Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue1.Jan2000:
Journal of Economic Dynamics & Control
24 (2000) 143}163
The dynamics of spatial pollution:
The case of phosphorus runo!
from agricultural land
Renan U. Goetz!,*, David Zilberman"
! Department of Economics, University of Girona, Girona, Spain
" Department of Agricultural and Resource Economics, University of California, Berkeley, USA
Received 7 April 1997; accepted 7 October 1998
Abstract
The paper analyzes, based on a land classi"cation system, the optimal management of
negative production externalities while taking into account the intertemporal and spatial
aspects of the problem. To incorporate both aspects simultaneously, a two-stage
modeling approach is proposed where the solution of the spatial problem ("rst stage) is
optimized over time (second stage). As a result, it is possible to relate long-run and
short-run supply and input demand functions. Attention is given to runo!s from agricultural land leading to the contamination of a water body, in particular, the one of
phosphorus and the eutrophication of lakes. The employment of a land classi"cation
system supports a full-information approach and allows to address the optimal management of mineral fertilizer and manure based on zonal taxes, zonal permits, and zonal
standards which all vary over time. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: C61; H23; Q24; Q25
Keywords: Space time-dependent optimal control; Pollution tax; Tradable phosphorus
permits; Zoning
* Correspondence address: Department of Economics, University of Girona, Avda. Lluis SantaloH ,
17071 Girona, Spain. Fax: #34/972/418032; e-mail: goetz@econ.udg.es.
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 7 - 0
144 R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163
1. Introduction
Negative production externalities have been the concern of many economic
studies. In the case where the pollutants accumulate over time, these studies
were conducted within a dynamic framework. Besides the intertemporal aspect,
the heterogeneity of space may be of prime consideration for the economic
analysis of negative production externalities. Prominent negative externalities
from agricultural production like the runo! of phosphorus (P) or the leaching of
nitrogen (N) share the intertemporal and spatial aspect.
Hochman et al. (1977) analyzed the properties of optimal tax incentives
schemes to control pollution in a spatial setting. Both aspects, time and space,
were integrated in a modelling approach by Tomasi and Weise (1994). They
focused their analysis on the optimal spatial allocation of agricultural land
versus residential land over time. Yet, the agricultural land is often exogenously
given, and the optimal management of nonpoint-source pollution needs to
concentrate on the agricultural sector itself. Therefore, we de"ne agricultural
activities, which allow us to address agricultural policy questions. We base our
analysis on the full-information approach by employing a land classi"cation
system which accounts for di!erences in environmental vulnerability of locations and by considering improved monitoring technology to account for
heterogeneity in residue (pollutant) generation (Khanna et al., 1998). In this way
nonpoint-source pollution can be viewed as point-source pollution provided
that an adequate monitoring technology and a classi"cation system exist. Most
importantly, however, we present a modelling approach which consists of the
simultaneous solution of the micro-level (agricultural production) and macrolevel (aggregate supply and demand) over space and time. In particular, it allows
us to derive relationships between long-run and short-run supply and input
demand functions.
2. Environmental vulnerability and space
Traditionally economists think of space in the form of some distance measure.
However, two locations next to each other and equally distanced from a receiving water body may demonstrate a completely di!erent vulnerability to negative
agricultural production externalities. In the case of agricultural surface runo!s,
the reason may be that one location is sloping downhill towards the receiving
water body and the other one uphill. Wu and Segerson (1995) noted the
importance of distinct site characteristics for classifying the environmental
vulnerability of a site to potential pollution. The distance of a location with
respect to a point of reference is only one aspect among many others. However,
not only do site characteristics have to be considered but also characteristics of
the receiving water body. Thus, it is possible to relate the potential pollution to
R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163 145
the water body (e.g., milligram of bioavailable P per liter) and take a "rst step in
evaluating the potential damage.
