Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Welfare and Efficiency
Frank Cowell: Microeconomics
August 2006
Public Goods
MICROECONOMICS
Principles and Analysis
Frank Cowell
Public Goods
Overview...
Frank Cowell: Microeconomics
The basics
Characteristics of
public goods
Efficiency
Contribution
schemes
The Lindahl
approach
Alternative
mechanisms
Characteristics of public goods
Frank Cowell: Microeconomics


Two key properties that we need to distinguish:
Excludability




Rivalness





You are producing a good.
A consumer wants some.
Can you prevent him from getting it if he does not pay?
Consider a population of 999 999 people all consuming 1
unit of commodity i.
Another person comes along, also consuming 1 unit of i.
Will more resources be needed for the 1 000 000?
These properties are mutually independent
They interact in an interesting way
Rival?
Excludable?
Frank Cowell: Microeconomics
Typology of goods: classic
definitions
[ Yes ]
[ No ]
[ Yes ]
[ No ]
pure
private
[??]
[??]
pure
public
How the characteristics interact
Frank Cowell: Microeconomics
Example:
bread
Example:
defence
Example:
flowers
Example:
bridge




Private goods are both rival and
excludable
Public goods are nonrival and
nonexcludable
Consumption externalities are nonexcludable but rival
Non-rival but excludable goods often
characterise large-scale projects.
Example: Bread
(E) you can charge a price for
bread
(R) an extra loaf costs more
labour and flour
Example: National defence
(E) you can't charge for units of
'defence‘
(R) more population doesn't
always require more missiles
Example: Scent from Fresh
Flowers
(E) you can't charge for the
scent
(R) more scent requires more
flowers
Example: Wide Bridge
(E) you can charge a toll for the
bridge
(R) an extra journey has zero
cost
Aggregating consumption:
Frank Cowell: Microeconomics


How consumption is aggregated over agents depends on
rivalness characteristic
Also depends on whether the good is optional or not
Private goods
xi =
S h=1x
nh
h
i
Optional public goods
xi = max h ( xih )
Non-optional public goods
xi = xi1 = xi2 =...
Pure rivalness means that you add
up each person’s consumption of
any good i.
Pure nonrivalness means that if
you provide good i for one person
it is available for all.
Pure nonrivalness means that if
one person consumes good i then
all do so.
Public Goods
Overview...
Frank Cowell: Microeconomics
The basics
Extending the
results that
characterise
efficient
allocations
Efficiency
Contribution
schemes
The Lindahl
approach
Alternative
mechanisms
Public goods and efficiency
Frank Cowell: Microeconomics
Jump to
“Welfare:
efficiency”


Take the problem of efficient allocation with
public goods.
The two principal subproblems will be treated
separately...




Characterisation
Implementation
Implementation will be treated later
Characterisation can be treated by introducing
public-goods characteristics into standard
efficiency model
Frank Cowell: Microeconomics
Efficiency with public goods: an
approach


Use the standard definition of Pareto efficiency
Use the standard maximisation procedure to
characterise PE outcomes...






Specify technical and resource constraints
These fix utility possibilities
Fix all persons but one at an arbitrary utility level
Then max utility of remaining person
Repeat for another person if necessary
Use FOCs from maximum to characterise the
allocation
Efficiency: the model
Frank Cowell: Microeconomics


Let good 1 be a public good, goods 2,...,n private
goods
Then agent h’s consumption vector is
(x1h, x2h , x3h, ..., xnh)
where x1 is the same for all agents h.
and x2h , x3h, ..., xnh is h’s consumption of good 2,3,...n

Agents 2,…,nh are on fixed utility levels uh

Differentiating with respect to x1 involves a
collection of nh terms
 good 1 enters everyone’s utility function.
Efficiency: the model
Frank Cowell: Microeconomics


Let good 1 be a public good, goods 2,...,n private goods
Then agent h’s consumption vector is
(x1h, x2h , x3h, ..., xnh)
where x1 is the same for all agents h.
and x2h , x3h, ..., xnh is h’s consumption of good 2,3,...n

