have implications for the economic relevance of wetlands, and for implementing economic policies
designed to control point and nonpoint source pollution.
In this paper we examine under what conditions wetlands are economically rational to use for
abatement of nonpoint nitrogen pollution, recog- nizing that agricultural run-off as well as the
nitrogen abatement capacity of wetlands are stochastic. Economic criteria for wetlands are es-
tablished using a simplified theoretical model of a watershed. In contrast to previous studies on wet-
lands, we extend the analysis by explicitly model- ing the uncertainty of nonpoint emissions and the
uncertainty of nitrogen abatement in wetlands. The implications of uncertainty on the conditions
for optimal nitrogen abatement in a watershed are discussed, and an example from southwestern
Sweden is used to illustrate and evaluate the theoretical findings.
2. The model
To establish economic criteria for construction of wetlands under uncertainty, consider a water-
shed in which there are two sources of nitrogen emissions. First, there are agricultural emissions
resulting in nonpoint source NPS emissions of nitrogen. Second, there is a waste water treatment
plant WWTP that emits nitrogen. Assume fur- ther that there are two options for controlling
emissions. Nitrogen load from agricultural land may be reduced by construction of wetlands that
are designed for nitrogen abatement. Point source emissions from the WWTP may be reduced by
investments in waste water treatment. This sim- plified watershed is illustrated in Fig. 1 below.
While it is recognized that NPS emissions are stochastic, the model is simplified by assuming
that emissions from the WWTP are deterministic. This assumption accommodates the plausible fact
that emissions from nonpoint sources are charac- terized by a high variability, compared to emis-
sions of the WWTP, due to climate, precipitation, and other factors Malik et al., 1993.
The level of nitrogen leakage from agricultural land depends on management practices and on
stochastic events such as temperature and precipi- tation Shortle and Dunn, 1986. Leakage is thus
stochastic and in this paper denoted by R 0 . Part of
the agricultural leakage enters wetlands, thus making nitrogen abatement in wetlands stochas-
tic. In addition, the abatement capacity of a wet- land depends on weather and is consequently
stochastic in itself Mitsch and Gosselink, 1993. Finally, since both abatement in wetlands and
leakage from agricultural land are stochastic, the total nitrogen load from the watershed to the sea
is a stochastic variable.
To model nitrogen retention in wetlands, a denitrification
model developed
in Bystro¨m
1998, is utilized.
1
Denoting retention abate- ment in wetlands by Q
0 and total nitrogen load by P
0 , both of which are stochastic, the following definitions are established:
R 0 =r+o
L −L
w
1 Q
0 =a+o
1
L
w
+ bR
0 s 2
P 0 =R0−Q0+Z
3 where,
Fig. 1. Watershed model displaying the structure of the prob- lem and the variables used.
1
Note that in this paper we use the linear statistical model presented in Bystro¨m 1998. It can also be noted that the
model used in this paper includes the estimated standard errors of Bystro¨m. The uncertainty of wetlands’ abatement
efficiency is thus incorporated as a noise term o
1
of Eq. 2 which is later used for defining the variance of total pollution
load. For empirical applications, the standard errors reported in Eq. 2 can be derived through the t-values reported in
Bystro¨m 1998.
r = Expected nitrogen L
=Maximum area of agricultural land.
leakage per hectare of agricultural land.
L
w
= Area of wetland. o
,o
1
= Random
variables with zero mean.
Z = Emissions from s [0,1] = Fraction of
agricultural run-off that WWTP.
enters wetlands. a,b = Retention
coefficients.
It is assumed in this paper that the random errors, o
and o
1
, are independent and normally distributed. By the notation chosen in Eq. 1 it is
clear that we assume that there can be no idle land in the watershed, all agricultural land is used
either as wetlands or for cultivation of crops.
2
The watershed is modeled from the perspective of a regulatory agency, where the objective is to
maximize economic return from the watershed area, subject to a pollution constraint. If environ-
mental damages of excessive nitrogen emissions are caused not only by a high level of expected
nitrogen load, but also by random peak flows, then monitoring the uncertainty of emissions be-
comes important for controlling environmental damages. Consequently, the distribution of emis-
sions needs to be reflected in the pollution con- straint. One approach to incorporate uncertainty
into standard models of constrained optimization is to require that the environmental pollution
constraint is achieved by a certain probability. By specifying an acceptable probability level, it is
possible to monitor the risk of violating the pollu- tion constraint. Recognizing that the abatement
target has to be achieved with a certain probabil- ity, the decision problem can be modeled as
follows: Max
L
w
,Z
L −L
w
p − CL
w
− CZ −Z
4 such that
ProbP 0 5P]a
5 where,
p = Per hectare profit
P = Pollution of crop cultivation on
constraint. agricultural land.
