- Email: [email protected]
The optimality and feasibility of uniform air pollution controls
JOURNAL
OF ENVIROKMENTAL
The Optimality
and
ECONOMICS
AND
Feasibility WILLIAIV
School
of Management,
State
of Uniform A. HAMLEN, University
301-312
5,
hlANAGEhlENT
Air
( 19%)
Pollution
Controls
JR.
of New
York at Bu#aZo
Received March 27, 1978 An optimal control problem is developed which minimizes both the social damages of pollution concentrations and the private costs of emission abatement subject to a differential equation relating emissions and concentrations. A specific use of the model is to analyze the optimality and feasibility of uniform controls on emissions and concentrations across an air shed.
In several recent articles externality theorists have derived optimal air pollution control strategies which yield “uniform” controls over the relevant air shed. Baumol [l] uses an elementary equilibrium model to obtain a single uniform tax on all emitters. Baumol and Oates [2] obtain a similar result for a minimum cost “satisficing” model and Hamlen [5] uses a Pareto-Optimal general-equilibrium model to derive a single uniform tax. T. Tietenberg [la, 131 has led the way in criticizing these results for the case of air pollution because the models used fail to take into consideration the variations in atmospheric diffusion within the local air shed. These variations, claims Tietenberg, will lead to nonuniform air pollution controls, whether they be of the direct or decentralized type. In order to analyze this added complexity to the air pollution control problem one can follow Tietenberg’s theoretical approach [la] of breaking the urban setting into distinct sectors. The diffusion of pollution emitted in one sector as it reaches another sector is described by a single coefficient, Thus each sector is associated with any other sector by a set of diffusion coefficients. Solving a minimum cost programming problem in which each sector must meet prespecified pollution concentration levels subject to the given diffusion coefficients and technical abatement constraints could be expected to lead to a solution of nonuniform controls on emitters located in different sectors with the variation in controls highly dependent on the variations of diffusion coefficients. This method relies on the use of atmospheric diffusion models which yield a set of specific diffusion coefficients for each emission point and receptor point. These are known as ‘%wtor diffusion models.” There is an alternative approach to the problem that will yield a more general understanding of the effect of diffusion variations within the air shed on the optimal and feasible air pollution controls. This approach makes use of another concept of atmospheric diffusion. A column of air with a prespecified unit of area and height H, passes across an air shed at the wind velocity u(a). An illustration of such a column of air is given in Fig. 1. The pollution concentration at some 301 00950696/78/0054-0301$02.00/O Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
302
WILLIAM
A. HAhfLEN,
JR.
distance x in the windward direction is given by a(z). It is obtained by assuming a homogeneous vertical dispersion within the column of air and by integrating the emission rate into the column over the distance x while integrating out some loss rate over the distance. This simple model is widely accepted as the basis of more complex models used for prediction purposes.* When pollution concentrations are to be predicted over time, the meteorological parameters are usually assumed to be spatially fixed over the air shed. When different concentrations are to be predicted across the air shed the values of the meteorological parameters are set at the average values for the period of interest but allowed to vary over the distance X. This latter method is relevant to the problem evoked by Tientenberg. The purpose of this paper is to present a formulation of this diffusion process and incorporate it into a reasonably acceptable partial equilibrium, optimal control problem. Although the complete model is well suited to analyses of many problems in location theory, the present application will center around the optimality and feasibility of uniform air pollution controls and the deviations from uniform controls. In the second section below the above diffusion process is described in more detail. The third section contains an optimal control problem which minimizes social damages as a function of the pollution concentration and private abatement costs as a function of emissions. The objective function is constrained by the differential diffusion equation relating emission and concentrations. ATMOSPHERIC
The spatial by Eq. (1) :
variation
DIFFUSION
in pollution 5’
y(x’) = where q(x’) Q(x) H(x) U(X)
= = = = L(x) = x =
ACROSS
/[
AN
concentration
Q(2) -~__
URBAN
AIR
SHED
over an air shed is governed
1 0 H(x)u(x) - L(2) dx,
(1)
the pollution concentration per unit volume at X’ ; the per unit area emission rate at Z; the mixing height at point x ; the wind speed at z ; the loss rate of pollution per unit volume of air at x; the distance in the windward direction beginning at some initial upwind point x = 0 and directly crossing the urban air shed to x = 2 such that ~‘e(0, xl.
Following the concepts found in the new urban economics literature (see Mills and Mackinnon [9]), the air shed will be assumed to contain a well defined central business district denoted by CBD. In addition it will be assumed that the non-CBD areas of the air shed uniformly encircle the CBD and exhibit homogeneous characteristics for equal distances from the CBD in any direction. Adjustments in the resulting analysis would thus be necessary for port cities on the east and west coasts and Great Lakes as well as for Solow’s Long Narrow City [ll]. The windward distance .%Ywill begin at some point upwind of the CBD and directly cross the CBD at x = O.Fiz. 1 See Hamlen
[S],
Lettau
[7],
Turner
[15],
and Leahey
[S]
as examples.
UNIFORM Height
\
\air
AIR POLLUTION of
CONTROLS
303
surface
shed .
FIG. 1. Moving
column of air.
