Chapter 5 Public Goods. Reading Essential reading –Hindriks, J and G.D. Myles Intermediate Public...

Chapter 5Public Goods

Reading• Essential reading

– Hindriks, J and G.D. Myles Intermediate Public Economics. (Cambridge: MIT Press, 2005) Chapter 5.

• Further reading– Andreoni, J. ‘Impure altruism and donations to public goods: a theory of warm-

glow giving’, Economic Journal (1990) 100: 464—477. – Abrams, B.A. and M.A Schmitz ‘The crowding out effect of government transfers

on private charitable contributions: cross sectional evidence’, National Tax Journal (1984) 37: 563—568.

– Bergstrom, T.C., L. Blume, L. and H. Varian ‘On the private provision of public goods’, Journal of Public Economics (1986) 29: 25—49.

– Bohm, P. ‘Estimating demand for public goods: an experiment’, European Economic Review (1972), 3: 55—66.

– Cornes, R.C. and T. Sandler The Theory of Externalities, Public Goods and Club Goods. (Cambridge: Cambridge University Press, 1996) [ISBN 0521477182 hcr] Chapters 6–10.

Reading– Cullis, J. and P. Jones Public Finance and Public Choice, (Oxford:

Oxford University Press, 1998) [ISBN 0198775792 pbk] Chapter 3. – Isaac, R.M., K.F. McCue and C.R Plott ‘Public goods in an experimental

environment’, Journal of Public Economics (1985), 26: 51—74. – Itaya, J.-I., D. de Meza and G.D. Myles ‘In praise of inequality: public

good provision and income distribution’, Economics Letters (1997), 57: 289—296.

– Oakland, W.H. ‘Theory of public goods’ in A.J. Auerbach and M. Feldstein (eds.), Handbook of Public Economics (Amsterdam: North-Holland, 1987) [ISBN 044487612X hbk].

– Samuelson, P.A. ‘The pure theory of public expenditure’, Review of Economics and Statistics (1954) 36: 387—389.

– Warr, P.G. ‘The private provision of a pure public good is independent of the distribution of income’, Economics Letters (1983) 13: 207—211.

• Challenging reading– Groves, T. and J. Ledyard ‘Optimal allocation of public goods: a solution

to the ‘free rider’ problem’, Econometrica (1977) 45: 783—809.

Reading

– Itaya, J.-I., D. de Meza and G.D. Myles ‘Income distribution, taxation and the private provision of public goods’, Journal of Public Economic Theory (2002) 4: 273—297.

– Foley, D.K. ‘Lindahl’s solution and the core of an economy with public goods’, Econometrica (1970) 38: 66—72.

– Laffont, J.-J. ‘Incentives and the allocation of public goods’, in A.J. Auerbach and M. Feldstein (eds.), Handbook of Public Economics. (Amsterdam: North-Holland, 1987) [ISBN 044487612X hbk].

– Milleron, J.-C. ‘Theory of value with public goods: a survey article’, Journal of Economic Theory (1972) 5: 419—477.

Introduction

• National defense: all inhabitants are simultaneously protected

• Radio broadcast: received simultaneously by all listeners in range of the transmitter

• These are both public goods

• If many consumers benefit from a single unit of provision the efficiency theorems do not apply

Definitions

• A pure public good satisfies:– Nonexcludability If the public good is supplied,

no consumer can be excluded from consuming it

– Nonrivalry Consumption of the public good by one consumer does not reduce the quantity available for consumption by any other

• A private good is excludable at no cost and is perfectly rivalrous

Definitions

• Goods can possess different combinations of rivalry and excludability

• Club goods are studied in chapter 6

• Common property resources are studied in chapter 7

• These are both examples of impure public goods

RivalrousNon-Rivalrous

Excludable

Non-Excludable

PrivateGood

CommonPropertyResource

ClubGood

PublicGood

Figure 5.1: Typology of goods

Private Provision

• Each consumer has an incentive to rely on others to provide the public good

• The reliance on others is called free-riding

• This leads to inefficiency since too little public good is provided

• All consumers will benefit from providing more public good

Private Provision

• Consider two consumers who allocate their incomes between a private good and a public good

