Water Quality Trading: Can We Get the Prices of Pollution Right?1
Yoshifumi Konishi
Faculty of Liberal Arts
Sophia University
Jay Coggins
Department of Applied Economics
University of Minnesota
Bin Wang
Department of Public Health
Pennsylvania State University
Draft: February 20, 2012
1 We
gratefully acknowledge nancial support from a U.S. Environmental Protection Agency 2005 Targeted Watershed Grant and a Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists
B. We also thank Scott Farrow, Qiuqiong Huang, Frances Homans, Jacob LaRiviere, and seminar participants at Kyoto University, University of Minnesota, and at the 2nd Congress of the East Asian Association
of Environmental and Resource Economics for their helpful comments.
1
Abstract: A substantial challenge has loomed in designing water-quality trading mechanisms: getting the prices right that account for spatially explicit damage relationships in
a watershed. This paper extends recent work by Hung and Shaw (2005) and Farrow et al.
(2005) by incorporating two important features of many watersheds: (i) branching rivers;
and (ii) nonlinear pollution damages. The mechanism of Hung and Shaw fails to achieve
the social optimum when there are critical zones in a branching river, but it is robust to
nonlinear damages. The mechanism of Farrow et al. fails when damages are nonlinear, but
is robust to branching. When the initial distribution of permits is not at the optimum, neither mechanism dominates. The former precludes efficient trades across branches while
the latter encourages inefficient trades. We also find that the efficiency loss due to not
getting the total supply of permits right is substantially larger than that from not getting
prices right.
2
1
Introduction
The idea of using water-quality trading (WQT) to aid in protecting water quality is appealing. The U.S. experience with its SO2 allowance market proved that markets for air
pollution can work. Should they not work for water pollution too? The U.S. Environmental Protection Agency (EPA) seems to think they can. It actively encourages states
to establish rules for water-quality trading (WQT).2 Currently, there are a total of 54 water quality trading programs in the United States, with eleven states having a state-wide
trading policy in place or in development and three more adopting watershed-specific
state trading programs (EPA, 2011). The results from these programs have, for the most
part, been disappointing. Several barriers to trading have emerged (see, for example,
EPA, 2008; King and Kuch, 2003; Morgan and Wolverton, 2005; Woodward and Kaiser,
2003).
One such barrier is the difficulty of getting the prices of pollution right (Farrow et al.,
2005; Hung and Shaw, 2005). By this we mean, following Muller and Mendelsohn (2009),
that each source faces a set of permit prices that reflect correctly the marginal damages
caused over the landscape by its own emissions and those of its trading partners. The
spatial relationship between the location of air emissions and the location of resulting
damages is well known (Mauzerall et al., 2005). Spatial dependence is likely even more
prominent for water pollution, where the attenuation and transport characteristics of numerous water pollutants are often critically dependent upon local hydrogeographic conditions at and downstream from each source (Todd and Mays, 2005; Schnoor, 1996).
2 The
Agency’s “Water-Quality Trading Policy” (EPA 2003) and “Water-Quality Trading Assessment
Handbook” (EPA 2004) are meant to help guide state and local environmental policy. Improving water
quality in a cost-effective manner has become a top EPA priority, in part due to a series of litigations concerning Section 303(d) of the Clean Water Act (CWA) since the 1980’s. The CWA requires all states, territories, and authorized tribes to develop lists of “impaired waters” every two years and to develop the
total maximum daily load (TMDL) for every impaired waterbody/pollutant. By the early 2000’s, EPA was
placed under court order, agreeing in a consent decree to enforce a TMDL in 27 litigated cases. A waterbody
is designated as “impaired” for a pollutant when it violates ambient water quality standards for that pollutant. As of September 2010, 39,988 waters were listed as “impaired.” A TMDL is the maximum amount of
a pollutant that a waterbody can receive and still meet water quality standards. It also allocates that load
among the various sources of the controlled pollutant.
3
Thirty years ago or so a lively literature arose in which various permit-trading schemes
were proposed and analyzed. Montgomery’s (1972) ground-breaking ambient pollution
system (APS) establishes a separate permit market for every receptor point. This system is
impractical, as firms would have to know the impacts of their emissions on all relevant receptors and participate in a number of downstream markets. Another early contribution
is by Atkinson and Tietenberg (1982), who considered a system of pollution offsets (POS)
in which each new or expanding source is required to buy offsets from existing sources
if their emissions violate an ambient environmental standard at any receptor point. The
emissions must be traded at the ratio of the two sources’ transfer coefficients. Under the
POS, the exchange rate must be calculated for each trade based on computer simulations.
A modified version of the POS has been applied to many bilateral water-quality trades
in recent years. In practice, the system has often resulted in sizable transaction costs,
because each bilateral trade must undergo intensive scientific evaluation and ad hoc negotiations with potential trading partners.3 The papers by Montgomery and by Atkingson
and Tietenberg belong to a large literature from that era in which a host of competing
arrangements for trading systems were proposed. Most were concerned, either explicitly
or implicitly, with trading for air quality.
A more recent literature, aimed specifically at water quality trading, focuses on designing tradable permit systems that address characteristics specific to water. Our focus is
upon two of the recent contributions, both of which restrict attention to point-source problems. Hung and Shaw’s (2005) trading-ratio system (TRS) takes into account an important
feature of rivers: water flows unidirectionally from upstream to downstream. The TRS
transforms ambient environmental standards into ambient zonal discharge constraints
according to physical transfer coefficients. Firms then participate in a single watershedwide market in which they can trade with each other (with certain restrictions) at the
3 In
this connection, the following comment on Oregon’s thermal-trading initiative makes the point:
“[T]he trade took considerable resources on the part of both Clean Water Services (CWS) and DEQ to develop. The effort would have been neither practical nor worthwhile for a source much smaller than CWS
to undertake” (Oregon DEQ, 2007).
4
predetermined trading ratios subject to the zonal discharge constraints. An essential feature of the TRS is that the trading ratios are based on physical transfer coefficients rather
than marginal damages, which can be more difficult to estimate. Though it offers several
advantages over APS or POS, the TRS has an important drawback. The unidirectional
nature of the transfer characteristics means that the TRS might not work as advertised for
branching river systems.
Farrow et al. (2005) proposed an alternative trading system, which was later applied
successfully in the study of air pollution markets by Muller and Mendelsohn (2009). Like
the TRS, Farrow et al.’s system allows firms to trade freely in a single watershed-wide
market at predetermined exchange rates, but the rates are based on the ratios of marginal
damages (hence, we refer to it as a damage-denominated trading ratio system or DTRS).
Because the DTRS does not rely on the unidirectional nature of the transfer characteristics, it is robust to branching. Its disadvantage, however, is that it may not be robust to
damages that are nonlinear in pollution levels. Nonlinear damages may result from either
nonlinear pollution-transport processes (Todd and Mays, 2005) or a nonlinear biological
response of aquatic species and human health to water pollution (Anderson et al., 2002;
Van Kirk and Hill, 2007) or both, even if economic agents’ marginal (dis)utility from water pollution is approximately constant. The quality of water and aquatic habitat at any
location in the river is typically a nonlinear function of concentrations of these pollutants,
often exhibiting some threshold effect. Numerous examples exist: brown trout may cease
growth at water temperatures above 18.7-19.5 celsius degrees (Elliott and Hurley, 2001)
and may not survive seven days above 24.7 0.5 celsius degrees (Elliott, 2000). The population size of cutthroat trout is a highly nonlinear function of selenium exposure and
was estimated to experience 90% declines at mean selenium concentrations exceeding 17
µg= g of dry weight (Van Kirk and Hill, 2007). And algae growth is often modeled as a
logistic function of nutrient concentrations (Anderson et al., 2002).4
4 It
appears that the effects of nonlinear damages have not yet been given the attention they deserve in
the context of water-quality trading, though their existence and importance have long been well recognized
5
This paper extends the work of Hung and Shaw (2005) and Farrow et al. (2005) by
incorporating into a single model the two important features noted above: (i) branching
rivers and (ii) nonlinear damages. We investigate the efficiency and cost-effectiveness properties of the two systems in a framework similar in nature to that of Muller and Mendelsohn (2009). In Section 2, we start by defining a social planner’s efficient decision program
in water-quality management for a generic watershed. Our planner minimizes the sum of
abatement costs and pollution damages by choosing a vector of emissions from stationary
point sources distributed across space in a watershed drained by a branching river. The
model generalizes Farrow et al. (2005) and Muller and Mendelsohn (2009) in a non-trivial
manner by accounting for nonlinear damages. We show that under some regularity conditions, there exists a cost-effectiveness program that implements the efficient outcome.
The result thus allows us to discuss the TRS and DTRS on the same efficiency grounds.
Sections 3 and 4 consider in turn the problems with the TRS and the DTRS. First, in
Section 3, we show that the TRS fails to achieve a cost-effective optimum and, hence, the
social optimum, when a critical zone (that is, a hot spot at which the pollution arriving
from upstream exceeds the zone’s concentration constraint) exists at a confluence of a
branching river. In this case the discharge constraint in the critical zone must be set to
zero and the constraint upstream of the critical zone must be tightened. At a confluence,
though, the required adjustment to the upstream discharge constraints becomes indeterminate. Thus, Hung and Shaw’s main result, that the TRS equilibrium achieves the
cost-effective outcome, is not robust to branching.
