Public Goods
•  Pareto Efficiency
•  Market Failure: Competitive markets are
not efficient
•  Solutions to the Free-Rider Problem: Clarke
Groves Mechanism
•  General Policy Recommendation: Separate
funding from who benefits
Slide 1
Private vs. Public Goods
•  Private goods are
–  excludable: If you don’t pay you won’t get the good.
–  rival: if you consume a certain amount of the good
there is less to consume for others.
•  Public goods are
–  non-excludable: If you don’t pay you can still get the
good.
–  non-rival: your consumption of the good does not
diminish the amount available for others.
Slide 2
Private vs. Public Goods
•  Remember that rivalness and excludability are the
keys to defining public vs. private goods
–  Just because government supplies a good does not
make it public
•  e.g. public education is largely a private good
–  Just because a good is not supplied by government does
not make it private
•  Many governments are doing nothing to reduce carbon
emissions; but reduction is clearly a public good.
Slide 3
Examples of Public Goods
• 
• 
• 
• 
• 
National defense
Public safety
Clean air
Street lights (a very local public good)
Child health, happiness (public good
primarily to their parents; to a lesser extent
to society)
Slide 4
Examples of Collective Goods
•  These goods are non-rival but excludable
–  Cable TV
–  Websites
•  Technically you could be charged to visit any website
(excludable)
•  e.g. MOOCS (massive open online course)
Slide 5
Examples of Commons Goods
•  These goods are non-excludable but rival
–  Fisheries
–  Well water
–  Other open access resources
Slide 6
Impure public goods
•  Goods occupying the middle ground between
these extremes: i.e., exhibit some excludability or
rivalness
–  Roads: Costly to exclude people (given current
technology) but as more and more people use it
consumption becomes rival (congested).
–  Polar bears: Rival when hunted; congestible when
viewed in person; non-rival and non-excludable when
enjoyed for pure existence value (i.e. when people get
utility from knowing polar bears exist even though they
can’t see them in person)
Slide 7
Examples of GE models
•  We will mostly work with simple examples
of general equilibrium models.
•  One input, two outputs, two consumers
•  The 2x2 production model with two
consumers: 2 goods, 2 inputs, 2 consumers
Slide 8
Assumptions for the 2x2
production model
•  Two inputs: Capital K, Labor L
•  Two consumption goods: private good x, pure
public good G
•  Person 1 and person 2 (consumers): strictly
convex indifference curves (decreasing MRS
between goods).
•  Producers: strictly convex isoquants (decreasing
marginal rate of technical substitution between
inputs). We also assume that production of the two
goods does not exhibit increasing returns to scale.
Slide 9
Problems society must solve:
•  How to allocate the existing stock of capital
and labor efficiently between the production
of good x and the production of good G.
•  How to distribute these goods efficiently
among the population once they are
produced.
Slide 10
Consumption Efficiency
•  A distribution of goods is consumption
efficient if it is not possible to reallocate
these goods and make at least one person in
the economy better off without making
someone else worse off.
•  Given a fixed amount of goods x and G,
both people will always consume the same
amount of the public good G, and hence we
can only make one person better off by
redistributing good x thus making the other
Slide 11
person worse off.
Production Efficiency
•  An allocation of inputs (K and L) is
production efficient if it is not possible to
reallocate these inputs and produce more of
at least one good in the economy without
decreasing the amount of some other good
that is produced.
Slide 12
Production Efficiency
max f ( L ,K ) + λ ( f ( L − L ,K − K ) − G)
L x ,K x ,λ
x
x
x
G
x
x
Interior solution :
∂f x
∂f G
−λ
=0
∂L x
∂LG
∂f x
∂f G
−λ
=0
∂K x
∂K G
G = fG ( L − Lx ,K − K x )
Slide 13
Condition for Interior Production
Efficiency
•  A given set of inputs available in an
economy should be allocated across sectors
until the marginal rate of technical
substitution for each pair of inputs is equal
in each sector.
Slide 14
The Production Possibility
Frontier with 2 Inputs
•  While the set of Pareto efficient allocations
in the Edgeworth box of production makes
all the efficient input combinations visible,
the PPF gives us all the combinations of
goods that are production efficient.
•  The PPF is the value function of the
problem that gives us production efficient
allocations as its solution.
