Markets in the Licenses and Efficient Pollution - Thesis | ECON 781, Papers of Economics

Download Markets in the Licenses and Efficient Pollution - Thesis | ECON 781 and more Papers Economics in PDF only on Docsity! JOURNAL OF ECONOMIC THEORY 5, m-a8 (1972) Markets in Licenses and Efftcient Pollution Control Programs* W. DAVID MONTGOMERY Division of‘ the Humanities and Social Sciences, Calijbrnia Institute of Technology, Pasadena, California 91109 Received May 19, 1972 1. INTRODUCTION Artificial markets have received some attention as a means of remedying market failure and, in particular, dealing with pollution from various sources. Arrow [I] has demonstrated that when externalities are present in a general equilibrium system, a suitable expansion of the commodity space would lead to Pareto optimality by bringing externalities under the control of the price system. Since his procedure is to define new commod- ities, each of which is identified by the type of externality, the person who produces it and the person who suffers it, his conclusion is pessimistic. Each market in the newly defined commodities involves but one buyer and one seller, and no forces exist to compel the behavior which would bring about a competitive equilibrium. On the other hand, many forms of pollution are perfect substitutes for each other. Sulfur oxide emissions from one power plant trade off in the preferences of any sufferer with sulfur oxide emissions from some other power plant at a constant rate. This fact leads to the possibility of establishing markets in rights (or “licenses”) which will bring together many buyers and sellers. Dales [2] has discussed a wide variety of such arrangements. Unfortunately, because of the elements of public goods present in most environmental improvements, it appears unlikely that markets in rights, containing many sufferers from pollution as participants, will lead to overall Pareto optimality. They can only serve the more limited, but still * Parts of this article appeared in my Ph.D. dissertation “Market Systems for the Control of Air Pollution,” submitted to the Department of Economics at Harvard University in May, 1971. Research on this thesis was partly supported under Grant No. AP-00842 from the Environmental Protection Agency to Walter Isard. I am also indebted to Kenneth Arrow and James Quirk for valuable advice. Needless to say, all errors are solely the responsibility of the author. 395 &py&hat 8 1972 by Academic Press, Inc. All rights of reproducrion in any form reserved. 396 MONTGOMERY valuable, function of achieving specified levels of environmental quality in an efficient manner. An example of this function is found in a proposal by Jacoby and Schaumburg [6] to establish a market in licenses (or “BOD bonds”) to control water pollution from industrial sources in the Delaware estuary. The purpose of the present article is to provide a solid theoretical foundation for such proposals. Markets such as those proposed by Jacoby and Schaumburg will be characterized in a general fashion, and it will be proved that even in quite complex circumstances the market in licenses has an equilibrium which achieves externally given standards of environmental quality at least cost to the regulated industries. Two types of license are discussed: a “pollution license,” and an “emission license.” The emission license directly confers a right to emit pollutants up to a certain rate. The pollution license confers the right to emit pollutants at a rate which will cause no more than a specified increase in the level of pollution at a certain point. Since a polluter will in general affect air or water quality at a number of points as a result of his emissions, he will be required to hold a portfolio of licenses covering all relevant monitoring points. All such licenses are free transferable. A main thesis of this article is that the market in pollution licenses will be more widely applicable than the market in emission licenses. 