Pub Econ Lecture 07 Public Goods

8/3/2019 Pub Econ Lecture 07 Public Goods

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Public Finance

Dr. Katie Sauer

Public Goods


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Ben and Jerry are deciding their consumption of cookies

and ice cream.

- let cookies be the numeraire

- look at choice of ice cream

Each will have individual demand for ice cream.

- as price falls, Qd rises- different Qd for each individual


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Market Demand is found by horizontally summing the

individual demand curves.

At $2, what is market demand?

2 + 1 = 3

Ben Jerry

The Market Outcome


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Price

Q cones3

$2

DB

+ DJ = D

= SMB

The market demand is the SMB

of ice cream consumption.

The market supply is the SMC of

ice creamproduction.S = SMC

The market outcome is

found where SMB = SMC.

Consumers demand

different quantities at the

sameprice.


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Optimal ConsumerChoice using the tangency condition

A consumers optimal choice is found at the tangency of- budget constraint

- indifference curve

That is, when:MRS = slope of budget constraint

MRS = - MUx BC slope = - px

MUy py

MRS is the rate you are willing to give up good y to get

one additional unit of good x.


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Ben:

MUBic = MRSB

ic,c = PicMUBc Pc

Jerry:

MUJic = MRSJic,c = Pic

MUJc Pc

In market equilibrium:

MRSBic,c = MRSJic,c = Pic

Pc


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An individual will adjust consumption until theirMRS

equals the market price ratio.

If the price of an ice cream cone is $2 and theprice of

cookies is $1, then the price ratio is 2.

This means:

In equilibrium, each individual is indifferent

between trading 2 cookies to get one ice cream

cone.

Individual marginal utility levels for each good can still

differ.


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Suppose Ben is consuming 1 cone and 1 cookie and his

marginal utilities are as follows:

MUic = 7

MUc 3

MRSic, c = 2.33

2.33 > 2

Ben would consume another ice cream cone.

- reduceMU ice cream


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Suppose Jerry is consuming 2 cones and 1 cookie andhis marginal utilities are as follows:

MUic = 3

MUc 2

MRSic, c = 1.5

1.5 < 2

Jerry would consume fewer ice cream cones.- raise MU ice cream


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Market Supply

On the supply side, equilibrium in competitive markets

requires that

Pic =MC

ic

Since we know that in equilibrium on the demand side

Pic must equalMRSic,c, we have:

MCic = MRSic,c


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The private market outcome is also the social-efficiency-

maximizing outcome.

With no market failures,

MRS at any quantity = SMB of that quantity

marginal value to individual = marginal value to society

MC at any quantity = SMC of that quantity

marginal cost to producers = marginal cost to society

SMB = SMC


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Optimal Provision of Public Goods

Suppose Ben and Jerry are choosing between cookies

(numeraire) and missiles.

**individuals cannot choose their own specificconsumption of a public good**

- whatever amount isprovided is consumed

equally by all

The choice therefore is over willingness to pay for a

given quantity.


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Ben

Jerry

For the 1st missile, Ben would be willing to pay $2

and Jerry would be willing to pay $4.

- total willingness to pay is $6


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Market Demand is found by vertically summing the

individual demand curves.

- market demand is SMB as well

Price

Q missiles1

$6

DB,J

= SMB

S = SMC

Supply =MC = SMC

Optimal level:SMB = SMC

5

$3


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Mathematically:

The SMB is the sum of each individuals valuation:

MRSBm,c + MRSJm,c = SMB

The SMC is the marginal cost ofproduction.

Social-efficiency-maximizing level:

MRSBm,c + MRSJm,c = MC


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Example: Suppose Ben and Jerry live far from others. It

is the 4th of July and they want to celebrate. They care

about the consumption of ice cream and fireworks.

- ice cream is aprivate good $1 per unit- fireworks are apublic good $1 per unit

- ice cream is the numeraire

Suppose they have identicalpreferences.