Second, in Section 4, we show that Farrow et al.’s DTRS is robust to branching, but
fails to achieve the cost-effective optimum (hence, the social optimum) in the presence
of nonlinear damages. Under the DTRS, the trading ratios across space must be fixed
prior to trading, which can give incorrect incentives for trading participants at the margin. Mathematically, this result is due to the non-existence of emissions vectors that can
in the economics literature (Helfand and House, 1995; Larson et al., 1996; Segerson, 1988).
6
satisfy the necessary conditions for the social optimum and Farrow et al.’s initial permit
allocation rule, which occurs precisely because of the nonlinearity of pollution damages.
The zest of our paper lies in Section 5. Our finding that the TRS falls down precisely
where the DTRS succeeds, and vice versa, suggests that the relative performance of the
two systems may depend on the distribution of sources in a watershed featuring both
branching rivers and nonlinear damages. We investigate this question by constructing a
small numerical model and perturbing the geographic distribution of pollution sources
in a watershed. Our simulation model is an illustrative adaptation of the National Water Pollution Control Assessment (NWPCA) model developed for EPA. That model was
used in Farrow et al. (2005) as well as in other regulatory applications. The model is well
grounded in hydrology and so is suitable for estimation of the water-quality impacts of
pollution in a complex watershed. Our simulation, though true to the NWPCA, is based
upon a simple river system with three sources.
Using our parameterized model, we consider a second-best scenario in which the total number of permits available at the initial allocation is optimal, but their distribution
among sources is not. This assumption helps us disentangle the sources of inefficiency if
either the TRS or the DTRS fails to achieve the social optimum. Because the total amount
of permits is optimal, any inefficiency must be attributed to problems with the trading
ratios. That is, sources are faced with incorrect price signals. Our simulation results
demonstrate that neither system dominates: each stumbles in its own way. On one hand,
the TRS can result in welfare loss because the trading ratios based on transfer characteristics may preclude some efficient trades across branches. On the other hand, the DTRS can
result in welfare loss because the fixed trading ratios based on marginal damages at some
emissions vectors may, due to incorrect marginal incentives, encourage inefficient trades.
Under the DTRS, sources may either over-abate or under-abate relative to the optimum.
In this sense, our results suggest the impossibility of getting the spatially explicit prices
right for WQT.
7
More encouraging, though, is the fact that the deadweight losses associated with either system are relatively modest so long as the total number of permits is optimal. Put
another way, the efficiency loss from failing to issue the correct number of permits is
much greater than the efficiency loss from failing to set the correct trading ratios. This
last is what is meant by getting prices right. Moreover, somewhat paradoxically, even
with perfectly competitive markets we show that issuing the correct number of permits
can also be essential for getting the prices of pollution right. We defer to the concluding
section a brief discussion of the implications of our results for nonpoint-source pollution.
2
A theoretical model of water-quality management
In this section we develop a static model of water-quality management in a generic river
basin. We show how the TRS and the DTRS can both be derived directly from this
more generic model. Thus, the two alternatives can be compared on the same efficiency
grounds within our framework.
Let e = (e1 , . . . , ei , . . . , e N ) be a vector of emissions of a single pollutant, where ei represents emissions from point source i, and let ē be a vector of baseline or uncontrolled
emissions. Index i serves the dual purpose of denoting a source and also its geographic
location. Clearly, ei
ēi for every i. Let x = ( x1 , ..., xm , . . . , x M ) be a vector of ambient
pollution levels, where xm denotes concentration at receptor m. Assume that there exists
a linear mapping T : R N ! R M describing the scientific relationship between e and
x. This linearity assumption has a long heritage in the economics literature (see Montgomery, 1972; Krupnick et al., 1983; McGartland and Oates, 1985; and Hung and Shaw,
2005). Let T be given by x = Te0 , where T is a M
N matrix of nonnegative transfer coef-
ficients.5 Let S : R M ! R, given by S(x), be a differentiable function that describes total
5 The literature on groundwater hydrology suggests that the mapping
T may not be linear (see Todd and
Mays, 2005). Thus, damage functions can be nonlinear for two different reasons. First, environmental harm
may be a nonlinear function of concentrations (our S). Second, concentrations at receptors may themselves
be a nonlinear function of emissions (our T). Our results apply to nonlinearity of either type, but we
8
economic damages as a function of the vector of ambient pollution levels. Assume that
∂S=∂xm > 0 for all m. It follows that total economic damages as a function of emissions
are differentiable and are given by D (e) = S( Te0 ). Define a vector of abatement levels
a = ē
e, where by definition ai 2 [0, ēi ]. Each source i is assumed, here and throughout
the paper, to have a twice-differentiable abatement cost function Ci ( ai ), with Ci0 > 0 and
Ci00 > 0.
Under these standard assumptions, an efficient program minimizes the sum of abatement costs and damages:
min
∑i=1 Ci (ai ) + D(ē
N
a).
(1)
Given that the Ci ’s and D are differentiable (and so continuous) and that ai 2 [0, ēi ] for all
i, the Wiestaurass theorem ensures that a solution to (1) exists. Denote this optimum ae f f .
In the earlier literature (Montgomery, 1972; Krupnick et al., 1983; McGartland and
Oates, 1985; Hung and Shaw, 2005), it was often assumed that a social planner solves not
(1) but rather an auxiliary cost-effectiveness program of the form:
min ∑i=1 Ci ( ai ) s.t. xm
N
X̄m and x = Te0
(2)
where X̄ = ( X̄1 , . . . , X̄M ) is a vector of environmental constraints on ambient pollution
levels, one for each receptor. Denote the solution to (2), by a HS . As we will see in Section 3,
the TRS attempts to solve the cost-effective program (2) rather than the efficient program
(1).
And we will see in Section 4 that Farrow et al. (2005) considered a different program
still. Their DTRS is aimed at minimizing the sum of abatement costs subject to a constraint on total damages, TD. Assuming that D is an additively separable, linear damage
maintain the assumption of linearity in T throughout the paper.
9
function of emissions, D (e) = ∑i di ei , Farrow et al. solve:
min
∑i=1 Ci (ai )
N
s.t. D (ē
a)
TD.
(3)
Muller and Mendelsohn (2009) observe that in order for the solution to (3) to coincide
with the solution to (1), the planner must set the constraint on total damages, TD, at the
efficient level. Let a FSCH denote the solution to (3).
Let us now establish the first result, which links Hung and Shaw’s TRS and Farrow et
al.’s DTRS.
Proposition 1: Provided that Ci ’s and D are continuous, the following are true:
(i) Given the efficient solution ae f f , there exists a constraint vector X̄e f f in terms
of pollution concentrations such that the solution a HS to the auxiliary program (2)
subject to X̄e f f is the optimal solution ae f f ;
(ii) Given the efficient solution ae f f , there exists a constraint value TD
ef f
in terms
of total damages such that the solution a FSCH to the auxiliary program (3) subject to
TD
ef f
is the optimal solution ae f f ;
(iii) The social planner requires no more information to implement program (2) than
to implement program (3) in achieving the efficient solution ae f f .
Proof : To see (i), given the efficient solution ae f f , let the constraint vector X̄e f f be defined
as
X̄e f f = T (ē
ae f f )0 .
Suppose by way of contradiction that a HS solves (2) subject to X̄ = X̄e f f , but a HS 6= ae f f .
Because D is originally an increasing function of pollution concentrations x, and because
xe f f = T (ē
ae f f )0 = X̄e f f , we have:
∑i Ci (ai
ef f
) + D (xe f f ) < ∑i Ci ( aiHS ) + D (x HS )
10
∑i Ci (aiHS ) + D(xe f f ),
where the first inequality follows from a HS 6= ae f f and the second inequality follows because x HS
xe f f implies D (x HS )
D (xe f f ). But this inequality implies that there exists
ef f
ae f f 6= a HS such that ∑i Ci ( ai ) < ∑i Ci ( aiHS ) with xe f f = X̄e f f , a contradiction to the
assumption that ae f f minimizes the sum of abatement costs subject to the environmental
constraint X̄e f f .
The proof of (ii) is analogous, with the constraint value TD
D (ē
ef f
defined as TD
ef f
=
a e f f ).
To establish (iii), note that in order for the solution to (3) to achieve ae f f , the planner
must set TD at the efficient level, which requires that the planner knows the two mappings T : R N ! R M and D : R M ! R. But this information is all that is required for the
regulator to find the optimal constraint vector X̄e f f . This completes the proof.
Proposition 1 establishes the practical equivalence of (2) and (3), the two alternative
cost-effectiveness programs. In practice, so long as the social planner has perfect knowledge of T and D, it does not matter whether environmental policy is set based upon X̄ or
upon TD. This does not mean, however, that the two trading mechanisms are equivalent.
As we shall see, the TRS and the DTRS present different informational requirements: the
TRS requires that transfer coefficients be estimated while the DTRS requires that ratios of
marginal damages be estimated. More importantly, we will describe a set of conditions
under which the TRS equilibrium may not achieve the solution to (2). We will also describe a (different) set of conditions under which the DTRS equilibrium may not achieve
the solution to (3). We show, therefore, that one cannot guarantee that the equilibrium
under the TRS is equivalent to that under the DTRS. In the following sections, we investigate these questions by incorporating (i) branching of a river in the mapping T and (ii)
nonlinear damages in the mapping S.
A word of caution is in order when interpreting our Proposition 1. The equivalence between the two cost-effectiveness programs assumes that the planner knows ae f f . This in
turn requires that she has complete information regarding the abatement cost functions.