1.1. The Applicable Pollution Control Problem Consider the following problem of pollution control: In a certain region there is a set of IZ industrial sources of pollution, each of which is fixed in location and owned by an independent, profit-maximizing firm. The prices of the inputs and outputs of these firms are fixed, because the region is small relative to the entire economy. Therefore any change in the level of output of a firm or industry in the region will have only a negligible impact on the output of the economy as a whole, and prices will be unaffected by output changes in the region. These firms are represented by a set of integers I == {l,..., n}. Some regional standard of environmental quality in terms of a single pollutant has been chosen as a goal by a resource management agency. This standard is denoted by a vector Q* = (ql*,..., qm*). If air pollution is the particular area of interest, qi* might be an annual average concen- tration of sulfur dioxide at point j in an air basin. If water pollution is involved, qi* might be a measure of dissolved oxygen deficit at point j on a river. Since there is only one pollutant present in the region, the elements of Q* represent concentrations of the one pollutant at various locations. The development of a decentralized system for achieving environmental goals at a number of different locations is the most im- portant contribution of this article. MARKETS IN LICENSES 399 Now consider the case in which the firm must adopt an emission level ei and adjusts its output in order to obtain maximum profit for the fixed level of emission. Define yic by The cost to firm i of adopting emission level ei is defined as the difference between its unconstrained maximum of profit and its maximum of profit when emissions equal e, . That is, F,(ei) = 2 P,(yi, - i?ir) - [G&Jo ,..., LiR, ei) - GAja ,...,Y~R , cd]. T (1.1) This cost is composed of two terms: the change in gross income from altering the output vector and the change in costs from setting emissions at a nonoptimal level (with an optimal adjustment of output).’ Consider the variation in Fj(eJ when a small change is made in ei . Differentiating totally with respect to ei , we find dFi(ei) = ‘-- 1 (pr - 2) $$ dei + -$$ de, . (1.2) c We have assumed that output levels adjust to maximize profit for a given level of ei . That is, yi, adjusts so that pT - aG,/ayi7 = 0 forj = l,..., r. Therefore [13], dFi(eJdei = aGi/aei . (1.3) It can further be shown that the convexity of Gi( y,, ,..., yiR , ei) implies the convexity of FJeJ. THEOREM 1.1. If Gi( yi, ,..., yjR , ei) is COnVf?x, Fi(ei) is a/So conuex. Proof. The proof is immediate from the definition of convexity. It is convenient to be able to use a single-valued function F,(eJ to associate with any emissions level its cost. The properties of Fi(ei) proved above allow us to conclude that any relevant conditions which are satisfied r Three general classes of techniques of emission reduction are available. First, emissions can be reduced by reducing the scale of output, or by altering the product mix of the firm. Second, the production process or the inputs used, such as fuels, can be altered. Finally, “tail-end” cleaning equipment can be installed to remove pollutants from effluent streams before they are released into the environment. All three of these techniques will commonly be found in combination. 400 MONTGOMERY by the partial derivative of Gi with respect to ei will be satisfied by the derivative of Fi . In particular, we can conclude that if the profit- maximizing firm has any choice of ei , it will minimize Fi(ei) subject to whatever costs or constraints we impose on it. Moreover, if Gi is convex, it follows that the conditions under which x:i Fi(ei) is minimized are the same as the conditions under which the total economic cost to firms of emissions control is minimized. 2. THE CHARACTERIZATION OF AN EFFICIENT EMISSION VECTOR The goal of management is limited to bringing about an emission vector which will result in air of quality Q* at least total cost to the region. Such an emission vector is called efficient, and designated E**. The concept of least cost to the community is also given a specific meaning: it is the minimum of the sum xi Fi(e,). With some risk of ambiguity, this sum is called “joint total cost.” To provide a reference to which later results can be compared, a general solution for the efficient emission vector can now be derived. The problem is to choose the vector E = (e, ,..., e,) to minimize CiFi(ei) subject to the constraints E>,O and EH < Q*, where Q* > 0, hii > 0 for all i, j. We will label this the “total joint cost minimum problem.” Our