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Achieving environmental standards with stochastic discharges
JOURNAL
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
10,
103- 111 (1983)
Achieving Environmental Standards with Stochastic Discharges’ BRIAN BEAVIS AND MARTIN WALKER Department
of Economics,
University
of Newcastle upon Tyne, England
Newcastle
upon Tyne NE1
7RU,
Received April 4, 1982; revised March, 198 1 The inherent stochastic nature of the levels and composition of environmental waste discharges are well known to environmental control agencies, who recognize that realistically environmental quality constraints must be formulated in probabilistic terms. Given the specification of environmental constraints in such terms, it is shown that the set of activity vectors which satisfy the constraint will, in general, be nonconvex. This militates against the determination of an efficient solution. A means of avoiding the potentially high computational and informational costs associated with the determination of an efficient solution in such cases is proposed.
1. INTRODUCTION
Given the inherent difficulty of measuring environmental damage resulting from pollution, it is not surprising that policy-makers have directed attention toward a policy of attaining exogenously determined environmental quality standards as a second-best solution. In practice, producers are frequently unable to predict with any great degree of accuracy the quantity and quality of wastes associated with specific levels of their productive activities. Moreover, waste treatment processes themselves are often subject to stochastic influences, particularly when biological treatment is involved. The inherent stochastic nature of the levels and composition of environmental waste discharges are well known to environmental control agencies, who recognize that realistically environmental quality constraints must be formulated in probabilistic terms. * In E.E.C. countries, for example, water pollution standards are often expressed in terms of allowable upper-bounds on the 95th percentile of the aggregate rate of discharge of pollutants to water courses. The present paper examines the problems involved in efficiently achieving exogenously determined environmental quality constraints, where the latter are expressed in probabilistic terms. The specification of environmental constraints in such terms introduces an important nonconvexity which militates against the determination of ‘We are grateful to the Department of the Environment for financial support.‘The Department is not necessarily in agreement with the views expressed. The paper has also benefited from comments by an anonymous referee. ‘The agencies typically are aware that the assimilative capacity of the environment is itself also subject to stochastic influences. We abstract from such problems in the present paper. 103 0095~0696/83 $3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved
104
BEAVIS
AND
WALKER
an efficient solution. A means of avoiding the potentially high computational and informational costs associated with the determination of an efficient solution in such cases is proposed. The proposal introduced can perhaps be regarded as a third-best solution to the pollution control problem in that it automatically guarantees that the environmental constraint is satisfied, it involves some loss in efficiency, but economizes on potentially high computational costs. It is also amenable to decentralization via a tax-price approach, and appropriate tax-prices are derived in the paper.
2. THE FIRM WITH STOCHASTIC
DISCHARGES
For simplicity, consider a firm which produces a single output, q, with a technology summarized by a conventional, twice continuously differentiable, strictly concave production function q = 4(x,), where x, is a vector of inputs, and q(0) = 0. In addition to producing output, the firm also generates wastes as a by-product of its production activities. The firm has the option of either discharging the wastes it generates directly to the environment or of treating the wastes before they are discharged. Specifically, the wastes discharged by the firm to the environment, which we denote by e, depend on the levels of inputs used in production, x,, the levels of inputs used in waste treatment, x2, and a continuous random variable 8, which has density function h(8) independent of x, and x2. Formally it is assumed that:
e = e(x,, x2, 81, where e is twice continuously differentiable, with de/ax, > 0, de/k, < 0, and that e is linearly homogeneous for a given level of 0; that is, constant returns to scale prevail in waste treatment. We also require e(0, x2, 8) = 0; if the firm does not engage in production it discharges no waste. If x2 = 0, the firm does not treat its wastes before they are discharged to the environment. The above simple model allows for the possibility that the production activities of the firm give rise to random waste generation, as well as for the possibility that waste treatment itself is an uncertain process. By varying the levels of inputs used in production and waste treatment the firm can alter its mean rate of discharge:
p = /e(x,, x2,
8)h(8)
de,
and the variance of its discharge rate: a2
=-
/[ e( x1,x2, e)]‘h(e)
de - p2.
