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Spatial economic theory of pollution control
JOURNAL
OF ENVIRONMENTAL
Spatial
ECONOMICS
AND
Economic
hlANAC;tMENT
Theory
3, 16-28 ( 1976)
of Pollution
Control
VIJAY K. MATHUR’ Departmen
of Economics, Cleveland State LiAw-sity,
Cleveland, Ohio 44115
Received April 9, 1974; revised September 18, 1975 In February 1972 the Nixon Administration proposed a tax on sulfur oxide emissions, beyond the Federal standards, of electric power plants. It was hoped that such a tax would discourage power plant locations in heavily polluted areas like urban areas. Assuming that such a tax varies over space because pollution is not invariant over space, the spatial theory of the firm is presented in this paper cast doubt on the efficacy of such a polltuion tax to achieve the desired goal of the administration. In particular it is shown that the costminimizing spatial firm would abate its pollution in response to a tax by either changing its location or by reducing waste through process change or by using a transportable abatement good if the tax savings due to the change in marginal abatement through location change per unit change in abatement is greater than or equal to transport rate on the abatement good. If the goal of the firm is to maximize profits, the desired outcome would follow if in addition to the fulfillment of the above condition the percentage change in the delivered price of raw material (situated at one end of the linear location space) per unit distance is greater than or equal to the percentage change in net marginal revenue, i.e., marginal revenue net of transport cost per unit of output. In simple terms, the conclusions of this paper raise doubts regarding the effectiveness of a pollution tax in curtailing pollution of a firm which is operating in a space economy. This outcome is peculiar to a spatial firm. Sufficient conditions also have been obtained when land input is included in the production function of the firm.
Many economists are in agreement that a pollution tax is the efficient method to achieve a desired level of pollution, as opposed to regulation.2 It is also contended that such a tax, ceteris paribus, will induce the firm operating in a spaceless economy to abate its wastes through whatever form of abatement practiced. However, there is one problem in a tax scheme. Tietenberg [9] has shown in his general equilibrium model that not all firms will face the same per unit tax in general. One reason for this outcome, out of others, is that marginal contribution of waste to pollution varies spatially. This conclusion is the starting point in this paper. Assuming that a tax on pollution varies spatially, it has yet to be demonstrated that the desired outcome will follow in the case of a firm operating in a space-economy. For example, it is taken for granted that a pollution tax would necessarily discourage the location of a firm in heavily polluted areas, namely, urban areas, where economic activities are concentrated. As we will show here, the above a priori belief materializes 1 The author is grateful to Diran Bodenhorn, Ralph C. D’Arge, and an anonymous referee for their comments, to W. J. Baumol for reviewing the earlier draft of this paper, and to Amir Khalili for his help during the course of this research. Of course, the responsibility for any error rests with the author. 2 For a detailed proof of this proposition see Baumol and Oates [I, pp. 522541. 16 Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
SPATIAL THEORY OF POLLUTION
17
onlyundercertainassumptions. We will utilizeaninterurban locationmodelof the firm to examine the general problem raised here. F@rsund [4, p. 25-J has also investigated the problem of pollution control with a spatial framework, but the main purpose of his general equilibrium model is to derive Pareto optimal conditions for distributing goods, services, and residuals between the given locations. Besides being a general equilibrium model in the Walrasian tradition with spatial dimensions as opposed to a micromodel like ours, he does not examine a “market system capable of implementing the optimal solution. . . . This involves analyzing the economic adjustment of consumers and producers including the choice of location” [4, p. 333. A similar type of general equilibrium model is discussed by Mathur [7] elsewhere. In this paper, we are interested in the economic adjustment problem of a polluting firm in space given the presence of a pollution tax which varies over space. In particular we will examine the following: 1. It will be proven that when a cost-minimizing firm has two abatement strategies available to it, in particular, (a) change of location to less polluted areas and (b) process change, a pollution tax would either induce the firm to decrease its waste (through process change) at a given location or discourage its location in heavily polluted areas given the amount of waste. This result will be shown to exist for a general production function. 2. The above model will be extended to include another abatement good (Z) available to the firm besides the above-mentioned abatement strategies. In the extended model it will be shown that a pollution tax will discourage the location of the firm in the urban area when the transport rate on Z is less than or equal to the tax savings due to the change in the marginal locational abatement per unit change in Z. It will also become evident that the same condition applies when we consider the effect of the tax on the use of waste or the use of Z to abate pollution. 3. We would also consider the issues above for a profit-maximizing firm. It will become evident that the outcome of the tax depends upon the fulfillment of another condition to the one in the extended cost model. 4. There is another issue which is germane to the present problem. Recently, Khalili, Mathur, and Bodenhorn [6], hereafter referred to as KMB, have shown that given a monocentric city (single market), no externalities and a homogeneous production function of degree II of the firm, the optimal location of the firm will move toward the market as output expands if and only if n > 1. It will be shown that the above result will hold even in the presence of external diseconomy of pollution when such an externality is not internalized (in particular, when pollution tax is zero). But this result need not hold when pollution tax is positive. 5. Our concern with the issues in this paper is within the framework of an interurban location model where the f.o.b. prices faced by the locator vary continuously over space because inputs and outputs must be transported at positive cost. The property that land prices decline with distance from the market is relevant only in an intraurban location model where consumers are evenly distributed over the space so that demand consideration becomes an intrinsic part of market area consideration. Therefore, the models in the main body of this paper exclude land. However, in Appendix B we will examine the issues of this paper for a cost-minimizing firm when land input is included within the framework of the model. The discussion is restricted to the cost minimizing model because our interest here is to see the further complexities caused by the inclusion of land.
18
VIJAY
K. MATHUR
FIG. 1. Location space.
In February 1972 the Nixon Administration proposed a tax on sulfur oxide emissions, beyond the Federal standards, of electric power plants. It was hoped that such a tax would discourage power plant locations in heavily polluted areas.3 As our models will show, such an outcome is possible under certain conditions. We will confine ourselves to the air pollution problem, but the models presented here are equally applicable to the water pollution problem as well where the firms dump their waste at one point. Section I presents the basic cost-minimizing location model of the firm, Sections 11.1 and III examine the first two issues; Section IV presents the profit-maximizing location model of the firm and examines issue 3. The effect of returns to scale on location is investigated in Section 11.2. Section V contains a summary and conclusions. I. BASIC COST MINIMIZING
MODEL
We will use a simple model where location is limited to the set of points along a line between an input site and an urban area which is a market for the firm’s output. This situation is depicted in Fig. 1. For a location point like k, h’is the distance from the market MO to k, whereas the distance from k to input site M1 is H - h, where H is the distance from Ma to M1. Assumptions
1. A one-plant firm produces a given level of homogeneous output (qO) by utilizing positive amounts of a transportable input M1 and a homogeneous input which we will call waste disposal (W). Such a treatment of waste is not new in the literature and hence need not be elaborated upon here.4 Therefore, qo = F(Ml,
W),
(1)
where F is continuous, twice differentiable, and convex to the origin. 2. The firm’s total cost includes: (a) Cost associated with input MI which is traded in a competitive market. Given its base price pl and transport rate rl its cost is Cpl + rl(H - h)]Mi. (b) Distribution cost of the output (r&q,,) where yq is the transport rate of shipping the output to the market. (c) A cost associated with the flow of net pollution concentration R *. Before defining this cost and R*, let us first define the components of R*. Assume that the flow of pollution concentration without abatement (R) depends upon the amount of waste disposal (W). R = R(W),
(2)
3 Some notion of the relationship between location, as measured by distance, and air pollution damage of power plants could be found in a recent study by Blomquist [2] for Winnetka, Illinois. According to him, “within 11,500 feet of the power plant a typical property loses 0.9 percent of its value for each 10 percent move closer to the plant” [2, p. lOO]. Note that this finding is based upon the fact that there were no disamenity sources adjacent to the plant and it burns relatively clean fuel as compared to other steam-electric power plants in Illinois. 4 For example, see Davis and Kamien [3, pp. 74-751 for a discussion on this subject.
