lecture 4 public goods

7/29/2019 Lecture 4 Public Goods

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Public goodsLecture 4 Tom Holden

Intermediate Microeconomics Semester 2

http://micro2.tholden.org/

ECO 2051 Intermediate Microeconomics 1
http://micro2.tholden.org/http://micro2.tholden.org/


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Externalities exercise from lecture 3.2 (1/3)

It takes 45 minutes to go around town on an uncongested road.

The road through town can get congested. It then takes+

minutes

to travel the direct route.

Assume 6000 commuters need to travel from one side of town to theother.

With no tolls how many will travel through town? The two journeys must take the same time for people to be prepared to

travel along either road. I.e. 45 =+

, so 4480 go the direct route, and

1520 people go around.

How many minutes will be spent travelling? 45 6000 = 270000 minutes, or 4500 hours.



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Suppose a social planner determines the allocation of routes.

How many people will go a) through town b) around town?

Social planner minimises total travelling time, which is

2 0

100 6000 45

FOC: 0 =

45, so 2240 people go through town and 3760 people

go around.

How many person-minutes would be spent travelling?

2240+

3760 45 = 219824 minutes.



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If each commuter values their time at x per minute what levelshould a toll on the road through town be set at to minimisecommuting time?

Equating the costs of the two roads we have that in equilibrium: 45 =+

where is the toll. We want = 2240 for efficiency, hence =

.

Are commuters better off? By how much? Before a consumer would always pay 45 (effectively), since they were

indifferent between the two roads. But they are still indifferent between thetwo roads, and the one around still costs 45, so they are no better off ifthe planner keeps the toll.

But the planner raises 50176 pounds in revenue, which can beredistributed, making everyone better off by the equivalent of about 8.36minutes.



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Motivating questions

What are public goods?

How do they relate to externalities?

When is a public good provided optimally?

How can we find out peoples preferences for public good provision?



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Reading

Varian, Chapter 36

Morgan Katz and Rosen, Chapter 18

Less good on this topic.



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Introduction

With externalities we saw how two parties could trade to a Paretooptimal solution.

However when there is more than one party experiencing theexternality it is necessary for agreement to be reached on the valueof the externality.

This is particularly true in the case of non-depletable externalities,

when all parties experience the externality equally.

In this case the externality problem becomes a public good problem.



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The definition of Public Goods

A pure public good has two features. It is:

Non-rival in consumption

All individuals who consume the good, consume the same amount althoughthey may value it differently.

In this case, how much should be provided?

Easy once you know peoples valuations, but how do we find them out?

Non-excludable

If I buy a public good, I cannot stop you from consuming it as well.

This generates a free-rider problem, individuals do not need to pay, therefore they donot reveal how much they value the good.



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Pareto efficient provision:to provide or not to provide (1/4)

Two flatmates called 1 and 2 are deciding whether to buy a TV forthe living room.

Their initial wealths are and respectively.

Their contribution to the TV are and respectively.

Their private consumptions are and respectively. Thus = and = .

They both consume units of the public good, the TV (either = 0 or =1).

They have utility functions given by , and , respectively.

The cost of the television is , so the television is provided if > .



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We can define and to be the most that the respective flatmatesare prepared to contribute in order to obtain the television.

These are reservation prices.

When = , the first flatmate must be just indifferent between having theTV and paying and not having the TV and paying nothing.

I.e. , 1 = , 0 .

Likewise, when = : , 1 = , 0 .



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Providing the TV is a weak Pareto improvement if:

, 1 , 0 and

, 1 , 0 .

But from the definition of the reservation wage this means providingthe TV is a weak Pareto improvement if:

, 1 , 1 and

, 1 , 1 .

So as utility is increasing in both arguments:

, i.e. and

, i.e. .



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So a necessary condition for providing the TV being efficient is thateach individual is contributing less than his reservation price.

A sufficient condition for it to be efficient to provide the TV is that , since if this holds we can find and suchthat = , and under these payments each individual must beat least as well off as without the TV.



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Free riding

Suppose each person has a wealth of 500 and values the TV at 100.

Suppose also that the TV costs 150.

It is Pareto optimal to buy the television.

but each individual has an incentive to free ride.

Look at the game below.