To assess the environmental vulnerability of a location based on a land
classi"cation system, this paper takes the P index, developed under the patronage of the U.S. Department of Agriculture } The Natural Resources Conservation Service } (Sharpley, 1995), as an example. The P index allows to assign each
location within the catchment of the receiving water body the potential P load
resulting from the cultivation of a particular crop with a given crop management
practice while taking account of the particular characteristics of the water body.
For the sake of concreteness, we consider the example of a lake. Phosphorus
loadings of a lake above the sum of the permanent settling rate of P in the
sediment and of the P losses through the out#ow of water lead to the euthropication of a lake. This will result in excessive algae growth and a lack of dissolved
oxygen (Spulber and Sabbaghi, 1994). As a result, the water quality of the lake
declines which, in turn, reduces the biodiversity and recreational and commercial bene"ts of the lake. It also increases the costs for water treatment to provide
drinking water or to supply water for manufacturing processes (Holmes, 1988;
Spulber and Sabbaghi, 1994).
The P loading from agricultural land at a speci"c location within the watershed depends on the water delivery ratio, the water body sensitivity, the land
management, and on characteristics of the land, such as the soil P content and
the erosion and runo! potential. The latter characteristics are incorporated in
the P index by considering site-speci"c information such as soil texture, permeability of the soil, rainfall, the length and gradient of the slope, the cultivated
crop itself, and crop management and tillage practices. The factor soil P content
includes available soil P and P absorption capacity. The water delivery ratio
re#ects the share of P runo! transferred from the edge of the "eld to the lake,
whereas the water body sensitivity accounts for factors such as the degree of
surface mixing, the depth of the lake, water residence time, and the development
of reducing conditions at the water sediment surface (Sharpley and Halvorson,
1994).
Let the P load for a given type of fertilizer i, associated with a particular crop,
be denoted by a (o), where o represents a derived P index attributing a potential
i
P load to each location within the watershed, regardless of the type of fertilizer.
Intuitively, one expects locations close to the tributaries of the lake or to the lake
itself to have high potential P loads. Therefore, we associate a small o with
a high P load. The economic loss resulting from the deterioration of the water
quality of the lake is captured by the C(2) (twice continuously di!erentiable)
damage function d(s), where s presents the concentration of bioavailable P in the
lake. We propose that d@(s)'0 and dA(s)50.
One can think of other classi"cation systems with respect to site-speci"c
environmental vulnerability; for example, leaching of nitrate into groundwater
or degradation of a biotope resulting from agricultural production and the
146 R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163
application of plant protection agents. Analogous to the land classi"cation
system with respect to P runo!, these systems would classify the land based on
a variety of factors, where the measurement of the distance from a particular
point of reference is only one among many other factors. In this paper we focus
our analysis on the problem of P runo!s from agricultural land. However, to
generalize our results, we also discuss the most important "ndings in the context
of nitrogen leaching into the groundwater.
3. The economic model
Examples in the United States (Boggess et al., 1993) and Switzerland (Amt fuK r
Umweltschutz, 1993) show that lakes are often euthropic where the number of
large animal units per hectare is high. This can be explained by the fact that
P runo!s from agricultural land, where manure has been applied, contains
a high fraction of soluble phosphorus which is immediately available for biological uptake by the algaes. Moreover, manure application rates are based on the
management of nitrogen, leading in most cases to an increase in soil P in excess
of crop requirements. This is due to the generally lower ratio of N : P added in
manure than what is taken up by crops1 (Sharpley and Halvorson, 1994).
This situation suggests focus on the analysis on crop production and the
stocking rate per hectares in order to "nd the optimal P concentration of the
lake. For this purpose, let a (o) denote the P runo!s associated with the
1
application of mineral fertilizer u(o) and a (o) with the application of animal
2
manure b(o)oy(o), where y(o) presents the number of large animal units, o'1
the amount of manure per large animal unit, and b(o)3[0, 1] the share of the
available manure being applied. We assume for the case of P runo!s that
a (o)'a (o), ∀o as a result of the high P content and high bioavailability of the
2
1
manure compared to mineral fertilizer. Thus, the dynamics of the lake with
respect to o can be stated as
sR "(a (o)u(o)#a (o)b(o)oy(o))g(o),
(1)
1
2
where a dot over the variables s or j, to be introduced later, indicates the
operator d/do, and g(o) presents a density function. Furthermore, we assume
that the entire agricultural land within the watershed is cultivated and is equal
to one by an appropriate normalization procedure.
