Public Goods

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Public Goods

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

Almost essential

Welfare and Efficiency

Almost essential

Welfare and Efficiency

PrerequisitesPrerequisites

August 2006August 2006

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...The basics

Efficiency

Contribution schemes

The Lindahl approach

Public Goods

Characteristics of public goods

Alternative mechanisms

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Characteristics of public goods Two key properties that we need to distinguish:Two key properties that we need to distinguish: ExcludabilityExcludability

You are producing a good.You are producing a good. A consumer wants some.A consumer wants some. Can you prevent him from getting it if he does not pay?Can you prevent him from getting it if he does not pay?

RivalnessRivalness Consider a population of 999 999 people all consuming 1 Consider a population of 999 999 people all consuming 1

unit of commodity unit of commodity ii. . Another person comes along, also consuming 1 unit of Another person comes along, also consuming 1 unit of ii. . Will more resources be needed for the 1 000 000?Will more resources be needed for the 1 000 000?

These properties are mutually independentThese properties are mutually independent They interact in an interesting wayThey interact in an interesting way

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Typology of goods: classic definitions

Rival?[ Yes ] [ No ]

pureprivate

[??]

[??] purepublic

[ Yes ]

[ No ]

Exc

lud

able

?

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

How the characteristics interact

Private goods are both rival and Private goods are both rival and excludableexcludable

Public goods are nonrival and Public goods are nonrival and nonexcludablenonexcludable

Consumption externalities are non-Consumption externalities are non-excludable but rivalexcludable but rival

Non-rival but excludable goods often Non-rival but excludable goods often characterise large-scale projects.characterise large-scale projects.

Example:defence Example:defence

Example: National defence(E) you can't charge for units of 'defence‘(R) more population doesn't always require more missiles

Example: National defence(E) you can't charge for units of 'defence‘(R) more population doesn't always require more missiles

Example:bread

Example:bread

Example: Bread(E) you can charge a price for bread (R) an extra loaf costs more labour and flour

Example: Bread(E) you can charge a price for bread (R) an extra loaf costs more labour and flour

Example:bridge

Example:bridge

Example: Wide Bridge(E) you can charge a toll for the bridge(R) an extra journey has zero cost

Example: Wide Bridge(E) you can charge a toll for the bridge(R) an extra journey has zero cost

Example:flowers

Example:flowers

Example: Scent from Fresh Flowers(E) you can't charge for the scent(R) more scent requires more flowers

Example: Scent from Fresh Flowers(E) you can't charge for the scent(R) more scent requires more flowers

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Private goods nh

xi xih

h=1

Non-optional public goods xi xi

1xi2

Aggregating consumption:

Pure rivalness means that you add up each person’s consumption of any good i.

Pure nonrivalness means that if one person consumes good i then all do so.

Optional public goods xi max h

( xih )

Pure nonrivalness means that if you provide good i for one person it is available for all.

How consumption is aggregated over agents depends on How consumption is aggregated over agents depends on rivalness characteristic rivalness characteristic

Also depends on whether the good is Also depends on whether the good is optionaloptional or not or not

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Efficiency



Public Goods

Extending the results that characterise efficient allocations


Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Public goods and efficiency

Take the problem of efficient allocation with Take the problem of efficient allocation with public goods.public goods.

The two principal subproblems will be treated The two principal subproblems will be treated separately...separately... CharacterisationCharacterisation ImplementationImplementation

Implementation will be treated laterImplementation will be treated later Characterisation can be treated by introducing Characterisation can be treated by introducing

public-goods characteristics into standard public-goods characteristics into standard efficiency modelefficiency model

Jump to “Welfare: efficiency”

Jump to “Welfare: efficiency”

