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HB31
.M415
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no.
working paper
department
of economics
POLLUTION PERMITS AND
COMPLIANCE STRATEGIES
Jean-Jacques Laffont
Jean Tirole
95-9
June, 1994
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
POLLUTION PERMITS AND
COMPLIANCE STRATEGIES
Jean-Jacques Laffont
Jean Tirole
June, 1994
95-9
MASSACHUSETTS US'!'; rUTG
OF TECHNOLOGY
MAR 2
9 1995
LIBRARIES
95-9
Pollution Permits and Compliance
Strategies*
and
Jean- Jacques Laifont^
June
Jean Tirole*
1994
16,
Abstract
The paper
analyzes the impact of spot and futures markets for
tradeable pollution permits on the potential polluters' compliance decisions.
Polluters can
else stop
buy permits,
markets induce excessive
market reduces
invest in pollution abatement, or
We
show that stand-alone spot
investment. The introduction of a futures
production or source out.
this incentive to invest,
to control pollution.
A menu
coupled with intertemporally bundled
Because of
its
but
is
not the optimal
way
of options on pollution rights, possibly
focus on long-run
sales, yields
demand
higher welfare.
elasticities
and rent
ex-
a variety of situations such as
demand-side management, public transportation, bypass in telecommunications, or forward sales by a private monopolist.
traction, this paper can be applied to
'The authors are
grateful to Lans Bovenberg, Carlo Carraro, Paul Joskow, Peter
Klibanoff and Y.H. Farzin for helpful discussions and to Fondazione Eni Enrico Mattei for
financial support.
tIDEI and
GREMAQ,
Toulouse, France
*IDEI, Toulouse, France;
CERAS,
Paris,
and MIT, Cambridge,
MA, USA
Introduction
1
Most developed countries have been experimenting with taxes, subsidies and markets
for pollution
example,
command approach
permits to replace the old
title
IV
US Clean
of the 1990
Amendments has
Air Act
emissions of sulfur dioxyde cutting pollution by
to pollution control. 1
more than a
set a
For
cap on global
half from 1995 on;
it
defines
a system of one-year spot and futures permits that are tradeable on the Chicago Board
of Trade.
2
The
and more generally the design of pollution control
futures price,
will
determine the polluters' incentives to build new plants or adopt alternative, pollutionreducing technologies, namely investments in scrubbers or in boilers using low-sulfur coal
or in other forms of fuel-switching (investments in gas or nuclear plants). Because such
investments are in part sunk,
penalties
on pollution so
it is
important for the polluters to correctly foresee future
as to plan their compliance strategy.
existing tradeable emissions permits
program
We
affects incentives to
how
study
first
the
reduce pollution, and
then derive the optimal mechanism.
While our primary motivation
mind that the
in
over to any
lies
with the control of pollution,
form of bypass
side
side
some long-term investment* and
that involves
take the long-run elasticity of
demand
demand
management
facing rational customers
demand
for the
fuel or electricity
effect of
when
They
is
into account.
good produced by the
in
Among
other forms,
for,
4
demand
the key decision
an alternative technology that reduces
utility.
For example, customers
heating or manufacturing equipment
is
therefore to
Consider for instance the topical
in the electricity sector.
whether to invest
expectations on bypass
carry
power (the government, a private firm) must
management programs have an important dynamic component;
their
must be borne
ideas developed in this section have wide applicability.
situations in which a party with market
issue of
it
also illustrated
if
may
switch to
they foresee high gas prices.
The
by customers purchasing automobiles
anticipating a deterioration in the quality of public transportation, by firms building
direct access to
a long-distance telephone company
if
they extrapolate high access charges
to the local loop, by the possibility of banks developing private clearing houses at
home or
abroad when concerned about a high usage cost of a Central-Bank-controlled settlement
'See, e.g.,
2
(1992).
3
Hahn-Noll (1983),
Our (1994a) paper
In a previous
paper
OECD
NRRI
For details, see, e.g.
(
offers
(1993),
and Tietenberg (1985).
and Wessler
on the US reform.
(1992), Public Utilities Report (1993), Rose-Burns (1993)
a
(brief) industrial organization perspective
Laffont-Tirole (1990)),
degree price discrimination by a regulated firm
we studied how bypass interferes with optimal secondbecause the model was static, it abstracted from the
;
commitment and intertemporal price discrimination.
management aims at providing, through, say, residential energy conservation audits,
information and advice to customers on how to take advantage of peak load pricing and how to select
issues of
4
(Pure) load
the most efficient energy.
1
system, as well as by
many
private sector applications.
Our framework
also applies to the
case of forward sales by a private monopolist.
Because of the wide applicability of our analysis, we want to abstract from the
economy and regulatory
cinating) political
functioning of
50 2
issues that are having
pollution permits markets in the US,
question of intertemporal pricing vis-a-vis a long-term
a deep impact on the
and instead focus on the general
demand function. For
some of our conclusions apply more straightforwardly
(fas-
this reason,
to the other situations described
above than to the SO2 pollution permits markets.
The
of spot
first
part of the paper (sections 2 through 4) analyzes the properties of a system
and futures markets
government
for pollution permits. Stand-alone spot
create excessive incentives for investment.
pollution permits market
that
is
number
sets at the beginning of each period the
is
The reason
markets
(in
which the
of permits for that period)
for this excessive
that the optimal price of pollution permits
is
bypass of the
a Ramsey price,
exceeds the marginal pollution cost in order to contribute to the reduction of
the overall budget deficit.
Agents do not internalize the
loss of
revenue they create by
bypassing the market and therefore invest too much. [The other applications of our model
may
help clarify this point. Consumers of electricity
who
invest in self-generation or in
energy-switching equipment do not internalize the loss of revenue they impose on the
power company, which
balance
its
is
forced to charge electricity above
its
marginal cost in order to
budget. Similarly, firms which install a direct link with long-distance telephone
companies to bypass the local exchange do not internalize the local telephone company's
wedge between
its
price
and marginal
cost. Last, the
not internalize the monopoly mark-up
of the
when they
customers of a private monopoly do
invest in order to forego
consumption
monopolized good.]
This incentive to overinvest can be reduced by the introduction of a futures market.
Suppose for example that the government
at a price slightly
period welfare loss
sells in
period
1
permits to pollute in period
2,
below that that prevails under stand-alone spot markets. The secondis
of the second-order. But the
commitment
to a lower second-period
price for permits discourages investment, yielding a first-order gain because of the
up on second-period permits.
We
then show that the allowance program
is
mark-
not by
itself
time consistent, and discuss ways of establishing time consistency.
The second
part of the paper (sections 5 and 6) derives welfare-improving reforms by
considering the optimal
instruments.
Under
mechanism under two
alternative assumptions about regulatory
"overall regulation", the regulator monitors not only the agents'
pollution, but also their investment in bypass
then consists in offering a
menu
and
their production.
The optimal
control
of options on pollution rights, the purchase price of
which decreases with the striking price (these options take the form of a bilateral contract
OTC
between the regulator and the agent similar to those offered on
markets.)
Under
"pure pollution regulation", the regulator (an environmental protection agency) monitors
only the agents' pollution.
It
then
may become
optimal to
the purchase of options
tie
with a lower price for current permits, giving rise to bundled sales of pollution permits.
Section 7 concludes by listing
some
desirable extensions.
From the Ramsey-boiteux
Technically, our paper builds on four literatures.
we borrow the idea that
pricing
must participate to the coverage of the
and therefore that the prices of pollution permits must exceed marginal
will
borrow from the durable- good literature
the government
5
when we introduce
analysis,
overall deficit,
cost.
We
then
futures markets; for,
then a monopoly issuer of long-term pollution permits. Our analysis
is
generalizes the standard durable-good one because the willingness to
pay of the buyers
of futures permits can change over time, and, mainly, because the buyers can bypass
through a private investment. Accordingly, we
such as the nonoptimality of "leasing", that
under commitment.
It differs
from
agents' second-period willingness to pay.
two respects.
First, the
model investment increases
Our model,
is
mechanism
as a simple
menu
investment
stochastically the
besides having a different economic
technically simpler. Second, the particular context of our
us to view the optimal
Besanko's model
Section 5.1, on overall
of spot markets.
their analysis in
eliminates pollution (bypass) while in their
is
is
obtain results of independent interest
on Baron-Besanko (1984)'s pionneering analysis of mechanism design
regulation, builds
interpretation,
will
model
leads
of second-period options (Baron and
couched in a regulatory framework.) Last, we use insights from the
theory of options in finance.
The model
2
.
Valuations for pollution:
•
of agents/potential polluters.
period.
profit
An
The model has two
periods,
Each agent can consume
agent's valuation for polluting one unit
is
6t
6
t
or
€
=
1
1,2,
and a continuum
unit of pollution in each
[0, 1].