(3)
However, if e is either multiplicatively or additively separable in 0, then the firm will be unable to vary its mean rate of discharge independently from the variance of its
STOCHASTIC
ENVIRONMENTAL
105
STANDARDS
discharge rate.3 The more general possibility that the firm can vary its mean rate of discharge independently from the variance of its discharge rate is assumed in the body of this paper. The model also assumes that the firm is a price-taker in both the output and input markets, and that its objective is the maximization of expected profits. For given levels of the mean and variance of the rate of waste discharges, fi and h2, respectively, the firm’s maximum expected profits, II*, can be taken as a function of the given levels of the mean and variance of its discharge rate. Under the above assumptions it can be shown that II* = II*@, 82) is strictly concave (see the Appendix). 3. THE ENVIRONMENTAL
CONTROL
AGENCY’S
PROBLEM
If the discharge rates of the individual firms are stochastic, then the Environmental Control Agency can realistically only formulate environmental quality standards in probabilistic terms. Such constraints take the following form:
where E and (Y are given parameters, (typically (Y will be small), and ej is the discharge rate of firm j. If the number of firms, n, is large, if the discharge rates of the individual firms are independent of each other, and if no single firm’s discharge rate is dominant in variance, then, from the Central Limit Theorem, the aggregate rate of waste discharge into the environment will be normally distributed with mean p = &j and variance cf2 = X(1,2, where pj and u; are the mean and variance, respectively, of the discharge rate of firmi. Under these conditions the environmental quality constraint Eq. (4) can be expressed in the form: E - CPj a ZdCu,‘,
(5)
where Z is given by lZ,1/~e-(‘/2)“’
du= 1 -a.
In many situations, the number of firms will not be sufficient for the Central Limit Theorem to be applicable. In such cases Chebyshev’s Inequality can be 3To see this, we have on taking the total differentials of the mean and variance of the discharge rate: j-&h(e)
de
1 [ dx,
+
jEh(e)
de
1 dxz,
where dx, and dx, are vectors of differentials, and setting dp = 0, gives: do2 = 2[/e(ae/ax,)h(e)
de]
If e(x,, x2, e) = g(x,, x2)e, then se/ax, 2g(x,, lag/ax2
x2x[ag/ax,jeh(e) jeh(e) de]
de] dx, + [ag/ax2jeh(e) dx2 = 0 SOthat da2 = 0.
dx, =
+ 2[le(ae/ax,)h(e)
de]
dx,.
Bag/ax,, and aejax, = eag/ax2, and do2 de] dx2). But dp = [ag/ax,jeh(e) de] dx,
A similar analysis can be performed for the case where: e(x,, x2, 0) = g(x,, x2) + 0.
= +
106
BEAVIS
AND
WALKER
applied. Assuming independence of discharge rates, the latter requires:
where k is a given constant. From Eq. (6) we clearly have: Prob. ( cej so that the environmental
> cpj
+ kixof)
G l/k*,
quality constraint Eq. (4) can be expressed in the form:
which has the same form as Eq. (5). In the remainder of this paper the analysis will be performed for the normal/large numbers case. It is comforting to know, though, that the mathematical structure of the problem extends to a much wider class of problems. It is assumed that the Agency seeks to establish the vector (CL,,. . . , p,,, a:, . . . , a:), which maximizes the sum of firms’ profits CIIj, subject to the environmental quality constraint (5). Given an appropriate constraint qualification, the following Kuhn-Tucker conditions are necessary for an optimal solution to the Agency’s problem:
aIIj --A pj[ap,1=o,y., u$$-&]-o,7, E
-
CPj
X[E - cpj
-
zJCD/2
- Z&$]
>,
0,
= 0,
(11) (12) 03) (14)
where h is the Lagrange multiplier. If the set of vectors (CL,,. . . , p,,, a:, . . . , CT,‘)which satisfy Eq. (5) were to form a convex set then conditions (9)-(14) would also be sufficient conditions for an optimal solution to the Agency’s problem, as the profit functions of the individual firms are by construction strictly concave functions of the mean and variance of their discharge rates. However, the set of vectors, (p,, . . . , p,,, a:,. . . , CT,‘) which satisfy Eq. (5) do not in general form a convex set. Proof: Let S denote the set of means and variances of discharge rates of the individual firms which satisfy the environmental constraint (5), and suppose that S
STOCHASTIC
ENVIRONMENTAL
107
STANDARDS
is convex. Let p = (p,, . . . , fi,, 6:, . . . , 6:) and 9 = (p,, . . . , &,, a:, . . . , 62) be elements of S, which satisfy Eq. (5) with equality, and suppose jj * 9. Since S is convex: xp + (1 - X)9 E s,
forOcX<
1;
that is, E-C[Afij+
(1 -x)/ij]-ZJC(h6;+(1
-X)q2)
>O.
(15)
Now set pi = fij in Eq. (5) and multiply through by X; then set p, = pj in Eq. (15) and multiply through by (1 - A), and add the two resulting equations to obtain E - c[Xli,
+ (1 - “)fij]
- z[JxzC&;
+ \/(l - x)2C52]
= o.