Eq. (1) can also be utilized to describe the dynamics of the concentration of
nitrate in the groundwater, where a (o), i"1, 2, presents a leaching function
i
1 For example, dairy, beef, swine, sheep and poultry manure has an average N : P ratio of 4.1 while
the N : P requirements of major grain crops is 7.3.
R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163 147
associated with fertilizer i. Mineral fertilizer carries nitrogen in its mineral form,
often in the form of nitrate. Thus, it is ready available by the plant, but the
mineral fertilizer is also more suspectable to the leaching of nitrate than manure,
where the nitrogen is predominately in its slow-releasing organic form. Therefore, the case of nitrate leaching suggests that a (o)(a (o), ∀o.
2
1
To complete the economic model, we like to introduce the C(2) crop production function given by q(z) with q@'0 and qA(0, and z,u(o)#b(o)oy(o). Let
the returns from livestock management be denoted by p y(o) and the associated
2
C(2) cost function by c (y(o)), where we assume c@ '0 and cA '0. The price
3
3
3
associated with the crop is denoted by p . Finally, we assume that a social
1
planner exists, who maximizes the present discounted net bene"ts from
agricultural production within the watershed of the lake while taking account of economic losses resulting from the accumulation of bioavailable P in
the lake.
4. The management of phosphorus runo4s from agricultural land
4.1. Spatial and intertemporal aspects
To obtain an analytical solution more easily, we propose the framework of
a two-stage optimal control problem. Basically, in the "rst stage we solve the
spatial problem by determining the optimal trajectories of u(o), b(o), y(o) and s(o)
for every o3[0, oN ], where oN indicates the upper limit of the P index. The value
function LB
LB
t"o
! a"0.
Lp
Lp
Lp
1
1
1
(25)
(26)
The results show that the short-run aggregate supply of livestock, as well as the
short-run aggregate demand for manure applied on the "eld and to be transported out of the watershed, is una!ected by an increase in the price for the crop.
However, the short-run aggregate demand for mineral fertilizer increases and,
consequently, the short-run aggregate supply of the crop goes up as well.
156 R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163
An increase in the returns of livestock management a!ects the short-run
aggregate supply and input demand functions in the following way:
oN
LyL
L>
"
g(o) do'0,
Lp
Lp
0 2
2
oN
LuL
L;
"
g(o) do(0,
Lp
Lp
0 2
2
P
P
(27)
oN
LB
LbK
LyL
a"
g(o) do'0,
oyL #bK o
Lp
Lp
Lp
2
0
2
2
B
PA
oN
L> LB
LB
t "o
! a"o 0g(o) do"0,
Lp
Lp
Lp
0
2
1
2
P
(28)
where the results of Eqs. (21) and (22) are used. An increase in p shows that the
2
short-run aggregate demand for mineral fertilizer decreases while the short-run
aggregate demand for manure applied on the "elds increases and the short run
aggregate demand for manure shipped out of the water catchment remains
unchanged. Therefore, the short-run aggregate supply of livestock increases. The
discussion of the e!ects of the variations in the P runo! coe$cients a and
1
a and the cost for mineral fertilizer c is presented in Appendix B.
