Nonconvexity induced by external costs on production: Theoretical curio or policy dilemma?

JOURNAL

OF ENVIRONMENTAL

ECONOMJCS

AND

MANAGEMENT

13,101-128 (1986)

Nonconvexity Induced by External Costs on Production: Theoretical Curio or Policy Dilemma? PAUL

BURROWS*

Department of Economicsand Related Studies, Vniversiv of York Heslington, York YOl 5DD, England Received July 19,1983; revised January 1985

1. INTRODUdTlON I. 1. Prevailing View This paper considers, with a degreeof scepticism, the sweeping policy conclusions that can be found in the theoretical literature concerning nonconvexities in the social production set induced by external costs on firms.’ This literature has provided proofs of the possibility that such nonconvexities exist, but these proofs have readily been interpreted as a basis on which to judge the advisability of policies to control external costs.2 Indeed, a number of authors have felt able to proceed from specific theoretical examples of the existenceof nonconvexity to the opinion that nonconvexity represents a general obstacle to policy. One of the m a in themes will be to demonstrate that the theoretical possibility of the existence of a nonconvexity at a m icro level does not imply the existence of a m icro- (or, a fortiori, macro-) nonconvexity in the real world. Nor does it imply that any nonconvexity that does exist in reality is relevant to policy. The intention is not to deny that nonconvexities may represent an obstacle to the efficient control of external costs in specific contexts. But in order to establish the real world policy relevance of a nonconvexity, it is necessaryto verify its existence e m p irically and to establish that it stands in a particular relationship to the no-policy starting point. 1.2. Existence and Relevance For the most part the analysis will be concerned with the convexity assumptions upon which are based the conventional analyses of bargaining and Pigovian tax *I am indebted to Richard Bamett, Thomas D. Cracker, Peter Lambert, and Ed Mishan for a number of valuable suggestions, but they have no guilt by association for any remaining errors. ‘The term “external cost induced nonconvexity” will often be abbreviated to “nonconvexity.” The terms convexity and nonconvexity (concavity) will relate to the production or preference sets, whether the associated functions (curves) are convex or concave. See Rothenberg (1960, pp. 435-436), and Chiang (1984, pp. 348-352). *This point wiII be supported by some quotes at the end of Section 1.2. The relevant literature is: Portes (1970), Baumol (1972, p. 317), Baumol and Oates (1975, pp. 102, 120), Starrett and Zeckhauaer (1974, p. 83), Fisher and Peterson (1976, p. 16), Gould (1977, p. 558), SIater (1975, p. 864), and Dasgupta and Heal (1979, p. 91). Starrett (1972, p. 190) and Starrett and Zeckhauser (1974, p. 73) do make explicit the distinction between what we shah call “relevant” and “irrelevant” nonconvexities. But their analyses of tax solutions assume relevance. Finally, a brief dissenting view on the importance of nonconvexity can be found in Page and Ferejohn (1974). 101 0095~06%/86 $3.00 Copyright Q 1986 by Academic Press, Inc. All rights of reproduction in any form reserved

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PAUL BURROWS

solutions to the social inefficiency which results from undepletable, nonseparable external costs imposed unilaterally on producers.3 In essence these analyses have concentrated on the situation in which the configuration of external damage costs and internal abatement cost yields an interior solution requiring nonzero, but only partial, abatement by the external cost generator. 4 Typically the strong assumption of marginal damage cost having a value of zero at zero pollution and rising with each increment in the pollution level, together with marginal abatement cost having a zero value at some positive pollution level and a positive value which increases with each reduction in pollution, is sufficient to rule out the possibility that there are comer solutions, or multiple interior solutions, which satisfy the first order Pareto optimal&y conditions. The aggregate social gain curve, the sum of the net benefit curves of the polluting and polluted activities, has a single interior peak.5 The conventional upward sloping marginal damage cost curve relies on two assumptions which together are sufficient, though neither is necessary: (1) increasing marginal physical damage as pollution rises; (2) increasing u&e of a marginal unit of physical damage as pollution rises. As far as the conventional marginal abatement cost curve is concerned the required assumption is that abaters face diseconomies of scale in abatement activities. It will be suggested that the recent partial equilibrium models of external cost-induced nonconvexity in the production set can be viewed as counter examples, at the micro level, to the conventionally assumed damage cost abatement cost configuration. The conventional analysis of the operation of markets and of control policies, especially Pigovian taxes and regulated standards, has probably reached the point of view that, in the case of pervasive external costs involving large numbers of pollutees and/or polluters, markets are unlikely to be socially efficient but policies offer the prospect of an improvement in social efficiency.6 This improvement might be achieved on the basis of investment in information relating to the whole of the social gain curve, or (more realistically at present?) through gradient iterative schemes that myopically seek the interior peak of the social gain curve by using information only on the impact of small changes in the pollution level. Iterative taxes or regulated standards, which make stepwise adjustments on the basis of marginal cost-benefit analyses (essentially a step is taken only if marginal damage cost exceeds marginal abatement cost), would yield a social gain at each step and converge on the interior peak, which is at the globally efficient pollution level under the conventional convexity conditions. 38ee Baumol and Oates (1975, Chaps. 3 and 4), Dick (1974, Chaps. 3-6) Pearce (1976, Chap. 5), and Burrows (1980, Chaps. 2-4). 4ExtemaI cost generators will be assumed to be polluters. The conventional analysis assumes social cost minimising behaviour by pollurees (see Burrows, 1980, p. 171, fn. 7), and we shall follow suit. 50n the derivation of the social gain curve (variously called the net-benefit or the aggregate or joint profit curve), see Gould (1977), Dasgupta and Heal (1979, Chap. 3), and Burrows (1980, Chap. 3. We shall follow the literature and assume that in a partial equilibrium analysis the sum of benefits (profits) and social gain coincide. In a general equilibrium analysis with nonconvexity this need not be so. See Section 2.3 below. 6There are a number of reservations to this generalisation which concern information problems, enforcement costs, path dependence on the production side (lock-in effect), etc., but perhaps it is not misleading for the purpose at hand. On the reservations see Baumol and Oates (1975, Part II), and Burrows (1980, Chap. 4).

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A nonconvexity will be described as “relevant” if it would block the convergence of such myopic iterative adjustments on the globally efficient pollution level, and as “irrelevant” if it would not.7 As, we shah see, a, convergence of the iteration on the globally efficient state will be blocked only if nonconvexities which exist at the m icrolevel are transmitted as a local peak in the aggregate social gain curve, and if the local peak is generated by a pollution level which lies in the policy-relevant range, namely, between the globally efficient level and the no-policy starting point.’ Under these conditions the iteration may converge on the local peak, which lacks the top level efficiency properties. Much of the subsequent discussion will be directed towards establishing the proposition that it is far from clear that external costs do generally yield relevant nonconvexities; yet even if they did the normative implications would not be simple. The reason is that a locally efficient state may be a significant improvement on the no-policy situation. To establish a case against an iteration to a local peak would require it to be demonstrated that the process of attaining the local peak would preclude (or at least raise the cost of) a feasible subsequent move to a better local peak or even to the globally efficient state. Clearly we must consider the sensein which even relevant nonconvexities can be regarded as an “obstacle to policy.” To oppose, say, Pigovian tax policies simply because there is a possibility of arriving at a local peak is to run the risk of allowing ourselves to be “paralyzed by councils of perfection”; as M ishan has expressed it, “The intellectually fastidious among us, patiently awaiting the evolution of techniques sufficiently refined to enable the community to realise an ‘ideal’ outcome, unwittingly give authority to the licence resulting from inadequate legislation, a licence that is making the status quo increasingly intolerable.“9 Yet such opposition appears to be the implication of the frequently voiced opinion that nonconvexities are both common and important in the real world. Here are some examples (with additions to quotes in brackets): the problem [of nonconvexity] is very real and potentially very serious in practice (Baumol. 1972, p. 317). we are left with little reason for confidence in the applicability of the Pigouvian approach, literally interpreted. We do nbtknowh& to calculate the required taxes and subsidies and we do-not know how to appror&iate’t&nl~y trial and error (Baumol, 1972, p. 318). ‘A “relevant” nonconvexity would also block the convergence of marginal bargains on the globally efficient level, but this paper is less concerned with this implication than with the policy implications. On the problems raised for otherwise ideal&d markets by relevant nonconvexities, see, for example, Starrett (1972, p. 190) and Starrett and Zeckhauser (1974, pp. 75-77). Cooter (1980, p. 501) shows that the over-supply of pollution rights by a competitive market, predicted by Starrett (1972) would not occur under polluter liability. ‘Note that even policy-relevant nonconvexity would not prevent the attainment of the globally efficient state if the policy-maker could base his selection of taxes or regulated standards on accurate information on the whole social gain curve: see Starrett (1972, p. 194) and Gould (1977, p. 563). A similar point applies to court damages awards, but individual awards are ill suited to coping with the problem of pervasive public bads (Cooter, 1980, p. 504). 9(a) The first quote is from Baumol(l972, p. 320) in another context. Baumol and Oates (1975, Chap. 10) do not question the relevance of nonconvexities, but proceed to support the pursuit of suboptimising standards, which does not avoid the problems raised by relevant nonconvexities. The justification given for this pursuit is that even marginal cost-benefit analyses are not feasible for an iterative tax (pp. 136-137). However, they then say (p. 138) that it may be possible to adjust the standards in the light of just such analyses, thereby approximating the Pigovian outcome! (b) See Mishan (1969, p. 81). This was in the context of objections to welfare propositions generally.

