Optimal Trading Ratios for Pollution Permit Markets∗
Stephen Holland†
Andrew J. Yates‡
September 16, 2013
Incentive-based environmental regulations, such as permit markets or emissions taxes,
have typically been designed to minimize the costs of achieving emissions targets.1 This
focus on reducing abatement costs simplifies program implementation by eliminating the
need to quantify marginal damages. However, advances in air and water quality modeling
now make it feasible to estimate these damages precisely and thereby to incorporate them
into program design. In fact, recent estimates from such models show that trading that does
not incorporate marginal damages leads to quite substantial efficiency losses (Muller and
Mendelsohn, 2009; Henry et al., 2011; Fowlie and Muller, 2013). This literature argues that
regulators should turn from the narrow criterion of minimizing abatement costs to the more
general criterion of efficiency.
In this paper we demonstrate a novel method for improving the efficiency of pollution
permit markets by optimizing the way in which emissions are exchanged through trade. In
our model, there is asymmetric information and pollution sources are differentiated by the
number of permits they are required to hold to emit one unit of pollution. When sources
trade permits, these differentiated requirements govern the exchange of emissions, and hence
we call them trading ratios. Montgomery (1972) analyzed trading ratios equal to the ratio
of marginal damages, and more recent literature argues that efficiency can be improved by
∗
We would like to thank seminar participants at the Energy Institute at Haas, UNC Greensboro, the
AERE summer 2012 conference, and the NBER summer 2012 conference, as well as Meredith Fowlie, Nicholas
Muller, William Pizer, and Rob Williams for helpful comments and suggestions.
†
Department of Economics, University of North Carolina at Greensboro.
‡
Department of Economics, University of North Carolina at Chapel Hill.
1
In practice, these regulations have proven quite successful (Carlson, et al., 2000; Ellerman, et al., 2000;
Keohane, 2006; Fowlie, et al., 2012).
1
using these marginal damage trading ratios rather than simple one-for-one trading (Farrow
et al., 2005; Muller and Mendelsohn, 2009; Henry et al., 2011; and Fowlie and Muller, 2013.)
Taking this as a point of departure, we ask if further efficiency improvements are possible.
The rather surprising answer is yes. We characterize the optimal trading ratios and show
that they generally depart from the marginal damage trading ratios.
To understand why this is the case, it is helpful to start with the concept of cost effectiveness, which is defined as achieving an environmental objective at least cost. If the
environmental objective is an emissions target, then one-for-one permit trading minimizes
abatement costs and is cost effective. If the environmental objective is a targeted level
of damages, then permit markets with marginal damage trading ratios are cost effective.2
However, permit markets are generally implemented to allow firms to flexibly respond to
private information about their abatement costs, which may not have been available to the
regulator when the regulator designed the program. With this asymmetric information between the regulator and the regulated firms, permit markets can no longer achieve the first
best (full information) distribution of pollution. In such a second-best environment, it is no
longer clear that ex post cost effectiveness is even a desirable goal. Optimal trading ratios
potentially trade an ex post loss in cost effectiveness for an ex ante efficiency gain.
Cost effectiveness is deeply entrenched in environmental policy.3,4 The goal of achieving
an environmental target at a lower cost is easily explained to policy makers and to the general
public. Moreover, the way in which permit markets minimize abatement costs through
trading is easily understood. This seductive simplicity, however, may have prevented policy
makers from exploring modifications to permit markets which increase efficiency but depart
from cost effectiveness.
Our analysis is motivated by four observations. First, regulators are currently grappling
with how to regulate nonuniformly mixed pollutants. Early cap-and-trade programs could
2
This is easily shown since minimizing market abatement costs subject to a constraint on aggregate
damages implies that the ratio of marginal abatement costs to marginal damages should be equal across all
sources. (Fowlie and Muller 2013).
3
One-for-one trading arises naturally from the property rights or Coasian view of emissions markets as
the assignment of a property right.
4
One-for-one trading treats all firms identically and thus reduces regulator discretion and the incentive
for a firm to attempt to obtain favorable treatment.
2
yield large gains by simply reducing the overall level of emissions.5 Since many low-cost
emissions reduction opportunities may have already been realized, current programs must
more carefully target emissions reductions to high-damage areas in order to pass a costbenefit test.6 Trading ratios modify emissions trading to target emissions reductions to high
damage areas. Our analysis provides a framework for implementing trading ratios optimally.
Second, recent analyses have estimated the costs of undifferentiated trading and found
them to be substantial (Muller and Mendelsohn, 2009; Henry et al., 2011). Most provocatively, Henry et al. study the celebrated SO2 Acid Rain cap-and-trade program established
by the 1990 Clean Air Act Amendments. Their estimates of the additional damages resulting
from undifferentiated trading relative to a command-and-control baseline substantially exceed the estimated abatement cost savings relative to a similar baseline in Carlson et al. As
a result, Henry et al. argue for trading which weights emissions by their marginal damages,
i.e., for marginal damage trading ratios. Optimal trading ratios can improve upon marginal
damage trading ratios.
Third, proposed cap-and-trade programs to limit greenhouse gas (GHG) emissions would
swamp existing cap-and-trade programs in size and scope. Ellerman and Buchner (2007)
compare the Acid Rain Program, which is the largest existing non-GHG program, with
the European Union Emissions Trading Scheme (EU ETS) and note that the EU ETS is
much larger even though it covers a smaller fraction of total emissions than the Acid Rain
Program.7 To address global climate change efficiently, similarly sized programs would need
to be implemented throughout the world and then linked together. The massive scale of
such programs implies that efficiency gains, which are small in relative terms, could be quite
large in absolute terms. Optimal trading ratios can increase efficiency even for uniformly
mixed GHG emissions.
5
The Acid Rain Program established by the 1990 Clean Air Act Amendments for regulating SO2 emissions;
the leaded gasoline trading program; and the RECLAIM program regulating NOx emissions in southern
California each had one-for-one trading over broad regions and a declining level of allowed emissions.
6
For example, the US EPA recently attempted to modify the Acid Rain Program with CAIR and later
CSPAR, which was then struck down by the courts. The new programs attempt to account for spatial
heterogeneity in damages mainly by prohibiting trades across regions.
7
The EU ETS covers approximately 11,500 sources, compared to about 3,000 for the U.S. SO2 program,
and the prepolicy emissions in the EU ETS were over two billion metric tons of CO2, versus sixteen million
(short) tons of SO2 in the U.S. program. In addition, the value of the allowances distributed under the EU
ETS is about $41 billion versus about $5 billion under the U.S. SO2 program.
3
Fourth, regulators have begun to incorporate trading ratios into emissions markets in
a variety of contexts.8 The NOx Budget Program uses trading ratios to restrict banking
through a “flow control” provision. The Clean Air Interstate Rule (CAIR) uses trading ratios to reduce uniformly the allowed emissions for each permit. The Cross State Air Pollution
Rule (CSPAR), which would have replaced CAIR but was invalidated by the courts, similiarly would have used trading ratios through an “assurance provision” to reduce uniformly
the emissions per permit if emissions exceed a threshold. The failed Waxman-Markey legislation addressing U.S. GHG emissions would have used trading ratios to implement costly
borrowing. Despite this growing use of trading ratios, optimal implementation of trading
ratios has not been studied.
To understand the intuition of optimal trading ratios, first consider uniformly mixed
pollution. Here it is well known that one-for-one trading of emissions between firms can be
more efficient than a no-trading structure in which all firms are simply given an endowment
of emissions and are not allowed to trade these endowments. But one-for-one trading may
not be the most efficient structure. We characterize the optimal trading ratios and show
that only in special cases will these optimal trading ratios imply one-for-one trading. Thus,
for example, some sources of GHG emissions should be given more favorable trading ratios
than others, i.e. be required to hold fewer permits for each unit of emissions.
The main reason for this result is the presence of asymmetric information about the costs
of reducing pollution between the sources and the regulator that designs the market. Asymmetric information means that the regulator can set the cap optimally in expectation but
cannot set the cap optimally ex post. The cap is either too tight, in the case of unexpectedly
high costs, or is set too loose, in the case of unexpectedly low costs. We show that the
regulator can effectively relax the cap when costs are high and tighten the cap when costs
are low by giving relatively favorable trading ratios to sources whose emissions are positively
correlated with the market price of permits. By allowing flexibility in emissions even with
a fixed cap, optimal trading ratios can improve efficiency even though they increase costs
ex post. In effect, the regulator trades the loss of ex post cost effectiveness for an ex ante
efficiency gain.
8
See Holland and Moore (forthcoming) for more details on each of these programs.
4
Next consider non-uniformly mixed pollution. When pollution is not uniformly mixed,
abatement cost minimization is clearly an inappropriate goal. Cost effectiveness at attaining
a targeted level of damages implies that trading should be at the marginal damage trading
ratios. This goal seems more appropriate. However, the asymmetric information implies
that the regulator cannot set the target level of damages optimally: ex post it is either too
high or too low. Optimal trading ratios allow for flexibility in the target level of damages,
and we show that marginal damage trading ratios are generally not optimal.
We also investigate the degree to which our results have practical importance for environmental policy. This is accomplished by numerical analysis of two pollution problems. The
first is a multi-country carbon emission market, and the second is a nitrogen trading market
for several rivers in North Carolina and Virginia. In both cases, we show that the optimal
trading ratios lead to efficiency improvements. The magnitude of these improvements varies
from significant to trivial, depending on the value of the parameters. These results suggest
that permit market designs should give careful consideration to the efficacy of the optimal
trading ratios.
Our analysis combines two prominent strands of the literature on incentive based regulations. The first strand follows the seminal work of Montgomery (1972), who compared
markets in “pollution licenses” and “emissions licenses” and argued that the former are generally more applicable. Importantly, Montgomery recognized that emissions licenses should
not be simply traded one-for-one if pollution differs across the sources and proposed trading
at the ratio of marginal damages.9 More recent work estimates marginal damages and argues
that trading should weight emissions by marginal damages (Muller and Mendelsohn, 2009;
Henry, Muller, and Mendelsohn, 2011; Fowlie and Muller, 2013).
The second strand of literature follows Weitzman (1974), who introduced the idea of
informational asymmetries in permit markets. Since the parameters of permit markets must
be set potentially years in advance, the regulator lacks information which will be available
to market participants when they make abatement decisions. This asymmetric information
has important implications for the choice of policy instruments and the resulting literature
9
Montgomery proposed trading at the ratio of the transfer coefficients. The ratio of the transfer coefficients
is exactly the ratio of marginal damages holding ambient concentrations at the other sites constant.
5
on “prices vs. quantities” is vast. However, this literature has not analyzed optimal trading
ratios in the context of asymmetric information.
Our contribution is most closely related to Fowlie and Muller (2013) who find that under uncertainty “differentiated policies may not welfare dominate simpler, undifferentiated
designs.” Since we analyze the optimal trading ratios, our optimal trading ratios can do
no worse than undifferentiated trading ratios. Moreover, our results show precisely the assumptions under which optimal trading ratios yield an efficiency gain relative to marginal
damages trading ratios.