2
1
6. Long-run analysis
Finally we consider the case ¹PR in order to study the comparative statics
of the long-run solution of problem (P2). Therefore, we propose to reformulate
problem (P2) in order to simplify the comparative static analysis. The revised
version of problem (P2) is given by
P
=
(
24 (2000) 143}163
The dynamics of spatial pollution:
The case of phosphorus runo!
from agricultural land
Renan U. Goetz!,*, David Zilberman"
! Department of Economics, University of Girona, Girona, Spain
" Department of Agricultural and Resource Economics, University of California, Berkeley, USA
Received 7 April 1997; accepted 7 October 1998
Abstract
The paper analyzes, based on a land classi"cation system, the optimal management of
negative production externalities while taking into account the intertemporal and spatial
aspects of the problem. To incorporate both aspects simultaneously, a two-stage
modeling approach is proposed where the solution of the spatial problem ("rst stage) is
optimized over time (second stage). As a result, it is possible to relate long-run and
short-run supply and input demand functions. Attention is given to runo!s from agricultural land leading to the contamination of a water body, in particular, the one of
phosphorus and the eutrophication of lakes. The employment of a land classi"cation
system supports a full-information approach and allows to address the optimal management of mineral fertilizer and manure based on zonal taxes, zonal permits, and zonal
standards which all vary over time. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: C61; H23; Q24; Q25
Keywords: Space time-dependent optimal control; Pollution tax; Tradable phosphorus
permits; Zoning
* Correspondence address: Department of Economics, University of Girona, Avda. Lluis SantaloH ,
17071 Girona, Spain. Fax: #34/972/418032; e-mail: goetz@econ.udg.es.
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 7 - 0
144 R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163
1. Introduction
Negative production externalities have been the concern of many economic
studies. In the case where the pollutants accumulate over time, these studies
were conducted within a dynamic framework. Besides the intertemporal aspect,
the heterogeneity of space may be of prime consideration for the economic
analysis of negative production externalities. Prominent negative externalities
from agricultural production like the runo! of phosphorus (P) or the leaching of
nitrogen (N) share the intertemporal and spatial aspect.
Hochman et al. (1977) analyzed the properties of optimal tax incentives
schemes to control pollution in a spatial setting. Both aspects, time and space,
were integrated in a modelling approach by Tomasi and Weise (1994). They
focused their analysis on the optimal spatial allocation of agricultural land
versus residential land over time. Yet, the agricultural land is often exogenously
given, and the optimal management of nonpoint-source pollution needs to
concentrate on the agricultural sector itself. Therefore, we de"ne agricultural
activities, which allow us to address agricultural policy questions. We base our
analysis on the full-information approach by employing a land classi"cation
system which accounts for di!erences in environmental vulnerability of locations and by considering improved monitoring technology to account for
heterogeneity in residue (pollutant) generation (Khanna et al., 1998). In this way
nonpoint-source pollution can be viewed as point-source pollution provided
that an adequate monitoring technology and a classi"cation system exist. Most
importantly, however, we present a modelling approach which consists of the
simultaneous solution of the micro-level (agricultural production) and macrolevel (aggregate supply and demand) over space and time. In particular, it allows
us to derive relationships between long-run and short-run supply and input
demand functions.
2. Environmental vulnerability and space
Traditionally economists think of space in the form of some distance measure.
However, two locations next to each other and equally distanced from a receiving water body may demonstrate a completely di!erent vulnerability to negative
agricultural production externalities. In the case of agricultural surface runo!s,
the reason may be that one location is sloping downhill towards the receiving
water body and the other one uphill. Wu and Segerson (1995) noted the
importance of distinct site characteristics for classifying the environmental
vulnerability of a site to potential pollution. The distance of a location with
respect to a point of reference is only one aspect among many others. However,
not only do site characteristics have to be considered but also characteristics of
the receiving water body. Thus, it is possible to relate the potential pollution to
R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163 145
the water body (e.g., milligram of bioavailable P per liter) and take a "rst step in
evaluating the potential damage.