Then using Eq. (15) yields:
which on squaring both sides and combining X(1 - X):
terms, gives, upon dividing
by
06) Squaring both sides of Eq. (16) and simplifying finally gives:
(17) which hold if, and only if, Csj2 = Dj2, which in turn implies that Cpj = Cfij as 9 and 7 both satisfy Eq. (5) with equality.4 Since it is clear that there exist points on the boundary of S for which CS2 * C6” and CrZ, f #, in general S will not be a convex set. The above proof depends only on the assumption that firm’s discharge rates are independent. Indeed, we are confident that even this requirement can be relaxed, but we have not attempted to do so. Hence, in general, we would expect the feasible set defined by an environmental constraint of the form (4) to be nonconvex. Given that the Kuhn-Tucker conditions, Eq. (9)-(14) are necessary but not sufficient conditions for a global solution to the Agency’s problem, the location of points which simultaneously satisfy the environmental quality constraint and globally maximize firms’ profits in the aggregate will be an extremely costly exercise, requiring the institution of some form of computational search procedure and detailed information concerning the profit functions of the individual firms. Moreover, the Agency’s problem is not amenable to solution via the adoption of a decentralized, tax-price scheme of the type originally proposed by Baumol/Oates [l], for if at a given tax-price level the environmental quality constraint is exactly achieved, given the nonconvexity of the constrained region, there can be no guarantee that aggregate firms’ profits are global& maximized at that tax-price level. vary
41t is worth noting that this condition must their mean rates of discharge independently
be satisfied whether of their variances.
or not individual
firms
are able to
108
BEAVIS
AND
WALKER
4. AN ALTERNATIVE
SOLUTION
In such circumstances it seems worthwhile considering alternative schemes for allocating the scarce environmental capacity among discharging firms which are less costly to implement, but which may entail some loss in efficiency. An appealing alternative is to restrict consideration to only those allocations which lie in the largest convex set enclosed by the set of allocations which satisfy the environmental quality constraint Eqs. (5) or (8). Such an approach automatically guarantees the appropriate level of environmental quality, but may entail a loss in efficiency in that aggregate firms’ profits will not necessarily be globally maximized. Nevertheless it would seem to offer the prospect of finding a “reasonably good” approximation to the Agency’s problem, and, additionally, allows for the adoption of a decentralized, iterative tax-price approach, thereby potentially avoiding the high computational and informational costs associated with the determination of a globally efficient solution. The environmental constraints appropriate to such an approach can be determined as follows. Let S denote the set of vectors (CL,,. . . , pL,, a:, . . . , u,‘) which satisfy the environmental constraint (5) and B(S) its boundary. Let S’ be the largest convex set enclosed by S and B( S’) its boundary, and define p = x11, and u2 = Cuj2. Along B(S) du2/dp -C 0 and d2u2/dp2 > 0, so that B(S) is strictly convex towards the origin in (p, u2)-space. Moreover, since S’ is the largest convex subset of S, in (pu2)-space S’ must have a linear boundary which is tangential to B(S) in that space. Let (p,O), (0, C2) E B(S’) and (fi, b2) E B(S’) n B(S). Then h2 = (E2 2fiE + fi2)/Z2; and since B(S) is linear: 52-62
62
-=-= P
2(E - P)
P-P
z2
’
where the last term is the slope of B(S’), which yields: /I = +(fi + E), a2 = ( E2 - p2)/z2.
(18) (19)
B(S’) can be determined by maximizing the product of p and E2, i.e., /-iti2 = (p + E)( E2 - p2)/z2.
(20)
A necessary and sufficient condition for Eq. (20) to be maximized is that: (-3fi
+ E)(P + E) = 0,
(21)
which, since fi = -E is meaningless here, implies ,b = +E.
(22)
Hence, B( S’) in (II, u2)-space is given by: 02
4E2 = --4E 3~2 p+gz27
(23)
STOCHASTIC
so that the environmental
ENVIRONMENTAL
constraint appropriate
109
STANDARDS
to the “third-best”
problem is:
(24 Maximization of aggregate firms’ profits subject to the environmental constraint, Eq. (24) provides the following Kuhn-Tucker conditions, which are necessary and sufficient, given an appropriate constraint qualification.
(25) (26)
(27)
4E2 - 12Expj X[4E2 - 12Ecpj
- 9Z=&r;
> 0,
(29)
- 9Z2&rj2]
= 0.
(30)
The above conditions can be satisfied by levying tax-prices at the rates 12X E and 9h Z2 on the mean and variance of the discharge rates, respectively, of the individual firms. Since X depends on E and (Y, the latter connection being via Z, reductions in the allowable levels of aggregate waste discharge, E, and in the probability that those levels are not exceeded, (Y,will increase both tax rates. As one would intuitively expect, a decrease in a! (i.e., an increase in the probability that the standard will be met) requires a higher tax on the variance relative to the mean. For given levels of E and (Y, however, the tax-price ratio is constant, so that in the absence of full information concerning the profit functions of individual firms, the adoption of a decentralized, iterative tax-price procedure along Baumol/Oates lines will not involve simultaneous iteration on both tax-price rates. It is especially worth emphasizing that the above approach requires separate taxes on the mean and the variance. Furthermore, separate taxes on the mean and variance will still be required even in the special case where the individual firms cannot vary their mean rates of discharge independently of their variances.’ 5. CONCLUSIONS
To focus on the effects of formulating environmental quality constraints in probabilistic terms, the paper abstracts from a number of features which are common to real-world environmental problems. Analysis of environmental problems ‘This can be seen by considering the special case where for each firm uj* = kjpj, and kj is a constant. In this case a constant per unit tax on p,, equal across all firms, would yield an optimal solution only where k, = k (all j).