11
A primary appeal of permit trading is that in many cases the policy can be put in place,
and an optimal outcome thence achieved, by a planner who has no information regarding
individual abatement cost functions. That may not be true here. The point is important
because the apparent advantage of DTRS over TRS is that DTRS can achieve the efficient
optimum provided that TD is set optimally. This advantage of DTRS disappears, though,
in view of our proposition, if the TRS equilibrium can itself achieve the optimum of the
alternative program (2). For then, the same optimum can be achieved either by TRS or
by DTRS. In determining under which conditions one is to be preferred over the other, it
is important for us to re-evaluate the equivalence between the TRS equilibrium and the
optimum of (2) under general conditions.
3
The Trading-Ratio System (TRS)
The Hung-Shaw TRS allocates tradable discharge permits beginning at the zone (and thus
the source) that is furthest upstream. Allocation proceeds from there on down the stream,
ensuring along the way that the concentration standard is met at each zone. This means
that for some sources low on the river, few permits, or even none, will be received in the
initial allocation.6 The Hung-Shaw system indexes zones so that m = 1 indicates the most
upstream source and M the most downstream source.7 For simplicity, Hung and Shaw
assume that there is one discharger in each zone. This means that, in our notation, the
set of zones fmg coincides with the set of polluting sources fi g. As they observe, this
does not jeopardize the generality of their results. Thus, in this section we shall use i to
denote both sources and zones (or receptors). Given the unidirectional flow of a river,
the transfer matrix T has a special characteristic: for any m and n with m > n, τ mn = 0,
6 The TRS allocation scheme, by design, privileges upstream sources over downstream sources. This
might create a certain amount of political resistence in practice, but it makes good economic sense. An
efficient outcome should “fill the river” with pollution up to the standard at each receptor. Failing to do
this will lead to higher aggregate abatement costs.
7 Indexes along two branches above their confluence, though important for bookkeeping purposes, have
no ordinal relationship to each other.
12
where τ mn is the element of T that measures the water-quality impact of of pollution from
zone m upon concentration at zone n. As do Hung and Shaw, we assume that each source
influences its own zone in a unitary fashion: τ ii = 1 for all i.
Given the ambient zonal pollution standards X̄ from program (2), the TRS regulator
uses the transfer coefficients in T to allocate zonal tradable discharge permits (TDPs) Z̄
so that the standards are met if no trade occurs. Starting from the most upstream zone,
define Z̄1 = X̄1 and, for j > 1, define Z̄ j = X̄ j
given j, we might find that τ ( j
1) j X̄ j 1
j 1
∑i=1 τ i j Z̄i . It is possible that, for a
> X̄ j . That is, the level of pollutant arriving from
upstream when the standard is exactly met there exceeds zone j’s standard even when
no pollution is emitted in zone j. In this case, zone j is called a critical zone. The HungShaw allocation scheme sets Z̄ j = 0 and, in turn, reduces the allocation of permits to the
upstream zone (or, possibly more than one upstream zone) to the point at which zone j
is no longer critical: Z̄ j
1
= ( X̄ j =τ ( j
1) j )
j 2
∑k=1 τ k j
1 Z̄k .
(See Hung and Shaw, p. 88,
whose slight subscript typo has been corrected here.)
The TRS allocation scheme ensures that the water-quality impacts of all upstream
zonal standards on a given zone are accounted for via the upstream transfer coefficients.
Note that in using the TRS procedure, the regulator takes as given the set of zones fi g,
the zonal environmental standards X̄, and the transfer coefficients T. Each discharger is
then allowed to trade freely in a watershed-wide permit market according to the transfer
coefficients T, so long as its emissions do not exceed the permits it holds.
Formally, each source i solves:
min
rki ,rsi ,rs j
Ci ( ai )
s.t. Z̄i
pi rsi + ∑ j p j r ji
(ēi
rki )
ai = rki + rsi
rsi =
n
13
i 1
(4b)
(4c)
∑ j =i + 1 r i j
rki , rsi , rs j
∑ j=1 τ ji r ji
(4a)
0,
(4d)
(4e)
where pi and p j are the market prices of permits from sources i and j, r ji is the amount of
pollution control purchased from source j to offset pollution at source i, rki is the amount
of pollution control from source i that is kept by source i to meet the zonal standard
Z̄i , and rsi is the amount of pollution control sold by source i. As Hung and Shaw observe, the TRS possesses two advantages over other trading schemes. The first is that
each discharger must participate in only a single watershed-wide permit market, so that
transaction costs are low. The second is that the regulator allocates initial zonal discharge
permits Z̄ in such a way that the ambient environmental constraints X̄ are satisfied exactly
at the initial allocation.
One can rewrite constraint (4b) to obtain Hung and Shaw’s trading constraint (their
equation 5):
ei
Z̄i + ∑ j=1 τ ji r ji
i 1
∑ j =i + 1 r i j ,
n
(5)
where ri j is the net amount of zonal discharge permits sold by source i to source j. This
constraint means that any discharger can buy permits only from upstream zones and sell
permits only to downstream zones. Because sources can trade permits at exchange rates τ,
in any TRS equilibrium (including the boundary case), for any j > i, the spatially explicit
prices of permits must satisfy
τ i j p j = pi .
(6)
The economic implications of this equality are substantial. Even if a high-cost source
is located upstream of, or on a different branch from, a low-cost source, this constraint
strictly prohibits any cost-minimizing trade between them: τ i j = 0 for i > j. This might
seem justifiable at first on the grounds that water flows downstream, so that any downstream pollution reduction or a reduction on a different branch has no effect on the concentration at the upstream location. However, the marginal damages of pollution from
the high-cost source can be larger than those of the low-cost source when damages are
nonlinear. In this case, increased abatement by the low-cost source in exchange for de-
14
creased abatement by the high-cost source might be Pareto improving. Because it prohibits the cross-branch or upstream sales required to achieve this improvement, the TRS
can fail to achieve the least-cost outcome. We shall return to this point in Section 5 when
presenting the results of our numerical work.
According to Proposition 1, the solution to program (2) also solves program (1), regardless of branching or nonlinear damages. The question is whether Hung and Shaw’s
TRS equilibrium is guaranteed to achieve the solution to program (2). Our next result,
Proposition 2, shows that the answer is no.8 There are situtions, not unusual in actual
practice, in which the outcome of the TRS is either indeterminate (the permit-allocation
scheme breaks down) or inefficient (it fails to solve program (2)).
We first turn our attention to an important property of the transfer coefficients. This
property is satisfied in Hung and Shaw’s numerical example, but is not otherwise noted
in their paper. The coefficients must be associative. Intuitively, this means that the amelioration or degradation of a unit of pollutant between zone i and zone i + 1 is the same
whether that unit was emitted at zone i or arrived there from upstream. Formally, associativity is defined as follows.
Definition: Given a matrix T = fτ i j g of transfer coefficients, say that T is associative if for
all i, k, and m, τ ki τ im = τ km . Say that T is non-associative if there exist i, k, and m for which
τ ki τ im 6= τ km .
Proposition 2: The equilibrium under the Hung-Shaw TRS does not achieve the cost-effective
solution to program (2) if (i) transfer coefficients are non-associative with τ ki τ im > τ km for some
i, k, and m or (ii) there exists a critical zone at the confluence of upstream branches.
Proof : To prove (i), suppose that T is non-associative and let i, m, and k be such that
8 We
have also shown that when there are multiple adjacent critical zones, the original TRS allocation
scheme breaks down. For this situation we have derived a modified version of the TRS in which permits
are allocated starting at the downstream-most source and proceeding upstream. Our modified version
achieves the optimal outcome in the face of adjacent critical zones. The proof of this claim is available upon
request.
15
τ ki τ im > τ km . Using transfer coefficients and the definition of xi , the constraint in program
(2) can be rewritten as
∑i τ im ei
Let
TRS
ef f
X̄m for all m.
be the set of emissions vectors that satisfy this constraint. On the other hand, let
be the set of emissions vectors that satisfy the trading constraint (5). Because each
polluter must obey this constraint, the TRS equilibrium solves program (2) only if the
ef f
constraint sets
vector e 2
Z̄m
TRS
and
TRS
are equivalent. We shall show that there exists an emissions
ef f .
that is not in
em + ∑im=11 τ im rim
TRS ,
For any e 2
∑in=m+1 rmi
(5) is satisfied. Thus, define Am =
0. Using the definition Z̄m = X̄m
∑im=11 τ im Z̄i ,
we have
∑i =1
m 1
em
τ im rim + ∑i=m+1 rmi + Am + ∑i=1 τ im Z̄i = X̄m .
n
m 1
Using the trading constraint (5) for Z̄i and rearranging terms, we obtain
∑i=1 τ im em + ∑i=m+1 rmi + Am ∑i=1
m
n
m 1
+ ∑i=1 τ im
τ im rim
∑k=1 τ ki rki + ∑k=i+1 rik
m 1
i 1
n
X̄m .
The last two terms of the left hand side can be further rearranged to yield
∑i =1
m 1
τ im rim + ∑i=1 τ im
∑k=1 τ ki rki + ∑k=i+1 rik + rim + ∑k=m+1 rik
m 1
= ∑i =1
m 1
i 1
∑k =1 (
i 1
m 1
n
τ ki τ im + τ km )rki + ∑i=1 τ im ∑k=m+1 rik ,
m 1
n
where we used the fact that indexes i and k are anonymous and thus are interchangeable.
Thus we obtain
∑i=1 τ im em + ∑i=m+1 rmi + Am + ∑i=1
m
n
m 1
+ ∑i =1
m 1
∑k =1 (
i 1
τ im ∑k=m+1 rik
n
τ im τ ki + τ km )rki
X̄m .
