Welfare Analysis in Partial Equilibrium.
Social welfare function: assigns social welfare value (real
number) to each profile of utility levels (u1, u2, ...uI ) :
W (u1, u2, ...uI )
(Utilitarian welfare).
Assume: W is strongly monotonic in its arguments.
For any given consumption and production levels of good
l, (x1, ...xI , q1, ..., qJ ), where
I
X
i=1
xi=
{(u1, u2, ..., uI ) :
i=1
ui ≤
I
X
i=1
qj ,the utility
j=1
vectors that are attainable are given by:
I
X
J
X
φi(xi)+ωm−
J
X
cj (qj )}.
j=1
As the boundary of this set expands, the maximum social
welfare W attainable on this set (through redistribution
of the numeraire good) increases (strictly).
Thus,
*For any strongly monotonic social welfare function W,
a change in the consumption and production of good l
leads to an increase in (the maximum attainable) social
welfare if and only if it increases the Marshallian surplus:
S(x1, ...xI , q1, ...qJ ) = [
I
X
i=1
φi(xi) −
J
X
cj (qj )].
j=1
Thus, social welfare analysis of changes in the consumption and production of good l can be carried out exclusively in terms of the Marshallian surplus.
Indeed, as we have seen, Pareto efficiency also requires
that the consumption and production of good l must
satisfy
max
S(x1, ...xI , q1, ...qJ )
(x1,...,xI )≥0
(q1,....qJ )≥0
I
J
X
X
s.t.
xi=
qj .
i=1
j=1
Consider a consumption and production vector of good
b1, ...x
bI , qb1, ...qbJ ) such that for yb =
l, (x
b1, ...x
bI ) solves:
(i) (x
max [
xi,i−1,..I
s.t.
I
X
i=1
qj ,j=1,...J
s.t.
J
X
j=1
i=1
bi
x
φi(xi)]
i=1
b xi ≥ 0, i = 1, ..I.
xi = y,
(ii) (qb1, ...qbJ ) solves
min
I
X
I
X
J
X
cj (qj )
j=1
b qj ≥ 0, j = 1, ..J.
qj = y,
We have seen that:
bi) = P (y)
b = B 0(y),
b ∀i such that x
bi > 0
φ0i(x
b ∀j such that qbj > 0,
c0j (qbj ) = C 0(y),
where P is the inverse aggregate demand function, B 0(.)
is the industry marginal benefit and C 0(.) is the industry
marginal cost (or the aggregate inverse supply function).
b1, ...x
bI , qb1, ...qbJ ) = [
S(x
I
X
φi(bxi) −
J
X
bj )]
cj (q
i=1
j=1
b − C(y)
b
= B(y)
Zyb
Zyb
B 0(y)dy − C(0) − C 0(y)dy
=
=
0
Zyb
0
Zyb
0
P (y)dy −
Zyb
0
C 0(y)dy − C(0)
= [ [P (y) − C 0(y)]dy] − S(0)
0
Note:
Zyb
[ [P (y) − C 0(y)]dy]
0
is the area between the aggregate demand and supply
surves and can be written as :
Zyb
[ [P (y) − C 0(y)]dy]
0
Zyb
b (y)]
b + [yP
b (y)
b − (C(y)
b − C(0))]
= [ [P (y) − yP
0
b + P S(P (y))
b
= CS(P (y))
where CS(p) and P S(p) denote the aggregate consumer
and producer surplus generated in a (hypothetical) market with price taking consumers and producers at price
market price p.
Therefore, in partial equilibrium analysis, social welfare
maximization, Marshallian surplus maximization and Pareto
efficiency are roughly equivalent in their implication for
the production and consumption of "the good" and eventually reduce to maximization of CS + PS.
Zyb
It is easy to see that [ [P (y) − C 0(y)]dy] is maximized
at the output where:
0
P (y ∗) = C 0(y ∗)
i.e., social marginal benefit equates industry’s marginal
cost.
As C 0(y) is inverse aggregate supply curve, this is also
the aggregate output consumed and produced in a competitive equilibrium (supply=demand).
Thus, competitive equilibrium outcome is equivalent to
Marshallian surplus maximization.