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Each individual will face

MUF = 1

MUic 1

or MUF = MUic

to determine how many ice cream cones and fireworks

each will buy.


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The socially optimal level of fireworks is:

MRSBF,ic + MRSJF,ic = 1

MUBF + MUJF = 1

MUBic MUJic

sincepreferences are identical, rewrite as:

2MUF = 1

MUic

orMUF = (1/2) MUic


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UnderlyingMathematics

Ben and Jerry live by themselves and far away fromothers.

- each has income of $100

They each choose consumption over a private good, friedchicken and a public good, fireworks.

- each are priced at $1 per unit

Both Ben and Jerrys preferences can be expressed asU = 2ln(C) + ln(F)

where F = FB

+ FJ


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Ben will maximize U = 2ln(CB) + ln(FB + FJ)subject to (1)CB + (1)FB = 100

Can rewrite as a lagrangian:

Jerry will do the same.

)100()ln()ln(2 BBJBB FCFFCL ! P


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Private Provision of a Public Good

How will each behave, given that the other will alsoprovide some of the fireworks?

Nash Bargaining = solve for own optimal

strategy, given the others behavior


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0)(

1!

!

x

xP

JBB FFF

L

3

2100 JB

FF

!

Ben:

Bens reaction

function:

3

2100 BJ

FF

!Jerrys reaction

function:

02

!!x

xP

BB CC

L

0100 !!x

xBB

B

FCL

P


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Combine the two reaction functions:

3

321002100

!

B

B

F

F

And solve:

!

3

42001003 BB

FF

BB FF 42003009 !

20!BF

20!

JF


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Total fireworkprovision will beFB

+ FJ= 20 + 20

= 40

How does this compare to the socially optimal amount?

- need to sum MRS

- MRS is ratio of marginal utilities


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)(1

JBB

FFFF

UMU

!

x

x!

BB

CCC

UMU

2!

x

x!

Ben:

)(2

100

)(22

1

,

JB

B

JB

B

B

JBB

CFFF

F

FF

C

C

FFMRS

!

!

!

Jerry:)(2

100,

JB

JJ

CF

FF

FMRS

!

Socially Optimal Level

)(2

100,

JB

BB

CFFF

FMRS

!


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Socially optimal:

sumMR

S and set equal to price ratio

)(2

)(200

)(2

)100()100(,,

JB

JB

JB

JBJ

CF

B

CFFF

FF

FF

FFMRSMRS

!

!

1

1

1

2

200!!!

C

F

p

p

F

F

)(2)(200

FF!

FF 2200 !

67.66!F Socially optimal amount


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The private market will under-provide the fireworks.private market = 40

socially optimal = 66.67

Suppose the government recognizes that the private

sector underprovides fireworks by 26.67. Suppose it

mandates that Ben and Jerry must each contribute

$13.34 toward public fireworkprovision.


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Ben will now maximize U = 2ln(CB) + ln(FB + FJ+26.67)

subject to (1)CB + (1)FB = 100 13.34

Can rewrite as a lagrangian:

Jerry will do the same.

)66.86()67.26ln()ln(2 BBJBB FCFFCL ! P


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0)67.26(

1!

!

x

xP

JBB FFF

L

3

232.33 JB

FF

!

Ben:

Bens reaction

function:

3

232.33 BJ

FF

!Jerrys reaction

function:

02

!!x

xP

BB CC

L

066.86 !!x

xBB

BFC

L

P


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Combine the two reaction functions:

3

3232.33232.33

!

B

B

F

F

And solve:

!

3

464.6632.333 BB

FF

BB FF 464.6696.999 !

67.6!BF

67.6!

JF


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Total fireworkprovision will be

FB + FJ +26.67= 6.67+6.67+26.67

= 40.01 = 40

The governmentprovision of fireworks has crowded outprivateprovision.

- complete crowd-out

Ben and Jerry have returned to their optimal level ofproduction and the socially optimal level has not been

achieved.

Documents

Pub Econ Lecture 07 Public Goods