SPATIAL
THEORY
(a)
FIG. 2. Pollution
19
OF POLLUTION
(b)
concentration
and abatement functions.
where R’ > 0; R” 3 0. Single primed variables represent first derivatives and double primed variables represent second derivatives, a convention followed throughout this paper. This function is shown in Fig. 2a. Furthermore, assume that in addition to abatement achieved through process change, i.e., using less W relative to IV1 to produce q. at different relative prices, the firm can also abate by moving its plant away from the heavily polluted market site MO, i.e., the urban area. This will be referred to as locational abatement. Then the amount of abatement A depends upon h. A = A(h),
(3)
where A’ > 0; A” < 0. Such a function is depicted in Fig. 2b. Since an urban area is a concentration of economic activities, we will expect a heavy pollution concentration at the market site. Pollution concentration is expected to decrease with the increase in distance from the market (h) given the same W as the environmental capacity to handle pollution concentration increases with increase in distance from the market. Therefore, R* = R(W) - A(h),
(4)
Since the function R(W) is convex and the function A(h) is concave by assumption, it follows that the function R* is convex. Assuming that a public authority in charge of pollution control imposes a pollution tax at a rate t on R*, the cost associated with R* is tR* = t[R(W) - A(h)]. Hence, the total cost of the firm is C = bl+
ri(H - h)]M,
+ r&o
+ t[R(W)
- A(h)].
(5)
Given assumptions (1) and (2), the firm wishes to minimize cost given by (5) subject to (1). Form the function, L = L-PI + rdfJ - h)lM,
+ rnhqo + CR(W)
where X is the Lagrange multiplier. dL/aM~ dL/aW
- A(h)] + Cqo - W~I,
WI,
The first-order conditions are
= [pl + ri(H - h)] - XF, = 0
(6)
= tR’ - XFw = 0
(7)
aL/ah = -rIMI
+ r,qo - tA’ = 0
dL!dX = qo - F(M,, W) = 0
(8) (9)
where F1 and Fw are first partial derivatives. Since the interpretation of first-order conditions is straightforward, we will not devote valuable space to them. The second-
20
VIJAY
K. MATHUR
order conditions for a minimum require that the bordered Hessian determinant negative and all the bordered principal minors of the Hessian be negative. II. COMPARATIVE
be
STATIC ANALYSIS
1. Effect of Pollution Tax on Location and Waste Disposal In the neighborhood of the optimum it can now be shown that (ah/i%) > 0 and (awl&) < 0. Let us take the first case. Using the total differentials of the system (6)-(9) with q. and t as exogenous variables and assuming dqo = 0; using Cramer’s rule; and simplifying, we obtain dh -=at
1 D
t Fww - -R” x
1
- 2FlFwFlw + Fw2F11
where D < 0 and is the bordered Hessian determinant and FI1, FIw, and F,, are second partials of F. From the second bordered principal minor of the Hessian we know that the first term in the brackets on the right-hand side of Eq. (10) is negative. Since rI, Fw, FI, and R’ > 0, it follows that (ah/dt) > 0. In other words, the higher the pollution tax, the farther away from the market is the optimal location of the firm. Similarly we obtain, dW -= at
1 ; R(rlFwA
- tFlR’A”).
(11)
Since D < 0; rl, F,, F,, A’, and R’ > 0; and A” < 0; it follows that (aWlat> < 0. This means that the firm will use less W relative to M1 as t increases. This is an abatement through process change. Note that these results are obtained without assuming a homogeneous production function of the firm as is usually assumed in the economics’ literature.5 2. Returns to Scale, Pollution Tax, and Optimal Location Assuming a homogeneous production function of degree n, KMB [4, p. lo] have recently proven that the optimal location of the firm will move toward the market 6 In an earlier communication with this author, Professor Ralph C. D’Arge pointed out that my model does not consider the fact that the greater the distance (H - h), the less the pollution concentration at the site of M1. Even though I was mainly interested in the problem of pollution control in heavily polluted areas, like urban areas, the incorporation of a net pollution concentration function for the site of M1 will not substantially alter my results. For example, take the first version of the model and let A1 = g(S) where S = (H - h), g’ > 0, g” < 0, dS/dh = - 1, and d%/dh2 = 0. It follows that A1 = A,(h), Ai’ < 0, and A1 ” 6 0. Therefore, RI*= Ri( W) - At(h) and Ro* = Ra( W) -Ao(h) where RI’, Rd > 0; R1”, Ra” > 0; A< > 0; and A 0” 6 0. Assuming R* = RI* + Ro*, total pollution cost of the firm is tR *. Incorporating such a pollution cost function in the model we obtain ah/at > 0 and a W/at < 0 if A,’ + Ao’ > 0. It is not difficult to see that this condition will be satisfied under the assumptions of the model. Since market site MO is the hub of economic activity and it is a heavily polluted area, we will expect that a movement away from the market would cause a greater reduction in pollution concentration (i.e., greater increase in abatement) at the margin as compared to marginal increment of concentration (i.e., marginal reduction in abatement) at the site of M,. In the event that one wishes to consider a net pollution concentration function at site k on the location line (Fig. 1) where Rk* = Rk( W), our results will be unchanged. Similar modifications can be made in the other models without disturbing the major thrust of the analysis.
SPATIAL
THEORY
21
OF POLLUTION
site as output increases if and only if )2 > 1. They obtained the proof based upon a model without pollution externality. We will examine this issue using the above model. Using total differentials of the system (6)-(9), and assuming dt = 0 and a homogeneous production function of degree n, we obtain the following (see Appendix A for details). ah
--
dqo
=
~
1
{X(FwFuv - F1Fww)Cr,qo(~ - 1)+ tA’1
DMI
+ tFX’(r,F~M~
-t tA’ - rQqo)i
(1’2)
where D < 0 and r,, t, A’ > 0, and R” 3 0. The optimal location of the firm will move toward the market when (ah/ago) < 0. This result can be obtained under the following alternative situations. (a) Assume that input M, is superior,6 i.e., F w F 1w - FIFww > 0. Then assuming y1 > 0, (ah/ago) < 0 if tA’ - r,qo > 0. This result implies that with any returns to scale the firm’s optimal location will move toward the market site when savings in distribution cost per unit distance is more than or equal to the increase in pollution cost per unit distance or t > (rQqo/A’). (b) The result obtained by KMB will hold with pollution externality when the firm is not forced to internalize the externality. In other words, when t = 0, dh/dqo < 0 ifandonlyifn> 1. III. EXTENDED
COST-MINIMIZING
MODEL
Here we will analyze the effect of a pollution tax on location, i.e., distance h, on waste disposal (W) and on the use of abatement good (Z) with a new abatement function and hence a new total cost function. Consider now an abatement function, A = A(h, Z),
(13)
where Ah, Az > 0; Ahh, Am < 0; and AU > 0. Single subscripted variables are first partials, double subscripted are second and cross partials of A. The new total cost function of the firm is C = Cp1 + rl(H - h)lM,
+ r&o
+ tCR(W) - A(h, -01 + (PZ + r.&)Z,
(14)
where pz and rz are the base price and transport rate of Z, respectively. Abatement good Z is sold in a competitive market at the market site. Minimizing (14) subject to (l), we obtain the following first-order conditions: CpI + rl(H - h)] - XF, = 0,
(1%
tR’ - XFw = 0,
(16)
+ rdo - tAh + rzZ = 0,
(17)
- tAz + (pz + rzh) = 0,
(18)
go = F(M1, W) = 0.
(19)
-rIMI
Once again the second-order conditions for a minimum require that the bordered Hessian determinant be negative and all the bordered principal minors be negative. 6 For a general production using two inputs, both inputs could be superior, but only one of the inputs could be inferior. In the case where M, is inferior, FwFllv - FIRvw < 0, W woxld be superior, i,e., FIFlw - FwFll > 0.