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency with public goods: an approach Use the standard definition of Pareto efficiencyUse the standard definition of Pareto efficiency Use the standard maximisation procedure to Use the standard maximisation procedure to

characterise PE outcomes... characterise PE outcomes... Specify technical and resource constraintsSpecify technical and resource constraints These fix utility possibilities These fix utility possibilities Fix all persons but one at an arbitrary utility level Fix all persons but one at an arbitrary utility level Then max utility of remaining personThen max utility of remaining person Repeat for another person if necessaryRepeat for another person if necessary

Use FOCs from maximum to characterise the Use FOCs from maximum to characterise the allocationallocation

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency: the model

Let good 1 be a public good, goods 2,...,Let good 1 be a public good, goods 2,...,nn private private goodsgoods

Then agent Then agent hh’s consumption vector is’s consumption vector is(x1

h, x2h , x3

h, ..., xnh)

where where x1 is the same for all agents is the same for all agents hh..and and x2

h , x3h, ..., xn

h is hh’s consumption of good 2,3,...’s consumption of good 2,3,...nn Agents 2,…,Agents 2,…,nnhh are on fixed utility levels are on fixed utility levels h

Differentiating with respect to Differentiating with respect to x1 involves a involves a

collection of collection of nnh terms terms good 1 enters everyone’s utility function.good 1 enters everyone’s utility function.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency: the model

Let good 1 be a public good, goods 2,...,Let good 1 be a public good, goods 2,...,nn private goods private goods Then agent Then agent hh’s consumption vector is’s consumption vector is

(x1h, x2

h , x3h, ..., xn

h)where where x1 is the same for all agents is the same for all agents hh..

and and x2h , x3

h, ..., xnh is hh’s consumption of good 2,3,...’s consumption of good 2,3,...nn

Agents 2,…,Agents 2,…,nnhh are on fixed utility levels are on fixed utility levels h

Problem is to maximise Problem is to maximise UU11((xx11,, xx2211,, xx33

11,, ...,..., xxnn11) subject to:) subject to:

UUh((xx11,, xx22hh,, xx33

hh,, ...,..., xxnnhh) ) ≥ ≥ h, , h = h = 2, …, 2, …, nnhh

f(qf) ≤ 0, f = f = 1, …, 1, …, nnff

xi ≤ qi + Ri , i= i= 1, …, 1, …, n n

Use all this to form a Lagrangean in the usual way…Use all this to form a Lagrangean in the usual way…

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Finding an efficient allocation

max L( [x ], [q], ) :=

U1(x1) + hh [Uh(xh) h]

f f f (q f)

+ i i[qi + Ri xi]

where

xh = (xx11,, xx22hh,, xx33

hh,, ...,..., xxnnhh)

xi = h xih , i = 2,...,n

qi = f qi f

Lagrange multiplier for each utility constraint

Lagrange multiplier for each utility constraint

Lagrange multiplier for each firm’s technology

Lagrange multiplier for each firm’s technology

Lagrange multiplier for materials balance, good iLagrange multiplier for materials balance, good i

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

FOCs For any good i=2,…,n differentiate Lagrangean w.r.t xi

h.

If xih is positive at the optimum then:

hUih (xx11,, xx22

hh,, xx33hh,, ...,..., xxnn

hh) = i

But good 1 enters everyone’s utility function. So, differentiating w.r.t x1:

nh

hUjh (xx11,, xx22

hh,, xx33hh,, ...,..., xxnn

hh) = 1 h=

Differentiate Lagrangean w.r.t qif. If qi

f is nonzero at the optimum then:

fif(qf) = i

Likewise for good j:fj

f(qf) = j

MU to household h of good i

MU to household h of good i

shadow price of good i

shadow price of good i

Sum, because all are benefited

Sum, because all are benefited shadow price of good 1shadow price of good 1

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Another look at the FOC...