104

PAUL BURROWS This problem [of nonconvexity] is no mere theoretical curiosity. We shall see that it produces some very real and difficult issues in the choice of policy (Baumol and Oates, 1975).” In the real world.. . the entire array of problems [generated by nonconvexity]. . . will be encountered (Starrett and Zeckhauser, 1974, p. 83). Some recent papers.. . show that such [nonconvexity] cases camtot be regarded as mere theoretically-possible curiosities; on the contrary, they are likely to be very important in the design of policies to deal with external effects (Gould, 1977, p. 558; emphasis added). For external diseconomies in general the case [of policy-relevant nonconvexity] that we have just analysed is more likely to be the rule than the exception (Dasgupta and Heal, 1979, p. 91).

2. EXTERNAL COSTS ON FIRMS Three different theories have been presented to rationalise the existence of nonconvexity somewhere in the social production set when external costs are imposed on firms: a firm shut-down theory, a pollution saturation theory, and a theory of multiplicative interdependence between firms’ production functions. Each of the theories takes as given the strict convexity of the polluting firm’s private production set. 2.1. Finn Shut-Down Consider the impact on the cost and profit functions of a single pollutee firm, B, resulting from an external damage cost generated by the production of a polluter firm, A. The level of pollution, E, enters firm B’s total cost function positively and total profit function negatively up to the level E,, at which B’s profit falls to zero and the pollutee goes out of business.I1 There is a sharp nonconvexity at Es, as shown by firm B’s profit functions in Figs. 1 and 2, quadrants (ii).12 The corresponding marginal damage cost curves (MD) in quadrants (i) have a discontinuity at E,, with zero marginal physical damage, and therefore marginal damage cost, above this pollution level, so that the conventional upward sloping marginal damage cost curve over the whole pollution range does not exist. With the profit function bounded below, the production function is not globally concave. “It is interesting, however, that the authors’ excellent second volume, devoted to environmental policy (Baumol and Oates, 1979) makes no mention of nonconvexity problems. “Assuming a rigid technology and therefore a fiied ratio of pollution, E, to firm A’s output, q”, the two profit functions can be written as

arm

97’ = r”(

&I, q”),

where -z 0 84” anB -=(J 8%

if a < q,4(4 if q4 2 qAts),

where arB/aq, is marginal damage cost; qAtsj is the level of A’s output that provokes B’s shut-down. i* On the sharp nonconvexity due to shut-down see Starrett (1972), and Starrett and Zeckhauser (1974, p. 73).

NONCONVEXITY

BY EXTERNAL

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105

Marginal %Ege

CoSt

FIGURE 1

(i)

Ee2 I

E~ 8

En2 t

E(or aA)

Ee2

E~

En2

E(or aA)

FIGURE 2

The implications of the sharp nonconvexity in this two-firm world depend on the positions of the marginal damage and marginal abatement cost (MAC) curves.13The following three cases can be distinguished. Case 1. The nonconvexity lies above the pollution level, E,, in Fig. 1, that would be chosen by the polhtter in the absence of an externality-intemalising policy. To the left of Es, the marginal damage and abatement cost curves and the social gain curve (the aggregate of IT* and &’ in quadrant (ii)) have the characteristics assumed in the conventional analysis. The only local peak, A4 at Eel, is globally efficient, and a myopic iteration starting from E,, does lead there, since M D > MAC over the range E,,, + E,,. The nonconvexity is policy-irrelevant even in the two-firm world. 13The marginal abatement cost (MAC) curve under rigid polluter technology is the marginal profit curve for the polluter.

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PAUL BURROWS

Case 2(a). The nonconvexity lies at a pollution level below the no-policy level E,, (Fig. 2). The point M at E,, is a local peak, but because the excess of abatement cost over (zero) damage cost in the pollution range Es + E,,(ced) is greater than the excess of damage cost over abatement cost in the range Ee2 + E,(ubc), it is socially efficient for firm B to shut down, and for the En2 level of pollution to prevail. The globally efficient point N will be retained under a myopic iterative policy, since the observed marginal damage is zero in the pollution range E n2 + Es, while MAC > 0, and no step from En2 is taken. Again the marginal cost-benefit analysis is sufficient for social efficiency even in the two-firm world, and the nonconvexity is policy-irrelevant. Case 2(b). The nonconvexity once again lies below the no-policy level E,,*, but now the configuration of MD and MAC is such that Fig. 2, quadrant (i), should be redrawn with abc larger than ted. In that case the local peak M would become globally efficient, and N would be reduced to a local peak at a lower social gain level than M. The starting point of a policy iteration would be Enz, the local peak N. But as in case 2(a), no iterative step would be taken with MD = 0, MAC > 0, so that the policy is not sufficient for global efficiency.r4 However, it is important to recognise that although the iterative policy fails to attain the globally efficient state, it neither worsens the situation relative to the no-policy starting point, nor prevents other means (if any exist) from being tried to generate the globally efficient state, nor prevents a move towards the global peak if more information later becomes available-l5 It might be thought that, unknown to the iteration-policy maker, there are opportunity losses incurred, but these are not true current opportunity losses given the limitation to local information, and they do not constitute a case against applying the marginal cost-benefit test. In general the implication of this analysis of the shut-down theory is that, even in the two-firm world, the marginal cost-benefit based iteration either improves social efficiency or at worst leaves it unchanged. Sharp nonconvexity does not introduce a risk that reduced social efficiency will result from the policy, even though it may be policy-relevant in the sense of blocking a potential improvement in social efficiency. This argument is strengthened when the shut-down type of nonconvexity is placed in a macroeconomic context, which the proponents of the theory have not done. The main point missed when exclusive attention is paid to the single pollutee case is that if there are many pollutees the shut-down of individual firms does not necessarily imply a discontinuity, or even a downward sloping segment, in the aggregate marginal damage cost curve. Curiously the earlier literature on nonconvexity16 recognised the fact that micrononconvexities are not necessarily reflected as macrononconvexities, but this fact has been lost in the modelling of externality-induced nonconvexity. If all pollutee firms were confronted with identical marginal damage cost curves and the same functions relating profit level to pollution, and if the number of firms 14Starrett and Zeckhauser (1974, p. 79): “there can be no assurance that the preferred equilibrium will be found.” 150f course the impact of pollution above E, may be irreversible (for example, the destruction of a trout farm on a permanently damaged freshwater ecosystem). But this impact is not the result of a partial-control policy. 16See Farrell (1959) and Rothenberg (1960). The nonconvexities they considered were not due to production externalities.

NONCONVEXITY

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(i)

Aggregate marginal

28’

2%"

26. 24.

I I

22. (ii)

20. 18. 16 14 12 10 8 6 4 2 0

Shut-ddwn olnt of lost 00 e lutee

FIGURE 3

affected by pollution remained constant, we could anticipate that the aggregate marginal damage cost curve would display a sharp nonconvexity at the one pollution level which brought about the shut-down of each and every pollutee firm.17 In these circumstances the implications just stated for the m icroanalysis could be extended to the macrolevel. However, these are very restrictive circumstances, and such a lack of diversity is unlikely to prevail in practice, so the aggregate marginal damage cost curve needs closer examination. l8 Consider an example in which there are N = 10 pollutee firms, and there is a constant rate of shut-down as pollution increases from 0 to 70 units (in particular we assume, in Fig. 3, that one more firm shuts down whenever pollution is increased by 10 units). The reason why firms shut down at different pollution levels is presumed to be the differences in their profit levels rather than in their marginal damage cost curves. The uniform individual marginal damage “In the pure public bad case assumed, the aggregate curve would be the vertical sum of the individual marginal damage curves. ‘RMas-Colell (1977) uses a diversification assumption to ensure that consumers’ discrete switching between commodities leads to regularity in aggregate behaviour.