Our work also contributes to the literatures on hybrid incentive-based mechnisms which
attempt to mitigate variance in permit prices through, for example, price floors or ceilings
supported by injections or withdrawals of permits.10 Optimal trading ratios can also help
to mitigate permit price variance since differentiated trading ratios imply that aggregate
emissions are not fixed even for a fixed supply of permits. If trading ratios relax aggregate
emissions when the permit price is relatively high, the variance of permit prices decreases.
In addition, our analysis contributes to the literature on environmental taxes. It is
well-known that emissions trading and environmental taxes can each correct environmental
externalities by pricing the externality. Optimal trading ratios imply that sources should
face different emissions prices even for uniformly mixed pollutant, i.e., trading ratios should
simply not reflect expected marginal damages. Similarly, we show that facility-specific taxes
generally should not equal expected marginal damages but should be adjusted to reflect
abatement cost uncertainty.11
Section 1 of the paper describes the model and derives the main results. Section 2 analyzes
the well-studied case where marginal damages and marginal abatement costs are linear.
Section 3 presents some simple models and numerical calculations of the main results, and
Section 4 estimates the gains from optimal trading ratios under a model of a uniformly-mixed
pollutant calibrated to global carbon markets and under a model of a nonuniformly-mixed
pollutant calibrated to water quality trading in the Neuse River basin.
10
For recent contributions to this literature see Fell and Morgenstern, 2010; Hasegawa and Salant, 2011;
Grüll and Taschini, 2011; and Stocking, 2012.
11
This has been observed before in a quadratic model by Chavez and Stranlund (2009). Also Weitzman
(1974) uses the optimal facility-specific taxes to derive his comparative advantage of prices vs. quantities
formula in a quadratic model, but does not present the actual values for these taxes.
6
1
Model
Assume there are n regulated facilities that pollute. The abatement costs for facility i are
Ci (ei ; θi ), where ei is the emissions from facility i and θi is a parameter that influences costs.
Because abatement costs are in terms of emissions, we define marginal abatement costs as
i
i
M ACi ≡ − ∂C
. We assume costs are convex in emission reductions, so that − ∂C
> 0 and
∂ei
∂ei
Ci00 ≡
∂ 2 Ci
∂e2i
> 0. From the point of view of the regulated facility, the cost parameter θi is
known when the abatement decision is made. From the point of view of the regulator, θi
is random variable with expected value θ̄i and variance σi2 . We use the expression “cost
shocks” to refer to various realizations of these random variables. Let E = (e1 , e2 , . . . , en ) be
the vector of emissions. These emissions cause damages, which are enumerated by a damage
function D(E). The marginal damage from facility i is M Di ≡
∂D
∂ei
> 0.
The regulator uses a permit market to ameliorate the damages from pollution. We assume
this permit market is competitive. Each facility is given an endowment of permits wi and
P
the total endowment is w =
wi . Regulated facilities face possibly different constraints
on the number of permits they must hold for each unit of emissions. These constraints are
described by a facility-specific variable ri that is chosen by the regulator. In particular, if
facility i emits ei units of pollution then they must hold ri ei permits. The ratio of rj to ri
reflects the rate at which emissions of facility i can be converted to emissions of facility j
through the trade of permits between the two facilities. If
ri
rj
= 1 for every i and j, then we
have one-for-one trading of emissions. We refer to the ri ’s as trading ratios.12
The choice variables for the regulator are nominally the trading ratios and the permit
endowments. However, because we assume the permit market is competitive, the market
equilibrium only depends on w and is independent of the distribution of the wi .13 Moreover,
the permit market equilibrium is unchanged if the trading ratios and the total endowment
are all multiplied by the same constant. Without loss of generality, then, we can normalize
the total endowment as convenient. In the theoretical analysis we normalize w to be equal
to one.
12
Alternatively one might call them holding requirements or compliance requirements.
The analysis is unchanged if the regulator distributes permits with any non-distortionary method such
as through auctioning.
13
7
Given a price p for permits, facility i selects emissions to minimize the sum of abatement
costs and expenditures in the permit market. Facility i’s problem is
min Ci (ei ; θi ) + p(ri ei − wi ).
ei
The first-order condition for ei is
∂Ci
= ri p.
∂ei
−
(1)
focf
An immediate consequence of this equation is that, if two facilities have different trading
ratios, then their marginal abatement costs will not be equal, i.e., the regulation is not
cost-effective. Under our normalization of the total permit endowment, the permit market
equilibrium condition is
X
ri ei = 1.
(2)
mc
i
The facilities’ response to the regulator’s choice of trading ratios is summarized by equations
(1) and (2). This is a system of n + 1 equations and n + 1 unknowns (each of the ei and p).
It is useful to describe the solution to these equations as a function of the vector of trading
ratios R and the vector of cost parameters Θ. Thus we have ei (R; Θ) and p(R; Θ).
To compute the optimal trading ratios, the regulator selects values for the trading ratios
to minimize the expected sum of abatement costs and damages. Thus the regulator’s problem
is to choose R to minimize
"
W≡E
#
X
Ci (ei (R; Θ); θi ) + D (E(R; Θ)) .
i
For this problem, it is useful to consider the derivative of the regulator’s objective with
respect to rj
"
#
X ∂Ci ∂D ∂ei
∂W
=E
+
,
∂rj
∂ei
∂ei ∂rj
i
(3)
derivew
(4)
focrg
as well as the corresponding first-order conditions
"
#
X ∂Ci ∂D ∂ei
∂W
=E
+
= 0 for j = 1, 2, . . . n.
∂rj
∂ei
∂ei ∂rj
i
8
The first-order conditions imply that, on average, the marginal abatement costs are equal to
marginal damages, where the average is weighted by the
∂ei
’s.
∂rj
There is not a simple closed
form solution to the first-order conditions, even in a standard case in which the abatement
cost functions and the damage function are quadratic.
Now consider the intuitive, but ultimately flawed, approach to selecting the trading ratios
based on the ratio of marginal damages. This corresponds to selecting r1 and rj (for j 6= 1)
such that:
∂D
E[ ∂e
]
rj
= ∂Dj for every j 6= 1.
r1
E[ ∂e1 ]
(5)
mdr
Now consider the system of equations defined by (1), (2), and (5). This is a system of 2n
equations and 2n unknowns (ei , p, and rj ). Let ei (r1 ; Θ) , p(r1 ; Θ) and rj (r1 ; Θ) be the
solutions to these equations as a function of r1 and the Θ vector. The regulator’s problem
in this case is to find the value for r1 that minimizes total expected costs:
"
min E
r1
X
#
X
Ci (ei (r1 ; Θ); θi ) + D(
ei (r1 ; Θ)) .
i
i
The first-order condition for r1 is
"
E
X ∂Ci
i
∂D
+
∂ei
∂ei
∂ei
∂r1
#
= 0.
(6)
The first-order condition implies that the regulator sets marginal abatement costs equal to
marginal damages on average where the average is weighted by the
∂ei
’s.14
∂r1
Let r̃1 be the
solution to (6) and let r̃j ≡ rj (r̃1 ; Θ). We refer to the vector R̃ = (r̃1 , r̃2 , . . . , r̃n ) as the
marginal damage trading ratios.
The main result of our paper is to show that the marginal damage trading ratios will
generally not be equal to the optimal trading ratios. This may seem a bit surprising, so
let us first give intuition for why it is indeed true before turning to a more formal analysis.
The intuition is perhaps the clearest for the special case of uniformly mixed pollution. Here
marginal damages are the same across facilities, so one might expect that the optimal trading
14
Although (6) appears similar to (4), note that there is no reason that (4) should should hold for every
j when (6) holds.
9
focr
ratios would be equal across facilities as well. To see why such one-for-one trading is, in
fact, not generally optimal for uniformly mixed pollution, consider the market equilibrium
condition (2). Evaluating this at the solution ei (R; θ) gives
X
ri ei (R; Θ) = 1.
(7)
i
Now suppose for the moment the market is indeed designed with one-for-one trading and let
r be the common value for the trading ratios. It follows from (7) that the sum of emissions is
equal to the constant 1r , i.e., equal to the effective permit endowment. In general, however,
when the trading ratios differ between facilities, the sum of emissions will not be constant, and
moreover, it will vary according to the realized values of Θ. This suggests that permit markets
that do not use one-for-one trading have an interesting and under-appreciated feature. In
these markets, facilities in aggregate have the flexibility to emit more (or less) pollution
depending on the actual values of their abatement cost functions, even though the total
supply of permits is fixed.
The regulator, in turn, can use this feature to improve the performance of the permit
market. Because of the uncertainty about abatement costs, the regulator does not know the
efficient quantity of pollution. Loosely speaking, when aggregate marginal abatement costs
are high, the efficient quantity of pollution is large. When the aggregate marginal abatement
costs are low, the efficient quantity of pollution is small. The regulator can engender a similar
relationship between emissions and abatement costs by optimally selecting the trading ratios.
Now return to the formal analysis of the general case of an arbitrary damage function.
To show that the optimal trading ratios will generally not be equal to the marginal damage
trading ratios, we utilize the structure of the regulator’s problem as well as the characteristics
of the marginal damage trading ratios to evaluate the derivative of the regulator’s objective
function W. This gives us our main result (all proofs are in the Appendix).
popt
Proposition 1. The derivative of the regulator’s objective function W with respect to rj ,
evaluated at the point R̃, is given by
∂W = COV (p, ej ) − COV
∂rj R̃
−1
A
X ai ∂D
i
r̃i ∂ei
!
, ej
10
+
X
i
∂D
∂D −1 ai p
COV r̃j
− r̃i
,A
∂ei
∂ej
r̃i Cj00
insight
cor:uniform
where the covariances are also evaluated at R̃ and where ai ≡ ri2 /Ci00 and A ≡
P
i
ai .
Proposition 1 shows that the derivative of W with respect to rj , evaluated at the marginal
damage trading ratios, can be written as the sum of n + 2 covariances. If this sum is positive,
then the derivative is positive, and the objective function can be improved by decreasing the
trading ratio rj below the marginal damage trading ratio r̃j . If this sum is negative, then the
derivative is negative, and the objective function can be improved by increasing rj above the
marginal damage trading ratio r̃j . If this sum is equal to zero, then the derivative is equal
to zero, and the first-order-condition (4) is satisfied at the point R̃. In this case, the optimal
trading ratio for facility j is in fact equal to the marginal damage trading ratio r̃j . Below
we fully analyze the properties of the derivative in Proposition 1. But at this point, it is
important to stress that there is no reason, in general, that the sum of the n + 2 covariances
should be equal to zero. In other words, the marginal damage trading ratios are generally
not optimal.
One interesting implication of Proposition 1 is that the the optimal trading ratios lead
to ex post inefficiency. Once again this is perhaps most clearly illustrated with the case of
uniformly mixed pollution. If trading ratios are not one-for-one, then by Equation (1), the
marginal costs are not equal. Aggregate abatement costs could be reduced by increasing
abatement from a low-cost facility and decreasing abatement from a high-cost facility. The
regulator tolerates this (second order) loss in ex post efficiency to obtain the (first order)
gain in ex ante efficiency from using the optimal trading ratios.