U1h(xh) 1

——— = —Ui

h (xh) i

An important rule for public goods:An important rule for public goods:

For private goods For private goods i, j i, j = 2,3,..., = 2,3,..., nn : :

nh

h=

Condition when good 1 is public and good Condition when good 1 is public and good ii is private is private

Sum over households of marginal willingness to pay = shadow price ratio of goods = MRT

Sum of marginal willingness to paySum of marginal

willingness to pay

Ujh(xh) j j

f(qf)——— = — = ——Ui

h (xh) i i

f(qf)

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Efficiency



Public Goods

Private provision of public goods?


Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The implementation problem

Why is the implementation part of the problem Why is the implementation part of the problem likely to be difficult in the case of pure public likely to be difficult in the case of pure public goods?goods?

In the general version of the problem private In the general version of the problem private provision will be inefficientprovision will be inefficient

We have an extreme form of the externality issueWe have an extreme form of the externality issue

We run into the Gibbard-Satterthwaite resultWe run into the Gibbard-Satterthwaite result

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Example

Good 1 - a pure public goodGood 1 - a pure public good Good 2 - a pure private goodGood 2 - a pure private good Two persons: A and BTwo persons: A and B

Each person has an endowment of good 2Each person has an endowment of good 2 Each contributes to production of good 1Each contributes to production of good 1

Production organised in a single firmProduction organised in a single firm

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

[[ –– ]] 1,13,0

0,32,2

[+]

[+]A

lfA

lf

BillBill

[+][+] [[––]]

Public goods: strategic view (1)If Alf reneges [–] then Bill’s best response is [–].

If Bill reneges [–] then Alf’s best response is [–].

Nash equilibrium

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

[[ –– ]] 0,03,1

1,32,2

[+]

[+]A

lfA

lf

billbill

[+][+] [[––]]

Public goods: strategic view (2)If 1 plays [–] then 2’s best response is [+].

If 2 plays [+] then 1’s best response is [–].

A Nash equilibrium

By symmetry, another Nash equilibrium

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Which paradigm?

Clearly the two simplified +/Clearly the two simplified +/– – models lead to models lead to rather different outcomes.rather different outcomes.

Which is appropriate? Will we inevitably end up Which is appropriate? Will we inevitably end up at an inefficient outcome?at an inefficient outcome?

The answer depends on the technology of The answer depends on the technology of production.production.

Also on the number of individuals involved in the Also on the number of individuals involved in the community.community.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

A Voluntary Approach (1)

Consider in detail the implementation problem Consider in detail the implementation problem for public goodsfor public goods

Logical to view the way individual action Logical to view the way individual action would work in connection with public goodswould work in connection with public goods

Begin with a simple contribution modelBegin with a simple contribution model

Take the case with Take the case with nnhh persons. persons.

Then see what the “classic” solution would Then see what the “classic” solution would look likelook like

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Each person has a fixed endowment of (private) Each person has a fixed endowment of (private) good 2: good 2: RR22

hh

And makes a voluntary contribution of some of And makes a voluntary contribution of some of this toward the production of (public) good 1: this toward the production of (public) good 1: zzhh = = RR22

h h –– xx22hh

This is equivalent to saying that he chooses to This is equivalent to saying that he chooses to consume this amount of good 2:consume this amount of good 2: xx22

hh

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Contribution of all households of good 2 is:Contribution of all households of good 2 is: nnhh

z z ==zzhh hh=1=1

This produces the following amount of good 1:This produces the following amount of good 1:xx11 ==zz

So the utility payoff to a typical household is:So the utility payoff to a typical household is:UUhhxx1 1 , x, x22

hh

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Suppose every household makes a “Cournot” assumption:Suppose every household makes a “Cournot” assumption: nnhh zzkk = =z z (constant)(constant) kk=1=1 kkhh

Given this and the production function agent Given this and the production function agent hh perceives perceives its optimisation problem to be:its optimisation problem to be: max max UUhhz z + + RR22

h h –– xx22hh , x, x22

hh This problem has the first-order condition:This problem has the first-order condition:

UU11hhxx1 1 , x, x22

hhzzz z + + RR22h h –– xx22

hh – – UU22hhxx1 1 , x, x22

hh

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


The FOC yields the condition:The FOC yields the condition: 11 U U11

hhxx1 1 , x, x22hh

———— ———— ————— —————zzhh zzhh U U22

hhxx1 1 , x, x22hh

MRT = MRT = MRSMRShh

HHowever, for efficiency we should have:owever, for efficiency we should have: 11 U U11

hhxx1 1 , x, x22hh

———— ———— hh ————— —————

zzhh zzhh U U22hhxx1 1 , x, x22

hh

MRT = MRT = h h MRSMRShh

Each person fails to take into account the “externality” Each person fails to take into account the “externality” component of the public good provision problemcomponent of the public good provision problem

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Outcomes with public goods

Production possibilities

0

Contribution equilibrium

Efficiency with public goods

MRT = MRSMRT = MRS

Myopic rationality underprovides public good... x*

x1

x2

x̂ MRT = MRSMRT = MRS

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Graphical illustrations

We can use two of the graphical devices We can use two of the graphical devices that have already proved helpful.that have already proved helpful.

The contribution diagram:The contribution diagram: Nash outcomesNash outcomes

PE outcomesPE outcomes

The production possibility curveThe production possibility curve

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Outcomes of contribution game

zb

b(·)

a(·)

Alf’s ICs in contribution space Alf’s reaction function

Alf assumes Bill’s contribution is fixed

Bill’s ICs in contribution space Bill’s reaction function

Likewise Bill’

Cournot-Nash equilibrium Efficient contributions

Cournot-Nash outcome results in inefficient shortfall of contributions.

za

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Efficiency



Public Goods

“Personalised” taxes?


Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

A solution?

Take the standard efficiency result for public goods: Take the standard efficiency result for public goods: jMRSj = MRT

This aggregation rule has been used to suggest an allocation This aggregation rule has been used to suggest an allocation mechanismmechanism

The “Lindahl solution” is tax-based approach.The “Lindahl solution” is tax-based approach.

However, it is a little unconventional.However, it is a little unconventional.

It suggests that people pay should taxes according to their It suggests that people pay should taxes according to their willingness to paywillingness to pay

The sum of the taxes covers the marginal cost of providing The sum of the taxes covers the marginal cost of providing the public good.the public good.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

An example

Good 1 - a pure public goodGood 1 - a pure public good Good 2 - a pure private goodGood 2 - a pure private good Two persons: Alf and BillTwo persons: Alf and Bill Simple organisation of production: A single Simple organisation of production: A single

firmfirm

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Willingness-to-pay for good 1

x1

x1

Ua(•)/Ua(•) 1 2

1 2 Ub(•)/Ub(•)

Plot Alf’s MRS as function of x1

WTP by Alf for x1

the more there is of good 1 the less Alf wants to pay for extra units

Bill’s MRS as function of x1

x1

MRS21(x1)a

MRS21(x1)b

WTP by Bill for x1

x1

Bill is less willing to pay for good 1 than Alf Use this to derive efficiency condition

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency

1/z

x1

x1

*x1

x1

Ua(•)/Ua(•) 1 2

1 2 Ub(•)/Ub(•)

1 2hUh(•)/Uh(•)

MRS for Alf and for Bill

Sum of their MRS as function of x1

MRT as function of x1

Efficient amount of x1

Consider these as demand curves for good 1

For a public good we aggregate demand “vertically”

MRS at efficient allocation.

MRS21(x1)a *

MRS21(x1)b *

hMRS21(x1)h *

Can we use these WTP values to derive an allocation mechanism?

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Lindahl solution

1/z

x1

x1

*x1

x1

Ua(•)/Ua(•) 1 2

1 2 Ub(•)/Ub(•)

1 2hUh(•)/Uh(•)

Efficient allocation of public good

Willingness-to-pay at efficient allocation.

