Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Welfare and Efficiency
Frank Cowell: Microeconomics
June 2006
Externalities
MICROECONOMICS
Principles and Analysis
Frank Cowell
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
A special type of
transaction
Production
externalities
Consumption
externalities
Connections
The nature of externality
Frank Cowell: Microeconomics


An externality is a kind “involuntary” transaction
A case where market allocation methods don’t
work




Agents cannot be excluded from the transaction using
conventional price mechanism
An example of “market failure”?
Externalities can be detrimental or beneficial
We will deal with two broad types:


Production externalities
Consumption externalities
Production externality
Frank Cowell: Microeconomics

One firm influences another’s production conditions




Model this as a parameter shift




If firm f’s output produces an externality…
…production function of firm k has f’s output as a parameter…
… or MC curve of firm k has f’s output as a parameter.
Example: networking


Affects other firms’ cost curves.
Not effect of wage or input price changes…
…externality is outside the market mechanism.
One firm’s activity creates pool of skilled workers from which
neighbouring firms may benefit.
Example: pollution

One firm’s activity (glue production) causes emissions that are to the
detriment of its neighbours (restaurants who must filter the air).
Consumption externality
Frank Cowell: Microeconomics

One agent’s consumption of a good directly affects
another


Alf’s consumption of good 1 is an argument of Bill’s
utility function
Related to the analysis of public goods


Public goods are non-excludable and non-rival
These properties are mutually independent

Consumption externalities are non-excludable but
rival

Example: Scent from fresh flowers


Nonexcludable: you can’t charge for the scent
Rival: more scent requires more flowers
Externality questions
Frank Cowell: Microeconomics

How can we model different types of externality?

How can we quantify an externality?

How can we value an externality?

How will the externality modify the efficiency
conditions?

How can we implement an efficient outcome if
there are externalities?
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
How production
externalities
work; how they
are evaluated
Production
externalities
Consumption
externalities
Connections
•Basics
•Efficiency
•Simple implementation
•Private initiative
Production: the framework
Frank Cowell: Microeconomics

There is a known collection of firms



Describe each firm’s activities using net-output vector.




Net output by firm f of good i is qif, i = 1,2,...,n
Usual sign convention
Net output vector is qf = (q1f, q2f, q3f, ..., qnf)
Firm f’s production possibilities are known





Indexed by f = 1,2, ..., nf.
Identities of firms exogenously determined
Implicit production function Ff(•)
Argument is net output vector is qf , and possibly other things.
Set of feasible net outputs given by Ff(qf) ≤ 0
Transformation curve given by net outputs such that Ff(qf) = 0
Now introduce externality...
Quantifying an externality
Frank Cowell: Microeconomics

Consider a polluting firm f.




When f produces good 1 it causes the pollution





could affect other firms k = 1, 2, ..., f – 1, f + 1, ..., nf.
the more f produces good 1, the greater the damage to k.
How much damage?

Jump to
“Multi-output
firm”
Case of a positive externality follows easily
Just reverse signs appropriately...
…and rename “victim” as “beneficiary.”
Consider the impact of pollution on firm k
Will enter the production function Fk(•)
Use the firm’s transformation curve...
Standard
diagram....
Externality: Production possibilities
Frank Cowell: Microeconomics
q2k
Fk(•)
Fk(•)
Production possibilities, firm k
>0
Production possibilities, if firm
f’s emissions increase
=0
Fk(•) < 0
low emissions
by firm f
If Fk(•) = 0 then an increase
high emissions
by firm f
in the negative externality
will result in Fk(•) > 0.
q1`k
Valuing an externality
Frank Cowell: Microeconomics


What is value to victim firm k of pollution by f ?
Need quantification of pollution:




Use same approach as for “value of an input”
Focus on impact of marginal amount:



Identify source of externality – production of good 1
Then use units of output of good 1.
How much impact on activity of firm k?
Need the derivative of production function Fk.
Measure effect in terms of a numéraire:


Here we take this to be good 2.
But could be any other good.
Production externality
Frank Cowell: Microeconomics

