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Public GoodsPublic Goods
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EC 221 Intermediate Microeconomics Saree Worawisutsarakul
Public Goods 1
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OverviewOverview
Public good: A good that must be provided in
the same amount to all theaffected consumers
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OverviewOverview
Examples:
Facilities provided bythe government
National defense
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Pareto Improvement toPareto Improvement toProvide the Public GoodProvide the Public Good
Consider
Two persons: Person 1, Person 2They decide whether or not topurchase a TV.
TV is a public good.
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Budget constraintsBudget constraints
Let
Each personsinitial wealth
Each personscontribution to the TV
Each personsprivate consumption
21,ww
21, gg
21,xx
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Budget constraints forpersons 1 and 2:
111 wgx =+
222 wgx =+
Budget constraintsBudget constraints
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where
They can acquire the TV iftogether they pay the cost .
cgg + 21
Budget constraintsBudget constraints
c costs of the TV
c
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Utility functionUtility function
Each persons utility function willdepend on:
his private consumption
the availability of the publicgood
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Utility functionUtility function
Utility functions for persons
1 and 2:
where No TV
TV is available
( )Gxu ,11( )Gxu ,22
=1
0G
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Reservation priceReservation price
Reservation price: The maximum amount of money
that each person is willing topay to have the TV available
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If each person
pays and gets TV
does not get TV
iii rwx =
ii wx =
Reservation priceReservation price
ir
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Each person is indifferent between
{paying + TV available}and{not having TV}
if
Reservation priceReservation price
ir
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These equations define foreach person.
( ) ( )0,1, 11111 wurwu =
( ) ( )0,1, 22222 wurwu =
Reservation priceReservation price
ir
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Two possible allocations:1) An allocation where the public
good (TV) is not provided:
Each person spends his wealthonly on his private consumption.
( )0,, 21 ww
Pareto improvementPareto improvement
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Pareto improvementPareto improvement
2) An allocation where the publicgood (TV) is provided:
where( )1,, 21 xx
111 gwx =222 gwx =
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Pareto improvementPareto improvement
Each persons privateconsumption is determined byhis wealth left after makingcontribution to the public good.
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Condition forCondition forPareto improvementPareto improvement
It will be a Pareto improvement toprovide the TV if
both persons are better offhaving the TV provided than nothaving it provided:
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(2)
(1)( ) ( )1,0, 1111 xuwu + 21
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NoteNote
2) The sufficient condition will
depend on because
depends on .( )21, rr( )
21
,ww
( )21,ww
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NoteNote
It is possible that:
for some Public good is provided.
for other
Public good is not provided.
( )21,wwcrr >+ 21
( )21,wwcrr
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NoteNote
In general
Whether or not the public goodshould be provided will depend on:
the distribution of wealth
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Having the public good providedwill be a Pareto improvement if
crr >+ 21
Private ProvisionPrivate Provision
of the Public Goodof the Public Good
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For the public good:
Each person may not have anincentive to reveal his truewillingness to pay.
Each person has an incentive topay as little as possible towardthe provision of the public good.
Private ProvisionPrivate Provisionof the Public Goodof the Public Good
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Private ProvisionPrivate Provisionof the Public Goodof the Public Good
People are attempting to freeride on each other.
Each person hopes that theother person will purchasethe public good.
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Free RidingFree Riding
Free riding game
Two players:
Two strategies:
Player B
Dont buy TV
Player A
Buy TV
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Free RidingFree Riding
Let
Each persons wealth = 500
Each persons valuefor the TV = 100
Cost of the TV = 150
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Free RidingFree Riding
Suppose: Each person cannot exclude the
other person from watching theTV.
Each person will decideindependently whether or not tobuy the TV.
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Payoff matrix
0,0100, 50Dont buy
50,10050,50Buy
Dont buyBuyBA
Free RidingFree Riding
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Free RidingFree Riding
Dominant strategy equilibrium:
(Dont buy, Dont buy)
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Free RidingFree Riding
Difference: Prisoners dilemma game The strategy that maximizes
the sum of the playerspayoffs is for both to makethe same choice.
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Free RidingFree Riding
Free riding game
The strategy that maximizesthe sum of the playerspayoffs is for only one to buythe TV.
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Free RidingFree Riding
Pareto improvement:
If A buys the TV and B makes apayment between 50 and 100,
both will be made better off.
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Pareto Efficient LevelPareto Efficient Level
of the Public Goodof the Public GoodConsider: Two persons have to decide howhow
muchmuch of the public good toprovide.