The “ Lindahl solution” suggests that people pay should taxes according to their willingness to pay

Combined “tax prices” pa + pb

just cover marginal cost of producing the amount x1* of the public good

Charge these WTPs as “tax prices “

pa

pb

pa + pb But what of individual

rationality?

But what of individual

rationality?

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The Lindahl Approach

let let ph is the “tax-price” of good 1 for person hh, set by , set by the government.the government.

The FOC for the household’s problem is:The FOC for the household’s problem is: UU11

hh((xx11, , xx22hh) )

1.1. ———————— = = pphh

UU22hh((xx11, , xx22

hh) )

For an efficient outcome in terms of the allocation of the For an efficient outcome in terms of the allocation of the two goods:two goods:

nnhh 11

22 pphh = = ———— hh=1=1 zzzz

Conditions 1,2 determine the set of household-specific Conditions 1,2 determine the set of household-specific prices { prices { ph}}

hMRSh = MRThMRSh = MRT

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The Lindahl Approach (1)

But where does the information come from for this But where does the information come from for this personalised tax-price setting to be implemented? personalised tax-price setting to be implemented?

Presumably from the households themselvesPresumably from the households themselves

In which case households may view the determination of In which case households may view the determination of the personalised prices strategically.the personalised prices strategically.

In other words In other words hh may try to manipulate may try to manipulate ph (and thus the (and thus the allocation) by revealing false information about his MRSallocation) by revealing false information about his MRS

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The Lindahl Approach (2)

Take into account this strategic possibilityTake into account this strategic possibility Then Then hh solves the utility- solves the utility-maximisation problemmaximisation problem::

choose (choose (xx11, , xx22hh) to max ) to max UUhh((xx11, , xx22

hh) ) subject to subject to 1.1. the budget constraint:the budget constraint:

pphhxx11 + + xx22hh RR22

hh

2.2. the following perceived relationship: the following perceived relationship: xx11 = = ((cc + + pphhxx11) )

But here But here pphh is endogenous: is endogenous: So this becomes exactly the problem of So this becomes exactly the problem of

voluntary contributionvoluntary contribution

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The Way Forward

Given that the Lindahl problem results in the same Given that the Lindahl problem results in the same suboptimal outcome as voluntary contribution suboptimal outcome as voluntary contribution (subscription) what can be done?(subscription) what can be done?

Public provision through regular taxationPublic provision through regular taxation

Change the problemChange the problem

Change perception of the problemChange perception of the problem

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics


Efficiency



Public Goods

Truth-revealing devices


Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

A restricted problem One of the reasons for the implementation One of the reasons for the implementation

problem is that one invites selection of a social problem is that one invites selection of a social state state , where , where is large. is large.

Sidestep the problem by restricting Sidestep the problem by restricting .. We would be changing the problemWe would be changing the problem But in a way that is relevant to many situationsBut in a way that is relevant to many situations

Suppose that there is an all-or nothing choice.Suppose that there is an all-or nothing choice. Replace the problem of choosing a specific amount of Replace the problem of choosing a specific amount of

good 1 from a continuum …good 1 from a continuum … ……by substituting the choice problem “select from by substituting the choice problem “select from

{NO-PROJECT, PROJECT} ”{NO-PROJECT, PROJECT} ”

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The Clark-Groves approach Imagine a project completely characterised by Imagine a project completely characterised by

the status-quo utility, the status-quo utility, the payment required from each member of the the payment required from each member of the

community if the project goes ahead community if the project goes ahead the utility to each person if it goes ahead.the utility to each person if it goes ahead.

For all individualsFor all individuals utility is separable and utility is separable and income effect of good 1 is zero: income effect of good 1 is zero: UUhhxx1 1 , x, x22

hhxx11+ x+ x22hh

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The C-G method (2)

Person Person hh has endowment of has endowment ofR2h of private good 2. of private good 2.