All of this assumes no externalities or other distortions
(taxes, subsidies etc).
Welfare loss due to distortions is measured by the change
in CS +PS i.e., the area between aggregate demand and
the supply (or industry MC curve).
Sometmes, called deadweight loss.
Example. Welfare loss due to a distortionary tax (in a
competitive market).
Sales tax on good l: t per unit paid by consumers.
Tax revenue returned to consumers through lump sum
transfer (non distortionary spending).
Let (x∗1(t), ..., x∗I (t), q1∗(t), ..., qJ∗ (t)) and p∗(t) be the
competitive equilibrium allocation and price given tax rate
t.
FOC:
φ0i(x∗i (t)) = p∗(t) + t, for all i such that x∗i (t) > 0.
c0j (qj∗(t)) = p∗(t), for all j such that x∗j (t) > 0.
Let
x∗(t) = x(p∗(t) + t) =
I
X
x∗i (t).
i=1
Market clearing:
x(p∗(t) + t) = q(p∗(t))
*
Easy to check that p∗(t) is strictly decreasing in t
and that (p∗(t) + t) is non-decreasing in t
Then, x∗(t) is non-increasing in t so that
x∗(t) ≤ x∗(0).
Strict inequality if cj strictly convex.
Let S ∗(t) = S(x∗1(t), ..., x∗I (t), q1∗(t), ..., qJ∗ (t)).
We have that
x∗Z(t)
S ∗(t) = [
0
[P (y) − C 0(y)]dy] − S(0)
W elf are change
= S ∗(t) − S ∗(0)
=
x∗Z(t)
[P (y) − C 0(y)]dy]
x∗(0)
which is negative since x∗(t) ≤ x∗(0) and P (y) >
C 0(y) for all y ∈ [0, x∗(0)).
Market failure:
Situations in which the some assumptions of the fundamental theorems of welfare economics do not hold
- usually the first theorem (i.e., market does not lead to
Pareto efficiency).
An important factor behind market failure: EXTERNALITIES.
An externality is present whenever a consumer’s utility
or the technological possibilities of a firm are directly affected by the actions of some other agents (consumers or
firms) in the economy.
Exclude effects mediated through prices.
A Simple Model of Bilateral Externality.
Two price taking agents, i = 1, 2.
Each agent i’s utility depends on her consumption of L
traded goods (x1i, ....xLi) as well as some other action
h ∈ R+ taken by agent 1:
ui(x1i, ....xLi, h) = x1i + gi(x2i, ....xLi, h)
= x1i + gi(x−1i, h)
where
x−1i = (x2i, ....xLi)
and gi is a differentiable function with
∂gi
6= 0.
∂h
The possible range over which h may be chosen is not
affected by the budget constraint.
Let the price of the numeraire good 1 be equal to 1.
Given prices p ∈ RL−1 and wealth wi, the indirect utility
of agent i for each level of h is given by
vi(p, wi, h) = max x1i + gi(x−1i, h)
s.t.
x1i + p • x−1i ≤ wi, x−1i ≥ 0.
equivalent to
vi(p, wi, h) = max[wi − p • x−1i + gi(x−1i, h)]
x−1i
The Walrasian demand for goods 2, ...L are independent
of wi so that
vi(p, wi, h) = wi − p • x−1i(p, h) + gi(x−1i(p, h), h)]
which can be written as being of the form:
vi(p, wi, h) = wi + φi(p, h)
As prices are assumed to be fixed and unaffected by the
choice of h by the agents, we can write φi(p, h) = φi(h).
The preferences of each agent regarding choice of h is
summarized by φi(h).
[The analysis remains unchanged if the two agents are
firms and φi(h) is the derived profit function.]
Assume: φi(h) is twice differentiable with φi”(h) < 0
(strictly concave).
In a competitive equilibrium in which the price vector
is p, both agents maximize utility subject to constraints
imposed by p and wealth wi.
Therefore, in equilibrium consumer 1 chooses h so as to
maximize φi(h):
φ0i(h∗) ≤ 0
= 0, if h∗ > 0..