exploration will proceed throughout this article on the assumption that Gi is convex. This implies that Fi(ei) is convex, and therefore that z:i Fi(ei) is also convex. It is also assumed that H is semipositive. The typical shape of FJe,) is illustrated in Fig. 1. Minimizing a convex function subject to linear constraints and non- negativity constraints is equivalent to finding the saddle point of an associated Lagrangean. Formally, (E**, U**) = (ef* ,..., e,**, uf* ,..., z&*) will be a saddle point of the expression with E* * > 0, U** > 0. The differential Kuhn-Tucker conditions for this saddle point are MARKETS IN LICENSES 401 These conditions are necessary and sufficient [7]. Moreover, it is easy to show that the minimum does in fact exist. THEOREM 2.1. E** and U** satisfying (2.1) and (2.2) exist. Proof. Since CiFi(cJ = 0 and CiFi(ei) 3 CiFi(Zi), for ei > 0, xj Fi(ei) is bounded from below. By hypothesis, the set is not empty. Therefore, xi I;,(e,) is defined on a nonempty closed set and bounded from below; therefore, it attains a minimum over the set Y for some element of Y. If x:i Fi(eJ is not strictly convex, then E** need not be unique. Since, however, minEEV xi Fi(eJ is unique, it does not matter what particular minimizer is chosen. Therefore, I shall refer to the vector which minimizes costs; the reader may interpret this reference as meaning “any element of the set of E which minimizes z:i Fi(eJ.” The following theorem is true if xi Fi(eJ is strictly convex. THEOREM 2.2. If E** minimizes xi Fi(eJ subject to EH < Q* and E>O,thenE** GE. Proof. Assume per contra that e?* > ?i for some i = i’. Then Fi$e:*) > Fi,(Ci*) and hi&,* > hijei, . Therefore and C hije’* + hi,jSi < 1 hije,“*. (2.3a) (2.3b) i#i’ 1 By (2.3b) the vector (et* ,..., r;i, ,..., eX*) satisfies EH < Q* and by (2.3a) E** does not minimize x3i Fi(ei). 3. MARKETS IN LICENSES We can now proceed to the construction of markets which, in equi- librium, lead to emission rates which satisfy the conditions of Theorem 2.1. A set of licenses are defined, such that the possession of licenses confers the right to carry out a certain average rate of emission. Consider the function A(Hi 9 &I, 404 MONTGOMERY That is, the relevant element of the diffusion matrix is taken to be a correct predictor of the amount which an average rate of emission at point i contributes to pollutant concentration at point j. Each firm is allowed to have an average rate of emission which produces no more pollution at any point than the amount which the firm is licensed to cause at that point. The firm will minimize li,(eJ + Cjpj(&i - li”j) subject to the licensing constraint. In the theorems which follow we use the convention that lj is a scalar, a total number of licenses allowing pollution at point j. Thus xi iii = I, . When Li = (Ii, ,..., lim) and L = (II ,..,, I,), xi Li = L. The strategy of proof is to define a market equilibrium relative to an initial allocation of licenses and to derive necessary and sufficient con- ditions for its existence. A subsidiary construction, called a “license- constrained joint cost minimum,“is defined and shown to exist. It generates a second set of necessary and sufficient conditions. It is shown that the emission vector and shadow prices which satisfy the conditions of a license-constrained joint cost minimum for given totals of licenses also satisfy the conditions of competitive equilibrium relative to any initial allocation of licenses in which the given totals are completely distributed among firms. An equilibrium license portfolio for each firm is constructed, and shadow prices on each firm’s licensing constraints are identified. To prove that a competitive equilibrium achieves the joint cost minimum defined in Section 2, we show that when license totals equal desired air qualities any emission vector and price vector which satisfy the equi- librium conditions also satisfy the conditions for efficiency. In the course of the proof the efficient emission vector is identified as the equilibrium emission vector and shadow prices on the air quality constraints in the overall joint cost minimum are identified as the prices of licenses. The equilibrium license portfolio has each element just equal to the pollution caused by the efficient rate of emission for the corresponding firm. The proof itself is rigorous and abstract. DEFINITION. A market equilibrium is an n + 2 tuple of vectors Li* >, 0, E* > 0, and P* > 0 such that Li* and E* minimize subject to iii - hijei > 0, j = l,..., m, for all i and which also satisfy the market clearing conditions MARKETS IN LICENSES 405 LEMMA 3. I. A market equilibrium exists if and only if there exist vectors <ui*, ,*--, dJ 2 0 i = I,..., n, (PI*,*.