CL
w
= Cost function a =
Probability by for construction of
which the pollution constraint should be
wetlands. met.
CZ −Z=Cost
Z =Initial WWTP
emissions. function for
reduction of WWTP emissions.
The probability a specifies the minimum proba- bility according to which the pollution constraint,
P, should be satisfied. If a equals 0.5, then pollution constraint 5 is equivalent to incorpo-
rating the deterministic constraint of reducing solely expected emissions. If, however, a is be-
tween 0.5 and 1, constraint 5 requires that the abatement target is achieved by a probability level
exceeding that of reducing only expected emis- sions. Thus, constraint 5 effectively specifies a
reliability constraint that the policy planner may address by imposing additional restrictions on
expected emissions or by improving the distribu- tion of total emissions.
A convenient approach for solving the decision problem specified by 4 and 5 is to use the
chance constrained programming approach which has been commonly applied for solving economic
programming problems under uncertainty e.g. Charnes and Cooper, 1964; McSweeny and
Shortle, 1990; Shortle, 1990. The problem formu- lation under this approach is useful for illustrating
the impact of uncertainty, and how the variance of emissions affects the optimal solution. The
method makes it possible to replace the proba- bilistic pollution constraint with its deterministic
equivalent, such that standard deterministic opti- mization methods can be applied. Following Mc-
Sweeny and Shortle 1990 and Paris and Easter
2
Furthermore, with the chosen modeling approach, the problem is analyzed as a steady-state problem in order to
focus the paper on uncertainty aspects. It would be interesting in a future paper to also include a time-perspective in the
analysis, which could provide a more complete picture of nitrogen retention in wetlands.
Table 1 Four possible cases describing wetlands’ impact on expected emissions and the uncertainty of emissions in a watershed
Abatement capacity of wetlands Wetlands effect on the variance of emissions
Not Affected Decreases
Increases Case 3
Case 2 Positive
Case 1 Nonenegative
Case 4 Case 4
Case 4
1985, the constraint in Eq. 5 is rewritten as a chance constraint
3
: EP + f
a
VP
12
5 P
6 where EP is the expected nitrogen load, VP =
VR
+
VQ − 2Co6Q,R is the variance of total nitrogen load and f
a
is a parameter that specifies the weight that should be attached to the variance
of emissions in order for the abatement target to be reached with a probability a Taha, 1976. In
this paper it is assumed that VP follows a normal distribution. For a given probability, the
value of f
a
is then obtained from the standard normal cumulative distribution. If uncertainty of
emissions is irrelevant to the problem, then f
a
is equal to zero i.e.a = 0.5. If, however, the uncer-
tainty is to be monitored, that is a \ 0.5, then f
a
is positive, thus including the variance into the pollution constraint. Consequently, pollution con-
straint 6 becomes stricter with increasing reli- ability requirements. It is obvious from Eq. 6
that there are two main strategies for reducing the nitrogen load such that the abatement target is
not violated. First, the strategy may address abatement efforts that are mainly directed to-
wards reducing the average, or expected, emis- sions, EP. Secondly, abatement measures that
are mainly designed to reduce uncertainty, and thus the variance, of emissions may be used. Since
WWTP emissions are assumed to be deterministic, a reduction of these emissions affects only the
expected nitrogen load. In this paper wetlands represents a strategy that affects the uncertainty
of nitrogen load to the sea. As a consequence, criteria for the economic relevance of wetlands
include not only the expected abatement and the construction costs thereof, as have been shown in
previous studies e.g. Bystro¨m, 1998; Gren, 1993, 1995, but more so the impact of wetlands on the
uncertainty of nitrogen load to the recipient.
Unless anything specific is known about wet- lands’ abatement capacity, or impact on the vari-
ability of nitrogen load, there are four possible cases that may characterize wetlands’ impact on
water quality in the watershed. These scenarios are displayed in Table 1.
Clearly, in the event that case 4 prevails, wet- lands are never economically rational in terms of
nitrogen abatement, irrespective of costs. How- ever, do any of the other situations rule out
wetlands as an economically relevant abatement measure? To address this question, let us examine
the optimal solution to the pollution problem graphically for the three first situations. Fig. 2a – c
display the optimal solutions and the constraints to the models. Each scenario is referred to by its
number.
