Public Economics

Chikako YamauchiGRIPS

Lecture 4

Public Goods

Outline

1. Definition of Public Goods2. Efficient Provision of Private

Goods3. Efficient Provision of Public Goods4. Numerical Example5. Problems in Achieving Efficiency

1. Definition of Public Goods

Introduction What is the difference between pizza and

national defense? two people cant consume a pizza simultaneously a person can be excluded from eating pizza

but consumption of national defense by one person does not diminish the consumption by another person

its difficult to exclude someone from consuming national defense

Definition of pure public good1. Consumption of a public good is nonrival

Once provided, the additional resource cost of another person consuming the good is zero.

2. Consumption of a public good is nonexcludable

preventing someone from consuming the good is either very expensive or impossible

Impure public goods: some characteristics of a public good

Examples Pure public goods

National defense Clean air House cleaning in an apartment with many roommates

Private goods Pizza Medical and health service Public housing

Impure public goods Lighthouse A scenic view

Pureness of public goods depend on state of technology

Lighthouse Nonrival: once the beacon is lit, one ship can take

advantage of it without impinging on another ships ability to do the same

Nonexcludable: no particular vessel can be excluded from taking advantage of the signal

Light house disturbed by new technology: a jamming device Excludable: unless you have obtain a special

receiver, you cannot receive the signal from lighthouse

Pureness of public goods depend on market condition & legal setting

A scenic view Pure public good when there are few people

(number of consumers is small) But increased usage (number of consumers is

larger) leads to congestion and decreased quality, so it becomes a rival good

If new technology and law allow charging entry fee, it will become an excludable good

Other examples like this include major roads and firework watching

People may value public goods differently

even though everyone consumes the same quantity of a public good, they do not necessarily value it equally Ex. Missile defense: some may have a

negative valuation if they think it makes the world more dangerous

Ex. Apartment cleaning: everyone benefits from a clean apartment, but some roommates may value a clean apartment more than others.

Abstract things can have public good characteristics

Examples honesty in business fairness in the income distribution a legal framework, law and order

An application to analysis of terrorism Where governments and economies function poorly,

sects often become major suppliers of social services, political action, and coercive force [Eli Berman, NBER working paper 10004, 2003; The Economist, Jan 15, 04]

Implication: improving law and order, transparency, fairness and social services is one way to combat religious militancy [Berman, Shapiro & Felter, Quarterly Journal of Economics, 2011 August]

Label of providers and nature of goods do not have to match

Private goods can be provided by the public sector Pizza: provided by private firms Medical care and housing: provided by either private

firms or the government

Public goods can be provided by the private sector garbage collection, fire protection, libraries, public

transportation government plans public good provision, but hires labor

and capital to actually produce the public good government can hire firms, which provide these public

goods

2. Efficient Provision of Private Goods

How do we derive aggregate demand for private goods? [Fig.4.1]

Each persons demand curve represents the willingness-to-pay for an additional unit of a good

For private good, holding P constant, add together individual quantities (Horizontal summation)

How do we derive aggregate demand for private goods? [Fig.4.1]

Each persons demand curve represents the willingness-to-pay for an additional unit of a good

Everyone pays the same price, 4, but individuals consume different quantities their taste, income, etc. are different. This is possible because fig leaves are private goods.

Equilibrium in private good marketEquilibrium where supply curve intersects aggregate demand curve

Sf

4

Pareto efficiency in private good market

Utility-maximizing individuals set MRSfa = Pf/Pa Since only relative prices matter for rational

choice, the price of apples can be arbitrarily set at any value. For simplicity, consider Pa=$1

Then, MRSfa = Pf Because demand curve shows the maximum

price consumers would pay (Pf) at each level of fig leave output, it also shows MRSfa at each level of fig leave output

Pareto efficiency in private good market

Profit-maximizing firms set MCk = Pk (k=f,a) And thus MRTfa = MCf / MCa = Pf / Pa Since Pa=$1, MCa=Pa=$1, and MRTfa = MCf = Pf Because supply curve shows the minimum price

(Pf) firms would sell at each level of fig leave output, it also shows MRTfa at each level of fig leave output

Combining the conditions derived for consumers and firms, at the equilibrium (where demand and supply curves cross), MRTfa = MRSfaAdam = MRSfaEve

This is the necessary conditions for Pareto efficiency

Thus, the equilibrium price and output are PE

Pareto efficiency in private good market

3. Efficient Provision of Public Goods

Efficient provision of public goods: Intuition

Consider a fireworks display as a public good it is nonrival and nonexcludable.

Suppose the fireworks display already has 19 rockets. For the 20th rocket:

Should we add a 20th rocket? We know MC. What is MB from adding another

rocket? ?