For example, 9t
is
obtained by producing one unit of output, thus creating one unit of pollution.
Alternatively,
it is
the firm's opportunity cost of sourcing out for a short-run input that
the agent knows her date-1 valuation 6 X but
economizes one unit of pollution. At date
1,
not yet her date- 2 valuation $2 which
learned at date 2 and
is
,
is
distributed according
to the conditional cumulative distribution function GjC^I^i)) with density ^2(^2^1)-
assume that a high
likely, in
5
6
the
first-period valuation
makes a high second-period valuation more
the sense of first-order stochastic dominance:
^- <
See Fudenberg-Tirole (1991, chapter 10) for a review of this literature.
analysis can be generalized to an arbitrary number of periods.
The
We
0.
In addition to being
change
realistic (the
in 6 reflecting for
example changes
in
demand
or outage of a plant
that forces greater use of dirtier plants), imperfect correlation also provides a rationale
The
for second-period trading of pollution permits.
first-period valuation
according to the cumulative distribution Fi(8\), with density fi(Oi).
—
assumption that /i/(l
welfare function.
technology. At cost
the usual
agents' individual valuation, and
i,
each agent chooses whether to invest in a pollution-eliminating
1,
the agent can obtain utility
of investment, the agent
must
2
without polluting. 7 In the absence
(as in period 1) pollute
one unit
Note that we assume that the investment pays
valuation.
We make
pollutes.
At date
• Investment:
distributed
nondecreasing in order to ensure concavity of the social
is
The government does not observe the
who
only observes
Fi)
is
off
in order to obtain her
only in period
2;
it
is
straightforward to solve the alternative case in which the investment can be used in both
Note
periods.
last that
the investment
not transferable (at least at a low cost) from
is
one agent to another.
The
private and social discount factor
• Pollution
we
let
damage
political
welfare.
public
of pollution,
Normalizing the mass of agents to be one,
who
of agents
where
D
t
(0)
S.
=
0,
pollute at date
D't
>
0,
D" >
t.
0.
D
Let
We
t
(n t ) denote
will abstract
economy considerations and assume that the government maximizes
The government
money
denoted
welfare criterion:
n t denote the fraction/number
the social
from
damage and
is
faces a
costs society $(1
-I-
shadow
cost of public funds A
>
0;
that
is,
social
raising $1 of
A) because of the distortionary taxation. Alternatively,
the taxes on pollution or the proceeds from sales of pollution permits could be used to
finance the fixed costs of public utilities, in which case A would stand for the
of their budget constraints.
principal
is
8
a private party.
shadow
cost
[The analysis applies equally well to a situation in which the
What
matters for our analysis
is
that the principal does not
fully value the agents' rents.]
In the
7
absence of investment and in a static context, the optimal mechanism consists,
is modeled somewhat rigidly. There are several extensions worth considone could allow a more continuous choice in abatement (in this model, say, through a costly
choice of a probability that the investment succeeds in eliminating pollution). Second, the abatement
cost could vary across agents (and possibly, in the overall regulation context, be unobserved by the gov-
The investment technology
ering. First,
might also depend on the agents' first-period type. Third, investments create purely private
we discuss investments which generate innovations in depollution
technology which are public goods.
d
This second possibility is, for example, relevant when allowances are freely allocated to utilities before
being traded. [We would not expect the shadow costs of all utilities' budget constraints to be equalized,
but this does not alter the nature of the argument.]
We should also point out that there is an intense debate in the US about who should benefit from the
sale of emissions allowances (see, e.g., Burkhart (1993)).
ernment);
it
benefits; in Laffont-Tirole (1994b),
as
n
known, either
well
is
so as to
in setting a price
maximize the
and price mechanisms
think of the
p or
in
social welfare function.
in
mechanism
choosing a number of emissions allowances
(There
our model because there
as a quantity
is
is
no distinction between quantity
no aggregate uncertainty.
mechanism when we introduce the
We
will
sale of future
=
allowances.) For an arbitrary distribution F(6) with density f(6), let n
N(p) =
_1
denote the static demand function and p = P{n) = iV (n) the static inverse
l
— F(p)
demand
Given a pollution damage function D(-), the optimal number of allowances
function.
is
given by:
max {_£(„) +
+
x)nP
+J\o- P)f{e)de)
=
denote the elasticity of demand, the optimal price for allowances
/jpf^
given by the standard Ramsey formula:
Letting
is
(i
77
The marginal
to D'/(l
+
A).
social cost of pollution,
W
l+A,'
p
when expressed
in cost of public funds,
We
equal
Equation (1) says that the Ramsey index for the "good pollution" sold
by a social welfare maximizing regulator subject to budget considerations
Ramsey
is
fraction (A/(l
+
A)) divided
by the
elasticity of
is
equal to a
demand.
conclude that in a static context, the social optimum can be implemented by a
market for pollution permits, with the price or the number of permits set
at their
Ramsey
level.
We could extend (although
Note that we have adopted a partial equilibrium approach.
we have not done
so) our analysis to
about partial equilibrium
for inputs that
may
the agents might
market
in
fire
is
that
it
a general equilibrium situation. 9 The main concern
may ignore the effect
themselves be taxed.
some
of pollution taxes
One mainly has
of their employees
when the tax
affected.
If this is
must be replaced by (presumably higher)
is
raised.
possibility that
In a perfect labor
not the case, the elasticities of our analysis
This correction
is
10
which
reflect a)
the layoff
demand and supply
functions.
superelasticities,
responsiveness to a pollution tax and b) the global labor
9
mind the
which the employees could find a similar job elsewhere, the tax income levied
on labor would not be
10
in
on the demand
of course irrelevant in the case of a private principal.
11
See Laffont-Tirole (1993, section 3.9) for an extension of regulatory models to general equilibrium.
See Laffont-Tirole (1993, chapter 3) for an exposition of the computation of such superelasticities in
the context of multiple outputs.
n Or
in case of
a (nonbenevolent) regulator who does not internalize the revenue from other taxes.
For example, the regulator might internalize the revenue from the pollution tax because this revenue is
earmarked for specific environmental projects, but not the revenue from the other taxes which are used
to reduce the
budget
deficit.
A
related point concerns the specific application of our
In the US,
70%
A
constraint.
of
S0 2
pollution
is
produced by regulated
to the
S0 2 market.
subject to a budget
utilities
reduction in the pollution tax relaxes the budget constraint and should
therefore be valued at the
shadow
cost of the budget constraint.
US
note that the extreme subsidy implied by the
lump-sum
model
transfer
from taxpayers to regulated
[Incidentally, let us
grandfathering approach amounts to a
utilities,
a policy that
in principle
is
ruled
out by regulatory statutes.]
Residual second-period
•
second-period
period
1
demand
demand curve when
and the others do not
to invest.)
The
residual
demand curve
=
is
/'/i(0i)
(1
+
>
6 choose to invest in
valuation agents have
more
incentive
- G2 (p2 \01 ))dei.
[1
Jo
P R (6) denote the Ramsey price given by
R
12
is, P {6) solves:
maxj-I> 2 (n 2 (p2 ,0)) +
introduce the residual
then given by:
Let
That
we
the agents with valuation 6\
we will see, higher
(as
n 2 (p 2 ,0)
For future reference,
curve:
AJnjfo.flJpa
+
(1) for
(2)
the residual
demand curve n 2 (-, 0).
f Wi) [/Va -PakfcC^i)^] dOA
.
Q
(3)
We
of
assume that the Ramsey price increases with 8 (which
demand
is
not too sensitive to
cost price for the residual
6).
demand
For future reference,
if
the elasticity
will also define the
marginal
9 have invested,
TA
commitment and knowing
R
the government would choose price P {9) in the
This marginal cost price increases with
>
the case
curve:
1
that agents with 6t
we
is
6.
In the absence of
second period.
Remark
rise to
:
Agents have unit demands
for pollution in this
model. Multi-unit demands give
second-degree price discrimination and, possibly, to marginal prices below marginal
costs in the presence of bypass (Laffont-Tirole (1990)).
Multi-unit
demands
also
make
allowance markets suboptimal even in a static context. In this sense our model depicts
the best case for markets.
12
We
assume that the second-period welfare function
is strictly
concave in P2.
Incentives to invest under a trading
3
program
Stand-alone spot markets and the introduction of a futures
3.1
market
The Clean Air Act Amendments
up a market
set
already fixed.
13
abatement, we
for
them
of 1990 created sulfur dioxide emissions allowances, and
Chicago Board of Trade. The number of allowances
at the
To analyze the
will first
effect of
is
a market program on investment in pollution
assume that the government
is
commit
able to
to the
number
of
allowances (nj, n 2 ), or equivalently (in this situation of perfect information about demand)
market prices {pi,p 2 )
to
at date
the government of issuing
for
1.
We will then consider the implications of the possibility
new permits
or of buying back existing ones at date 2.