Slide 15
From Edgeworth Box of Production to PPF
Good G
capital
x=12,
G=28
Good x
Good G
28
x=24,
G=20
x=30,
G=10
labor
PPF
20
10
12
24
30
Good x
Slide 16
The Slope of the PPF
•  With good x on the horizontal axis and good G on
the vertical axis, the absolute value of the slope of
the PPF represents the Marginal Rate of
Transformation (MRT) of good G for good x; it
indicates how many units of good G the economy
would have to sacrifice (by transferring inputs
from the production of good G to the production
of good x) in order to produce 1 more unit of good
x.
Slide 17
Units of x forgone to produce one
more unit of G
•  Similarly, the marginal rate of
transformation (MRT) of good x for good
G, indicates how many units of good x the
economy would have to sacrifice (by
transferring inputs from the production of
good x to the production of good G) in
order to produce 1 more unit of good G.
Slide 18
Product Mix Efficiency
•  Which combination of goods will give us Paretoefficiency?
•  We can have efficiency in production and in
consumption (trivially satisfied with one private
and one public good), and yet there is still room
for a Pareto improvement, because we are
producing too much of one good and not enough
of the other.
•  Product mix efficiency puts together both sides,
consumers and producers.
Slide 19
Solving for Pareto Efficient
Allocation
max u1 ( x1,G)
s.t. u2 ( x 2 ,G) = u2
s.t. x1 + x 2 = f x ( Lx ,K x )
s.t. G = f G ( LG ,KG )
s.t. Lx + LG = L
s.t. K x + KG = K
Slide 20
Using PPF, the problem is
max u1 ( x1,G)
s.t. u2 ( x 2 ,G) = u2
s.t. x1 + x 2 = x
s.t. G = G( x )
max u1 ( x1,G(x)) + φ ( u2 ( x − x1,G( x )) − u2 )
x1 ,x,φ
Slide 21
Pareto Efficiency with PG
•  First Order Conditions are necessary and
sufficient
∂u1
∂u2
−φ
=0
∂x1
∂ (x − x1 )
⎛ ∂u2
∂u1 ∂G
∂u2 ∂G ⎞
+ φ⎜
+
⎟ =0
∂G ∂x
⎝ ∂ (x − x1 ) ∂G ∂x ⎠
Slide 22
Samuelson Condition (1954)
•  Combining both equations from previous
slide
∂u1
∂u2
∂x
∂G
∂
G
+
=−
∂u2
∂u2
∂G
φ
∂ (x − x1 ) ∂ (x − x1 )
∂u1
∂ u2
Since
−φ
=0
∂x1
∂ (x − x1 )
and (x − x1 ) = x 2
∂u1 ∂u2
∂G + ∂G = − ∂x
∂u1 ∂u2
∂G
∂x1 ∂x 2
Slide 23
The Samuelson Condition
1’s
MRS
2’s
MRS
MRT of x
for G
∂u1 ∂u2
∂G + ∂G = − ∂x
∂u1 ∂u2
∂G
∂x1 ∂x 2
MRT tells us
the marginal
cost of
producing the
public good
to society in
terms of the
units of the
private good
society must
sacrifice.
MRS of person i tells us the marginal benefit of G expressed in
units of good x; person i is willing to give up MRS units of
Slide 24
good x for one more unit of G
The Samuelson Condition
•  Efficient provision of public goods requires
that the sum of the marginal rate of
substitution of the private good for the
public good across all individuals is equal to
the marginal rate of transformation of the
private good for the public good.
Slide 25
Pareto-efficient conditions for an
economy with public goods
1. Allocate private goods until the point at which the
marginal rate of substitution between any two
private goods is equal and equal to the marginal
rate of transformation between these goods.
2. For efficient production, the marginal rate of
technical substitution of the inputs to production
of all goods must be equal.
3. Wherever public goods exist, the sum of the
marginal rates of substitution (for all people in
society) of private for public goods must equal the
marginal rate of transformation between these Slide 26
goods.
More on efficiency
•  In the case of an economy with only private goods, the benefit to
society of the last unit of a private good provided (expressed as the
willingness to forgo units of another good) is equal to the benefit of the
one person in society who receives this last unit. If there are some
people who receive a higher benefit from the last unit than others, we
don’t have Pareto efficiency. Hence, the marginal benefit of a private
good must be the same across all people for the allocation to be
efficient.