110
BEAVIS
AND
WALKER
must in practice take into account those spatial aspects relevant to the problem under consideration. Such considerations can easily be incorporated into the mode16, but their introduction has no affect on the method of analysis nor on the conclusions obtained. We have argued that when environmental quality constraints are expressed in the form: Prob. (Ce, 2 E) < (Y,then, in general, we expect the feasible set to be nonconvex. The proposed solution to alleviating the problems caused by nonconvexity is in keeping with the so-called “standards-and-prices” approach to environmental problems, in that it is designed to ensure that the environmental quality constraint is satisfied, albeit at the expense of some loss in aggregate firms’ profits. An alternative way of overcoming the nonconvexity problem would be to allow the environmental constraint to be sacrificed at the expense of some gain in aggregate firms profits. To apply this approach one would simply take the convex-hull of the set defined by the environmental quality constraint in Eq. (5). Adoption of the latter approach, however, would raise the question of why an exogenously determined, environmental quality standard was considered in the first place. We are aware that if the number of firms is small then the adoption of a decentralization tax-price approach, whether of the iterative variety or not, is likely to run into the familiar problem, which arises even in the absence of uncertainty, that firms may no longer regard their influence on tax-prices as negligible. APPENDIX
Given the assumptions regarding firm technology presented in Section 2, we show below that the firm’s maximum expected profits are a strictly concave function of the mean and variance of its discharge rate. Proof Let x = (X,, x~), and i = (i,, i,) be the profit maximizing input vectors associated with the given mean and variance of the discharge rate pairs (j& a2), and - (fi, a2), respectively. From the strict concavity of the production function for O
n*(xz
+(1 - X)X) >, MI*(x)
+ (1 - x)rI*(x),
(1)
from which it follows that: rI*( x/z + (1 - X)/% XC2 + (1 - X)$)
> xn*(ic
8’) + (1 - h)rI*(j&
- _ 52) (2)
Hence, strict concavity of II*(fi,
S2) requires:
p(Ax + (1 - X)X: e) Q hp(7:
e) + (1 - X)p(X:
8)
(3)
and 02(xx 60n
how
+ (1 - h)X: 8) 4 h&(x:
this may be done
see, for example,
Beavis
e) + (1 + +2(x: and Walker
[2] or Tietenberg
e) [3].
(4)
STOCHASTIC
ENVIRONMENTAL
111
STANDARDS
where, for example, p(X : (9) and o*(X : 19) denote the mean and variance of the discharge rate, respectively, which correspond to the input levels X. The linearly homogeneity of e in X, and x2 implies: &IX
+ (1 - x)X: e) = xcL@: e> + (1 - A)~(?:
e)
(5)
so that Eq. (3) is satisfied. Now suppose
u2(ii~ + (1 - x)X: e) > h*(z: e>+ (1 - A)u2(i: e).
(6)
The linear homogeneity of e, implies
u*(~x + (1 - h)K: e) = A2a2(x:e) + (1 - qZu2(i: e) +2h(l
- A)Cov.[e(X:
e), e(a:
e)],
(7)
where Cov. denotes covariance. Using Eq. (7) in Eq. (6) gives, upon simplification, a2(~:
e) + u2(?: e>- 2pu(z: e)+:
e) < 0,
(8)
where p = Cov.[e(Z: e), e(x: B)]/e(x: 8)e(x: e), and ~(3: 8) and ~(2: 6) are t_he standard deviations of the discharge rate associated with the input levels X, and f, respectively. The smallest possible value for the left-hand side of the above inequality occurs when pjs at its maximum, which is unity. But for p = 1, Eq. (8) reduces t0 [~(x: e) + U(X: e)]* c 0, which is impossible. Hence, Eq. (4) must hold. The strict concavity of lI*(@, 6’) is therefore established. REFERENCES 1. W. J. Baumol and W. E. Oates, The use of standards and prices for the protection of the environment, Swedish J. Econ. 73 (1971), 42-54. 2. B. Beavis and M. Walker, Interactive pollutants and joint abatement costs: Achieving water quality standards with effluent charges, J. Enoiron. Econ. Manog. 6 (1979), 275-286. 3. T. H. Tietenberg, Controlling pollution by price and standard systems: A general equilibrium analysis, Swedish J. Econ. 75 (1973), 193-203.