Whether the government can commit or not
2,
the government optimally chooses in period
demand curve
ni(pi)
=
1
—
to modify the
Ramsey
the
1
number of permits
F\(pi) (see equation (l
price corresponding to the
The
)).
in period
focus of our analysis will
naturally be the choice of the second-period price P2.
Consider
first
the agents' investment decision at date
period price P2, an agent with valuation
8E{62 \0X )
6\ invests if
-i>
6E[max(92
Given an expected second-
1.
and only
- p2
,
0)|0 X J.
Using first-order stochastic dominance, we see that equation
such that the agent invests
if
and only
if
>
9t
9 (p2 ). u
m
nonincreasing: the higher the expected price, the higher
market.
We
will
assume that
in the relevant
do not), and therefore the cutoff
is
range
<
if
m
( .4 )
defines a cutoff 9 (p 2 )
Furthermore, the cutoff
<
(some bypass and others
1
strictly decreasing.
Including the investment costs, the second-period welfare can be written
Wa(ft, **(*))
is
the incentive to bypass the
is
9*
(4)
= -D 2 (n 2 (p2 ,9'(p2 )))
+ ^)n 2 {p2 ,e
+
(l
+
[^
/i(*i)
as:
(5)
m
(p 2 ))p2
JO* - p2 )<7 2 (02 |0i)<W2 ]
<Mi
[JPa
1
+
13
In fact, the
bill
O'M*^-*
states that an allowance
The motivation
is
d6x
.
not a property right and can be "limited, revoked and
such ill-defined property rights seems to be the desire to
government to react to macroeconomic news about demand and supply (Hausker
(1992, p. 556)). There is no unanticipated macroeconomic shock in our basic model, and we will
accordingly assume that allowances carry a clear and irrevocable property right. See also the discussion
on aggregate shocks in this section and the next.
14
The derivative of the difference between the left- and right-hand sides of equation ( 4 ) with respect
otherwise modified."
leave flexibility to the
to 0j
is
equal to -*/„"'
^d02 > 0.
for granting
Let us define
rf
= P R (0'(p?))
and
p
The Ramsey
price
p*
is
MC _
pMOpigO )}
show that by committing
gives
lease pollution
to a lower second-period price of pollution permits
commitment can be complemented
by selling pollution rights on a futures market.
is
and would
2.
the government can improve welfare. Such a
p2
1
the gov-
p^
would use marginal cost pricing at date
1
in period
if
That is, p2 is the price that would prevail if only spot markets were
c would obtain if it were known that the government (suboptimally)
rights each period.
We now
rfe
the rational-expectations price that would obtain
ernment could not commit to a second-period price
set up. Similarly,
<
We denote the futures
in period
price 8p2 so that
,
the period-two price of the advance pollution permit. Equivalently, the government
away
to all agents options to pollute at price
these two interpretations breaks
see section
p2
in period 2 [the equivalence
down when the government's
credibility
is
between
in question:
4].
Proposition
1
Under commitment,
second-period price p\
the optimal emissions allowances
€ (P^'P?)- In
program
particular, the introduction of a futures
yields a
market
lowers the second period price from p 2 to p\ and reduces bypass.
Proof: Let us differentiate (5):
dW
dW
dW2 d9'
dp2
dp 2
36 dfa
2
where
(a)
f£ >
p,
<
P*(0'(p2 )), «fr
<
if
p2 <
P MC (e*(p2 )),
and j£
<
0.
< pfc Then 0'(p2 > 9 m (j%c ) and so ^(^(ft)) > P MC {6"
= P2°- Hence pa < P MC (0'(p2)) and therefore
> 0, a contradiction.
Suppose first that ft
(P2°))
(b)
if
2
Suppose next that P2
increases with 6,
.
)
^
>
p?.
From our
> p? = P*(r(p?)) >
So p2 also exceeds the Ramsey price
^<
assumption that the Ramsey price
we have
p2
Therefore
earlier
0,
and
^^ <
for the
(since
**(*•(*»)).
corresponding residual demand curve.
p2 >
P MC (0"(p2)))-
D
.
Intuitively, the
of
commitment
government commits to a lower second-period price than
in the
absence
investment and limit inefficient bypass. Because
in order to discourage
>
pollution permits are sold at a price exceeding the marginal social cost (pf
a reduction in bypass increases welfare.
The
10
P?
),
be best grasped by
intuition for this can
looking at the rational-expectations equilibrium of a stand-alone spot market system.
The
and
regulator then optimizes over the spot market price in period 2,
(this is the
envelope theorem). Because, furthermore, the cutoff type
is
8W /dp2 =
so
2
indifferent
between
bypassing and using second-period permits, the only effect on welfare of a change in the
cut off 6
—
=
Welfare increases with the cutoff
Let us
damage
the direct effect on goverment revenue and on pollution
is
now come back
[{
1
+ X )P2
:
-D>]—
because p*
> g&a&Lpg))
to the case of a futures market.
.
With a
futures market, the
regulator can always duplicate the stand-alone spot market solution by committing to
price pf.
But he
can do better by lowering the futures price p? below p£
in general
order to lower the cutoff 6* fa).
He however
does not go as far as selling permits at price
equal to the second-period marginal cost, because at this point there
to lowering the cutoff. This yields the intuition for Proposition
It
is
sometimes suggested
create excessive bypass.
side
in the regulatory
1.
arena that prices above marginal cost
that "the fact that electricity prices throughout
parts of the United States currently are above marginal costs by
that consumers are overconserving."
Our dynamic
The same
point
is
made
analysis further points at
The government can organize a
this context:
no longer a gain
is
For example, Costello (1992), discussing the value of demand-
management programs, argues
pollution allowances.
in
and
many
in itself suggests
here in the context of
a way of limiting bypass
futures market for permits
and
sell
in
a large
enough number of permits so as to somewhat discourage investment.
Discussion
3.2
Our treatment
•
deserves a
number
The spot market as a guide
Our
spot
is
to investment in pollution abatement.
on the number of advance allowances n 2 to be put
number of allowances n has no
at its
ment
focus
of comments.
x
Ramsey
in the
level.
More
on investment and therefore
is
The
optimally set
generally, the spot price for allowances could affect invest-
same way current
in electricity conservation.
effect
in the market.
electricity prices are often
thought to guide investments
This effect could operate through two channels
in
our model.
First,
investments might pay
off
already in period
plements a low second- period price in the
A
1.
low first-period price then com-
Second, the government
fight against bypass.
might have private information about the social cost of pollution or about the aggregate
demand
In a situation in which the government
for permits.
second-period prices, the government
may
unable to commit to
is
try to signal a low second-period price through
a low first-period price.
Time pattern of pollution and permits
•
We
can easily analyze the temporal evolution of pollution
the case in which the pollution
agents'
E
demand
G2{0\0\) for
6l
damage is
As we
all 6.
that (under
commitment and
That pollution decreases over time
demand
agents with high
in pollution
for all n,
and the
also constant, F(9)
is
=
for
>
A small) nj
n\ and p\
>
p\.
a logical consequence of investments in pollution-
is
for permits disappear
due to the
is
facts that,
from the market because they invest
abatement, and, second, that a commitment to a low futures price reduces
excessive bypass.
While we
time-decreasing price
may
The reason why we
ance assumptions
pollution
may
raise the
demand
find these
two reasons compelling, their implication of a
not necessarily be appealing.
feel time-increasing prices
may be
may be
desirable
violated. Consider the pollution
is
GNP
per capita
may
Second, scientific advances or observations
may
increase over time for three reasons. First, growth in
for pollution control.
that the time invari-
damage. The marginal cost of
bring bad news about environmental damage. And, perhaps
in
prices in
not necessarily good assumptions.
That permits become cheaper over time
abating technologies.
= D 2 (n)
absence of investment
will discuss, these are
and permits
levels
time- invariant, Di(n)
for pollution rights in the
One can then show
first,
price.
more
cynically,
an increase
environmental awareness or the organization of powerful environmental interest groups
may
force otherwise well-informed governments to
(in this
view, the
politicians
tion
damage function
and not as the
t
(-) is
to be interpreted as the cost of pollution to
social cost of pollution). Third, for
damage depends on the stock
pollution,
D
pay more attention to the environment
some
pollutants the pollu-
rather than the flow, and the marginal costs of current
namely the present discounted value of the associated future pollution
ments, increases over time. Last, we note that the assumption of a time-invariant
for pollution
permits (absent investment)
over time, and there
is
is
also strong.
incre-
demand
The composition of GNP changes
no reason why the environmental impact of production remains
constant.
•
Aggregate uncertainty about the cost of future pollution damage.