In any Pareto optimal allocation, it must be the case that
h = ho must solve:
max[φ1(h) + φ2(h)].
h≥0
To see this note that if this is not the case, then
- one can choose a different value of h to increase the
joint welfare
- and then compensate the person who loses as a result
of this change by a transfer of wealth (numeraire good)
so that the welfare of one person can be increased without
reducing the welfare of the other person,
and this would violate Pareto optimality.
FOC:
φ01(ho) ≤ −φ02(ho)
= −φ02(ho), if ho > 0.
Thus, if externality is present so that φ02(h) 6= 0 at all
h ≥ 0, the equilibrium value h∗ is not Pareto optimal
unless ho = h∗ = 0.
Suppose ho > 0, h∗ > 0. Then,
φ01(h∗) = 0
φ01(ho) = −φ02(ho)
If externality is negative - in particular, φ02 < 0, then
φ01(ho) > 0 so that strict concavity of φ1 implies h∗ >
ho.
If externality is positive - in particular, φ02 > 0, then
φ01(ho) < 0 so that strict concavity of φ1 implies h∗ <
ho.
Traditional Solutions:
1. Government Intervention: Quotas and Taxes.
Suppose externality is negative so that h∗ > ho.
- Regulate quantity of h directly and set a maximum
permissible level (ceiling) or quota at ho. [Command and
control]
- Impose a tax on externality generating activity: Pigouvian taxation.
Set tax of th per unit of h on agent 1.Consumer 1 then
maximizes:
max φ1(h) − thh
h≥0
which has the FOC:
φ01(h) ≤ th
= th, if h > 0.
So if tax is set at a level so that
th = −φ02(ho)
then h = ho solves consumer 1’s maximization problem
(uniquely).
Note Pigouvian tax = marginal externality at the optimal
solution = amount consumer 2 is willing to pay to reduce
h marginally from its optimal level ho.
Taxation internalizes an externality.
[However, if ho = 0, then any tax > −φ02(ho) also ensures Pareto optimality.]
All of the above holds when externality is positive so that
h∗ < ho.
In that case the quota is replaced by a minimum quantity
restriction on h and the tax is a subsidy per unit of h.
Even in the case of negative externality, one can subsidize
reduction in h (instead of taxing h) to achieve optimality.
For example, a subsidy of sh to agent 1 for each unit of
reduction below h∗.
Agent 1 maximizes
φ1(h) + sh(h∗ − h)
= [φ1(h) − hsh] + shh∗
which is equivalent to imposing a tax per unit of h equal
to sh and combine this with a lump sum subsidy of shh∗.
So setting sh = −φ02(ho), restores optimality.
Similarly, for a positive externality where h∗ < ho,a tax
on reduction of h below ho can work to reduce optimality.
Note: important to tax externality generating activity directly.
Also note: though direct quantity control (quotas) and
Pigouvian taxes/subsidies are equivalent above, they are
not necessarily so if government has less than full information about benefits and costs of the externality to
individual agents.
2. Decentralized Solution: Bargaining with Enforceable Property Rights.
Establish enforceable property rights over externality generating activity.
Take the case of negative externality.
Suppose we give right to externality free environment to
agent 2.
Agent 1 can try to get permission from agent 2 to generate any level of h > 0 in return for a payment (in terms
of the numeraire good).
Bargaining process.
Take a very simple process: agent 2 makes agent 1 an
offer which agent 1 can either accept or reject.
The offer specifies payment T in return for permission to
generate some level of h.
If agent 1 rejects the offer, no further negotiation takes
place and h = 0.
Agent 1 will accept the offer as long as
φ1(h) − T ≥ φ1(0).
So, in a SPNE, of the bargaining game, agent 2 will set
(h, T ) so as to
max [φ2(h) + T ]
h,T ≥0
s.t. φ1(h) − T ≥ φ1(0)
and this is equivalent to
max[φ2(h) + φ1(h) − φ1(0)]
h≥0
which implies h = ho, the socially optimal level.
What if agent 1 is given property right over externality
generating activity?
Agent 2 can make an offer to pay agent 1 an amount
y in return for reducing the level of externality to some
h < h∗.
Agent 1 will agree to do so only if
φ1(h) + y ≥ φ1(h∗).