Next consider a special case in which the marginal damage function
∂D
,
∂ei
evaluated at the
marginal damage trading ratios, is non-stochastic for every i. This special case obviously
applies when damages are linear in emissions because marginal damages are constant. But
it also applies to uniformly mixed pollution because aggregate emissions don’t vary.15 For
this special case, the derivative in Proposition 1 simplifies considerably.
Corollary 1. Suppose that
∂D
,
∂ei
evaluated at R̃, is non-stochastic for every i. Then the
derivative of the regulator’s objective function W with respect to rj , evaluated at the point
15
For uniformly mixed pollution, the marginal damage trading ratios are all equal, so r̃i = r̃. From (2),
aggregate emissions are constant and equal to 1/r̃. So all marginal damage functions are non-stochastic.
11
R̃, is given by
∂W = COV (p, ej )|R̃ .
∂rj R̃
Under the conditions of Corollary 1, the optimal trading ratios are equal to the marginal
damage trading ratios if and only if COV (p, ej ), evaluated at R̃, is equal to zero for every
j. Although both the cases of uniformly mixed pollution and linear damages have received
considerable attention in the literature, Corollary 1 appears to be a novel insight.16 Below
we give simple numerical examples in which the covariance is indeed not equal to zero, for
both uniformly mixed and linear damages (see Section 3).
Proposition 1 and Corollary 1 show that the optimality of marginal damage trading ratios
will generally depend on COV (p, ej ). In particular, Corollary 1 shows that the regulator
should decrease rj below r̃j , i.e., offer facility j a favorable trading ratio, if and only if
COV (p, ej ) is positive. Since the permit price is driven by the marginal abatement costs,
the permit price will be high when marginal abatement costs are high across facilities. This
is precisely the situation in which the aggregate permit endowment should be relaxed. By
giving favorable trading ratios to facilities whose emissions are large when the permit price
is high, the regulator can, in essence, relax the aggregate emissions constraint in the event
of high prices and hence improve efficiency.
1.1
Optimal Facility-Specific Taxes
We have shown that marginal damage trading ratios are generally not optimal and that
the regulator can improve the performance of the market by adjusting the trading ratios to
account for the possibility of abatement cost shocks. This raises the question as to whether
pricing mechanisms, such as pollution taxes, should be set to expected marginal damages or
whether they too should be adjusted to account for the possibility of abatement cost shocks.
Suppose ti is the tax per unit of emissions for facility i, and T is the vector of facilityspecific taxes. As is well-known, the facility will equate its marginal abatement costs and
16
Rabotyagov and Feng (2010) observe that the ratio of abatement costs may not be equal to the ratio of
delivery coefficients. In their model, however, they consider an exogenous emission constraint.
12
the tax, so the first-order condition for ei is
−
∂Ci
= ti .
∂ei
(8)
focftax
Let the solution to this equation be ei (T ; Θ). The regulator selects the facility-specific taxes
to minimize the expected sum of abatement costs and damages. Thus the regulator’s problem
is to choose T to minimize
"
W ≡E
#
X
Ci (ei (T ; Θ); θi ) + D (ei (T ; Θ)) .
i
The first-order condition of the regulator’s objective with respect to tj is17
"
#
X ∂Ci ∂D ∂ei
∂D
−1
∂W
=E
+
= E −tj +
=0
∂tj
∂ei
∂ei ∂tj
∂ei
Cj00
i
(9)
focRgTax
(10)
eq:opttax
Solving for tj implies that
h
tj =
E
∂D
/Cj00
∂ei
i
∂D
=
E
+
∂ei
E 1/Cj00
COV
∂D
, 1/Cj00
∂ei
E 1/Cj00
.
Since in general there is no reason the covariance term in (10) should equal zero, it is generally
not the case that optimal facility-specific taxes should equal expected marginal damages.18
Thus optimal facility-specific taxes should be adjusted to account for the possibility of abatement cost shocks.19
The optimal facility-specific taxes in (10) are related to the theory of optimal taxation
first studied by Ramsey (1927). In optimal Ramsey taxation, larger taxes are applied to
more inelastic markets. Note that 1/C 00 is related to the abatement cost elasticity.20 Thus if
marginal damages are high when the abatment cost elasticity is high, then the second term in
∂e
∂ei
−1
The second equality follows from (8) and from differentiating (8), which implies ∂tjj = C
=0
00 and ∂t
j
j
for i 6= j.
18
Chavez and Stranlund (2009) obtain a similar result in a model with quadratic abatement cost and
damage functions.
19
In the special case of constant marginal damages, ∂D/ei is non-stochastic so the second term in (10) is
zero and the optimal tax is simply marginal damages.
20
The abatement cost elasticity is (1/C 00 )(P/e).
17
13
(10) is positive and the optimal facility-specific tax exceeds expected marginal damages (i.e.,
is larger in the inelastic market). This intuition is illustrated graphically in the Additional
Appendix C.
2
A Linear-Quadratic Example
We use an example with specific functional forms to give more explicit conditions under
which the optimal trading ratios will be different from the marginal damage trading ratios.
We call this the linear-quadratic example because the abatement cost function
λi
Ci (ei ; θi ) =
2
θi
− ei
λi
2
(11)
acf
(12)
foclq
is quadratic and the marginal abatement cost function
−
∂Ci
= θi − λi ei
∂ei
is linear. We interpret θi as the intercept and λi as the slope of the marginal abatement cost
function. It is convenient to collect the λi into the diagonal matrix Λ. The damage function
is quadratic as well. We have
1
D(E) = E t VE,
2
(13)
where V is a symmetric matrix with entries vij . We also assume that the random variables
θi are independent.
A distinct advantage of the linear-quadratic example is that we can obtain simple closedform expressions for p and ej . These, in turn, enable us to evaluate the various covariances in
Proposition 1. For convenience we consider separately the cases of uniformly mixed pollution
and non-uniformly mixed pollution.
2.1
Uniformly Mixed Pollution
We characterize uniformly mixed pollution with the condition vij = v for every i, j. It follows
immediately from (5) that the marginal damage trading ratios imply one-for-one trading.
14
dam
What about the optimal trading ratios? From Corollary 1, we know that the answer depends
on the various COV (p, ej ). Only in special cases will all these covariances be equal to zero,
so only in special cases will the optimal trading ratios be equal to the marginal damage
trading ratios. The precise conditions under which this occurs are described in the first of a
series of results for the linear-quadratic example.21
pops2
Result 1. In the linear-quadratic example, suppose that pollution is uniformly mixed. Then
the optimal trading ratios are equal to the marginal damage trading ratios if and only if
σj2
λj
is the same for every facility j.
We see that, in the linear-quadratic example, the condition that all covariances be equal
to zero is essentially a requirement that the abatement cost functions exhibit a specific type
of homogeneity. In particular, the ratio of the variance of the cost parameter to the slope
of the marginal abatement cost function must be the same across all facilities. If this is
the case, then the optimal trading ratios are equal to the marginal damage trading ratios.
If, however, the ratios of the variance of the random variables to the slope of the marginal
abatement cost vary across facilities, then the optimal trading ratios are different from the
marginal damage trading ratios.
Result 1 suggests that as the dispersion of the
σj2
λj
increases, the welfare advantage of using
the optimal trading ratios rather than the marginal damage trading ratios should increase
as well. This is indeed the case, although the appropriate measure of dispersion is a bit
complicated. As a preliminary, let xj =
values using
1
λj
σj2
λj
and then define the weighted average x̄w of these
as the weight for observation xj . We have
x̄w ≡
1
P
1
i λi
!
X 1
xj =
λ
j
j
1
P
1
i λi
!
X σj2
j
λ2j
.
With this in hand, we can approximate the welfare advantage of using optimal trading ratios.
linear
Result 2. In the linear-quadratic example, suppose that pollution is uniformly mixed. Then,
to a first-order approximation, the welfare advantage of the optimal trading ratios relative to
21
The proofs of these results are in the Additional Appendix A.
15
the marginal damage trading ratios is proportional to
1
P
1
i λi
!
X 1
| xj − x̄w | .
λj
j
We see that the welfare advantage of the optimal trading ratios is proportional to the
weighted average of the absolute deviations of xj from their weighted average x̄w . In the
special case of Result 1, xj = x̄w for every j and the welfare advantage is zero. In the
special case in which all the slopes are the same (λj = λ for every j) but the σj2 ’s vary,
then the weighted averages become simple averages, and this dispersion measure simplifies
considerably to the mean absolute deviation of the σj2 .
In summary, the results for uniformly mixed pollution show that heterogeneity of abatement costs (through differences in the ratio of variance to slope of marginal abatement cost)
leads to a wedge between the optimal trading ratios and the marginal damage trading ratios.
The greater the degree of this heterogeneity, the greater the welfare advantage of the optimal
trading ratios.
2.2
Non-Uniformly Mixed Pollution
When pollution is non-uniformly mixed, the damages are heterogeneous across sources. We
want to isolate the effect of this damage heterogeneity on the difference between the optimal
trading ratios and the marginal damage trading ratios. So we utilize an additional assumption that eliminates the abatement cost heterogeneity that was found to have an effect on
this difference for uniformly mixed pollution. Accordingly, we have
σi2 = σ and λi = λ for every i.
(14)
From our discussion of uniformly mixed pollution, we know that (14) implies that COV (p, ej ) =
0. This does not mean, however, that the optimal trading ratios will be equal to the marginal
damage trading ratios. Corollary 1 does not apply in this case because pollution is nonuniformly mixed. We must consider the general expression in Proposition 1 and this expression has other covariances beside COV (p, ej ). In particular, these other covariances
16
noh
are functions of marginal damages, and we want to investigate the degree to differences in
marginal damages cause the derivative in Proposition 1 to be different from zero.
To do this, we first define the optimal emission standards e∗i as the solutions to
min E
ei
hX
i
Ci (ei ; θi ) + D(E) .
(15)
Let E ∗ be the vector of optimal emission standards. With these in hand, we can specify
exactly when there is no wedge between the optimal trading ratios and the marginal damage
trading ratios.
pops3
Result 3. In the linear-quadratic example, suppose that pollution is non-uniformly mixed,
equation (14) holds and V is invertible. Then the optimal trading ratios are equal to the
marginal damage trading ratios if and only if E ∗ is proportional to E[Θ].
We see that if the optimal emission standards are proportional to the expected values
of the random variables, then the optimal trading ratios are equal to the marginal damage
trading ratios. It is possible, but unlikely, that this condition holds. To see this, consider
the special case in which E[θi ] = 1 for every i. Here all the expected values are the same, so
E[Θ] is a constant vector. This, along with (14), completely eliminates all heterogeneity in
the variables that define abatement costs. From Result 3, it follows that the optimal trading
ratios are equal to the marginal damage trading ratios if and only if E ∗ is a constant vector.