We
have assumed that the social cost of pollution tomorrow
10
is
known, and more
generally that the government can perfectly
may
tomorrow. In practice, the government
to adjust the
number
compute today the optimal
is
much
and may want
face substantial uncertainty
of permits once the uncertainty
show that the pollution damage
level of pollution
is
resolved. Scientific advances
may
greater or lower than one expected. Similarly,
the price of clean technologies (such as gas)
may
fluctuate.
This uncertainty does not
invalidate the general point that the government wants to sell forward permits in order
to guide investment.
section
does exacerbate the time-consistency issue, as we discuss in
[It
4.]
Suppose that the second-period pollution damage
variable k
number
is
is £> 2 ("2i«)i
The optimal
publicly revealed at the beginning of period 2.
of second-period permits,
ti
2 {k),
must as
and that the random
contingent
in the noncontingent case trade off the
second-period welfare and the effect on investment behavior. Let p2 (n) denote ' ne secondperiod price that will prevail (clear the spot market).
Given the equilibrium cutoff
0*,
p2 (n) and n 2 (n) are linked by the following relationship:
n 2 (p2 (K),0
m
)
= nl(K).
Let the government commit to intervening in the second-period spot market (by buying
number
or selling) in order to support a total
cutoff
is
where the
and
n 2 (it)
in state of nature
k.
first
-i=
expectation in (6)
is
The
E[max(B2 -rf(«),0)in
(6)
with respect to 6 2 and the second with respect to
Price pl(n) contributes, for each k, to the determination of the investment
behavior as summarized by the cutoff
Let
W
2 ( P2 ,0',k)
6'.
= -D2 {n 2 {p2, 0*),k)
+{l
+ \)MP2,9 m )P2
+ ff /i(«i)[/i(*» - P2)92(e2 \e
+ S}.M0i)[E(62 \6
denote the state-contingent second-period welfare.
constraint (6).
The
1
2
(p;(*c),
dp2
*-,«)
1
)d02 ]d0 1
)-i]dO1
And
first-order condition with respect to
dW
As
it.
then given by
E{e2 \P)
62
of permits
let
p 2 (it)
fi
denote the multiplier of
is:
= li[l-G2 (p;(K)\6-))>0.
in the case of certainty, the second-period price in
each state of nature
is
lower than
the ex post optimal (Ramsey) second-period price, in order to discourage investment.
11
Assume
further that a higher k raises the marginal pollution
Then a standard
0).
damage (d2 .D 2 /3n 2 3/c >
revealed preference argument applied to the Lagrangian shows that
the second-period pollution nj decreases with
k.
Last, note that price reductions below the ex post optimal price are
damage
the lower the pollution
—
is,
the lower
is.)
it
The
point
is
effective,
that the probability
on not investing, the cutoff type uses a second-period permit, namely
that, conditionally
[1
(that
more
The
G^CpSC*)!^")] decreases with k.
cutoff type
remark
price reductions in high-pollution states. This
therefore
is
will
more appreciative
have implications
of
for the next
section.
4
Is
Even
if
the allowance program time consistent?
the agents' property rights are well specified, the government
number
alter the
=
of allowances nj
n 2 (p2 ,#"(p2 )) °y
either selling
may
in period 2
new ones
or buying
back existing ones.
Even though the number
period residual
The reason
is
demand
of allowances
n 2 exceeds the Ramsey
curve, the government has
level given the second-
no incentive to buy back allowances.
that ex post capital gains created by a reduction in the
number of allowances
goes to the owners of allowances, while those linked with ex ante reduction of the number
raise the price received
by the government. To see
allowances at price p£, the government buys back nj
satisfying
n2
=
n 2 (p2,0*(P2))- The marginal change
[~D'2
The
first
term
— n2
suppose that, after issuing n"2
allowances at a price p^
in welfare
is
equal to
in this expression
is
the efficiency effect and, for P2
The second term
is
> P2
i
ls
strictly
the distributional
with the transfer from the government to the owners of allowances. So,
Second-period welfare after a buy back of n 2
w(p2)
—
n 2 allowances at the new market price pi
= -d 3 (n 2 (p2, *•(!$))) + a
P; n ;
[
+•/
+ mmp,, r(A)) - •»;]]
e 2 g 2 (e 2 \e 1 )de 2 de l
fiWi)
JO
/0
Jp,
Jvi
with
n2
=
n a <p,,*-(p5))
= /
'
/,(*,)[1
- C(p>|«,)]«0i.
Jo
At date
2,
P2
op*
buybacks are suboptimal.
I5
>
15
+ (1 + A)*] p- - \(n; - n,).
negative as price exceeds social marginal cost.
loss associated
this,
the cut off type 9'{p 2 )
The marginal change
is
and the regulator optimizes over p2then given by the formula in the text.
fixed
in welfare *jj£ is
12
is
:
number
In contrast, a small increase in the
period welfare.
resist
As
in the
of allowances at
=
r*2
n\ raises second-
standard durable-goods model, the issuer of permits cannot
imposing capital losses on the owners of permits. So, the allowance program
is
not
time consistent.
How
can the government restore confidence that
The standard solution
allowances in the future?
consists of leasing instead of selling, does not
invest in bypass technologies.
spot market.
But, as
work
it
will
not flood the market with
to the durable-goods problem,
in a context in which
which
some consumers
Here, "leasing" amounts to selling allowances only on the
we have
seen, the absence of a futures
market induces excessive
bypass.
In contrast, a price support policy (the equivalent of a
"money-back guarantee" or
"most-favored-nation clause" in industrial organization) allows the government to solve
its
time consistency problem. Suppose that the government promises to reimburse the n^
owners of allowances for any capital
new
issues of allowances. For p?
decrease
<
loss {p\
P21 * ne
that they incur as the result of
marginal change in welfare due to a unit price
is
-[-D'2 +
as p\
— p?)^ >
+
{\
< P R (0*(pD). By insuring owners
\)p^-\n <V
(7)
2
against capital losses, the government internalizes
16
the cost of devaluing existing property rights.
An
alternative
commitment device
pollution permits at price p\.
That
is,
striking price pj. Agents then do not
is
to give
away
in period
1
free options to purchase
each agent receives at price
pay
for
R=
an option with
an asset that the government
tempted to "expropriate" by flooding the market with new copies of
this asset.
Ex
the government would actually like to raise the permits price from pj to P^i^iPi))
However,
this
move
is
opposed by agents who were granted
in period
1
be
will later
post,
>
P2-
the right to pollute
at price p%.
We summarize
the analysis of this section
Proposition 2 The allowance program
would
like to sell
new permits
in the
is
in:
not time consistent in that the government
second period. The standard method to achieve the
commitment outcome, namely
leasing (stand-alone spot markets) is ineffective because of
the possibility of investment.
The commitment outcome, though, can
be achieved either
through a price support policy or through a first-period giveaway of options to pollute at
striking price
16
p!J.
Another commitment mechanism might, as usual, be a reputation
13
for not issuing
new
allowances.
•
Time consistency
in the presence of aggregate uncertainty.
Suppose now that the second-period pollution damage
D
2
(n 2 ,tz)
state contingent
is
and that the realization of the random variable k occurs at the beginning of period 2
subsection 3.2).
It is
straightforward to obtain time consistency
instance, the government can give
buy pollution permits
to
nature
at price
away
P2 {k)
«
is
verifiable.
For
in period 1 options that will enable the agents
(see subsection 3.2 for its definition) in state of
Alternatively the government could
/c.
if
(see
commit
to a state-contingent price support
policy.
Time
consistency
pose that k
is
is
harder to achieve when the state of nature
is
to understate the pollution
new permits. One may then think about
patibility for the
not verifiable. Sup-
The government may then be tempted
learned by the government only.
understate k, that
is
damage
in order to flood
the market with
price support policies that restore incentive
government. Suppose that in period
1
the government
to
sells
n2
=
comnl(R)
futures permits, where k corresponds to the highest possible pollution damage. Let the
government
P?
<
P21
set
a price p2 and commit to a refund policy rfa) per permit for any price
with r (p2)
=
(The h 2 futures permits are sold
say.
market given the refund policy and the
want to
sell
new permits
at a
at a price that clears the
fact that, in equilibrium the
government must
market clearing price p 2 (k) in each state of nature
k.)
Can
a refund policy implement the commitment outcome? Using the notation of subsection
3.2, in state of
nature
/c,
the government chooses pi so as to maximize:
W
2 (P2,
The
0', «)
-
A[r(p2 )
+ p 2 ]n 2
.
analysis in subsection 3.2 suggests choosing r(-) such that
\[r'(p2 )
+
l]n 2
= n[l-G2 (p2 \d-)]
or
r'(ft)
= ^[1 - GMP)] ~
I-
Recall that in the absence of aggregate uncertainty, the price support rule satisfied r'(p2 )
— 1.
Here
more
it is
in high-
concave in p2 because a second-period price reduction deters investment
,
than
Last, for the price support policy r(-) just
in low- pollution states.
obtained to yield the commitment outcome
it
second-period objective function be concave.
sufficiently
=
concave in P2
(say,
because
D2
is
must be the case that the government's
Because (— r(p 2 ))
very convex.)
