression is
L = UA(x, yA - ysA) + [UB(x, yB - ysB) - K]
Public Goods and
Resource Allocation
• The first-order conditions for a maximum
are
L/ysA = U1Af’ - U2A + U1Bf’ = 0
L/ysB = U1Af’ - U2B + U1Bf’ = 0
• Comparing the two equations, we find
U2B = U2A
Public Goods and
Resource Allocation
• We can now derive the optimality
condition for the production of x
• From the initial first-order condition we
know that
U1A/U2A + U1B/U2B = 1/f’
MRSA + MRSB = 1/f’
• The MRS must reflect all consumers
because all will get the same benefits
Failure of a
Competitive Market
• Production of x and y in competitive
markets will fail to achieve this allocation
– with perfectly competitive prices px and py,
each individual will equate his MRS to px/py
– the producer will also set 1/f’ equal to px/py
to maximize profits
– the price ratio px/py will be too low
• it would provide too little incentive to produce x
Failure of a
Competitive Market
• For public goods, the value of producing
one more unit is the sum of each
consumer’s valuation of that output
– individual demand curves should be added
vertically rather than horizontally
• Thus, the usual market demand curve
will not reflect the full marginal valuation
Inefficiency of a
Nash Equilibrium
• Suppose that individual A is thinking
about contributing ysA of his initial yA
endowment to the production of x
• The utility maximization problem for A is
then
choose ysA to maximize UA[f(ysA + ysB),yA - ysA]
Inefficiency of a
Nash Equilibrium
• The first-order condition for a maximum
is
U1Af’ - U2A = 0
U1A/U2A = MRSA = 1/f’
• Because a similar argument can be
applied to B, the efficiency condition will
fail to be achieved
– each person considers only his own benefit
The Roommates’ Dilemma
• Suppose two roommates with identical
preferences derive utility from the number
of paintings hung on their walls (x) and the
number of chocolate bars they eat (y) with
a utility function of
Ui(x,yi) = x1/3yi2/3
(for i=1,2)
• Assume each roommate has $300 to
spend and that px = $100 and py = $0.20
The Roommates’ Dilemma
• We know from our earlier analysis of
Cobb-Douglas utility functions that if each
individual lived alone, he would spend 1/3
of his income on paintings (x = 1) and 2/3
on chocolate bars (y = 1,000)
• When the roommates live together, each
must consider what the other will do
– if each assumed the other would buy
paintings, x = 0 and utility = 0
The Roommates’ Dilemma
• If person 1 believes that person 2 will
not buy any paintings, he could choose
to purchase one and receive utility of
U1(x,y1) = 11/3(1,000)2/3 = 100
while person 2’s utility will be
U2(x,y2) = 11/3(1,500)2/3 = 131
• Person 2 has gained from his free-riding
position
The Roommates’ Dilemma
• We can show that this solution is
inefficient by calculating each person’s
MRS
U i / x
yi
MRSi 

U i / y i 2 x
• At the allocations described,
MRS1 = 1,000/2 = 500
MRS2 = 1,500/2 = 750
The Roommates’ Dilemma
• Since MRS1 + MRS2 = 1,250, the
roommates would be willing to sacrifice
1,250 chocolate bars to have one
additional painting
– an additional painting would only cost them
500 chocolate bars
– too few paintings are bought
The Roommates’ Dilemma
• To calculate the efficient level of x, we
must set the sum of each person’s MRS
equal to the price ratio
y1 y 2 y1  y 2 px 100
MRS1  MRS2 