Because expected marginal damages are different across facilities (which is the definition of
non-uniformly mixed pollution) and expected marginal abatement costs are identical, only in
very special circumstances will the optimal emission standards be all the same.22 Hence the
optimal trading ratios will generally be different from the marginal damage trading ratios.
We now discuss the actual values for the marginal damage trading ratios and the optimal
trading ratios. Because pollution is non-uniformly mixed, the marginal damage trading
ratios will not imply one-for-one trading. Intuitively, of course, they should reflect, well,
marginal damages. But marginal damages vary depending on the level of emissions. Our
22
More explicitly, the optimal emissions standards will be all the same if and only if row sums of V are
all the same. In the more general case in which E[Θ] is not a constant vector, E ∗ is proportional to E[Θ] if
and only if E[Θ] is an eigenvector of (Λ + V)−1 .
17
eq:OptEmStd
next result specifies the appropriate emissions level that is consistent with our definition of
marginal damage trading ratios.
rpropmd
Result 4. In the linear-quadratic example, suppose that pollution is non-uniformly mixed.
Then the marginal damage trading ratios are given by
R̃ =
VE ∗
(E[Θ] − ΛE ∗ )
=
.
E[p]
E[p]
(16)
propmd
We see that the marginal damage trading ratios are proportional to the marginal damages,
evaluated at the optimal emission standards. Equivalently, the marginal damage trading
ratios are proportional to the expected marginal abatement costs, evaluated at the optimal
emissions standards.
Next we analyze the optimal trading ratios. Consider an arbitrary trading ratio vector
R. Our next result gives an explicit equation for the value of the regulators objective as a
function of R.
recta
Result 5. In the linear-quadratic example, suppose that pollution is non-uniformly mixed
and (14) holds. Then we have
W=
X
2
Ci (E[ei ]; θ̄i ) + D(E[E]) +
σ
2λ2
λ+
X
i
t
vii −
R VR
Rt IR
!
,
(17)
where I is the identity matrix.
The regulator wants to select R to minimize this expression (the expected emissions of
pollution E[ei ] and E[E] depend on this choice). To analyze this choice, we first consider
two extreme cases. First, suppose that the variance is equal to zero. In this case, the
optimal trading ratios are equal to the marginal damage trading ratios because there is no
uncertainty, and so all the covariances in Proposition 1 are zero. Now suppose that the
variance is large. Then the regulator’s objective is approximately equal to the σ 2 term in
(17). Thus the regulator wants to select the values for ri to maximize
Rt VR
.
Rt IR
18
ecta
This is the well known problem of maximizing the ratio of quadratic forms. The solution
follows from Kaiser and Rice (1973). Let ν be the largest eigenvalue of V. The optimal
vector R is equal to the eigenvector of V corresponding to this eigenvalue. In less extreme
cases, the optimal trading ratios are reflect the trade-off between these two benchmarks.
As the variance increases, the optimal trading ratios approach the eigenvector of V. As
the variance decreases, the optimal trading ratios approach the marginal damage trading
ratios.23
In summary, the results for non-uniformly mixed pollution show that the inherent heterogeneity of damages in non-uniformly mixed pollution generally leads to a wedge between
the optimal trading ratios and the marginal damage trading ratios.
3
Numerical Calculations
sec:uniform
In this section we illustrate Corollary 1 graphically and with several simple numerical calculations.
3.1
Uniformly Mixed
sec:NumExU
We start with uniformly mixed pollution. The fact that Corollary 1 applies to this case seem
especially surprising, since the intuition for using one-for-one trading (i.e. marginal damage
trading ratios) seems quite strong. After all, whenever trading ratios are not equal, marginal
abatement costs are not equal. The regulator, however, tolerates this ex post efficiency loss
to obtain an ex ante efficiency gain.
We illustrate this with a simple numerical example of the linear-quadratic model with
two facilities. The example is illustrated in Figures 1 & 2 and Table 1. Facility 1’s marginal
abatement costs are known with certainty. Facility 2’s cost shock can either be high (H)
or low (L) with equal probability. For these numerical calculations, we normalize the total
Suppose that E[Θ] is an eigenvector of (Λ+V)−1 . Then it follows from Result 3 that the optimal trading
ratios are equal to the marginal damage trading ratios for any value of σ 2 . In addition, from the proof of
Result 3, it follows that E ∗ is an eigenvector of V and so from (16), R̃ is an eigenvector of V as well. Now
consider Result 5. Since the optimal trading ratios are equal to R̃ for any σ 2 , then R̃ must maximize (17)
for any σ 2 . It follows that R̃ must in fact be the eigenvector corresponding to the largest eigenvalue of V.
In this case the two benchmarks are the same, and they are both equal to R̃.
23
19
permit endowment such that the marginal damages trading ratios are unity, i.e., R̃ = (1, 1).
Figure 1 and Panel A of Table 1 show the outcomes in the resulting permit market with
marginal damage (one-for-one) trading ratios. Figure 1 shows the marginal abatement costs
for each facility and the two market marginal abatement costs corresponding to the high
and low outcome.24 The first-best (full information) outcome occurs at the intersection of
the appropriate market marginal abatement costs and the marginal damage. Because of
asymmetric information, the first-best outcome is not obtained. In the event of a high cost
shock, the total permit endowment is too small, marginal abatement costs exceed marginal
damages, and there is a deadweight loss relative to the first-best outcome. This deadweight
loss is indicated by the the upper triangle. In the event of a low cost shock, the total
endowment is too large, marginal abatement costs are less than marginal damages, and the
deadweight loss is the lower triangle. Panel A of Table 1 illustrates this numerically. Notice
that marginal abatement costs for both facilities equal the permit price, and aggregate
emissions are unchanged across the two shocks.
We are, of course, primarily interested in the covariance between prices and emissions.
Inspection of either the data in Panel A of Table 1 or the relationship between the points in
Figure 1 reveals that COV (p, ej )|R̃ 6= 0 for either facility. In fact, the covariance of emissions
and prices are negative for Facility 1, but positive for Facility 2. Applying Corollary 1
shows that efficiency can be increased by increasing Facility 1’s trading ratio, but decreasing
Facility 2’s trading ratio.
The optimal trading ratios are illustrated in Figure 2 and Panel B of Table 1.25 The
optimal trading ratio for Facility 1 is larger than the marginal damage trading ratio. It
follows that the cost of emissions are higher for Facility 1 and so its emissions are lower for
both cost shocks. This is reversed for Facility 2. Aggregate emissions under the optimal
trading ratios are different than under the marginal damage trading ratios. Importantly,
optimal trading ratios lead to emissions that are larger in the case of the high cost shock,
but smaller with a low cost shock. This is a first order efficiency gain at the expense of a
second-order cost-effectiveness loss. Correspondingly, as shown in Panel A and B of Table 1,
24
The market marginal abatement cost is found by horizontal summation of the facilities marginal abatement costs.
25
The optimal trading ratios are found by numerical optimization. The code is available on request.
20
deadweight loss decreases when optimal trading ratios are used.26
Table 1: Marginal damage and optimal trading ratios with uniformly mixed pollution
Panel A: Marginal damage trading ratios.
r1 = r2 = 1; DWL=2.08
M AC1
Low
12.5
High
17.5
price
12.5
17.5
M AC2
12.5
17.5
e1
7.5
2.5
e2
2.5
7.5
e1 + e2
10
10
Panel B: Optimal trading ratios
r1 = 1.03; r2 = 0.98; DWL=1.92
M AC1
Low
12.89
High 17.89
1
3.2
price M AC2
12.55 12.23
17.41 16.97
e1
7.11
2.11
e2
2.77
8.03
e1 + e2
9.87
10.14
The example is parameterized by M AC1 = 20 − e1 ;
M AC2L = 15 − e1 ; M AC2H = 25 − e1 ; M D = 5 + e1 + e2
where high and low costs occur with equal probability.
Total permits are normalized to 10.
Non-uniformly Mixed
Corollary 1 also applied to non-uniformly mixed pollution, provided that damages are linear.
Consider another example, in which the two facilities and their abatement costs are identical
to those used in the example above, but in which damages are linear and differ across the
two facilities. Facility 1 has high marginal damages (M D1 = 14) and Facility 2 has low
marginal damages (M D2 = 10).
Table 2 illustrates the results for the marginal damage trading ratios and the optimal
trading ratios. From (5), the marginal damage trading ratios satisfy r2 = 10/14∗r1 . Panel A
of Table 2 shows that the optimal value for r1 is 1.22, so that r2 = 0.87. Thus the high damage
facility (Facility 1) pays a relatively high price for its emissions and the low damage facility
(Facility 2) pays a relatively low price for its emissions. Panel B of Table 2 shows the optimal
trading ratios. Since COV (p, e1 ) < 0, the optimal trading ratio for Facility 1 is raised to
26
Because the market is not cost effective ex post, the market marginal abatement cost is not simply
the horizontal sum of the facility marginal abatement costs. Thus the deadweight loss cannot be simply
illustrated in Figure 2.
21
tab:ExUni
1.27. On the other hand, since COV (p, e2 ) > 0, the optimal trading ratio for Facility 2 is
lowered to 0.84. The optimal trading ratios reduce the deadweight loss from the asymmetric
information by about 5% and increase the responsiveness of the aggregate emissions to the
cost shock.
Table 2: Marginal damage and optimal trading ratios with non-uniformly mixed pollution:
linear damages with M D1 = 14, M D2 = 10
Panel A: Marginal damage trading ratios.
r1 = 1.22; r2 = 10/14 ∗ r1 = 0.87; DWL=4.22
Low Cost
High Cost
M AC1
11.6
16.4
price
9.56
13.4
M AC2
8.31
11.7
e1
8.36
3.64
e2
6.69
13.3
e1 + e2
15.1
16.9
Panel B: Optimal trading ratios.
r1 = 1.27; r2 = 0.84; DWL=3.99
Low Cost
High Cost
1
4
M AC1
12.07
16.66
price M AC2
9.49
7.93
13.09 10.95
e1
e2 e1 + e2
7.93 7.07
15
3.34 14
17.4
The example is parameterized by M AC1 = 20 − e1 ; M AC2L =
15 − e1 ; M AC2H = 25 − e1 ; where high and low costs occur with
equal probability. Total permits are normalized to 16, which
would be the optimal emissions with 1:1 trading.
Policy Applications
We have established that the optimal trading ratios will generally be different from the
marginal damage trading ratios, even for uniformly mixed pollution. We now illustrate the
potential policy implications of this observation by considering two permit trading applications.