The variables, Q(X), H(X), U(Z) and L(z) may all vary over 5 and thus determine the variation in q(x). It is a reasonable assumption that the wind speed of the column of air is constant both as it approaches and leaves the CBD. This, of course, does not exclude a turbulent effect on the wind due to the rougher surface of the urban area. The wind direction is taken as the “prevailing” direction for the average time of interest. The daily average represents an important air pollution control period. Longer averages arc usually calculated from the estimates of daily average. The height of the column is called the mixing height (or mixing depth) and is usually taken as the upper bound within which polluted air will rise and then become trapped. Generally, in an urban environment there exists a “heatisland” effect which makes the ground level urban temperature warmer than that of the surrounding nonurban air mass. Thus the height to which polluted urban parcels of air rise depends on both the extent of the heat island effect and the actual “temperature profile” (change of temperature with height) of the surrounding air mass. In Fig. 2 a simple diagram of the mixing height is presented. The group level temperature of the nonurban air mass is given by 2’0 and the initial temperature of a warmer parcel of polluted urban air by To+. Since the polluted parcel is warmer than the surrounding temperature it begins to rise. As it rises it decreases in temperature at a constant rate known as the dry adiabatic lapse rate (0.98” c/m) (see Hamlen, 1976). When the parcel reaches a surrounding temperature equal to its own it tends to stop rising. This height is usually taken as the mixing height. In Fig. 2 it is shown as either HA or Ho depending on which respective temperature profile of the surroundin, w nonurban air is used. The temperature profile A corresponds to a bright day of sunshine for which the temperature of the surrounding nonurban air mass decreases with height. Tem-
WILLIAM
304 Height
above
the
A. HAMLEN,
JR.
surface
adiabatic
lapse
rate
/
FIG. 2. The mixing height.
perature profile B corresponds to an inversion situation. It occurs when the surface is cooler than the air above the surface and results in a low mixing height (inversion height) as shown in Fig. 2. This yields the worst pollution situations. Since the federal standards on air pollution deal primarily with these worst situations, the inversion condition is the relevant temperature profile to consider. One useful result becomes immediately apparent with the use of Fig. 2. If the heat island effect, is reduced and To+ moved closer to To, the mixing height will decrease. H. and K. Lettau [S] have found that the increased urban heat island effect of the inner city or CBD is attributable to the effective “storm sewer” system and the resulting effective drainage-off of most rain water so that evaporational cooling is significantly reduced for the urban area in comparison with the rural surrounding.2 Thus as the column of air approaches the CBD this effect would tend to increase H(X) and reduce it after the column moves away from the CBD. Hamlen [3,4] has found that another significant effect that reduces the mixing height away from the CBD is the increased surface roughness due to higher buildings. This tends to pivot slightly the temperature profile B toward B’ in Fig. 2. Both effects, however, increase the mixing height as the column of air approaches the CBD and decrease it thereafter.3 We can condense the two terms H(x) and U(X) in Eq. (1) to form a single term k(z) = H(x)u(a)
(2)
with dlc/dz > (<) 0
for
2’ < (>) 8.
(3)
The general loss function L(x) in (1) is more accurately defined by L[~(z)]. As the column of air increases in pollution concentration the natural counterforce of Fickian (isotropic) diffusion is engendered. This diffusion occurs in the vertical and crosswind directions while being insignificant relative to the wind speed in the windward direction. The diffusion process refers to the physical fact that molecules moving randomly tend to equalize the concentrations in adjoining parcels of air at a rate direct.ly proportional to the negative of the concentration gradient. The column of air moving directly across the CBD will in general be expected to exhibit a higher concentration than adjoining columns of air. There2 This result is found in section 6 of the paper by H. and K. Lettau. However, it was described to this author through personal communication with H. Lettau. *It has been known since the earliest diffusion model by D. B. Turner [14] that the worst pollution potential exists in the nonurban settings, during an inversion situation.
UNIFORM
Social
AIR
POLLUTION
305
CONTROLS
Damages
FIG.
3.
Social damage function.
fore a net flow of molecules in the crosswind directions will occur. In addition a net flow of molecules in the vertical direction above the mixing height will occur due to Fickian diffusion. The following conditions on L[Q(x)] are assumed: dL/dq(x)
> 0;
d2L,‘dq(:r)2 > 0.
(4)
The variables p(z) and Q(z) will be discussed further in the next section. The differential equation of (1) will be used in the analysis of the following section or* dq/dz = cj = &(x)/k(a) OPTIMAL
AND
FEASIBLE
- L/-q(x)].
(5)
CONTROLS
Optimal air pollution controls can be designed around a number of different strategies. The most acceptable approach from a theoretical standpoint is one with a utility analysis such as described by Baumol-Oates [Z, p. 361. This provides for the optimal level of air pollution to be determined within the solution. On a practical basis the utility effects are usually related to measurable monetary effects in the local economy. A simple expression of this is to divide the monetary effects into social damages caused by any level of pollution and private abatement costs caused by any reduction in emissions. The damage function is given by: Social Damages = O[I,(.C) ; e(z)]
(61
with aD/aq(z)
> 0;
a2D/aq(x)2 > 0.
This damage function is shown in Fig. 3. Damages increase at an increasing function of the pollution level. The parameter O(X) represents a shift parameter to account for nonhomogeneous characteristics at varying distances from the CBD. For example, if the damage to human health is of paramount importance, the damages would be expected to increase at distances closer to the CBD and yield the following assumptions of the derivations :
aD/ae,
d2D/dtP > (<) 0
for
it: < (>)a.
(7)
The private abatement costs across the air shed must be evaluated in terms of both per unit area costs and point-source costs. It will be assumed that the abatement cost for any single point source emission increases at an increasing rate for decreases in the emission level as shown in Fig. 4. Following the technique 4This is equivalent to the basic dynamic waste accumulation equation used by
V. Smith
[lo].