•  However, in the case of public goods, everybody is forced to consume
the same amount of the public good, but the marginal benefit for each
person of consuming this amount may differ at an efficient allocation.
The marginal benefit to society of providing this amount of public
good is equal to the sum of the marginal benefits received by all
people.
•  This implies that the marginal rate of substitution of different members
of society for a public good (in terms of a private good) need not be
the same for efficiency to hold!
Slide 27
Exercise
•  Suppose there are 2 consumers, Ara and Bahar,
and two goods, good x and good G. Ara’s utility
function is given by uA(xA, G) =xAG and Bahar’s
utility function is given by uB(xB, G) =xB1/4G3/4.
Both goods are produced with labor and capital
and the PPF is given by G(x) = 8 - x.
Slide 28
Exercise Cont’d
•  Suppose Ara receives one unit of good x.
How many units of good x must Bahar
receive and how many units of G should be
produced for the allocation to be Pareto
efficient?
Slide 29
Answer
•  We need to ensure that the Samuelson condition is
satisfied.
•  MRSAra x for G+ MRSBahar x for G=MRTx for G
•  We know that xB = x– 1, G = 8 - x
•  xA/(8-x) + 3xB /(8-x) = 1
•  1/(8-x) + 3(x-1) /(8-x) = 1
•  -2 +3x=8-x  x= 2.5, G = 5.5, xB = 1.5.
Slide 30
Exercise Cont’d
•  How does the Pareto efficient product mix
change as we let Ara reach higher levels of
utility?
Slide 31
Answer
max uB ( x B ,G) = x G
1/ 4
B
3/4
s.t. x A G = uA
s.t. x A + x B = x
s.t. G = 8 − x
max x
x B ,x,φ
1/ 4
B
(8 − x )
3/4
+ φ (( x − x B )(8 − x ) − uA )
Slide 32
Answer
•  We need to ensure that the Samuelson
condition is satisfied.
•  MRSAra x for G+ MRSBahar x for G = MRTx for G
•  We know G = 8-x, and uA = xA(8-x)
•  xA/(8-x) + 3xB /(8-x) = 1
•  uA/(8-x)2 + 3(uA-8+x) /(8-x)2 = 1
•  4uA -24 + 3x=(8-x)2
Slide 33
Answer
•  We find the Pareto efficient amount of x as
a function of uA.
•  Implicitly given by
•  4uA - 24 + 3x(uA)=(8-x(uA))2
•  And G(uA) = 8 – x(uA).
Slide 34
Answer Ct’d
•  From last two equations (totally
differentiated with respect to x and uA)
•  dx/duA = 4/(2x+19) >0,
•  dG/duA = - 4/(2x+19) <0
•  As we want to achieve a higher utility for
Ara, the product mix shifts in favour of
more of the private good and less of the
public good.
Slide 35
Efficient Provision of Public
Goods with Q-linear Prefs
•  Let’s assume that both consumers have
quasi-linear utility functions of the form
ui ( x i ,G) = x i + γ i (G)
γ i '> 0,γ i ''< 0.
Slide 36
The Samuelson Condition
∂u1 /∂G ∂u2 /∂G
∂x
+
=−
∂u1 /∂x1 ∂u2 /∂x 2
∂G
∂x
γ 1 ' (G) + γ 2 ' (G) = −
∂G
Slide 37
Demand for the public good
•  Note that if we were to set-up the standard
utility maximization problem for each
consumer, we find
pG
γ 1 ' (G) =
px
pG
γ 2 ' (G) =
px
Slide 38
Vertically adding demand
•  Suppose px= 1. Then pG = γi’(G)
•  For a given y, pG is equal to the amount of
good x the individual is willing to give up
for one more unit of good G.
 The Samuelson Condition on LHS in the
case of quasi-linear preferences is adding
individual demands for the public good
vertically.
Slide 39
Price of G/ Marginal
Willingness to pay
γ1’(G1) + γ2’(G1)
γ1’(G1)
Social Marginal Benefit Curve
γ2’(G1)
Demand 2
Demand 1
G1
Public good
Slide 40
Efficient Provision of Public
Good
•  We can draw the MRT in the above graph.
It tells us the marginal cost of producing the
public good to society in terms of the units
of the private good society must sacrifice.