Does either player have a dominant strategy?


Individual A

Buy Dont Buy

Individual B Buy 25, 25 -50, 100

Dont Buy 100, -50 0, 0


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Public goods: Continuous choice case (1/2)

Everything is as before except the decision is on how much money tospend on the TV.

The cost of the television is , where is its quality.

When utility functions are increasing, the following condition issufficient for Pareto efficiency:

Given consumer 2s level of utility, consumer 1 is as well-off as possible.

Why do we not need an equivalent condition with the roles of the two individualsreversed?

Thus we just need to choose , and , to maximise , subject tothe constraints that:

, = ( is some fixed utility level), and:

= (the budget constraint).

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Public goods: Continuous choice case

Choose , and , to maximise , subject to the constraints that

, = and = .

Lagrangian:

= , ,

FOCs:

FOC : 0 =

, hence =

.

FOC : 0 =

, so =

.

FOC : 0 =

, thus:

=

.

Hence:

= .

I.e. MRS MRS = MC

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Interpretation of the solution (1/2)

MC is the opportunity cost of spending money on the public good

MRS measures the willingness to pay for an extra unit of the publicgood.

We can think of this as the goods marginal benefit (MB).

Thus the solution can be written as MB MB = MC.

So, the optimality condition is that the marginal social benefit isequal to the marginal social cost.

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Interpretation of the solution (2/2)Diagram

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MRS

0

MRS

MRS

MRS MRS

MC Efficient provision of a

public good: The sum of

the MRS must equal the

marginal cost.

Private good: consumers

would set their own MB

equal to the marginal cost,

i.e. MRS = MRS =

MC.


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The intuition for optimality

Approach: look at situations where this condition is not met andshow that they can be improved upon.

Say MC = 1, MRS =

and MRS =

.

MRS is how much of the private good would be required tocompensate for 1 less unit of the public good.

If the public good is reduced by 1 unit this will cost 1, it takes only to compensate both individuals for the loss.

So its optimal to reduce the amount of the public good thats provided.

Alternatively, if MRS =

and MRS =

then the totalmarginal benefit of an extra unit of public good is worth more than itcosts.

So its optimal to increase the amount of the public good thats provided.



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Examples (1/2)

A town has one private good (bottles of beer) and one public good (size of the town skating rink in square metres).

Beer costs 1, and the skating rink costs 5 per square metre.

There are 500 citizens all with an income of at least 5000.

All citizens have the same quasi-linear utility function.

, =

.

How do we find the Pareto optimal size of the skating rink?

(Should get = 80.)



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Examples (2/2)

Suppose in the fishery/factory example from lecture 3.2, there weretwo fisheries.

As before, the steel firm has total cost

, .

The first fishery one has total cost , .

The second fishery has total cost , .



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Free riding in the continuum case

Assume = 1.

Given individual 2 contributes some fixed level units of public good,

individual 1 will maximize , subject to the constraint

that 0.

With an interior solution, the FOC says: 0 =

, i.e. MRS = 1.

Likewise, MRS = 1.

So MRS MRS > MC = 1.

I.e. the public good is under supplied.

When the constraint binds,

< 0, so MRS < 1, but we must

still have MRS = 1, so there is still too little of the public good unless

MRS = 0, which only happens when

= 0.



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Solutions: Public intervention

The government may subsidise private firms to provide the good.

The government may just provide the good itself.

Has to decide how much (if any) of the good ought to be provided.

Raises the question of preference revelation.

Strong incentive to lie on a survey.

E.g. pretend not to care about the good, in order to escape paying for it.

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Voting on public goods (1/3)

Suppose that a majority-rule vote is held on whether to install trafficsignals at several street corners.

Assume:

3 individuals.

Each signal costs 300 to install, which is split equally between the threeindividuals.

Individuals will thus only vote for a signal if they think the signal is worth atleast 100.

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Will the signal be installed?

For which corners is the outcome efficient?

Problem with yes-no votes: the vote indicates only whether the public goodis worth more or less than a certain amount.

But in fact the intensity of preferences matters too.

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Value to Each Voter,

Signal Location Hayley Nancy Chris Value to

Society,

Outcome of

Vote

Corner A 50 100 150 300 YES

Corner B 50 75 250 375 NO

Corner C 50 100 110 260 YES


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We saw in lecture 3.1 that voting may lead to intransitive socialpreferences.