WILLIAM
306
A. HAMLEN, JR.
FIG. 4. Private cost function.
used by Smith [lo] in a temporal model, the per unit area will be assumed to contain N(X) similar point source emitters, Qp(x), all of which have equivalent cost functions. For example, a particular area might consist of a steel mill with fifteen stacks emitting equal levels of pollution. The total area emissions and area costs of abatement would then be equal to fifteen times the emissions and costs of one source. In addition a shift parameter y(z) can be included in the point source cost function to represent systematic changes in industries for increased distances from the CBD. This gives the following cost function and assumptions on the derivatives : Total private
costs = TC[Q(X)]
= N (x)C[Q,(X)
; y(z)]
(8)
with ac __ < 0, ac, (4
a2c ___ aQ,W
> 0,
and
dC __ > 0. Y (2)
At this point in the analysis there will be no a priori assumption made on the derivatives of the shift parameter over the distance x. Replacing Q(X) in (5) by N(x)&,(x), the optimal air pollution control problem can be defined as Maximize
-
iN(xK'CQ,(x); r(z)1 + Rub>; ~b)lldz
(94
subject to
(9b) m> where Q,(X) = the control variable n(x) = the state variable. Problem (9) is the spatial equivalent to the standard fixed time, variable end point problem of optimal control, where the Hamiltonian is strictly concave in the admissible control function Qp(x) and yields the following necessary condi-
UNIFORM
tions for an optimal
AIR POLLUTION
CONTROLS
307
solution:
iWx), &p(x), ~1 = - (N(x)CCQ,(x) ; -t(x)1 + Nqb) ; eb>lt + B(z)(CN(x)Q,(2)lIc(x)l - -WMl~ dq p,-=---dX
SfzL dX
N(x)&,(x) k(x) dD -+dq (xl
-uq(x)l
OW (lob)
aL
(1Oc)
dq (xl
~ = -N(x)[dC/dQ,(z)l+ a&, (xl B(X) = 6
CB(X)N(X)I~(-Vl
= 0
(104 We>
where B(X) = the multiplier associated with the differential equation; (lob) and (1Oe) is the transversality condition. There are several ways of analyzing these necessary conditions. Examination of B(x) would yield insight into the decentralized control of emissions by taxes or subsidies. However, equivalent results will be obtained with the direct controls on pollution concentration and emissions and these will be examined here. The problem evoked by Tietenberg is that controls, whether direct or decentralized, cannot be uniform over the air shed. Thus the phase diagram of interest is with respect to the q(x) and Qp(a) axes containing the locus of dq/dn: = 0 and dQ,/dx = 0. Using the feasibility conditions, Eq. (lob), @ = 0 results in the following 1 rium locus, given that the inverse function rule for L is k(x), Q&)1 eclul'1% satisfied : q(x) = L-l[N(~)/k(~);
e(x)].Q,(.rj.
(11)
Given that dL/dq(z) > 0, it can be seen that the phase diagram of (11) is represented by Fig. 5 with the dynamic adjustment reactions described by the arrows. Points above the locus imply that new emissions are being sufficiently lost through Fickian diffusion and concentrations will decrease over the distance x. The opposite case occurs for points below the locus. s(x)
Q(x) FIG.
5. The
feasibility
conditions.
WILLIAM
308
A. HAMLEN,
JR.
4(x) i(x) =0 1: FIG. 6. Optimality
From the optimality
condition
conditions
z < O.Sg,
(1Od) :
B (a) = [K/rXJ,(x)]k
(x).
Therefore, dB/dx Setting
= k(x)
a2c -4,
CPC
aQ*(xl"
+
.f?
ac --.--
dk(x)
1 [a&,(x) ax 1 +
a&,(x)~Y (2) dx
(12)
(12) equal to (10~) yields: &, =
i
[D’(q(x);e(x)
+ L’(q(x))]/k(x)
- C’dk/dx
where D’ = aD/aq(x),
L’ = c?L/dq(x),
aC’ - ___.h(x)
dy dx
(13)
C’ = aC/aQ,(x).
First by setting d-y/dx = 0 it is possible to analyze the case where there are no systematic variations in the types of industries over the distance to the CBD. Setting Q, = 0, the [q(s), QP(x)] 1ecus depends on the value of dlc/dx. Using condition (3) the problem must be divided into two stages ; the distance x = 0 to z = 0.5x and then the distance x = 0.5z to 12:= J?. In the former, case 1, the column of air is approaching the CBD and the phase diagram is given in Fig. 6. It is clear that since q(x) and Q(r) must be nonnegative, the optimal control of emissions requires permitting increased emissions at closer distances to the CBD, or Q, > 0. Figures 5 and 6 are combined in Fig. 7 to give the full
Figure Ehase
Diagram
7 x < .5X
FIG. 7. Phase diagram z < 0.52.
UNIFORM
AIR POLLUTION
FIG. 8. Optimality
condition
CONTROLS
309
z > 0.59.
feasible and optimal dynamic adjustments with increasing q(z) and Q,(z) over the distance x. This also proves that the control standards cannot be uniform over the distance 0 to O.SJ?. The same result is obtained by Tietenberg. In the second case where the column of air moves away from the CBD and k(z) decreases with distance the [q(a), QP(z)] locus for Q, = 0 is given in Fig. 8 along with the dynamic adjustment equations. This is then combined with Fig. 5 to form a complete phase diagram in Fig. 9. It is found that there exists a stable branch which would lead to asymptotically uniform controls with @ = 0, Q, = 0. It is also possible that any combination of nonuniform controls can be obtained depending on the initial values of q(O.52) and &,(0.53). Next, it is interesting to examine the effect of dropping the assumption that &y/&c = 0. Using (13) it can be seen that, if dr/dz > 0, the Q = 0 locus in Fig. 7 can be shifted to the right and possibly cross the q(2) axis. In Fig. 10 two possible shifts are illustrated, & = 0 and Q’ = 0. Only with a very strong shift such as Q’ = 0 is a stable branch found leading to optimal and feasible uniform controls on the upwind side of the CBD. This corresponds to the case of industries with higher marginal emission costs being near the CBD. On the downwind side of the CBD the curve & = 0 would be shifted to the right (such as in Fig. 9) with the same results as discussed above. Next consider a modification in problem (9). Assume that some upper bound on emissions is imposed such that the following constraint is added :
Q(x) FIG. 9. Phase diagram z > 0.52.