22
VIJAY
K.
MATHUR
Hence Eqs. (15))(19) yield optimal values of M1, W, h, Z, and h. We can now evaluate the effect of taxes on h, Z, and Win the neighborhood of the optimum. Totally differentiating the system of Eqs. (15)-(19) where q. and t are exogenous variables and assuming dqo = 0, we obtain the following solutions which concern us. ah -= at
az -= at
i {XXCtAdzz D*
+ A.&a
?- [A.&tAhhX D*
-I- r?Fd)
aw -- = L at
- tAu)]
+ rdR’FIFwAzz},
•k XAh(rz - tAhz)X],
(F12R’[t2A,,hAZZ - (rz - tAhz)2]
D*
- rZ&dtAdzz
+ Az(rz - t&Z)]),
(22)
where D* < 0, is the bordered Hessian determinant and X = 2FlFwFlw
- F12 Fww - f R” - Fw2F11. ( x >
Since we know from the second-order condition that XX > 0, XtAhhX + rlzFW2 < 0, and t2AhhAzz - (r~ - tAhz)2 > 0, it follows, given the properties of pollution concentration function (2) and abatement function (13), that ;>
0; E
> 0;
and
ky
< 0 if (rz - tAhZ) 6 0.
These results indicate that the cost-minimizing firm would ceteris paribus abate its pollution concentration as t increases by either changing its location away from the market or by increasing the use of abatement good Z or by changing its process which uses relatively less amounts of waste disposal (W) if the tax savings due to the change in marginal abatement from h per unit change in Z, (tAhZ) is greater than or equal to transport rate (rz) on Z. If we assume rz = 0, we will always have the desired effect of a tax on all forms of abatement. We can imagine such a possibility when Z is a public abatement good and the public authority who provides such a good charges only the base price of Z to the firm no matter where the firm is located in our linear space. IV. PROFIT-MAXIMIZING MODEL In this section we will consider the effect of pollution tax on location (h), on the use of abatement good Z and on waste (W) for profit-maximizing tim in a spaceeconomy. In addition to the cost function of Eq. (14) the firm is assumed to face the following demand function. PO = PO@)
where po’ < 0 for all q > 0. Furthermore, for all q > 0. The firm intends to: Max: p&k
- r,hq - CPI + rl(H -
(23) it is assumed that E = 2po’ + po”q < 0
h)lM, - CR(W) - A@,231-
(Pz
+ rzh>Z,
Subject to: q = F(M,, W). Form the Lagrange function L* = p&)q
- r&q - bpl + rl(H - h)]Ml
- CR(W) - A(& Z>l -(Pz + r&P + dq - F(MI, WI,
SPATIAL THEORY OF POLLUTION
23
where p is the Lagrange multiplier. The first-order conditions for a maximum are -b~+rdH--)I-ctF,=O, -tR’ rlMl
(24) - /LF,+,= 0,
(25)
- r,q + IAh - rzZ = 0,
(26)
tAz p0’4
+
+ rzh) = 0,
(Pz
PO -
r,h
+
P =
(27) (28)
0,
q - F(M,, W) = 0.
(29)
The second-order conditions for a constrained maximum are that bordered principal minors of the Hessian alternate in sign starting with a positive second-bordered principal minor. Hence, Eqs. (24)-(29) yield optimal values of M1, W, h, Z, q, and p. We now evaluate the effect of pollution tax on h, Z, and W in the neighborhood of the optimum. After taking the total differentials of the system (24)-(29) and using t as an exogenous variable we obtain the following bordered Hessian determinant p which is negative as required by second-order conditions.
r=
-P*Fl, @WI ‘0
--crFlw -GFww + tR”) 0
0
0 0
-FI
-Fw
6 0 t&h (t&z
0 0
0 0
-Fl --Fw
(tAhz - rz)
0 0
0
-r, 0 E
0
1
- rz) - rq
0
tAzz
* (30)
1 0
We have presented p here because it will simplify the analysis. Using Cramer’s rule and noting that t is the only exogenous variable in the system we obtain the following solutions. dh 1 - = yy { -R’tAzzU dt dZ
-z-r
+ [(tAhz - rz) - tAhAzz]Vl},
(31)
1 - - C(tAu - rz)(U 4- AhV1) - AzV2],
dW 1 = 7 { R’V, -k [t-&&z dt
- (t&z
- rz)]U}.
(32) (33)
Assuming that the second-order conditions are satisfied it follows from the thirdbordered principal minor and fourth-bordered principal minors, respectively, that VI < 0 and V,, V, > 0. The term contained in U is U = Ed’$‘w
- w,(FwFl1 -
F~‘Iw)
- ,uQ’~w
(34)
where E < 0, p < 0 and assuming W as superior input, FwFll - FIFlw < 0. Therefore, dh/dt > 0, dZ/dt > 0, and dW/dt < 0 if (tAhz - rz) 3 0 and U < 0. Once again if we assume rz = 0, then we require U < 0. The condition for U may be restated in another way. Given the assumptions of the model U < 0 when pFlw(r,Fl - rJ < 0 or rq 6 rl/Fl.
(35)
24
VIJAY
K.
MATHUK
It follows from the first-order conditions (24) and (28) that F1 = -pl*/p where I-‘,* is the delivered price of input MI and -P = po’q + p. - r,]h, which can be labeled as net marginal revenue (NMR). Therefore, after proper substitution, condition (35) becomes r,/NMR < rl ;I)~*. (36) The solution can be restated as follows. dh z > 0,
dZ
dw
-;; > 0 and
dt
< 0
if (tA h% - r;l) 3 0 and r,INMR
,< rl,/pl*.
The first condition is the same as we obtained in our extended cost-minimizing model. The second condition means that the percentage change in the delivered price of input M, with respect to distance should be greater than or equal to the percentage change in net marginal revenue with respect to distance. It should be noted that when dh = 0 the firm will always reduce its waste (W) or increase the use of abatement good (Z) when t changes. Therefore, the above results are peculiar to the firm operating in a space-economy, where it faces locational change as a possible alternative in its abatement strategy. It should further be noted that this result for a profit-maximizing firm holds irrespective of the size of the price elasticity of demand as long as it is greater than one. V. SUMMARY
AND
CONCLUSION
Most of the models of pollution control are based upon the theory of the firm where the firm operates in a spaceless economy, e.g., see Baumol and Oates [l] and Tybout [8]. In this paper we have integrated space into the conventional theory of the firm in order to examine the effect of a pollution tax on waste disposal in heavily polluted areas like most urban areas and on different forms of abatement. This integration gives us some new insights into the effectiveness of a pollution tax as a pollution control device assuming that such a tax varies over space. Our basic cost-minimizing spatial model of the firm shows that when the firm can reduce its pollution concentration either by reducing its input W (waste disposal) through process change or by moving its location to a relatively less polluted area, a pollution tax of any magnitude produces a desirable result. But if we consider in addition an abatement good Z in the cost-minimizing model which is available to the firm and has positive transport rate (rz), the mere presence of a pollution tax (t) is not enough to either discourage location of the firm in heavily polluted areas or to encourage the firm to use a relatively cleaner process to produce its output or to use the abatement good Z. The tax has to be of such a magnitude that t 3 (rZ/Ahz). Including the input land with declining price from the market imposes a further condition related to the technology of the firm in addition to the above condition for t on the efficacy of pollution tax to curtail pollution (See Appendix B). When we consider profit-maximizing behavior of the firm with only two inputs in the production function and demand elasticity greater than one, the pollution tax produces the desirable result when t 3 rz/A12Z and r,/NMR ,< rl/pl*. In summary, a tax on sulfur oxide emissions of power plants will not necessarily discourage their location in urban centers as hoped by the Nixon Administration; rather, the desired results depend upon the fulfillment of certain of the above conditions (or condition). Furthermore, the basic cost model shows that given a positive
SPATIAL
THEORY
25
OF POLLUTION
pollution tax, economies of scale may not necessarily lead to the concentration of economic activity at the market site, a result contrary to the one arrived at by KMB
L-61. APPENDIX
A
Using total differentials of the system of Eqs. (6)(9), Cramer’s rule, we obtain, --xF11 ah -=-
1 --xFm
agO
D
assuming dt = 0, and using
--Flw -(XFww
- tR”)
0
-Fl
0
-Fw
-rl
0
-rn
0
-Fl
--Fw
-1
0
(37)
,
where D < 0 and is the bordered Hessian determinant. Multiply the first row and first column by M, and second row and second column by W, respectively, of (37); add the second row to the first and the second column to the first of the resultant determinant; using the following properties of a homogeneous production function, MJI M12Fn
+
2M1
+
WFlw
WFW
=
+ W2Fww
(38)
nqo,
= n(n -
l)qo,
(39)
we obtain, -An@-l)qo+tR”W2
--X(FWWW~+MIWFUV)
0
-nq0
0
-WFw
+ tR” W2 ah
1
ago
DM12 W2
-(XFww-tR”)W2
-X(f’wwW2+M~WFw1)
(40)
+ tR” W2 0
-r&f1
-ra
0
-1
0
- WF,
-90
Multiplying the fourth column of (40) by -x(n - 1) and adding it to the first, multiplying the second row by - 1 and adding it to the first, and using properties (39) and MIFlw + WFww = (n - l)Fw of a homogeneous production function, we obtain 0 ah -= agO
I
tR” W2
DM12 W2
-AMI -(XFww
WFlw
0
- MIFI
- tR”)W2
0
- WFW
-YY
0
-1
o/
0
-rlMl
- WF,
-nq0
.