*, Pm*) 3 0 such that Fi’(ei*) + C uzhij > 0, ei” ki’(ei*) + C UGh,] = 0, (3.2a) j pi* - 24; 3 0, c c,: - ui”;:] = 0, (3.2b) l,?;: - hijei* > 0, C ::[I: - hiiei*] = 0, (3.2~) for all i and T (1: - c!d < 0, $ pj* [; (1: - C’J] = 0. (3.24 Proof. First we characterize the vectors Li* and es* which minimize cost for the firm. Minimizing a function is equivalent to maximizing its negative; and the negative of a convex function is concave. Therefore, we can state the problem of the firm as one of maximizing the concave function --Fi(ei) - &j*(lij - $j). Form the Lagrangean From the Kuhn-Tucker theorem the following conditions are necessary and sufficient for the constrained maximum; where #Jj(lz ,..., ii*, , f?j*, 24: ,..., UX) = +i*. +*/aei < 0, a+i*/alij < 0, ej* * (a$i*/&?i) = 0, C 1: * (a&*/iYij) = 0, 1 24: - (afji*/aUij) = 0. Performing the indicated differentiation gives 3.2a to 3.2~ which must be satisfied for all i. Equation (3.2d) repeats the market clearing condition. 406 MONTGOMERY DEFINITION. A license-constrained joint cost minimum is a vector E** which minimizes C Fi(ei) subject to EH < Lo and E >, 0. In making this definition we assume that some arbitrary vector of licenses Lo is issued by the management agency. We must assume that the set (E ) EH < Lo and E > 0} is not empty. Then the same argument used in Section 2 to establish the existence of a joint cost minimum will establish the existence of a license-constrained minimum. We now can use the following lemma to prove existence of an equilibrium on the pollution license market. LEMMA 3.2. An emission vector E** is a license-constrained joint cost minimum ifand only if there exists a vector (UT*,..., uz*) > 0 such that Fi’(eT*) + C uj**hij 3 0, 7 @* [E;‘(e?*) + T uT*hij] = 0, (3.3a) j IjO - C hije:* > 0, (3.3b) z C uj** [lj” - C hijeT*] = 0. j E Proof. The proof is as in Lemma 3.1. The market equilibrium will exist for any distribution of licenses such that 1: > 0 and z:i l:j = Ijo. THEOREM 3.1. A market equilibrium of the pollution license system exists for xi l,Oj = lj’. Proof. We proceed constructively by using (3.3a) and (3.3b) to show that ei* = e,F*, I: = hijeT*, pi* = uj** and u$ = up* for all i satisfy (3.2a)-(3.2d). Equation (3.2a). Since Fi’(e:*) + Cj uT*hij 2 0 for all i, and ef* > 0, it follows from that T e,* [Fi’(ef *) + 1 uT*h,j] = 0 j e,* [Fi’(et*) + C uF*hl = 0 j for all i. Therefore e?* and uf* satisfy 3.2a for all i. 9 D MARKETS IN LICENSES 409 purchases of licenses) can then be represented as the minimization of the sum F,(eJ + Cj hijpj*e, . The emission rate ei* in Fig. 1 is the minimizer of this sum. Theorem 3.1 states that there exist prices which clear markets for licenses when each firm chooses license holdings, and emission rates ei*, to minimize cost. COST 0 e: ci EMISSIONS i ------___ E FIG. 1. A-Fi(ei) + XI pjhdjei ; B-Cj pjhifei ; C-fi(ei) f xj Phei - Cr PJ,P, ; D-Fi(ei); E---G p& . The initial allocation of licenses is equivalent to a lump sum subsidy, and is independent of emission level. Therefore, this subsidy can be represented as a horizontal line, cjpj*l& , in Fig. 1. The curve Fi(ei) + xfpj*hijei - Cjpi*fE is the net cost function which represents the actual cost of emission control and licenses. Note that ei* is inde- pendent of the size of the subsidy. Because of this result, the management agency can distribute licenses as it pleases. Considerations of equity, of administrative convenience, or of political expediency can determine the allocation. The same efficient equilibrium will be achieved. It should, however, be noted that in assuming the convexity of Fi(ei) we impose certain conditions relating to nonnegative profits. Let 7Ti be the (maximal) profit earned before the introduction of a licensing system, and let Si(et) be the profit earned when emissions are set at rate ei . Then by (1.1) we have Gi(ei) = fi6 - Fi(ei). In the long run the firm will only stay in business if ;;i(ei) > 0. In this