(7)
Because transfer coefficients are non-associative with τ ki τ im > τ km , the last term on the
16
left side of (7) is negative. If this term is sufficiently large in absolute value, the sum of
the last four terms on the left side of (7) can be negative. In this case, we can rewrite (7) as
∑i=1 τ im em
m
X̄m + Mm ,
where Mm > 0. This means that there exists e 2
(i.e. e 2
=
TRS
that does not satisfy ∑i τ im ei
X̄m
e f f ).
To see (ii), suppose that e HS is the solution vector for program (2). By assumption, we
must have a critical zone at the confluence receptor m: ∑m
fm
1i
τ (m
1i )m X̄m 1i
> X̄m where
1i gi is the collection of indices immediately upstream of zone m, along two or more
branches. We know that at the optimum, given our assumption that Ci0 > 0,
X̄m =
∑m
1i
τ (m
HS
1i ) m e m 1i .
Suppose, without loss of generality, that there are only two zones upstream of the critical
confluence, say, m
1i = a, b. By assumption,
X̄m = τ am e aHS + τ bm ebHS ,
On the other hand, without knowing individual sources’ cost functions, information on
e HS is not available to the TRS regulator. Thus she must, without knowledge of e HS ,
allocate zonal discharge load standards Z̄’s such that Z̄m = 0 and
X̄m = τ am Z̄ a + τ bm Z̄b .
Because there is only one constraint equation for two zonal standards, the allocation is
indeterminate. It is trivial to see that if the TRS allocates Z̄’s in such a way that, for
example, e aHS > Z̄ a and Z̄b = ( X̄m
τ am Z̄ a )=τ bm > ebHS , then the trading equilibrium
17
can never achieve e HS . Similar arguments apply when there are more than two upstream
zones. This completes the proof.
Proposition 2 implies that the TRS cannot always be relied upon to deliver the efficient
outcome even if the ambient environmental constraints X̄ are set optimally. One might ask
whether the conditions are likely to be met in practice. Non-associativity is unlikely to be
a serious concern. In many cases, perhaps most cases, a linear T is a good approximation
and so associativity is guaranteed. We return to this point in Section 5.9
We believe that the second condition, in which a critical zone lies at a confluence
of branches, is not at all unusual. In a branching river, confluence zone m is critical if
∑m
1i
τ (m
1i )m X̄m 1i
> X̄m , where fm
1i gi is the collection of indices directly upstream
of zone m, along all contributing branches. Economic activity and population both tend
to concentrate around the confluence of rivers. The water quality there is often important
for both aquatic species and people living nearby. Thus a zone of confluence might be
more likely than others to be critical.
Moreover, the TRS mechanism also has a practical disadvantage. Consider a branchless river system. Here the TRS equilibrium achieves the efficient outcome, if it achieves
it at all, with no trade. To see this, note that as in the proof of Proposition 1, the efficient
environmental constraints are found by setting X̄e f f = T (ē
ae f f )0 . Then as Hung and
Shaw show, in a branchless watershed the constraint set arising from Z̄ is equivalent to
that arising from X̄e f f and the TRS equilibrium achieves the cost-effective outcome. But
9 Associativity is violated in the following hydrological model of pollutant flow in a groundwater aquifer.
Todd and Mays (2005) model the concentration of a pollutant at distance δ and time t from a point source
as:
X (d) = X0 τ (δ ),
and
τ (δ ) =
1
2
1
erf
d
p
vt
2 tD
+ exp
dv
D
1
erf
d + vt
p
2 tD
,
where erf( ) is the Gauss error function, D the dispersion coefficient, v the average linear velocity, and X0
the pollution concentration at the point source. The transfer coefficients τ (δ ) derived from this model are
not associative. See also Sado et al. (2010), who apply the TRS to a set of point sources on the Passaic River
in New Jersey. Their transfer coefficients do not quite satisfy the associativity property.
18
because the cost-effective outcome must coincide with the efficient outcome, which also
coincides with the initial allocation, and because it is assumed that there is only one discharger in each zone, this implies that discharging pollution so as to satisfy Z̄ exactly,
without engaging in any trade, is also cost-minimizing. Put another way, the regulator
cannot implement the efficient optimum in a decentralized manner. This claim, whose
proof we omit, is stated in the following result.
Proposition 3: Suppose that in program (2), zonal environmental constraints X̄ are set at the
efficient levels and that there is only one discharger in each zone. Then if the TRS trading achieves
the cost-effective optimum of program (2), it is achieved with no trade.
4
The Damage-denominated Trading-Ratio System (DTRS)
The DTRS of Farrow et al. is similar to the TRS of Hung and Shaw in that both are innovative schemes for controlling water quality through trade in permits to emit pollution
satisfying a set of trading ratios. There the similarity ends.
The fundamental regulatory constraint in the DTRS is a single limit on aggregate monetary damages, here denoted TD, rather than a set of physical environmental standards.
The trading ratios are themselves based upon marginal damages, rather than upon physical transfer coefficients. Each source i’s marginal damage di is calculated by integrating
its contribution to monetary damages over that source’s “zone of influence.” Having calculated marginal damages for each source, the regulator distributes permits L̄i (in terms
of emissions at the point of discharge) in such a way that aggregate damages meet the
overall monetary constraint at the initial allocation: ∑i di L̄i = TD. Trade is allowed between any two sources, but only according to the ratio of their marginal damages. The
aggregate limit on damages will be satisfied in the face of any permissible trade at these
ratios.
19
Given the vector d of marginal damages and a vector e of emissions, Farrow et al. (and
also Muller and Mendelsohn 2009) assume that aggregate damages are linear: D (e) =
∑in=1 di ei . It is this quantity that must not exceedTD. The assumed linearity of the damage
function means that the di ’s do not depend upon emissions from other sources.
Each source i solves the following cost-minimization program:
min
rki ,rsi ,rs j
pi rsi + ∑ j p j rs j
Ci ( ai )
s.t. (ēi
dj
∑ j di r s j
rki )
ai = rki + rsi
rki , rsi , rs j
(8a)
L̄i
(8b)
(8c)
0,
(8d)
where pi and p j are the market prices for a permit from source i and j, rs j is the amount of
pollution control purchased from source j to offset pollution at source i, rki is the amount
of pollution control from source i that is kept by source i to meet the emissions standard
L̄i , and rsi is the amount of pollution control sold by source i.
Note that substituting ei = ēi
ai and rki = ai
rsi into (8b), one obtains an analogue
of (5), the Hung-Shaw trading constraint:
ei
L̄i + ∑ j
dj
r
di s j
rsi .
(9)
This constraint means that each polluting source can trade with any source, according
to the marginal damage ratios, so long as the level of its discharge does not exceed the
sum of the original discharge limits L̄i and the net purchase of damage-denominated
permits ∑ j (d j =di )rs j
rsi . Because sources can trade permits at the exchange rates d j =di ,
the spatially explicit prices of permits in the equilibrium (including the boundary case)
satisfy the analogue of (6):
dj
p = pi .
di j
20
(10)
Note that unlike in the TRS, one can be sure that di 6= 0 in practice for all i: a source for
which this is not true would not be part of the trading system. Therefore, each source
can trade with any other source, including those located upstream or downstream or on
different branches of the river.
Farrow et al. (2005) derive the first-order necessary (and sufficient) conditions for each
source’s optimization problem, from which the following interior equilibrium condition
is derived:
Ci0 ( ai )
d
p
= i = i.
0
C j (a j )
dj
pj
(11)
This condition is, however, an incomplete characterization of a market equilibrium. First,
there will be n
1 equations for (n
2) + n2 unknowns f ai , rki , rs j gi, j . Second, as Farrow
et al. observe, (11) holds only when source i is a net buyer and source j is a net seller (and
vice versa). Proposition 4 confirms that equation (10) must hold for any equilibrium,
interior or boundary. It also provides the complete set of conditions that must be satisfied
at an interior equilibrium.
Proposition 4: Suppose that marginal abatement cost Ci0 ( ai ) = MCi ( ai ) is strictly increasing for every source i. Then for given baseline emissions fēi g, initial permits f L̄i g, and trading ratios fdi g, an interior market equilibrium of Farrow et al.’s DTRS mechanism is a vector
f pi , ai , rki , rsi , rs j gi, j that solves the following system of equations:
ai = Ri ( pi ) ,
∑ di [ēi
(12a)
Ri ( pi )
L̄i ] = 0,
pj
pi
=
for all i, j,
di
dj
where Ri ( pi ) is an abatement decision function given the price pi . The vector frki , rsi , ∑ j
21
(12b)
(12c)
dj
di rs j gi, j
is uniquely determined by the following response functions:
8
>
< L̄i if ēi
rki =
>
: 0 if ēi
8
>
< L̄i
rsi =
>
: 0
∑j
Ri ( pi )
Ri ( pi )
i
L̄i
(12e)
Ri ( pi ) > L̄i
if ēi
if ēi
Ri ( pi )
(12d)
Ri ( pi ) > L̄i
ēi + Ri ( pi ) if ēi
8
>
< 0
dj
r =
di s j >
: ē
L̄i
L̄i if ēi
Ri ( pi )
L̄i
(12f)
Ri ( pi ) > L̄i
Proof : First, we show a stronger version of condition (10): under the DTRS system, prices
must satisfy (12c) in any equilibrium regardless of whether each source is a net buyer or
a net seller. To see this, note that if (12c) does not hold, say, if
pj
pi
> ,
di
dj
then unlimited arbitrage profits are available to any source k 6= i, j who buys permits
from source j and sells them to i. Market demand for permits from j is infinite while the
number of permits available from j is finite at L̄ j . Note that whether they can actually buy
and sell permits or not is irrelevant: they simply demand permits taking prices as given.