4.1
Uniformly Mixed: Carbon Trading
Consider a stylized global carbon trading market. Ackerman and Bueno (2011) determine
simple two-parameter functions that characterize the cost of reducing carbon emissions for
various geographic regions of the world. To apply these to our model, we interpret our
22
tab:ExMD1
facilities as regions and write Ackerman and Bueno’s functions in terms of emissions rather
than emission reductions. This gives
−
ai (bi (1 + θi ) − ei )
∂Ci
=
,
∂ei
ei
where ai and bi are constants determined by Ackerman and Bueno. We interpret bi (1 + θi )
as the stochastic business-as-usual (BAU) emissions. For simplicity we model θi as a three
point symmetric distribution with zero expectation so that θi takes on the values {−ki , 0, ki }
with probabilities {ρi , 1 − 2ρi , ρi }. For example, there is a probability ρi that BAU emissions
increase by ki percent over their expected value. We also assume that the θi are independent
across regions. To complete the model we specify the marginal damage function as
X
∂D
=β
(ei − bi ) + s,
∂ei
where β (the slope of marginal damage) comes from Newell and Pizer (2003) and s (the
social cost of carbon at expected BAU) comes from USEPA (2010).
For tractability, we focus on the industrial sectors of the four regions with the largest
emissions: China, Europe, South/South East Asia, and the U.S.A. The results are given in
Tables 3 and 4. For a given distribution of θ, we calculate the total costs (expected sum of
abatement costs and damages) under the optimal trading ratios and the marginal damage
trading ratios.27
In Table 3, we consider symmetric abatement cost shocks (the tail probabilities ρi and
the percentage change in BAU emissions ki are the same across regions). Since China has the
largest BAU emissions, shocks to Chinese abatement costs drive the carbon price. Hence,
Chinese emissions covary positively with price under marginal damages trading ratios, and
Corollary 1 implies that efficiency can be improved by lowering China’s trading ratio. Indeed,
China’s optimal trading ratios are below one in each scenario, whereas the trading ratios for
the other regions exceed one in each scenario.
We can use the data in Table 3 to further illustrates the mechanism by which optimal
trading ratios increase efficiency by comparing the variance of the prices under optimal and
27
The Mathematica code for the optimizations is available upon request.
23
marginal damage trading ratios. Optimal trading ratios improve efficiency by decreasing the
variance of prices. This reduction occurs because the variance of emissions increases (the
variance is zero under marginal damage trading ratios but non-zero under optimal trading
ratios). The columns of Table 3 have different amounts of uncertainty, and the table shows
that as uncertainty decreases (either through a decrease in ki or ρi ), the reduction in the
variance of prices decreases.
Finally, Table 3 shows that optimal trading ratios reduce deadweight loss, but cannot
eliminate it. For the largest uncertainty (ki = 0.5 and ρi = 50%), optimal trading ratios
reduce the deadweight loss by $0.5 billion or about 5%. For lower levels of uncertainty, the
gains from optimal trading ratios are more modest.
In Table 4, we consider asymmetric cost shocks. Here China’s abatement costs are
uncertain and the other regions’ abatement costs are known. The gains from using optimal
trading ratios are more dramatic than in the symmetric case of Table 3. With a high level
of uncertainty about China’s abatement costs (kChina = 0.5), optimal trading ratios reduce
the deadweight loss by about 22% or around $2 billion per year.
4.2
Non-Uniformly Mixed: Nitrogen Trading
Our non-uniformly mixed policy application considers the trading of Nitrogen emission permits between waste-water treatment plants (WWTP) located in North Carolina and Virginia. The data for this application is described in detail in Yates et al (2013) and Doyle
et al (2013). Briefly, there are 51 waste-water treatment plants spatially distributed along
a river system that connects to a coastal estuary (see Figure 3). The abatement costs of
WWTP i are consistent with the linear-quadratic example and furthermore the slope of all
marginal abatement cost functions are equal to the common value λ. The value for λ is
obtained from engineering cost estimates for reducing pollution at generic WWTP.
There is a distinct advantage to using the linear-quadratic structure for policy applications of this type. The equations characterizing equilibrium in the permit market are linear
in emissions, so that expected total costs of a given policy are a functions of the first and
second powers of the random variables θi . Thus we do not need to make explicit distributional assumptions about the random variables. Rather we just need to specify the mean
24
Table 3: Carbon Trading with Optimal Trading Ratios: Symmetric Scenarios
co
ρi = 50%
ki = 0.2
ki = 0.5 ki = 0.33 ki = 0.2 ki = 0.1 ρi = 25% ρi = 10%
Optimal trading ratios
China
Europe
S/SE Asia
U.S.A.
St. Dev. Price
Marginal Damage TR
Optimal TR
Expected Price
Marginal Damage TR
Optimal TR
Total Cost
First Best
Marginal Damage TR
Optimal TR
Deadweight Loss
Marginal Damage TR
Optimal TR
Percent Reduction
1
2
0.877
1.082
1.133
1.063
0.946
1.037
1.055
1.030
0.981
1.013
1.019
1.011
0.995
1.003
1.005
1.003
0.990
1.007
1.010
1.006
0.996
1.003
1.004
1.002
38.534
37.649
25.694
25.463
15.418
15.371
7.709
7.704
10.902
10.886
6.895
6.891
74.022
73.838
74.022
73.997
74.022
74.023
74.022
74.023
74.022
74.024
74.022
74.023
305.699
321.374
320.665
305.699
312.554
312.424
305.699
308.148
308.132
305.699
306.309
306.308
305.699
306.921
306.917
305.699
306.188
306.187
15.674
14.966
4.5%
6.855
6.725
1.9%
2.449
2.432
0.7%
0.610
0.609
0.16%
1.222
1.218
0.33%
0.488
0.488
0.13%
The permit endowment is set so that the marginal damage trading ratios are 1. This represents
approximately a 50% reduction from BAU emissions.
Costs and deadweight loss (DWL) in billions of dollars. Prices in 2007 dollars per ton carbon
25
Table 4: Carbon Trading with Optimal Trading Ratios: Asymmetric Scenarios
kChina = 0.5 kChina = 0.33 kChina = 0.2 kChina = 0.1
Optimal trading ratios
China
Europe
S/SE Asia
U.S.A.
St. Dev. Price
Marginal Damage TR
Optimal TR
Expected Price
Marginal Damage TR
Optimal TR
Total Cost
First Best
Marginal Damage TR
Optimal TR
Deadweight Loss
Marginal Damage TR
Optimal TR
Percent Reduction
1
2
3
0.798
1.146
1.154
1.152
0.903
1.071
1.073
1.072
0.965
1.026
1.027
1.027
0.991
1.007
1.007
1.007
30.426
27.113
20.285
19.312
12.171
11.966
6.085
6.060
74.023
73.635
74.022
73.968
74.022
74.031
74.022
74.028
305.699
315.270
313.194
305.699
309.934
309.516
305.699
307.220
307.166
305.699
306.079
306.076
9.571
7.495
21.7%
4.235
3.817
9.9%
1.521
1.467
3.6%
0.380
0.377
0.9%
The permit endowment is set so that the marginal damage trading ratios are 1. This
represents approximately a 50% reduction from BAU emissions.
Costs and deadweight loss (DWL) in billions of dollars. Prices in 2007 dollars per ton
carbon.
For each column, the tail probabilities are ρChina = 50% for China and ρROW = 0% for
the other three regions.
26
cob
and variance of the distributions. The expected value for the cost parameters θ̄i are based
on the size of the WWTP. The variances, σi2 , are free parameters, but for convenience, they
are scaled proportionately to the expected values, so that a single parameter η captures the
“percent error” in the random variables.28
Damages are measured at 96 sites along the river system and in the estuary. The damage
function is given by
1
D(E) = (AE + Y )t B(AE + Y ).
2
In this expression, E is the vector of emissions of nitrogen from the 51 WWTP, Y is the
vector of background levels of Nitrogen from non-point sources at the 96 measurement sites,
B is a diagonal matrix with entries bjj (interpreted as the slope of marginal damage at
site j), and A is a transfer matrix that maps emissions from the WWTP through the river
system to the measurement sites. The elements of Y and A are determined by matching
the location of the WWTP and measurement sites to the output of the USGS maintained
SPARROW model (Hoos and McMahon 2009). For simplicity we assume that the bjj = b,
and that b is within the range determined by Yates et al (2013).
We determine the optimal trading ratios and the marginal damage trading ratios as a
function of the parameters η and b.29 The parameter η measures the magnitude of the uncertainty about abatement costs. The parameter b measures the magnitude of the severity
of damages from emissions. The results for all parameter combination are shown in Table 5. Figure 3 shows the marginal damage trading ratios and the optimal trading ratios
corresponding to the first column in this table. The ratio of the largest trading ratio to
the smallest is consistently larger for the optimal trading ratios than for the marginal damage trading ratios. The optimal trading ratios decrease the price dispersion relative to the
marginal damage trading ratios. They also increase the dispersion in total emissions. Notice
that the expected prices for the marginal damage trading ratios cases do not depend on the
value for η, because the marginal damage trading ratios do not account for uncertainty on
η E[θi ]
The standard deviation of each random variable is 100
( 2 ), so that it is very likely that a realization of
the random variable lies within η percent of the expected value (for a normal random variable the probability
is 0.95).
29
Because the abatement costs and damages are both quadratic, the marginal damage trading ratios are
equal to the marginal abatement costs evaluated at the optimal standards.
28
27
the abatement cost side. In contrast, the optimal trading ratios do depend on the value for
η.
The percent reduction in deadweight loss from using optimal trading ratios rather than
marginal damage trading ratios varies quite a bit according to the values for the parameters.
But the percentage reduction is generally substantial and greater than in the carbon example.
The reason for this is that there are a few WWTP with very large variances relative to the
other WWTP.30 In the Appendix, we show that simply taking the WWTP with the largest
variance and artificially reducing its variance to be equal to the average variance changes
the reduction in deadweight loss from 78 percent to 28 percent. Furthermore, by assuming
that all WWTP have the same variance, the reduction in deadweight loss approaches, but
does not quite equal, zero. Even in this case there is still room for a small improvement by
using the optimal trading ratios rather than the marginal damage trading ratios.
5
Conclusion
We analyze a model of asymmetric information between a regulator and polluters and show
that optimal policies are not based simply on expected marginal damages. In the context
of cap-and-trade, we find that optimal trading ratios (trading ratios which maximize ex
ante efficiency) generally depart from marginal damage trading ratios. The gains from
optimal trading ratios depend on the sum of various covariances. In simple cases, such as
uniformly mixed pollution or constant marginal damages, the gain is simply determined by
the covariance of the price and a facility’s emissions. In particular, if a facility’s emissions
covary positively with the market price of permits, then the regulator can improve efficiency
by giving the facility a relatively favorable trading ratio. Intuitively, this favorable trading
ratio allows additional emissions (despite a fixed cap) in precisely the case when the cap is
set too tight from an ex post perspective.
In the case of an emissions tax, our results imply that the regulator can improve ex ante
efficiency by setting facility-specific taxes according to a Ramsey-like rule which adjusts the
30
These WWTP have large output, and it is assumed that the expected value is proportional to output,
and the variance is proportional to expected value.