So, in SPNE of the bargaining game, agent 2 will set
(h, y) so as to
max φ2(h) − y
h,y≥0
s.t. φ1(h) + y ≥ φ1(0)
and this is equivalent to
max[φ2(h) + φ1(h) − φ1(0)]
h≥0
which again implies h = ho, the socially optimal level.
Coase theorem (Coase, 1960): If trade of the externality
can occur, bargaining will lead to an efficient outcome no
matter how property rights are allocated.
Better than government intervention as it does not require government to have information about individuals.
But does not work well if individuals have incomplete
information about each other.
Also, transaction costs in decentralized bargaining where
many agents are involved.
3. Market for right to generate externality.
Missing market - at the heart of the externality problem.
Suppose:
- property rights are well defined and enforceable
- competitive market for right to engage in externality
generating activity exists.
Again, take the case of negative externality.
Suppose agent 2 has right to externality free environment.
Agent 2 can sell units of the right to generate externality
in a competitive market.
Agent 1 can buy.
Let ph be the price per unit.
Both agents price taking.
Agent 1’s demand:
max φ1(h) − hph
h
which has the FOC:
φ01(hd) ≤ ph
= ph, if hd > 0.
Agent 2’s supply:
max φ2(h) + hph
h
which has the FOC:
φ02(hs) ≤ −ph
= −ph, if hs > 0.
In competitive equilibrium:
and
b
hd = hs = h
b ≤ p ≤ −φ0 (h)
b
φ01(h)
h
2
b = p = −φ0 (h),
b if h
b > 0.
φ01(h)
h
2
b maximizes [φ (h) + φ (h)] i.e.,
This implies h
1
2
b = ho
h
and the rights are traded at equilibrium price
p∗h = φ01(ho) = −φ02(ho).
Public Goods.
A public good is a commodity for which use of a unit of
the good by one agent does not preclude its use by other
agents.
No rivalry in consumption.
If commodity not desirable:
Public bad: consumption of a unit of the good by one
agent does not decrease its consumption by other agents
* Non-depletable: consumption by one individual does
not affect supply available for other agents.
Knowledge, air quality...
Intermediate cases: consumption by one affects availability to others to some degree.
Ex. Congestion effects.
Partially depletable.
If entirely non-depletable: pure public good.
If entirely depletable: pure private good.
Public goods: excludable or non-excludable.
Excludable public goods are those for which it is possible
to exclude some individuals from accessing or consuming
the public good.
Non-excludable: technologically impossible or very costly
to do so.
We focus on non-excludable pure public goods.
Model.
I price taking consumers, L traded private goods and
one public good.
Partial eqm: quantity of public good has no effect on the
demand or prices of the L traded goods.
Each consumer i’s utility:
ui(x1i, ...xLi, x) = x1i + gi(x−1i, x)
where x is the provision of (consumption of) the public
good and x−1i = (x2i, ...xLi).
Let the price of the numeraire good 1 be set equal to 1,
and let p be the price vector of goods 2, ...L.
Then, consumer i solves
max[x1i + gi(x−1i, x)]
s.t.p • x−1i ≤ wi, x−1i ≥ 0.
where wi is the wealth of consumer i (in terms of the numeraire good) available for expenditure on private goods
2, ...L.
Note wi may include profits from shares of firms and may
therefore depend on p.
The demand for goods 2, ...L, is independent of wi.
Given p, the indirect utility for each given level of x:
vi(p, wi, x) = wi − p • x−1i(p, x) + gi(x−1i(p, x), x)
= wi + φi(p, x)
and fixing the prices of goods 2, ...L, we can write φi(p, x) =
φi(x) as the derived utility to individual i from the public
good (willingness to pay for the public good).
We assume φi(x) is twice differentiable with φ0i(x) > 0,
φi”(x) < 0 on R+.
Cost of producing or supplying q units of the public good
is c(q) units of the numeraire good.
We assume c(q) is twice differentiable with c0(q) > 0
and c”(q) > 0 at all q ≥ 0.
Note: for public bad whose reduction is costly: φ0i(x) <
0, c0(q) < 0.