The project specifies a payment The project specifies a payment zh for each person for each person conditional on the project going ahead.conditional on the project going ahead.

Total production of good 1 is Total production of good 1 is ((z) where) where z := := h zh

Social states states Social states states = {= {0 , , 1} where} where 0 : : (0)(0)= 0= 0 1 : : ((z))= 1= 1

Measure the welfare benefit to each person by the Measure the welfare benefit to each person by the compensating variation CVcompensating variation CVh . .

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

R2 – zb b

Project payoffsx2

a

R2 – za a

0 1

′

R2a

′

Billx2

b

R2b

0 1

Alf

Consumption space for Alf and Bill

x1

x1

Endowments and preferences

Outcomes if project goes ahead

Compensating variation for Alf, Bill

°

°

Alf would like the project to go ahead.

Bill would prefer the opposite.

The elements of

CV is positive for Alf... ...negative for Bill But sum is positive

Should project go ahead?

Should project go ahead?

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

A criterion for the project

Let CVLet CVhh be the compensating variation for be the compensating variation for household household hh if the project is to go ahead. if the project is to go ahead.

Then clearly an appropriate criterion overall isThen clearly an appropriate criterion overall is nnhh

CVCVhh > 0 > 0 hh=1=1

Gainers could compensate losersGainers could compensate losers But how do we get the right information on CVs?But how do we get the right information on CVs?

Introduce a simple, powerful conceptIntroduce a simple, powerful concept

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Use announced information

Approve the project only if this is positiveApprove the project only if this is positive nnhh

CVCVhh > 0 > 0 hh=1=1

If person If person kk is pivotal, then impose a penalty of this size is pivotal, then impose a penalty of this size nnhh

CVCVhh

hh=1=1 hhkk

Theorem: a scheme which Theorem: a scheme which approves a project if and only if announced CVs is approves a project if and only if announced CVs is

non-negative, and non-negative, and imposes the above penalty on any pivotal household imposes the above penalty on any pivotal household

will guarantee that truthful revelation of CVs is a dominant will guarantee that truthful revelation of CVs is a dominant strategy.strategy.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The pivotal person

Pick an arbitrary person Pick an arbitrary person hh..

What would be the sum of the announced CVs if he What would be the sum of the announced CVs if he were eliminated from the population?were eliminated from the population?

If this sum has the opposite sign from that of the full If this sum has the opposite sign from that of the full sum of the CVs, then sum of the CVs, then hh is is pivotalpivotal. Adding him swings . Adding him swings the result.the result.

We use this to construct a mechanism.We use this to construct a mechanism.

Consider the following tableConsider the following table

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

[[ N

oN

o ]] Nil forgone gains of others

costs imposed on

othersNil

[Yes]

[Yes]

Decision

Decision

Everyone else says:Everyone else says:

[Yes][Yes] [[NoNo]]

Public goods: revelationTwo possible states

Payoff table

Agent h decision

An exampleAn example

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Example: model Amount of public good is 0 or 1Amount of public good is 0 or 1

if public good is produced cost is shared equallyif public good is produced cost is shared equally population of size population of size NN each pay 1/ each pay 1/ N N of total of total

Agents’ valuations differ Agents’ valuations differ valuation of valuation of hh is net of contribution to public good is net of contribution to public good vvhh = a = a ++ [[ h h − 1]− 1] [[b b – – aa ] / [ ] / [N N − 1]− 1] , , hh = 1,2,…, = 1,2,…,NN assumeassume b b > 0 > > 0 > aa

Mean valuation is Mean valuation is ½½[[aa + + bb]] project should go ahead if project should go ahead if aa + + bb > 0> 0 assume, however, that assume, however, that aa + + bb < 0 < 0