Thus equilibrium prices must adjust in such a way that (12c) is satisfied. Therefore, each
source i faces the same effective price in all zonal markets j, pi = p j (di =d j ). It is irrelevant,
then, from which sources it buys permits or to which sources it sells. It follows that source
i abates such that MCi ( ai ) = pi . Because MCi ( ai ) is strictly increasing, the interior optimal
abatement ai = MC
1(p
i)
is unique. Thus pi = p j ddi for all j. Now, let us construct an
j
excess demand function. Given an arbitrary price pi of permits, source i would choose
abatement Ri such that MCi = pi . Thus, Ri is a well-defined function. The excess demand
for permits from source i is zi ( pi ; ēi , L̄i ) = ēi
22
Ri ( pi )
L̄i . If zi > 0, then i must buy
permits. If zi < 0, it sells its excess permits. All permits sold to and purchased from i
must be exchanged at the ratio di =d j with permits from any source j. This means that the
common units of exchange are di zi . Thus, the market clears in equilibrium if equations
(12a)-(12f) hold. For a given vector fēi , L̄i , di gi and n sources, this gives us n equations
for n unknown prices f pi gi . Thus the equilibrium is exactly identified. The proof of the
expressions for frki , rsi , ∑ j
dj
di rs j gi, j
is obvious and thus omitted. This completes the proof.
This characterization of market equilibrium turns out to be useful for the simulations
in Section 5. There may be non-trivial boundary equilibria in which ai = 0 or ai = ēi .
These boundary cases can be dealt with by defining Ri ( pi ) = 0 if MCi ( ai ) > pi for all
ai 2 [0, ēi ] and Ri ( pi ) = ēi if MCi ( ai ) < pi for all ai 2 [0, ēi ]. The rest of the equilibrium
conditions are intact. We now ask whether the equilibrium of the DTRS achieves the
cost-effective solution of program (3).
Our next result is analogous to Proposition 2. According to Proposition 1, the solution
to program (3) also solves program (1), regardless of branching or nonlinear damages.
The question is whether the DTRS equilibrium is guaranteed to achieve the solution to
program (3). Proposition 5 shows that the answer is no. If the aggregate damage function
is nonlinear in concentrations, then the DTRS equilibrium does not minimize costs.
Proposition 5: Suppose that aggregate environmental damages are a nonlinear function of pollution concentration, such that at the efficient solution ee f f we have
D (ee f f ) 6=
∑i
∂D (ee f f ) e f f
ei .
∂ei
Then the DTRS equilibrium does not achieve the cost-effective solution of program (3).
Proof : We offer a proof for the case of an interior optimum of (3). Note that at the interior optimum, the emission vector ee f f must satisfy the necessary (but not sufficient)
23
condition:
ef f
MCi ( ai )
∂D (ee f f )=∂ei
=
for all i, j,
ef f
∂D (ee f f )=∂e j
MC j ( a j )
ef f
where ei
= ēi
ef f
ai . On the other hand, according to Proposition 4, at an interior
equilibrium, we have
ef f
MCi ( ai )
di
for all i, j.
=
ef f
dj
MC j ( a j )
Thus, in order for the trading equilibrium to achieve the cost-effective solution, the regef f
ulator must evaluate the exchange rates (the d’s) at the optimum: di
= ∂D (ee f f )=∂ei .
Under the DTRS system, the regulator allocates L̄’s in such a way that:
∑i di
ef f
L̄i = TD = D (ee f f ).
(13)
We now ask if there exists some initial allocation L̄, satisfying (13), such that the resulting equilibrium would achieve the cost-effective solution. We claim that such an allocation does not exist. Suppose, by contradiction, there exists such an allocation L̄ and that
the resulting trading equilibrium is also efficient: e DTRS = ee f f . Because the equilibrium
must satisfy the market-clearing condition (12b), we have
∑i di
ef f
∑i di
e f f DTRS
ei
.
L̄i =
(14)
However, because the aggregate damage function is nonlinear, we have
∑i di
ef f ef f
ei
6= D (ee f f ).
(15)
Combining (13), (14), and (15), we see that
∑i di
ef f
L̄i =
∑i di
e f f DTRS
ei
= D ( e e f f ) 6 = ∑i di ei .
24
ef f ef f
which contradicts that e DTRS = ee f f . This completes the proof.
Because Farrow et al.’s original system assumes damages are a linear function of pollution levels, the result that the DTRS breaks down under the assumption of nonlinear
damages should not come as a surprise. However, an important point is that with nonlinear damages the DTRS fails to achieve the efficient (and the cost-effectiveness) solution
even if the regulator evaluates the exchange rates di ’s at the efficient allocation. Whether
this is significant in practice is an empirical matter.
As is evident from the proof, the DTRS breaks down because the initial allocation of
permits L̄ follows Farrow et al.’s original allocation rule (13). A natural question arises:
what would happen if one were to use a different allocation rule? For example, the regulator could allocate permits so that S( L̄1 , . . . , L̄n ) = TD. Here one encounters an insuperable difficulty: there is no allocation rule the regulator could rely upon in this case. Indeed,
the problem is similar to that of the TRS. To see this, suppose that the regulator agreed
upon the desired level of aggregate damage TD. Because the damage function is nonlinear, there will inevitably exist many vectors L̄ such that S( L̄1 , . . . , L̄n ) = TD. The regulator’s problem is indeterminate. (Recall that D is the composition function D (e) = S( Te).)
In the following section, we investigate how the TRS and DTRS perform, relative to
the efficient solution as well as to each other, if the initial allocation of permits follows
such an arbitrary rule.
5
A numerical model
The results of the previous two sections imply that in numerous watersheds of practical
interest, water quality trading based on either TRS or DTRS may not achieve the efficient
outcomes. The numerical exercise reported here is designed to give an indication of the
welfare losses associated with the shortcomings of the two systems, relative to each other
25
and also relative to the social optimum. We do this by constructing a numerical model.
Though the model is fairly sophisticated in its constituent parts, incorporating the relevant scientific aspects of a realistic watershed, in order to illuminate our key question we
have kept it relatively small. The specific parameter values are hypothetical but realistic.
Consider a second-best scenario in which the regulator has imperfect information regarding polluters’ abatement costs, but has perfect information regarding environmental
damages. Together, these conditions mean that the fundamental constraints X̄ (for the
TRS) or TD (for the DTRS) cannot be set at the efficient levels. There can be infinitely
many ways to allocate initial permits under such a scenario, all meeting the constraint
in the absence of trade. We assume that the total number of permits in the initial allocation is equal to the socially optimal level. This assumption enables us to disentangle
the sources of inefficiency, as it implies that TRS and DTRS could potentially achieve the
social optimum. Put another way, if either the TRS or the DTRS fails to achieve the social optimum, it is not because the supply of permits is incorrect relative to the optimum.
Indeed, the theoretical premise of ideal permit trading is that, in the absence of market
failures or other distortions, the trading outcome does not depend on the initial distribution of permits. Here we have no market imperfections, but the premise is violated
anyway.
5.1. The Water Quality Model
The Environmental Protection Agency has developed a theoretically acceptable and
practical method of modeling water quality impacts, the NWPCAM. Farrow et al. (2005)
consider a water-quality impact model based on the hydrologic assumptions used in NWPCAM. We follow the same strategy.
Water pollutants, such as phosphorus, nitrogen, and heat, decay as water carrying
them flows downstream through a river system. Let xmi be source i’s contribution to the
pollution concentration at location m downstream. Then xmi is a function of the emissions
26
ei at source i, stream flow Q, and an exponential decay term:
xmi =
8
>
< 0
>
:
ei
Q
if m is not downstream of i
exp
k̂δ mi
,
(16)
if m is downstream of i
where k̂ is the decay parameter and δ mi is a distance in river miles between source i and
location m.10 The pollution concentration at location m is the sum of contributions from
all upstream sources: xm = ∑i xmi . This hydrological model incorporates both the unidirectional flow and the natural decay as important characteristics of a river system. Furthermore, the model can also incorporate branching in the river.11
It is clear that the impact of changes in concentration at location i on concentration at
any downstream location j is linear:
de f
τij =
dx j
= exp( k̂δ i j ),
dxi
(17)
where the τ i j are the transfer coefficients. Note that these coefficients satisfy the associative property.12
10 In
Farrow et al., the function takes the form
ei
exp
Q
kθ t
20
δ mi
U
,
where k is the nominal decay rate, θ is a coefficient reflecting sensitivity of k to the mean temperature t,
and U is the stream velocity. Because units do not have any significant meaning in this simulation, we
have simplified this relationship by replacing kθ (t 20) =U with k̂, which is assumed to be constant and is
evaluated at average hydrological parameters.
11 To see this, suppose two sources, i and j, are located along two different branches upstream of a confluence. The impacts of emissions from i and j on pollution concentrations at location k below the confluence
are expressed as xki and xk j .
12 Using (17), we can write:
τ i j τ jk = exp
k̂δ i j exp
k̂δ jk ,
= exp
k̂[δ i j + δ jk ] ,
= exp
k̂δ ik = τ i j ,
where the last equality follows because δ ik = δ i j + δ jk by assumption.