In any Pareto optimal allocation, the provision of the
public good must solve:
max[{
q≥0
I
X
i=1
φi(q)} − c(q)]
To see this, suppose to the contrary that there is a PO
allocation where qb units of the public good are produced
and for some q 6= qb
I
X
i=1
i.e.,
I
X
i=1
φi(q) − c(q) >
I
X
i=1
b − c(q)
b
φi(q)
b >
φi(q) − (c(q) − c(q))
I
X
i=1
b
φi(q)
which implies that if we modify the PO allocation by
changing the quantity of the public good (produced and
consumed) from qb to q and pass on the difference in
production cost to the consumers as change in their consumption of the numeraire good (leaving everything else
unchanged), then the total utility of all individuals increase.
By a suitable transfer of the endowment of numeraire
good between, we can now ensure that there is another
allocation which makes everyone better off relative to the
initial PO allocation. A contradiction.
Therefore, the FOC for socially optimal quantity of public
good q o:
I
X
i=1
φ0i(q o) ≤ c0(q o), if q o = 0,
= c0(q o), if q o > 0.
At an interior solution, the sum of consumers’ marginal
benefits from the public good (marginal willingness to
pay for the public good) must equal the marginal cost of
production.
Private Provision of Public Goods.
Assume market exists for public good and each consumer
i decides how much of the public good xi she wants to
buy at its market price p.
The total amount of the public good purchased is
x=
I
X
xi.
i=1
Assume a single price taking competitive firm produces
the public good at cost c(.).
[Alternatively, J price taking firms with industry cost
function c(.).]
At a competitive equilibrium with price p∗,each consumer
i’s purchase x∗i solves:
max[φi(xi +
xi
X
k6=i
x∗k ) − p∗xi]
taking the quantity of the public good purchased by other
agents as given.
To see this recall that the indirect utility for any level
of total consumption of the public good x (given wi,the
wealth available for expenditure on private goods, and
the price vector p of private goods, which is independent
of the consumption of the public good) is given by
vi(p, wi, x) = wi − p • x−1i(p, x) + gi(x−1i(p, x), x)
= wi + φi(p, x) = wi + φi(x)
and given the equilibrium consumption of the public good
by other agents,
x = xi +
X
x∗k
k6=i
and the wealth available for expenditure on private goods
is
b i − pxi
wi = w
b i is the total income of consumer i (endowment
where w
+ share income) so that agent i must choose xi to maximize
b i − pxi + φi(xi +
w
which yields the above.
X
k6=i
x∗k )
First order necessary and sufficient condition:
φ0i(x∗i +
X
x∗k ) = φ0i(x∗)
k6=i
≤ p∗, if x∗i = 0,
= p∗, if x∗i = 0,
where
x∗ =
I
X
i=1
x∗i .
Firm’s supply q ∗ satisfies
max[p∗q − c(q)]
q≥0
so that first order necessary and sufficient condition:
p∗ ≤ c0(q ∗), if q ∗ = 0
= c0(q ∗), if q ∗ > 0.
Market clearing:
q ∗ = x∗.
Thus, if q ∗ = x∗ > 0,then x∗k > 0 for some k so that
φ0k (x∗) = p∗ = c0(q ∗)
and since φ0i(x∗) > 0 for all i,
I
X
φ0i(q ∗) > c0(q ∗).
i−1
In contrast, the socially efficient (Pareto optimal) level of
provision of the public good xo satisfies
I
X
i−1
φ0i(q o) ≤ c0(q o), if q o = 0,
= c0(q o), if q o > 0.
so that
q o > q ∗.
Market provides too little of the public good.
Reason: purchase of the public good by one agent creates
positive externality for others.
Individual’s private purchase decision does not take into
account this externality.
Free-rider problem.
Suppose that consumers are ordered in their marginal
benefit from the public good i.e.,
φ01(x) < φ02(x) < ..... < φ0I (x), ∀x ≥ 0.
Then,
φ0i(x∗) = p∗
can hold for at most one consumer i and since φ0k (x∗) ≤
p∗ for all k 6= i , i = I.
Therefore, only the consumer who derives the largest marginal benefit from the public good will provide at and the
total provision of the public good would be exactly equal
to what it would be if agent I was the only consumer in
the economy.
The other agents free-ride perfectly.
Government Intervention.
Quantity based intervention: directly provide x∗ (funded
through lump sum taxes).