To assess the environmental vulnerability of a location based on a land
classi"cation system, this paper takes the P index, developed under the patronage of the U.S. Department of Agriculture } The Natural Resources Conservation Service } (Sharpley, 1995), as an example. The P index allows to assign each
location within the catchment of the receiving water body the potential P load
resulting from the cultivation of a particular crop with a given crop management
practice while taking account of the particular characteristics of the water body.
For the sake of concreteness, we consider the example of a lake. Phosphorus
loadings of a lake above the sum of the permanent settling rate of P in the
sediment and of the P losses through the out#ow of water lead to the euthropication of a lake. This will result in excessive algae growth and a lack of dissolved
oxygen (Spulber and Sabbaghi, 1994). As a result, the water quality of the lake
declines which, in turn, reduces the biodiversity and recreational and commercial bene"ts of the lake. It also increases the costs for water treatment to provide
drinking water or to supply water for manufacturing processes (Holmes, 1988;
Spulber and Sabbaghi, 1994).
The P loading from agricultural land at a speci"c location within the watershed depends on the water delivery ratio, the water body sensitivity, the land
management, and on characteristics of the land, such as the soil P content and
the erosion and runo! potential. The latter characteristics are incorporated in
the P index by considering site-speci"c information such as soil texture, permeability of the soil, rainfall, the length and gradient of the slope, the cultivated
crop itself, and crop management and tillage practices. The factor soil P content
includes available soil P and P absorption capacity. The water delivery ratio
re#ects the share of P runo! transferred from the edge of the "eld to the lake,
whereas the water body sensitivity accounts for factors such as the degree of
surface mixing, the depth of the lake, water residence time, and the development
of reducing conditions at the water sediment surface (Sharpley and Halvorson,
1994).
Let the P load for a given type of fertilizer i, associated with a particular crop,
be denoted by a (o), where o represents a derived P index attributing a potential
i
P load to each location within the watershed, regardless of the type of fertilizer.
Intuitively, one expects locations close to the tributaries of the lake or to the lake
itself to have high potential P loads. Therefore, we associate a small o with
a high P load. The economic loss resulting from the deterioration of the water
quality of the lake is captured by the C(2) (twice continuously di!erentiable)
damage function d(s), where s presents the concentration of bioavailable P in the
lake. We propose that d@(s)'0 and dA(s)50.
One can think of other classi"cation systems with respect to site-speci"c
environmental vulnerability; for example, leaching of nitrate into groundwater
or degradation of a biotope resulting from agricultural production and the
146 R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163
application of plant protection agents. Analogous to the land classi"cation
system with respect to P runo!, these systems would classify the land based on
a variety of factors, where the measurement of the distance from a particular
point of reference is only one among many other factors. In this paper we focus
our analysis on the problem of P runo!s from agricultural land. However, to
generalize our results, we also discuss the most important "ndings in the context
of nitrogen leaching into the groundwater.
3. The economic model
Examples in the United States (Boggess et al., 1993) and Switzerland (Amt fuK r
Umweltschutz, 1993) show that lakes are often euthropic where the number of
large animal units per hectare is high. This can be explained by the fact that
P runo!s from agricultural land, where manure has been applied, contains
a high fraction of soluble phosphorus which is immediately available for biological uptake by the algaes. Moreover, manure application rates are based on the
management of nitrogen, leading in most cases to an increase in soil P in excess
of crop requirements. This is due to the generally lower ratio of N : P added in
manure than what is taken up by crops1 (Sharpley and Halvorson, 1994).
This situation suggests focus on the analysis on crop production and the
stocking rate per hectares in order to "nd the optimal P concentration of the
lake. For this purpose, let a (o) denote the P runo!s associated with the
1
application of mineral fertilizer u(o) and a (o) with the application of animal
2
manure b(o)oy(o), where y(o) presents the number of large animal units, o'1
the amount of manure per large animal unit, and b(o)3[0, 1] the share of the
available manure being applied. We assume for the case of P runo!s that
a (o)'a (o), ∀o as a result of the high P content and high bioavailability of the
2
1
manure compared to mineral fertilizer. Thus, the dynamics of the lake with
respect to o can be stated as
sR "(a (o)u(o)#a (o)b(o)oy(o))g(o),
(1)
1
2
where a dot over the variables s or j, to be introduced later, indicates the
operator d/do, and g(o) presents a density function. Furthermore, we assume
that the entire agricultural land within the watershed is cultivated and is equal
to one by an appropriate normalization procedure.