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PAUL BURROWS

curves MD, in quadrant (i)19 are aggregated as follows: for the first 10 units of pollution, a’b’ is the vertical addition of segment ub for 10 firms; at pollution unit 10, one firm shuts down, producing the kink at b’, b”, in the aggregate curve; for units 11-20, b’c’ is the vertical addition of segment bc for nine firms; at pollution unit 20, a second firm shuts down, and subsequently cd aggregates for eight firms to c’d ‘, and so on. The result is an aggregate marginal damage cost curve (whose kinks at b’, c’, etc., disappear as N + cc) which is upward sloping to the pollution level (ES) at which the last four surviving pollutee firms shut down. If the marginal abatement cost curve is as shown in Fig. 3, quadrant (ii), there are three firm shut-down points, e’, f’, g’, that lie between the iteration starting level, E,, and the global peak at E, (i.e., the “problem,” case 2(b), on p. 106 applies). Nevertheless, the micrononconvexity proves no obstacle to a myopic iteration to E,, and this yields the social gain indicated by the shaded area. The policy implications of case 2(b) would apply at the macrolevel only if the shut-down of the last surviving pollutee firm occurred at a pollution level Es below E,,. In other words, only if the pollution control policy starts from a position in which all of the pollutees have been driven away, and the observed level of aggregate marginal damage is zero, does the shut-down nonconvexity obstruct a socially efficient iterative policy under the conditions illustrated in Fig. 3. Such an extreme starting point does not seem likely to prove a regular experience in pollution control policy. The analysis of the macro-shut-down nonconvexity is more complicated when the shut-down rate varies at different pollution levels and the number of firms affected, N, is not fixed. The points a’, b’, . . . , h’ in Fig. 3, quadrant (ii), will lie on a positively sloped locus if the multiple of the micromarginal damage cost level and the number of pollutee firms still operating successively rises for the pollution units just prior to each shut-down (units 9,19,29.. . ). A sufficient, but not necessary, set of conditions for the multiple to increase as pollution rises is that the microdamage cost curves are positively sloped, that the shut-down rate does not increase, and that the numbers of affected firms rises with the pollution level.2o It appears, therefore, that the conditions for a macrononconvexity, induced by shut-downs, that would obstruct an iteration to the globally efficient pollution level may be not at all general. There is a sufficient lack of correspondence between the existence of a micro-shut-down nonconvexity and the policy relevance of a macrononconvexity for the case against an iterative policy to be questioned, particularly in view of the fact that at worst any losses associated with such a policy, if based on accurate marginal evaluations, are in the nature of opportunity losses rather than a loss of efficiency relative to the no-policy situation.21 “Clearly the MD, curve shown is a potential marginal damage cost curve and is the actual curve only for a firm which continues in production. For the first firm to close, the actual marginal damage curve is ab, with the sharp nonconvexity at pollution unit 10. *O(a) Clearly, each of these three conditions is not necessary; for example, the shut-down rate may rise but be outweighed by the increase in the number of affected firms, so that the number of surviving pollutees increases with the pollution level. (b) It is imaginable that a particular combination of the three variables could lead to a smooth nonconvexity (downward sloping MD) in the aggregate. However, this possibility gives rise to considerations similar to those relating to the saturation theory to be considered in section 2.2. 211t has been argued by Starrett and Zeckhauser (1974, p. 75) that certain abatement methods can introduce a sharp micrononconvexity. A demonstration that this also at worst implies an opportunity loss for the policy maker is presented in a discussion paper, with the same title as this article, available from the author.

NONCONVEXITY

BY EXTERNAL

Ee h

PRODUCTION

En

!

I

I I

Ee

ER

En

COSTS

109

E

Total profits and

SOClOl win

(ii)

E

Empirical evidence to support the proposition that in practice sharp nonconvexities are widespread and give rise to serious policy problems is hard to come by. The examples offered in the literature (such as cows and crops) are picturesque but somewhat strained, reviews of the sketchy available evidence on external damage costs rarely mention pollutee shut-downs,” and evidence on the causes of plant relocation scarcely supports pollution as a major contributory factor.23 It seems, therefore, that we do not have either theoretical or empirical grounds for accepting the strong statements quoted at the end of Section 1, as far as sharp production nonconvexities are concerned 2.2. Saturation

The firm shut-down theory assumes that once the pollutee firm closes, extra units of pollution cause no physical damage. The saturation theory says that, even in the absence of a shut-down, an extra unit of pollution need not be associated with marginal physical damage which exceeds that of the previous unit. Thus a farmer will not indefinitely suffer increasing marginal physical damage as an invasion by his neighbour’s cattle increases, since eventually even conscientiously trampling marginal cows will find undamaged crops increasingly hard to come by.” The eventual decline in the marginal productivity of the tramplers yields a downward sloping 22F~r example, see Saunders (1976, Chap. 6), Pearce (1978, Chaps. 3 and Pearce (1978, p. 101) does mention three fii closures due to cumulative Wabigoon river in Canada. 230n relocation decisions see Luttrell(1962) and Cooper (1975, pp. 38-42). 24Starrett and zeckhauser (1974): “Casual consideration of technological suspect that this would be the case” (p. 75). where “technological” apparently

5), and Tibansky mercury pollution

(1975). of tbe

factors might lead us to means physiological!

110

PAUL

BURROWS

Eeh Ep

Ea En

Total

wd”fIt!

social galn (Ii)

FIGURES

segment of the micro marginal physical damage curve; the production function for tramplers is not globally concave. If the value of units of physical damage is constant, the implied marginal damage cost curve is as shown in quadrants (i) of Figs. 4 and 5, with MD + 0, as E + 00. The smooth nonconvexity lies to the right of pollution level E, in both figures, ER being associated with the point of inflexion on the rrB curve in quadrants (ii).25 As with the sharp nonconvexity, the implications of smooth nonconvexity in the two-firm world depend on the positions of the marginal damage and marginal abatement cost curves. The following four cases can be distinguished. Case l(a). The nonconvexity lies at a pollution level above the polluter’s chosen level, i.e., E, > E,. The conventional convexity assumptions hold in the policy-relevant range, so the nonconvexity is policy-irrelevant. Case l(b). The nonconvexity lies below the polluter’s chosen pollution level, i.e., E, < E,,, but as in Fig. 4, marginal damage cost nevertheless exceeds marginal abatement cost at all pollution levels in the relevant range En to E,. The resulting social gain curve has a single, interior peak M in quadrant (ii), so that again the nonconvexity does not alter significantly the conventional analysis. An iteration *9he

profit functions implicit in the figures are

99 = “yq”, %a), aZnB G

3Oa.s E>< ER.

