Public Goods Public Economics, June 27, 2014 Muneta Yokomatsu 1
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
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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
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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
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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
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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…
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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
– …
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Government’s coordination of supply of public goods can be
justified, if those public goods are under-supplied in market.