t2 public goods - university of victoria -...

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

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.

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)

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)

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

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.

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.

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 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 13

�

maxLx ,Kx ,λ

fx Lx,Kx( ) + λ fG L − Lx,K − Kx( ) −G( )Interior solution :∂fx∂Lx

− λ∂fG∂LG

= 0

∂fx∂Kx

− λ ∂fG∂KG

= 0

G = fG L − Lx,K − Kx( )

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.

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.

From Edgeworth Box of Production to PPF

Good x Good G

capital

labor

x=30, G=10

x=24, G=20 x=12,

G=28

Good x

PPF

Good G

28

20

10

12 24 30

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.

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 Pareto-efficiency?

•  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.

Solving for Pareto Efficient Allocation

Slide 20

�

maxu1 x1,G( )s.t. u2 x2,G( ) = u2s.t. x1 + x2 = fx Lx,Kx( )s.t. G = fG LG,KG( )s.t. Lx + LG = Ls.t. Kx + KG = K

Using PPF, the problem is

Slide 21

�

maxu1 x1,G( )s.t. u2 x2,G( ) = u2s.t. x1 + x2 = xs.t. G = G x( )maxx1 ,x,φ

u1 x1,G(x)( ) + φ u2 x − x1,G x( )( ) − u2( )

Pareto Efficiency with PG

•  First Order Conditions are necessary and sufficient

Slide 22

�

∂u1∂x1

−φ ∂u2∂(x − x1)

= 0

∂u1∂G

∂G∂x

+ φ ∂u2∂(x − x1)

+∂u2∂G

∂G∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0

Samuelson Condition (1954) •  Combining both equations from previous

slide

Slide 23

�

∂u1

∂Gφ

∂u2

∂(x − x1)

+

∂u2

∂G∂u2

∂(x − x1)

= −∂x∂G

Since ∂u1

∂x1−φ ∂u2

∂(x − x1)= 0

and (x − x1) = x2

∂u1

∂G∂u1

∂x1

+

∂u2

∂G∂u2

∂x2

= −∂x∂G

The Samuelson Condition

Slide 24

�

∂u1∂G∂u1∂x1

+

∂u2∂G∂u2∂x2

= −∂x∂G

MRT of x for G

2’s MRS

1’s MRS

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 good x for one more unit of G

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.


•  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 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!

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) =xB

1/4G3/4. Both goods are produced with labor and capital and the PPF is given by G(x) = 8 - x.

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?

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.

Exercise Cont’d

•  How does the Pareto efficient product mix change as we let Ara reach higher levels of utility?

Answer

Slide 32

�

maxuB xB ,G( ) = xB1/ 4G3 / 4

s.t. xAG = uA

s.t. xA + xB = xs.t. G = 8 − x

maxxB ,x,φ

xB1/ 4 8 − x( )3 / 4 + φ x − xB( ) 8 − x( ) − uA( )

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

Slide 36

�

ui xi,G( ) = xi + γ i G( )γ i '> 0,γ i ' '< 0.


Slide 37

�

∂u1 /∂G∂u1 /∂x1

+∂u2 /∂G∂u2 /∂x2

= −∂x∂G

γ1' G( ) + γ 2 ' G( ) = −∂x∂G

Demand for the public good

•  Note that if we were to set-up the standard utility maximization problem for each consumer, we find

Slide 38

�

γ1' G( ) =pGpx

γ 2 ' G( ) =pGpx

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

Demand 1 Demand 2

Public good

γ2’(G1)

γ1’(G1)

γ1’(G1) + γ2’(G1)

Social Marginal Benefit Curve

G1

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

Demand 1 Demand 2

Public good

Social Marginal Benefit Curve

Efficient G

Social Marginal Cost Curve

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.

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


•  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.

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.

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).

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)

•  Both people face prices of px, pG and have incomes of I1, and I2.

�

ui xi,G( ) = xiαG1−α

Efficient Provision

•  (a) Write down the conditions that describe the set of Pareto efficient allocations.

•  The optimality condition is given by:

�

∂u1 /∂G∂u1 /∂x1

+∂u2 /∂G∂u2 /∂x2

=pGpx

1−α( )x1αG

+1−α( )x2αG

=pGpx

px (x1 + x2) + pGG = I1 + I2

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( )x1αG

+1−α( )x2αG

=pGpx

⇒ px x1 + x2( ) =αpGG1−α( )

px (x1 + x2) + pGG = I1 + I2

⇒pGG1−α( ) = I1 + I2 ⇒ G* =

1−α( ) I1 + I2( )pG

,x* =α I1 + I2( )

px

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.

�

maxxi ,Gi ,λ

xiα Gj + Gi( )1−α

+ λ Ii − pxxi − pGGi( )

FOCs αxiα −1 Gj + Gi( )1−α

− λpx = 0

……… 1−α( )xiα Gj + Gi( )−α − λpG = 0

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.

Slide 55

�

xi =αpG1−α( )px

G j + Gi( )px

αpG1−α( )px

G j + Gi( ) + pGGi = Ii

Gi G j( ) =1−α( )Ii −αpGG j

pG

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.