•  With strictly concave PPF this marginal cost
is increasing; with linear PPF this marginal
cost is constant.
Slide 41
Price of G/ Marginal
Willingness to pay
Social Marginal Benefit Curve
Social Marginal Cost Curve
Demand 2
Demand 1
Efficient G
Public good
Slide 42
Summary
•  (Interior) Efficient provision of the public good
requires the Samuelson condition to hold; the sum
of marginal rates of substitution of the private
good for the public good must equal the marginal
rate of transformation.
•  With quasi-linear preferences, this condition boils
down to adding individual demand for the public
good vertically and finding its intersection with
the social marginal cost curve.
Slide 43
Competitive Equilibrium
•  Now that we understand Pareto efficiency in
a general equilibrium model with
production and a public good, we want to
find out if the competitive equilibrium in
this model is Pareto efficient.
•  First discuss what happens in the
competitive equilibrium, then determine
whether it’s Pareto efficient.
Slide 44
What is the correct assumption
about consumer behaviour?
•  However, there is a problem, when we discuss
public goods. What is the correct assumption
about consumer behaviour?
•  Naïve consumer is not aware of the nature of
public goods and hence treats the good like a
private good.
•  A more sophisticated consumer understands the
nature of the public good and wonders how many
units to buy, given that units of the good bought
by other consumers can also be consumed.
Slide 45
Naïve Consumer
•  In the first approach, the answer is
straightforward. Each consumer treating
both goods as private will choose to buy the
goods where MRS = price ratio.
•  If consumers are price takers, they all face
the same prices and therefore all their
marginal rates of substitution will be equal
in equilibrium.
Slide 46
Production Efficiency
•  Firms maximize profits and hence minimize
costs. From profit maximization,
MRTS=factor price ratio (because cost
minimization is a necessary condition for
profit maximization).
–  If all firms are price takers in the factor
markets, then all firms have equal MRTS.
–  The competitive equilibrium satisfies
production efficiency.
Slide 47
Samuelson Condition Violated!
•  Firms maximize profits. From profit maximization
of competitive firms (if the number of firms is
sufficiently large) P=MC.
–  For any two goods, consumers set MRS=px/pG.
–  From profit max a firm produces an amount of x where
px=MCx, and a firm produces an amount of G where
pG=MCG.
–  This implies px/pG=MCx/MCG, but MCx/MCG=MRT
and therefore MRS=MRT, not sum of MRS=MRT!
–  The competitive equilibrium is not efficient.
Slide 48
Private Contribution to the Public
Good Game
•  Next, we consider more sophisticated consumer
behaviour.
•  Consumers realize that units of the public good
purchased by other consumers will be available
for their consumption as well as units purchased
by themselves.
•  We start with an example in which both
consumers have identical Cobb-Douglas utility
functions.
Slide 49
Exercise: Private Provision vs.
Efficient Provision
•  In this exercise we contrast the private
provision of the public good with the
efficient public good provision. We can
think of the private provision as a game:
each person decides how much to contribute
to the public good. In the end all the
contributions collected are used to purchase
the public good (as many units as the
contributions buy).
Slide 50
The model
•  Suppose we have two people, person 1 and
person 2 with the following utility function
over a private good (x) and a public good
(G)
α 1−α
ui ( x i ,G) = x i G
•  Both people face prices of px, pG and have
incomes of I1, and I2.
Slide 51
Efficient Provision
•  (a) Write down the conditions that
describe the set of Pareto efficient
allocations.
•  The optimality condition is given by:
∂u1 /∂G ∂u2 /∂G pG
+
=
∂u1 /∂x1 ∂u2 /∂x 2 px
(1− α ) x1 (1− α ) x 2
pG
+
=
αG
αG
px
px (x1 + x 2 ) + pG G = I1 + I2
Slide 52
Efficient Provision Cont’d
•  Solve for the efficient quantity of the public
good G.
•  We need the optimality condition and the
joint budget constraint:
(1− α ) x1 (1− α ) x 2
pG
+
=
αG
αG
px
αpG G
⇒ px ( x1 + x 2 ) =
(1− α )
px (x1 + x 2 ) + pG G = I1 + I2
(1− α )( I1 + I2 ) * α (I1 + I2 )
pG G
*
⇒
= I1 + I2 ⇒ G =
,x =
(1− α )
pG
px
Slide 53
Equilibrium Private Provision of
the Public Good
•  Now suppose both agents try to maximize
their utility given the contribution of the
other agent to the public good. That is, we
want to find each agent’s best response
function.
