Eep101/econ 125 lecture 9Public Goods

David Zilberman

Overview• Heterogeneity, Non-rivalry, and Market

Failure

• Non-excludability and Market Failure

• Optimal Provision with Homogeneous

Individuals

• Private Market Outcome for a Non-

excludable Public Goods

• Mechanisms for Providing the Socially

Optimal Level of Public Goods

• The Specification of Congestion Costs in

Public Goods Models

Public goods are goods or services that

can be consumed by several individuals

simultaneously

Non-rivalry-Multiple individuals can

consume the same good without diminishing

its value.

Non-excludability, an individual cannot

be prevented from consuming the good .

Examples, fresh air, a public park, a

beautiful view, national defense .

Original demand-1P

Q

Demand of one individualAnd the supply

Q1

D=a-bQMC=c-dQ

Q1 =a−cb+d

D1

Horizontal demand -2P

Q

Demand of one individualAnd the supply

Q1

D2H=a-Qb/2MC=c-dQ

Q2H =a−c

.5b+d

D1

Q2

Horizontal demand -large NP

Q

Demand of one individualAnd the supply

Q1

DINH=a-Qb/NMC=c-dQ

QNH =a−c

1 / Nb+d

DH1

DHN

DH2

Q2 QN

Vertical aggregation-2P

Q

Demand of two individualsAnd the supply

Q1

D2=2a-2bQMC=c-dQ

Q1 =2a−c2b+d

D1

D2

Q2

Vertical aggregation-3P

Q

Demand of three individualsAnd the supply

Q1

D3=3a-3bQMC=c-dQ

Q1 =3a−c3b+d

D1

D2

Q2

D3

Q3

Vertical vs horizontal aggregation

Horizontal demand lead to infinitely elastic demand at P=a. Vertical leads to infinitely in-ealstic at Q=a/b

Vertical aggregation is used to find demand for public goods

As the population increases the demand becomes more inelastic.

The demand becomes Dn=na-nb The equilibrium quantity is When there are infinite individuals the quantity

demanded is equal to b/a and the demand is inelastic

Qn=na−cnb−d

Provision of public goods Free riding- most people will wait that others will

pay for the public and then they will free ride One justification for government intervention is

provision of public goods Bigger communities will have larger public good

provision

Dn

D2n

Qn Q2no

v1v2 s2s1

The cost to community with n peoplev1s1QnoThe cost to community with2n peoplev2s2Q2no

Provision mechanism Public provision by taxes Donations, fund raising( nature conservancy) Volunteer activity( duck unlimited,Siera club) Tax deduction-for contribution to public good

generating organizations Advertisement and naming (TV,concert hall) Special institutions-

The church Army Art patron (giving back to the community)

Non rivalry with excludability

If access to a good with non rivalry is possible it can be provided privately

First by a monopoly- the entry fee aoc will capture all the surplus but outcome is efficient

MC

Dn

D1

Qn

a

oc

Alternative mechanism -gov’t provision fee cover costs Fee in the area OMQn/N Consumer surplus SMO

MC

Dn

D1

Qn

a

Oc

M

S

A third mechanism-concessionaire earns competitive profit Fee in the area RMQnO/N Consumer surplus SMR Producers surplus RMO

MC

Dn

D1

Qn

a

Oc

M

S

R

Column3 Column4a=demand intercept

b=demadn slope

c =Mc intercewpt

D=mc slope Column1

10 1 4 2

Number

of quanitymonopoly price

cost covering price

concessionaire price

social surplus

1.00 2.00 18.00 12.00 16.00 6.00

2.00 4.00 32.00 16.00 24.00 32.00

3.00 5.20 38.48 15.95 24.96 67.60

4.00 6.00 42.00 15.00 24.00 108.00

5.00 6.57 44.12 13.89 22.53 151.14

6.00 7.00 45.50 12.83 21.00 196.00

7.00 7.33 46.44 11.87 19.56 242.00

8 7.60 47.12 11.02 18.24 288.80

Simulation a good with non rivalry and excludability

Heterogeneity of preferences

Dless

Dmore

MC1

MC2

G

A and C optimal outcomes when only buyers that like the product more intensively are purchasing tickets. The price of ticket under monopoly are AOQcC and AOQGGC

A

QcOQG

Heterogeneity of preferences

Dless

Dmore

DtotalMC1

MC2

G

B

G and B optimal outcomes

In case of public good if marginal cost are high the benefits of the less interested group are important for determining the optimal outcome (Point G)

C

A

QcO Qb

Heterogeneity of preferences

Dless

Dmore

DtotalMC1

MC2

G

B

In case of excludabilityMonopoly will provide optimal outcome at G but will charge AGLO as entry fee so that half the population will be excluded

IN case of MC1 the monopoly will offer Qc

C

A

QcO Qb L

Example Two equal sized groups

Dmore =20-X Dless -10-X Joint demand

P =30-2X for X<10 =20-X for 20>X>10

Case 1:MC=.5X

• What is Optimal X* • Try 30-2X=.5X (point A)• X=12 is not feasible• Try 20-X (point B)• =.5X socially• optimal X=20/1.5=13.33• Fee by monopoly• 26.66*13.33/2=311 A B

Case 2 :MC=2X To find optimal X solve 30-2X=2X Optimal X=7.5 Benfits for people who like the product (20+20-7.5)*7.5/2=121.75 People who like less 12.5*7.5/2=46.675 Monopoly ignores second group. solves 20-X=X Optimal policy monopoly for the rich X=6.666 Fee $111 Alternative 1 regulated concessionaire providing

7.5 units for 46.675-income 93.35 Cost 56.25 Alternative 2 pay equals to cost divided by buyers

Excludability with heterogeneity

Differentiated provision- people pay different fees for different products

Exclusive vs public beach Hunting licenses Different housing accommodations in

national parks Allow to raise fund and pay and address

equity considerations

public good when part of the public does not care

2 groups-one does not care for D1=10-X D2=0 MC=X Should government provide public good? In this case the groups that like the product

may provide it through collective action-clubs Optimal X=5 Costs 25/2= 12.5/N will be paid by users

Clubs:Optimal size Benefits depend on amenity size X and

number of users N B(N,X) Cost increases with X Optimality problem Optimal decision rules

MAX

X,NNB(N .X) −c(X)

N∂B(N.X)∂X

−∂c(X)∂X

=0

B(N.X) +N∂B(N.X)

∂N=0

NMB=MCMarginal benefits of size=Marginal cost of congestion

Case 2 MC=2X30-2X=2XOptimal X=7.5Benfits for people who like the product(20+20-7.5)*7.5/2=178.125Peole who like less12.5*7.5/2=46.675Optimal policy monopoly for the rich X=6.666

Case 3Dmore 30-XMC=4XOptimal policy in X=8Monopoly price to more52*8/2=208Monopoly price to less12*8/2=48In this case the monopolywill solve30-X=4XX=6 and charge 54*6/2=162Which means under provisionOf public good