Agents 2,…,nh are on fixed utility levels uh

Problem is to maximise U1(x1, x21, x31, ..., xn1) subject to:
 Uh(x1, x2h, x3h, ..., xnh) ≥ uh, h = 2, …, nh
f f
 F (q ) ≤ 0, f = 1, …, nf Technological feasibility
 qfi is net output of good i by firm f
 xi ≤ qi + Ri , i= 1, …, n Materials Balance
Finding an efficient allocation
Frank Cowell: Microeconomics
max L( [x ], [q], l,
Lagrange multiplier for
m,each
k)utility
:= constraint
Lagrange multiplier
h(xhfor
h]
U1(x1) +

l
[U
)

u
h h technology
each firm’s
Lagrange
f (q f) multiplier for
 f mf Fmaterials
balance, good i
+ i ki[qi + Ri  xi]
where
xh = (x1, x2h, x3h, ..., xnh)
xi = h xih , i = 2,...,n
qi = f qi f
FOCs
Frank Cowell: Microeconomics
For any good i=2,…,n differentiate Lagrangean w.r.t xih.
MU to household h
shadow price
h
of good i
of good
If xii is positive at the optimum then:

lhUih (x1, x2h, x3h, ..., xnh) = ki
But good 1 enters everyone’s utility function. So,
differentiating
w.r.t x1:
Sum, because
all

are benefited
nh
 lU
h
shadow price of good 1
j
h (x , x h, x h, ..., x h)
1 2
3
n
= k1
h=1

Differentiate Lagrangean w.r.t qif. If qif is nonzero at the
optimum then:
mfFif(qf) = ki

Likewise for good j:
mfFjf(qf) = kj
Another look at the FOC...
Frank Cowell: Microeconomics

For private goods i, j = 2,3,..., n :
Ujh(xh)
kj
Fjf(qf)
——— = — = ——
Uih (xh)
ki
Fif(qf)
 Condition
Sum of marginal
willingness to pay
nh

h=1

when good 1 is public and good i is private
U1h(xh)
k1
——— = —
Uih (xh)
ki
An important rule for public goods:
Sum over households of marginal willingness to
pay = shadow price ratio of goods = MRT
Public Goods
Overview...
Frank Cowell: Microeconomics
The basics
Private provision
of public goods?
Efficiency
Contribution
schemes
The Lindahl
approach
Alternative
mechanisms
The implementation problem
Frank Cowell: Microeconomics

Why is the implementation part of the problem
likely to be difficult in the case of pure public
goods?

In the general version of the problem private
provision will be inefficient

We have an extreme form of the externality issue

We run into the Gibbard-Satterthwaite result
Example
Frank Cowell: Microeconomics



Good 1 - a pure public good
Good 2 - a pure private good
Two persons: A and B



Each person has an endowment of good 2
Each contributes to production of good 1
Production organised in a single firm
Public goods: strategic view (1)
[+]
2,2
0,3
[–]
If Bill reneges [–] then
Alf’s best response is [–].
3,0
1,1
[+]
[–]
Alf
Frank Cowell: Microeconomics
If Alf reneges [–] then
Bill’s best response is [–].
Bill
Nash equilibrium
Public goods: strategic view (2)
[+]
2,2
1,3
Alf
If 2 plays [+] then 1’s
best response is [–].
A Nash equilibrium
By symmetry, another
Nash equilibrium
[–]
Frank Cowell: Microeconomics
If 1 plays [–] then 2’s best
response is [+].
3,1
0,0
[+]
[–]
bill
Which paradigm?
Frank Cowell: Microeconomics

Clearly the two simplified +/– models lead to
rather different outcomes.

Which is appropriate? Will we inevitably end up
at an inefficient outcome?

The answer depends on the technology of
production.

Also on the number of individuals involved in the
community.
A Voluntary Approach (1)
Frank Cowell: Microeconomics

Consider in detail the implementation problem
for public goods

Logical to view the way individual action
would work in connection with public goods

Begin with a simple contribution model

Take the case with nh persons.

Then see what the “classic” solution would
look like
A Voluntary Approach (2)
Frank Cowell: Microeconomics

Each person has a fixed endowment of (private)
good 2:


And makes a voluntary contribution of some of
this toward the production of (public) good 1:


R2h
zh = R2h – x2h
This is equivalent to saying that he chooses to
consume this amount of good 2:

x2 h
A Voluntary Approach (3)
Frank Cowell: Microeconomics

Contribution of all households of good 2 is:
nh
z=
Sz
h
h=1

This produces the following amount of good 1:
x1 = f(z)

So the utility payoff to a typical household is:
Uh(x1 , x2h)
A Voluntary Approach (4)
Frank Cowell: Microeconomics

Suppose every household makes a “Cournot” assumption:
nh
S
zk =`z (constant)
k=1
kh


Given this and the production function agent h perceives
its optimisation problem to be:
 max Uh(f(`z + R2h – x2h ) , x2h)
This problem has the first-order condition:
 U1h(x1 , x2h) fz(`z + R2h – x2h ) – U2h(x1 , x2h) = 0
A Voluntary Approach (5)
Frank Cowell: Microeconomics