Firm k may be affected by
others' output of good 1:
Fk(qk;
q11, q12 , ..., q1k-1,
q1
Characteristics of production
generates inefficiency
k+1...)
vanishes if there
is no externality
net output
of firm k
 Now
this is positive for a negative
evaluate the marginal
impact
of
externality:
it is shifting
some firm f on others:“inwards” firm k’s feasible set.
1 Fk(•)
e21f := – ——k ———
f
F
q
2
1
k=1
nf
Marginal product of
good 2 for firm k
Direct impact of f on production
possibilities of firm k
…evaluated in terms of good 2…
…and summed over all k
Value of the marginal externality
imposed through production by f of
good 1.
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
Deriving the
conditions for a
PE allocation
Production
externalities
Consumption
externalities
Connections
•Basics
•Efficiency
•Simple implementation
•Private initiative
Externality and efficiency
Frank Cowell: Microeconomics
Jump to
“Welfare:
efficiency”


Take the problem of efficient allocation with
externality
Two main subproblems are treated separately...



Characterisation uses standard efficiency model



Characterisation
Implementation
introduce production/consumption externality features
examine impact on the FOCs
Implementation may follow on from this...
The approach
Frank Cowell: Microeconomics

Use a maximisation procedure to characterise efficiency:




So problem is to maximise U1(x1) subject to:




Specify technical and resource constraints
Fix all persons but one at an arbitrary utility level
Then max utility of remaining person
Uh(xh) ≥ uh, h = 2, …, nh
Ff(qf; q11,q12 ,...,q1f1,q1f+1...) ≤ 0, f = 1, …, nf
xi ≤ qi + Ri , i= 1, …, n
where



xh = (x1h, x2h, x3h, ..., xnh)
xi = h xih , i = 1,...,n
qi = f qi f
materials'
balance
technical
feasibility
Lagrangean method:
Frank Cowell: Microeconomics

Introduce Lagrange multipliers:





lh for each utility constraint
mf for each firm’s technology constraint
ki for materials’ balance on good i.
Then maximise
U1(x1) + hlh [Uh(xh)  uh]
 f mf F f (qf; q11,q12 ,...,q1f1,q1f+1...)
+ i ki[qi + Ri  xi]
First-order conditions for an interior maximum:
only good 1
 lhUih (xh) = ki, i = 1,...,n
generates an
k
nhf F (•)
externality
 mf Fif(qf) + mf
——— = k1
k=1
q1f
 mfFif(qf) = ki , i = 2,3,...,n

From the FOC...
Frank Cowell: Microeconomics




Consider tradeoff between goods 1 and 2
From the first of the FOCs:
U1h(xh)
k1
——— = —
U2h(xh)
k2
Use the definition of e21f . Then other FOCs give
F1f(qf)
k1
——— – e21f = —
F2f(qf)
k2
This is the efficiency criterion:


Instead of the condition “MRT=shadow price ratio”…
…we have a modified marginal rule.
Frank Cowell: Microeconomics
Efficiency with production
externality
Production possibilities
If externality is ignored...
q2f
~
qf
F1 k1
— = — + externality
F2 k2
^qf
Taking account of
externality...
Produce less of good 1
F1 k1
— =—
F2 k2
for efficiency...
q1`f
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
Corrective taxes
and other
devices...
Production
externalities
Consumption
externalities
Connections
•Basics
•Efficiency
•Simple implementation
•Private initiative
Implementation
Frank Cowell: Microeconomics



Use the efficiency criterion for guidance on
policy design
The simple marginal rule suggests a method of
implementation
We can use it to modify the market mechanism:



MRT – producer prices
MRS – consumer prices
…how to connect the two of these?
Towards a policy rule
(2)
value of
shadow prices
externality
Frank Cowell: Microeconomics


F1f(qf)
k1
Take the modified FOC ——— – e21f = —
F2f(qf)
k2
(private) marginal cost
of producing 1
Rearrange:
F1f(qf)
k1
——— = — + e21f
F2f(qf)
k2
consumer prices
F1f(qf)
p1
——— = — + e21f
F2f(qf)
p2

Introduce the market:

t = – e21f
Corrective tax (negative
f(qf)
F
p1
1
externality) or subsidy
——— = — – t
(positive externality):
f f
F2 (q )
p2
Production externality: policy
Frank Cowell: Microeconomics

From the FOC a simple corrective tax can be designed




Alternative 1: merger.