Price-based intervention: taxes and subsidies:
Subsidy to each consumer i:
si =
X
φ0k (q o)
k6=i
per unit of the public god purchased. In that case, at any
price p of the public good, each consumer i maximizes:
φi(xi +
X
k6=i
xk ) − (p − si)xi
so that FOC:
φ0i(xi +
X
k6=i
xk ) ≤ p − si, if xi = 0
= p − si, if xi > 0.
so that (assuming q o > 0) we have a competitive equilibrium at price po = c0(q o), xo =
I
X
xoi = q o,and
i=1
o
choosing i for which xi > 0 we have from the FOC:
X
X
0
o
o
φi(xi +
xk ) +
φ0k (q o)
k6=i
k6=i
I
X
=
φ0k (q o) = po
k=1
so that
I
X
φ0k (q o) = c0(q o)
k=1
which ensures Pareto optimality.
All of this requires that the government know the benefits
derived by consumers from the public good.
Lindahl Equilibria.
For each consumer i, design a market for the public good
as experienced by consumer i.
Assume that you can price a consumer for all the public
good she enjoys.
Each consumer i’s consumption of the public good is a
distinct good with its own price pi.
Given price pi each consumer decides on her total consumption xi of the public good so as to maximize:
max[φi(xi) − pixi]
xi≥0
FOC:
φ0i(xi) ≤ pi
= pi, if xi > 0.
Think of the firm as producing a bundle of I goods, one
for each consumer, (q1, ..., qI ) subject to a technological
constraint
q1 = ... = qI
which reflects the fact that by its very nature, you can’t
produce different levels of public good consumption for
different consumers.
Thus, in effect, firm solves:
max[(
q≥0
I
X
i=1
piq) − c(q)]
with FOC:
I
X
i=1
pi ≤ c0(q)
= c0(q), if q > 0.
In market equilibrium with price p∗∗
i , the optimal consumption of the public good by each consumer must equal
the total production of the public good:
∗∗
x∗∗
i =q
and the FOCs imply:
I
X
i=1
φ0i(q ∗∗) ≤
I
X
i=1
0 ∗∗
p∗∗
i ≤ c (q )
and
I
X
φ0i(q ∗∗) =
i=1
I
X
0 ∗∗
∗∗
p∗∗
i = c (q ), if q > 0,
i=1
which means that
q ∗∗ = q o
and the market equilibrium is Pareto optimal.
This type of equilibrium with personalized markets for the
public good is called a Lindahl equilibrium.
In essence, each consumer is priced for her entire consumption of the public good, so there are no externalities
or free riding.
However, it requires excludability.
Should be able to exclude consumers from consumption
of the public good if they don’t pay according to their
personalized price.
Also, single buyer in each personalized market makes the
price taking assumption unrealistic.
Multilateral Externalities.
Numerous parties generate and are affected by externalities.
Depletable externalities (private, rivalrous) : experience
of the externality by one agent reduces the amount felt
by other agents. Somewhat like a private good.
Non depletable externalities are like public goods. Ex.
Air pollution.
Lindahl Equilibria.
For each consumer i, design a market for the public good
as experienced by consumer i.
Assume that you can price a consumer for all the public
good she enjoys.
Each consumer i’s consumption of the public good is a
distinct good with its own price pi.
Given price pi each consumer decides on her total consumption xi of the public good so as to maximize:
max[φi(xi) − pixi]
xi≥0
FOC:
φ0i(xi) ≤ pi
= pi, if xi > 0.
Think of the firm as producing a bundle of I goods, one
for each consumer, (q1, ..., qI ) subject to a technological
constraint
q1 = ... = qI
which reflects the fact that by its very nature, you can’t
produce different levels of public good consumption for
different consumers.
Thus, in effect, firm solves:
max[(
q≥0
I
X
i−1
piq) − c(q)]
with FOC:
I
X
i=1
pi ≤ c0(q)
= c0(q), if q > 0.