The FOC yields the condition:

1
————
fz(Sh zh )

MRT
= MRSh
However, for efficiency we should have:



U1h(x1 , x2h)
= —————
U2h(x1 , x2h)
1
————
fz(Sh zh )
=
S
U1h(x1 , x2h)
h —————
U2h(x1 , x2h)
MRT = Sh MRSh
Each person fails to take into account the “externality”
component of the public good provision problem
Outcomes with public goods
Frank Cowell: Microeconomics
x2
 Production possibilities
 Efficiency with public goods
 Contribution equilibrium

^x
MRT = MRS


x*
MRT = SMRS
x1
0
Myopic rationality
underprovides public
good...
Graphical illustrations
Frank Cowell: Microeconomics

We can use two of the graphical devices
that have already proved helpful.

The contribution diagram:


Nash outcomes

PE outcomes
The production possibility curve
Outcomes of contribution game
Frank Cowell: Microeconomics
 Alf’s ICs in contribution space
 Alf’s reaction function
 Bill’s ICs in contribution space
 Bill’s reaction function
 Cournot-Nash equilibrium
 Efficient contributions
zb
ca(·)
 Alf assumes
Bill’s contribution
is fixed



 Likewise Bill’





cb(·)
za
 Cournot-Nash
outcome results
in inefficient
shortfall of
contributions.
Public Goods
Overview...
Frank Cowell: Microeconomics
The basics
“Personalised”
taxes?
Efficiency
Contribution
schemes
The Lindahl
approach
Alternative
mechanisms
A solution?
Frank Cowell: Microeconomics

Take the standard efficiency result for public
goods: Sj MRSj = MRT

This aggregation rule has been used to suggest an
allocation mechanism

The “Lindahl solution” is tax-based approach.

However, it is a little unconventional.

It suggests that people pay should taxes according
to their willingness to pay

The sum of the taxes covers the marginal cost of
providing the public good.
An example
Frank Cowell: Microeconomics
Good 1 - a pure public good
 Good 2 - a pure private good
 Two persons: Alf and Bill
 Simple organisation of production: A single
firm

Willingness-to-pay for good 1
Frank Cowell: Microeconomics
 Plot Alf’s MRS as function of x1
 WTP by Alf for x1
 Bill’s MRS as function of x1
U1a(•)/U2a(•)
 WTP by Bill for x1
a
MRS21(x1)
x1
x1 the more there is of
good 1 the less Alf
wants to pay for extra
units
U1b(•)/U2b(•)
b
MRS21(x1)
x1
x1
Bill is less willing to
pay for good 1 than
Alf
Use this to derive
efficiency condition
Ua1(•)/Ua2(•)
Efficiency
Frank Cowell: Microeconomics
a
MRS21(x*1)

x1
 MRS for Alf and for Bill
 Sum of their MRS as function of x1
 MRT as function of x1
 Efficient amount of x1
Ub1(•)/Ub2(•)
b
MRS21(x*1)
 MRS at efficient allocation.

x1
For a public good we
aggregate demand “vertically”
1/fz
ShUh1(•)/U2h(•)
Can we use these WTP
values to derive an allocation
mechanism?
h
ShMRS21(x*1)
x*1

Consider these as demand
curves for good 1
x1
Ua1(•)/Ua2(•)
pa
Frank Cowell: Microeconomics

x1
Lindahl
solution
 Efficient allocation of public good
 Willingness-to-pay at efficient
allocation.
 Charge these WTPs as “tax prices “
Ub1(•)/Ub2(•)
pb

x1
1/fz
Combined “tax prices” pa + pb
just cover marginal cost of
producing the amount x1* of
the public good
ShUh1(•)/U2h(•)
pa +
But what of
individual
rationality?
pb
x*1

 The “ Lindahl solution”
suggests that people pay
should taxes according to their
willingness to pay
x1
The Lindahl Approach
Frank Cowell: Microeconomics


let ph is the “tax-price” of good 1 for person h, set by
the government.
The FOC for the household’s problem is:
1.

U1h(x1, x2h)
———— = ph
U2h(x1, x2h)
For an efficient outcome in terms of the allocation of the
two goods:
S MRSh = MRT
nh
2.
Sp
h=1

1
h
h
= ——
fz(z)
Conditions 1,2 determine the set of household-specific
prices { ph}
The Lindahl Approach (1)
Frank Cowell: Microeconomics

But where does the information come from for this
personalised tax-price setting to be implemented?

Presumably from the households themselves

In which case households may view the determination of
the personalised prices strategically.