1−α
α
max x i (G j + Gi ) + λ ( Ii − px x i − pG Gi )
x ,G ,λ
i
i
FOCs αx
α −1
i
(G
j
+ Gi
(
)
1−α
− λpx = 0
……… (1− α ) x G j + Gi
α
i
)
−α
− λpG = 0
Slide 54
Solving for best response
function
•  Want to know what is the optimal amount of
public good purchased by person i if person j
contributes Gj
•  So we are looking for a solution Gi(Gj)
•  Use FOCs and b.c. to get rid of xi.
αpG
xi =
G j + Gi
(1− α ) px
αpG
px
G j + Gi + pG Gi = Ii
(1− α ) px
(
)
(
( )
Gi G j =
)
(1− α ) Ii − αpG G j
pG
Slide 55
Efficient and Private Provision
•  Draw a diagram with person 1’s
contribution to the public good on the x-axis
and person 2’s contribution to the public
good on the y-axis.
•  We will see in the diagram why the Nash
Equilibrium is not efficient.
•  Think about the Cournot Duopoly game.
Slide 56
Nash Equilibrium
•  First present analytical results
•  Then draw a graph using a symmetric setup
Slide 57
Nash Equilibrium
•  Nobody has an incentive to deviate.
G1 =
G2 =
G1 =
G2
(1− α )I1 − αpG G2
pG
(1− α )I2 − αpG G1
pG
(1− α )( I1 − αI2 )
2
1−
α
(
) pG
I1 − αI2 )
(
=
(1 + α ) pG
I2 − αI1 )
(
=
(1 + α ) pG
Slide 58
Nash Equilibrium is not efficient
•  Sum up private contributions in NE and
you’ll see they are lower than the efficient
G.
G1 + G2
I1 − αI2 ) ( I2 − αI1 )
(
=
+
(1 + α ) pG (1 + α ) pG
(1− α )( I1 + I2 ) * (1− α )(I1 + I2 )
G1 + G2 =
<G =
(1 + α ) pG
pG
Slide 59
The best response functions
•  Price of x =1, price of G = 2, Income of
each person = 200, alpha =1/2
Contribution
of person 2
100
Best response of person 1
50
Nash Equilibrium
Best response of person 2
50
100
Contribution of
person 1
Slide 60
Diagram
Along red line:
efficient amount of
public good
Contribution
of person 2
100
Utility of person 1
increases
Both people
better off
than in NE.
50
Utility of
person 2
increases
50
100
Contribution of
Slide 61
person 1
Neutrality result of inefficient
provision of G
•  What would happen in the Nash equilibrium if we
were to change the distribution of income between
the two people, leaving their combined income the
same?
•  We have seen that the the sum of contributions in
the NE is given by
(1− α )( I1 + I2 )
G1 + G2 =
(1 + α ) pG
•  A redistribution of income holding combined
income constant does not change the amount of
Slide 62
public good provided.
Extra $ all spent on public good
•  How come the level of public good does not
change?
•  Surely the contribution of both people
changes.
•  We can find out by totally differentiating
each person’s equilibrium strategy with
respect to the changes in each person’s
income, such that dI1+dI2 = 0
Slide 63
Poorer contributor reduces, richer
contributor increases contribution
dG1
dI1 − αdI2 ) dI1 + αdI1
(
=
=
=
dG2
dI2 − αdI1 )
(
=
=
(1 + α ) pG
(1 + α ) pG
(1 + α ) pG
1
dI2
pG
1
dI1
pG
Since dI1= - dI2, the increase in contribution due to the
increase in income by one person is offset by the
decrease in the equilibrium contribution by the other
person in the exact same amount. Total level of the
public good is unchanged.
Slide 64
Neutrality Result of Inefficient
Provision
•  It can be shown that this neutrality result
holds even if people have different
preferences. They don’t have to be C-D
either. For any utility functions, as long as a
redistribution of income results in both
people contributing privately to the public
good, the total amount of the public good
purchased remains unchanged.
•  Bergstrom et al. (1986).