310
WILLIAM
A.
HAMLEN,
Stable
FIG.
10. Phase
diagram
JR.
Branch
a~ < 0.59,
&/dx
> 0.
This will in turn modify the result (10~) to the following: &!L ClX
dD -+--
dL
aq (xl
&I (xl
P.
(1Oc)’
As long as the optimal and feasible q(x) is less than Q t,he multiplier p will be zero. However, when q(x) reaches Q a re-examination of (13) shows that &, < 0. A number of situations can be analyzed again assuming d-y/dx = 0, and x < 0.5x. The emissions and pollution concentration will increase as the CBD is approached but suddenly reach an unstable condition when q(z) = Q. This is shown in Fig. 11. When x > 0.5x, two cases may occur depending on whether q is less than or greater than the level of q(x) where Q! = 0, @ = 0. Both cases are illustrated in Fig. 12 and left to the reader for further analysis. CONCLUSION The above presentation is only an introduction to the many possible insights which can be gained with the use of the above model. For example, comparisons between different urban settings can be made by changing N(z) and dH/dx, corresponding to larger cities or cities with worse meteorological conditions. Also analyses of the decentralized taxes is possible through study of the optimal and feasible values of B (x). It should be emphasized that the results of other diffusion models, such as vector models, should be similar since the evaluation of a diffusion model is first
FIG. .ll.
Phase
diagram
with
(I, z < 0.58.
UNIFORM
AIR POLLUTION
CONTROLS
311
Q(x) FIG. 12. Phase
diagram
with
Q and
g’, z > 0.55.
made on its ability to predict pollution concentrations for given emission levels and meteorological parameters. Other diffusion models have the same basic foundations and the one described in this study has been widely and successfully used. With respect to Tietenberg’s specific problem, it has been shown that uniform controls either of the concentration levels or the emission levels are unlikely but not impossible over some region of the urban setting. The above model can also be made more elaborate by replacing the cost and damge functions by the utility functions of the individuals living and working within the region. The sum of the weighted ut.ility functions such as used by Hamlen [5] would then be a function of the distance z. This is a more accurate approach than recommended by Tietenberg [13] in that the level of pollution experienced by each individual would be unique as well as the contribution of each emitter. ACKNOWLEDGMENTS This author initiated continued work under York. A communication diffusion process.
the idea for this a grant from The from H. Lettau
paper while working under a grant from A.T.&T. and Research Foundation of the State University of New was beneficial in clarifying an important point of the
REFERENCES 1. W. 2. 3. 4. 5. 6. 7.
8.
Baumol, On Taxation and the Control of Externalities, Amer. Econ. Rev. 62, 307-322 (1972). W. Baumol and W. Oates, “The Theory of Environmental Policy,” Prentice Hall, Inc., Englewood Cliffs (1975). W. Hamlen, A Model to Predict the Pollution Potential with the Use of Airways Surface Observation data, Atmospheric Environment 10, 855-863 (1976). W. Hamlen, Clean Air as a Public Utility, Working Paper No. 194, School of Management, State University of New York at Buffalo (1974). W. Hamlen, The Quasi-Optimal Price of Undepletable Externalities, The Bell J. Econ. 8, No. 1 (1977). D. M. Leahey, An Advective Model for Predicting Air Pollution within an Urban Heat Island with Applications to New York City, J. Air Poll&. Control Ass. 7, 22 (1972). H. Lettau, 2. Physical and Meteorological Basis for Mathematical Models of Urban Diffusion Processes. Proceedings of Symposium on Multiple-Source Urban Di$usion Models, Research Triangle Park: E.P.A. (1970). H. Lettau and K. Lettau, Regional Climatonomy of Tundra and Boreal Forests in Canada, Climate of the Arctic, pp. 209-221 (1975), P roceedings of the 24th Alaska Science Conference, University of Alaska, 1973.
312
WILLIAM
A. GAMLEN, JR.
9. E. Mills and J. MacKinnon, Notes on the New Urban Economics, The Bell J. Econ., 593-601 (1973). 10. V. Smith, Dynamic Waste Accumulation: Disposal Versus Recycling, &art. J. Econ. 86, 600-616 (1972). 11. R. Solow and W. Vickrey, Land Use in a Long Narrow City, J. Econ. Theory 3, No. 4,430-447 (1971). 12. T. H. Tietenberg, On Taxation and the Control of Externalities : Comment, Amer. Econ. Rev., 462466 (1974). 13. T. H. Tietenberg, The Quasi-Optimal Price of Undepletable Externalities: Comment, The Bell J. Econ. (forthcoming). 14. D. B. Turner, A Diffusion Model for an Urban Area, J. Appl. Meteor., 3, No. 1 (1964). 15. D. B. Turner, Workbook of Atmospheric Dispersion Estimates. Research Triangle Park, North Carolina: E.P.A. (1970).