14
is
convex,
W
2
must be
Comparing markets and optimal pollution con-
5
trol: overall
We saw
regulation
that the introduction of a futures market reduces inefficient bypass and improves
on a setting of sequential spot markets. Yet, and unlike
yield optimal pollution control. It
The second-period
optimal.
pollution level,
it
it
investment.
Perhaps
to give the intuition
price pi plays several roles:
allocates (in
guides investment.
may be useful
Clearly,
an ex post
it
in the static case,
efficient
it
markets do not
why markets
are not
determines the second-period
way) the pollution among agents, and
would be useful to have another instrument to control
more instruments can
less obviously,
also help extract the agents'
rent.
To
it
see
why a futures
price p2
is
a poor rent extraction instrument,
us
let
first recall
that
corresponds to a free option with exercise price p2- At the very least the government
could
sell this
option at price Rq
=
o\E[max(0 2
buy the option and welfare would increase by
uniform striking price
of willingnesses to
menu
offering a
is
not optimal.
—
AiZo-
P2>0)|0]
17
Still,
>
0-
AH
agents would
charging for an option with a
The government can exploit the temporal correlation
pay by practicing a form of (intertemporal) nonlinear pricing and
Agents with a high first-period valuation reveal their type
of options.
by demanding an expensive option with a low striking price; because they are
have a high willingness to pay
invested), they are eager to
Section
still
in period 2
and therefore to use a permit
pay a low striking
6 will derive the optimal
(if
likely to
they have not
price.
(commitment) mechanism under the assumption
that the environmental regulator observes only the agents' pollution level, and not their
investment or production. This case corresponds to pure environmental regulation. Pure
environmental regulation
is
directly
comparable with the market system studied
in pre-
vious sections, because both posit that pollution levels are the only control instruments.
This section analyzes the simpler case of overall regulation in which the regulator can
observe the agents' pollution
abstract
17
to
level,
investment and production. 18
We
first
mechanism design problem and then give a simple interpretation
Actually,
Ro + 6R
it is
clear that
such that type
increases welfare by
second-order
6\
by decreasing p?
=
XSR because
is still
present the
of our findings.
and raising the price of the option R from Ro
between buying the option and not buying it, one
pi around the optimal commitment price has only a
slightly
indifferent
the change in
effect.
18
In our model, the observation of period- 2 production and pollution would give the relevant information
about investment. So, one might as well assume that both or none of the two are observed. The overall
regulation case is simpler to analyze because the investment choice does not introduce an extra incentive
constraint.
15
Mechanism design
5.1
At an abstract
the government sets up at date
level,
makes announcements
and
X
of his first-
2
1
a mechanism, in which each agent
and second-period valuations,
and 2 respectively. The agent's second-period announcement
8™
(' ne
The mechanism
agent did not invest).
probability Xi(0i) of being allowed to produce/pollute in period
investing, probabilities
2
and
and
superscripts "i"
and I?'(^ii^?')
fr°
m
"ni" refer to the fact that the agent invested in period
-
Ti(e 1 Ji)}g2 (e2 \$ 1 )dB2 ]
- T^
i
{e 1
,^
i
on the
B\,
principle implies that there
is
no
+
1
We
or not.
an agent with type
X
:
- A) Wo !^? A. *T)
1
(i
if
)]g2 (e2 \e1 )de2 )}
where the first-period announcement
contingent on
in period
2.
a) Agents' behavior. Let U{9\) denote the utility or rent of
is
a probability y(#i) of
'he agent to the government, where the
adopt the convention that transfers are made in period
ment
1,
B\
specifies a
x 2 (6i,0 2 ) and xJ'A.fl™) °f being allowed to produce
transfers 7^(^i, ^2)
1
contingent on whether
is
The second-period announcement can thus be denoted
investment actually occurred.
(investment took place) or
in periods
is
contingent on
0\,
and the second-period announce-
realization of the investment
loss of generality in focusing
and on 62
.
The
revelation
on mechanisms that induce
the agent to reveal truthfully. Using the envelope theorem and an integration by parts,
the slope of the rent function
uWx) =
is
given by: 19
x l (e 1 )-s[y(e 1
)Jo
-^-(e2 \e 1 )x2 (e 1 ,e 2 )d0 2
(8)
x
+
The agent
is
(i
- ,(*))
^i(e2 \e
Jo
willing to participate in the
1
)xT(e1 ,e2 )de2 ).
program only
the agent's rent increases with his type (see equation
19
The second-order
2
.
This obtains
in #i for all
2
2.
(which
as well (which holds
The second-order
Appendix
all 0\
it is
Because
the individual rationality
To guaran-
necessary and sufficient that x"'(0i,02) and x'2 (0i,8 7 ) be
A
_
2
increasing in 6 2 for
for all 0\.
if
.
is
))>
>
conditions of incentive compatibility of this problem are complex.
tee truthful revelation of 6 2 in period 2
nondecreasing in
8
(
U{6\)
if
'
is
if
1
1-W-^W)
+A
fx (B x )
the case for instance
^//jffl
is
ft(«s|0i)
if
A
is
not too large) and nondecreasing
nonincreasing in 6\ and
conditions of incentive compatibility in period
.
16
1
-£- nonincreasing
in #i).
are complex and are dealt with in
constraint can be written as:
U(0)
The
agents'' total rent
0.
(9)
therefore:
is
=
/* u(e 1 )f1 (e l )de 1
f u (<M[i - F (e )]de
1
The government's
Welfare maximization.
1
1.
Jo
Jo
b)
=
optimization program
-8D 2 {ti\\ - y ($M xr(e
/oMWflj) +
1
,e2 ) 92 (e 2 \e l )de 2 ]f1 (e 1 )de 1 )
+ * J? *2*j(*i, «2)*(*»l«i W)
+(1
+
+(1
- yiOiWti ^T(6 U 62 )g2 (62 \6 1 )d62 ]f
A)
is:
y(0i)(~i
1
(6 1 )d6 1
1
-\fo U(61 )[l-F1 (6 1 )]d0 1 )},
where
U
(&i) is given
components:
by equation
(
8
Note that
).
the social cost of pollution
+
of production (weighted at
1
received by the government
if
damage
because
A,
it
in
social welfare
is
divided into three
both periods, the expected value
would be equal to the monetary transfer
the latter could fully capture the agents' rents), and the
cost of leaving rents to the agents (proportional to A.) Let d t
=
D[(ni) and d2
=
D'2 (n 2 )
denote the social marginal costs of pollution. Pointwise maximization of social welfare
yields the following first-order conditions.
The maximization with
(a rewriting of equation
(
respect to x\(6i) gives the
^
1- )):
„(„ =
„
,
=
+
now standard monopoly formula
j^-l^i
do,
otherwise.
Let
^-TTA + TTA-^rFor types
who do
not invest, the maximization over
=
We
6\
is
x™ (61,62)
yields:
otherwise.
assume that the function 62
(which
(11)
the case for instance
—
if
A
^^"^N —rjfcjgy
62
—
is
not too large), and increasing in 6\ as
17
*s
increasing in 62 for
well.
all
[These
conditions will be used to ensure that agents' second-period second-order conditions are
satisfied.]
We
can then define a single-valued nonincreasing function p 2 (-) by:
who do
For types
l-fi(flO(-ffr(P2(gl)|g|))
m)
!
1
+A
l
^(ft^,)^)
/,(*)
maximization over x 2 (6 l ,62 ) yields the same condition
invest, the
=
We
Last,
=1
an agent invests
S[r
if
if
A
and only
is
if
62
>
^2(^1) (in the absence of investment) or
i9l)
e2 g2 (02 \e 1 )de2
+
^T (l-G
2 (p2 (e 1 )\e l ))}
T
(16)
if
extract the agents' rent.
Note that condition
(
16)
is
increasing in
6\
(which
his type exceeds 6*.
is,
The reader
will
check that the cutoff
the regulator restricts investment in order to better
20
in
(
16) has
cost (besides the
> p2 {6\)
side of the condition in
not too large.) Then there exists a cutoff 9* such that
best level. That
its first
those types 62
(15)
-
otherwise.
the case in particular
On
and only
if
assume that the left-hand
and one
P!(9i)
respect to the investment decision y(0i), using the fact that the
-i +
if
=
efit
*
*(*«.)i».>
the agent has invested):
(if
y{9x)
exceeds
—
l-F,(»,)HfrteWI'.))
m-7iw
we optimize with
92(^1)
We
otherwise.
A
agent produces in period 2
>
cost:
define a new, nonincreasing function q2 (-) by:
„,,,_
feW)
#2
j
'
under no investment, except that production no longer entails a pollution
as
is
A
= JL- +
~ 1+A
(0\
P2(l)
a simple interpretation. The investment has one ben-
monetary investment
that polluted; and
it
cost
i).