case the cost function will have the 410 MONTGOMERY form Fi(ei) = min(ii, - iii(eJ, ii*). An upper bound, equal to 75$ , is placed on costs incurred by the firm. This upper bound destroys the convexity of the cost function unless t;;(O) < fii . Such an assumption is implicit in the assumption that F,(eJ is convex. It would appear that the need to purchase licenses imposes a cost on the firm additional to the cost of emission control J’Jei). Even though this cost sums to zero for all firms taken together, it may be positive for some individual firms and negative for others. Fortunately, we can prove the following theorem, namely, that if Fi(0) < fii , then even if a firm is allocated no licences initially (i.e., lf$ = 0 for some i and alli), it can still earn nonnegative profits at any levels of emissions and license holding. THEOREM 3.4. IfF,(O) < ifi,Fi(ei*) + Cjpj*Iz d +i . Proof. We have proved that Fg(ei*) + C pj*lz = Fi(ei*) + C pj*hiie.i*. 3 3 If Fi(ei*) + &pi*hfjei* > iii , then Fi(ei*) + Cjpi*hiiei* > Fi(0). But ei = 0 and lij = 0 satisfy ei < n(Hi , ti), so that ei* does not minimize cost subject to the licensing constraint. This contradiction establishes the theorem. This demonstration completes the discussion of pollution licenses. We began by showing that for any vector of licenses Lo which implies feasible concentration levtls at each monitoring point there exists a competitive equilibrium in the license market. We then showed that the concentrations which result from the equilibrium will be less than or equal to the levels permitted by the vector of licenses and that joint total costs are minimized subject to this constraint. Finally we showed that when Lo = Q*, the problem of achieving desired air quality standards at minimum cost is solved by the market in pollution licenses. The major generalization provided by this theorem is that it establishes the possibility of achieving environmental goals at a number of geographic points while maintaining the advantages of a market system. Thus one important objection to the use of economic incentives, that they could lead to change in the pattern of emissions such that although air quality improvements at one point are achieved, it is at the expense of deteriorating air quality elsewhere, is laid to rest. Moreover, we discovered that the fixed totals could be allocated arbitrarily among firms. Overall convexity and the possibility, for each firm, of absorbing all costs of abatement in profits were assumptions necessary for the operation of the system when no information on cost functions is available. MARKETS IN LICENSES 411 We turn now to an alternative licensing system. It will turn out that this system of emission licenses is interesting because it provides a means of linking up the proposal to issue transferable licenses with other proposals for achieving efficient solutions in a decentralized manner. 3.2. The Market in Emission Licenses The effluent charge is a tax which a firm must pay on each unit of pollutant which it emits into a water course. A corresponding charge for air pollution control might be called an emission charge. In order to calculate a charge which will lead to efficiency in air pollution control, the manager must solve in advance the overall cost-minimization problem. It is not difficult to show that the correct tax on emission by firm i is equal to the shadow price on its emissions determined by the minimization of joint total cost. The tax is Cj ui*hij , where uj* is the value of the Lagrange multiplier on the j-th quality constraint evaluated at the optimum. But in order to calculate such a tax the manager must know the cost functions of each firm. It is, of course, possible to obtain that information in an iterative process by varying the tax. This is a cumbersome and politically unattractive procedure, and it has been shown by Marglin [IO] that the information transferred to the regulatory authority by such a procedure is as great as the information needed to set quantity standards for each firm. That is, whenever it is possible to calculate the correct tax it is possible to achieve E** in the initial allocation. A licensing scheme does not require such prior or iterative gathering of information. The market makes the necessary calculations independently in the course of reaching