Slide 65
Increasing the number of
contributors
•  What happens to the level of provision if the
number of contributors increases?
•  It is straight forward to analyze a special
case of the previous set-up. Let’s assume we
have n identical individuals facing identical
budget constraints.
•  Then the Nash Equilibrium is symmetric:
everybody contributes the same amount.
Slide 66
Increasing the number of
contributors
•  To put this more formally,
Gi
(∑
j ≠i
)
Gj =
(1− α )Ii − αpG ∑ j ≠i G j
pG
G1* = G2* = ... = Gn*
Gi* =
G =
*
i
(1− α )Ii − αpG (n − 1)Gi*
pG
(1− α )Ii
(1 + α (n − 1)) pG
Slide 67
Increasing the number of
contributors
•  Comparative Statics: change n
for n ≥ 2 :
Gi* =
(1− α )Ii
(1 + α (n − 1)) pG
Individual contributions
∂Gi*
−αpG (1− α ) Ii
=
<0
decrease with n
2
∂n (1 + α (n − 1)) pG
n (1− α ) Ii
Total
*
*
G = nGi =
(1 + α (n − 1)) pG
contributions
increase
1 + α ( n − 2)) pG
(1 + α (n − 1)) pG − αpG
∂G *
(
= (1− α ) Ii
= (1− α ) I
>0
2
2
with n
∂n
(1 + α (n − 1)) pG
(1 + α (n − 1)) pG
Slide 68
BBV and increasing n
•  Note that BBV look at cases where they
increase n at the same time as they leave
total wealth in society the same. In this case,
it can be shown that more equal distribution
of wealth is lowering the public good
provision (Theorem 5).
•  We considered the case of increasing the
number of agents and at the same time
adding Ii to the wealth of the economy with
each agent that we added.
Slide 69
BBV and government provision
•  BBV also look at the possibility of
government providing some amount of the
public good, while individuals still
contribute voluntarily
•  Govt taxes individuals to finance public
good purchases.
•  Question of crowding out: Are private
contributions reduced by the same amount
Slide 70
as government contributes?
Crowding out
•  Theorem 6.
•  Assumptions: taxes are collected from
contributors and non-contributors. As
government provides some amount of the
public good, there is less than a dollar for
dollar reduction in private contributions 
partial crowding out.
Slide 71
Theorem 6 in detail
•  Complete crowding out: taxes collected
from people who contributed before and
taxes are smaller than contribution.
•  Partial crowding out if:
–  a) some of the contributors are taxed more than
their private contribution
–  b) non-contributors are taxed as well as
contributors
Slide 72
Conclusion
•  In the presence of public goods, private provision
of public goods is often inefficient.
•  We have also encountered the problem of freeriding: People try to get out of paying for a public
good knowing that if it is paid for by others, they
will still be able to consume the public good.
–  Note that while equilibrium contributions may be
greater than zero, we generally expect them to be too
low.
•  We will next talk about solutions to the provision
Slide 73
of public goods.
Achieving an Efficient Provision
of Public Goods
•  The Demand-Revealing Mechanism or
Groves-Clarke Mechanism
–  Exercise to see how the mechanism works
–  Weaknesses of the Groves- Clarke Mechanism
Slide 74
Theoretical Solutions to the
Public Good Problem
•  People have an incentive to lie about their true
preferences in order to get out of paying for the
public good while they can still benefit from its
consumption.
•  We would therefore need a scheme that will give
people an incentive to reveal the truth about their
public good preferences to the government.
•  To find such a scheme we will now take a look at
the work of Groves and Clarke.
Slide 75
Edward H. Clarke
•  Discovered the demand revealing process as a
University of Chicago graduate student during the
late 1960s. He has written extensively on the
application of demand revealing processes which
were noted in the 1996 Nobel Prize awards in
economics. He is currently a senior economist
with the Office of Management and Budget in the
area of government regulatory management.
Slide 76
The Demand-Revealing
Mechanism
•  A demand-revealing mechanism creates the
incentive for people to reveal their public
good preferences in a truthful manner.
•  Here is how the demand-revealing
mechanism works.
Slide 77
5 easy steps…
1.  A distribution of cost shares is announced.
2.  Given these cost shares each person
reports the net benefit from consuming a
certain amount of the public good (net
benefit is equal to total benefit minus cost
share). People may or may not tell the
truth.