It
eliminates the pollution of
allows production by types in [92(^1 )»?2(^i)]-
the other hand, an agent's rent grows faster with 8 t in case of investment because
second-period production becomes more
We still need to
tion
program
is
likely.
check that the global second-order condition for an agent's maximiza-
satisfied.
Last, let I{92 \6x)
=
We will
do so when we describe the implementation.
—^nra y
denote the informativeness of
B\
about the second-
period needs for pollution rights. As the informativeness of 6\ goes uniformely to zero for
20
See appendix
1
18
every 6\ (the types are intertemporally independent in the limit), the optimal mechanism
converges to a mechanism which can be implemented by a sequence of spot markets with
a supply of pollution rights determined each period by the regulator. Furthermore, either
all
firms should invest or none.
When
the uncertainty about future needs
post efficiency of period 2 spot markets
large
it is
intuitively clear that the ex
very attractive. Alternatively one can study
is
when the existence
the design of policies
is
of spot markets each period
a constraint.
is
This constraint implies ex post productive efficiency conditionally on the number of pollution rights.
It
excludes the price discrimination involved in the optimal mechanism.
Consequently we can expect too much rent to be given up to firms in period
vestment
is
less (costly)
2.
controlled by the regulator, investment will be decreased in period
production in period
2.
When
investment
is
1
When
in-
to induce
not controllable, investment can
be similarly decreased indirectly by committing to a larger (than ex post optimum)
level
of pollution rights (and a smaller price of pollution rights). See section 6.
5.2
Implementation
In the optimal
mechanism the regulator uses the
correlation of types across periods to
decrease the cost of asymmetric information by committing to period-two allocations
which are not ex post
efficient (since
p 2 (0 and
g 2 (-)
In contrast markets yield ex post efficient allocations
the optimum.
The optimal mechanism
depend on the
first-period type).
and therefore cannot implement
price-discriminates for the period-two rights to
produce and to pollute on the basis of the first-period type.
In period 1, the
optimum can be implemented by a market
for first-period pollution
permits at a uniform price p* given by (11) and a set of options markets
who commit not
to invest, the regulator offers a
menu
of options {p?,
:
For the agents
Rfa}, where p 2
is
the striking price to be paid in period 2 for polluting in period 2 and
R(p 2 )
price (in period 1) of an option with exercice price p? (where R{.)
a decreasing function
defined below). For the agents
where q2
of options {q2 , 5(92)})
in period 2
is
and S(q2 )
is
who commit
is
is
is
the purchase
to invest the regulator offers a second
menu
the striking price or usage tax to be paid for producing
the purchase price of an option with exercice price q2 (where S(.)
a decreasing function).
Agents with higher
likely to
today.
0j
value
know that
(conditionally on not investing) they are
more
need a pollution permit tomorrow, and buy better (lower striking price) options
Similarly, agents with a higher willingness to
pay
in
period
1
buy lower usage
taxes.
The purchase
price of options to pollute
is
19
computed so that an agent who does not
q
invest (namely, with type #i
<
with exercice price pi{6i) defined
6') picks the option
by (13). Let us compute the first-period price R(p2) of an option with striking price p 2
and that induces
Assuming that the
this choice.
first-period allocation
by the spot market of pollution permits with price pj, the agent's
(discounted at period
1) is, for
Utf x jx =
)
The
I'
if
we
-
is
2 (p 2 (e 1 )\e1 ))]i/2 (e1 )
define 9\(p 2 ) as the inverse function of
=
=
.
o.
p2 {Bi) defined by
- G2 (p3 \e1 (p2 ))) <
-6(l
:
:
«Mh)) - *(i - G
R'(p 2 )
9^
(O 2 -P2(Oi))92(02\ei)de 2
.
first-order condition of incentive compatibility
[
Or,
+6
implemented
utility level in period 2
a type-#i agent claiming he has type
-R(P2(0i))
is
(13),
0.
(17)
Note that the second-order condition of incentive compatibility amounts
to
'dG2
'{^MOiWi^pWi) >
or
pWi) <
The
£/(0,0)
function R(.)
=
is
0.
obtained from (17) and the binding participation constraint
0.
who commit
Similarly, the "second-period" utility of those
type
a
>
6")
is
to invest (that
is,
who have
:
1
v2 (e u h) = -siqM) -i + s f
(e2
.
-
2 {h))92{e 2 \e x
)de2
leading to
S'{q2 )
=
-S[\
- G3 (q2 \ei(q2 )j\
with
<fe(0,(<fe))
= q2
with the "pasting condition"
m
v(e ,e
The optimal mechanism
is
m
)
= u(0\e m
illustrated in Figure
FIGURE
1
).
1.
HERE
Let us reemphasize that pollution permits of period 2 cannot be traded as agents
with different first-period valuations must face different second-period striking prices and
therefore the second-period allocation cannot be efficient.
We
summarize our analysis
in:
20
e2 t
l+Jt
e.
no investment
Figure
1
Proposition 3 Under
tracts
overall regulation, the
depending on whether the agent chooses
government
two menus of option con-
offers
In each menu, a higher-valuation
to invest.
agent chooses a more expensive option with a lower striking price. The allocation of pollution is not ex post efficient.
investment than in the
d)
when
Remark
pollution
less
first best.
we have
In this section
:
government taxes investment and induces
Last, the
derived the
benchmark of an optimal mechanism
and investment are controllable and when the regulator can commit
tertemp orally. To analyze the performance of markets for pollution rights,
useful to have three other benchmarks,
namely the case where the regulator
it
in-
would be
still
controls
investment but cannot commit, the case where he cannot control investment and can
commit, and
finally the case
where both problems occur.
Unfortunately the analysis of limited commitment or renegotiation proof commitment
is
quite complex because of the ratchet effect (see chapters 9 and 10 in Laffont-Tirole
(1993)).
A
thorough analysis in the case of a continuum of types
scope of this paper. In section
6,
we
like
here
is
beyond the
mechanism when investment
discuss the optimal
is
unobservable.
Comparing markets and optimal control: pure en-
6
vironmental regulation
We now
assume that the
pollution
federal
government (or one of
its
and has no control over investment and production. Unlike
government then cannot levy a tax on production nor can
vestment;
it
can (and
options must
now
will) still issue
in section
?? the
directly control (or tax) in-
it
nontradeable options on pollution permits, but these
play the triple role of allocating pollution rights, guiding investment
and extracting the agents'
lies
agencies) can only regulate
rents.
A
complete analysis of pure environmental regulation
We
outside the scope of this paper.
concerning this situation.
We
set
up the
content ourselves with making a few points
incentive problem
and then look
at
two polar
cases.
The government
sets
makes announcements
6\
up
at the beginning of date 1 a
and 6 2 of his
first-
invest.
It
also splits the agents into
For types
6\
to pollute in period
1
and a transfer
complement of 0i), who do not
mechanism
two groups.
£ 0i, the mechanism
in
which each agent
and second-period valuations,
2 respectively. Without loss of generality, the
truthfully.
mechanism
specifies
elicits
Those
mechanism
21
and
1
the agents' information
in the subset of types
t
a probability ii(0i) of being allowed
t(6i) to the regulator.
invest, the
in periods
For types
&i
€ 0J
specifies probabilities x\(0\)
(the
and
12(^1,^2) of being able to pollute in periods
to the regulator.
The
We
and a transfer T(6i,6 2 )
2, respectively,
adopt the convention that these transfers are date-2 transfers. 21
rent U(8\) of an agent with type 6\
= max{
U(0i)
and
1
is:
+ max[0,-i + 6E(92 \91 )],
mtt[x 1 (91 )91 -6t(91 )]
max[xi(§i)9i
+ max[E(max 6x2(81,62)62 - 6T(0
fceej
(18)
,6 2 )\8i),
l
St
- i - 6ndnT(9 ly 92 )]}.
SE($a\$i)
h
The
first
strategy for the agent
Even though on
is
to apply (or not) for a date-1 pollution right only.
the equilibrium path, agents
who apply
for a period 1 pollution right
only do invest (wlog, since not investing and not producing in period 2
not investing and having an option at a striking price above
1),
is
we must
equivalent to
consider the
possibility that the agent elects not to invest.
The second
strategy consists in claiming one
entitled to choosing in {x 2 (6 x , ')iT(9 u -)}
he
is
m
is
not investing (9\ € 0f), and thus be
period
2.
Note that the agent can claim
not investing and actually invest. Let 0* be defined by 6E(9 2 \9*)
envelope theorem and integrating by parts, the gradient of the rent
the expected present discounted
'
Ufa) =
x 1 (61 )
x 1 (e1 )
-
amount
for
is
=
i.