OF ENVIROKMENTAL
The Optimality
and
ECONOMICS
AND
Feasibility WILLIAIV
School
of Management,
State
of Uniform A. HAMLEN, University
301-312
5,
hlANAGEhlENT
Air
( 19%)
Pollution
Controls
JR.
of New
York at Bu#aZo
Received March 27, 1978 An optimal control problem is developed which minimizes both the social damages of pollution concentrations and the private costs of emission abatement subject to a differential equation relating emissions and concentrations. A specific use of the model is to analyze the optimality and feasibility of uniform controls on emissions and concentrations across an air shed.
In several recent articles externality theorists have derived optimal air pollution control strategies which yield “uniform” controls over the relevant air shed. Baumol [l] uses an elementary equilibrium model to obtain a single uniform tax on all emitters. Baumol and Oates [2] obtain a similar result for a minimum cost “satisficing” model and Hamlen [5] uses a Pareto-Optimal general-equilibrium model to derive a single uniform tax. T. Tietenberg [la, 131 has led the way in criticizing these results for the case of air pollution because the models used fail to take into consideration the variations in atmospheric diffusion within the local air shed. These variations, claims Tietenberg, will lead to nonuniform air pollution controls, whether they be of the direct or decentralized type. In order to analyze this added complexity to the air pollution control problem one can follow Tietenberg’s theoretical approach [la] of breaking the urban setting into distinct sectors. The diffusion of pollution emitted in one sector as it reaches another sector is described by a single coefficient, Thus each sector is associated with any other sector by a set of diffusion coefficients. Solving a minimum cost programming problem in which each sector must meet prespecified pollution concentration levels subject to the given diffusion coefficients and technical abatement constraints could be expected to lead to a solution of nonuniform controls on emitters located in different sectors with the variation in controls highly dependent on the variations of diffusion coefficients. This method relies on the use of atmospheric diffusion models which yield a set of specific diffusion coefficients for each emission point and receptor point. These are known as ‘%wtor diffusion models.” There is an alternative approach to the problem that will yield a more general understanding of the effect of diffusion variations within the air shed on the optimal and feasible air pollution controls. This approach makes use of another concept of atmospheric diffusion. A column of air with a prespecified unit of area and height H, passes across an air shed at the wind velocity u(a). An illustration of such a column of air is given in Fig. 1. The pollution concentration at some 301 00950696/78/0054-0301$02.00/O Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
302
WILLIAM
A. HAhfLEN,
JR.
distance x in the windward direction is given by a(z). It is obtained by assuming a homogeneous vertical dispersion within the column of air and by integrating the emission rate into the column over the distance x while integrating out some loss rate over the distance. This simple model is widely accepted as the basis of more complex models used for prediction purposes.* When pollution concentrations are to be predicted over time, the meteorological parameters are usually assumed to be spatially fixed over the air shed. When different concentrations are to be predicted across the air shed the values of the meteorological parameters are set at the average values for the period of interest but allowed to vary over the distance X. This latter method is relevant to the problem evoked by Tientenberg. The purpose of this paper is to present a formulation of this diffusion process and incorporate it into a reasonably acceptable partial equilibrium, optimal control problem. Although the complete model is well suited to analyses of many problems in location theory, the present application will center around the optimality and feasibility of uniform air pollution controls and the deviations from uniform controls. In the second section below the above diffusion process is described in more detail. The third section contains an optimal control problem which minimizes social damages as a function of the pollution concentration and private abatement costs as a function of emissions. The objective function is constrained by the differential diffusion equation relating emission and concentrations. ATMOSPHERIC
The spatial by Eq. (1) :
variation
DIFFUSION
in pollution 5’
y(x’) = where q(x’) Q(x) H(x) U(X)
= = = = L(x) = x =
ACROSS
/[
AN
concentration
Q(2) -~__
URBAN
AIR
SHED
over an air shed is governed
1 0 H(x)u(x) - L(2) dx,
(1)
the pollution concentration per unit volume at X’ ; the per unit area emission rate at Z; the mixing height at point x ; the wind speed at z ; the loss rate of pollution per unit volume of air at x; the distance in the windward direction beginning at some initial upwind point x = 0 and directly crossing the urban air shed to x = 2 such that ~‘e(0, xl.
Following the concepts found in the new urban economics literature (see Mills and Mackinnon [9]), the air shed will be assumed to contain a well defined central business district denoted by CBD. In addition it will be assumed that the non-CBD areas of the air shed uniformly encircle the CBD and exhibit homogeneous characteristics for equal distances from the CBD in any direction. Adjustments in the resulting analysis would thus be necessary for port cities on the east and west coasts and Great Lakes as well as for Solow’s Long Narrow City [ll]. The windward distance .%Ywill begin at some point upwind of the CBD and directly cross the CBD at x = O.Fiz. 1 See Hamlen
[S],
Lettau
[7],
Turner
[15],
and Leahey
[S]
as examples.
UNIFORM Height
\
\air
AIR POLLUTION of
CONTROLS
303
surface
shed .
FIG. 1. Moving
column of air.