• The consumers take prices as fixed • Each consumer derives a benefit from the

provision of the other• This introduces strategic interaction into

the decision processes• The Nash equilibrium has to be found

Private Provision

• Let be the provision of consumer h

• Fig. 5.1 shows the preferences of consumer 1

• Assume consumer 2 provides

• The utility of consumer 1 is maximized at

• Varying traces out the locus of choices for consumer 1

hg

1g

2g

2g

BudgetConstraint

1g

2g

1g2g

Figure 5.2: Preferences and choice

Private Provision

• Fig. 5.2 constructs the locus of choice for consumer 2

• If consumer 1 chooses to provide consumer 2 chooses

• The locus of chooses is given by the solid line

• This is the best-response function

2g

1g

2gBudgetConstraint

1g

2g

1g

Figure 5.3: Best reaction for 2

Private Provision

• The Nash equilibrium is where the choices of the two consumers are the best reactions to each other

• Neither has an incentive to change their choice

• This occurs at a point where the best-response functions cross

• The equilibrium choices are and 2g 1g

2g

2g

1gFigure 5.4: Nash equilibrium

1g

Private Provision

• The private provision equilibrium is inefficient

• But it is privately rational • A simultaneous increase

in provision by both consumers gives a Pareto improvement

• Pareto-efficient allocations are points of tangency between indifference curves

Locus of ParetoEfficient Allocations

Set of Pareto-Improvements

1g

2g

2g

1g

Figure 5.5: Inefficiency of equilibrium

Efficient Provision

• At a Pareto-efficient allocation the indifference curves are tangential

• This does not imply equality of the marginal rates of substitution because the indifference curves are defined over quantities of the public good purchased by the two consumers

• Instead the efficiency condition involves the sum of marginal rates of substitution and is termed the Samuelson rule

Efficient Provision• The tangency condition is

• Calculating the derivatives

• The marginal rate of substitution is

.1

2

.1

2

21 ||constUconstU dg

dg

dg

dg

22

2

1

11

Gx

G

G

Gx

UU

U

U

UU

hx

hGh

xGU

UMRS ,

Efficient Provision

• The tangency condition then becomes

• This is the Samuelson rule– The sum of marginal rates of substitution is equated

to the marginal rate of transformation between public and private goods

– The marginal rate of substitution measures the marginal benefit to a consumer of another unit of public good

– The marginal rate of transformation is the marginal cost of another unit

12,

1, xGxG MRSMRS

Efficient Provision

• For two private goods the efficiency condition is

• Why the difference?– An additional unit of a private good goes to either

consumer 1 or consumer 2– Efficiency is achieved when both place the same

marginal value upon it– An additional unit of public good benefits both

consumers– The marginal benefits are therefore summed

2,

1, jiji MRSMRS

Allocation through Voting

• The level of public good provision is frequently determined by voting

• Political parties promise different levels of provision

• Majority voting determines which party wins

• Need to assess whether this attains efficiency


• There is a population of H voters• The cost of the public good is shared equally• Consumer h has income • The utility function is

• Each consumer votes for the value of G that maximizes utility

hM

G

H

GMUGxU hh ,,

Privategood

Publicgood

Publicgood

Utility

1G HGmG


• Rank the consumers by income so

• Fig. 5.5 shows that the preferred levels of public good satisfy

• Assume an odd number of voters

• The median voter will be decisive

• Their choice will win the vote

HMMM ....21

Figure 5.6: Allocation through voting

HGGG ....21

mG


• The choice of the median voter satisfies

• The necessary condition can be written as

• So voting achieves efficiency only if

G

H

GMU mm

G,max

HMRS m 1

H

h

hm

H

MRSMRS

1


• Can any prediction be made?– Income has a long right tail– If MRS falls with income the median MRS is

greater than mean– This implies voting results in Gm exceeding the

efficient level

• There is no guarantee that voting will achieve efficiency

Personalized Prices

• With private goods consumption is adjusted to equate marginal valuation with market price

• With public goods it is not possible for consumers to adjust consumption

• This suggests adjusting prices to match the valuations of the fixed quantity

• This is the basis of personalized pricing

Private good

Public good

Price Same Different

Quantity Different Same

Table 5.1: Prices and quantities

Personalized Pricing

• Personalized pricing can be achieved by setting the share of the public good financed by each consumer