Slide 78
3. 
4. 
5. 
More steps
Based on their reports in 2, the level of public good that
maximizes the sum of reported net benefits of all people
is provided.
Based on their reports in 2, a tax for person 1 is
calculated as follows: Find the sum of net benefits for
each quantity of the public good without person 1. If the
level of public good at which the sum of net benefits
without person 1 is maximized changes from the level
determined in 3, person 1’s tax is equal to the difference
of the sum of net benefits at the new optimal level and
the level determined in 3 without taking person 1’s net
benefit into account for any of the quantities of the
public good.
Repeat step 4 for all the other people.
Slide 79
Example
Consider a community comprising three
individuals: Alice, Brenda, and Chip. They have
quasi-linear preferences with the private good
entering their utility functions linearly. A
benevolent planner is undertaking the provision of
a pure public good. The preferences of community
members are given in the following table.
Slide 80
Person
Total Benefit of Public Good
Alice
Brenda
Chip
Quantity
1
2
3
60
110 150
80
120 140
120 200 270
4
180
150
330
•  The numbers in the table represent the total
benefits accruing to the person named at left for
the unit numbered above. So for example, Brenda
would receive a total benefit of 120 were two units
of the public good produced instead of one unit.
All terms are in dollars.
Slide 81
Question a)
•  Assume that the good can be provided at a
constant marginal cost of $120. What is the
socially efficient level of the good?
Slide 82
Answer a)
•  The socially efficient level of the public good
maximizes social net benefit, that is, total social
benefit minus total cost. Because we have only
four choices of quantities, we can directly
calculate the social net benefit (SNetB )and then
pick the quantity of the public good that yields the
highest SNetB.
•  More generally, SNetB is maximized, where the
sum of marginal benefits equals the marginal cost.
If we cannot use fractions, then pick the quantity
that has a social marginal benefit closest to but
still greater than marginal cost.
Slide 83
Answer a)
•  Finding highest SNetB.
Quantity
P
e
r Alice
s Brenda
o
n Chip
1
2
3
4
60
110 150 180
80
120 140 150
120
200 270 330
SB
260
430 560 660
Total Cost
120
240 360 480
SNetB
140
190 200 180
SNetB
highest,
therfore
optimal
quantity
is 3
Slide 84
Answer a) Another way of
determining social optimum
70
130
60
100
•  Table gives marginal benefit for each person and
social marginal benefit (add up marginal benefit at
each quantity). At a quantity of 3 we have SMB >
MC but at 4 units we have SMB<MC. Therefore
the efficient amount of the public good is 3 units.
Slide 85
Question b)
•  Suppose we announce that all three
members have to share the cost of the public
good equally, that is, everybody pays $40
per unit but there is an extra tax on a person
whenever this person’s net benefit changes
the group decision. Calculate the net benefit
of each person (total benefit minus cost for
each person) and fill in the table below.
Slide 86
Answer b)
Quantity
P
e
r
s
o
n
1
2
3
4
Alice
20
30
30
20
Brenda
40
40
20
-10
Chip
80
120
150
170
SNetB
140
190
200
180
3 units of public good will be picked given
these reports of the three people
Slide 87
Question c)
•  Calculate the Groves-Clarke taxes for each
person.
Slide 88
Answer c)
•  In the absence of Alice, three units of the public good
would be provided. Because Alice’s valuation of the public
good does not change the group choice, her tax would be
0.
P
e
r
s
o
n
Quantity
1
2
Brenda 40 40
Chip
80 120
SNetB 120 160
3 units of public good will be picked given
these reports of Brenda and Chip
3
20
150
170
4
-10
170
160
Slide 89
Answer c)
•  If we do not count Brenda’s marginal benefit, the quantity of the
public good would be four since this maximizes the sum of net
benefit from Alice and Chip. Brenda therefore needs to pay a tax in
the amount of the difference in the sum of net benefits of Alice and
Chip if the quantity of the public good is changed from 4 to 3 units.
That is, Brenda’s tax is equal to 190 –180 = 10.
P
e
r
s
o
n
Quantity
1
2
Alice 20 30
Chip 80 120
SNetB 100 150
3
30
150
180
4
20
170
190
4 units of public good will be picked given
these reports of Alice and Chip
Slide 90
Answer c)
•  Finally, without Chip the quantity would be
2 not 3. Chip needs to pay a tax of
70-50=20.