In Fig. 2a, IC is an isocost curve that describes
all combinations of wetlands and WWTP abate- ment that yield a constant total cost. If both
construction costs for wetlands and the costs of WWTP reductions are linear, then IC
is linear. IC
1
– IC
3
are defined analogously although they depict different total costs. Costs increase as the
IC curve shifts out from the origin, since costs increase in construction of wetlands, and with
increased reduction of WWTP emissions. The slope of the IC-curves is equal to the relative
marginal cost between emission reductions in WWTPs and construction of wetlands. PC
– PC
3
denote pollution
constraints. The
PC-curves define all combinations of wetlands and WWTP
3
For a description of the general approach and the tech- nique of chance constrained programming CCP, see Taha
1976 for a comprehensive explanation. For previous applica- tions, see Shortle 1990, McSweeny and Shortle 1990, Paris
and Easter 1985, and for the original text on CCP methods, see Charnes and Cooper 1964.
Fig. 2. Optimal allocation of abatement under three scenarios.
abatement that correspond to a specific pollu- tion constraint. PC
denotes the pollution con- straint when uncertainty of emissions is not
considered a = 0.5, f
a
= 0. The optimal alloca-
tion of abatement efforts can be observed in the figures as the point where an IC-curve is tan-
gent to the pollution constraint, or where the intersection between the curves yields the lowest
total costs. For the base case when only ex- pected nitrogen loads are considered f
a
= 0,
the optimal solution is given by Z , L
w
in Fig. 2a.
When the pollution constraint is to be satisfied with a probability greater than 0.5, the PC-curve
shifts out, since including the variance now re- quires increasing abatement efforts. Fig. 2a de-
picts case 1, when construction of wetlands does not affect the uncertainty of emissions. In this
case there are no possibilities to affect the vari- ance of emissions, and thus the only possibility is
to reduce expected emissions such that the modified pollution constraint, PC
1
, is not vio- lated. It can be noted that the curvature and the
slope of pollution constraint PC
1
is identical to the original PC
, since neither WWTP reductions nor construction of wetlands affect the distribu-
tion of nitrogen load. The optimal allocation in case 1 is given by Z
1
, L
w 1
. Fig. 2b depicts case 2. Under this scenario
construction of wetlands reduces the variability of total nitrogen load. This means that wetlands
become more efficient with respect to fulfilling the abatement target, compared to case 1 see also
Eq. 6. If the variance-reducing effect of wet- lands is higher for the first units of wetlands
constructed, then the pollution constraint PC
2
is convex as shown in the figure this is formally
shown in the appendix to this paper. Hence, a higher emission reduction is required in the waste
water treatment plant in order to substitute for one unit of nitrogen reduction in wetlands com-
pared to case 1. The optimal combination of wetlands and WWTP reductions is now given by
L
w 2
, Z
2
. The total cost is in this case given by IC
2
, which is lower than the total costs in the first case of Fig. 2a.
The scenario under case 3 is shown in Fig. 2c. In this case construction of wetlands increases the
uncertainty of emissions. If the rate of augmenta- tion in variance increases as larger areas of wet-
lands are
constructed, then
the pollution
constraint in case 3 is convex as indicated in Fig. 2c this is shown in the appendix. The effect on
the pollution constraint as defined by Eq. 6 is in this case ambiguous and it is unclear whether
construction of wetlands contributes to solving the pollution problem. In this scenario wetlands
are less efficient compared to the situation when construction of wetlands does not affect the vari-
ance, fewer units of WWTP reductions are re- quired to substitute for one unit of nitrogen
reduction in wetlands. The three cases above depict the possible cases
for wetlands in theory. In practice it can be expected that one of these situations prevails. But
in which of the cases are wetlands economically relevant to use for nitrogen abatement, and in
which case is the relevance of wetlands unambigu- ous? Under case 4, wetlands are clearly not eco-
nomically relevant, since wetlands neither perform abatement, nor improve the distribution of emis-
sions. Therefore case 4 can be ruled out directly. In case 3 wetlands have a positive abatement
capacity, but construction of wetlands also in- creases the variability of total emissions, which
makes the overall abatement effect uncertain, at least when the abatement target is to be achieved
with some degree of certainty a \ 0.5, f
a
\ 0. In
case 1 and 2 the expected abatement capacity of wetlands is positive. Moreover, in case 2 the
abatement capacity, or the economic relevance, of wetlands increases with the introduction of a reli-
ability requirement since wetlands in case 2 also reduce the uncertainty of emissions. In this case
the abatement performed by wetlands is unam- biguous, and the abatement capacity is not dimin-
ished if stricter reliability constraints are imposed. Three criteria for the economic relevance of using
wetlands for nitrogen abatement can now be formulated.
The abatement capacity of wetlands must be positive and increasing in wetland area.
The use of wetlands for nitrogen abatement must not increase the uncertainty, or variance,
of total nitrogen load.
Given that the two first conditions are fulfilled, we also require that wetlands have sufficiently
low abatement costs to be considered as a viable measure for pollution reduction in nitro-
gen abatement programs. The relative prices are given by the slope of the IC-curves in Fig.
2a – c.
3. Economic policy and wetlands