ad

where +h

< 0, and

NONCONVEXITY

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PRODUCTION

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111

starting from E,, would proceed to the global peak at E,, so the nonconvexity is policy-irrelevant. The likelihood of this case proving to be the common one in practice would be enhanced by the availability of low cost process-switching as an alternative method of abatement to adjustments in the polluter A’s output.26 If reductions in the pollution level from E,, were initially attainable at low marginal cost with marginal cost rising at an increasing rate as the pollution level falls, the risk of an intersection of MAC and the downward sloping segment of M D would be reduced.27 Case 2(a). If saturation leads to a downward sloping M D segment which is steep enough over some of its range for marginal damage cost to fall below marginal abatement cost within the pollution range E, to E,, the nonconvexity may block a move to a globahy efficient state. In Fig. 5, quadrant (i), the M D and MAC curves are drawn such that the excess of M D over MAC (the two black areas) is less than the excess of MAC over M D (hatched area). The implied social gain curve has interior local peaks at M and N, of which N is globally efficient at the pollution level Eq. In this case a myopic iteration starting from E, will move the system to E,, since M D > MAC in the Eq to E,, range. The nonconvexity again proves to be policyirrelevant. Case 2(b). The one case in which the smooth nonconvexity is policy-relevant in the two-firm world is when the size of the black areas exceeds that of the hatched area (if Fig. 5 were redrawn). The social gain curve would then have two interior local peaks, M and N, but the one at M , pollution level E,, would be globally efficient. This is analogous to the sharp nonconvexity case 2(b), the main difference being that there the starting point En2 (in Fig. 2) was a local peak level of pollution, whereas here even to attain the local peak N, a move is required to pollution level E,. This move would in fact be undertaken in an iteration, but the nonconvexity would block the further move to M , since MAC > M D for some of the steps en route. The iterative policy, therefore, does still improve social efficiency compared with E,, (contrast the sharp nonconvexity case 2(b)), but there remains the opportunity loss of failing to reach M . This case is distinguishable from previous failures to reach the global peak, because here the iteration does involve a move away from the starting point; however, it is a move in the direction of the global peak.*’ Multiple interior local peaks would be generated by marginal damage curves which had multiple discontinuous segments, but such complex cost relationships have not yet been rationalised in the literature. 29 The question which the simple cost configuration of case 2(b) raises is whether a moue to an interior local peak is to be resisted. 260n the derivation of a MAC curve for process-switching abatement see Burrows (1980, pp. 17-27). *‘There is some evidence that process-switching does display these characteristics. See Kneese and Shultze (1975, Chap. 2) (vide: “As a virtually universal phenomenon, the greater the percentage of pollutants already removed from a industrial process, the higher will be the cost of removing an additional amount” (p. 19)), and Hanke and Gutmanis (1975). *‘With zero control as a starting point it does not seem possible that policies to reduce external costs will run the risk of a perverse move away from the global peak. On this potential problem with nonconvex sets, not in the context of external costs, see Baumol (1977, p. 146) and Heal (1973, pp. 230-234). 29Multiple maxima could result from strong reciprocal externalities between many activities (see Baumol and Oates, 1975, pp. 116-117), but whether these are common is a moot point.

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BURROWS

What if such a move were made, and it were later discovered that Shangri-la existed at an even lower pollution level? Then, in retrospect having proceeded to the global peak via the local peak, rather than directly, may be seen to have incurred the cost of (say) a temporarily altered technology which is abandonned in order to meet later a more stringent control level. This is a lock-in cost of the initially ill-informed decision. A risk-averse policy maker (and pollution abaters) might wish to build in technological flexibility when proceeding to the local hill, although flexibility is not costless. But a refusal to climb the local peak could be justified only if the ex ante expected net present value of the social gain from a policy of remaining at the no-policy position, with the possibility of a subsequent move to the global peak, exceeded that from a move to the local peak with the possibility of a subsequent move to the global peak being made at higher cost. The implications of this analysis of the saturation theory are strengthened if this type of nonconvexity is placed in a macroeconomic context. They can be stated briefly because they are in some respects analogous to those of the macroeconomic analysis of sharp nonconvexity (p. 106). One point of difference, however, is that the saturation theory relates to the marginal physicuf damage curve, rather than directly to the marginal damage cost curve. The assumption that the value of units of physical damage is constant (p. 110) does not necessarily carry over to an aggregative analysis. If pollution damage to the output of particular products were significant in relation to the quantity of goods marketed, then the value of a marginal unit of the declining output of each pollutee would rise. Consequently a declining marginal physical product curve can co-exist with a rising marginal damage cost curve for an individual pollutee, and a fortiori for a pollutee group, when a pollutant is pervasive in its effects. A second consideration in a macroeconomic analysis is that when pollutants are widespread and serious in their effects (as the shut-down and saturation theories presume), their damage consequences will be determined in part by the assimilative capacity of the environment. Examples used to illustrate the saturation theory, such as cows and crops, involve direct damage. Pollutants more typically are conveyed to pollutees by some medium, such as watercourse or the atmosphere, which has a limited capability to degrade or disperse pollutants. 3oAs the pollution level rises this capability increasingly falls short of the quantity of pollutants conveyed, thereby raising the probability that the physical damage function will be convex at least in the policy-relevant range.31 The other factor that can prevent downward sloping segments of micro marginal physical damage curves from generating a similar aggregate curve is a positive relationship between the pollution level and the numbers of pollutees.32 This relationship can be produced by a spectrum of pollution sensitivities of pollutee firms, with decreasingly sensitive firms being affected as pollution increases and by the increase in geographical spread of pollution damage as the pollution level rises. In Fig. 6, it is assumed for simplicity that at intervals of 10 pollution units, for one reason or the other, one new firm enters the class of pollutees; the firms’ marginal damage cost curves are MD,, . . . , MD, in quadrant (i), where firm 1 is the first 30!+e Pearce (1976, Chap. 4). 3’!ke Page and Ferejohn (1974, p. 458). Declining marginal assimilative capacity does not, however, deny the possibility that at some pollution level diminishing marginal physical damage will set in. 32“Similar” in the sense of having a negative slope in the same pollution range.

NONCONVEXITY

BY EXTERNAL

PRODUCTION

30 t 40 t 50 T 60 EP.I ERZ h3

70

COSTS

113

Indlvldual

(i)

7 6t

20

80

Aggregate

yr3r;;al casts

Aggregate

b T

80

90

MD

loo

E

FIGURE 6

affected and firm 6 the last.33At pollution level ER1, firm l’s marginal damage cost begins to decline due to saturation, and subsequently other firms’ curves do so at 10 unit intervals (E,,, etc.). Aggregating these curves vertically, it is apparent, from quadrant (ii), that the micrononconvexities for firms 1 to 3 lie below the no-policy pollution level E,,. Yet these do not prove to be policy-relevant, as the aggregate nonconvexity exists only at E, and above,34 and an iteration from E, to E, would yield a social gain equal to the shaded area. For nonconvexity to be policy-relevant, E, -C E, is necessary (though not of course sufficient). The implications of these macroeconomic considerations are similar to those relating to sharp nonconvexity (p. 106-108). The lack of correspondence between the existence of a micro saturation nonconvexity and the policy relevance of a macrononconvexity is sufficient to question the validity of general statements to the effect that nonconvexity is a pervasive obstacle to control policy. In particular instances relevant nonconvexities will occur and threaten serious opportunity losses at least. Unfortunately, with the current dearth of estimated pollution dose-response 33 Each of the marginal damage cost curves is assumed to be similar in shape of Fig. 6, but this is not necessary for the argument. 34Thus the cows, roaming on from the first farmer’s devastated fields, vent their fury on crops further away. Eventually, however, exhaustion sets in, even for the more athletic cows, and the trampling productivity of marginal cows dwindles above E,.

114

PAUL BURROWS

curves for impacts on firms, the policy-relevance of saturation nonconvexity in production cannot be either refuted or corroborated by firm, nonimpressionistic empirical evidence. 2.3. Multiplicative

Interdependence

An interesting model presented by Baumol, Bradford, and Oates centres on the proposition that a sufficiently strong external cost necessarily introduces a nonconvexity into the social production possibility set, if the externality arises only when the levels of the polluting and polluted activities are both nonzero. The purpose of the analysis that follows is to identify the nature of the nonconvexity, and to consider the conditions under which it exists, as well as those under which it would be policy-relevant. This will enable us to evaluate the authors’ claim that their proposition is both general and policy-relevant.37 The essential feature of the model is a multiplicative interdependence between the polluter’s and pollutee’s cost functions. Specifically the pollutee’s function contains an external cost component whose magnitude is equal to the product of a parameter, w, which measures the strength of the externality, and the multiple of the two outputs. 38 The parameter w is assumed to be constant which implies that, for a given level of pollutee output, the external cost is proportional to the polluter’s output, i.e., the ceteris paribus marginal external cost of the polluting activity is constant. In addition, the model assumes the technologies and locations of both firms to be fixed. It can be shown that if the strength of the externality is great (w is large), then a social production possibility set that would be convex in the absence of the externality may be made nonconvex. 39 This result is illustrated in Fig. 7. In quadrant (i) DNB is the no-externality production possibility curve (PPC), with a no-externality general equilibrium at N( qA8, qBm), where both firms maximise profits with p A = MC”, PB = MCB, and PA/PB = MRTAB. With the prevailing ratio of goods prices the individual profits (rrITA,nB) and aggregate profits (as functions of polluter output) are as shown in quadrant (ii). If all of the resources of this two-firm economy were used by firm A, then firm B would earn zero profit and firm A earn B’W, if all go to B then the respective profit levels of A and B are zero and OK Aggregating the profit functions yields the single-peaked social gain curve VN’W for the convex world, where N’ is globally efficient. 350n the crude point cost estimates available for damage to firms’ materials, crops, etc., see Pearce (1978, Chaps. 3 and 5). The fiendish difficulty of reaching empirically supportable conclusions on the shape of total damage cost functions can be seen from the extensive survey in Pearce (1982). Against expectation this is particularly true for damage costs on production, given the currently available information. The evidence for damage to people is a little firmer: noise, air pollution and morbidity/mortality, and radiation are examples where the balance of available evidence appears to support the conventional convexity assumption. 36See Baumol and Oaks (1975, p. 108). gee also Baumol (1964, 1972) and Baumol and Bradford (1972). In the context of a constant returns model the words “sufficiently strong” could be omitted from this proposition (see Baumol and Oates, 1975, p. 106). The constant returns assumption is so restrictive that we shaU concentrate on the nonlinear model whose implications apply a fortiori to the linear model. 37See Baumol and Oates (1975, pp. 108122-123). 3*See Baumol and Oates (1975, pp. 108,122-123). 39See page 124 of the Appendix.