Merging the firms “internalises” the externality.
Combined firm takes into account interdependence of production.
Alternative 2: public issue of “pollution rights”




Called “Pigovian” (from A.C. Pigou’s Economics of Welfare)
Needs information about production functions.
Both for victim and perpetrator.
Again the externality is internalised.
Polluter takes account of true its activity because of new market
Equilibrium price determined as for the Pigovian tax.
However could there be a purely private solution?
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
Development of
a “pseudo
market”
Production
externalities
Consumption
externalities
Connections
•Basics
•Efficiency
•Simple implementation
•Private initiative
Private solution: A model

Frank Cowell: Microeconomics

Efficient outcome through individual initiative?
Assume that there are just two firms and two goods.




Firm 1’s output of good 1 imposes costs on firm 2.
Full information:





The second of these assumptions is unimportant.
First assumption may be important.
Each firm knows the other’s production function.
Externality is common knowledge.
Activity can be monitored.
Communication is costless.
Firm 2 (victim) has an interest in communicating


Does this by setting up a financial incentive for firm 1.
How should this be structured?
The victim’s problem
Frank Cowell: Microeconomics




Firm 2 offers firm1 a side-payment (Bribe) b.
This payment needs to be accounted for in the
computation of profits.
It can be treated as a control variable for firm 2.
Optimisation problem of firm 2 (the victim) is:
n
max
S piqi2 − b− m2 F2(q2, q11)
{q2, b} i=1

Solve this in the usual way…
The victim’s problem: interpretation
Frank Cowell: Microeconomics

Firm 2 designs incentive for firm 1



Incentive scheme captures costs to firm 2





a “side-payment schedule”
or “conditional bribe function.”
Slope equals marginal cost of pollution.
The higher is the level of the polluting output...
...the lower is the level of the conditional bribe.
Should influence actions of perpetrator (firm 1)
Analyse firm 1’s behaviour in same framework
Solving the victim’s problem
Frank Cowell: Microeconomics




FOC for net outputs of firm 2 is
pi − m2 Fi2 (q2, q11) = 0
FOC for the side payment b is:
dF2(q2, q11) dq11
− 1 + m2 ─────── ── = 0
dq11
db
Using the definition of the externality:
dq11
− 1 + m2F22(q2, q11) e211 ── = 0
db
Rearranging the FOC then gives:
db
── = m2F22(q2, q11) e211 = p2 e211
dq11
The perpetrator’s problem
Frank Cowell: Microeconomics




For firm 2’s “schedule” to work, firm 1 has to know about it
It rationally incorporates this into its profit calculation
It will note that the bribe is conditional on a variable under
its own control
The optimisation problem for firm 1 is:
n
max
q1

S piqi1 + b(q11) − m1F1(q1)
i=1
Again solve this in the usual way…
Solving the problem
Frank Cowell: Microeconomics
Feedback effect from 1’s net
 FOC for net outputs
ofonfirm
1 offer
is:
output
2’s bribe
d b(q11)
p1qi1 + ───── − m1 F11(q1)
dq11


=0
p2 − m1 F21(q1) = 0
Substituting in for the slope of the bribe function:
F11(q1)
p1
──── = ── + e211
F21(q1)
p2
This condition same as FOC for efficiency!
Private solution: result
Frank Cowell: Microeconomics

Bribe function has internalised the externality



FOC conditions same as before




Firm 2 conditions side-payment on observable output of
good 1
Firm 1’s responds rationally to the side-payment.
Private solution induces an efficient allocation
Implements the same allocation as the Pigovian tax
But no external guidance is required.
It should be independent of where the law places
the responsibility for the pollution (Coase’s result)
Private solution: difficulties
Frank Cowell: Microeconomics

Solution makes important informational
requirements



It requires a special notion of participation.