Imagine there are three voters A, B and C, and three levels of publicgood, X, Y, and Z.

In the aggregate, it appears X is preferred to Y, Y is preferred to Z andZ is preferred to X. Standard Condorcet paradox.

But if X, Y and Z are different levels of public good provision, is itreally plausible that people would have all of these preferences?

A B C

1st preference X Y Z

2nd preference Y Z X

3rd preference Z X Y


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Single-peaked preferences

Collective intransitivity is avoided if all preferences are singlepeaked.

This rules out all or nothing preferences such as those held byperson C (assuming X, Y and Z are ordered as we would expect).



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With single-peaked preferences

Suppose preferences are instead as given below:

Voter Cs preferences are now single-peaked.

In this case Y>Z, Z>X and Y>X, no issues of collective rationality.


A B C

1st preference X Y Z

2

nd

preference Y Z Y3rd preference Z X X


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Do single-peaked preferences really help?

Honest voting on expenditure with single-peaked preferences leadsto choosing the amount preferred by the median-voter.

Imagine there was a pro-spending and an anti-spending political party, andthink about where the parties would like the indifferent voter to be.

That half the population want more expenditure and half want lessdoes not mean it is an efficient outcome.

Maybe the half that want more all want a large amount more, whereas theother lot only want a little less.

And in any case, there is no reason people should vote honestly.

With most voting mechanisms individuals have an incentive to misrepresenttheir preferences.



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Demand revelation mechanisms

There are problems with voting as we already know (Arrow etc.).

Just asking people their valuation will encourage them:

To overvalue if their payment doesnt vary in their reported valuation, and,

To undervalue if their payment does vary in their reported valuation, andthey believe the good will be provided anyway.

The Vickrey-Clarke-Groves mechanism is a general solution to thisproblem, at least when individual preferences are quasi-linear.



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The Vickrey-Clarke-Groves mechanism (1/3)

Suppose we want to choose between a suite of different alternatives, , , .

For example, these could be different levels of public good provision.

Or they could be allocations of students to rooms.

Or they could be allocations of a single object to one person out of (as inan auction).

Assume that person values outcome at .



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Step 1:

Ask all individuals to report their valuations of each alternative (i.e. , , , ).

Step 2: Find the alternative , , that maximises

.

Step 3:

Charge individual a fee of: max ,,

> 0.

AKA: Clarke tax

If an individual has no effect on (they are not pivotal) this will be zero.



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Why should people report truthfully?

Suppose agent reports as their valuation function.

Then the chosen will maximise: .

And agent s value from participating is:

max

,, .

They cannot affect the final term, so they would like to choose their report

such that the thing the social planner maximises corresponds with thefirst two terms.

But setting will do this!



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Example

Person Cost share Value Net value Fee

A 100 50 -50 0

B 100 50 -50 0

C 100 250 150 100

TV costs 300 which will be split equally.

It is Pareto optimal to provide in this case.

Individuals A and B are not pivotal. To become pivotal individual A would need to exaggerate net cost to -100.

This would save 50, but he would need to pay a fee of 100 so its not worthdoing.

Individual C is pivotal and pays a fee. To save on the fee, theyd need to become not pivotal by understating their

valuation.

But in this case, theyd lose their net valuation of 150 and only save 100.



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Problems with the VCG mechanism

It only works with quasi-linear preferences, otherwise payment ofthe fee would change individuals demands.

Although the allocation of public goods is optimal the overall picture

is not, because the fees are all deadweight losses.

In large (infinite) populations the revenues can be returned withoutaffecting incentives.



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Another VCG example

Voter Option F Option G

1 60 20

2 10 80

3 20 0Total 90 100

What fees do the three voterspay under the VCG mechanism?

Demonstrate that they are

incentive compatible.



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Summary

We answered two key questions in this topic:

What is the optimal allocation of public goods?

Important to see the difference between the conditions for public and private goods.

How can preferences be discovered so that optimality can be delivered?

Voting aspects are closely related to our lecture on welfare.

Important to understand the VCG Mechanism.

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lecture 4 public goods

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