NONCONVEXITY

BY EXTERNAL

'A6.5

'A3

'A8

PRODUCTION

COSTS

115

'Al0

Total t%s (iii)

T!m;;;

%,f'ts ",%"'

= '::.:~~~e~B;..t~,

Q~6.5 f I

'A3 , / Le.-' .r,

/+--+, /

Q~8 IN"

'Al0 I I / r\-(no

'A Social

gain externality)

FIGURE I

The impact of a strong multiplicative externality is to leave points D and B unaltered in quadrant (i), but to move all other points on DNB downwards to below the chord DB. This produces a groba& nonconvex production set ODGB.40 The maximum vertical distance between DNB and DGB, at qA6,+ represents the largest loss of pollutee output due to the externality. In quadrant (iii) the total damage curve, with qB variable, peaks at this output, and is zero at qA = 0 and at qA10 (where qB = 0). Up to the limit imposed by the resource constraint any increase in polluter output, with pollutee output fixed, leads to positive, constant marginal damage (total damage curves TD ( qsiO), TD ( qszo), etc.). But increases in polluter output from D with the resource constraint binding create initially positive but declining marginal damage, and above q46.5 declining total damage due to the fall in the pollutee output that is available to be destroyed by the externality. The profit curves associated with this nonconvex production set are shown in quadrant (iv). The polluter’s profit curve s ” is, of course, unaltered if based upon the same goods price ratio as at N. At this price ratio, however, the pollutee’s profit curve becomes T B (externality) which coincides with the original 7rB curve only at point V’; the 4See Baumol and Oates (1975,

p. 107).

116

PAUL BURROWS

pollutee can make a profit only if the polluter’s output is less than qA2. The resulting aggregate profit curve, rrA + rrE (externality), reflects the nonconvexity in the production set. It has a minimum point at output qf, at which DGB and the price ratio line are tangential in quadrant (i). It also has two local peaks, V’ and IV’, either of which may be the global peak (as drawn, V’ is the global peak at the N price ratio: D is on a higher price ratio line than B in quadrant (i)). Before considering the policy relevance of the nonconvexity, it should be emphasised that this demonstration of the existenceof a global nonconvexity relies on the particular assumptions of the model; it is not general: (A) Multiplicative form. The demonstration does not hold for all classes of externality. In fact, as shown in Section 2 of the Appendix, it holds only for the particular class of multiplicative interdependence, which is a sub-set of nonseparable externalities. There is some disagreement in the literature as to whether externalities are more commonly of the separable or the nonseparable type. Baumol and Oates believe that separable externalities are not common,41 whereas Nath and Tisdell and Walsh take the opposite view. 42 Certainly, however, there does not appear to be an empirical basis for claiming the generality of the multiplicative external damage cost function.43

A global nonconvexity will occur in the (B) A large and constant w coefficient. model of multiplicative interdependence only if w is large and constant. The authors recognise that in their model, with w constant, the global nonconvexity depends upon the externality being sufficiently strong to outweigh the effect on the curvature of the production possibility curve of diminishing returns in the polluting and polluted industries. 4 However, it is also true that the global nonconvexity will occur only if the externality is sufficiently strong at every level of the polluter’s output. A multiplicative externality that is “strong” at some levels of the polluter’s output but not others (i.e., w varies) would lead only to a local nonconvexity (as is demonstrated in the Appendix below, Sect. 3), which has important implications for its policy relevance. The range of polluter output within which the local nonconvexity would occur depends on the form of the relationship between w and the polluter’s output. 45 The conventional assumption is probably that w would be large only at high levels of the output, which would imply that if a nonconvexity existed it would be limited to such levels. (C) Fixed technology and location. The effect of an externality on the curvature of the production possibility curve depends on the nature of the interdependence between the polluter’s and pollutee’s production functions, which can be altered by a change in the technology or the location of either firm. It has been demonstrated by Smith that a flexible technology can “reduce” the nonconvexity, while Baumol and Oates have shown that spatial separation of the firms can “limit the magnitudes of 41See Baumol and Oates (1975, p. 103, fn. 4). Their view is apparently based on the argument that separability implies that the addition to the pothttee’s fixed cost remains if he goes out of business, whereas the normal interpretation is that all fixed costs go to zero when the firm goes out of business. 42See Nath (1969, p. 85) and Walsh and T&dell (1973, p. 454 et seq. 43This author has been unable to trace any cost function estimates that verify the multiplicative externality form. “See Baumol and Oates (1975, pp. 103 and 108 refer to the externality being sufficiently strong). The condition that must be satisfied for convexity is given in the Appendix (Sect. 2, p. 124) below. 45Equation (11) in the Appendix.

NONCONVEXITY

BY EXTERNAL

PRODUCTION

COST8

117

FIGURE 8

the nonconvexities . . . but does not prevent them.“46 The most persuasive interpretation of the words “reduce” and “lim it the magnitudes” appears to be that technological change, or relocation, can convert a global nonconvexity into a local nonconvexity or restrict a local nonconvexity to a smaller range of the polluter’s output, rather than eliminate altogether an existing nonconvexity. This can be illustrated in relation to Fig. 8, where DNB is the no-externality production possibility curve. If the polluter A has available to him a technological switch, or new location, which eliminates the externality but, at any qA/qB ratio, raises the MCA/MCB ratio, then the production set is ODHE. 48 This is clearly more restricted than ODNB, which would be the attainable set only if there were no interdependence between the production functions. 49 If, in the presence of the externality but the absence of any switch/relocation option, the locally nonconvex production set ODAB would prevail, then the effect of the availability of the option is to extend the production set from ODAB to ODHVB. As a result the qA range of the nonconvexity is reduced from RB to VB, where R is the point of inflexion on DAB. This result will prove significant for the analysis of policy relevance, to which we now turn. The analysis of policy relevance will concentrate on the implications of local nonconvexities, but occasional reference will be made to the special case of global nonconvexity. We shall seek to substantiate two propositions: . that the model of multiplicative interdependence does not establish the policy relevance of a nonconvexity even if one exists; . that even when a nonconvexity is policy-relevant the cost it imposes is, in many cases, likely to be lim ited to an opportunity cost, which may not deter the policy maker from undertaking a myopic iteration.

%ee Smith (1975, p. 294) and Baumol and Oates (1975, p. 112). 470~r interpretation of the significance of technological or locational flexibility would be strengthened by zero cost technical innovations or relocation, for then the globally convex no-externality production possibility curve (DNB in Fig. 7) would be attainable. 4*On any ray from the origin in Fig. 8, the slope of DHE exceeds that of DNB. 49Figure 8 assumes that only the polluter’s technology/location is flexible, since this is the case that typically arises in evaluations of the impact of pollution control policies. However, the argument here carries over to more complex production sets with pollutee technology switch/relocation feasible.