27
Given the hydrological relationship between emissions and ambient pollution concentrations, an important question remains as to how we should model the relationship
between pollution concentrations and (monetary) damages. In this regard, Farrow et al.
assumed that marginal damages D are constant at each location m: ∂D =∂xm = WTP
Hm
where WTP is the constant per-capita marginal damages from changes in water quality
and Hm is the population size at location m. According to this specification, damages from
each source’s emissions are given by Di (ei ) = di ei , where di is constant and independent
of emissions levels ei :
M
di =
∑
WTP
Hm
τ mi
m=1
1
.
Q
To justify the constant marginal willingness to pay (WTP), Farrow et al. argue that water
quality is inversely related to pollution concentrations (that is, ∂Wm =∂xm =
1) and that
“the household marginal willingness to pay for a small improvement in water quality,
WTP, is constant . . . over the range of water quality conditions considered in this study.”
(2005, p.197).
In the non-market valuation literature, however, it is often found that individuals obtain utility from recreational and aesthetic values of water quality rather than directly
from pollution concentration levels. That is, individuals care if the river water is swimmable, drinkable, and fishable, and if the river water provides habitat for important
aquatic species. The quality of water and aquatic habitat at any location in the river is
typically a nonlinear function of concentrations of these pollutants. As noted in Section
1, the science of aquatic systems, including that for fish and algae, indicates that the underlying biology can be highly nonlinear as a function of water quality. In such cases,
economic damages may be expected to be nonlinear too.
Willingness to pay may not be linear either. In many watersheds in the U.S., water
quality is already impaired, so that a further increase in pollution concentrations might
cause serious water-quality degradation. Even if WTP for a small change in quality is
constant over a range of concentration levels, it might be much bigger for the same small
28
change above a particular concentration threshold. Indeed, our communications with water practitioners reveal that their primary concern is closely related to this nonlinear biological response around hotspots. We stress here that this concern is one of the important
political factors that has plagued water-quality trading in many watersheds. Whether the
practice of replacing the true nonlinear relationship with a linear approximation is critical
is an empirical question.
To model the nonlinear biological responses that allow for the type of threshold effects
highlighted above, we consider a logistic damage response to pollution concentrations at
each location m:
Sm ( xm ) =
b
1 + exp( a( xm
c))
for all m = 1, . . . , M,
(18)
where a is a damage-sensitivity parameter, b a scale parameter, and c a concentration
threshold. The total economic damages are then given by D (e) = ∑m Sm ( xm ). Logistic
models are commonly used in biology and ecology for modelling the response of species’
mortality or population size to pollution. Though in biology, the parameters a and c
often depend on a variety of environmental factors, for simplicity of analysis we treat
them as constants that do not vary by time or location. The parameter b is a scaling
parameter that transforms biological damages into monetary economic damages, which
we also assume are constant. With these assumptions, marginal aggregate damages with
respect to emissions from any source i depend not only upon emissions from that source
but also upon emissions from other sources, both downstream and upstream of i (that is,
∂D =∂ei = ∑m (∂Sm =∂xm ) (∂xm =∂ei )). When there is a branch in a river system, marginal
damages also depend on the emissions from sources located along the other branches.
The parameter values of the model can vary widely by pollutant and watershed. Because our goal is to obtain generic efficiency properties of the two trading systems, we
decided to choose representative parameter values for the water-quality model (16) and
29
then choose a set of parameters a, b, and c that generate interior optima for at least two out
of three sources given the assumed cost parameters (see below). The value of k̂ = 0.005
is chosen based on three parameters: the mean of the decay rates for seven representative
water pollutants (U.S. EPA, 2002), the average water temperature of 20 Celsius degrees,
and the stream velocity of 1.5 miles per hour. The scale parameter b and the threshold
parameter c are important in generating interior solutions. We thus started with arbitrary
values a = 5 and c = 5 and then searched for the associated value of b. The parameters
used for the simulation are summarized in Table 1.
[Table 1. Water Quality Model Parameters]
5.2. Simulation Scenarios
We assume that the river has a main stem M and a single branch B. The river has a
maximum length of 200 river miles along the main stem and 150 river miles along the
branch: m M 2 [0, 200] and m B 2 [0, 150]. The confluence occurs at m M = 100 (or equivalently, m B = 50). There are three polluting sources. Source 1 is located in the most
upstream point of the main stem (m M = 0), source 2 at the confluence (m M = 100 or
m B = 50), and source 3 at the most upstream point of the branch (see Figure 1). The firms
(or sources) have quadratic abatement cost functions of the form Ci ( ai ) = (1=α i )
ai2 ,
with ai 2 [0, ēi ] and α i > 0.
[Figure 1. Hypothetical River]
For a given initial allocation of permits, we first compute the social optimum, and then
ef f
allocate permits in an equal amount to each polluting source: L̄1 = L̄2 = L̄3 = ∑ ei =3.
Under the TRS, this means that zonal environmental standards, the X̄’s, are allocated so
that X̄1 = L̄1 , X̄3 = L̄3 , X̄2 = L̄2 + τ 12 X̄1 + τ 32 X̄3 . Once again, we do not assume that
30
ef f
the regulator knows ei
ef f
or ∑ ei . We chose this allocation rule because we are interested
in the relative performance of the two trading systems under conditions that can be compared to the social optimum. Under the TRS, the correct transfer coefficients are known to
the regulator and are announced to the polluters. Under the DTRS, the regulator does not
know the social optimum, and so she evaluates the di ’s at the initial allocation.13 In each
of these setups, we simulate the trading outcomes for two sets of cost parameters: (A)
α 1 = 7.5, α 2 = 15, α 3 = 7.5 and (B) α 1 = 15.0, α 2 = 7.5, α 3 = 15.0. For each case, we also
compute the social economic costs at the initial allocation of permits, as the no-trading
baseline.
5.3. Simulation Results
Case A: α 1 = 7.5, α 2 = 15, α 3 = 7.5
In this case, a low-cost firm is located downstream of two high-cost firms. At the socially optimal outcome, source 2 abates all of its emissions while source 1 emits more
than source 3 despite the fact that they have the same marginal costs and the same baseline emissions (Table 2). This occurs because marginal damages at the optimum increase
more in source 3’s emissions than in source 1’s emissions. The TRS mechanism, as noted
above, precludes upstream firms from buying permits from downstream firms. It also
precludes firms located on different branches above a common confluence from engaging in any trades. Because the potential seller (the low-cost firm) is located downstream,
no trade can occur between the downstream firm and the upstream firms. Moreover, at a
social optimum efficient trades should occur between the two upstream firms. Yet again
no trade between them is allowed under the TRS. As a result, firms incur higher abatement costs under the TRS than at the optimal outcome. On the other hand, the DTRS
does allow trades among any of the three sources. Because of this, the DTRS performs
substantially better than the TRS (and therefore, the no-trading baseline).
13 We
also experimented with various choices of di ’s. The results were not sensitive to these values.
31
[Table 2. Simulation Results]
The problem with the DTRS, however, is that in this case the damage-denominated
trading coefficients turn out to be poor approximations to the true marginal damages at
the optimum. This point is demonstrated in Figure 2. The figure plots marginal damages
as a function of each source’s emissions, holding other sources’ emissions at the optimum.
Figure 2 also shows each source’s marginal cost and trading coefficient. The social optimum occurs where marginal damages from each source are equated with its marginal
cost and the overall constraint is satisfied. Interestingly, the equilibrium does not occur
where each source’s marginal cost equals its trading coefficient di . This is because each
source makes its abatement/trading decision so that its marginal cost equals the spatially
explicit price it faces, pi = (d j =di ) p j . Thus, when the di ’s are poor approximations to
actual marginal damages, the trading outcome under DTRS does not equate marginal
damages with marginal costs.
Another problem with the DTRS is that, in equilibrium, the sum of emissions can
ef f
exceed the sum of initial emissions permits: ∑ eiDTRS > ∑ ei . This occurs because
neither the individual polluters nor the equilibrium market-clearing condition are constrained by ∑ eiDTRS
ef f
∑ ei . As explained in the proof of Proposition 4, the common
unit of exchange is di ei instead of ei itself, and therefore, the market clears instead with
∑ di eiDTRS
ef f
∑ di L̄i where the initial distribution of permits is given by ∑ L̄i = ∑ ei
in
our simulation. Because sources can emit more than the socially efficient amount, environmental damages are higher, but abatement costs are lower, at the DTRS equilibrium
than at the social optimum. Lastly, note that Figure 2 shows that the first-order conditions
are not sufficient in two ways. First, for source 3, its marginal cost curve intersects with
the marginal damage curve at two points. Second, though not plotted, each source has
infinitely many marginal damage curves corresponding to different emissions levels by
other sources.
32
Case B: α 1 = 15.0, α 2 = 7.5, α 3 = 15.0
In this case, a high-cost firm is located downstream of two low-cost firms. At the
social optimum, source 3 abates all of its emissions while source 2 emits the most (Table
2). Efficient trades should occur under the TRS, because this system allows the high-cost
downstream source to buy permits from the two low-cost upstream sources. Indeed, this
is exactly what happened in the simulation. Each firm is allocated 18.5 units of discharge
permits initially in both the TRS and the DTRS. In the TRS equilibrium, firm 2 bought
14.4 units from source 3 to increase its emissions to 32.9 while firm 3 sold 18.5 units at the
trading ratio τ 32
0.78 (i.e. 18.5
τ 32 = 14.4 units for source 2). Note that under the
TRS, the downstream firm has a choice of buying permits from either source 1 or source
3. Therefore, source 2 buys from the partner with the lowest effective permit price.
It follows then that the effective equilibrium prices are equalized across space: p1 =
τ 12 p2 = τ 32 p2 = p3 . At this equilibrium price, source 1 has no incentive to sell its permits to source 2 and thus ends up emitting exactly at the initial allocation. The trading
outcome in the DTRS is similar. Source 3 abates its emissions down to zero and sells its
permits mostly to source 2. A difference occurs, because source 3 also sold its permits
to source 1. As we have noted, under the DTRS firms located in different branches are
allowed to trade, and the equilibrium prices satisfy p1 = (d1 =d2 ) p2 = (d1 =d3 ) p3 . It turns
out that at these equilibrium prices, it is cheaper for source 1 to buy permits from source 3.