• The Lindahl mechanism asks each consumer to announce public good demand as a function of share

• The shares are adjusted until all consumers demand the same quantity

• If the demands honestly reflect preferences the equilibrium is efficient


• The tax shares for the two consumers are 1 and 2

• The shares satisfy 1 + 2 = 1

• The budget constraint of h is xh + hGh = Mh

• Gh is chosen to maximize utility

• Equilibrium shares ensure G1 = G2= G*

• Efficiency is achieved

1 2

2G1G

*G*G

Reactionof 1

Reactionof 2

Figure 5.7: Lindahl equilibrium


• The choice problem is

• This has necessary condition

• Summing over consumers

• The allocation satisfies the Samuelson rule

hhhhhG

GGMUh ,max

hhx

hG

U

U

1212

2

1

1

x

G

x

G

U

U

U

U


• Personalized pricing suffers from two significant drawbacks– There are practical difficulties of

implementation when there are many consumers

– The Lindahl mechanism is not incentive compatible and consumers have an incentive to announce false demand functions


• Assume consumer 1 is honest

• If consumer 2 were also honest the equilibrium would be eL

• The equilibrium can be moved to eM if consumer 2 announces a false demand function

• Allocation eM maximizes the utility of consumer 2 given the demand function of consumer 1

1 2

2G1G

Le

Me

Figure 5.8: Gaining by false announcement

Mechanism Design

• Consumers will make false announcements if this is advantageous

• This will distort the outcome

• Mechanism design is the search for allocation mechanisms that cannot be manipulated

• A preference revelation mechanism ensures true preferences are revealed

Mechanism Design

• Understatement– The benefit of the

public good is vh = 1– Cost of 1 is met by

those reporting rh = 1 – Announcements either

rh = 0 or rh = 1 – Provided if r1 + r2 ≥ 1– Nash equilibrium is rh =

0, h = 1, 2– No provision

Player 1

Player 2

0

0

1

1

00

0

0

1

1

21 2

1

Figure 5.9: Announcements and payoffs

Mechanism Design

• Overstatement– Benefit of the public

good is v1 = 0, v2 = ¾– Cost of 1 is shared

equally– Reports are r1 = 0 or 1,

r2 = ¾ or 1 – Provide if r1 + r2 ≥ 1– Equilibrium r1 = 0, r2 =

1– Inefficient provision

Figure 5.10: Payoffs and overstatement

Announcementof Player 1

Announcementof Player 2

0

1

1

00

41

21

21

21

41

41

43

Mechanism Design

• The Clarke-Groves mechanism ensures – True values are revealed– The public good is provided only when it

should be

• The allocation of cost is taken as given• Consumers report their net benefits

(benefit – cost)• Public good is provided if sum of net

benefits is positive

Mechanism Design

• If the public good is provided side payments are made

• These side payments reflect the fact that extracting the truth is costly

• The side payments internalize the net benefit of the public good to other players

Mechanism Design

• Net benefits are vh = -1 or vh = 1 (the mechanism must work for both)

• Reports are rh = -1 or rh = 1

• The public good is provided if r1 + r2 ≥ 0

• If provided the payoffs are v1 + r2 for player 1 and v2 + r1 for player 2

• The rh in the payoffs are the side payments

Player 1

Player 2

-1

-1

+1

+1

00

11 v

11 v11 v

12 v

12 v 12 v

Figure 5.11: Clarke-Groves Mechanism

Mechanism Design

Player 1

Player 2

-1

-1

+1

+1

0

2

Player 1

Player 2

-1

-1

+1

+1

0 20

0

11 v 11 v

0 2

Figure 5.12: Payoffs for Player 1

• When v1 = -1 a truthful report is weakly dominant for player 1• When v1 = 1 player 1 is indifferent between truth and false

statement• The mechanism ensures there is no incentive not to be

truthful

Mechanism Design

• The mechanism ensures truthful reports and efficient provision of the public good

• The drawback is the cost of the side payments

• If v1 = v2 =1 the total cost of the side payments is 2

• The side payments must be financed from outside the mechanism

More on Private Provision

• The private provision model predicts inefficiency

• The model also makes additional predictions that can be contrasted to evidence

• These predictions also have implications for government policy

More on Private Provision• Consider a transfer of

from consumer 1 to consumer 2

• The transfer shifts the indifference curves from the solid to the dashed

• The point delivers the same utilities after the transfer as the point did before

