public good 4 slides

7/25/2019 Public Good 4 Slides

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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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Public Goods 2

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Budget constraints forpersons 1 and 2:

111 wgx =+

222 wgx =+


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where

They can acquire the TV iftogether they pay the cost .

cgg + 21


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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Public Goods 3

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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 =


ir

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Each person is indifferent between

{paying + TV available}and{not having TV}

if


ir

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These equations define foreach person.

( ) ( )0,1, 11111 wurwu =

( ) ( )0,1, 22222 wurwu =


ir


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Public Goods 4

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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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2) An allocation where the publicgood (TV) is provided:

where( )1,, 21 xx

111 gwx =222 gwx =

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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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Public Goods 5

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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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Public Goods 8

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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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Public Goods 9

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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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Let

Each persons wealth = 500

Each persons valuefor the TV = 100

Cost of the TV = 150

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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




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Public Goods 10

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Dominant strategy equilibrium:

(Dont buy, Dont buy)

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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 riding game

The strategy that maximizesthe sum of the playerspayoffs is for only one to buythe TV.

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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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Public Goods 11

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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 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


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Constraint:

( ) 2121 wwCcxx +=++( ) ( ) 212211 wwgxgx +=+++



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Public Goods 12

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The total amount of the theirprivate consumption and thepublic good is equal to the totalamount of their wealth.

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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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Person 1s problem:

( ) 222 ,s.t. uGxu =( ) 2121 wwGcxx +=++

( )GxuGxx ,max 11,, 21

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Condition for Pareto

efficiency:( )GMCMRSMRS =+ 21



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Public Goods 13

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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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Public Goods 14

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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 personscontribution tothe public good

21,ww

21,xx

21, gg

Free Riding ProblemFree Riding Problem



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Public Goods 15

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Cost of providing the public good:

( ) GGc =

( ) 1==dG

dcGMC

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Total amount of the public goodprovided:

Person s utility function:21 ggG +=

( )Gxu ii ,i

21 gg +=

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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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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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Public Goods 16

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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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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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Denote:

Person s endowmentjii gwe +=

ie

iji



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Public Goods 17

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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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Public Goods 18

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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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In general

Equilibriumamount of thepublic good

Efficientamount of thepublic good

public good 4 slides

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