As a result, source 1’s equilibrium emissions are slightly higher under the DTRS than under the TRS while source 2’s equilibrium emissions are lower under the DTRS than under
the TRS. The extra trade between source 1 and source 3, however, decreased efficiency
slightly, compared to the TRS equilibrium. This is because the damage-denominated
trading coefficients, the di ’s, are poor approximations of the true marginal damages at
the optimum, as shown in Figure 2. In this case, therefore, the DTRS encouraged inefficient trades. Note, however, that both TRS and DTRS reduce deadweight loss relative to
the no-trading baseline, and closely approximate the social optimum in this case.
33
Observation 1: Neither the TRS nor the DTRS dominates in performance. The TRS can preclude
efficient trades while the DTRS can encourage inefficient trades.
[Figure 2. Marginal Damages, Marginal Costs, Trading Coefficients]
5.4. Price vs. Quantity
On one hand, our numerical analysis suggests the impossibility of getting prices right
in watersheds having certain characteristics. Neither the TRS nor the DTRS succeeds
in providing the correct price signals for water-quality trading. On the other hand, our
analysis also indicates that in some cases, the equilibria approximate the social optimum
quite closely. We obtained these results by setting the total supply of permits equal to the
socially optimal level. A natural question then is, which type of inefficiency is larger: not
getting the prices of pollution right or not getting the quantity of permits right? We investigate this question by simulating the trading outcomes for varying levels of the supply of
permits (in percentage reduction from the baseline discharge level) and allocating permits
in equal number to each discharger. The result is shown in Figure 3.
First, the relative performance of the two systems varies, in an unsystematic way, with
the supply of permits. In Case A (α 1 = 7.5, α 2 = 15, and α 3 = 7.5) where a low-cost
source is located downstream of two high-cost sources, the DTRS performed substantially
better than TRS when the total supply of permits was kept to the socially optimal level.
This is because the TRS prohibited any trade from taking place. However, when the total
supply of permits is reduced to 60-70% of the baseline discharge level, the TRS performs
better than the DTRS despite the fact that no trading still takes place under the TRS. This
occurs because the efficiency loss due to the TRS precluding trading was outweighed
by the efficiency loss due to the DTRS encouraging inefficient trades, which increased
environmental damages substantially relative to no trade. In contrast, in Case B (α 1 =
15.0, α 2 = 7.5, and α 3 = 15.0), in which a high-cost source is located downstream of
34
two low-cost sources, the DTRS performed slightly better than the TRS for all levels of
initial permit supplies. In this case, TRS and DTRS provide similar price signals so that
the magnitude of the efficiency loss due to environmental damages is similar between the
two systems (see the left panel of Case B in Figure 3). However, because the DTRS offers
more flexibility in trading, it reduces abatement costs a bit more than does the TRS. This
effect dominates the relative performance of the two systems.
Second, total economic costs C + D do not exhibit a simple convex relationship with
respect to the total supply of permits under the two systems. This is because neither environmental damages nor abatement cost has a simple relationship to the supply of permits. Despite the fact that environmental damages are defined as a decreasing function
of emissions or pollution concentrations, environmental damages in the trading equilibrium are not necessarily a decreasing function of the reduction in the total supply of permits,
and analogously, despite the fact that abatement costs are a convex function of abatement levels, total abatement costs are not necessarily a convex function of the reduction
in the total supply of permits. These effects are especially strong in the DTRS, because the
damage-denominated trading coefficients are based on marginal damages at the initial
allocation, and thus depend endogenously on the initial supply of permits. Somewhat
counter-intuitively, these trading coefficients can either decrease or increase efficiency relative to the TRS. On one hand, the trading coefficient can decrease efficiency by providing
incorrect trading margins, which adversely affects environmental damages. On the other
hand, however, the trading coefficients can improve efficiency by providing flexibility for
trading partners, which reduces abatement costs.
Lastly, at least in the current model, getting the quantity of permits right appears to
be more important than getting the prices of permits right. In Case A, the estimated efficiency losses are only 18.2% and 0.1% of the total economic damages, respectively, for
TRS and DTRS when the socially optimal number of permits are distributed. (In Case B,
the corresponding results are 3.6% for TRS and 0.03% for DTRS.) In contrast, the max-
35
imum efficiency losses due to mis-specifying the total supply of permits are 80.9% and
97.8% of the total economic damages, respectively, for TRS and DTRS. (In Case B, the corresponding results are 80.3% for TRS and 78.1% for DTRS.) It is important to emphasize
that this result is not a direct consequence of the logistic damage response we assumed in
(18). Rather it stems from the multiple effects of mis-specifying the quantity of pollution,
including the incorrect price signals that arise from it.
Observation 2: Under both the TRS and the DTRS, deadweight loss due to incorrect price signals
may not be large relative to deadweight loss due to an incorrect quantity of permits. However, it is
still possible that the incorrect price signals can lead the DTRS equilibrium to perform worse than
the no-trading baseline.
[Figure 3. Total Supply of Permits and Relative Performance of TRS and DTRS]
6
Discussion
This paper examined the efficiency properties of two recently developed water-quality
trading models, the trading ratio system (TRS) proposed in Hung and Shaw (2005) and
the damage-denominated trading ratio system (DTRS) proposed in Farrow et al. (2005).
These two models emerged as potential water quality trading models that address both
spatially explicit damages and transaction costs. We showed that both trading systems
fail to achieve the efficient optimum (and the cost-effectiveness optimum) under general
conditions that are likely to hold in numerous watersheds. More specifically, the TRS fails
when the river has critical zones in a branching river whereas the DTRS fails when the
pollution damages are nonlinear in either emissions levels or pollution concentrations.
We derived these results under the first-best scenario in which the regulator knows the
36
efficient vector of environmental constraints (for TRS) and the efficient damage constraint
(for DTRS).
Furthermore, in a second-best scenario where the regulator cannot set these constraints
at the efficient levels, neither system dominates in terms of efficiency, because the TRS excludes efficient trades while the DTRS promotes inefficient trades. These results indicate,
in this sense, the impossibility of getting the spatially explicit prices of pollution right under either system. However, our computational results do indicate the possibility that the
two systems may still approximate the socially efficient optimum sufficiently closely if
the total allowable permits are set initially at levels sufficiently close to the optimum. The
(maximum) efficiency loss due to mis-specifying the total supply of permits was much
larger than that from mis-specifying the spatial prices of permits. Interestingly, the magnitude of inefficiency due to incorrect signals may also depend on the total supply of
permits. Thus our paper suggests the importance of getting the quantity of pollution
right even while striving to get the spatial prices of pollution right.
Though we kept the assumptions of our model as general as possible with respect to
water pollution and watershed characteristics, like Hung and Shaw and also Farrow et
al. we ignored the problem of nonpoint-source pollution (NSP). Nonpoint sources can of
course play an important role in a watershed. It is usually difficult and costly to identify
and monitor the level of NPS pollution-causing activity (or discharge levels), because
land-use practices (for example, fertilizer application) or land use itself (for example,
buildings and parking lots) are the major sources of such pollution. If the discharge from
each source is difficult to identify, neither trading nor direct control would achieve the efficient outcome. However, in recent years substantial efforts have been devoted to transforming nonpoint sources into point sources. Scientists of various disciplines continue
to improve their ability to identify and monitor pollution levels from nonpoint sources.
As our understanding of NPS improves, the present study could have important implications for the optimal management of nonpoint-source pollution.
37
In the case of nonpoint pollution, the business of getting the spatial prices of pollution right becomes even more difficult for two reasons. First, because water pollution can
travel through multiple nonpoint sources before it reaches the river, and because the number of nonpoint sources is often quite large, the spatial dependence of marginal damages
from each source’s pollution is likely to be exacerbated. Second, each nonpoint source’s
discharge is likely to affect pollution concentration levels at multiple receptors. In such
a case, allowing sources to trade at the exchange rates based on the transfer coefficients
between the receptor points along the river, as in the TRS, would likely result in substantial deadweight loss for two reasons. First, because each source’s emissions can affect
pollution concentrations at multiple receptors for some of which transfer coefficients can
be zero (across branches, for example), trading based only on the non-zero transfer coefficients may result in inefficient trades. Second, just as with point-source pollution, the TRS
can also preclude efficient trades from taking place by restricting the exchange among
sources whose emissions affect pollution concentrations at receptors located on separate
branches. On the other hand, the DTRS would be relatively robust to nonpoint-source
pollution. As long as the discharge level from each nonpoint source can be identified,
pollution damages still being the function of pollution concentration levels at receptor
points, the marginal damage from each nonpoint source can be calculated. Because trading ratios based on the marginal damages are the correct exchange rates for the nonpoint
sources, the DTRS could potentially offer the correct trading incentives. The problem
arises, however, when damages are highly nonlinear. As we have demonstrated, evaluating the marginal damages at any allocation (including the optimum) and fixing the
trading ratios at that evaluation point would encourage inefficient trades to take place by
giving incorrect trading incentives. The number of trading sources is likely to be large in
the case of NSP, and so the error from pre-fixing the trading ratios might result in substantial deadweight loss.