In market equilibrium with price p∗∗
i , the optimal consumption of the public good by each consumer must equal
the total production of the public good:
∗∗
x∗∗
i =q
and the FOCs imply:
I
X
i=1
φ0i(q ∗∗) ≤
I
X
i=1
0 ∗∗
p∗∗
i ≤ c (q )
and
I
X
φ0i(q ∗∗) =
i=1
I
X
0 ∗∗
∗∗
p∗∗
i = c (q ), if q > 0,
i=1
which means that
q ∗∗ = q o
and the market equilibrium is Pareto optimal.
This type of equilibrium with personalized markets for the
public good is called a Lindahl equilibrium.
In essence, each consumer is priced for her entire consumption of the public good, so there are no externalities
or free riding.
However, it requires excludability.
Should be able to exclude consumers from consumption
of the public good if they don’t pay according to their
personalized price.
Also, single buyer in each personalized market makes the
price taking assumption unrealistic.
Multilateral Externalities.
Numerous parties generate and are affected by externalities.
Depletable externalities : experience of the externality by
one agent reduces the amount felt by other agents.
Like a private good.
Non depletable externalities are like pure public goods.
Ex. Air pollution.
Assume agents who generate externalities are different
from those who experience them. (Simplifying)
In particular, suppose externality is generated by firms
and experienced by consumers.
Also, assume externality generated by firms is homogenous
(consumers do not care about source of externality).
Price taking agents.
L traded goods with price vector p.
J firms that generate externality in the process of production.
Given p, the profit function of firm j can be defined as a
function of the externality hj ≥ 0 it generates: π j (hj ).
I > 1consumers with quasi-linear utility.
Given price vector p, the preferences of consumer i with
e experienced by her is given
respect to the externality h
i
e ).
by her derived utility φi(h
i
e )+w .
Her indirect utility is actually φi(h
i
i
Assume: ∀j = 1, ...J, i = 1, ...I, π j , φi are twice differentiable with π j ”(.) < 0, φi”(.) < 0.
To fix ideas, assume that the externality is negative i.e.,
φ0i(.) < 0, π 0j > 0.
Depletable Externality.
At any unrestrained competitive equilibrium where there
is no market for the externality, each firm j generates
externality h∗j so that
π 0j (h∗j ) ≤ 0, if h∗j = 0,
= 0, if h∗j > 0.
e o, i =
Any PO allocation involves the levels (hoj, j = 1, ..J, h
i
1, ...I) that solve:
max[
s.t.
I
X
i=1
e =
h
i
I
X
hj ,e
hi i=1
J
X
e )+
φi(h
i
J
X
π j (hj )]
j=1
hj .
j=1
The constraint reflects the depletable nature of the externality - one more unit of the externality experienced by
one person implies there is one unit less of the externality
to be experienced by others.
If a PO allocation does not solve the above problem, then
one can modify the profile of externality generated and
consumed so as to increase
I
X
i=1
Since
J
X
e )+
φi(h
i
J
X
π j (hj ).
j=1
π j (hj ) is returned to consumers as share in-
j=1
come the total indirect utility
I
X
i=1
e ) + w ) of all
(φi(h
i
i
individuals can be increased i.e., a higher utility possibility frontier can be reached which violates the initial
hypothesis of Pareto optimality.
If μ is the multiplier for this constraint, FOC (necessary
& sufficient) :
e o) ≤ μ
φ0i(h
i
e o > 0.
= μ, if h
i
μ ≤ −π 0j (hoj),
= −π 0j (hoj), if hoj > 0.
If well defined and enforceable property rights can be
specified over externality and competitive market for the
externality exists (with I, J being large enough so that all
agents are price takers) with price =μ, then Pareto optimality can be attained by decentralized decision making.
Note: here it does not matter whether
- consumers are awarded the property rights so that each
firm has to buy units of the externality generated by it
from the consumers
OR
- firms are awarded the property rights so that each consumer has to buy units of reduction in the externality
generated by firms.
Non-depletable Externality.
Each consumer experiences an externality level equal to
J
X
hj .
j=1
Essentially, the externality generated by firms is pure public bad.
In any Pareto optimal allocation, the level of externality
generated (hoj, j = 1, ...J) satisfies:
max
I
X
hj i=1
φi(
J
X
hk ) +
k=1
J
X
π j (hj )]
j=1
FOC (necessary and sufficient):
I
X
i=1
φ0i(
J
X
k=1
hok ) ≤ −π 0j (hoj)
= −π 0j (hoj), if hoj > 0.