(41)
Evaluating the determinant of (41) and simplifying we get -
FJww)(r,nq,,
-
rIMI)
+ FdR”[ry(nqo
-
WFW)
- rlMIJ}.
(42)
26
VIJAY
K.
MATHUR
Using the first-order condition rlMr = r,qo - tA’, the property (38) and simplifying, the Eq. (42) reduces to
ah --=-dqo
1 DMI
(qFwFw
- F1Fww)Cr,qo(n - 1) + fA’1 + tFJ?‘(r,M$‘~
APPENDIX
+ ~4’ - rgqo)).
(43)
B
There are two ways to introduce land (Mz) into our cost-minimizing models of the firm. Consider a two-input case where in addition to input of waste (IV) we have land in the production function and its associated cost in the cost function of the firm, Therefore, cost function of the basic model, Eq. (5), and the extended model, Eq. (14), will not have cost associated with Ml but land cost. Assume that the city is bounded by two end points where market is at point MO, and Ml represents the other boundary point of the linear space (See Fig. 1). Further assume that the price of land is given by the expression pz* = pz + r2(H - h) wherep, is the price at the boundary point M, and r2 represents the rate at which price of land declines as distance h increases from the market site. Hence the cost of land is p2*M2. Introducing land in this manner in our basic and extended cost models of the firm, our results will hold as before. Another way to introduce land into the cost-minimizing models is to treat it as an additional input besides inputs Ml and Win the production function and its associated cost in the cost functions of the firm. This is the case we will discuss here and will use the extended cost-minimizing model with land cost included in the cost function given by Eq. (14). Assume that the price of land, (44)
PS = pdh),
where pz’ < 0 and pz” = 0. Therefore, the total cost is given by C’
= Cp1+ rdH - h)lM1+ pdh)M2+ t[WV - A(h,Z>l + r&q0 +(PZ + r&P.
Assume the following production differentiability and convexity.
function of the firm with the usual properties of qo = F(MI, Ma W).
Minimizing (45) subject to (46) we obtain the following addition to (15), (16), and (18).
(46) first-order
pz(h) - XFz = 0, -rIMI
(45)
-I- pz’M2 i- r,qo - tAh -i- rZ
conditions
in (47)
= 0,
(48)
qo - F(MI, Mz, W) = 0,
(49)
where x once again is the Lagrange multiplier and is positive. After taking the total differentials of the systems consisting of Eqs. (15), (16), (18), and (47))(49), treating t and q. as exogenous variables and rearranging the equations we obtain the following
SPATIAL
THEORY
27
OF POLLUTION
bordered Hessian determinant G which is negative.
G =
- XFll -XF21
--xF12 --xF22
-XFwl --i-l 0 -Fl
-XFw2 PZl 0 -Fz
-r1
0
P2’ 0 -tAhh
0 0
- ~FIW -
hF2w
- (XFww - tR”) 0 0 --Fw
trZ
-
(Fz - tAhz)
tAZh)
0
-
tAzz
0
--F1 --F2
-Fw 0 0 0
. (50)
Second-order conditions for a minimum are assumed to be met. Assuming dqo = 0, we obtain the following solutions.
ah 1 = ; (-R’tAzzK [tAhAZZ + (rZ tAhZ)]Jl], at a2 1 --- = z [-(rz - tAhz)(R’K+ Add + AZ&], at -
aw 1 = ; ( -R’Js + [tA*AZZ + (rz - tAhz)]AzK).
at
(51)
(52)
(53)
It follows from the third-bordered principal minor and fourth-bordered principal minors of G, respectively, that J1 < 0 and Jz, J3 < 0. The term represented by K is K = ~{FwCrdF2’22 - F#Iz) - PZ’(F~FH - FlFdl
- (p2’R + r-32) X(FJ2w - F2’1w)l.
(54)
Since inputs are assumed to be superior, (FlFz2 - F,F,,) < 0 and (F,F,, - F,F12) 0, (az/at) > 0, and (awlat) < 0 are (rz - tAhZ) < 0 (55) and tp2’Fl + r,F2)(F&w - F2Flw) 3 0 (56) The condition (56) can be restated in another way by utilizing the first order conditions, pz = XF, and p,* = XFI where pl* = pl + rl(H - h). It is, trdpl*)
3
-(Pz'/P~>.
(57)
The interpretation of (55) is the same as before, but condition (57) implies that the percentage change in the delivered price of input M1 must be greater than or equal to the percentage change in the price of land. If we use a Cobb-Douglas production function or a nonhomogeneous production function of the following type (see Henderson and Quandt [S, p. 61]), q = il!M~~M~PWP- pM~p+‘M2p+‘w~+l,
(58)
the term (FIFZW - FzFlw) in Eq. (54) is equal to zero. Hence, in that event we obtain the appropriate signs of Eqs. (51)-(53) if (rz - tAhZ) < 0. REFERENCES 1. W. J. Baumol and W. E. Oates, The use of standards and prices for protection of the environment, Swed. J. Econ. 73,42-54 (1971). 2. G. Blomquist, The effect of electric power plant location on area property value, Land Econ. 51, 97-100 (1974).
28
VIJAY
K. MATHUR
3. 0. A. Davis and M. I. Kamien. Externalities, information, and alternative collective action, Or Joint Economic Committee. United States Congress, “The Analysis and Evaluation of Public Expenditure: The PPB System,” Vol. 1, pp. 67- 86, U. S. Government Printing Office, Washington, D. C. (1969). 4. F. R. Forsund, Allocation in space and environmental pollution, SW&. J. &O/I. 74, 19-34 (1972). 5. J. M. Henderson and R. E. Quandt, “Microeconomic Theory: A Mathematical Approach,” McGraw-Hill, New York (1971). 6. A. Khalili, V. K. Mathur, and D. Bodenhorn, A generalization of location and the theory of production, J. Econ. Theory 9, 467-475 (1974). 7. V. K. Mathur, Models of air pollution damage control in a region, in “Proceedings of the Conference on Urban Economics” (Mattila and Thompson, Eds.), pp. 2740, Department of Economics, Wayne State University, Detroit (1973). 8. R. A. Tybout, Pricing of pollution and other negative externalities, BeN. J. Econ. Munageme~~t Sri. 3, 254-266 (1972). 9. T. H. Tietenberg, Specific taxes and pollution control, Quurt. J. Ecorz. 87, 503-522 (1973).
