Public Goods
What is a Public Good
• A good with shared benefits
• The “public-ness” of a good has to do with
its’ innate features and the legal
framework
• Not whether it is publicly or privately
financed
• Not because supply is determined by
public policy
Public Goods
• A public good is a good that is nonexcludable and non-rival in consumption
• Whereas only one person at a time
benefits from a private good, multiple
people benefit from a public good at the
same time.
Taxonomy of Goods
Degree of Rivalry (congestibility)
Excludable
Non-Excludable
Degree of Excludability
Rival
Private Goods
Markets very
efficient
Commons Goods
Tragedy of the
commons
Non-Rival
Club goods
Markets relatively
efficient
Public Goods
Free-rider
problem
Congestibility
• Public goods can be pure public goods or
congestible.
Pure public good
Congestible public good
MB
MB
MBn
MBn
n
n
Market provision of Public Goods
• Markets often struggle when it comes to
providing public goods
• Free-riding means somebody is attempting to
benefit from a public good without paying for it.
• Markets can and do provide some public goods
• non-rivalry is not cause of difficulty in providing
public goods, but rather non-excludability is
• If everyone attempts to free ride, the public good
is not provided
Public and Private Goods
• For private goods, to aggregate demand
curves (willingness to pay) we sum the
individual demand curves horizontally
• For public goods, we sum the individual
demand curves vertically
• The difference comes from rivalry—while
only one person can consume a private
good, any number of people can consume
a public good
P or
MB
P2+P1
Demand
Supply
If this good is a private
good, we sum the demand
curves horizontally to derive
the market demand curve
Market Demand
P2
P1
If this good is a public
good, we sum the
demand curves vertically
to derive the market
demand curve
In principle, the supply
curve for a public good
is the same as for any
other good
D2
D1
Q
P or
MB
Supply
The efficient quantity of the
public good to be provided is
where demand=supply, or
where the sum of the
marginal benefits equals the
marginal cost
P2+P1
n
 MB
i 1
P2
Demand
P1
D2
D1
Q*
i
 MC
While this works for private
goods, you run into problems
trying to create markets for
public goods—rather than
telling people they can buy as
much as they want at the
market price, you are asking
them to pay how much Q* is
worth to them!
Q
Deception!
• If the police were funded entirely by asking
citizens to pay them what the citizens
thought the police were worth to them,
what would happen?
• People would misrepresent their personal
benefits
S
$
$alot
D
alot
Imagine this is societies
supply and demand for some
public good, and that your
contributions make up a very
small part of total
contributions. If you underrepresent your demand, what
happens to the demand
curve?
Did you see it move? It did, I
swear!
What happens when everybody
acts this way?
Q
Free-Riding
• Nearly all individuals have incentive to free
ride on the provision of a public good
• The amount of the public good actually
provided will probably be somewhere near
the amount demanded by the person with
the highest willingness to pay
• Especially in large groups, free riding will
lead to massive under provision of the
public good
Public goods as Prisoners’
Dilemma
• 2 Players
• Strategies are to contribute or not to
contribute to the public good
• Public good works like this. Players can
choose to invest 10 dollars in public good.
If they do so, each player gets a benefit of
$7.50 each
Public goods as Prisoners’
Dilemma
Player 2
Contribute
Not Contribute
15
Contribute
17.50
Pareto Superior
15
7.50
Player 1
7.50
10
Nash
Equilibrium
Not Contribute
17.50
10
Public goods as Prisoners’
Dilemma
• Role of government is to move from “state
of nature” (that is, the Nash equilibrium) to
the Pareto preferred outcome
• Compulsory taxation and public financing
of the project
Supply and types of public goods
• So far, we have assumed (at least
implicitly) that the supply of a public good
is additive in nature
• Quantity supplied=Σ of contributions
• If this is the case, free-riding is rational but
inefficient
Other types of public goods
• Weakest Link public goods—a good such that
the level of the public good that prevails is the
amount purchased by the individual who
purchased the least.
• Volunteer public goods—a good such that, once
a single person has provided it, it is provided for
everybody (a.k.a. best shot public goods).
• Volunteer is the opposite of a weakest link public
good.
Volunteer public goods
• Volunteer public goods will tend to be
provided, despite the fact that free riding in
this case is rational
• Examples:
• Security
• Stopping to help a broken down motorist
• Free riding is rational and efficient
Volunteer public goods
• The amount of the public good received by
any one person is identical to the quantity
received by anyone else
• If zi is the contribution made by the ith
member of society, and Gi is the amount of
the public good consumed by that member
• G=max{z1,z2,…zn}=G1=G2=…=Gn
Why is free riding efficient?