The variables, Q(X), H(X), U(Z) and L(z) may all vary over 5 and thus determine the variation in q(x). It is a reasonable assumption that the wind speed of the column of air is constant both as it approaches and leaves the CBD. This, of course, does not exclude a turbulent effect on the wind due to the rougher surface of the urban area. The wind direction is taken as the “prevailing” direction for the average time of interest. The daily average represents an important air pollution control period. Longer averages arc usually calculated from the estimates of daily average. The height of the column is called the mixing height (or mixing depth) and is usually taken as the upper bound within which polluted air will rise and then become trapped. Generally, in an urban environment there exists a “heatisland” effect which makes the ground level urban temperature warmer than that of the surrounding nonurban air mass. Thus the height to which polluted urban parcels of air rise depends on both the extent of the heat island effect and the actual “temperature profile” (change of temperature with height) of the surrounding air mass. In Fig. 2 a simple diagram of the mixing height is presented. The group level temperature of the nonurban air mass is given by 2’0 and the initial temperature of a warmer parcel of polluted urban air by To+. Since the polluted parcel is warmer than the surrounding temperature it begins to rise. As it rises it decreases in temperature at a constant rate known as the dry adiabatic lapse rate (0.98” c/m) (see Hamlen, 1976). When the parcel reaches a surrounding temperature equal to its own it tends to stop rising. This height is usually taken as the mixing height. In Fig. 2 it is shown as either HA or Ho depending on which respective temperature profile of the surroundin, w nonurban air is used. The temperature profile A corresponds to a bright day of sunshine for which the temperature of the surrounding nonurban air mass decreases with height. Tem-
WILLIAM
304 Height
above
the
A. HAMLEN,
JR.
surface
adiabatic
lapse
rate
/
FIG. 2. The mixing height.
perature profile B corresponds to an inversion situation. It occurs when the surface is cooler than the air above the surface and results in a low mixing height (inversion height) as shown in Fig. 2. This yields the worst pollution situations. Since the federal standards on air pollution deal primarily with these worst situations, the inversion condition is the relevant temperature profile to consider. One useful result becomes immediately apparent with the use of Fig. 2. If the heat island effect, is reduced and To+ moved closer to To, the mixing height will decrease. H. and K. Lettau [S] have found that the increased urban heat island effect of the inner city or CBD is attributable to the effective “storm sewer” system and the resulting effective drainage-off of most rain water so that evaporational cooling is significantly reduced for the urban area in comparison with the rural surrounding.2 Thus as the column of air approaches the CBD this effect would tend to increase H(X) and reduce it after the column moves away from the CBD. Hamlen [3,4] has found that another significant effect that reduces the mixing height away from the CBD is the increased surface roughness due to higher buildings. This tends to pivot slightly the temperature profile B toward B’ in Fig. 2. Both effects, however, increase the mixing height as the column of air approaches the CBD and decrease it thereafter.3 We can condense the two terms H(x) and U(X) in Eq. (1) to form a single term k(z) = H(x)u(a)
(2)
with dlc/dz > (<) 0
for
2’ < (>) 8.
(3)
The general loss function L(x) in (1) is more accurately defined by L[~(z)]. As the column of air increases in pollution concentration the natural counterforce of Fickian (isotropic) diffusion is engendered. This diffusion occurs in the vertical and crosswind directions while being insignificant relative to the wind speed in the windward direction. The diffusion process refers to the physical fact that molecules moving randomly tend to equalize the concentrations in adjoining parcels of air at a rate direct.ly proportional to the negative of the concentration gradient. The column of air moving directly across the CBD will in general be expected to exhibit a higher concentration than adjoining columns of air. There2 This result is found in section 6 of the paper by H. and K. Lettau. However, it was described to this author through personal communication with H. Lettau. *It has been known since the earliest diffusion model by D. B. Turner [14] that the worst pollution potential exists in the nonurban settings, during an inversion situation.
UNIFORM
Social
AIR
POLLUTION
305
CONTROLS
Damages
FIG.
3.
Social damage function.
fore a net flow of molecules in the crosswind directions will occur. In addition a net flow of molecules in the vertical direction above the mixing height will occur due to Fickian diffusion. The following conditions on L[Q(x)] are assumed: dL/dq(x)
> 0;
d2L,‘dq(:r)2 > 0.
(4)
The variables p(z) and Q(z) will be discussed further in the next section. The differential equation of (1) will be used in the analysis of the following section or* dq/dz = cj = &(x)/k(a) OPTIMAL
AND
FEASIBLE
- L/-q(x)].
(5)
CONTROLS
Optimal air pollution controls can be designed around a number of different strategies. The most acceptable approach from a theoretical standpoint is one with a utility analysis such as described by Baumol-Oates [Z, p. 361. This provides for the optimal level of air pollution to be determined within the solution. On a practical basis the utility effects are usually related to measurable monetary effects in the local economy. A simple expression of this is to divide the monetary effects into social damages caused by any level of pollution and private abatement costs caused by any reduction in emissions. The damage function is given by: Social Damages = O[I,(.C) ; e(z)]
(61
with aD/aq(z)
> 0;
a2D/aq(x)2 > 0.
This damage function is shown in Fig. 3. Damages increase at an increasing function of the pollution level. The parameter O(X) represents a shift parameter to account for nonhomogeneous characteristics at varying distances from the CBD. For example, if the damage to human health is of paramount importance, the damages would be expected to increase at distances closer to the CBD and yield the following assumptions of the derivations :
aD/ae,
d2D/dtP > (<) 0
for
it: < (>)a.
(7)
The private abatement costs across the air shed must be evaluated in terms of both per unit area costs and point-source costs. It will be assumed that the abatement cost for any single point source emission increases at an increasing rate for decreases in the emission level as shown in Fig. 4. Following the technique 4This is equivalent to the basic dynamic waste accumulation equation used by
V. Smith
[lo].