Imposed on both firms.
There may be an incentive for firms to misrepresent costs,
leading to loss of efficiency.
What determines the set of participants?
What if there is free entry?
It focuses only on marginal impacts


If the polluter is allowed to sell pollution rights there
could be problems with this private sector “solution”
This is similar to the nonconvexity problem
A fundamental nonconvexity
Frank Cowell: Microeconomics
Production possibilities…
q2
If firm 1’s pollution could
drive the other out of business
 The optimal point?

If polluter can sell pollution
rights indefinitely...
~
q

0
^q
q1
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
Interactions
between
consumers
Production
externalities
Consumption
externalities
Connections
Consumption externality
Frank Cowell: Microeconomics

Household ℓ affected by others’ Characteristics of goods
generates inefficiency
consumption of good 1:
consumption
of household ℓ
Uℓ(xℓ;

vanishes if there
x11,x12 , ..., x1ℓ1, x1ℓ+1,...) is no externality
Now evaluate the marginal impact
of some household h on others:
1
e21h:= ——ℓ ———
h
U
x
2
1
ℓ=1
nh
Uℓ(•)
MU of good 2
for household ℓ
Direct impact of h on utility of ℓ…
…evaluated in terms of good 2…
…and summed over all ℓ
 Gives the value of the marginal
externality imposed through
consumption by h of good 1.
Lagrangean method:
Frank Cowell: Microeconomics


Use same method as for production externalities
Introduce Lagrange multipliers:





lh for each utility constraint
mf for each firm’s technology constraint
ki for materials’ balance on good i.
Then maximise
U1(x1;,x12 , x13, ...)
+ hlh [Uh(xh; x11,x12 , ..., x1h-1, x1h+1,...)  uh]
 f mf F f (qf)
+ i ki[qi + Ri  xi]
only good 1
First-order conditions for an interior maximum:
generates the
externality
nh
U1ℓ(•)
 lhU1h (x1;,x12 , x13, ...) + lh
——— = k1
ℓ=1
x1h
 lhUih (x1;,x12 , x13, ...) = ki , i = 2,3,...,n
f f
 mfFi (q ) = ki , i = 1,2,...,n

FOC has a similar interpretation...
Frank Cowell: Microeconomics


From the FOC for production:
F1f(qf)
k1
——— = —
F2f(qf)
k2
Substituting in the value of the externality we also have
U1h(xh)
k1
——— + e21h = —
U2h (xh)
k2

Again we have a modified marginal rule

Again it can give us useful guidance on policy
Negative consumption externality
Frank Cowell: Microeconomics
Production possibilities
Competitive equilibrium (with
consumption externality)
x2
U1h
F1
— = — – externality
U2h
F2
Efficiency with consumption
externality
Produce less of good 1
U1h
F1
— = —
U2h
F2
for efficiency...
x1`
Towards a policy rule
value of
shadow prices
externality
Frank Cowell: Microeconomics


Take the modified FOC
h willingness to pay
for 1 in terms of 2
Rearrange:
U1h(xh)
k1
——— + e21h = —
U2h (xh)
k2
U1h(xh)
k1
——— = — – e21h
U2h (xh) k2
Producer
prices


Introduce the market:
A Pigovian tax/subsidy (for
negative/positive
externalities)
U1h(xh)
p1
——— = — – e21h
U2h (xh) p2
U1h(xh)
p1
——— = — + t
U2h (xh) p2
t = −e21h
.
Externalities
Overview...
Frank Cowell: Microeconomics
The nature of
externality
Lessons and
applications
Production
externalities
Consumption
externalities
Connections
Externalities: lessons
Frank Cowell: Microeconomics



The analysis of externality is not a peripheral issue
in microeconomics
Connects to other key topics...
Industrial organisation:



Production externalities and industry supply
Merger as a solution to inefficiency with externality
Public goods:

An extreme form of consumption externality
Externalities: summary
Frank Cowell: Microeconomics

Characterisation
problem:
modify the MRS = MRT rule by the
marginal cost of externality

Implementation
problem:
For production externalities –
encourage private resolution
through extended markets?
Otherwise introduce a
tax/subsidy corresponding to
the marginal cost of externality