• Consider 4 individuals who incur costs to
provide this public good:
• z1=$12
• z2=$18
• z3=$7
• z4=$13
• Because G=max{z1,z2,…zn}=G1=G2=…=Gn, and
max{z1,z2,…zn}=$18, everybody gets $18 worth
of the public good and the other $32 spent is
wasted!
Who will volunteer?
• If individuals are heterogeneous, you will
likely see the least-cost individual or
greatest-benefit individual volunteer.
• Example: Say a woman is being mugged
on a busy street—who will help her?
• Likely the person with the most to gain, or
person with lowest cost
• Otherwise, you might wind up in a game of
“chicken”
Volunteer goods and the game of
chicken
• 2 Players
• Each can decide to provide the volunteer
good or not
• The volunteer good is worth $20 to each
individual, and it costs $5 to provide it.
• If the volunteer good is not provided, each
gets a payoff of $5
Volunteer goods and the game of
chicken
Player 2
Provide
Free Ride
15
20
Provide
15
15
Player 1
15
5
Free Ride
20
5
Multiple Nash Equilibria!
Multiple equilibria?
• Both are Nash equilibria because, once you get
there, neither individual has an incentive to
change what he or she is doing.
• This type of game is also referred to as a
chicken game or a hawk-dove game.
• In a game such as this one, the players do not
have dominant pure strategies, and it is best for
them to play a mixed strategy, which involves
randomly choosing among the pure strategies.
Mixed Strategies
• In a mixed strategy equilibrium, each player
randomizes between each possible strategy in
such a way to make the other player indifferent
between all of his choices
• What is the intuition?
• If you contribute every time, the other guy will
never contribute, and you’d like to free ride
• If you never contribute, you run the risk of not
having the good produced, which you do want.
Solving the Mixed Strategy
Equilibrium
• Let P2 be the probability that player 2
provides the public good.
• Player 1 earns 15P2+(1-P2)15 if he
provides the public good himself
• Player 1 earns 20P2+(1-P2)5 if he does not
• Player 2 knows this, and must keep him
indifferent between providing and not
providing, so:
Solving the Mixed Strategy
Equilibrium
• 15P2+(1-P2)15= 20P2+(1-P2)5 and we
solve for P2=2/3
• By the same logic, P1=2/3 as well
• What is likelihood that the public good gets
provided?
• 8/9 (4/9 that they both provide it, 2/9 that 1
does and 2 doesn’t, and 2/9 that 2 does
and 1 doesn’t)
Understanding the Mixed Strategy
Equilibrium
Player 2
Provide
Provide
Free Ride
15
20
(A)
(C)
15
15
Player 1
Free Ride
20
15
5
(A)
(B)
5
As the cost of providing
the public good rises (that
is, as C rises relative to A),
the likelihood of provision
falls.
As the loss associated
with the good not being
provided falls (that is, B
rising relative to A and C),
the likelihood of provision
also falls.
Which equilibrium?
• Which of the three equilibria will prevail in
a simultaneous move game?
• Signalling/Pre-commitment
• Focal points
• Social Norms
• Sequential Decisions and last mover
Volunteer Public Goods in a
Sequential Game
Player 1
Player 2
15
15
15
20
Use backward induction—Player 1
moves first, and determine what player 2
would do in either situation in order to
determine what he should do!
5
20
15
N.E.
5
Player 1 knows that if he provides, player 2
will free ride, and player 1 gets a payoff of
10. If 1 doesn’t provide, player 2 will, and
player 1 gets a payoff of 20
Volunteer Public Goods in a
Sequential Game
Player 1
Player 2
15
15
15
20
How does this differ from the simultaneous
move game? The nature of the information
sets!
5
20
15
N.E.
5
The oval represents an “information set,”
or the notion that player 2 doesn’t know
which of those nodes he is at.
Weakest-Link public goods
• The amount of the public good received by
any one person is identical to the quantity
received by anyone else
• If zi is the contribution made by the ith
member of society, and Gi is the amount of
the public good consumed by that member
• G=min{z1,z2,…zn}=G1=G2=…=Gn
Weakest Link public goods
• Volunteer public goods will tend to be provided,
because free riding in this case is irrational
• Examples:
• Levees
• Technological standards
• Health/disease control
• Airport security
• Free riding is inefficient and irrational
Why is free riding irrational?
• Consider 4 individuals who incur costs to
provide this public good:
• z1=$12
• z2=$18
• z3=$7
• z4=$13
• Because G=min{z1,z2,…zn}=G1=G2=…=Gn, and
max{z1,z2,…zn}=$7, everybody gets $7 worth of
the public good. If the consumer who put in $7
actually valued it at (say) $12, free riding has
harmed him!