Quantity
P
e
r Alice
s
o Brenda
n
SNetB
1
2
3
4
20
30
30
20
40
40
20
-10
60
70
50
10
2 units of public good will be picked given
these reports of Alice and Brenda
Slide 91
Table with Tax
Quantity
P
e
r
s
o
n
1
2
3
4
Tax
Alice
20
30
30
20
0
Brenda
40
40
20
-10
10
Chip
80
120
150
170
20
SNetB
140
190
200
180
Overall net benefit with Groves-Clarke taxes for each person is
for Alice: 30, for Brenda: 20-10=10, for Chip: 150-20=130.
Slide 92
Question d)
•  Brenda is the one who prefers two units to
three units under this payment scheme. Has
she an incentive to lie?
Quantity
P
e
r
s
o
n
1
2
3
4
Alice
20
30
30
20
Brenda
40
40
20
-10
Chip
80
120
150
170
SNetB
140
190
200
180
Slide 93
Answer d)
•  Suppose she changes the choice from three
to two units by stating the following net
benefit schedule.
Quantity
P
e
r
s
o
n
1
2
3
4
Alice
20
30
30
20
Brenda
b1
b2
b3
b4
Chip
80
120
150
170
SNetB
100+ 150+ 180+ 190+
b1
b2
b3
b4 Slide 94
•  We assume 150+b2 is greater than any of the other
sums of net benefits.
•  Then only two units will be supplied.
•  Brenda is now paying a tax for changing the
quantity from 4 to 2 units: 4 units would be
supplied without her and the difference in net
benefit of the other two is 190-150 = 40.
•  This means Brenda’s net benefit from getting two
units is 40 but then she would have to pay a tax of
40, which implies an overall net benefit of 0. This
is lower than if she tells the truth and 3 units of the
public good are provided. Then she receives a net
benefit of 20 minus her tax of 10, that is, 20 –
10=10.
Slide 95
No incentive to lie
•  Brenda cannot make herself better off if she
lies.
•  Note that this is true no matter which lie she
tells as long as the reported values b1, b2, b3,
b4 result in a group choice of 2 units.
•  How much Brenda benefits depends on her
true net benefits for the different units and
the reports of all the other people.
Slide 96
Exercise
•  Generalize above argument and show that
given the other people’s reports nobody has
an incentive to lie.
Quantity
P
e
r
s
o
n
1
2
3
4
Alice
a1
a2
a3
a4
Brenda
b1
b2
b3
b4
Chip
c1
c2
c3
c4
SNetB
a1+b1
+c1
a2+b2
+c2
a3+b3
+c3
a4+b4
+c4
Slide 97
1. 
Weaknesses of the GrovesClarke Mechanism
The Groves-Clarke tax doesn’t really generate a Pareto
efficient outcome. The level of the public good will be
optimal, but the private consumption could be greater.
We are taking away money from the people by imposing
the tax that they otherwise could spend on private
consumption (see Brenda’s and Chip’s situation). These
taxes have to be taken away completely, i.e. the money
collected through Groves-Clarke taxes would have to be
destroyed. On the other hand, the more people are
involved in the decision making process the less likely it
is that their valuation of the public good will change the
group decision. In this case Groves-Clarke taxes will
only rarely be collected.
Slide 98
• 
Weaknesses of the GrovesClarke Mechanism
There are equity and efficiency tradeoffs. While
it is efficient to supply the level of the public
good determined by the Groves- Clarke
mechanism, there are some people who might be
strictly worse off with this level of public good
than with a different amount (Brenda and Chip).
The Groves- Clarke mechanism implements a
solution that is potentially a Pareto improvement
over the private provision of public goods, but
we cannot compensate the losers to achieve an
actual Pareto-improvement.
Slide 99
Weaknesses of the GrovesClarke Mechanism
•  Another weaknesses of the Groves- Clarke
mechanism is that it only is strategy-proof
(people have incentive to report truthfully)
if they have quasi-linear utility functions.
Slide 100
How does the government decide
on the level of Public Good?
•  General elections, public goods funded out
of general revenues which are raised
independently of how people feel about
public goods levels (by and large)
Slide 101