OF ENVIRONMENTAL
Spatial
ECONOMICS
AND
Economic
hlANAC;tMENT
Theory
3, 16-28 ( 1976)
of Pollution
Control
VIJAY K. MATHUR’ Departmen
of Economics, Cleveland State LiAw-sity,
Cleveland, Ohio 44115
Received April 9, 1974; revised September 18, 1975 In February 1972 the Nixon Administration proposed a tax on sulfur oxide emissions, beyond the Federal standards, of electric power plants. It was hoped that such a tax would discourage power plant locations in heavily polluted areas like urban areas. Assuming that such a tax varies over space because pollution is not invariant over space, the spatial theory of the firm is presented in this paper cast doubt on the efficacy of such a polltuion tax to achieve the desired goal of the administration. In particular it is shown that the costminimizing spatial firm would abate its pollution in response to a tax by either changing its location or by reducing waste through process change or by using a transportable abatement good if the tax savings due to the change in marginal abatement through location change per unit change in abatement is greater than or equal to transport rate on the abatement good. If the goal of the firm is to maximize profits, the desired outcome would follow if in addition to the fulfillment of the above condition the percentage change in the delivered price of raw material (situated at one end of the linear location space) per unit distance is greater than or equal to the percentage change in net marginal revenue, i.e., marginal revenue net of transport cost per unit of output. In simple terms, the conclusions of this paper raise doubts regarding the effectiveness of a pollution tax in curtailing pollution of a firm which is operating in a space economy. This outcome is peculiar to a spatial firm. Sufficient conditions also have been obtained when land input is included in the production function of the firm.
Many economists are in agreement that a pollution tax is the efficient method to achieve a desired level of pollution, as opposed to regulation.2 It is also contended that such a tax, ceteris paribus, will induce the firm operating in a spaceless economy to abate its wastes through whatever form of abatement practiced. However, there is one problem in a tax scheme. Tietenberg [9] has shown in his general equilibrium model that not all firms will face the same per unit tax in general. One reason for this outcome, out of others, is that marginal contribution of waste to pollution varies spatially. This conclusion is the starting point in this paper. Assuming that a tax on pollution varies spatially, it has yet to be demonstrated that the desired outcome will follow in the case of a firm operating in a space-economy. For example, it is taken for granted that a pollution tax would necessarily discourage the location of a firm in heavily polluted areas, namely, urban areas, where economic activities are concentrated. As we will show here, the above a priori belief materializes 1 The author is grateful to Diran Bodenhorn, Ralph C. D’Arge, and an anonymous referee for their comments, to W. J. Baumol for reviewing the earlier draft of this paper, and to Amir Khalili for his help during the course of this research. Of course, the responsibility for any error rests with the author. 2 For a detailed proof of this proposition see Baumol and Oates [I, pp. 522541. 16 Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
SPATIAL THEORY OF POLLUTION
17
onlyundercertainassumptions. We will utilizeaninterurban locationmodelof the firm to examine the general problem raised here. F@rsund [4, p. 25-J has also investigated the problem of pollution control with a spatial framework, but the main purpose of his general equilibrium model is to derive Pareto optimal conditions for distributing goods, services, and residuals between the given locations. Besides being a general equilibrium model in the Walrasian tradition with spatial dimensions as opposed to a micromodel like ours, he does not examine a “market system capable of implementing the optimal solution. . . . This involves analyzing the economic adjustment of consumers and producers including the choice of location” [4, p. 333. A similar type of general equilibrium model is discussed by Mathur [7] elsewhere. In this paper, we are interested in the economic adjustment problem of a polluting firm in space given the presence of a pollution tax which varies over space. In particular we will examine the following: 1. It will be proven that when a cost-minimizing firm has two abatement strategies available to it, in particular, (a) change of location to less polluted areas and (b) process change, a pollution tax would either induce the firm to decrease its waste (through process change) at a given location or discourage its location in heavily polluted areas given the amount of waste. This result will be shown to exist for a general production function. 2. The above model will be extended to include another abatement good (Z) available to the firm besides the above-mentioned abatement strategies. In the extended model it will be shown that a pollution tax will discourage the location of the firm in the urban area when the transport rate on Z is less than or equal to the tax savings due to the change in the marginal locational abatement per unit change in Z. It will also become evident that the same condition applies when we consider the effect of the tax on the use of waste or the use of Z to abate pollution. 3. We would also consider the issues above for a profit-maximizing firm. It will become evident that the outcome of the tax depends upon the fulfillment of another condition to the one in the extended cost model. 4. There is another issue which is germane to the present problem. Recently, Khalili, Mathur, and Bodenhorn [6], hereafter referred to as KMB, have shown that given a monocentric city (single market), no externalities and a homogeneous production function of degree II of the firm, the optimal location of the firm will move toward the market as output expands if and only if n > 1. It will be shown that the above result will hold even in the presence of external diseconomy of pollution when such an externality is not internalized (in particular, when pollution tax is zero). But this result need not hold when pollution tax is positive. 5. Our concern with the issues in this paper is within the framework of an interurban location model where the f.o.b. prices faced by the locator vary continuously over space because inputs and outputs must be transported at positive cost. The property that land prices decline with distance from the market is relevant only in an intraurban location model where consumers are evenly distributed over the space so that demand consideration becomes an intrinsic part of market area consideration. Therefore, the models in the main body of this paper exclude land. However, in Appendix B we will examine the issues of this paper for a cost-minimizing firm when land input is included within the framework of the model. The discussion is restricted to the cost minimizing model because our interest here is to see the further complexities caused by the inclusion of land.
18
VIJAY
K. MATHUR
FIG. 1. Location space.
In February 1972 the Nixon Administration proposed a tax on sulfur oxide emissions, beyond the Federal standards, of electric power plants. It was hoped that such a tax would discourage power plant locations in heavily polluted areas.3 As our models will show, such an outcome is possible under certain conditions. We will confine ourselves to the air pollution problem, but the models presented here are equally applicable to the water pollution problem as well where the firms dump their waste at one point. Section I presents the basic cost-minimizing location model of the firm, Sections 11.1 and III examine the first two issues; Section IV presents the profit-maximizing location model of the firm and examines issue 3. The effect of returns to scale on location is investigated in Section 11.2. Section V contains a summary and conclusions. I. BASIC COST MINIMIZING
MODEL
We will use a simple model where location is limited to the set of points along a line between an input site and an urban area which is a market for the firm’s output. This situation is depicted in Fig. 1. For a location point like k, h’is the distance from the market MO to k, whereas the distance from k to input site M1 is H - h, where H is the distance from Ma to M1. Assumptions
1. A one-plant firm produces a given level of homogeneous output (qO) by utilizing positive amounts of a transportable input M1 and a homogeneous input which we will call waste disposal (W). Such a treatment of waste is not new in the literature and hence need not be elaborated upon here.4 Therefore, qo = F(Ml,
W),
(1)
where F is continuous, twice differentiable, and convex to the origin. 2. The firm’s total cost includes: (a) Cost associated with input MI which is traded in a competitive market. Given its base price pl and transport rate rl its cost is Cpl + rl(H - h)]Mi. (b) Distribution cost of the output (r&q,,) where yq is the transport rate of shipping the output to the market. (c) A cost associated with the flow of net pollution concentration R *. Before defining this cost and R*, let us first define the components of R*. Assume that the flow of pollution concentration without abatement (R) depends upon the amount of waste disposal (W). R = R(W),
(2)
3 Some notion of the relationship between location, as measured by distance, and air pollution damage of power plants could be found in a recent study by Blomquist [2] for Winnetka, Illinois. According to him, “within 11,500 feet of the power plant a typical property loses 0.9 percent of its value for each 10 percent move closer to the plant” [2, p. lOO]. Note that this finding is based upon the fact that there were no disamenity sources adjacent to the plant and it burns relatively clean fuel as compared to other steam-electric power plants in Illinois. 4 For example, see Davis and Kamien [3, pp. 74-751 for a discussion on this subject.