WILLIAM
306
A. HAMLEN, JR.
FIG. 4. Private cost function.
used by Smith [lo] in a temporal model, the per unit area will be assumed to contain N(X) similar point source emitters, Qp(x), all of which have equivalent cost functions. For example, a particular area might consist of a steel mill with fifteen stacks emitting equal levels of pollution. The total area emissions and area costs of abatement would then be equal to fifteen times the emissions and costs of one source. In addition a shift parameter y(z) can be included in the point source cost function to represent systematic changes in industries for increased distances from the CBD. This gives the following cost function and assumptions on the derivatives : Total private
costs = TC[Q(X)]
= N (x)C[Q,(X)
; y(z)]
(8)
with ac __ < 0, ac, (4
a2c ___ aQ,W
> 0,
and
dC __ > 0. Y (2)
At this point in the analysis there will be no a priori assumption made on the derivatives of the shift parameter over the distance x. Replacing Q(X) in (5) by N(x)&,(x), the optimal air pollution control problem can be defined as Maximize
-
iN(xK'CQ,(x); r(z)1 + Rub>; ~b)lldz
(94
subject to
(9b) m> where Q,(X) = the control variable n(x) = the state variable. Problem (9) is the spatial equivalent to the standard fixed time, variable end point problem of optimal control, where the Hamiltonian is strictly concave in the admissible control function Qp(x) and yields the following necessary condi-
UNIFORM
tions for an optimal
AIR POLLUTION
CONTROLS
307
solution:
iWx), &p(x), ~1 = - (N(x)CCQ,(x) ; -t(x)1 + Nqb) ; eb>lt + B(z)(CN(x)Q,(2)lIc(x)l - -WMl~ dq p,-=---dX
SfzL dX
N(x)&,(x) k(x) dD -+dq (xl
-uq(x)l
OW (lob)
aL
(1Oc)
dq (xl
~ = -N(x)[dC/dQ,(z)l+ a&, (xl B(X) = 6
CB(X)N(X)I~(-Vl
= 0
(104 We>
where B(X) = the multiplier associated with the differential equation; (lob) and (1Oe) is the transversality condition. There are several ways of analyzing these necessary conditions. Examination of B(x) would yield insight into the decentralized control of emissions by taxes or subsidies. However, equivalent results will be obtained with the direct controls on pollution concentration and emissions and these will be examined here. The problem evoked by Tietenberg is that controls, whether direct or decentralized, cannot be uniform over the air shed. Thus the phase diagram of interest is with respect to the q(x) and Qp(a) axes containing the locus of dq/dn: = 0 and dQ,/dx = 0. Using the feasibility conditions, Eq. (lob), @ = 0 results in the following 1 rium locus, given that the inverse function rule for L is k(x), Q&)1 eclul'1% satisfied : q(x) = L-l[N(~)/k(~);
e(x)].Q,(.rj.
(11)
Given that dL/dq(z) > 0, it can be seen that the phase diagram of (11) is represented by Fig. 5 with the dynamic adjustment reactions described by the arrows. Points above the locus imply that new emissions are being sufficiently lost through Fickian diffusion and concentrations will decrease over the distance x. The opposite case occurs for points below the locus. s(x)
Q(x) FIG.
5. The
feasibility
conditions.
WILLIAM
308
A. HAMLEN,
JR.
4(x) i(x) =0 1: FIG. 6. Optimality
From the optimality
condition
conditions
z < O.Sg,
(1Od) :
B (a) = [K/rXJ,(x)]k
(x).
Therefore, dB/dx Setting
= k(x)
a2c -4,
CPC
aQ*(xl"
+
.f?
ac --.--
dk(x)
1 [a&,(x) ax 1 +
a&,(x)~Y (2) dx
(12)
(12) equal to (10~) yields: &, =
i
[D’(q(x);e(x)
+ L’(q(x))]/k(x)
- C’dk/dx
where D’ = aD/aq(x),
L’ = c?L/dq(x),
aC’ - ___.h(x)
dy dx
(13)
C’ = aC/aQ,(x).
First by setting d-y/dx = 0 it is possible to analyze the case where there are no systematic variations in the types of industries over the distance to the CBD. Setting Q, = 0, the [q(s), QP(x)] 1ecus depends on the value of dlc/dx. Using condition (3) the problem must be divided into two stages ; the distance x = 0 to z = 0.5x and then the distance x = 0.5z to 12:= J?. In the former, case 1, the column of air is approaching the CBD and the phase diagram is given in Fig. 6. It is clear that since q(x) and Q(r) must be nonnegative, the optimal control of emissions requires permitting increased emissions at closer distances to the CBD, or Q, > 0. Figures 5 and 6 are combined in Fig. 7 to give the full
Figure Ehase
Diagram
7 x < .5X
FIG. 7. Phase diagram z < 0.52.
UNIFORM
AIR POLLUTION
FIG. 8. Optimality
condition
CONTROLS
309
z > 0.59.
feasible and optimal dynamic adjustments with increasing q(z) and Q,(z) over the distance x. This also proves that the control standards cannot be uniform over the distance 0 to O.SJ?. The same result is obtained by Tietenberg. In the second case where the column of air moves away from the CBD and k(z) decreases with distance the [q(a), QP(z)] locus for Q, = 0 is given in Fig. 8 along with the dynamic adjustment equations. This is then combined with Fig. 5 to form a complete phase diagram in Fig. 9. It is found that there exists a stable branch which would lead to asymptotically uniform controls with @ = 0, Q, = 0. It is also possible that any combination of nonuniform controls can be obtained depending on the initial values of q(O.52) and &,(0.53). Next, it is interesting to examine the effect of dropping the assumption that &y/&c = 0. Using (13) it can be seen that, if dr/dz > 0, the Q = 0 locus in Fig. 7 can be shifted to the right and possibly cross the q(2) axis. In Fig. 10 two possible shifts are illustrated, & = 0 and Q’ = 0. Only with a very strong shift such as Q’ = 0 is a stable branch found leading to optimal and feasible uniform controls on the upwind side of the CBD. This corresponds to the case of industries with higher marginal emission costs being near the CBD. On the downwind side of the CBD the curve & = 0 would be shifted to the right (such as in Fig. 9) with the same results as discussed above. Next consider a modification in problem (9). Assume that some upper bound on emissions is imposed such that the following constraint is added :
Q(x) FIG. 9. Phase diagram z > 0.52.