2x 2x
2x
py 0.20
• This means that
y1 + y2 = 1,000x
The Roommates’ Dilemma
• Substituting into the sum of budget
constraints, we get
0.20(y1 + y2) + 100x = 600
x=2
y1 + y2 = 2,000
• The allocation of the cost of the
paintings depends on how each
roommate plays the strategic financing
game
The Roommates’ Dilemma
• If each person buy 1 painting
Person 1’s utility is
U1(x,y1) = 21/3(1,000)2/3  126
Person 2’s utility is
U2(x,y2) = 21/3(1,000)2/3
 126
The Roommates’ Dilemma
Person 2
Buy
Buy
Not Buy
126,126 100,131
Person 1
Not Buy
131,100
0,0
Lindahl Pricing of
Public Goods
• Swedish economist E. Lindahl
suggested that individuals might be
willing to be taxed for public goods if they
knew that others were being taxed
– Lindahl assumed that each individual would
be presented by the government with the
proportion of a public good’s cost he was
expected to pay and then reply with the
level of public good he would prefer
Lindahl Pricing of
Public Goods
• Suppose that individual A would be
quoted a specific percentage (A) and
asked the level of a public good (x) he
would want given the knowledge that this
fraction of total cost would have to be
paid
• The person would choose the level of x
which maximizes
utility = UA[x,yA*- Af -1(x)]
Lindahl Pricing of
Public Goods
• The first-order condition is given by
U1A - AU2B(1/f’)=0
MRSA = A/f’
• Faced by the same choice, individual B
would opt for the level of x which satisfies
MRSB = B/f’
Lindahl Pricing of
Public Goods
• An equilibrium would occur when
A+B = 1
– the level of public goods expenditure
favored by the two individuals precisely
generates enough tax contributions to pay
for it
MRSA + MRSB = (A + B)/f’ = 1/f’
Shortcomings of the
Lindahl Solution
• The incentive to be a free rider is very
strong
– this makes it difficult to envision how the
information necessary to compute
equilibrium Lindahl shares might be
computed
• individuals have a clear incentive to understate
their true preferences
Important Points to Note:
• Externalities may cause a
misallocation of resources because of
a divergence between private and
social marginal cost
– traditional solutions to this divergence
includes mergers among the affected
parties and adoption of suitable
Pigouvian taxes or subsidies
Voting
• Voting is used as a social decision
process in many institutions
– direct voting is used in many cases from
statewide referenda to smaller groups and
clubs
– in other cases, societies have found it
more convenient to use a representative
form of government
Majority Rule
• Throughout our discussion of voting, we
will assume that decisions will be made
by majority rule
The Paradox of Voting
• In the 1780s, social theorist M. de
Condorcet noted that majority rule
voting systems may not arrive at an
equilibrium
– instead, they may cycle among alternative
options
The Paradox of Voting
• Suppose there are three voters (Smith,
Jones, and Fudd) choosing among
three policy options
– we can assume that these policy options
represent three levels of spending on a
particular public good [(A) low, (B) medium,
and (C) high]
– Condorcet’s paradox would arise even
without this ordering
The Paradox of Voting
• Preferences among the three policy
options for the three voters are:
Smith
Jones
Fudd
A
B
C
B
C
A
C
A
B
The Paradox of Voting
• Consider a vote between A and B
– A would win
• In a vote between A and C
– C would win
• In a vote between B and C
– B would win
• No equilibrium will ever be reached
Single-Peaked Preferences
• Equilibrium voting outcomes always
occur in cases where the issue being
voted upon is one-dimensional and
where voter preferences are “singlepeaked”
Single-Peaked Preferences
We can show each voters preferences in
terms of utility levels
Utility




A

 Fudd
 Jones


B
C
For Smith and Jones,
preferences are singlepeaked
Fudd’s preferences have
two local maxima
Smith
Quantity of
public good
Single-Peaked Preferences
If Fudd had alternative preferences with a
single peak, there would be no paradox
Utility





A
 Fudd
 Jones
Option B will be chosen
because it will defeat
both A and C by votes 2
to 1