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
10,
103- 111 (1983)
Achieving Environmental Standards with Stochastic Discharges’ BRIAN BEAVIS AND MARTIN WALKER Department
of Economics,
University
of Newcastle upon Tyne, England
Newcastle
upon Tyne NE1
7RU,
Received April 4, 1982; revised March, 198 1 The inherent stochastic nature of the levels and composition of environmental waste discharges are well known to environmental control agencies, who recognize that realistically environmental quality constraints must be formulated in probabilistic terms. Given the specification of environmental constraints in such terms, it is shown that the set of activity vectors which satisfy the constraint will, in general, be nonconvex. This militates against the determination of an efficient solution. A means of avoiding the potentially high computational and informational costs associated with the determination of an efficient solution in such cases is proposed.
1. INTRODUCTION
Given the inherent difficulty of measuring environmental damage resulting from pollution, it is not surprising that policy-makers have directed attention toward a policy of attaining exogenously determined environmental quality standards as a second-best solution. In practice, producers are frequently unable to predict with any great degree of accuracy the quantity and quality of wastes associated with specific levels of their productive activities. Moreover, waste treatment processes themselves are often subject to stochastic influences, particularly when biological treatment is involved. The inherent stochastic nature of the levels and composition of environmental waste discharges are well known to environmental control agencies, who recognize that realistically environmental quality constraints must be formulated in probabilistic terms. * In E.E.C. countries, for example, water pollution standards are often expressed in terms of allowable upper-bounds on the 95th percentile of the aggregate rate of discharge of pollutants to water courses. The present paper examines the problems involved in efficiently achieving exogenously determined environmental quality constraints, where the latter are expressed in probabilistic terms. The specification of environmental constraints in such terms introduces an important nonconvexity which militates against the determination of ‘We are grateful to the Department of the Environment for financial support.‘The Department is not necessarily in agreement with the views expressed. The paper has also benefited from comments by an anonymous referee. ‘The agencies typically are aware that the assimilative capacity of the environment is itself also subject to stochastic influences. We abstract from such problems in the present paper. 103 0095~0696/83 $3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved
104
BEAVIS
AND
WALKER
an efficient solution. A means of avoiding the potentially high computational and informational costs associated with the determination of an efficient solution in such cases is proposed. The proposal introduced can perhaps be regarded as a third-best solution to the pollution control problem in that it automatically guarantees that the environmental constraint is satisfied, it involves some loss in efficiency, but economizes on potentially high computational costs. It is also amenable to decentralization via a tax-price approach, and appropriate tax-prices are derived in the paper.
2. THE FIRM WITH STOCHASTIC
DISCHARGES
For simplicity, consider a firm which produces a single output, q, with a technology summarized by a conventional, twice continuously differentiable, strictly concave production function q = 4(x,), where x, is a vector of inputs, and q(0) = 0. In addition to producing output, the firm also generates wastes as a by-product of its production activities. The firm has the option of either discharging the wastes it generates directly to the environment or of treating the wastes before they are discharged. Specifically, the wastes discharged by the firm to the environment, which we denote by e, depend on the levels of inputs used in production, x,, the levels of inputs used in waste treatment, x2, and a continuous random variable 8, which has density function h(8) independent of x, and x2. Formally it is assumed that:
e = e(x,, x2, 81, where e is twice continuously differentiable, with de/ax, > 0, de/k, < 0, and that e is linearly homogeneous for a given level of 0; that is, constant returns to scale prevail in waste treatment. We also require e(0, x2, 8) = 0; if the firm does not engage in production it discharges no waste. If x2 = 0, the firm does not treat its wastes before they are discharged to the environment. The above simple model allows for the possibility that the production activities of the firm give rise to random waste generation, as well as for the possibility that waste treatment itself is an uncertain process. By varying the levels of inputs used in production and waste treatment the firm can alter its mean rate of discharge:
p = /e(x,, x2,
8)h(8)
de,
and the variance of its discharge rate: a2
=-
/[ e( x1,x2, e)]‘h(e)
de - p2.