Eq. (1) can also be utilized to describe the dynamics of the concentration of
nitrate in the groundwater, where a (o), i"1, 2, presents a leaching function
i
1 For example, dairy, beef, swine, sheep and poultry manure has an average N : P ratio of 4.1 while
the N : P requirements of major grain crops is 7.3.
R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163 147
associated with fertilizer i. Mineral fertilizer carries nitrogen in its mineral form,
often in the form of nitrate. Thus, it is ready available by the plant, but the
mineral fertilizer is also more suspectable to the leaching of nitrate than manure,
where the nitrogen is predominately in its slow-releasing organic form. Therefore, the case of nitrate leaching suggests that a (o)(a (o), ∀o.
2
1
To complete the economic model, we like to introduce the C(2) crop production function given by q(z) with q@'0 and qA(0, and z,u(o)#b(o)oy(o). Let
the returns from livestock management be denoted by p y(o) and the associated
2
C(2) cost function by c (y(o)), where we assume c@ '0 and cA '0. The price
3
3
3
associated with the crop is denoted by p . Finally, we assume that a social
1
planner exists, who maximizes the present discounted net bene"ts from
agricultural production within the watershed of the lake while taking account of economic losses resulting from the accumulation of bioavailable P in
the lake.
4. The management of phosphorus runo4s from agricultural land
4.1. Spatial and intertemporal aspects
To obtain an analytical solution more easily, we propose the framework of
a two-stage optimal control problem. Basically, in the "rst stage we solve the
spatial problem by determining the optimal trajectories of u(o), b(o), y(o) and s(o)
for every o3[0, oN ], where oN indicates the upper limit of the P index. The value
function LB
LB
t"o
! a"0.
Lp
Lp
Lp
1
1
1
(25)
(26)
The results show that the short-run aggregate supply of livestock, as well as the
short-run aggregate demand for manure applied on the "eld and to be transported out of the watershed, is una!ected by an increase in the price for the crop.
However, the short-run aggregate demand for mineral fertilizer increases and,
consequently, the short-run aggregate supply of the crop goes up as well.
156 R.U. Goetz, D. Zilberman / Journal of Economic Dynamics & Control 24 (2000) 143}163
An increase in the returns of livestock management a!ects the short-run
aggregate supply and input demand functions in the following way:
oN
LyL
L>
"
g(o) do'0,
Lp
Lp
0 2
2
oN
LuL
L;
"
g(o) do(0,
Lp
Lp
0 2
2
P
P
(27)
oN
LB
LbK
LyL
a"
g(o) do'0,
oyL #bK o
Lp
Lp
Lp
2
0
2
2
B
PA
oN
L> LB
LB
t "o
! a"o 0g(o) do"0,
Lp
Lp
Lp
0
2
1
2
P
(28)
where the results of Eqs. (21) and (22) are used. An increase in p shows that the
2
short-run aggregate demand for mineral fertilizer decreases while the short-run
aggregate demand for manure applied on the "elds increases and the short run
aggregate demand for manure shipped out of the water catchment remains
unchanged. Therefore, the short-run aggregate supply of livestock increases. The
discussion of the e!ects of the variations in the P runo! coe$cients a and
1
a and the cost for mineral fertilizer c is presented in Appendix B.
2
1
6. Long-run analysis
Finally we consider the case ¹PR in order to study the comparative statics
of the long-run solution of problem (P2). Therefore, we propose to reformulate
problem (P2) in order to simplify the comparative static analysis. The revised
version of problem (P2) is given by
P
=
(