28
Table 5: Nitrogen Trading with Optimal Trading Ratios
Uncertainty Percent Error
Slope of MD
b = 30
Trading ratios
Max Marginal Damage
3.553
0.063
Min Marginal Damage
Max Optimal
6.553
Min Optimal
0.015
St. Dev. Price
Marginal Damage TR
2.171
0.525
Optimal TR
Expected Price
Marginal Damage TR
2.739
Optimal TR
1.635
St. Dev. Total Emissions
Marginal Damage TR
0.038
0.055
Optimal TR
Expected Total Emissions
Marginal Damage TR
4.842
4.846
Optimal TR
Total Cost
First Best
9.773
Marginal Damage TR
9.870
Optimal TR
9.798
Deadweight Loss
Marginal Damage TR
0.097
Optimal TR
0.024
Percent Reduction
74.9%
1
2
3
4
η = 10
b = 60 b = 90
3.580
0.064
5.088
0.035
tab:neuse
b = 30
η=5
b = 60
b = 90
3.607
0.066
4.598
0.048
3.553
0.063
5.009
0.035
3.580
0.064
4.214
0.055
3.607
0.066
3.956
0.061
2.157
0.829
2.143
1.080
1.085
0.420
1.079
0.669
1.072
0.820
5.329
3.907
7.778
6.280
2.739
2.024
5.329
4.618
7.778
7.179
0.038
0.048
0.038
0.044
0.019
0.024
0.019
0.020
0.019
0.019
4.780
4.783
4.791
4.722
4.842
4.844
4.780
4.781
4.719
4.720
19.240
19.336
19.281
28.419
28.512
28.471
9.771
9.796
9.782
19.236
19.260
19.253
28.412
28.436
28.432
0.095
0.040
57.8%
0.094
0.053
43.8%
0.024
0.010
56.9%
0.024
0.016
31.4%
0.023
0.019
18.6%
The permit endowment is set to the sum of the optimal pollution standards.
Costs and deadweight loss (DWL) in millions of dollars per year.
Prices in dollars per pound.
Emissions in millions of pounds per year.
29
expected marginal damages for the covariance of marginal damages with the slope of the
marginal abatement costs.
Our theoretical analysis shows that it is possible for a regulator to improve efficiency using
optimal trading ratios. However, whether the regulator should implement optimal trading
ratios depends crucially on whether the benefits of optimal trading ratios are sufficient to
offset any additional regulatory costs which might arise from a more complicated regulatory
scheme. To estimate the magnitude of possible benefits, we compare optimal trading ratios
to marginal damage trading ratios in two policy environments: a global carbon trading
market and a nitrogen trading market for watersheds in North Carolina and Virginia. The
results from these calibrations show that the gains vary from significant to trivial depending
on the value of the parameters.
Our analysis presents policymakers with two related challenges. First, the idea of cost
effectiveness—meeting an environmental objective at least cost—may be an inappropriate
guide to policy. Cost effectiveness as a guiding principle is simple, intuitive, and readily
explained. However, our analysis shows that in the second-best world with asymmetric information the principle of cost effectiveness is misleading. In fact, in this environment which
explicitly recognizes the regulator’s informational disadvantage, we show that efficiency can
be improved by departing from the cost effective policy. Effectively, the regulator can attain
a gain in ex ante efficiency by allowing some ex post inefficiency. Thus cost effectiveness is
not necessarily an appropriate guide.
Second, our analysis demonstrates how regulators can use optimal trading ratios to improve efficiency. However, optimal trading ratios require much more careful analysis of
abatement cost uncertainty than is required by conventional analysis. In particular, guided
by the Weitzman analysis, abatement cost uncertainty has primarily been utilized in the
choice of instrument (taxes v. cap-and-trade). Weitzman’s intuitive results show that the
superior instrument can be determined by comparing the relative slopes of the marginal
abatement cost and the marginal damage functions. Once the superior instrument has been
determined, regulators simply focus on expected costs and damages. On the contrary, our
analysis suggests a need for a more ambitious analysis of abatement cost uncertainty. In
particular, our analysis shows how uncertainty about abatement costs and damages should
30
be incorporated into the design of the policy instrument and not just the choice of the policy
instrument. Designing policy with optimal trading ratios requires much more careful analysis
of uncertainty than conventional analysis requires.
31
Appendix
Before proving Proposition 1, we first prove a Lemma about the marginal damage trading
ratios.
psb
Lemma 1. When using the marginal damage trading ratios, the regulator selects R̃ such
that
"
E p − A−1
X ai ∂D
i
where ai ≡ ri2 /Ci00 and A ≡
P
i
#
ri ∂ei
=0
ai
Proof of Lemma 1.
From (1) and (2), the equilibrium for marginal damage trading ratios is defined by the n−
P
i
1
1 equations ∂C
/ri = ∂C
/r1 for each i 6= 1 and by the equation i ri ei = 1. Differentiating
∂ei
∂e1
the n − 1 equations with respect to r1 gives
i
1
− ∂C
− ∂C
Ci00 ∂ei
C100 ∂e1
∂ei ri
∂e1
+
=
+
ri2 r1
ri ∂r1
r12
r1 ∂r1
since ∂ri /∂r1 = ri /r1 from [5]. By substituting in p this implies that
C 00 ∂e1
Ci00 ∂ei
= 1
ri ∂r1
r1 ∂r1
for each i 6= 1. Differentiating
P
i ri ei
X
(18)
eq:LemDiff1
(19)
eq:LemDiff2
= 1 implies that
ri
i
∂ei X ri
+
ei = 0
∂r1
r
1
i
which implies that
X C 00 ri ∂e1
−1 X ∂ei
∂e1 C100 X ri2
=
ri
=
ri 1 00
=
r1
∂r1
r1 Ci ∂r1
∂r1 r1 i Ci00
i
i
where the first equality comes from
P
i ri e1
= 1 and rearranging (19), the second equality
follows from substituting in (18), and the third equality follows from algebra. Solving this
32
equation shows that
"
#−1
∂e1
−1 X ri2
−a1
= 00
= 2 A−1
00
∂r1
C1
Ci
r1
i
which implies
"
#−1
∂ei
ri −1 X ri2
−ai −1
=
A
=
00
00
∂r1
r1 Ci
C
r
r
1
i
i
i
(20)
eq:LemDer
(21)
eq:LemFOC
from (18).
Now the first order condition from the regulator’s problem in (6) implies that
"
0=E
X ∂Ci
∂D
+
∂ei
∂ei
i
"
=E
X
i
#
∂ei
∂r1
∂ei X ∂D ∂ei
−pri
+
∂r1
∂ei ∂r1
i
"
p X ∂D ai −1
=E
−
A
r1
∂e
r
r
i
i
1
i
#
#
where the first equality simply restates (6), the second equality follows from the definition
of p, and the third equality follows from subsituting in (19) and (20). Multiplying through
by r1 establishes the result. Proof of Proposition 1.
From (1) and (2), the equilibrium for the optimal trading ratios is defined by the n − 1
P
∂Cj
i
equations ∂C
/r
=
/r
for
each
i
=
6
j
and
by
the
equation
i
j
i ri ei = 1. Differentiating
∂ei
∂ej
the n − 1 equations with respect to rj gives
∂C
− ∂ejj
Cj00 ∂ej
Cj00 ∂ej
Ci00 ∂ei
p
=
+
=
+
ri ∂rj
rj2
rj ∂rj
rj
rj ∂rj
for each
i 6= j
(22)
eq:PropDiff
where the first equation follows from differentiating and the second equation follows from
P
the definition of p. Differentiating i ri ei = 1 with respect to rj implies that
X
i
ri
∂ei
+ ej = 0
∂rj
33
(23)
eq:PropDiff
which implies that
X
−ej =
ri
i
∂ei X ri2 p X ri2 Cj00 ∂ej
=
+
∂rj
Ci00 rj
Ci00 rj ∂rj
i
i6=j
p X ri2
prj Cj00 ∂ej X ri2
−
+
rj i Ci00
Cj00
rj ∂rj i Ci00
=
where the first equality follows from rearranging (23), the second equality follows from subsituting in (22), and the third equality follows from algebra. Solving this equation implies
that
"
#−1 X r2
Cj00 ∂ej
rj p
p
i
=
− ej −
00
00
rj ∂rj
Ci
Cj
rj
i
(24)
eq:PropDer1
(25)
eq:PropDer2
(26)
eq:OptTRFOC
which implies from (22) that
"
#−1 X r2
Ci00 ∂ei
rj p
i
=
− ej
ri ∂rj
Ci00
Cj00
i
i 6= j.
for each
The derivative of the regulator’s objective with respect to rj is as in (4)
"
#
X ∂Ci ∂D ∂ei
∂W
=E
+
∂rj
∂e
∂ei ∂rj
i
i
"
=E
X
i

"
= E pej +
X ∂D X
i

∂ei X ∂D ∂ei
−pri
+
∂rj
∂ei ∂rj
i
"
X r2
i
= E pej +
00
C
i
i
∂ei
#−1 "
i
ri2
Ci00
#−1 rj p
− ej
Cj00
#

ri
∂D p 
−
00
Ci
∂ej Cj00
(
)
#
∂D 2
∂D
∂D
X
X
X
r
r
r
r
j ∂ei i
p
∂ej i
∂ei i 
−
− ej
00
00
00
Cj
Ci
Ci
Ci00
i
i
i

 

#−1 "
#−1
"
∂D
2
X
X
X X r2
r
pri
∂D
∂D
ri
∂ei i  
i
= E
(r
− ri
) + p −
ej
00
00 00 j
00
00
C
C
C
∂e
∂e
C
C
i
j
i
j
i
i
i
i
i
i
i
=E
"
X
i
A−1
ai p
∂D
∂D
(rj
− ri
) +
ri Cj00 ∂ei
∂ej
p − A−1
X ai ∂D
∂e
i
i
ri
! #
ej
where the first equality is a restatement of the FOC in (4), the second equality follows
34
from the definition of p, the third equality from subsituting (23), (24), and (25), and the
fourth and fifth equalities follow from algebra.
To derive the covariance formula, recall that COV (X, Y ) = E[XY ]−E[X]E[Y ]. Applying
h i
h i
∂D
∂D
this formula repeatedly to (26) and noting that r̃j E ∂ei = r̃i E ∂e
by the definition of
j
P
∂D
marginal damage trading ratios and that E[p − A−1
r̃ /Ci00 ] = 0 by Lemma 1 establishes
∂ei i
that:
X
∂D
∂D −1 ai p
∂W =
COV r̃j
− r̃i
,A
+ COV
00
∂rj R̃
∂e
∂e
r̃
C
i
j
i
j
i
p − A−1
X ai ∂D
i
r̃i ∂ei
!
, ej
which implies the result.
Proof of Corollary 1.
Because each marginal damage function is non-stochastic, we have
∂D
/r̃i
∂ei
=
∂D
/r̃1
∂e1
every i. Substituting these expressions into the formula in Proposition 1 gives:
∂W = COV (p, ej ) − COV
∂rj R̃
= COV (p, ej ) − COV
A
−1
X ai ∂D
r̃1 ∂e1
i
∂D
∂e1
r̃1
where the third equality follows since r̃1 and
!
, ej
+
X
i
−1 ai p
COV 0, A
r̃Cj00
!