What is the likely outcome?
• If individuals are heterogeneous, you will
likely see an amount provided equal to the
amount wanted by the lowest value
demander.
• Unless the decisions are made
sequentially, you wind up in a coordination
game.
Weakest link goods and
coordination games
• 2 Players
• Each can decide to provide the public
good or not
• The weakest link good is worth $55 to
each individual, and it costs $15 to provide
it.
• If the volunteer good is not provided, each
gets a payoff of $20
Weakest link goods and
coordination games
Player 2
Provide
Free Ride
40
20
Provide
40
5
Player 1
5
20
Free Ride
20
20
Multiple Nash Equilibria!
Coordination games v. HawkDove/Chicken games
• In a coordination game, both players want
to play the same strategy
• In a hawk-dove or chicken game, both
players want to play different strategies.
• Both types are going to have multiple
equilibria, both in pure and mixed
strategies.
Solving the Mixed Strategy
Equilibrium
• Let P2 be the probability that player 2
provides the public good.
• Player 1 earns 40P2+(1-P2)5 if he provides
the public good himself
• Player 1 earns 20P2+(1-P2)20 if he does
not
• Player 2 knows this, and must keep him
indifferent between providing and not
providing, so:
Solving the Mixed Strategy
Equilibrium
• 40P2+(1-P2)5 = 20P2+(1-P2)20 and we
solve for P2=3/7
• By the same logic, P1=3/7 as well
• What is likelihood that the public good gets
provided?
• 9/49 (16/49 that neither provide it, 12/49
that 1 does and 2 doesn’t, and 12/49 that
2 does and 1 doesn’t)
Which equilibrium?
• Which of the three equilibria will prevail in
a simultaneous move game?
• Again, the answers may be found through
such mechanisms as Signalling/Precommitment, Focal points, Social Norms,
and Sequential Decisions
Weakest Link Public Goods in a
Sequential Game
Player 1
Player 2
40
40
N.E.
5
20
20
20
5
20
Use backward induction—Player 1
moves first, and determine what player 2
would do in either situation in order to
determine what he should do!
Player 1 knows that if he provides, player 2
will as well, and player 1 gets a payoff of
40. If 1 doesn’t provide, player 2 won’t
either, and player 1 gets a payoff of 20
Weakest Link Public Goods in a
Sequential Game
• If the game is sequential, and all agents
are homogenous, the public good will be
produced efficiently.
• If agents are heterogeneous, however, we
may still not get the efficient amount of the
public good.
• Consider a 2 person society where one
person wants $1,000,000 worth of the
public good, the other wants $2.
National Defence as a public good
• War (or the threat of warfare) may be enough to
justify a government.
• Consider a game between 2 countries
• Each can spend $1T on military or not
• Without spending, GDP in each country is $5T
• If one spends and the other doesn’t, the spender
can invade and take money—we’ll assume that
they destroy $1T worth of stuff and take $3T
home with them
Military Spending as Prisoners’
Dilemma
Country 2
Not Spend
Spend
$5T
Not Spend
$8T
Pareto Superior
$5T
$1T
Country 1
$1T
$4T
Nash
Equilibrium
Spend
$8T
$4T
Despite the fact that both countries would prefer the equilibrium without militaries, it is
unstable. If country 1 arms, country 2 has to arm or else they’ll get a payoff of $1T. If
country 1 doesn’t arm, country 2 has an easy target for improving their welfare.
National Defence as a public good
• But doesn’t the same logic apply to individuals
within a country, deciding to volunteer for service
or not (or to contribute money to defence)?
• Everybody in the country would prefer to free
ride on the contributions of everyone else.
• Without some coercive taxation and public
financing of military, the country will wind up
overrun by their neighbours!
Public Finance and Public Supply
• For most prisoners’ dilemma type goods,
and many weakest-link type goods, there
is some potential for a government to
improve on the anarchic outcome by
overcoming the free rider problem
• 2 methods—public finance and public
supply
Public Finance and Public Supply
• Public Finance—government raises
revenues, and then contracts out to private
firms.
• Public Supply—government raises
revenues, and uses these revenues to
produce the public good themselves.
• Overcoming free riding problem makes a
case for public finance, but it does not
necessarily make a case for public supply!
Information and Public Goods
Information and Public Goods
• From here, we’ll mostly stick to discussing
public goods that are prisoners’ dilemmas
in nature.
• For these goods, the quantity of the public
good supplied is the sum of individual
contributions.