Note: This is exactly the condition for optimal provision
of the public good if we interpret −π 0j as a firm’s marginal
cost of production of public good.
[In our specific discussion of public good, we took J = 1].
As in the case of public good, if we have a competitive
market for the externality we will not get the Pareto optimal level of externality.
Free rider problem.
This is a fundamental difference of the multilateral externality problem from the bilateral case.
Recall that in an unrestrained competitive eqm, each firm
j generates externality h∗j so that
π 0j (h∗j ) ≤ 0, if h∗j = 0,
= 0, if h∗j > 0.
Suppose we give property rights to firms and let consumers buy from any firm each unit of externality reduction below its unrestrained level.
If consumers buy zero from firm j, it would generate h∗j
amount of externality.
Given price p∗∗ for unit reduction in externality, firm j
chooses h∗∗
j so as to solve:
max ∗ π j (hj ) + (h∗j − hj )p∗∗
hj ∈[0,hj ]
and so FOC:
∗∗
π 0j (h∗∗
j ) ≤ p
= p∗∗, if h∗∗
j > 0.
Each consumer i buys reduction of amount x∗∗
i so as to
solve:
max φi(
xi
J
X
j=1
h∗j − xi −
X
k6=i
∗∗x
x∗∗
)
−
p
i
k
so that FOC:
J
X
X
0
∗
∗∗
∗∗
hj − xi −
x∗∗
−φi(
k ) ≤ p
j=1
k6=i
= p∗∗, if x∗∗
i > 0.
In equilibrium,
J
X
(h∗j − h∗∗
j )=
I
X
∗∗
x∗∗
i =x
j=1
i=1
Suppose x∗∗ > 0.Then, x∗∗
i > 0 for some i. In that case
−φ0i(
J
X
j=1
h∗j − x∗∗) = p∗∗
i.e.,
φ0i(
J
X
j=1
∗∗
h∗∗
j ) = −p
and since φ0i < 0, ∀i,
I
X
i=1
φ0i(
J
X
j=1
∗∗
h∗∗
j ) < −p
≤ −π 0j (h∗∗
j ), j = 1...J
This violates Pareto optimality.
Market based solution can work only if we have personalized markets for externality as in the Lindahl equilibrium
concept.
But that, as we have seen, is problematic.
Similar free rider problems would also arise in any bargaining solution.
Government intervention can secure efficiency (as for other
pure public goods).
Quotas: Government can regulate a quota equal to hoj
on externality generation. (ex. emission standards).
Taxes: Impose a tax per unit of externality generation:
t=−
I
X
φ0i(
i=1
J
X
hok )
k=1
Each firm j will then set hj so as to solve:
max π j (hj ) − thj
hj ≥0
which yields FOC:
π 0j (hj ) ≤ t
= t, if hj > 0
and comparing to necessary and sufficient condition for
Pareto efficiency in externality generation, we have hj =
hoj.
Both taxes and quota require government to know utility
and profit functions of consumers & firms.
Mixed solution: Tradeable Externality permit.
Government gives ho =
J
X
hok permits to firms.
k=1
Each permit allows a firm to generate a unit of externality.
Total amount of externality generated is at the optimal
level.
Question: will each firm generate the optimal level of
externality?
Permits distributed arbitrarily among firms.
Firm j given hj permits.
Firms then trade these permits in a competitive market.
Let p∗h be the equilibrium price.
Each firm j’s demand for permits (and its externality
generation) at this price is given by solving:
max π j (hj ) + p∗h(hj − hj )
hj
so that from the FOC:
π 0j (hj ) ≤ p∗h
= p∗h, if hj > 0.
Market clearing:
J
X
j=1
hj = ho.
Comparing to the FOC for Pareto optimality, it can be
shown that the above equations can be satisfied only if
p∗h = −
I
X
φ0i(ho)
i=1
and
hj = hoj.
This scheme is useful when government has limited information on individual profit functions of firms & cannot
say which firm should bear how much of the externality
reduction.
Of course, it requires government to be able to compute
the aggregate externality level ho.