SPATIAL
THEORY
(a)
FIG. 2. Pollution
19
OF POLLUTION
(b)
concentration
and abatement functions.
where R’ > 0; R” 3 0. Single primed variables represent first derivatives and double primed variables represent second derivatives, a convention followed throughout this paper. This function is shown in Fig. 2a. Furthermore, assume that in addition to abatement achieved through process change, i.e., using less W relative to IV1 to produce q. at different relative prices, the firm can also abate by moving its plant away from the heavily polluted market site MO, i.e., the urban area. This will be referred to as locational abatement. Then the amount of abatement A depends upon h. A = A(h),
(3)
where A’ > 0; A” < 0. Such a function is depicted in Fig. 2b. Since an urban area is a concentration of economic activities, we will expect a heavy pollution concentration at the market site. Pollution concentration is expected to decrease with the increase in distance from the market (h) given the same W as the environmental capacity to handle pollution concentration increases with increase in distance from the market. Therefore, R* = R(W) - A(h),
(4)
Since the function R(W) is convex and the function A(h) is concave by assumption, it follows that the function R* is convex. Assuming that a public authority in charge of pollution control imposes a pollution tax at a rate t on R*, the cost associated with R* is tR* = t[R(W) - A(h)]. Hence, the total cost of the firm is C = bl+
ri(H - h)]M,
+ r&o
+ t[R(W)
- A(h)].
(5)
Given assumptions (1) and (2), the firm wishes to minimize cost given by (5) subject to (1). Form the function, L = L-PI + rdfJ - h)lM,
+ rnhqo + CR(W)
where X is the Lagrange multiplier. dL/aM~ dL/aW
- A(h)] + Cqo - W~I,
WI,
The first-order conditions are
= [pl + ri(H - h)] - XF, = 0
(6)
= tR’ - XFw = 0
(7)
aL/ah = -rIMI
+ r,qo - tA’ = 0
dL!dX = qo - F(M,, W) = 0
(8) (9)
where F1 and Fw are first partial derivatives. Since the interpretation of first-order conditions is straightforward, we will not devote valuable space to them. The second-
20
VIJAY
K. MATHUR
order conditions for a minimum require that the bordered Hessian determinant negative and all the bordered principal minors of the Hessian be negative. II. COMPARATIVE
be
STATIC ANALYSIS
1. Effect of Pollution Tax on Location and Waste Disposal In the neighborhood of the optimum it can now be shown that (ah/i%) > 0 and (awl&) < 0. Let us take the first case. Using the total differentials of the system (6)-(9) with q. and t as exogenous variables and assuming dqo = 0; using Cramer’s rule; and simplifying, we obtain dh -=at
1 D
t Fww - -R” x
1
- 2FlFwFlw + Fw2F11
where D < 0 and is the bordered Hessian determinant and FI1, FIw, and F,, are second partials of F. From the second bordered principal minor of the Hessian we know that the first term in the brackets on the right-hand side of Eq. (10) is negative. Since rI, Fw, FI, and R’ > 0, it follows that (ah/dt) > 0. In other words, the higher the pollution tax, the farther away from the market is the optimal location of the firm. Similarly we obtain, dW -= at
1 ; R(rlFwA
- tFlR’A”).
(11)
Since D < 0; rl, F,, F,, A’, and R’ > 0; and A” < 0; it follows that (aWlat> < 0. This means that the firm will use less W relative to M1 as t increases. This is an abatement through process change. Note that these results are obtained without assuming a homogeneous production function of the firm as is usually assumed in the economics’ literature.5 2. Returns to Scale, Pollution Tax, and Optimal Location Assuming a homogeneous production function of degree n, KMB [4, p. lo] have recently proven that the optimal location of the firm will move toward the market 6 In an earlier communication with this author, Professor Ralph C. D’Arge pointed out that my model does not consider the fact that the greater the distance (H - h), the less the pollution concentration at the site of M1. Even though I was mainly interested in the problem of pollution control in heavily polluted areas, like urban areas, the incorporation of a net pollution concentration function for the site of M1 will not substantially alter my results. For example, take the first version of the model and let A1 = g(S) where S = (H - h), g’ > 0, g” < 0, dS/dh = - 1, and d%/dh2 = 0. It follows that A1 = A,(h), Ai’ < 0, and A1 ” 6 0. Therefore, RI*= Ri( W) - At(h) and Ro* = Ra( W) -Ao(h) where RI’, Rd > 0; R1”, Ra” > 0; A< > 0; and A 0” 6 0. Assuming R* = RI* + Ro*, total pollution cost of the firm is tR *. Incorporating such a pollution cost function in the model we obtain ah/at > 0 and a W/at < 0 if A,’ + Ao’ > 0. It is not difficult to see that this condition will be satisfied under the assumptions of the model. Since market site MO is the hub of economic activity and it is a heavily polluted area, we will expect that a movement away from the market would cause a greater reduction in pollution concentration (i.e., greater increase in abatement) at the margin as compared to marginal increment of concentration (i.e., marginal reduction in abatement) at the site of M,. In the event that one wishes to consider a net pollution concentration function at site k on the location line (Fig. 1) where Rk* = Rk( W), our results will be unchanged. Similar modifications can be made in the other models without disturbing the major thrust of the analysis.
SPATIAL
THEORY
21
OF POLLUTION
site as output increases if and only if )2 > 1. They obtained the proof based upon a model without pollution externality. We will examine this issue using the above model. Using total differentials of the system (6)-(9), and assuming dt = 0 and a homogeneous production function of degree n, we obtain the following (see Appendix A for details). ah
--
dqo
=
~
1
{X(FwFuv - F1Fww)Cr,qo(~ - 1)+ tA’1
DMI
+ tFX’(r,F~M~
-t tA’ - rQqo)i
(1’2)
where D < 0 and r,, t, A’ > 0, and R” 3 0. The optimal location of the firm will move toward the market when (ah/ago) < 0. This result can be obtained under the following alternative situations. (a) Assume that input M, is superior,6 i.e., F w F 1w - FIFww > 0. Then assuming y1 > 0, (ah/ago) < 0 if tA’ - r,qo > 0. This result implies that with any returns to scale the firm’s optimal location will move toward the market site when savings in distribution cost per unit distance is more than or equal to the increase in pollution cost per unit distance or t > (rQqo/A’). (b) The result obtained by KMB will hold with pollution externality when the firm is not forced to internalize the externality. In other words, when t = 0, dh/dqo < 0 ifandonlyifn> 1. III. EXTENDED
COST-MINIMIZING
MODEL
Here we will analyze the effect of a pollution tax on location, i.e., distance h, on waste disposal (W) and on the use of abatement good (Z) with a new abatement function and hence a new total cost function. Consider now an abatement function, A = A(h, Z),
(13)
where Ah, Az > 0; Ahh, Am < 0; and AU > 0. Single subscripted variables are first partials, double subscripted are second and cross partials of A. The new total cost function of the firm is C = Cp1 + rl(H - h)lM,
+ r&o
+ tCR(W) - A(h, -01 + (PZ + r.&)Z,
(14)
where pz and rz are the base price and transport rate of Z, respectively. Abatement good Z is sold in a competitive market at the market site. Minimizing (14) subject to (l), we obtain the following first-order conditions: CpI + rl(H - h)] - XF, = 0,
(1%
tR’ - XFw = 0,
(16)
+ rdo - tAh + rzZ = 0,
(17)
- tAz + (pz + rzh) = 0,
(18)
go = F(M1, W) = 0.