118

PAUL

BURROWS

FIGURE 9

A preliminary point to note is that, in contrast to the case of global nonconvexity, if the production set is only locully nonconvex there can still be a well-behaved interior global peak, such as A on DAB in Fig. 9(a). A is sustainable as a value maximum by positive output prices. SoTo elucidate the significance of this point it is necessary to distinguish between a nonconvexity at high polluter output levels (cases 1 and 2 in the Appendix, Sect. 3) and a nonconvexity at lower polluter output levels (cases 3 and 4), and between interior and corner global peaks. Cases 1 and 2: Interior global peak. In Fig. 9(a), IV,, IV,, IV, are a set of Samuelsonian social indifference curves that intersect or touch the segment AB of the production possibility curve only once, so that a move from any point on AB towards A yields a social gain. 51 An important element in the analysis of policy relevance is the assumption that is made about the no-policy starting point, an element which has been ignored in the literature due to the concentration of effort on establishing the existence of a nonconvexity somewhere in the production set. It will be assumed here that the starting point is where the ratio of the marginal private costs of the two activities, the ratio of their goods’ prices, and the marginal rate of substitution in consumption are all equal, P
NONCONVEXITY

BY EXTERNAL

PRODUCTION

COSTS

119

as through A, competitive firms would have no incentive to raise the level of qA, because the price of q” to the producer (for a given MCA) has not risen relative to that of q8. The induced change in consumption pattern in favour of qs results from the rise in PA/PB to the consumers. The fact that A is the global peak does not preclude the possibility that social indifference curves below W, in Fig. 9(a) pass through more than one point on AB (not shown in the figure). If this happens then one of the social indifference curves will be tangential to AB from below, a local valley, say, at G. If G does not lie between S, and A, then S,A iteration is not obstructed; whereas if it does, then no iterative steps would be taken towards A, but once again the impact of a policy relevant nonconvexity is in the nature of an opportunity 10~s.~~ Finally, on interior peaks, if the social indifference curves have greater curvature than the nonconvex segment of the production possibility curve, it is possible for one of the social indifference curves to be tangential to that segment from above at an interior point. With a globally nonconvex production set this tangency would be the global peak.56 But if the nonconvexity is only local, there can be another tangency with the convex segment either on the same or on a different social indifference curve. With a redrawn indifference map, these points could be H and F in Fig. 9(a). The point F could be the global peak, in which case if the starting point is to the right of F, then intemalisation could induce a move to F; the nonconvexity is not an obstacle. However, if H were the global peak and F a local peak, there would be a valley between the starting point and H if the starting point lay to the right of F. There is consequently the possibility of an opportunity costwith the iteration being blocked at the local peak F. Cases 1 and 2: Corner global peak. If the global peak is at a comer, such as B in Fig. 9(b), then two situations can arise: (i) The social indifference curves are steeper than the production possibility curve at all levels of qA, and a fortiori steeper than the marginal private cost ratio line at any qA (Fig. 9(b)), a case that can arise also under global nonconvexity. In this case no point to the left of B can be a no-policy equilibrium at which PA/P’ = MPCA/MPCB = MRSAB, so the starting point is S, = B. No iterative itemalisation policy offers a marginal social gain, so the system stays at the global peak. But clearly no physical damage would be observed at S,, so this configuration is inadequate as a description of pollution control problems. (ii) The social indifference and production possibility curves are such as to yield an interior local peak at A together with a comer global peak at B, as in Fig. 9(c), a case that cannot occur with global nonconvexity. In between these peaks lies a valley at G. The no-policy starting point will lie either between A and G or at B, but the latter is no problem. 57 In the former case, say S,, the myopic policy yields marginal social gains summing to W, - W, in the move from S, to A. This is the

55A myopic view would perceive a social gain from an increase in qA, but in the context of pollution cunrrol policy this seems likely to manifest itself in a lack of control rather than a positive inducement to pollute more. 56See Baumol and Oates (1975, p. 121). 57Between G and B the social indifference curves are necessarily steeper than MPC A/MPCB at any level of qa , and a move to B would be made without policy.

120

PAUL BURROWS

only case, in all of the examples of production nonconvexity that have been considered, in which a move further away from the global peak may result from myopic iteration, so that it is analytically a special case even within a model where the externality is presumed to generate a nonconvexity. Nevertheless an analytical special case might be empirically relevant, so two further questions need to be considered. Would a move further away from a global peak ultimately cause a social loss, even though it initially led to a social gain? Would the problem be avoided under flexible technology or location? In relation to the first question, authors who have pointed out that gradient optimisation methods may move the system up a local hill but further away from the global peak seem to have taken it for granted that this is a “bad thing.“5a Clearly the move from S, to A when B is the global peak (Fig. 9(c)) involves the opportunity loss IV, - IV,, but this is less than the opportunity loss W, - IV, involved in staying at St. The case against the movement further away from the global peak must rest on the possibility that subsequently it will be discovered that B is the global peak, and that the output changes involved in the move from A to B will incur greater adjustment costs than those for a move from S, to B would have done. The frictionless competitive economy models which characterise the literature on externality-induced nonconvexity do not explain the resource cost of misdirected myopic moves. However, even if there is a risk that later on scrapping, redundancy, and other adjustment costs would be incurred by a move from S, to A, the (ex ante) expected social gain from the move may well exceed that from staying at S, until more global information is available, given the high risk associated with the latter strategy. This would not apply where more global information is available. For example, if we feel sure that separating smokers and nonsmokers on trains is globally efficient, whereas controlling the smoking rate similarly in all carriages is only locally efficient, then there is little point in taxing smokers iteratively (even if we could) rather than enforcing nonsmoking in some compartments. But the analysis in this paper has taken it for granted that it is severe limits on access to global information that provides the basis for iterative schemes. The case against myopic intemalisation is consequently not clear even in the particular case, under fixed technology, in which there is a risk of more than opportunity costs being incurred. When account is taken of technological/locational flexibility, the case is weakened further. We have seen (p. 117) that in Fig. 8 where A and B are, respectively, local and global peaks, technological/locational flexibility restricts the nonconvexity to VB. Smith has demonstrated that when an externality is intemalised the polluter’s reaction will be to switch to pollution-saving technology.59 Perfect intemalisation would lead to the selection of the lowest social cost method of 58See Baumol and Oates (1975, p. 115), Heal (1973, p. 230), and Dasgupta and Heal (1979, p. 90). 59See Smith (1975, p. 294). In the context of Baumol and Oates’ model we can rewrite the pollutee’s resource demand equation (see Appendix below, Eq. (1)) as

f-i3= CB(%, 4S)T

as/a(

PE/PL)

< 0,

where s = emissions per unit of qA, and PE, PL are the prices of environmental and “other” inputs. Then with low cost technology switches, s + 0 as PE/PL rises, and the second derivative of the production possibility curve becomes unambiguously negative for the production set that is relevant if the new techndogy is adopted (ODHVE in Fig. 9). However, the production set with a choice of the new technology, 0 DH VB, remains locally nonconvex.

NONCONVEXITY

BY EXTERNAL

PRODUCTION

COSTS

121

FIGURE10

abatement, where this takes the form of a change of technology or of location.60 Consequently the response to intemalisation from S, will not be a move along DS, if a low cost abatement technology (location) is available. In this two-good world the adoption of the new technology releasesresources previously made unproductive by pollution, making higher levels of both qa and qA feasible. In Fig. 8 the response will be a move from S, to the new global peak H; the valley moves from G to V so that the nonconvexity is no longer policy-relevant. Each step in the iteration indicates marginal social gains. Inevitably, different instruments of intemalisation (Pigovian taxes, zoning restrictions, and subsidies or regulations relating to the choice of technology) need to be evaluated, but the implications of instrument choice for the nonconvex production set are not dissimilar to those with a convex set. The general implication of this analysis is that low cost technology/location switches increase the likelihood of a nonconvexity being policy-irrelevant, and this is true both in the case just considered, where myopic intemahsation m ight lead to resource costs under fixed technology/location, and in the (more likely?) cases with an interior global peak, where myopia at worst leads to opportunity losses. Moreover, the point carries over to the eventuality of a global nonconvexity; a relevant global nonconvexity can be made irrelevant as the process of intemalisation induces a change of technology. Cases 3 and 4: Interior local peak. A local nonconvexity at low levels of polluter output raises no new issues of principle, so its implications can be selectively and briefly summarised. In Fig. 10 the fixed technology/location production possibility curve is DAB, and the set of social indifference curves as drawn yields a global peak at A,, and a local peak at D. From the starting point S, the nonconvexity is irrelevant to an iteration to A. Even if any of the price ratio lines associated with partial intemalisation at steps in the iteration happened to pass through a point on AD (as well as through a point on AS,), the firms would have no incentive to raise the qB/qA supply ratio beyond that at A. Consumer resistance, combined with the fundamental convexity of the private production set, would arrest the resource 6oSee Burrows (1980, Chaps. 2 and 3). This statement apparently contradicts Meade’s suggestion that “a marginal price-cum-tax mechanism” may not give rise to socially efficient polluter relocation” (Meade, 1973, pp. 90-93). However, his result derives not from nonconvexity but from setting a tax rate that is optimal only if a change in location is not socially efficient. If the tax were set equal to actual damage at any level of pollution, then the polluter would relocate if, and only if, the total cost of relocation were less than the total damage in the absence of policy. Crude approximations to such an information-intensive policy may be necessary, but this is independent of the question of convexity.