Using the
as usual equal to
of production:
M8inM,
+ S^E(92 \01
for
)
61
ee
1
n[6\l],
^t(91 ) + 6f^-^(92 \91 ))x 2 (9u 92 )d92
for
(19)
9,
6 0f.
Last, the agent's individual rationality constraint can be written
U(9i)
The
difficulty of the
>
maLx{0,6E(6 2 \6 1 )
- i).
(20)
problem stems not so much from the apparent complexity of the
incentive compatibility condition (18), which can be replaced by condition (19) together
with an ex post verification of global optimality for the agent.
rationality condition (20) creates trouble, as
the interval
is
[0, 1].
As
is
well
known the
it
may a
priori
Rather, the individual
bind at different values
in
analysis of type -dependent reservation utilities
complex.
21
The
reader might be concerned that the regulator would find out in the following way whether the
An agent who invests and announces he has not invested could be asked to pollute at
with some probability. To rule this out, we assume that the agent can always discard the pollutionabatement investment and behave like an agent who has not invested. Conversely, can an agent who does
not invest disguise as one who invests? The answer is yes since the agent can always elect not to produce
and therefore not pollute. This strategy of course may have a cost, which will be formalized when we
agent invested.
least
look at the incentive compatibility condition on investment.
22
some
In order to disentangle
a)
effects of the
non observability of investment and
production from those of an individual rationality constraint binding above the lowest
valuation,
we
first
look at the case in which (20) takes the more familiar form
tf(0i)>O
As we
we can
for all
X
:
.
ignore the bypass option as far as individual rationality
is
con-
cerned, as long as the high valuation types value first-period production enough (that
is, if
will see,
the optimal price for first-period permits
low enough) that they would not contemplate
is
refusing to interact with the pollution regulator.
We guess
(19).
The
the solution by ignoring some of the constraints that are implicit in (18) and
ignored constraints will be checked ex post. Note
that (19) implies that
first
for all 0!
Uifix)
as
x 2 can take any value between
x2 (0i,.)
in
= *i(*i) +
=
1
for 0i
S 0i D
constraint
is
[0 ,l]).
binding only at
0i
and
^(*»l*i))*i(«i,*i)«tfa
(So,
1.
x 2 (0i,.)
=
for 0\
While (21) implies neither
f
our optimization program.
"
(
6Jl
Positing for the
=
moment
(21)
€ ©i H
(18) nor (19),
[0,0*]
and
we use
(21)
that the individual rationality
(which must also be checked ex post), we also have
:
1
l/(0i )
=
f" U{6)d0.
Jo
We
can now build on the mechanism design of section
5.
The
objective function, the
incentive constraint and the individual rationality constraint are the
The only
difference
is
we
the agent invests. So,
and cannot prevent second-period production
necessarily have
x«(0i,.)
=
=
l
*ft(0i)
= O.
(22)
are the implications of the lack of control aver investment ?
0.
Equation
3/(00
=
(16),
when
modified, yields the optimal cutoff 0®
lh)
1
-i
if
+ S[J^
9 2 gi (ea ]fi l }d$ i
_A^°1-F1 (01
1+A /,(0,)
=
Note that 0®
>
0"
:
if
:
the fictitious case in which the regulator can control investment but
ft(0i)
as in section 5.
that the regulator has fewer instruments. Namely, an environmental
regulator does not control y(0i) directly,
What
same
+
)^)^
50,
Jo
^j(l-G
l
2|
1}
2
Let us consider
is
constrained to
:
9 {p»(9 l )\$ l ))]
~
otherwise.
(23)
the fact that the regulator loses control over production
investment occurs makes investment socially
less desirable.
23
when
We must now look for conditions under which the environmental regulator can de facto
power to do
control investment even though he has no statutory
purposes,
it
move
convenient to
is
(which follows the lines of section
The
ously
regulator
pollution permits at price p\ to agents
is,
who
simultane-
the environmental regulator bundles some
The second-period
-and second-period rights.
5.2.
into the implementation of the optimal regulation
That
options.
For expositional
5.2).
sells first-period
buy second-period
so.
first
options {p2,R{p2)} satisfy, as in section
:
= -s[i-G2 {p2 (el )\e1 ))].
R(P2(0i))
Now,
regulator
let
us assume that pj
sell
<
0®
(tied first-period pollution rights are cheap). Let the
unbundled first-period rights at price pf such that
e*-pl- R(p2 (B 9 )) + S f
1
(02
:
- P2 (e9 ))g2(02\0 9 )d02 = 0® -pf - i + *£(0 2 |0®).
(24)
P?>Pl
It is
shown that
easily
if
pf
<
0®,
(25)
no agent wants to bypass without buying a
first-
period pollution permit, and therefore the individual rationality constraint takes the
miliar
result
form U(0i)
>
<
Provided that pf
for all 0i.
0®,
Appendix
3 proves our
fa-
main
:
The bundled first-period permit
b)
We
is
cheaper that
have examined one polar case
of investment
by tying the purchase
price. Controlling
investment
is
in
its
unbundled counterpart.
which the regulator can control the
level
of second-period pollution rights to the first-period
then costless because agents have a large stake in period
1.
The
other extreme case
in period 1 that
we can
briefly consider
noboby purchases permits. The
is
when the
role of the
pollution costs are so high
IR
contraint
is
then quite
transparent.
Suppose (without
U(0)
is
=
0,
type
0,
loss of generality for this reasoning) that SE(02\O)
and actually
to leave a rent to those
all
types bypass.
who purchase
22
The only way
—i =
0.
Then
if
to avoid complete bypass
pollution rights.
Optimizing social welfare respect to 0® and pj(0i) we obtain the two following results
(see Appendix 4). There is more bypass that in the case where the marginal firm 0®
can be controlled through the purchase of a first-period pollution permit (case
22
This results from the slope of
U
a).
under bypass weakly exceeding that under any permit system.
24
To
counteract this excessive bypass, the striking prices of options offered in period one are
below marginal cost.
Desirable extensions
7
We
can summarize the way investments in private pollution abatement technologies are
affected
in
23
by policy instruments with the following graph giving the marginal agent engaging
the bypass activity.
Overall regulation
Futures market
+
'
1
.
Stand-alone
Pure environmental
First best
spot markets
regulation (case a)
The
Stand-alone
spot markets
Futures market
regulator does not observe investments.
strument
:
The
:
is
regulator does not observe investments.
period
He can
pollution rights for period 2 with the
1
His only in-
the supply of pollution rights each period.
offer in
commitment
of not changing the supply of pollution rights in period
First best
The
:
regulator observes investments
and has complete
2.
infor-
mation about the agents'.
The
Overall regulation
regulator observes investments but has incomplete in-
formation about the agents' valuations.
supply of pollution rights, can
offer in
tions of pollution rights in period 2
He can
period
1
menus of op-
and can make
Pure environmental
The
regulation
plete information about the agents' valuations.
:
control the
transfers.
regulator does not observe investments and has incom-
The
reg-
ulator only observes pollution, but, unlike under "futures
market''
We
,
can
conclude with a (non-exhaustive)
offer nonassignable, contingent permits.
list
of desirable extensions. First, in our model
the futures market guides investment by constraining the future spot price. In practice, the
futures market also supplies information about this spot price.
"price discovery"
could formalize this
by allowing the distribution of first-period valuations
of agents to be ex ante uncertain to the regulator.
agents' investment cost
23
One
Laffont-Tirole (1990) has
is
known.
We
in the
population
Second, we have assumed that the
could more generally allow the investment cost
shown how nonlinear
prices used to fight bypass
lower than marginal costs.
25
may
entail marginal prices
to differ
among
agents,
be particularly relevant
and
for
also be subject to
the
S0 2
moral hazard. 24
market, in which about seventy percent of the
pollution comes from public utilities. Third, overall regulation,
be performed by a single regulator, but rather (as
several agencies with conflicting objectives. Fourth,
has the means to achieve
political
its social goals.
the case for
it
US
we have assumed
is
prevails,
need not
electric utilities)
by
that the government
In practice, the government
We
hope that future research
interesting extensions.
As
when
US
may have
to obtain
sometimes offered to defend
Analyses of support- building for reform (Dewatripont-Roland (1992))
might be relevant here.
24
is
support from interest groups (this motivation
grandfathering.)
This extension would
in Laffont-Tirole (1993,
p86-103).
26
will
develop these (and other)
References
Baron, D. and D. Besanko [1984] "Regulation and Information in a Continuing Relationship"! Information Economics and Policy, 1:447-470.
Burkart,
M.
[1993] "Action
ties Fortnightly,
June
and Auction: The
502
Emissions Allowances", Public
Utili-
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Costello, K. [1992] "Comments on: 'Demand-Side
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of an Irrev-
Dewatripont, M. and G. Roland [1992], "Economic Reform and Dynamic Political Constraints", Review of Economic Studies, 59: 703-730.
