Public Goods

Public Economics, June 27, 2014

Muneta Yokomatsu

1

Public Goods

• Non-rivalness (非競合性)

The use of a good by one individual does not limit the amount of the good available for consumption by others.

• Non-excludability (排除不可能性)

It is impossible to exclude any individuals from consuming a good, even if they do not pay for its use.

* Public goods are an example of a particular kind of

consumer externality.

2

Classification of goods

3

Excludable Non-excludable

Rivalrous

Non-rivalrous

Private goods Common goods

Club goods Public goods

food, clothing, cars, computer, …

fish stocks, timer, coal, …

expressway, private parks, knowledge, airport lounge,

…

national defence, embankment, …

Public goods and market failure

• Non-rivalness (非競合性)

– Under-supplied (if the goods are supplied in market.)

• Non-excludability (排除不可能性)

– Free-rider problem

4

Marginal evaluation of private good

5

Private Good x

Price p

x*

p*

Da Db

Da+ Db

Aggregated demand

Supply curve

P = MC(x)

Da(p*)+Db(p

*) = x*

p* = WTPa = WTPb

Price p = Willingness-to-pay for marginal consumption of the good

Pareto efficient supply of private goods

6

Production possibilities frontier

Edgeworth box

X1

X2

2

1

2

1

2

1

2

1

p

p

xf

xf

xu

xu

xu

xu

BB

BB

AA

AA

MRSA = MRSB = MRT = p1/p2

Marginal evaluation of public good

7

Public Good x x*

p*

Da

Db

WTPa+WTPb Supply curve

P = MC(x)

Da(p*) = Db(p

*) = G*

WTP* = WTPa + WTPb

Social Willingness-to-Pay

= Sum of WTP

Everyone can consume the

same amount.

Price p = Willingness-to-pay for marginal consumption of the good

Price p

Pareto-efficient supply of public goods

8

AAGxx

xGuBA

,max,,

subject to *, BBB uxGu

BABA wwGcxx

Consumption of private goods

Consumption of public goods

Cost function of public

goods

Initial allocation of wealth

Pareto-efficient supply of public goods

• Lagrangian

• First-order conditions

9

GcxxwwuxGuxGuL BABABBBAA *,,

0

A

A

A x

u

x

L

0

B

B

B x

u

x

L

0

G

Gc

G

u

G

u

G

L BA

G

Gc

x

uG

u

x

uG

u

B

B

B

A

A

A

MRSA MRSA MC =

MRT

Pareto efficient supply of public goods

• Samuelson condition (サミュエルソン条件)

• In case of private goods,…

MRSA = MRSB = ... = MRT

10

MRTMRSi

i

Summation of MRS (marginal rate

of substitution; 限界代替率 )

MRT (marginal rate

of transformation; 限界変形率)

If Household A privately provide the public goods…

• For simplicity,

• Utility maximisation problem of Household A

11

GGc

BABA ggggG

ABAAgx

xgguAA

,max,

subject to

AAA wgx

First order condition

A

A

A

A

x

u

g

G

G

u

1

A

AAA

x

u

G

uMRS

Given that Household A purchases (produces) public goods…

12

xA wA

G=gA

wB = xB

Public goods

Public goods

Private goods

Private goods

A’s indifferent curve

B’s indifferent curve

A’s budget line (slope = -1)

B’s budget line (slope = -1)

• Household B can consume public goods at G = gA even if he does not pay at all.

• Therefore, the optimum strategy of Household B may be that he does not purchase public goods at all by himself.

Household A Household B

Free rider problem

A free rider is the one who enjoys benefits of goods without paying for them.

Example

– Surface road

– Dam

– …

13

Government’s coordination of supply of public goods can be

justified, if those public goods are under-supplied in market.