B
C
Smith
Quantity of
public good
The Median Voter Theorem
• With the altered preferences of Fudd, B
will be chosen because it is the
preferred choice of the median voter
(Jones)
– Jones’s preferences are between the
preferences of Smith and the revised
preferences of Fudd
The Median Voter Theorem
• If choices are unidimensional and
preferences are single-peaked, majority
rule will result in the selection of the
project that is most favored by the
median voter
– that voter’s preferences will determine
what public choices are made
A Simple Political Model
• Suppose a community is characterized
by a large number of voters (n) each
with income of yi
• The utility of each voter depends on his
consumption of a private good (ci) and
of a public good (g) according to
utility of person i = Ui = ci + f(g)
where fg > 0 and fgg < 0
A Simple Political Model
• Each voter must pay taxes to finance g
• Taxes are proportional to income and
are imposed at a rate of t
• Each person’s budget constraint is
ci = (1-t)yi
• The government also faces a budget
constraint
n
g   ty i  tny A
i 1
A Simple Political Model
• Given these constraints, the utility
function of individual i is
Ui(g) = [yA - (g/n)]yi /yA + f(g)
• Utility maximization occurs when
dUi /dg = -yi /(nyA) + fg(g) = 0
g = fg-1[yi /(nyA)]
• Desired spending on g is inversely
related to income
A Simple Political Model
• If G is determined through majority rule,
its level will be that level favored by the
median voter
– since voters’ preferences are determined
solely by income, g will be set at the level
preferred by the voter with the median level
of income (ym)
g* = fg-1[ym/(nyA)] = fg-1[(1/n)(ym/yA)]
A Simple Political Model
• Under a utilitarian social welfare
criterion, g would be chosen so as to
maximize the sum of utilities:
 A g  y i

SW  U i    y   A  f g   ny A  g  nf g 
n y
1


n
• The optimal choice for g then is
g* = fg-1(1/n) = fg-1[(1/n)(yA/yA)]
– the level of g favored by the voter with
average income
Voting for Redistributive Taxation
• Suppose voters are considering a lumpsum transfer to be paid to every person
and financed through proportional
taxation
• If we denote the per-person transfer b,
each individual’s utility is now given by
Ui = ci + b
Voting for Redistributive Taxation
• The government’s budget constraint is
nb = tnyA
b = tyA
• For a voter with yi > yA, utility is
maximized by choosing b = 0
• Any voter with yi < yA will choose t = 1
and b = yA
– would fully equalize incomes
Voting for Redistributive Taxation
• Note that a 100 percent tax rate would
lower average income
• Assume that each individual’s income
has two components, one responsive to
tax rates [yi (t)] and one not responsive
(ni)
– also assume that the average of ni is zero,
but its distribution is skewed right so nm < 0
Voting for Redistributive Taxation
• Now, utility is given by
Ui = (1-t)[yi (t) + ni] + b
• The individual’s first-order condition for a
maximum in his choice of t and g is now
dUi /dt = -ni + t(dyA/dt) = 0
ti = ni /(dyA/dt)
• Under majority rule, the equilibrium
condition will be
t* = nm /(dyA/dt)
The Groves Mechanism
• Suppose there are n individuals in a
group
– each has a private and unobservable
valuation (ui) for a proposed taxationexpenditure project
The Groves Mechanism
• The government states that, if
undertaken, the project will provide each
person with a transfer given by
t i  v i
i
• If the project is not undertaken, no
transfers are made
The Groves Mechanism
• The problem for voter i is to choose his
or her reported net valuation so as to
maximize utility
utility  u i  t i  u i  v i
i
The Groves Mechanism
• Each person will wish the project to be
undertaken if it raises utility
• This means
u i  v i  0
i
• A utility-maximizing strategy is to set vi =
ui
The Clarke Mechanism
• This mechanism also envisions asking
individuals about their net valuation of a
public project
– focuses on “pivotal voters”
• those whose valuations can change the overall
evaluation from positive to negative of vice versa
The Clarke Mechanism
• For these pivotal voters, the Clarke
mechanism incorporates a Pigouvian-like
tax (or transfer) to encourage truth telling
– for all other voters, there are no special
transfers
Important Points to Note:
• Public goods provide benefits to
individuals on a nonexclusive basis no one can be prevented from
consuming such goods
– such goods are usually nonrival in that
the marginal cost of serving another
user is zero
Important Points to Note:
• Private markets will tend to
underallocate resources to public
goods because no single buyer can
appropriate all of the benefits that
such goods provide
Important Points to Note:
• A Lindahl optimal tax-sharing scheme
can result in an efficient allocation of
resources to the production of public
goods
– computing these tax shares requires
substantial information that individuals
have incentives to hide
Important Points to Note:
• Majority rule voting may not lead to
an efficient allocation of resources to
public goods
– the median voter theorem provides a
useful way of modeling the outcomes
from majority rule in certain situations
Important Points to Note:
• Several truth-revealing voting
mechanisms have been developed
– whether these are robust to the special
assumptions made or capable of
practical application remain unresolved
questions