Nash Equilibrium

•  First present analytical results •  Then draw a graph using a symmetric setup

Slide 57

Nash Equilibrium •  Nobody has an incentive to deviate.

Slide 58

�

G1 =1−α( )I1 −αpGG2

pG

G2 =1−α( )I2 −αpGG1

pG

G1 =1−α( ) I1 −αI2( )1−α 2( )pG

=I1 −αI2( )1+ α( )pG

G2 =I2 −αI1( )1+ α( )pG

Nash Equilibrium is not efficient

•  Sum up private contributions in NE and you’ll see they are lower than the efficient G.

Slide 59

�

G1 + G2 =I1 −αI2( )1+ α( )pG

+I2 −αI1( )1+ α( )pG

G1 + G2 =1−α( ) I1 + I2( )1+ α( )pG

< G* =1−α( ) I1 + I2( )

pG

The best response functions •  Price of x =1, price of G = 2, Income of

each person = 200, alpha =1/2

Contribution of person 1


100

50

Best response of person 1

50

100

Best response of person 2

Nash Equilibrium

Diagram



100

50

50

100

Along red line: efficient amount of public good

Utility of person 1 increases

Utility of person 2 increases

Both people better off than in NE.

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

•  A redistribution of income holding combined income constant does not change the amount of public good provided. Slide 62 �

G1 + G2 =1−α( ) I1 + I2( )1+ α( )pG

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

Slide 64

�

dG1 =dI1 −αdI2( )1+ α( )pG

=dI1 + αdI11+ α( )pG

=1pGdI1

dG2 =dI2 −αdI1( )1+ α( )pG

=1pGdI2

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.

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


•  To put this more formally,

Slide 67

�

Gi j≠ iG j∑( ) =1−α( )Ii −αpG j≠ iG j∑

pGG1* = G2

* = ...= Gn*

Gi* =

1−α( )Ii −αpG (n −1)Gi*

pG

Gi* =

1−α( )Ii1+ α(n −1)( )pG


•  Comparative Statics: change n

Slide 68

�

for n ≥ 2 :

Gi* =

1−α( )Ii1+ α(n −1)( )pG

∂Gi*

∂n=

−αpG 1−α( )Ii1+ α(n −1)( )2 pG

< 0

G* = nGi* =

n 1−α( )Ii1+ α(n −1)( )pG

∂G*

∂n= 1−α( )Ii

1+ α(n −1)( )pG −αpG1+ α(n −1)( )2 pG

= 1−α( )I 1+ α n − 2( )( )pG1+ α(n −1)( )2 pG

> 0

Individual contributions decrease with n

Total contributions

increase with n

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 as government contributes? Slide 70

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 free-riding: 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 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

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.

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.

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.

More steps 3.  Based on their reports in 2, the level of public good that

maximizes the sum of reported net benefits of all people is provided.

4.  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.

5.  Repeat step 4 for all the other people.

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.

Total Benefit of Public Good

•  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.

Quantity 1 2 3 4

Alice 60 110 150 180 Brenda 80 120 140 150 Pe

rson

Chip 12 0 200 270 330

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?

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.

Answer a) •  Finding highest SNetB.

Quantity Person

1 2 3 4 Alice 60 110 150 180 Brenda 80 120 140 150 Chip 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

Answer a) Another way of determining social optimum

•  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.

60 100

70 130

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.

Answer b) Quantity

Person

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

Question c) •  Calculate the Groves-Clarke taxes for each

person.

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.

Quantity Person

1 2 3 4 Brenda 40 40 20 -10 Chip 80 120 150 170 SNetB 120 160 170 160

3 units of public good will be picked given these reports of Brenda and Chip

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.

Quantity Person

1 2 3 4 Alice 20 30 30 20 Chip 80 120 150 170 SNetB 100 150 180 190

4 units of public good will be picked given these reports of Alice and Chip

Answer c) •  Finally, without Chip the quantity would be

2 not 3. Chip needs to pay a tax of 70-50=20.

Quantity

Person

1 2 3 4

Alice 20 30 30 20

Brenda 40 40 20 -10

SNetB 60 70 50 10

2 units of public good will be picked given these reports of Alice and Brenda

Table with Tax Quantity

Person

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.

Question d) •  Brenda is the one who prefers two units to

three units under this payment scheme. Has she an incentive to lie?

Quantity

Person

1 2 3 4

Alice 20 30 30 20

Brenda 40 40 20 -10

Chip 80 120 150 170

SNetB 140 190 200 180

Answer d) •  Suppose she changes the choice from three

to two units by stating the following net benefit schedule.

Quantity

Person

1 2 3 4

Alice 20 30 30 20

Brenda b1 b2 b3 b4

Chip 80 120 150 170

SNetB 100+b1

150+b2

180+b3

190+b4

•  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.

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

Person

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

Weaknesses of the Groves- Clarke Mechanism

1.  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.


•  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.


•  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

t2 public goods - university of victoria -...

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