122

PAUL BURROWS

switch from qA to qB at A. A different indifference map, with social indifference curves having greater curvature than the nonconvex segment of DAB, could yield tangencies F and H (with the relevant Wcurve above DAB) either of which may be the global peak, the other being a local peak (cf. p. 119 above). At a point in between, such as G, there will be another tangency, with the relevant W curve below DAB at that point, i.e., a valley. If F is the global peak the worst that can happen is that an iteration to the global peak is blocked (if S, is to the right of G). But if H is the global peak and S, is to the left of G, then a myopic iteration could move the system to the local peak F, that is further away from the global peak. As with cases 1 and 2, we have a particular situation in which there is a risk of resource cost, but the argument which we previously presented (p. 120 et seq) against using this possibility as a general basis for opposing myopic internalisation applies here also.(‘l Cases 3 and 4: Corner local peak. If D is the global peak, again the worst that can happen is that a valley between S, and D blocks the iteration to the global peak. But if the social indifference curves are relatively flat and intersect DAB only once, then the iteration will proceed to the global peak; the nonconvexity is policy-irrelevant. Finally, as with cases 1 and 2, technological/locational flexibility may convert any relevant nonconvexity in cases 3 and 4 into an irrelevant one by narrowing the polluter-output range of the nonconvexity. The general conclusions to be drawn from this analysis of a nonconvexity due to multiplicative interdependence are that

(a) the model does not establish the policy relevance of a nonconvexity; it establishes only the possibility of the existence of a nonconvexity; (b) even where a nonconvexity proved to be relevant, the cost of this relevance, in many cases, would be only an opportunity cost which may be little deterrent to the policy maker given the necessity of decision-taking with poor information, and the prospect of social gains being derived from a move to a local peak; (c) in the (rare?) event that there would be a risk of resource costs being imposed by policy under fixed technology/location, this risk may be substantially mitigated by the impact of a myopic intemalisation policy on the choice of technology or location. 3. CONCLUSIONS

The argument in this paper began from the proofs that it is possible that external costs on firms create a nonconvexity somewhere in the social production set. It has attempted to demonstrate the distance between such proofs and the proposition that such nonconvexities provide a general obstacle to myopic, iterative external cost control policies. Ultimately a judgment against such iterations would require evidence that nonconvexities are policy-relevant, and that the net present value of the expected return to myopic policy adjustments falls short of the return to delaying control policies until more global information becomes available. None of this argument denies the possibility that nonconvexities may be important in specific cases; as yet there is insufficient evidence to judge them to be a general obstacle to control policies. %rguably

it applies a fortiori, since cases 3 and 4 are empirically less plausible than 1 and 2

NONCONVEXITY

BY EXTERNAL

PRODUCTION

123

COSTS

APPENDIX: LOCAL AND GLOBAL NONCONVEXITY IN THE PRODUCTION SET A. 1. General Convexity Condition Nonconvexity in the Baumol et al. model derives from the cost function specification. The general form of the functions used by Baumol and Oates (1975, pp. 108-109) is

rE = C%lA, 4eh

‘A = CA(4A),

0)

where rA, re are, respectively, the input demands of the polluter and pollutee, qa, qB their respective outputs. The equation of the production possibility curve is

r = ‘A + ‘B = CA(qA) + CB(qA9qB).

(4

Differentiating twice, we have

(CA”+ cjy2 d*qB

-(cA”A

+

c;~)c;

+

(cA”B

+

c/4)(ci

-=

dd

+

‘:)

-

‘i&l

CB B

(3)

(csB)*

(Baumol and Oates’ FXJ.(6)), where Cf = (dCA(qA))/dqA, etc. If (d2q8)/dq: for all qA, then the production set displays strict global convexity.

< 0

A.2. Externalities and nonconvexity To derive the implications of externalities for the convexity of the production set, we can distinguish three cases: (a) Multiplicative interdependence. The form of Eqs. (1) used by Baumol and Oates is

4:

rA = -

2

(4)

rB =

2 qB T +

wqAqB*

(5)

rB =

4B +

wqAqB,

(5’)

Alternatively

where w = a coefficient measuring the strength of the effect of A’s output on B’s costs, and is assumed constant; w > 0 for an external cost. Finding the partial derivatives of the particular production possibility equation that results from (4) and

124

PAUL BURROWS

(5)’ and inserting them in (3) yields:

d2qB -0 + WA)+ wL4 + VE) -= 4: 0 + w”)2 ’

(6)

which is negative if 2w2q, + wq, < 1, as it will be if w = 0, but not if w is a large, positive constant. This multiplicative interdependence is a nonseparable externality; the marginal cost of the pollutee is raised by w > 0, since Cl = 1 + wqA. (b) Separable externality. The convexity of the production set is unaffected if the externality is separable. The pollutee cost function is (7) Combining Eqs. (4) and (7) yields the second derivative of the production possibility curve: d%* -= 4:

-1

(8)

whether the externality exists or not (w > 0 or w = 0), and regardless of the size of w. The marginal cost of the pollutee’s output is independent of the polluter’s output (C,” = 1). (c) Nonseparable, nonmultiplicative externality. Baumol and Oates (1975) appear to identify nonseparable externalities with a multiplicative form of the pollutee’s cost function.62 But it can be shown that the two are not synonymous. Consider the meaning and implication of the nonmultiplicative pollutee cost function: rE = 4B + w7.

(9)

The externality enters the pollutee’s costs as an addition to marginal cost if qA > 0 (then Ci = 1 + w), and is therefore nonseparable, but the addition is independent of the size polluter’s output. The dependence of Ci on qA is a step function; the Ci curve shifts up parallel as qA goes from zero to one unit, but there is not a family of curves for different qA. An example of such an externality is where a laundry is obliged to rewash sheets made dirty by an air polhttant, but the rewashing is the same however many smuts land on the sheets. The second derivative of the production possibility curve for Eqs. (4) and (9) is

d2qB -- 1 -= 1+w’ 4: which is negative even for large values of w. Therefore even a strong nonseparable externality does not necessarily generate nonconvexity. 62 Compare their fn. 4, p. 103, with their Appendix A. Davis and Whinston (1962) take the same view.

NONCONVEXITY

BY EXTERNAL

PRODUCTION

125

COSTS

A.3. Multiplicative Externality and the Fom of a Nonconvexity The demonstration of the possibility of global nonconvexity for large values of w in the case of multiplicative interdependence requires the assumption that w is large at all levels of the polluter’s output. Once this strong assumption is dropped, a nonconvexity, if it exists, may be only local. This can be shown by postulating w =f(q,4)

01)

Four cases can be distinguished according to the signs of f ’ and f ‘I, ignoring the cases of f’ or f” equal to zero (Baumol and Oates’model assumed the f’ = f ” = 0 case). Combining Eqs. (4), (57, and (11) yields a production possibility curve whose second derivative is

Term

Term 1 J-l l(1

+/“(q+)q&

+ Zf’(q,)q,)(l

2

j--l +f(qA)q”)

+

2(/‘(4A)%l

+/(44))14r

+/‘(4A)qAqR

+f(q4)4R)

,(l +/GTh)~, Term

3

(14

0

The sign of Eq. (12) in the four cases: Casel:f’>O,

f”>0:63

f(qJ + 0 as qA --, 0 f’(q,J + small, positive, or + 0 f “( qA)qA -+ 0, unless f” + cc faster than qA + 0. Ignoring the extreme case f ” --, cc, we find that as qA + 0: Term

necessarily negative: + - 1 as qA + 0, if f ‘( qa) + 0, or + - (1 + 2f ‘(qA)qe),

if

f ‘(qA) + finite,

small.

Therefore the second derivative tends to - 1 (the no-externality case) as qA tends to zero (if f ‘(qA) + 0). At low levels of qA, convexity is assured, so that the nonconvexity, if it exists, is local. 63Note that the linear case f’ > 0, f” = 0 is similar.