(3)
However, if e is either multiplicatively or additively separable in 0, then the firm will be unable to vary its mean rate of discharge independently from the variance of its
STOCHASTIC
ENVIRONMENTAL
105
STANDARDS
discharge rate.3 The more general possibility that the firm can vary its mean rate of discharge independently from the variance of its discharge rate is assumed in the body of this paper. The model also assumes that the firm is a price-taker in both the output and input markets, and that its objective is the maximization of expected profits. For given levels of the mean and variance of the rate of waste discharges, fi and h2, respectively, the firm’s maximum expected profits, II*, can be taken as a function of the given levels of the mean and variance of its discharge rate. Under the above assumptions it can be shown that II* = II*@, 82) is strictly concave (see the Appendix). 3. THE ENVIRONMENTAL
CONTROL
AGENCY’S
PROBLEM
If the discharge rates of the individual firms are stochastic, then the Environmental Control Agency can realistically only formulate environmental quality standards in probabilistic terms. Such constraints take the following form:
where E and (Y are given parameters, (typically (Y will be small), and ej is the discharge rate of firm j. If the number of firms, n, is large, if the discharge rates of the individual firms are independent of each other, and if no single firm’s discharge rate is dominant in variance, then, from the Central Limit Theorem, the aggregate rate of waste discharge into the environment will be normally distributed with mean p = &j and variance cf2 = X(1,2, where pj and u; are the mean and variance, respectively, of the discharge rate of firmi. Under these conditions the environmental quality constraint Eq. (4) can be expressed in the form: E - CPj a ZdCu,‘,
(5)
where Z is given by lZ,1/~e-(‘/2)“’
du= 1 -a.
In many situations, the number of firms will not be sufficient for the Central Limit Theorem to be applicable. In such cases Chebyshev’s Inequality can be 3To see this, we have on taking the total differentials of the mean and variance of the discharge rate: j-&h(e)
de
1 [ dx,
+
jEh(e)
de
1 dxz,
where dx, and dx, are vectors of differentials, and setting dp = 0, gives: do2 = 2[/e(ae/ax,)h(e)
de]
If e(x,, x2, e) = g(x,, x2)e, then se/ax, 2g(x,, lag/ax2
x2x[ag/ax,jeh(e) jeh(e) de]
de] dx, + [ag/ax2jeh(e) dx2 = 0 SOthat da2 = 0.
dx, =
+ 2[le(ae/ax,)h(e)
de]
dx,.
Bag/ax,, and aejax, = eag/ax2, and do2 de] dx2). But dp = [ag/ax,jeh(e) de] dx,
A similar analysis can be performed for the case where: e(x,, x2, 0) = g(x,, x2) + 0.
= +
106
BEAVIS
AND
WALKER
applied. Assuming independence of discharge rates, the latter requires:
where k is a given constant. From Eq. (6) we clearly have: Prob. ( cej so that the environmental
> cpj
+ kixof)
G l/k*,
quality constraint Eq. (4) can be expressed in the form:
which has the same form as Eq. (5). In the remainder of this paper the analysis will be performed for the normal/large numbers case. It is comforting to know, though, that the mathematical structure of the problem extends to a much wider class of problems. It is assumed that the Agency seeks to establish the vector (CL,,. . . , p,,, a:, . . . , a:), which maximizes the sum of firms’ profits CIIj, subject to the environmental quality constraint (5). Given an appropriate constraint qualification, the following Kuhn-Tucker conditions are necessary for an optimal solution to the Agency’s problem:
aIIj --A pj[ap,1=o,y., u$$-&]-o,7, E
-
CPj
X[E - cpj
-
zJCD/2
- Z&$]
>,
0,
= 0,
(11) (12) 03) (14)
where h is the Lagrange multiplier. If the set of vectors (CL,,. . . , p,,, a:, . . . , CT,‘)which satisfy Eq. (5) were to form a convex set then conditions (9)-(14) would also be sufficient conditions for an optimal solution to the Agency’s problem, as the profit functions of the individual firms are by construction strictly concave functions of the mean and variance of their discharge rates. However, the set of vectors, (p,, . . . , p,,, a:,. . . , CT,‘) which satisfy Eq. (5) do not in general form a convex set. Proof: Let S denote the set of means and variances of discharge rates of the individual firms which satisfy the environmental constraint (5), and suppose that S
STOCHASTIC
ENVIRONMENTAL
107
STANDARDS
is convex. Let p = (p,, . . . , fi,, 6:, . . . , 6:) and 9 = (p,, . . . , &,, a:, . . . , 62) be elements of S, which satisfy Eq. (5) with equality, and suppose jj * 9. Since S is convex: xp + (1 - X)9 E s,
forOcX<
1;
that is, E-C[Afij+
(1 -x)/ij]-ZJC(h6;+(1
-X)q2)
>O.
(15)
Now set pi = fij in Eq. (5) and multiply through by X; then set p, = pj in Eq. (15) and multiply through by (1 - A), and add the two resulting equations to obtain E - c[Xli,
+ (1 - “)fij]
- z[JxzC&;
+ \/(l - x)2C52]
= o.