• The best response function also shifts

1g

2g

1g1g

2g

2g

Figure 5.13: Effect of income transfer

21 ˆ,ˆ gg

21 ˆ,ˆ gg


• Consumer 1 reduces contribution to the public good by

• Consumer 2 raises contribution by

• Total public good provision remains at

• Consumption of private good does not change

• The equilibrium is invariant to the transfer

1g

2g

1g1g

2g

2g

Figure 5.14: New equilibrium

21 ˆˆ ggG


• Assume H identical consumers

• Let be provision of all consumers but one

• Symmetry of equilibrium implies

• As H increases the equilibrium moves up the reaction function

• The contribution of each individual tends to zero

G

g

2H3H

Pareto-Efficient

ReactionFunction

Figure 5.15: Additional consumers

G

1

H

Gg


• The predictions of the model have been tested using experiments

• The typical experiment gives participants a fixed income to spend

• Income can be divided between a public good and a private good

• The private good has higher private benefit and the public good a higher social benefit


• Fig. 5.16 shows typical payoffs

• It is individually rational to spend all income on the private good

• It is socially optimal to spend all income on the public good

• The Nash equilibrium of the one-shot game has no investment in the public good

Social Benefit

Private Benefit

Public GoodPrivate Good

5

5

1

10

Figure 5.16: Public good experiment


• In experiments average contribution to the public good is 30 to 90 percent of income

• Most observations fall in the 40 to 50 percent range

• Among students the contribution to the public good is lowest for economists and falls with number of year of economics

• Repeating the game results in lower contributions in later rounds


• A number of explanations have been proposed for these results– The players may not use non-Nash conjectures. This

response has the problem of being arbitrary– Consumers may derive a warm glow from making a

contribution. The warm glow raises the benefit and so raises contribution

– Social interaction can be admitted. This could be a social custom or a consideration of fairness

Fund-Raising Campaigns

• The model used so far has a single round of voluntary contributions

• If consumers observe the equilibrium is inefficient they may wish to have a second round of contribution

• The same logic can be applied repeatedly suggesting efficiency may be approached afer numerous rounds

• This logic is now assessed using a fund-raising game


• In the fund-raising game a target level of funds must be achieved before a public good can be provided

• The game has an infinite horizon • The target for funds is C• There are two identical players (X and Y) who

derive benefit B from the public good• The good is socially desirable with B < C < 2B• Both players have discount rate d for delaying

completion for one period• The players alternate in making contributions


• In a contribution campaign the contributions are paid at the time they are made– This form of campaign is used if no credible

commitment can be made• Assume it is now the turn of X to make a

contribution at time T• The maximal contribution X will make to finish

the campaign is xT = [1 – ]B• At T – 1 the maximum contribution Y will make

(knowing X will finish the campaign at T) is yT–1 = [1 – 2]B

• This reasoning can be continued back in time


• This process is shown in Fig. 5.17

• Summing these contributions gives

[1 – ]B + [1 – 2]B + 3[1 – 2]B + …= B• The total contributions

never exceed the individual valuation

• The contribution campaign will never finance the public good

Time

TotalContributions

...

Tx

2Tx

4Tx3Ty

1Ty

C

T

Figure 5.17: A contribution campaign


• In the subscription campaign donation pledges are made

• Contributions are only made when and if pledges are sufficient to finance the public good

• This alters the strategic structure and the amount raised is equal to the total valuation of the contributors

• Start at time T with X about to make a pledge• The maximum X will pledge to finish the

campaign is xT = [1 – ]B


• The maximum that Y will pledge at T – 1 knowing that X will complete the campaign is yT–1 = [1 – 2]B

• Summing these pledges gives [1 – ]B + [1 – 2]B + [1 – 2]B + …= 2B• It is therefore possible to fund the project

because C < 2B• This result shows that allowing contributions to

be repeated may remove the inefficiency but requires the ability to make binding commitments

Chapter 5 Public Goods. Reading Essential reading –Hindriks, J and G.D. Myles Intermediate Public...

Documents

Transcript of Chapter 5 Public Goods. Reading Essential reading –Hindriks, J and G.D. Myles Intermediate Public...