A−1
X
ai , ej
= COV (p, ej )
i
∂D
∂e1
35
are non-stochastic.
for
Table 6: Nitrogen Trading with Optimal Trading Ratios: Artificial Reduction in Size of
Largest WWTP
Baseline Convert
η = 10
all
b = 30 to average
Trading ratios
Max Marginal Damage
Min Marginal Damage
Max Optimal
Min Optimal
St. Dev. Price
Marginal Damage TR
Optimal TR
Expected Price
Marginal Damage TR
Optimal TR
St. Dev. Total Emissions
Marginal Damage TR
Optimal TR
Expected Total Emissions
Marginal Damage TR
Optimal TR
Total Cost
First Best
Marginal Damage TR
Optimal TR
Deadweight Loss
Marginal Damage TR
Optimal TR
Percent Reduction
1
2
3
4
Convert
3 largest
to average
Convert
2 largest
to average
Convert
largest
to average
3.553
0.063
6.553
0.015
5.025
0.077
5.048
0.077
4.045
0.063
5.753
0.071
4.218
0.069
6.022
0.048
4.755
0.098
6.878
0.012
2.171
0.525
0.844
0.843
0.958
0.557
0.918
0.532
0.816
0.465
2.739
1.635
2.103
2.101
2.057
1.610
1.972
1.538
1.750
1.347
0.038
0.055
0.025
0.025
0.020
0.023
0.023
0.026
0.037
0.040
4.842
4.846
4.847
4.848
3.410
3.413
3.584
3.587
4.107
4.110
9.773
9.870
9.798
8.440
8.461
8.461
6.328
6.348
6.343
6.362
6.382
6.376
6.443
6.462
6.457
0.097
0.024
74.9%
0.021
0.021
.003 %
0.020
0.014
28.5%
0.020
0.014
28.5%
0.020
0.014
28.7%
The permit endowment is set to the sum of the optimal pollution standards.
Costs and deadweight loss (DWL) in millions of dollars per year.
Prices in dollars per pound.
Emissions in millions of pounds per year.
36
tab:neusear
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39
Figures
Figure 1: Marginal Damage Trading Ratios with Uniformly Mixed Pollution
Source 1 Emissions
Source 2 Emissions
fig1
Market Emissions
And Cap
𝑀𝑀
𝑝𝐻
𝑝𝐿
𝑒1𝐻 𝑒1𝐿
𝑀𝑀𝐶1
𝑀𝑀𝐶 𝐻
𝑀𝑀𝐶2𝐻
𝑒2𝐿 𝑒2𝐻
𝑀𝑀𝐶2𝐿
𝐸�
𝑀𝑀𝐶 𝐿
Figure 2: Optimal Trading Ratios with Uniformly Mixed Pollution
Source 1 Emissions
Source 2 Emissions
fig2
Market Emissions
And Cap
𝑀𝑀
𝑝𝐻
𝑝𝐿
𝑒1𝐻 𝑒1𝐿
𝑀𝑀𝐶1
𝑀𝑀𝐶2𝐻
𝑒2𝐿 𝑒2𝐻
40
𝑀𝑀𝐶2𝐿
𝑒1𝐿 + 𝑒2𝐿 𝐸� 𝑒1𝐻 + 𝑒2𝐻
Figure 3: Location of WWTP; Marginal Damage and Optimal Trading Ratios
fig3
ROANOKE
CHOWAN /
PASQUOTANK
NEUSE
TAR
Trading Ratios
3.3
MD Ratios
Optimal Ratios
0 12.5 25
50
75
41
100
Miles
¹
Additional Appendices
A
Proofs of Results for the Linear-Quadratic Example
aa:LinQuad
Proof of Result 1.
Combining (1) and (12) gives
ej =
θj − rj p
.
λj
Substituting this into (2) and solving for p gives
X
rj θj
−1
p=A
−1 .
λj
(27)
eq:ejlin
(28)
plq
(29)
vp
(30)
pdun
Now using (27), we have
1
1
(COV (p, θj ) − COV (p, rj p)) =
(COV (p, θj ) − rj V AR(p)) .
λj
λj
COV (p, ej ) =
Evaluate each of these in turn. We have
−1 2
V AR(p) = (A )
X
V AR
ri θi
λi
= (A−1 )2
X σ 2 r2
i i
λ2i
Using (28) and the fact that the random variables are independent gives
COV (p, θj ) = A−1 COV (
X rj θj
λj
, θj ) = A−1 (
rj 2
σ ),
λj j
Substituting in gives
A−1 rj
COV (p, ej ) =
λj
X σ 2 r2
σj2
i i
− A−1
λj
λ2i
.
This equation implies that the covariances, evaluated at any R, will all be equal to zero if
σ2
and only if λjj is equal to the same constant for every j. The desired result now follows from
Corollary 1. Proof of Result 2. Consider the first-order Taylor series expansion of W at the point R̃. We
have
X ∂W
W(R) ≈ W(R̃) +
· (rj − r̃j ),
∂rj
j
where it is understood for this discussion that the partial derivatives are evaluated at R̃.
Now let R = R̃ + (s1 δ, s2 δ, . . . , sn δ) where δ is small and si is a sign variable that takes on
values of either 1 or −1 as follows. If ∂W
> 0 then sj = −1. If ∂W
< 0 then sj = 1. Thus R
∂rj
∂rj
is a vector that is close to R̃ and, following the discussion of Proposition 1 in the main text,
42
the point R has been constructed so that it has lower total expected costs than the point
R̃. So to a first order approximation, the welfare improvement from using optimal trading
ratios is
X ∂W
X ∂W
X ∂W W(R) − W(R̃) ≈
· (δsj ) = δ
· (sj ) = −δ
(31)
∂rj .
∂r
∂r
j
j
j
j
j
owi
Next we evaluate the partial derivatives. We have
X σ 2 r̃2 ∂W
A−1 r̃ σj2
i
−1
= COV (p, ej ) =
−A
,
∂rj
λj
λj
λ2i
where the first equality comes from Corollary 1 and the second comes from evaluating (30)
at the marginal damage trading ratios R̃ = (r̃, r̃, . . . , r̃). Next we note that
X r̃2
A−1 =
i
!−1
=
λi
r̃2
1
P
1
i λi
.
Substituting in and simplifying gives us
σj2
1 X σi2
−P 1
λj
λ2i
i λi
i
1
∂W
= P 1
∂rj
r̃
i λi λ j
!
1
=
r̃
P
1
i λi
(xj − x̄w ) .
λj
Substituting this into (31) gives
X 1
.
W(R) − W(R̃) ≈ −δ
(x
−
x̄
)
j
w
P 1
λj
j r̃
i λ
i
=
X 1
−δ
P |xj − x̄w | .
1
λ
j
r̃
j
i λi
The desired result follows directly. Proof of Result 3.
Let E ∗ be the vector of optimal emissions standards. It is straightforward to show that
E ∗ = (Λ + V)−1 E[Θ].
(32)
From Proposition 1 we have
∂W = COV (p, ej ) − COV
∂rj R̃
−1
A
X ai ∂D
i
r̃i ∂ei
!
, ej
+
X
i
∂D
∂D −1 ai p
COV r̃j
− r̃i
,A
∂ei
∂ej
r̃i Cj00
From (14) and the proof of Result 1, we know that COV (p, ej ) = 0 for every j. Thus the
first term in this equation is zero. The last term is zero as well, because for the quadratic
43
eigen
P
∂D
model marginal damages are ∂e
=
k vj,k ek , and, in addition, ai is non-stochastic. So the
j
last term can be written as the sum of covariances of ej and p, which of course are equal to
zero. So we have
!
X ai ∂D
A−1 X
∂D
∂W −1
= −COV A
, ej = −
r̃i COV
, ej
∂rj R̃
r̃i ∂ei
λ i
∂ei
i
Substituting in the expression for marginal damage gives
∂W A−1 X X
r̃k r̃j σ 2
A−1 X X
1
=−
r̃i
r̃i
vi,k COV (ek , ej ) = −
vi,k 2 COV [θk , θj ] − P 2 .
∂rj R̃
λ i
λ i
λ
r̃i
k
k
where the second equality follows from (47). Now, because the random variables are independent, the COV (θk , θj ) will be equal to σ 2 when k = j and 0 otherwise. So we have
!
!
2
−1 2
X
X
XX
A−1 X
r̃
r̃
σ
∂W A
σ
r̃
k
j
j
=− 3
vi,k P 2
vi,k r̃i r˜k .
r̃i vi,j σ 2 −
=−
r̃i vi,j − P 2
3
∂rj R̃
λ
r̃
λ
r̃
i
i
i
i
i
k
k
Now let’s consider the first-order conditions
∂W = 0.
∂rj R̃
We can now write this system of n equations in terms of the n variables ri using matrix-vector
notation. This gives
!
R̃t VR̃
A−1 σ 2
VR̃ − R̃
−
= 0.
λ3
R̃IR̃
It follows immediately that R̃ is a solution to the first-order conditions if and only if R̃ is an
eigenvector of V .
Next we show that R̃ is an eigenvector of V if and only if E[θ] is an eigenvector of
(Λ + V)−1 .
The first step is to show that if E[Θ] is an eigenvector of (Λ + V)−1 then R̃ is an eigenvector of V. Because E[Θ] is an eigenvector of (Λ + V)−1 , equation (32) implies
E ∗ = φE[Θ]
(33)
eqq
E[Θ] − ΛE ∗ = VE ∗ .
(34)
standard
for some scalar φ. Now write (32) as
Substituting in for E ∗ on the left-hand side gives us
E[Θ] − φΛE[Θ] = VE ∗ .
44
Because Λ is diagonal with all entries equal to λ, we can write this as
(1 − λφ)E[Θ] = VE ∗ .
Using (33) this becomes
1 − λφ ∗
E = VE∗ .
φ
This implies that E ∗ is an eigenvector of V. Now, from (16), we have
R̃ =
1 − λφ ∗
VE ∗
=
E .
E[p]
φE[p]
Thus R̃ is proportional to E ∗ , so it is an eigenvector of V as well.
The second step is to show that that if R̃ is an eigenvector of V, then E[Θ] is an eigenvector
of (Λ + V)−1 . Because R̃ is an eigenvector of V, we have
VR̃ = ν R̃
for some scaler ν. Using (16), we have
V
which can be written as
V
VE ∗
VE ∗
=ν
E[p]
E[p]
VE ∗
E∗
−ν
E[p]
E[p]
= 0.
Because V is invertible, this implies
E∗
VE ∗
=ν
,
E[p]
E[p]
which implies that E ∗ is an eigenvector of V. This means we can write (34) as
E[Θ] − ΛE ∗ = νE ∗ .
Because Λ is diagonal with all entries equal to λ this becomes
E[Θ] = (λ + ν)E ∗ .
Substituting the definition of E ∗ from (32) gives
E[Θ]
= (Λ + V )−1 E[Θ],
λ+ν
which implies that E[Θ] is an eigenvector of (Λ + V )−1 .
Finally, we observe that (32) implies that E[Θ] is an eigenvector of (Λ + V )−1 if and only
if E ∗ is proportional to E[Θ].