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Case 2: f’ > 0, f” < 0:

fbl*) + 0 as qA -+ 0 f’k) + large, positive, or + + cc f “(qA)qA + 0, unless f” + - 00 faster than qa + 0. Ignoring the extreme cases f’ + cc, f” + - cc, we find that as qA + 0: Term Term Term

0

1 + --(I + 2f ‘(qdqd

2 +O I7 0

3 + 1.

Therefore the second derivative tends to - (1 + 2f ‘(qA)qB) as qA tends to zero. If a nonconvexity exists it is only local. Case 3: f’ c 0, f” > 0: f(qJ -+ wo + large, negative, or + - 00 in the limit as qa + 0 f’(qJ f “(qA)qA + 0, unless f” + oo faster than qa + 0. Ignoring the extreme cases f ’ + - cc, f ” + cc, we find that as qA + 0: Term Term Term

0 0 0

1 + -0

+ 2f’(q,)q,)

2 + 2wiq, 3 + 1.

The sign of term

1 is ambiguous, whereas 2 is positive. Therefore the sign of cl the second derivative is ambiguous, but it may Pe positive at low qA levels, especially if w. is large. However, the implied nonconvexity is not necessarily global, since as qA becomes large qB + 0, and if f (qA) and f ‘( qA) + 0 as qB --) 0, then the second derivative + - 1. In this case a local nonconvexity would be observed only at low levels of the polluter’s output. But if f(qA), f ‘(qA) do not tend to zero at high polluter output levels, then global nonconvexity can occur with this w function. Note, however, that the global nonconvexity requires w to be large at UN levels of polluter output. If instead the function were characterised by, say, f” > 0,

for qA 5 q%

f(qJ+Oasq, f’ < 0, f” > 0,

4 for qA > 42,

f’ ’ 0,

NONCONVEXITY

BY EXTERNAL

PRODUCTION

COSTS

127

where q: is an arbitrary level of qA, then global nonconvexity would not exist since case 1 would be relevant to low levels of qA. Case 4: f’ < 0, f” < 0:

fk4) + wo as qA + 0 f’k) + small, negative, or -+ 0 f”(qA)qA

-+ 0, unless f” + - 00 faster than q,, + 0.

Ignoring the extreme case f” + - co, we find that as qa --+ 0:

Term u1+-(l+2f’h.h) Term 2 + 2$q, 0

Term 3 + 1. 0 The difference between this result and that for case 3 is that term 1 is now more 0 likely to be negative, since f’(qA) tends to become small, so local convexity is more likely. At high levels of qA, qB + 0 and f(qA) + 0, local convexity of the production set is guaranteed (assuming f( qA) + 0 in an output range which lies within the feasible production set). The implication of the analysis of these four cases is that Baumol and Oates’ Proposition 2, 64 that a sufficiently strong externality which arises only when the level of each activity is nonzero must produce a nonconvexity in the social production set, is valid only for a local nonconvexity if w = f(qA), and w is large at some levels of qA and not others. REFERENCES W. J. Baumol, External economies and second-order optimality conditions, Amer. Econ. Rev. 54, 358-372 (June 1964). W. J. Baumol, On taxation and the control of externalities, Amer. .&on. Rev. 62, 307-322 (June 1972). W. J. Baumol, “Economic Theory and Operations Analysis,” Prentice-Hall, Englewood Cliffs, N.J. (1977). W. J. Baumol and D. F. Bradford, Detrimental externalities and non-convexity of the production set, Economicu 39, 160-176 (May 1972). W. J. B@umol and W. E. Oates, “The Theory of Environmental Policy,” Prentice-Hall, Englewood Cliffs, N.J. (1975). W. J. Baumol and W. E. Oates, “Economics, Environmental Policy, and the Quality of Life,” Prentice-Hall, Englewood Cliffs, N.J. (1979). P. Burrows, Pigovian taxes, polluter subsidies, regulation, and the size of a polluting industry, Gun. J. Econ. 12,494-501 (August 1979). P. Burrows, “The Economic Theory of Pollution Control,” Martin Robertson, Oxford, and MIT Press, Cambridge, Mass. (1980). A. C. Chiang, “Fundamental Methods of Mathematical Economics,” 3rd ed., McGraw-Hill, Tokyo (1984). 64See Baumol and Oates (1975, p. 108).

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M. J. Cooper, “The Industrial Location Decision Making Process,” Birmingham University Centre for Urban and Regional Studies, Occasional papers, Number 34 (1975). R. D. Cooter, How the law circumvents Starrett’s nonconvexity, J. &on. Theory 22,499-504 (1980). P. S. Dasgupta and G. M. Heal, “Economic Theory and Exhaustible Resources,” Cambridge Economic Handbooks, Cambridge (1979). 0. A. Davis and A. B. Whinston, Externalities, welfare and the theory of games, J. Polir. Econ. 7, 241-262 (June 1962). D. T. Dick, “Pollution, Congestion and Nuisance,” Lexington Books, Heath, Lexington (1974). M. J. Farrell, The convexity assumption in the theory of competitive markets, J. Polir. Econ. 67,377-391 (1959). A. C. Fisher and F. M. Peterson, The environment in economics: A survey, J. Econ. Lit. 14,1-33 (1976). C. J. Goetz and J. M. Buchanan, External diseconomies in competitive supply, Amer. Econ. Reu. 61, 883-890 (December 1971). J. R. Gould, Total conditions in the analysis of external effects, &on. J. 87, 558-564 (September 1977). S. H. Hanke and I. Gutmanis, Estimates of industrial waterboume residuals control costs: A review of concepts, methodology and empirical results, in “Cost-Benefit Analysis and Water Pollution Policy” (H. M. Peskin and E. P. Se&in, Eds.), The Urban Institute, Washington, D.C. (1975). G. M. Heal, “The Theory of Economic Planning,” North-Holland, Amsterdam (1973). A. V. Kneese and C. L. Shuhze, “PoIlution, Prices and Public Policy,” The Brookings Institution, Washington, D.C. (1975). W. F. Luttrell, “Factory Location and Industrial Movement,” Cambridge Univ. Press, Cambridge (1962). A. Mas-Colell, Indivisible commodities and general equilibrium theory, J. Econ. Theory 16, 443-456 (1977). J. E. Meade, “The Theory of Economic Externalities,” Sijthoff, Leiden (1973). E. J. Mishan, “Welfare Economics: An Assessment,” North-Holland, Amsterdam (1969). S. K. Nath, “A Reappraisal of Welfare Economies,” Routledge & Kegan Paul, London (1969). R. T. Page and J. Ferejohn, Externalities as commodities: Comment, Amer. Econ. Rev. 64, 454-459 (1974). D. W. Pearce, “Environmental Economics,” Longman, London (1976). D. W. Pearce, “The Valuation of Social Cost,” Allen & Unwin, London (1978). D. W. Pearce, “Benefits of Environmental Policies as Avoided Damage,” for O.E.C.D. (mimeo) (1982). H. M. Peskin and E. P. Se&in (Eds.), “Cost-Benefit Analysis and Water Pollution Policy,” The Urban Institute, Washington, DC. (1975). R. D. Portes, The search for efficiency in the presence of externalities, in “Unfashionable Economics: Essays in Honour of Lord Balogh (P. Streeten, Ed.), Weidenfeld and Nicholson, London (1970). J. Rothenberg, Non-convexity, aggregation, and Pareto optimality, J. Polit. Econ. 68,435-468 (October 1960). P. J. W. Saunders, “The Estimation of Pollution Damage,” Manchester Univ. Press, Manchester (1976). M. Slater, The quality of life and the shape of the marginal loss curves, Econ. J. 85, 864-872 (December 1975). V. K. Smith, Detrimental externalities, nonconvexities and technical change, J. Public Econ. 4, 289-295 (1975). D. A. Starrett, Fundamental nonconvexities in the theory of externalities, J. Econ. TheoT 4, 180-199 (1972). D. A. Starrett and R. Zeckhauser, Treating external diseconomies-Markets or taxes? in “Statistical and Mathematical Aspects of Pollution Problems” (J. W. Pratt, Ed.), Dekker, New York (1974). D. P. Tihansky, Historical development of water pollution control cost functions, J. Water Poht. Control Fed. 46, 813-833 (1974). D. P. Tihansky, A survey of empirical benefit studies, in “Cost-Benefit Analysis and Water Pollution Policy” (H. M. Peskin and E. P. Se&in,: Eds.), The Urban Institute, Washington, DC. (1975). C. Walsh and C. Tisdell, Non-marginal externalities as relevant and as not, Econ. Rec. 49, 447-455 (September 1973).