38
Table 1. Water Quality Model Parameters
Parameters
Units
k
Q
a
b
c
mile‐1
ft3/s
none
none
mg/L
decay rate
stream flow
damage parameter
damage scale parameter
concentration threshold
Values
0.005
10
5
6.7
5
Table 2. Simulation Results
Case A: α1= 7.5, α2=15.0, α3= 7.5
e2
e1
No Trading
TRS
DTRS
Optimum
23.7
23.7
48.9
42.0
23.7
23.7
0.0
0.0
Case B: α1= 15.0, α2= 7.5, α3= 15.0
e2
e1
No Trading
TRS
DTRS
Optimum
18.5
18.5
20.5
21.0
18.5
32.9
31.4
34.5
e3
23.7
23.7
34.4
29.0
e3
18.5
0.0
0.0
0.0
Damage
Cost
Total
511
511
530
60
1,942
1,942
1,589
1,787
2,454
2,454
2,119
1,848
Damage
Cost
Total
10
10
9
42
1,771
1,710
1,716
1,655
1,782
1,720
1,725
1,697
Figure 1. Hypothetical River Basin
Figure 2. Marginal Damages, Marginal Costs, Trading Coefficients
Case A: a1 = 7.5, a2 = 15.0, a3 = 7.5
35
35
MD1
30
30
d1
15
10
d3
25
25
20
20
15
15
10
MC1
5
0
20
40
60
80
100
0
MD3
10
MC2
5
0
30
MD2
25
20
35
d2
0
20
40
60
80
MC3
5
100
0
0
20
e2
(e1 = e1eff, e3 = e3eff)
e1
(e2 = e2eff, e3 = e3eff)
40
60
80
100
e3
(e1 = e1eff, e2 = e2eff)
Case B: a1 =15.0, a2 = 7.5, a3 = 15.0
30
35
MD1
25
30
MD2
30
25
MD3
25
20
20
20
15
15
15
10
MC1
MC2
5
5
d1
0
0
20
40
60
e1
(e2 = e2eff, e3 = e3eff)
10
10
80
100
0
5
d2
0
20
40
60
e2
(e1 = e1eff, e3 = e3eff)
MC3
d3
80
100
0
0
20
40
60
80
e3
(e1 = e1eff, e2 = e2eff)
100
Figure 3. Total Supply of Permits and Relative Performance of TRS and DTRS
Case A: a1 = 7.5, a2 = 15.0, a3 = 7.5
Damages, D
4000
No Trading
DTRS
TRS
Optimum
3500
3000
Abatement Costs, C
4000
4000
No Trading
DTRS
TRS
Optimum
3500
3000
3500
3000
2500
2500
2500
2000
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0
30
40
50
60
70
80
90
100
0
30
40
50
60
70
80
90
Total Economic Costs, C + D
100
0
30
No Trading
DTRS
TRS
Optimum
40
50
60
70
80
90
100
Reduction in Total Supply of Permits (% of Business‐As‐Usual Discharge)
Case B: a1 =15.0, a2 = 7.5, a3 = 15.0
Damages, D
4000
No Trading
DTRS
TRS
Optimum
3500
3000
Abatement Costs, C
4000
4000
No Trading
DTRS
TRS
Optimum
3500
3000
3500
3000
2500
2500
2500
2000
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0
30
40
50
60
70
80
90
100
0
30
40
50
60
70
80
90
Total Economic Costs, C + D
100
0
30
No Trading
DTRS
TRS
Optimum
40
Reduction in Total Supply of Permits (% of Business‐As‐Usual Discharge)
50
60
70
80
90
100
References
[1] Anderson, D.M, P.M. Glibert, J.M. Burkholder. (2002). Harmful Algal Blooms and
Eutrophication: Nutrient Sources, Composition, and Consequences. Estuaries, 25(4),
704–726
[2] Atkinson, S.E., T.H. Tietenberg. (1982). The Empirical Properties of Two Classes of
Designs for Transferable Discharge Permit Markets. Journal of Environmental Economics and Management, 9(2), 101-121.
[3] Baumol, W.J., W.E. Oates. (1988). The Theory of Environmental Policy. 2nd Edition.
Cambridge: Cambridge University Press.
[4] Breetz, H.L., Karen Fisher-Vanden, Laura Garzon, Hanna Jacobs, Kailin Kroetz, and
Rebecca Terry. (2004). Water quality trading and offset initiatives in the U.S.: A comprehensive survey. Dartmouth College, New Hampshire.
[5] Elliott, J.M. (2000). Pools as refugia for brown trout during two summer droughts:
trout responses to thermal and oxygen stress. Journal of Fish Biology, 56, 938-948.
[6] Elliott, J.M. and M.A. Hurley. (2001). Modelling growth of brown trout, Salmo trutta,
in terms of weight and energy units. Freshwater Biology, 46, 679-692.
[7] Environomics. (1999). A summary of U.S. effluent trading and offset projects. Prepared for U.S. EPA Office of Water.
[8] Fang, Feng, K. William Easter, and Patrick L. Brezonik. (2005). Point-nonpoint source
water quality trading: a case study in the Minnesota River Basin. Journal of the American Water Resources Association, 41(3), 645-658.
[9] Farrow, R.S, Shultz, M.T., Celikkol, P. Van Houtven, G.L. (2005). Pollution Trading in
Water Quality Limited Areas: Use of Benefits Assessment and Cost-Effective Trading
Ratios. Land Economics, 81(2), 191-205.
39
[10] Helfand, G.E. and B.W. House. (1995). Regulating Nonpoint Source Pollution under
Heterogeneous Conditions. American Journal of Agricultural Economics, 77(4), 10241032.
[11] Hung, Ming-Feng and Daigee Shaw. (2005). A trading-ratio system for trading water
pollution discharge permits. J. of Env. Economics & Mgmt., 49, 83-102.
[12] King, Denis M. (2005). Crunch time for water quality trading. Choices, 20(1), 71-75.
[13] King, Denis M. and Peter J. Kuch. (2003). Will nutrient credit trading ever work? An
assessment of supply and demand problems and institutional obstacles. The Environmental Law Reporter. Washington, DC: Environmental Law Institute.
[14] Mauzeralla, D.L., B. Sultanc, N. Kima, and D.F. Bradford. (2005) NOx emissions from
large point sources: variability in ozone production, resulting health damages and
economic costs. Atmospheric Environment 39, 2851–2866.
[15] Montgomery, W. David. (1972). Markets in licenses and efficient pollution control
programs. Jour. of Economic Theory, 5, 395-418.
[16] Morgan, Cynthia and Ann Wolverton. (2005). Water quality trading in the United
States. Working paper #05-07, U.S. EPA National Center for Environmental Economics, Wathington DC: EPA.
[17] Muller, N.Z., Mendelsohn, R. (2009). Efficient Pollution Regulation: Getting the
Prices Right. American Economic Review, 99(5), 1714-1739.
[18] Oregon Department of Environmental Quality. Water quality credit trading in
Oregon:
A case study report. Last updated July 27, 2007. Available online:
http://www.deq.state.or.us/wq/trading/docs/wqtradingcasestudy.pdf.
40
[19] Sado, Y., R.N. Boisvert, and G.L. Poe. (2010). Potential cost savings from discharge
allowance trading: A case study and implications for water quality trading. Water
Resour. Res., 46, W02501.
[20] Schnoor, J.L. (1996). Environmental modeling: Fate and transport of pollutants in water,
air, and soil. Wiley: New York.
[21] Segerson, K. (1988). Uncertainty and Incentives for Nonpoint Pollution Control. J. of
Env. Economics & Mgmt., 15, 87-98.
[22] Shortle, J.S. and J.W. Dunn. (1989). The Relative Efficiency of Agricultural Source
Water Pollution Control Policies. American Journal of Agricultural Economics, 68(3),
668-677.
[23] Taff, Steve J. and Norman Senjem. (1996). Increasing regulators’ confidence in pointnonpoint pollutant trading schemes. Water Resources Bulletin, 32(6), 1187-1193.
[24] Tietenberg, Tom H. (1st ed.1985; 2nd ed. 2006). Emissions trading: principles and
practice. Washington DC: Resources for the Future. 2nd edition.
[25] Todd, D.K. and L.W. Mays. (2005). Groundwater Hydrology. John Wiley & Sons, Inc.
3rd edition.
[26] U.S. EPA. (2002). Estimation of national economic benefits using the National Water
Pollution Control Assessment Model to evaluate regulatory options for concentrated
animal feeding operations. U.S. EPA Office of Water, Washington DC: EPA. EPA-821R-03-009.
[27] U.S. EPA. (2003). Water quality trading policy. U.S. EPA Office of Water, Washington
DC: EPA.
41
[28] U.S. EPA. (2004). Water quality trading assessment handbook: Can water quality
trading advance your watershed’s goal? U.S. EPA Office of Water, Washington DC:
EPA.
[29] Woodward, R.T., R.A. Kaiser, and A.M. Wicks. (2002). The structure and practice of
water quality trading markets. Journal of the American Water Resources Association.
[30] Van Kirk, R.W., Hill, S.L., (2007). Demographic model predicts trout population response to selenium based on individual-level toxicity. Ecological Modeling 206, 407420.
42