45
Proof of LQ Result 4.
From the proof of Lemma 1, the first-order condition for the marginal damage trading
ratio r1 is given in (21) as
#
"
X ∂Ci ∂D ∂ei
+
.
0=E
∂e
∂e
∂r
i
i
1
i
∂ei
i
From (20) we have ∂r
= r−a
A−1 . In the linear-quadratic model, the second derivative of
1
1 ri
the abatement cost function is non stochastic, so ai = ri2 /λi is non stochastic which implies
∂ei
is also non stochastic, and hence we can write the first-order condition as
that ∂r
1
X ∂Ci ∂D
∂ei
0=
E
.
+E
∂ei
∂ei
∂r1
(35)
fwlq1
(36)
fwlq2
(37)
fwlq6
Using (1) we can write this as
0=
Now consider
X
∂D
E [−p] ri + E
∂ei
∂ei
.
∂r1
∂D
r̃i = E
/E [p] .
∂ei
Clearly these trading ratios are marginal damage trading ratios. Furthermore, under these
trading ratios, the term in parentheses in (36) is equal to zero. Thus the term in parentheses
in (35) is zero. In other words, when evaluated at R̃, we have
∂Ci
∂D
E
+E
= 0.
(38)
∂ei
∂ei
fwlq3
Combining (38) and (37) gives
∂Ci
r̃i = −E
/E [p] .
∂ei
(39)
finally
Now, using the structure of the linear-quadratic problem with quadratic damages, we
can write (38) as a system of equations in matrix form as
−E[Θ] + ΛE[E(R̃, Θ)] + V E[E(R̃, Θ)] = 0,
where Λ is a diagonal matrix with entries λi . It follows that
E[E(R̃, Θ)] = (Λ + V )−1 E[Θ.]
(40)
For the linear-quadratic example, the optimal emissions standards defined from (15) are
46
fwlq4
the solutions to the first-order conditions
E[Θ] − ΛE ∗ = V E ∗ ,
(41)
oes
E ∗ = (Λ + V )−1 E[Θ].
(42)
fwlq5
E ∗ = E[E(R̃, Θ)].
(43)
fwlq6b
from which it follows that
Combining (40) and (42) gives
Next, we note that
∂Ci
E
= −E[θi ] + λi E[ei (R̃, Θ)] = −E[θi ] + λi e∗i ,
∂ei
where the second equality follows from (43). So from (39), we have
r̃i = (E[θi ] − λi e∗i ) /E[p].
Thus the marginal damage trading ratios are proportional to the expected marginal abatement cost evaluated at the optimal emissions standards. The desired equation follows from
(41). Proof of LQ Result 5.
From (27) and (28) we have
ei =
and
θi − ri p
.
λ
(44)
eee
(45)
vp
(46)
eqra2
(47)
eq:Covee
1 X
p = P 2(
θi ri − λ).
ri
It is useful to calculate several variances and covariances. First,
X
1
σ2
V AR[p] = P 2 2
V AR[θi ri ] = P 2
( ri )
ri
Because the θi are independent, we have
X
1
rj σ 2
COV [θj , p] = P 2 COV [θj ,
θi ri ] = P 2
ri
ri
and
COV [ei , ej ] =
=
1
COV [θi − ri p, θj − rj p]
λ2
1
(COV [θi , θj ] − ri COV [θj , p] − rj COV [θi , p] + ri rj V AR[p])
λ2
1
ri rj σ 2
= 2 COV [θi , θj ] − P 2 .
λ
ri
47
Note that (47) holds even if i = j.
To evaluate the regulator’s objective, we begin by evaluating expected abatement costs.
From (11), expected abatement costs for i are
"
2 #
θi
λ
− ei
E[Ci (ei ; θi )] = E
2
λ
2
θi
λ
θi
− ei +
E
− ei
λ
2
λ
2
r p λ θ̄
λ
i
i
= V AR
+
− E[ei ]
2
λ
2 λ
λ
= V AR
2
ri2 σ 2
P + Ci (E[ei ]; θ̄i ).
=
2λ ri2
(48)
eq:Ecosts
where the third equality follows from (44).
Now carry out a similar manipulation of the damage function D(E) = 12 E t VE where V
is a symmetric matrix with elements vij . Writing this as
D(E) =
we have
E[D(E)] =
=
1 XX
vij ei ej ,
2 i j
1 XX
vij E[ei ej ]
2 i j
1 XX
vij (COV [ei , ej ] + E[ei ]E[ej ])
2 i j
1 XX
vij COV [ei , ej ] + D(E[E])
2 i j
XX ri rj σ 2
vij COV [θi , θj ] − P 2 + D(E[E])
ri
i
j
!
2 XX
X
σ
σ2
vii − P 2
vij ri rj + D(E[E])
ri i j
i
σ2 X
Rt VR
= 2
vii − t
+ D(E[E])
2λ
R IR
=
=
1
2λ2
=
1
2λ2
where I is the identity matrix and Rt is the transpose of R.
Summing (48) over i and adding it to (49) establishes the result. 48
(49)
eq:Edam
B
Carbon Trading Computation
aa:carbon
Ackerman and Bueno (2011) describes the adaptation of the well-known McKinsey abatement cost curves for 2030 (from McKinsey & Company) for use in their integrated assessment
model (IAM) of global climate change named the Climate and Regional Economics of Development (CRED) model. The McKinsey curves are controversial for their estimates of
substantial quantities of negative cost abatement opportunities. Ackerman and Bueno circumvent this difficulty by fitting a simple two-parameter functional form through the positive
cost portion of the McKinsey abatement cost curves. The functional form is ai A/(bi − A)
where A is abatement and ai and bi are the fitted parameters. The function form goes
through the origin by assumption.
Figure 4 shows an example of the fitted functional form and the McKinsey abatement
cost curve for the industrial sector in S/SE Asia. Note that the curve asymptotes to bi (the
vertical line on the right). Thus bi can be thought of as BAU emissions, i.e., the maximum
possible abatement. Note also that ai is the marginal abatement costs when emissions
(equivalently abatement) are half of bi .
We use the equations by converting them to functions of emissions (rather than abatement) and by introducing stochastic BAU emissions. Thus the marginal abatement cost
curve is
ai (bi (1 + θi ) − ei )
∂Ci
=
.
−
∂ei
ei
Integrating the marginal abatement cost curve yields the total abatment costs
Ci (ei ; θi ) = ai ei − ai bi (1 + θi ) ln(ei )
Note that abatment costs would be infinite if emissions were zero. Also note that the
integration yields an unspecified constant of integration, so we calculate deadweight loss as
the difference between the first-best, full-information outcome and the policy of interest.
The appeal of this functional form is twofold. First, the function form is simple, and
i
asymptotes to
we can introduce stochasticity in a transparent way. Second, because − ∂C
∂ei
zero, we need not worry about the boundary condition of nonnegative emissions. As long as
the price is positive, we safely have an interior solution and the second order conditions are
satisfied.
To reduce the dimensions of the problem, we focus on the industrial sectors in the four
regions with the largest estimated bi ’s. These four regions are reported in Table 7. Table 7
reports the fitted coefficients for 2030 along with the actual emissions from 2006-2008. The
bi are reasonable approximations of BAU emissions.
The calculations are done in Mathematica. With four regions and three independent outcomes of each random variable, there are eighty-one possible states of the world to evaluate.
To calculate the first-best, full-information outcome, we calculate the optimal emissions cap
for each of the eighty-one states. Because marginal damage trading ratios for a uniformly
mixed pollutant imply one-for-one trading, the marginal damage trading ratios can be calculated by optimizing the emissions cap that minimizes the sum of expected abatement costs
and damages.
Calculation of the optimal trading ratios requires optimization over a four-dimensional
49
vector of trading ratios. We first calculate the equilibrium that would result from trading for
a given vector of trading ratios. We then optimize the the trading ratio vector to minimize
the sum of expected abatement costs and damages.
Use of McKinsey abatement cost curves for climate economics modeling
WP-US-1102
Figure 4: McKinsey Marginal Abatement Cost Curve and Ackerman-Bueno Approximation
Figure 1
McK
MAC: S/SE Asia - Industry
$400
Marginal abatement cost (US$/tC)
$300
$200
$100
$0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-$100
-$200
-$300
-$400
-$500
Abatement potential, 2030 (GtC)
The estimated values of A and B are presented in Table 1.
Table 1
aa:Tax
CMarginal
Illustrating
Optimal Facility-Specific Taxes
Abatement Cost Curve Coefficients
Figure 5 illustrates optimal facility-specific environmental taxes from (10). The figure ilLand“high”
Use Sector
Industryabatement
Sector
lustrates marginal damages and
and “low” marginal
costs for a single
A costs in theBlow-cost state
A are M AC1LB, then the slope of the
facility. If marginal abatement
marginal
equal across the
the COV 0.28
term in (10) is zero;
Africa abatement costs are 23.12
0.60two states;
64.50
andChina
the regulator should set the
facility-specific
tax
equal
to
expected
marginal
damages, as
36.84
0.20
66.39
2.64
illustrated.
However,
abatement costs
low-cost state 0.57
are M AC2L , then the
Russia/Eastern
Europeif marginal9.68
0.10 in the 36.80
slope
of the marginal abatement
cost curve is
greater in
the low-cost state;
Europe
92.43
0.40
101.91
0.83 the COV term
in L.
(10)
is
positive;
and
the
regulator
should
set
the
facility-specific
tax
above the expected
America/Caribbean
2.28
0.92
51.53
0.37
marginal damage.
Middle East
6.79
0.05
46.53
0.37
Other high-income
30.54
0.09
107.41
0.54
S/SE Asia
26.23
1.36
53.90
1.28
U.S.A.
75.41
0.16
64.74
1.39
Units: A in $ / tC, B in GtC
Use of (1) rather than the empirical curves simplifies cost calculations. For any carbon price p, (1) can
be inverted to yield the quantity of abatement available at MAC(q) < p
(2)
50
Table 7: Coefficients for Industry Marginal Abatement Cost Curves Coefficients and FossilFuel Emissions
China
Europe
S/SE Asia
U.S.A.
ai
66.39
101.91
53.90
64.74
MACCoef
bi
Emissions
2.64
2.02
0.83
1.19
1.28
0.56
1.39
1.72
Coefficients from Ackerman and Bueno
(2011).
The ai coefficient is the marginal abatement cost when emissions are half of
BAU in $ per ton C. The bi coefficient is
the maximal possible abatement, which
can be interpreted as BAU in GtC.
Emissions (from fossil fuels and cement
manufacture) are from the World Bank
database averaged across 2006-08 in
GtC.
Figure 5: Optimal Facility-Specific Taxes
𝑀𝐷
𝑀𝐷
𝐻
𝑡 = 𝐸[𝑀𝐷]
𝑀𝐴𝐶1𝐿
𝑀𝐷
𝑀𝐴𝐶 𝐻
𝐿
𝑀𝐴𝐶2𝐿
𝑒𝐿
emissions
𝑒𝐻
51
fig:tax