Then using Eq. (15) yields:
which on squaring both sides and combining X(1 - X):
terms, gives, upon dividing
by
06) Squaring both sides of Eq. (16) and simplifying finally gives:
(17) which hold if, and only if, Csj2 = Dj2, which in turn implies that Cpj = Cfij as 9 and 7 both satisfy Eq. (5) with equality.4 Since it is clear that there exist points on the boundary of S for which CS2 * C6” and CrZ, f #, in general S will not be a convex set. The above proof depends only on the assumption that firm’s discharge rates are independent. Indeed, we are confident that even this requirement can be relaxed, but we have not attempted to do so. Hence, in general, we would expect the feasible set defined by an environmental constraint of the form (4) to be nonconvex. Given that the Kuhn-Tucker conditions, Eq. (9)-(14) are necessary but not sufficient conditions for a global solution to the Agency’s problem, the location of points which simultaneously satisfy the environmental quality constraint and globally maximize firms’ profits in the aggregate will be an extremely costly exercise, requiring the institution of some form of computational search procedure and detailed information concerning the profit functions of the individual firms. Moreover, the Agency’s problem is not amenable to solution via the adoption of a decentralized, tax-price scheme of the type originally proposed by Baumol/Oates [l], for if at a given tax-price level the environmental quality constraint is exactly achieved, given the nonconvexity of the constrained region, there can be no guarantee that aggregate firms’ profits are global& maximized at that tax-price level. vary
41t is worth noting that this condition must their mean rates of discharge independently
be satisfied whether of their variances.
or not individual
firms
are able to
108
BEAVIS
AND
WALKER
4. AN ALTERNATIVE
SOLUTION
In such circumstances it seems worthwhile considering alternative schemes for allocating the scarce environmental capacity among discharging firms which are less costly to implement, but which may entail some loss in efficiency. An appealing alternative is to restrict consideration to only those allocations which lie in the largest convex set enclosed by the set of allocations which satisfy the environmental quality constraint Eqs. (5) or (8). Such an approach automatically guarantees the appropriate level of environmental quality, but may entail a loss in efficiency in that aggregate firms’ profits will not necessarily be globally maximized. Nevertheless it would seem to offer the prospect of finding a “reasonably good” approximation to the Agency’s problem, and, additionally, allows for the adoption of a decentralized, iterative tax-price approach, thereby potentially avoiding the high computational and informational costs associated with the determination of a globally efficient solution. The environmental constraints appropriate to such an approach can be determined as follows. Let S denote the set of vectors (CL,,. . . , pL,, a:, . . . , u,‘) which satisfy the environmental constraint (5) and B(S) its boundary. Let S’ be the largest convex set enclosed by S and B( S’) its boundary, and define p = x11, and u2 = Cuj2. Along B(S) du2/dp -C 0 and d2u2/dp2 > 0, so that B(S) is strictly convex towards the origin in (p, u2)-space. Moreover, since S’ is the largest convex subset of S, in (pu2)-space S’ must have a linear boundary which is tangential to B(S) in that space. Let (p,O), (0, C2) E B(S’) and (fi, b2) E B(S’) n B(S). Then h2 = (E2 2fiE + fi2)/Z2; and since B(S) is linear: 52-62
62
-=-= P
2(E - P)
P-P
z2
’
where the last term is the slope of B(S’), which yields: /I = +(fi + E), a2 = ( E2 - p2)/z2.
(18) (19)
B(S’) can be determined by maximizing the product of p and E2, i.e., /-iti2 = (p + E)( E2 - p2)/z2.
(20)
A necessary and sufficient condition for Eq. (20) to be maximized is that: (-3fi
+ E)(P + E) = 0,
(21)
which, since fi = -E is meaningless here, implies ,b = +E.
(22)
Hence, B( S’) in (II, u2)-space is given by: 02
4E2 = --4E 3~2 p+gz27
(23)
STOCHASTIC
so that the environmental
ENVIRONMENTAL
constraint appropriate
109
STANDARDS
to the “third-best”
problem is:
(24 Maximization of aggregate firms’ profits subject to the environmental constraint, Eq. (24) provides the following Kuhn-Tucker conditions, which are necessary and sufficient, given an appropriate constraint qualification.
(25) (26)
(27)
4E2 - 12Expj X[4E2 - 12Ecpj
- 9Z=&r;
> 0,
(29)
- 9Z2&rj2]
= 0.
(30)
The above conditions can be satisfied by levying tax-prices at the rates 12X E and 9h Z2 on the mean and variance of the discharge rates, respectively, of the individual firms. Since X depends on E and (Y, the latter connection being via Z, reductions in the allowable levels of aggregate waste discharge, E, and in the probability that those levels are not exceeded, (Y,will increase both tax rates. As one would intuitively expect, a decrease in a! (i.e., an increase in the probability that the standard will be met) requires a higher tax on the variance relative to the mean. For given levels of E and (Y, however, the tax-price ratio is constant, so that in the absence of full information concerning the profit functions of individual firms, the adoption of a decentralized, iterative tax-price procedure along Baumol/Oates lines will not involve simultaneous iteration on both tax-price rates. It is especially worth emphasizing that the above approach requires separate taxes on the mean and the variance. Furthermore, separate taxes on the mean and variance will still be required even in the special case where the individual firms cannot vary their mean rates of discharge independently of their variances.’ 5. CONCLUSIONS
To focus on the effects of formulating environmental quality constraints in probabilistic terms, the paper abstracts from a number of features which are common to real-world environmental problems. Analysis of environmental problems ‘This can be seen by considering the special case where for each firm uj* = kjpj, and kj is a constant. In this case a constant per unit tax on p,, equal across all firms, would yield an optimal solution only where k, = k (all j).