• n=the number of people contributing
• zi=the contribution of individual i
Information and Public Goods
• The quantity of the public good G available
to the entire population of n people is:
n
G   zi
i 1
• The individual contributions are to be
provided through coercive taxation.
Information and Public Goods
•
Two simplifying assumptions have been
made so far, at least implicitly:
1. All individuals value the public good by
the same amount
2. The government has full information on
how much each individual values the
public good
Heterogenous Individuals
• If individuals have different valuations,
efficiency dictates that each person pays
his/her marginal valuation of the good in
taxes:
• Ti=MBi for all I
• Thus, the tax T simply replaces the price
that people won’t truthfully pay
Imperfect information
• What if, however, the government doesn’t have
full information?
• Individuals have no incentive to accurately reveal
their marginal valuations
• Government can not determine efficient amount
of the public good!
• Recall efficiency condition:
n
 MB
i 1
i
 MC
n
 MB
i
i 1
 MC
P,
MB
P=MC
n
 MB
i
i 1
G*
Q
Imperfect information
• If the government doesn’t have full
information then the government confronts
the identical information problem that was
the whole reason for wanting a government
to finance the public good!
Optimal pricing and the Lindahl
solution
• Assume 2 individuals, person 1 and
person 2
• They each share the cost of the public
good, and pay a proportion of the cost of
the public good, s1 and s2, where s1=1-s2
(alternately, Σsi=1)
• Thus, if the per-unit price is P, person 1
pays P1=s1P
Lindahl Pricing
Cost shares
s1=1, s2=0
MB2
s2*
This is the demand curve for
person 1. Note that if he has to
pay for the entirety of the public
good, he doesn’t want any of it!
As the portion of the good he
has to pay for falls, he wants
more of it!
This is the demand curve for
person 2.
s1
s2
MB1
s1*
s1=0, s2=1
G*
Q
The two together are like
supply and demand—any
money spent by 1 (1’s demand)
is essentially supply for person
2!
This solution is not only efficient,
but also Pareto optimal!
Information and Lindahl Pricing
Cost shares
s1=1, s2=0
MB2
s2*
Does either person have incentive
to free ride? What if person 1
misrepresents his MB curve?
His share falls, and the output
falls. Does person 1 benefit by
doing this?
His welfare falls by this purple
area…
s1
s2
But increases by this red area.
MB1
s1*
s1=0, s2=1
G*
Q
In fact, it looks like he’d prefer
to misrepresent his MB curve
even more!
Information problems
• It should be obvious by now that, if
individuals would free ride on the provision
of a public good without government
intervention, they’ll do it if the government
gets involved as well.
• So, how to get them to reveal honestly?
Information Revelation
• How can we get individuals to reveal their
true MB information?
• What if we assure people that we will
finance the project through general
taxation?
• Result: overstating benefits and
overprovision!
• One solution: Clarke tax—individuals have
incentive to truthfully reveal
Cost-Benefit Analysis
• Many reasons for using CBA
• Individuals have no incentive to accurately
reveal preferences
• Suppliers (both private and public) may
have incentives to misrepresent their costs
Cost-Benefit Analysis
• Goal of CBA is to maximize social welfare, W
n
W   Bi (G )  C (G )
i 1
• Presumably (or perhaps hopefully) W>0
• If we take the derivative of W with respect to G
and solve for the first order condition (i.e.
maximize W w.r.t. G), we get:
n
 MB
i 1
i
 MC
Cost-Benefit Analysis
C(G)
The goal of CBA is to:
1. Determine what the total
benefit curve looks like…
$
n
 B (G)
i 1
i
2. Determine what the total
cost curve looks like…
3. And determine the
optimal quantity of the
public good (the quantity of
G that maximizes the
distance between TB and
TC)…
G*
G
Cost-Benefit Analysis
We can also show this
using marginal valuations:
$
Max W= ΣMB-MC
MC
ΣMB
G*
G
Estimating the value of life
• How much do you value your life?
• The value of human life is not infinite
• While we don’t ask this question directly,
we do use market prices to try to estimate
the value of a human life…
Estimating the value of life
Say there are two identical
Let’s assume that the
houses:
market price of the first
The only difference between the
house is $40,000 more than
two is that the second one is
the second house, and that
located next door to a nuclear
the probability of dying is
power plant.
1% more at the second
house:
And we all know who is the
safety inspector at this particular
nuclear power plant…
V=$40,000/1%=$4,000,000
Which house are
people willing to
pay more to live at?
Why?