Define Define zzhh := := ½½NN[[aa + + bb] ] − − vvhh

measures the sum-of-valuations if measures the sum-of-valuations if hh is excluded. is excluded. Suppose Suppose vv1 1 < < vv2 2 < 0 and that < 0 and that zz1 1 > 0, > 0, zz2 2 < 0< 0

both agents 1 and 2 would prefer no projectboth agents 1 and 2 would prefer no project agent 1 is pivotal if reports truthfully (agent 1 is pivotal if reports truthfully (zz11 is opposite sign to is opposite sign to aa + + bb ) ) agent 2 is not pivotal if reports truthfully (agent 2 is not pivotal if reports truthfully (zz22 is same sign as is same sign as aa + + bb ) )

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Example: choices If agent 1 declares… If agent 1 declares…

v v = = vv11 then outcome is no project then outcome is no project reverses sign of willingness to pay – so must pay penaltyreverses sign of willingness to pay – so must pay penalty gets payoff of –zgets payoff of –z11

v v < < vv11 then outcome and payoff are as above then outcome and payoff are as above v v > > vv11 then then

if if v v −− vv11 is small, outcome and payoff are as above is small, outcome and payoff are as above if if v v −− vv11 is large, project goes ahead and payoff is is large, project goes ahead and payoff is vv11

If agent 2 declares… If agent 2 declares… v v = = vv22 then outcome is no project and gets payoff of 0 then outcome is no project and gets payoff of 0 v v < < vv22 then outcome and payoff are as above then outcome and payoff are as above v v > > vv22 then then

if if v v −− vv22 is small, outcome and payoff are as above is small, outcome and payoff are as above if if v v −− vv22 is large, outcome reversed and payoff is is large, outcome reversed and payoff is vv22 ++ zz22

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Example: outcomes Payoff to agent 1 is… Payoff to agent 1 is…

––zz11 if declares if declares vv11 ––zz11 or or vv11 otherwiseotherwise but zbut z11 + + vv11 = = ½½NN[[aa + + bb] < 0 so that –z] < 0 so that –z11 > > vv11 … … … … so declaring so declaring vv11 is optimal is optimal

Payoff to agent 2 is… Payoff to agent 2 is… 0 if declares 0 if declares vv22 0 or 0 or vv22 or or vv22 ++ zz22otherwise otherwise but but vv22 < 0 and < 0 and zz22 < 0 … < 0 … … … so declaring so declaring vv22 is optimal is optimal

Overall outcomeOverall outcome Each has incentive to report truthfullyEach has incentive to report truthfully More resources are paid (in penalties) than are necessary to More resources are paid (in penalties) than are necessary to

produce the public goodproduce the public good

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The C-G model – assessment Strengths: Strengths:

only uses announced informationonly uses announced information elicits truth-tellingelicits truth-telling

Drawbacks: Drawbacks: Restriction to Ziff preferencesRestriction to Ziff preferences Does not ensure budgetary balanceDoes not ensure budgetary balance

The tipping mechanism can be used as the The tipping mechanism can be used as the foundation for more interesting design problems. foundation for more interesting design problems.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Summary

A big subject. A few simple questions to A big subject. A few simple questions to pull thoughts together:pull thoughts together:

What is the meaning of “market failure”?What is the meaning of “market failure”? Why do markets “fail”? Why do markets “fail”? What’s special about public goods?What’s special about public goods?

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Public goods: summary

Characterisation Characterisation problem: problem:

replace the MRS = MRT

rule by MRS = MRT

Implementation Implementation problem: problem:

The Lindahl "solution" may not be a solution at all if people can manipulate the system.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Public goods

The externality feature of public goods makes it The externality feature of public goods makes it easy to solve the characterisation problemeasy to solve the characterisation problem

Implementation problems are much harder.Implementation problems are much harder.

Intimately associated with the information Intimately associated with the information problem.problem.

Mechanism design depends crucially on the type Mechanism design depends crucially on the type of public good and the economic environment of public good and the economic environment within which provision is made.within which provision is made.

Documents

Public Goods