In other words h may try to manipulate ph (and thus the
allocation) by revealing false information about his MRS
The Lindahl Approach (2)
Frank Cowell: Microeconomics


1.
Take into account this strategic possibility
Then h solves the utility-maximisation problem:

choose (x1, x2h) to max Uh(x1, x2h) subject to
the budget constraint:

2.
the following perceived relationship:



phx1 + x2h  R2h
x1 = f(z, phx1)
But here ph is endogenous
So this becomes exactly the problem of
voluntary contribution
The Way Forward
Frank Cowell: Microeconomics

Given that the Lindahl problem results in the same
suboptimal outcome as voluntary contribution
(subscription) what can be done?

Public provision through regular taxation

Change the problem

Change perception of the problem
Public Goods
Overview...
Frank Cowell: Microeconomics
The basics
Truth-revealing
devices
Efficiency
Contribution
schemes
The Lindahl
approach
Alternative
mechanisms
A restricted problem
Frank Cowell: Microeconomics


One of the reasons for the implementation
problem is that one invites selection of a social
state qQ, where Q is large.
Sidestep the problem by restricting Q.



We would be changing the problem
But in a way that is relevant to many situations
Suppose that there is an all-or nothing choice.


Replace the problem of choosing a specific amount of
good 1 from a continuum …
…by substituting the choice problem “select from
{NO-PROJECT, PROJECT} ”
The Clark-Groves approach
Frank Cowell: Microeconomics

Imagine a project completely characterised by




the status-quo utility,
the payment required from each member of the
community if the project goes ahead
the utility to each person if it goes ahead.
For all individuals



utility is separable and
income effect of good 1 is zero:
Uh(x1 , x2h) = y(x1) + x2h
The C-G method (2)
Frank Cowell: Microeconomics



Person h has endowment of R2h of private good 2.
The project specifies a payment zh for each person
conditional on the project going ahead.
Total production of good 1 is f(z) where


Social states states Q = {q0 , q1} where
q0 : f(0) = 0
 q1 : f(z) = 1
Measure the welfare benefit to each person by the
compensating variation CVh .


z := Sh zh
Project payoffs
x2a
Consumption space for Alf and Bill
Frank Cowell: Microeconomics
Endowments and preferences
R2a q°
R2a – z a
Outcomes if project goes ahead

Alf
The elements of Q
q′
Compensating variation for Alf, Bill

0

x2b
R2b
x1
1
Bill
Alf would like the project
to go ahead.
Bill would prefer the
opposite.

q°
R2b – z b



q′

CV is positive for Alf...
...negative for Bill
But sum is positive
Should project
go ahead?
0
1
x1
A criterion for the project
Frank Cowell: Microeconomics

Let CVh be the compensating variation for
household h if the project is to go ahead.

Then clearly an appropriate criterion overall is
nh

S CVh > 0
h=1

Gainers could compensate losers
But how do we get the right information on CVs?

Introduce a simple, powerful concept

Use announced information
Frank Cowell: Microeconomics

Approve the project only if this is positive
nh

S CVh > 0
h=1

If person k is pivotal, then impose a penalty of this size
nh

S CVh
h=1
hk

Theorem: a scheme which
 approves a project if and only if announced CVs is
non-negative, and
 imposes the above penalty on any pivotal household
will guarantee that truthful revelation of CVs is a dominant
strategy.
The pivotal person
Frank Cowell: Microeconomics

Pick an arbitrary person h.

What would be the sum of the announced CVs if he
were eliminated from the population?

If this sum has the opposite sign from that of the full
sum of the CVs, then h is pivotal. Adding him swings
the result.

We use this to construct a mechanism.

Consider the following table
Public goods: revelation
[Yes]
[Yes]
[No]
Decision
Frank Cowell: Microeconomics
Everyone else says:
[No]
Two possible states
Agent h decision
Payoff table
S costs
Nil
imposed on
others
S forgone
gains of
others
Nil
An example
Summary
Frank Cowell: Microeconomics

A big subject. A few simple questions to
pull thoughts together:

What is the meaning of “market failure”?
Why do markets “fail”?
What’s special about public goods?


Public goods: summary
Frank Cowell: Microeconomics


Characterisation
problem:
Implementation
problem:
replace the MRS = MRT
rule by S MRS = MRT
The Lindahl "solution"
may not be a solution at
all if people can
manipulate the system.
Public goods
Frank Cowell: Microeconomics

The externality feature of public goods makes it
easy to solve the characterisation problem

Implementation problems are much harder.

Intimately associated with the information
problem.

Mechanism design depends crucially on the type
of public good and the economic environment
within which provision is made.