110
BEAVIS
AND
WALKER
must in practice take into account those spatial aspects relevant to the problem under consideration. Such considerations can easily be incorporated into the mode16, but their introduction has no affect on the method of analysis nor on the conclusions obtained. We have argued that when environmental quality constraints are expressed in the form: Prob. (Ce, 2 E) < (Y,then, in general, we expect the feasible set to be nonconvex. The proposed solution to alleviating the problems caused by nonconvexity is in keeping with the so-called “standards-and-prices” approach to environmental problems, in that it is designed to ensure that the environmental quality constraint is satisfied, albeit at the expense of some loss in aggregate firms’ profits. An alternative way of overcoming the nonconvexity problem would be to allow the environmental constraint to be sacrificed at the expense of some gain in aggregate firms profits. To apply this approach one would simply take the convex-hull of the set defined by the environmental quality constraint in Eq. (5). Adoption of the latter approach, however, would raise the question of why an exogenously determined, environmental quality standard was considered in the first place. We are aware that if the number of firms is small then the adoption of a decentralization tax-price approach, whether of the iterative variety or not, is likely to run into the familiar problem, which arises even in the absence of uncertainty, that firms may no longer regard their influence on tax-prices as negligible. APPENDIX
Given the assumptions regarding firm technology presented in Section 2, we show below that the firm’s maximum expected profits are a strictly concave function of the mean and variance of its discharge rate. Proof Let x = (X,, x~), and i = (i,, i,) be the profit maximizing input vectors associated with the given mean and variance of the discharge rate pairs (j& a2), and - (fi, a2), respectively. From the strict concavity of the production function for O
n*(xz
+(1 - X)X) >, MI*(x)
+ (1 - x)rI*(x),
(1)
from which it follows that: rI*( x/z + (1 - X)/% XC2 + (1 - X)$)
> xn*(ic
8’) + (1 - h)rI*(j&
- _ 52) (2)
Hence, strict concavity of II*(fi,
S2) requires:
p(Ax + (1 - X)X: e) Q hp(7:
e) + (1 - X)p(X:
8)
(3)
and 02(xx 60n
how
+ (1 - h)X: 8) 4 h&(x:
this may be done
see, for example,
Beavis
e) + (1 + +2(x: and Walker
[2] or Tietenberg
e) [3].
(4)
STOCHASTIC
ENVIRONMENTAL
111
STANDARDS
where, for example, p(X : (9) and o*(X : 19) denote the mean and variance of the discharge rate, respectively, which correspond to the input levels X. The linearly homogeneity of e in X, and x2 implies: &IX
+ (1 - x)X: e) = xcL@: e> + (1 - A)~(?:
e)
(5)
so that Eq. (3) is satisfied. Now suppose
u2(ii~ + (1 - x)X: e) > h*(z: e>+ (1 - A)u2(i: e).
(6)
The linear homogeneity of e, implies
u*(~x + (1 - h)K: e) = A2a2(x:e) + (1 - qZu2(i: e) +2h(l
- A)Cov.[e(X:
e), e(a:
e)],
(7)
where Cov. denotes covariance. Using Eq. (7) in Eq. (6) gives, upon simplification, a2(~:
e) + u2(?: e>- 2pu(z: e)+:
e) < 0,
(8)
where p = Cov.[e(Z: e), e(x: B)]/e(x: 8)e(x: e), and ~(3: 8) and ~(2: 6) are t_he standard deviations of the discharge rate associated with the input levels X, and f, respectively. The smallest possible value for the left-hand side of the above inequality occurs when pjs at its maximum, which is unity. But for p = 1, Eq. (8) reduces t0 [~(x: e) + U(X: e)]* c 0, which is impossible. Hence, Eq. (4) must hold. The strict concavity of lI*(@, 6’) is therefore established. REFERENCES 1. W. J. Baumol and W. E. Oates, The use of standards and prices for the protection of the environment, Swedish J. Econ. 73 (1971), 42-54. 2. B. Beavis and M. Walker, Interactive pollutants and joint abatement costs: Achieving water quality standards with effluent charges, J. Enoiron. Econ. Manog. 6 (1979), 275-286. 3. T. H. Tietenberg, Controlling pollution by price and standard systems: A general equilibrium analysis, Swedish J. Econ. 75 (1973), 193-203.