CBA and Time
• Benefits and Costs of public goods
generally extend over a period of time
• Most goods have costs now and benefits
later (generally any time any building is
undertaken)
• Some goods have benefits not and costs
later (environmental damage, for example)
• To value a good over time, we need to
choose a discount rate
Discount rates
• A discount rate, r, tells us how much we
value the present relative to the future
• If r=0, then we care about the future just
as much as today
• If r=∞, then we care only about today
• The choice of discount rates is very
important in determining whether or not a
project gets funded.
Two methods of time valuation
• Choose a discount rate and calculate the
NPV (net present value)
• Calculate the IRR (internal rate of
return)—the discount rate at which the
project breaks even
• In both cases, higher is better
Risk and Uncertainty
• With time comes risk and uncertainty:
• Will a project be relevant in 10-20 years?
• What will the costs of a project be in 20
years?
• What is the appropriate amount of risk for
a government to take on?
Other considerations
• Equity and distributional effects
• General caveat: It is more efficient to do
pure transfers than to create an inefficient
public good for distributional reasons.
Locational Choice
• “Voting with your feet”
• Different communities/governments create
different bundles of public goods financed
through taxation.
• Individuals move to the community with
the best mix of taxes and goods for them.
• Logic of federalism yields efficient supply
of many public goods by creating a
“market” for public goods.
Public Financing for Public
Goods
Public Goods—how much
• Unfortunately, we may never know how much is
efficient!
• Governments lack the required information to
make efficient choices and implementing Lindahl
solutions
• CBA, Clarke taxes, and federalism provide
potential means of figuring out information
• We’ll proceed, assuming that the government
has figured out the efficient quantity.
Public Goods—How to pay for
them?
• Lots of methods, but all boil down to taxes.
• Issuing government debt is little more than deferred
taxation (Ricardian Equivalence or fiscal illusion?)—
however, can be important to ensure that future
beneficiaries of public goods pay for them.
• Inflation is a tax on people who hold currency
(seigniorage tax)
• All taxes reduce welfare via two components—transfers
(recall rent seeking, however) and DWL
• The government receives the transfer
• The DWL is never to be seen again!
Excess Burden/Dead Weight Loss
Consumer Surplus
S-T
This area is
the tax
revenue
raised by the
government
P
PD
Taxes drive a “wedge”
Note that it does not matter
between the price paid by
if the tax is levied on the
consumers and the price
suppliers…
received by suppliers—buyers
pay more than sellers receive
Or the consumers…
S
Here we consider a per-unit
The tax incidence and
tax
deadweight loss are
PD is the price paid by
identical
demanders, and …
PS is the price received
by sellers
And here’s
the dead
weight loss
tax
P*
PS
QT is the new amount sold
D
QT
Producer Surplus
Q*
Q
D-T
Excess Burden/Dead Weight Loss
• While it is possible for individuals to
receive some benefit from the transfer, it is
impossible to receive any benefit from the
DWL
• DWL is greater the more elastic the supply
and/or demand curve are
• As the tax rate increases, the dead weight
loss increases by disproportionately more
Excess Burden/Dead Weight Loss
If we start with a tax with
this DWL
P
S
A small increase in the
tax rate adds this amount
of DWL
From this point, a samesized increase in the tax
rate adds this amount of
DWL
D
Q
Other sources of inefficiency
•
•
•
•
•
•
Administrative costs
Enforcement costs
Costs of compliance
Avoidance costs
Evasion costs
Rent-Seeking Costs
Why is all this important?
• For every one dollar the government
spends, they take far more than a dollar
out of the economy.
• How much is called the marginal cost of
government funds
• What is the MCF? Estimates range from
$1.2 to well over $4!
• Often depends on how many types of cost
are considered.
Taxes without excess burden
• Property taxes
• Taxes on inelastic goods
• Lump-sum/poll taxes
Who pays the tax?
• Tax incidence refers to the question of who
bears the burden of a tax
• Government can not influence tax
incidence!
• Tax incidence is determined by the relative
elasticities of the supply and demand
curves
• Whichever is more inelastic will bear more
of the burden of the tax
Tax Incidence
Here, demand is relatively inelastic
as compared to supply. Hence,
demanders bear most of the
burden of the tax
Here, demand is relatively elastic
as compared to supply. Hence,
suppliers bear most of the burden
of the tax
Burden borne
by demanders
Burden borne
by demanders
t
Burden borne
by suppliers
t
Burden borne
by suppliers
Surplus revenue
• Sometimes government have more money than
they know what to do with!
• Cut taxes—this reduces DWL, improves
efficiency, and generally increases incomes
• Repay debt—this reduces the tax burden of
future generations
• Why not just spend it on something else—this
often leads to the flypaper effect and ratchets.