Fudenberg, D. and
J.
Tirole [1991]
Game
Theory, Cambridge:
MIT
Press.
Hahn, R. and R. Noll [1983], "Barriers to Implementing Tradeable Air Pollution Permits:
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Hausker, K. [1992], "The Politics and Economics of Auction Design in the Market for
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LafFont, J.- J.
and
J.
Tirole [1990] "Bypass
and Creamskimming"
,
American Economic
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Laffont, J.- J.
and
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J.
Tirole [1993]
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A
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Laffont, J.- J.
and
J.
Tirole [1994b] "
A
Note on Environmental Innovation
",
",
Fondazione
Eni Enrico Mattei, discussion paper.
NRRI
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mimeo
Tietenberg, T.H. [1985] Emissions Trading, Resources for the Future, Washington D.C.
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27
Appendix
1
Proof that asymmetric information decreases bypass
:
Condition (16)
"p 2 (#i)" replaces
differs
"
first
best condition in that "92(^1)" replaces "0",
-" as second-period cutoff,
1 T"
To show that the
from the
and the
last
term
in (16)
is
added.
A
incentive to invest
is
lower than in the
first
best start from the
first
best
condition
^° + ITI
-'W
,
,
,
Add
A
P
•
,
the negative term
1
-F
—
x
:
1
+
A
/1
r&
dG2
(1
- G(
7TI
))
=
Jn
-^r— dO\
/
JnVx) der
and change variables to obtain
+ ,£
At
g2
w
_,
=
,2
and P2
+ _*_<, _
ow, +
Go
=
1
+A
this expression
Take the derivatives with respect to P2 and
^^/;,
is
§&*.
(XI)
negative.
q%.
We obtain
and
-^9252(92)
From the second order
negative.
Since (Al)
<
condition of footnote 14 and the definition of p2 (#i), (A2)
Therefore the expression (Al) with p2
is
0.
increasing in 6 X the cutoff value
is
=
P2{9\) a nd g2
=
92(^1)
is
is
negative.
higher.
Q.E.D.
28
Appendix
2
:
Second-order conditions of the agents' first-period optimisa-
problem (section
tion
5).
To study the second-order
conditions of the agents' first-period optimization problem,
convenient to use the implementation developed in section
it is
P2{6l)
=
I
K
>
R(p2(0*))
is
chosen so as to
Let
if*>f.
*(4)
make type
5.2.
6* indifferent
between investing and not
investing.
The
agent's optimization problem boils
down
to
:
1
max {[B x - rf]x! ft) - R{h) + S f
As
is
.
[1
- G2 {ei \e1 )}d62 \.
well-known, second-order conditions are satisfied
if
an increase in 6 X does not
decrease the derivative of the objective function with respect to
& SLi§
L
)^2' Sufficient conditions for this to be the case are
)(^
£i(.)
b)
y(.)
c)
p?(.)
d)
q2 (.) nonincreasing.
a)
b), c)
holds
holds
and
if
^i
and
if
d) together
—
+A
—
imply that
—
-
p?(.) is nonincreasing.]
nondecreasing
is
h\ s \)
the left-hand side of (16)
d)
hold
is
nondecreasing, which
if
a
2
1
is
:
nonincreasing
large.
c)
—
nondecreasing
1
b)
namely x x (0i)
nondecreasing
a)
[Note that
0\,
non decreasing
+A
i-/W) (-»(*!'»))
/,(*,)
in 6 X for each 9?.
29
g2 (62 \0 1 )
is
the case
if
A
is
not too
An Example
:
G 2 (92 \9 = 1 - e"*'* F {6 =
l )
p 2 (6 1 )
=
92(0i )
=
1
6X
d 2 6i
2A0!
=
1 )
l
,
+ 0j -
A
if
1+2A
1
0i
<
1+2A
--
0i
0*
1/2
1+2A
Figure 2
The dotted curve
in figure 2 pictures the effect of
an increase
in the social cost of
funds.
Turning to second-order conditions in period
2
A
1
1-F (6
1
+A
/i(0i)
2,
l )
the sufficient condition
- 30,
92 (0 2 |0i)
nondecreasing in B 2 takes the form
1-
A
(1-*,)
1+A
Bx
30
>
or 0i
>
A.
For small
As
#1
there
is
a problem illustrated by the bunching obtained in figure
for the second-order conditions in period 1, c)
and d) always hold and
a)
2.
and
b)
hold for A small enough.
Q.E.D.
31
Appendix 3
Proof that unbundled permits are more expansive than bun-
:
dled ones under pure environmental regulation.
We
look for a solution where V#i G [0,0®] the agent does not invest and asks for
pollution rights in period 2, V0i
6
[0®, 1] the agent invests.
V0x€[O,0®]
V0 a 6
[5®,
1]
okw-jf
[* ( #o+*j£[-f|iw^)]#i d9x
Optimizing social welfare as in section
W.CIM]
x
.
we obtain
5,
^
lW = l«».> T
+
^i^
T
(AI)
Let pj and pi{9\) the prices denned by equations (Al) and (A2).
Maximizing over 6® yields
('
-*•
+ *[
9)
B2 dG2 {92 \9^)
jf
1
+A
+
JJf-(l
dOt™
Vo
/i(^i)
- G,(ft(*)|*»))]
2
'
Substituting from (A2)
-*
+ *[
jf^ 9 dG (9 \9*) +6[p (9*)(l - G2{p»(P)\0*))]
2
2
2
2
or
1
1
{S
*
f
JO
92 dG2 (92 \9°)
-i}-6
'
f
Jj>i(8 9 )
(92
- P2 (9 9 ))dG2 (02 \0 9
)
(
+
f "'(-§(« |O)^>0.
l
32
(A3)
In terms of the implementation
mechanism the indifference of agent 0® can be rewritten
0*-p\- R(p2 (9°)) + S
I'
(02
- P2 (9 9 ))dG2 (92
Jp,(09)
1
= 0®-p* + 6 f
Jo
9 2 dG2 (92 \9 9 )
-i
or
P°
~ P'x =
{&
C 9 dG
2
2 (92
\9*)
- i) -
is t\
+R(p 2 (9 9 )) >
from (A3) and R{p 2 {9®))
>
(02
- p2 (69 ))dG2 (62 \0»)\
o.
0.
Q.E.D.
33
MIT LIBRARIES
3 9080 00922 6975
Appendix 4
:
Pure environmental pollution
permits (case b of section
A
Let
in the
6).
the (second-period) rent which
is
given
up to type
defined by (see figure 3, which depicts the agent's utility
— and when he does not —
flatter curve)
f (£„, (- w) *
A+
(
or
r
The
cost of rents
f
is
a+
e»
,
ft)
r (L>
a
-
x
ll (jf (jf ("
"
A
i ™>/»
invests
:
dGi
(
\
(
-
^«*>)**)««>>
^A))*)*)«W =
is
:
/
dG2 (e2 \e1 )d62 dF (e
1
e
+
+
(i
(1
+ A)*
j£
)\
1
+ \)sf
JO
1
e2 dG2 (02\Ox)de 2 dF1 (0 1 )
f
Jpa(«i)
[
£
6 2 dG 2 {6 2 \9,)
- ^dF (0 l
l
)
:
sd292 (p2 (0 l )\e 1 )f1 (e 1 )
-
(i
+
x)sP2 (e l )g2 (p2 (e 1 )\e 1 )fi (0 1 )
34
is
— steeper curve
*)*-jf jTC-^*"*')**
n (0i)
max - ^a(/*
get
when he
(~Bf)<*W**-
•
The optimization problem
We
The marginal type 9®
0.
now
-
(
absence of first-period
-\6F1 (o
1
)(-^ye
2 \o 1 )
=o
or
,
a ,
P2(Ul)
=
d2
7—
r —
+A
1
Furthermore
F (e
x
.
.
1
»
1
,
1)
/o
{-^{Meim
'
7TTTTT-:
^
+ XMOi)
g2 (P2(6i)\0i)
d
2
<
~1 + A"
:
-
Sd2 (l
-
(1
- G2 (p2 (& 9 )K))M0 9 ) +
1
(1
+ A)*
f
O2 dG2 (p 2 (O*)\6®)M0®)
1
+
X)S
f
Jo
6%
- xs
J^
+ (1 + \)ifi(B*)
e 2 dG 3 (O 2 \O*)fl (0°)
d
\- -^{e
2
r))d9 2 F^) =
o
or
P3{ " 9)
-i
+ 6j
02dG2 (e2 \6*)
+ 6^2j(l - G2
{
?2 {9®)\0®))
i
siope
J
=U
^
dG2
1
»
"aS
slope
=
£ /
Jp2{ei)
"
Figure 3
•906
b
35
ex
—
7tt-
d#i
Date Due
teas
I
SFP.
19
m
021997
Lib-26-67