in detail Bawa and some additional results on optimal pollution control policies are obtained. Reference Bawa, V. S., "On Optimal Pollution Control Policies"(unpublishedpaper).
CHAPTER
14
Taxes versus subsidies: a partial analysis
~
We can rest assured that firms and municipalities that are asked to reduce their damage to the environment will look to state and federal agencies for financial assistance. Such a request may seem uncomfortably analogous to the case of a holdup man who appeals to his victims to finance the costs of his going straight. Sometimes, however, a persuasive case can be made in terms of equity. What of the firm that built its smoking factories well away from the centers of population only to find itself surrounded by inhabitants a few decades later? Is it really the comp~ny that is responsible for the damage generated by its emissions of smoke? We must admit to feeling that too much has probably been made of such cases in the literature, and that there usually is some presumption against rewarding government agencies and private enterprises for the damage they have done to the environment in the past. But whatever the virtues of the matter, the issue is a real one. There will continue to be calls for subvention of industrial activities that may otherwise find themselves at a competitive disadvantage and of local agencies whose budgets are already under heavy strain. The central question here is whether or not it is possible to attain an optimal pattern of resource use through a program of subsidies rather than fees. In Chapter 4, we showed that there is a set of Pigouvian taxes that will sustain optimal levels of externality-generating activities in a competitive system. Can this also be achieved by some specified set of payments? The literature has occasionally suggested an affirmative answer to this question. Some writers (including one of the present authors)' have argued that the public authority can use either the stick or the carrot to induce socially desirable patterns of behavior. In recent years, however, a short series of articles has shown that, on any reasonable interpretation, this is simply untrue. Kamien, Schwartz, and Dolbear 2 have demonstrated
See W. J. Baumol, Welfare Economics and the Theory of the State, 2nd ed. (Cambridge, Mass.: Harvard University Press, 1965), p. 104. 2 M. I. Kamien, N. L. Schwartz, and F. T. Dolbear, "Asymmetry between Bribes and
I
Charges,"
Water Resources
Research
II, No.1
(1966), 147-57.
211
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..,
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that where the polluter recognizes the effects of his actions on the regulatory authority, a subsidy scheme may make it profitable for the firm to start off by polluting more than it would have otherwise in order to qualify for larger subsidy payments.3 Wenders, moreover, has suggested that, where there is this sort of interaction between the polluter's behavior and regulatory standards, there is less of an inducement for new pollution-abatement technology from a system of subsidies than a program of taxes.4 Consider a firm that is evaluating a pollution-reducing innovation. If the intrpduction of the new technique (and the resulting lower level of waste emissions) is likely at some future time to induce the public authority to reduce fiscal incentives, then the decision of the firm may well depend upon whether the agency is employing taxes or subsidies. In the former case, the prospective tax reduction would promise increased profits to the firm and thus encourage the introduction of the new technology, but under a system of subsidies, the change in fiscal incentives would take the form of a reduction in the future rate of payments from the agency and hence reduce the profitability of the innovation.
Bramhall and Mills5 have pointed out what to us seemsto be the most
important distinction between the two types of stimuli: the fact that an enterprise that would be unprofitable under a tax may be made profitable by a subsidy. Whereas a tax will typically drive firms out of a competitive industry and so generally lead to a decrease in its output, a subsidy may increase entry and induce an expansion in .competitive outputs. We shall explore this issue in some depth in this chapter and will contend that it is far more significant than a casual reading of the literature would suggest. We will show, for example, that, under pure competition, although a subsidy will tend to reduce the emissions of the firm, it is apt to increase the emissions of the industry beyond what they would be in the absence of fiscal incentives! Moreover, paradoxically, the more the subsidy suc3
In this case, the firm need not be very large for this sort of interdependence to arise. The pollution benchmark will presumably have to be set for each firm in light of its product line, its output level, and its inherited plant and equipment. As with price-control mecha. nisms, it would not be surprising to see the firm's benchmark pollution level, s., against which improvement is to be measured, set on the basis of its emissions during some arbitrarily chosen period. The firm might then have much to gain by emitting a great deal of pollution during that period to increase the value of the base level of its subsidies.
ceeds in limiting the emissions of the firm, the more it may stimulate those of the industry. Similar problems may well arise under oligopoly where the relevant exit and entry may, preponderantly, take the form of the opening and closing of plants rather than firms. Before turning to these propositions, however, it is important to note the element of truth in the contention that there can be equivalence between the carrot and the stick. We will show formally that, in principle, there does exist a program(s) of subsidies that can sustain optimal levels of polluting activities. But the very character of this program suggests immediately that, although it may be an interesting theoretical construct, it is virtually inconceivable that any such program would ever be adopted in practice. We will see that any plausible systems involve fundamental asymmetries between fees and subsidies. For expository convenience, we will for the most part deal only with detrimental externalities so that, according to the analysis of Chapter 4, the appropriate instrument for the achievement of Pareto optimality is always a set of taxes. We will find it convenient in this discussion to deal with just one polluting industry and with the firms that compose it. Thus, for most of this chapter, we leave our general-equilibrium framework and turn temporarily to a partial analysis. One more matter remains to be settled before getting to the substance of our discussion: the nature of the subsidy program we will consider. This is not as obvious as it may seem on first thought. Several different types of subsidy programs have in fact been proposed and their effects may well differ considerably. For example, some proposals have called for a tax credit for investment in pollution-control equipment or for some other device to help cover some proportion of the cost. However, as Kneese and Bower point out, such a subsidy is, at least in principle, likely to prove quite ineffective in stimulating pollution abatement.6 For, if the equipment adds to a firm's costs and contributes nothing to its revenues, the absorption of k percent of the cost by a government agency cannot
6 See A. V. Kneese and B. T. Bower, Managing Water Quality: Economics, Technology, Institutions (Baltimore: Johns Hopkins Press, 1968), pp. 175-78. They point out that various legislative proposals introduced in Congress offer this type of subsidy in a variety of forms including rapid tax write-offs and tax credits. They argue that aside from the fact that such subsidies can never by themselves make abatement investments profitable, they suffer from at least three other defects: First, they increase the "excess burdens" imposed by the tax system; second, this sort of arrangement rewards only the installation of particular types of equipment (for example, treatment equipment), and, hence, may not induce the adoption of the most efficient pollution-control methods; and, third, this type of subsidy aids only firms that are profitable enough to invest and may not be very helpful to marginal concerns. We may note, however, that, from the point of view of efficiency, failure to rescue marginal firms may well be desirable socially.