(19)
-rIMI
Once again the second-order conditions for a minimum require that the bordered Hessian determinant be negative and all the bordered principal minors be negative. 6 For a general production using two inputs, both inputs could be superior, but only one of the inputs could be inferior. In the case where M, is inferior, FwFllv - FIRvw < 0, W woxld be superior, i,e., FIFlw - FwFll > 0.
22
VIJAY
K.
MATHUR
Hence Eqs. (15))(19) yield optimal values of M1, W, h, Z, and h. We can now evaluate the effect of taxes on h, Z, and Win the neighborhood of the optimum. Totally differentiating the system of Eqs. (15)-(19) where q. and t are exogenous variables and assuming dqo = 0, we obtain the following solutions which concern us. ah -= at
az -= at
i {XXCtAdzz D*
+ A.&a
?- [A.&tAhhX D*
-I- r?Fd)
aw -- = L at
- tAu)]
+ rdR’FIFwAzz},
•k XAh(rz - tAhz)X],
(F12R’[t2A,,hAZZ - (rz - tAhz)2]
D*
- rZ&dtAdzz
+ Az(rz - t&Z)]),
(22)
where D* < 0, is the bordered Hessian determinant and X = 2FlFwFlw
- F12 Fww - f R” - Fw2F11. ( x >
Since we know from the second-order condition that XX > 0, XtAhhX + rlzFW2 < 0, and t2AhhAzz - (r~ - tAhz)2 > 0, it follows, given the properties of pollution concentration function (2) and abatement function (13), that ;>
0; E
> 0;
and
ky
< 0 if (rz - tAhZ) 6 0.
These results indicate that the cost-minimizing firm would ceteris paribus abate its pollution concentration as t increases by either changing its location away from the market or by increasing the use of abatement good Z or by changing its process which uses relatively less amounts of waste disposal (W) if the tax savings due to the change in marginal abatement from h per unit change in Z, (tAhZ) is greater than or equal to transport rate (rz) on Z. If we assume rz = 0, we will always have the desired effect of a tax on all forms of abatement. We can imagine such a possibility when Z is a public abatement good and the public authority who provides such a good charges only the base price of Z to the firm no matter where the firm is located in our linear space. IV. PROFIT-MAXIMIZING MODEL In this section we will consider the effect of pollution tax on location (h), on the use of abatement good Z and on waste (W) for profit-maximizing tim in a spaceeconomy. In addition to the cost function of Eq. (14) the firm is assumed to face the following demand function. PO = PO@)
where po’ < 0 for all q > 0. Furthermore, for all q > 0. The firm intends to: Max: p&k
- r,hq - CPI + rl(H -
(23) it is assumed that E = 2po’ + po”q < 0
h)lM, - CR(W) - A@,231-
(Pz
+ rzh>Z,
Subject to: q = F(M,, W). Form the Lagrange function L* = p&)q
- r&q - bpl + rl(H - h)]Ml
- CR(W) - A(& Z>l -(Pz + r&P + dq - F(MI, WI,
SPATIAL THEORY OF POLLUTION
23
where p is the Lagrange multiplier. The first-order conditions for a maximum are -b~+rdH--)I-ctF,=O, -tR’ rlMl
(24) - /LF,+,= 0,
(25)
- r,q + IAh - rzZ = 0,
(26)
tAz p0’4
+
+ rzh) = 0,
(Pz
PO -
r,h
+
P =
(27) (28)
0,
q - F(M,, W) = 0.
(29)
The second-order conditions for a constrained maximum are that bordered principal minors of the Hessian alternate in sign starting with a positive second-bordered principal minor. Hence, Eqs. (24)-(29) yield optimal values of M1, W, h, Z, q, and p. We now evaluate the effect of pollution tax on h, Z, and W in the neighborhood of the optimum. After taking the total differentials of the system (24)-(29) and using t as an exogenous variable we obtain the following bordered Hessian determinant p which is negative as required by second-order conditions.
r=
-P*Fl, @WI ‘0
--crFlw -GFww + tR”) 0
0
0 0
-FI
-Fw
6 0 t&h (t&z
0 0
0 0
-Fl --Fw
(tAhz - rz)
0 0
0
-r, 0 E
0
1
- rz) - rq
0
tAzz
* (30)
1 0
We have presented p here because it will simplify the analysis. Using Cramer’s rule and noting that t is the only exogenous variable in the system we obtain the following solutions. dh 1 - = yy { -R’tAzzU dt dZ
-z-r
+ [(tAhz - rz) - tAhAzz]Vl},
(31)
1 - - C(tAu - rz)(U 4- AhV1) - AzV2],
dW 1 = 7 { R’V, -k [t-&&z dt
- (t&z
- rz)]U}.
(32) (33)
Assuming that the second-order conditions are satisfied it follows from the thirdbordered principal minor and fourth-bordered principal minors, respectively, that VI < 0 and V,, V, > 0. The term contained in U is U = Ed’$‘w
- w,(FwFl1 -
F~‘Iw)
- ,uQ’~w
(34)
where E < 0, p < 0 and assuming W as superior input, FwFll - FIFlw < 0. Therefore, dh/dt > 0, dZ/dt > 0, and dW/dt < 0 if (tAhz - rz) 3 0 and U < 0. Once again if we assume rz = 0, then we require U < 0. The condition for U may be restated in another way. Given the assumptions of the model U < 0 when pFlw(r,Fl - rJ < 0 or rq 6 rl/Fl.
(35)
24
VIJAY
K.
MATHUK
It follows from the first-order conditions (24) and (28) that F1 = -pl*/p where I-‘,* is the delivered price of input MI and -P = po’q + p. - r,]h, which can be labeled as net marginal revenue (NMR). Therefore, after proper substitution, condition (35) becomes r,/NMR < rl ;I)~*. (36) The solution can be restated as follows. dh z > 0,
dZ
dw
-;; > 0 and
dt
< 0
if (tA h% - r;l) 3 0 and r,INMR
,< rl,/pl*.
The first condition is the same as we obtained in our extended cost-minimizing model. The second condition means that the percentage change in the delivered price of input M, with respect to distance should be greater than or equal to the percentage change in net marginal revenue with respect to distance. It should be noted that when dh = 0 the firm will always reduce its waste (W) or increase the use of abatement good (Z) when t changes. Therefore, the above results are peculiar to the firm operating in a space-economy, where it faces locational change as a possible alternative in its abatement strategy. It should further be noted that this result for a profit-maximizing firm holds irrespective of the size of the price elasticity of demand as long as it is greater than one. V. SUMMARY
AND
CONCLUSION
Most of the models of pollution control are based upon the theory of the firm where the firm operates in a spaceless economy, e.g., see Baumol and Oates [l] and Tybout [8]. In this paper we have integrated space into the conventional theory of the firm in order to examine the effect of a pollution tax on waste disposal in heavily polluted areas like most urban areas and on different forms of abatement. This integration gives us some new insights into the effectiveness of a pollution tax as a pollution control device assuming that such a tax varies over space. Our basic cost-minimizing spatial model of the firm shows that when the firm can reduce its pollution concentration either by reducing its input W (waste disposal) through process change or by moving its location to a relatively less polluted area, a pollution tax of any magnitude produces a desirable result. But if we consider in addition an abatement good Z in the cost-minimizing model which is available to the firm and has positive transport rate (rz), the mere presence of a pollution tax (t) is not enough to either discourage location of the firm in heavily polluted areas or to encourage the firm to use a relatively cleaner process to produce its output or to use the abatement good Z. The tax has to be of such a magnitude that t 3 (rZ/Ahz). Including the input land with declining price from the market imposes a further condition related to the technology of the firm in addition to the above condition for t on the efficacy of pollution tax to curtail pollution (See Appendix B). When we consider profit-maximizing behavior of the firm with only two inputs in the production function and demand elasticity greater than one, the pollution tax produces the desirable result when t 3 rz/A12Z and r,/NMR ,< rl/pl*. In summary, a tax on sulfur oxide emissions of power plants will not necessarily discourage their location in urban centers as hoped by the Nixon Administration; rather, the desired results depend upon the fulfillment of certain of the above conditions (or condition). Furthermore, the basic cost model shows that given a positive
SPATIAL
THEORY
25
OF POLLUTION
pollution tax, economies of scale may not necessarily lead to the concentration of economic activity at the market site, a result contrary to the one arrived at by KMB
L-61. APPENDIX
A
Using total differentials of the system of Eqs. (6)(9), Cramer’s rule, we obtain, --xF11 ah -=-
1 --xFm
agO
D
assuming dt = 0, and using
--Flw -(XFww
- tR”)
0
-Fl
0
-Fw
-rl
0
-rn
0
-Fl
--Fw
-1
0
(37)
,
where D < 0 and is the bordered Hessian determinant. Multiply the first row and first column by M, and second row and second column by W, respectively, of (37); add the second row to the first and the second column to the first of the resultant determinant; using the following properties of a homogeneous production function, MJI M12Fn
+
2M1
+
WFlw
WFW
=
+ W2Fww
(38)
nqo,
= n(n -
l)qo,
(39)
we obtain, -An@-l)qo+tR”W2
--X(FWWW~+MIWFUV)
0
-nq0
0
-WFw
+ tR” W2 ah
1
ago
DM12 W2
-(XFww-tR”)W2
-X(f’wwW2+M~WFw1)
(40)
+ tR” W2 0
-r&f1
-ra
0
-1
0
- WF,
-90
Multiplying the fourth column of (40) by -x(n - 1) and adding it to the first, multiplying the second row by - 1 and adding it to the first, and using properties (39) and MIFlw + WFww = (n - l)Fw of a homogeneous production function, we obtain 0 ah -= agO
I
tR” W2
DM12 W2
-AMI -(XFww
WFlw
0
- MIFI
- tR”)W2
0
- WFW
-YY
0
-1
o/
0
-rlMl
- WF,
-nq0
.