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Pareto efficiency conditionPareto efficiency condition
Let
Each personsprivate consumption
Each persons
contribution tothe public good
21,xx
21, gg
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Quality ofthe public good
Cost for the publicgood of quality G
G
( )Gc
Pareto efficiency conditionPareto efficiency condition
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Pareto efficiency conditionPareto efficiency condition
Constraint:
( ) 2121 wwCcxx +=++( ) ( ) 212211 wwgxgx +=+++
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Pareto efficiency conditionPareto efficiency condition
The total amount of the theirprivate consumption and thepublic good is equal to the totalamount of their wealth.
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Pareto efficiency conditionPareto efficiency condition
Pareto efficient allocation: An allocation where one person
is as well-off as possible giventhe other persons utility level
Lets fix Person 2s utility at .2u
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Pareto efficiency conditionPareto efficiency condition
Person 1s problem:
( ) 222 ,s.t. uGxu =( ) 2121 wwGcxx +=++
( )GxuGxx ,max 11,, 21
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Pareto efficiency conditionPareto efficiency condition
Condition for Pareto
efficiency:( )GMCMRSMRS =+ 21
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Pareto efficiency conditionPareto efficiency condition
The sum of the two personsbetween the private good
and the public good equals themarginal cost of providing an
extra unit of the public good.
MRS
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Interpretation forInterpretation for
Pareto efficiency conditionPareto efficiency conditionConsider:
marginal willingnessto pay for an extra unitof the public good
MRS
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Interpretation forInterpretation forPareto efficiency conditionPareto efficiency condition
Discrete public good
If The public good is provided.
( )GMCMRSMRS >+ 21
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Interpretation forInterpretation forPareto efficiency conditionPareto efficiency condition
Different levels ofpublic good
If
It is appropriate to providemore of the public good.
( )GMCMRSMRS >+ 21
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Efficient amountEfficient amount
of the public goodof the public goodSum and vertically.
The ( ) curve
Condition for Pareto efficiency:
1MRS 2MRS
21 MRSMRS +
MCMRSMRS =+ 21
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Efficient amountEfficient amount
of the public goodof the public good Efficient amount of the
public good
The two persons must consumethe same amount of the public
good.
G
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Efficient amount of theEfficient amount of thepublic goodpublic good (Varian: Figure 36.1)(Varian: Figure 36.1)
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LetEach persons wealth
Each personsprivate consumption
Each personscontribution tothe public good
21,ww
21,xx
21, gg
Free Riding ProblemFree Riding Problem
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Free Riding ProblemFree Riding Problem
Cost of providing the public good:
( ) GGc =
( ) 1==dG
dcGMC
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Free Riding ProblemFree Riding Problem
Total amount of the public goodprovided:
Person s utility function:21 ggG +=
( )Gxu ii ,i
21 gg +=
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Free Riding ProblemFree Riding Problem
Person 1s problem:
Person 1 has to choose and ,guessing that
111s.t. wgx =+
( )2111,
,max11
ggxugx
+
1x 1g
22 gg =
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Free Riding ProblemFree Riding Problem
Person 2s problem:
Person 2 has to choose and ,guessing that
222s.t. wgx =+
( )2122,
,max22
ggxugx
+
2x 2g
11 gg =
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Free Riding ProblemFree Riding Problem
Optimal condition:
If each person purchases bothgoods, then the betweenthe public good and the privategood should be 1 for each person.
11 =MRS
12 =MRS
MRS
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Free Riding ProblemFree Riding Problem
There is one extra constraintextra constraint oneach persons contribution:
Each person can only increaseincrease
contribution, or make nonocontribution at all.
0,0 21 gg
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Free ridingFree riding
Example:
Person 2 is free riding Person 1scontribution to the public good.
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Free ridingFree riding
Denote:
Person s endowmentjii gwe +=
ie
iji
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Person 1Person 1
Person 1s contribution:
Person 1s optimal choice is where:
His indifference curve istangent to his budget line.
11 =MRS
Gg =1
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Person 1Person 1s optimal choices optimal choice
(Varian: Figure 36.2 A)(Varian: Figure 36.2 A)
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Person 2Person 2
Person 2s endowment:
Since Person 2 can only increasethe public good:
His budget line is the bold line.
Gwe += 22
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Person 2Person 2
Person 2s optimal choice is to
free ride on Person 1scontribution;
consume at his endowment.
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Person 2Person 2s optimal choices optimal choice(Varian: Figure 36.2 B)(Varian: Figure 36.2 B)
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Free ridingFree riding
In general
Equilibriumamount of thepublic good
Efficientamount of thepublic good