equilibrium. For this reason we are led to consider licensing schemes as superior to taxation. The natural correlate of emission charges is a system of emission licenses. An emission license confers on the firm holding it the right to emit pollutants at a certain rate. It is not always desirable to allow such rights to be transferred on a one-for-one basis: the desirable rule governing exchange of emission rights is that a firm may be allowed to emit up to a level which causes pollution equal to that which would be caused if each firm from which it obtained rights emitted to the maximum extent permitted by the rights which it has given up. We must differentiate rights to emit by the location at which they permit emissions to take place. Then Ik , k = I,..., n, is a quantity of licenses to emit at location k. It is sufficient to allow k to run over the set of firms I since each firm is in a fixed location. Let Zilc represent the quantity of licenses allowing emissions at location k held by firm i. If the exchange of such licenses between polluters at different locations is to be permitted, some rule must be stated regarding the right to emit 414 MONTGOMERY THEOREM 3.5. A market equilibrium in emission licenses exists. Proof. In Theorem 2.2 it was shown that an emission vector minimizing joint total costs subject to the air quality constraints exists, and that in consequence E** and V** satisfying (2.1) and (2.2) exist. Let licenses be issued initially so that k z for all j. Then we show that E ** is an emission vector and V** a price vector satisfying (3.5a)-(3.5d). We begin by proving the following proposition: P.l. If CI, h,Jko 2 xi hije:*, then there exist I& such that xi I$ < lrco, I& >, 0, and & hk& > h,jef* for all i and k. Letting Li = (Ii, ,..., It,) and Hi be the i-th row of the matrix H we may write the inequalities which must have a nonnegative solution in matrix form as (L,* ,..., L,*) rH * . . -Hij < (-H,e:* ,..., -H,ez*, Lo). It is a theorem [3] that either these inequalities or the following inequalities have a nonnegative solution: < 0. We can write (1) as - ?$ hijxlj + x,+li 2 0 G = l,..., n), - $ hijx,$ + x,+~~ 3 0 (i = I,..., n) (1) (2) MARKETS IN LICENSES 415 and (2) as We assume that there do exist nonnegative solutions denoted with superscript O’s to (1) and (2). Let us multiply each line of (1) through by Zio and sum the result over i, giving for all k. Comparing this inequality with (2) we find - i g hijei*xij < - f f hdil:Xkj a i=l j=l i-1 j=1 Since xi hijei* < xi hiili” by hypothesis, and - i 2 hiiei*$j < - i 2 hijef*x$ i=l j-1 j=j j-1 for all k. We remove the minus signs and reverse the inequality, giving for all k. Therefore it must be true for some i that 2 h<jxfj > C hi&j j i for all k. Therefore, it must be true for k = i, which implies This contradiction establishes that there is no nonnegative solution to inequalities (1) and (2) and P. 1 is proved. 416 MONTGOMERY We can now proceed line by line to show that E** and II** satisfying (2.1) and (2.2) also satisfy (3.5a)-(3.5d). Equation (3.5a). From (2.1), ef* and z$* satisfy (3.5a) for all i. Equation (3.5b). Let plc* = Cj z.$*& and U; = z$* for all i. Then they satisfy (3.5b) since plc* - Cj u@,~ = 0 for all i and k. Equation (3.5~). Let & hkj xi l,ok = qj*. Then by (2.2), and by P.l there exist I& 3 0 such that for all i. If > holds for some i and j, and UT* = 0. Therefore, (3.5~) is satisfied with U$ = UT*. Equation (3.5d). If & lFk > x:i Ii*, for some k and hkj > 0, and u?* = 0 for all j. If hki = 0 for that k and some j, then for the corresponding j, u:*hkj = 0. In either case pk* = Cj uT*hkj = 0 and (3.5d) is satisfied. We reverse the direction of inference to prove that if xi hijli’ = qj*, the competitive equilibrium emission vector is efficient. We assume in addition that the rank of H is m: this involves no significant loss of generality since any constraint matrix can be made to satisfy the condition by striking out redundant constraints. The operation of eliminating redundant constraints does not change the set Y of emission vectors which satisfy the constraints. THEOREM 3.6. If XI, hkjlk’ = qj*, E* minimizes xi Fi(eJ subject to EH < Q*, E > 0. Proof. First we note that in proving Theorem 3.4 we established that (3.5a)-(3.5d) are satisfied, for all i, by UQ = uj**, and that the rank of H