(41)
Evaluating the determinant of (41) and simplifying we get -
FJww)(r,nq,,
-
rIMI)
+ FdR”[ry(nqo
-
WFW)
- rlMIJ}.
(42)
26
VIJAY
K.
MATHUR
Using the first-order condition rlMr = r,qo - tA’, the property (38) and simplifying, the Eq. (42) reduces to
ah --=-dqo
1 DMI
(qFwFw
- F1Fww)Cr,qo(n - 1) + fA’1 + tFJ?‘(r,M$‘~
APPENDIX
+ ~4’ - rgqo)).
(43)
B
There are two ways to introduce land (Mz) into our cost-minimizing models of the firm. Consider a two-input case where in addition to input of waste (IV) we have land in the production function and its associated cost in the cost function of the firm, Therefore, cost function of the basic model, Eq. (5), and the extended model, Eq. (14), will not have cost associated with Ml but land cost. Assume that the city is bounded by two end points where market is at point MO, and Ml represents the other boundary point of the linear space (See Fig. 1). Further assume that the price of land is given by the expression pz* = pz + r2(H - h) wherep, is the price at the boundary point M, and r2 represents the rate at which price of land declines as distance h increases from the market site. Hence the cost of land is p2*M2. Introducing land in this manner in our basic and extended cost models of the firm, our results will hold as before. Another way to introduce land into the cost-minimizing models is to treat it as an additional input besides inputs Ml and Win the production function and its associated cost in the cost functions of the firm. This is the case we will discuss here and will use the extended cost-minimizing model with land cost included in the cost function given by Eq. (14). Assume that the price of land, (44)
PS = pdh),
where pz’ < 0 and pz” = 0. Therefore, the total cost is given by C’
= Cp1+ rdH - h)lM1+ pdh)M2+ t[WV - A(h,Z>l + r&q0 +(PZ + r&P.
Assume the following production differentiability and convexity.
function of the firm with the usual properties of qo = F(MI, Ma W).
Minimizing (45) subject to (46) we obtain the following addition to (15), (16), and (18).
(46) first-order
pz(h) - XFz = 0, -rIMI
(45)
-I- pz’M2 i- r,qo - tAh -i- rZ
conditions
in (47)
= 0,
(48)
qo - F(MI, Mz, W) = 0,
(49)
where x once again is the Lagrange multiplier and is positive. After taking the total differentials of the systems consisting of Eqs. (15), (16), (18), and (47))(49), treating t and q. as exogenous variables and rearranging the equations we obtain the following
SPATIAL
THEORY
27
OF POLLUTION
bordered Hessian determinant G which is negative.
G =
- XFll -XF21
--xF12 --xF22
-XFwl --i-l 0 -Fl
-XFw2 PZl 0 -Fz
-r1
0
P2’ 0 -tAhh
0 0
- ~FIW -
hF2w
- (XFww - tR”) 0 0 --Fw
trZ
-
(Fz - tAhz)
tAZh)
0
-
tAzz
0
--F1 --F2
-Fw 0 0 0
. (50)
Second-order conditions for a minimum are assumed to be met. Assuming dqo = 0, we obtain the following solutions.
ah 1 = ; (-R’tAzzK [tAhAZZ + (rZ tAhZ)]Jl], at a2 1 --- = z [-(rz - tAhz)(R’K+ Add + AZ&], at -
aw 1 = ; ( -R’Js + [tA*AZZ + (rz - tAhz)]AzK).
at
(51)
(52)
(53)
It follows from the third-bordered principal minor and fourth-bordered principal minors of G, respectively, that J1 < 0 and Jz, J3 < 0. The term represented by K is K = ~{FwCrdF2’22 - F#Iz) - PZ’(F~FH - FlFdl
- (p2’R + r-32) X(FJ2w - F2’1w)l.
(54)
Since inputs are assumed to be superior, (FlFz2 - F,F,,) < 0 and (F,F,, - F,F12)
3
-(Pz'/P~>.
(57)
The interpretation of (55) is the same as before, but condition (57) implies that the percentage change in the delivered price of input M1 must be greater than or equal to the percentage change in the price of land. If we use a Cobb-Douglas production function or a nonhomogeneous production function of the following type (see Henderson and Quandt [S, p. 61]), q = il!M~~M~PWP- pM~p+‘M2p+‘w~+l,
(58)
the term (FIFZW - FzFlw) in Eq. (54) is equal to zero. Hence, in that event we obtain the appropriate signs of Eqs. (51)-(53) if (rz - tAhZ) < 0. REFERENCES 1. W. J. Baumol and W. E. Oates, The use of standards and prices for protection of the environment, Swed. J. Econ. 73,42-54 (1971). 2. G. Blomquist, The effect of electric power plant location on area property value, Land Econ. 51, 97-100 (1974).
28
VIJAY
K. MATHUR
3. 0. A. Davis and M. I. Kamien. Externalities, information, and alternative collective action, Or Joint Economic Committee. United States Congress, “The Analysis and Evaluation of Public Expenditure: The PPB System,” Vol. 1, pp. 67- 86, U. S. Government Printing Office, Washington, D. C. (1969). 4. F. R. Forsund, Allocation in space and environmental pollution, SW&. J. &O/I. 74, 19-34 (1972). 5. J. M. Henderson and R. E. Quandt, “Microeconomic Theory: A Mathematical Approach,” McGraw-Hill, New York (1971). 6. A. Khalili, V. K. Mathur, and D. Bodenhorn, A generalization of location and the theory of production, J. Econ. Theory 9, 467-475 (1974). 7. V. K. Mathur, Models of air pollution damage control in a region, in “Proceedings of the Conference on Urban Economics” (Mattila and Thompson, Eds.), pp. 2740, Department of Economics, Wayne State University, Detroit (1973). 8. R. A. Tybout, Pricing of pollution and other negative externalities, BeN. J. Econ. Munageme~~t Sri. 3, 254-266 (1972). 9. T. H. Tietenberg, Specific taxes and pollution control, Quurt. J. Ecorz. 87, 503-522 (1973).