20 20 20$5 $6 $4Adam EveMC MB MB

Efficient provision of public goods: Intuition

MB is sum of willingness to pay because the additional rocket will be consumed by both (nonrival, nonexcludable)

MB from adding another rocket = $4+$6 = $10 This is higher than MC = $5

Yes, we should add another rocket More generally, public goods should be provided

until:iMB MC

Adam

Eve

Market

Adam is willing to pay $6 for the 20th rocket

Eve is willing to pay $4 for the 20th rocket

Their group willingness to pay is $10, so the group willingness to pay schedule in C, DA+Er, must be 10 at r = 20

Other points on DA+Er are determined by repeating this procedure for each output level [Vertical Summation]

The efficient quantity of rockets is found where the sum of A & Es willingness to pay for an additional unit just equals the marginal cost of producing a unit. It occurs at r=45, p=$6

Efficient provision of public goods: graphical illustration

Efficiency in public goods market

Again, Adam and Eves marginal willingness to pay can be interpreted as their MRS

Thus, the sum of the prices they are willing to pay is the sum of their MRS: MRSraAdam + MRSraEve

From the production standpoint, price still represents MRTra Thus, the equilibrium is characterized by:

MRSraAdam + MRSraEve = MRTra

the sum of individual marginal benefits equals the marginal cost

4. Numerical Example

Numerical example Consider 2 individuals, Adam and Eve who

have the following inverse demand curves, and face a marginal cost curve below.

How do we derive the equilibrium if the good is a private good?

How about if the good is a public good?

P QA A 100 12

P QE E 200MC Q 2

3

Numerical Example: private good

If the good was a private good, then the aggregate demand curve is:

With a private good, everyone pays the same price.

200 2 200A E A EQ Q Q P P

200 2 200 400 3400 13 3

A E

A E

P P PQ P P P

P Q

Aggregate demand for the

private good

Numerical Example: private good

In a competitive market, P=MC

Approximately 133 units of the private goodare provided at a price of $88. Adam consumes around 22 units and Eve consumes around 111 units.

400 23 3 3

400 800133 P 883 9

QP MC Q

Q

Numerical Example: public good

If the good is a public good, the aggregate willingness to pay is:

With a public good, everyone consumes the same quantity.

1100 2002A E A E

P P P Q Q

33002

A EQ Q Q

P Q

Aggregate willingness to pay

for the public good

Numerical Example, public good Efficient provision would require: P=MC

Efficient provision would imply that Adam & Eve consume 138.46 units of the public good.

Private market may not arrive at this allocation The equilibrium under the assumption that the good

is a private good: Q=133, P=88

3 23002 3

138.46 P 92.3

P MC Q Q

Q

5. Problems in Achieving Efficiency

Does the 1st theorem apply to public goods?

Competitive market will provide private goods efficiently

Does competition lead to efficient provision of public goods as well?

People may have incentives to hide their true preferences, or willingness to pay for a public good ..because if Adam can get Eve to pay for the

public good, he can use his income for other purposes and still enjoy the public good.

Free Rider Problem This incentive to let others pay for the public

good while still enjoying the benefits is known as the free rider problem.

If everyone understates his/her willingness to pay, the private market will fall short of providing the efficient amount of the public good.

This incentive to free ride occurs because the public good is nonrival and nonexcludable: A person gets to consume the good even if he

does not pay for it.

Free Rider Problem Return to the public goods numerical example.

Suppose Adam chooses to free ride, and Eve therefore provides her optimal amount.

Eve chooses:

After Eve contributes 120 units of the public good, Adam does not provide any additional contributions, because the marginal benefit to Adam of the 120th unit is less than the marginal cost.

120E EP MC Q Q

Solutions to the free rider problem

Governmental intervention If government can

find everyones true preferences force people to pay for public goods using coercive power

(through taxation) Then government can prevent the free rider problem

Private collective action There are instances when free riding does not occur,

and individuals act collectively without coercion Volunteer fund raising for libraries, music halls, etc.

The importance of the free rider problem is an empirical question

Experiment to test the importance of the free rider problem

Laboratory experiments on college students A class with 10 students Each student is given 10 tokens Each student can decide how many tokens to keep

and how many to contribute to the group exchange For any tokens the student keeps, that student

receives 500 yen per token. For any tokens the student contributes, every student

in the class receives 300 yen each.

Ones contribution is thus nonexcludable and nonrival a pure public good

Public goods experiment: theory

If all ten students contribute all ten tokens, each receives 10 tokens * 10 students * 300 yen = 30,000 yen Total economic pie = 30,000 yen * 10 students = 300,000 yen

If students maximizes their personal utility You cannot control others behavior, so it is the best to keep all

of your tokens, regardless of what other people do. If no one contributes, then each student receives: 10 tokens * 500 yen = 5,000 yen Total economic pie = 10 students * 5,000 yen = 50,000 yen

Individual rational choice implies the free-rider problem and under-provision of public goods

Public goods experiment: results

On average, people contribute 50% Some free riding exists, but it is not the case that free

riding leads to zero or trivial amounts of public goods Other findings

Contribution reduces when the game is repeated and larger reward from keeping tokens is used

Contribution increases with prior communication A caution

Results may not purely reflect preferences for contribution for public goods, but also from a warm glow feeling of satisfaction from giving