310
WILLIAM
A.
HAMLEN,
Stable
FIG.
10. Phase
diagram
JR.
Branch
a~ < 0.59,
&/dx
> 0.
This will in turn modify the result (10~) to the following: &!L ClX
dD -+--
dL
aq (xl
&I (xl
P.
(1Oc)’
As long as the optimal and feasible q(x) is less than Q t,he multiplier p will be zero. However, when q(x) reaches Q a re-examination of (13) shows that &, < 0. A number of situations can be analyzed again assuming d-y/dx = 0, and x < 0.5x. The emissions and pollution concentration will increase as the CBD is approached but suddenly reach an unstable condition when q(z) = Q. This is shown in Fig. 11. When x > 0.5x, two cases may occur depending on whether q is less than or greater than the level of q(x) where Q! = 0, @ = 0. Both cases are illustrated in Fig. 12 and left to the reader for further analysis. CONCLUSION The above presentation is only an introduction to the many possible insights which can be gained with the use of the above model. For example, comparisons between different urban settings can be made by changing N(z) and dH/dx, corresponding to larger cities or cities with worse meteorological conditions. Also analyses of the decentralized taxes is possible through study of the optimal and feasible values of B (x). It should be emphasized that the results of other diffusion models, such as vector models, should be similar since the evaluation of a diffusion model is first
FIG. .ll.
Phase
diagram
with
(I, z < 0.58.
UNIFORM
AIR POLLUTION
CONTROLS
311
Q(x) FIG. 12. Phase
diagram
with
Q and
g’, z > 0.55.
made on its ability to predict pollution concentrations for given emission levels and meteorological parameters. Other diffusion models have the same basic foundations and the one described in this study has been widely and successfully used. With respect to Tietenberg’s specific problem, it has been shown that uniform controls either of the concentration levels or the emission levels are unlikely but not impossible over some region of the urban setting. The above model can also be made more elaborate by replacing the cost and damge functions by the utility functions of the individuals living and working within the region. The sum of the weighted ut.ility functions such as used by Hamlen [5] would then be a function of the distance z. This is a more accurate approach than recommended by Tietenberg [13] in that the level of pollution experienced by each individual would be unique as well as the contribution of each emitter. ACKNOWLEDGMENTS This author initiated continued work under York. A communication diffusion process.
the idea for this a grant from The from H. Lettau
paper while working under a grant from A.T.&T. and Research Foundation of the State University of New was beneficial in clarifying an important point of the
REFERENCES 1. W. 2. 3. 4. 5. 6. 7.
8.
Baumol, On Taxation and the Control of Externalities, Amer. Econ. Rev. 62, 307-322 (1972). W. Baumol and W. Oates, “The Theory of Environmental Policy,” Prentice Hall, Inc., Englewood Cliffs (1975). W. Hamlen, A Model to Predict the Pollution Potential with the Use of Airways Surface Observation data, Atmospheric Environment 10, 855-863 (1976). W. Hamlen, Clean Air as a Public Utility, Working Paper No. 194, School of Management, State University of New York at Buffalo (1974). W. Hamlen, The Quasi-Optimal Price of Undepletable Externalities, The Bell J. Econ. 8, No. 1 (1977). D. M. Leahey, An Advective Model for Predicting Air Pollution within an Urban Heat Island with Applications to New York City, J. Air Poll&. Control Ass. 7, 22 (1972). H. Lettau, 2. Physical and Meteorological Basis for Mathematical Models of Urban Diffusion Processes. Proceedings of Symposium on Multiple-Source Urban Di$usion Models, Research Triangle Park: E.P.A. (1970). H. Lettau and K. Lettau, Regional Climatonomy of Tundra and Boreal Forests in Canada, Climate of the Arctic, pp. 209-221 (1975), P roceedings of the 24th Alaska Science Conference, University of Alaska, 1973.
312
WILLIAM
A. GAMLEN, JR.
9. E. Mills and J. MacKinnon, Notes on the New Urban Economics, The Bell J. Econ., 593-601 (1973). 10. V. Smith, Dynamic Waste Accumulation: Disposal Versus Recycling, &art. J. Econ. 86, 600-616 (1972). 11. R. Solow and W. Vickrey, Land Use in a Long Narrow City, J. Econ. Theory 3, No. 4,430-447 (1971). 12. T. H. Tietenberg, On Taxation and the Control of Externalities : Comment, Amer. Econ. Rev., 462466 (1974). 13. T. H. Tietenberg, The Quasi-Optimal Price of Undepletable Externalities: Comment, The Bell J. Econ. (forthcoming). 14. D. B. Turner, A Diffusion Model for an Urban Area, J. Appl. Meteor., 3, No. 1 (1964). 15. D. B. Turner, Workbook of Atmospheric Dispersion Estimates. Research Triangle Park, North Carolina: E.P.A. (1970).