!I
!
'I
4
J. T. Wenders, "Methods of Pollution Control and the Rate of Change in Pollution
Abatement Technology," Water Resources Research II {June, 1975), 343-6. 5 D. E. Bramhall and E. S. Mills, "A Note on the Asymmetry between Fees and Payments," Water Resources Research II, No.3 (1966), 615-16. On this see also the papers by A. M. Freeman, "Bribes and Charges: Some Comments," Water Resources Research Ill, No. I (1967),287-88; and T. D. Tregarthen, "Collective Supply Problems in the Allocation of an Air Basin," paper delivered to the Economics of Pollution Section of the 1971 Annual Meetings of the Western Economic Association, Simon Fraser University, August, 1971.
turn its acquisition into a profitable proposition. So long as k is less than 100 percent, the installation of the equipment will lose money for the firm, and its attractiveness to management will remain doubtful, except perhaps as a public-relations gesture or as a pure act of conscience by the businessman. The type of subsidy with which we will be concerned in most of this chapter is of quite another sort. It involves a payment to the firm based on the reductions in its output of a pollutant or in some other sort of damage to the environment. That is, taking s to be the firm's output of the pollutant, and s* to be the base (benchmark) against which improvement is to be measured,' the subsidy payment can be described by the relationship g(s*-s), where dgjd(s*-s) > 0 (that is, the payments to the firm increase with the amount by which it decreases its emissions). In the bulk of our discussion we will assume that the subsidy payment per unit reduction in emissions is constant, so that the payment becomes v(s*-s), (1)
pk(Yk)
=the price of its product
= the
total emission of pollutant
ck(Yk>Ok)= total production cost t = the tax rate per unit of emission
Sk(Yk> Ok)
and where we assume
s~=
_Yk >0, a
pk(Yk)
ask
s:= _ <0. a
ak
ask
(3)
Similarly, it is clear that if the firm is instead offered the subsidy (I), its profit function becomes8 7rk= Yk 2
- Ck(Yk'
ak)
+ v[sZ -Sk(Yk>
ak)].
(4)
The equilibrium of the individual firm9
where v and s* are constants. Expression (I) immediately indicates one fundamental difference between programs of taxes and subsidies. With taxes, we need concern ourselves with only one parameter, the tax rate, but a system of subsidies requires that we specify values for two parameters: the unit subsidy (v) and the benchmark level of emissions (s*). In the subsidy programs with which we will concern ourselves, payments are made only to firms that are actually engaged in an activity that is (potentially) polluting. The firm that closes its doors ceases to receive any such payments, and no subvention is given to a firm that is considering entry into the area but has not actually done so. These are features we would expect to characterize any real subsidy program. Their critical significance for the analysis will become clear presently. 1 The formal subsidy relationship and the general case
It is convenient to begin by comparing directly the subsidy profit function (4) with the tax-profit function (2); this comparison immediately yields a significant result about the relative effects of the two types of fiscal incentives on the equilibrium of the individual firm. We see at once that if v = t, the two profit functions differ only by the constant quantity vs*. If the company is a profit maximizer and continues to engage in the same types of activity under either fiscal program, we see that the choice between a tax and subsidy system will not affect any of its decisions one iota. Whatever values of its decision variables it will find most profitable in the one case will also maximize profits in the other.1O There is another way that this conclusion has been described in the literature. The subsidy program (I) has been interpreted as equivalent to a tax on pollution, vs (with v being the per-unit tax rate), plus a lump-sum
8 We should note that the profit function (4) for the firm receiving a subsidy for the reduction of emissions can be taken to represent the profit function in the general case encompassing all three of the relevant possibilities: a subsidy program, a tax program, or the
absence of either. The function, as it stands, is the subsidy relationship. By setting v
= 0,
Assume that firm k is subject to a fixed Pigouvian tax per unit of emission. Its profit function is 7r= Yk where Yk = the output produced by firm k ak = its abatement outlay
7 Note that s' may, but need not be, based on observation example, its previous levels of smoke emission). of the firm's past behavior (for
pk(Yk)
- ck(Yk>
ak)
-
tsk(Yk>ad
(2)
we at once obtain the case with neither taxes nor subsidies. Finally, setting s' = 0, we are left with the pure tax case, with the firm having vs deducted from its profits and thus paying the tax rate v per unit of emission. This observation about the generality of (4) will prove useful to us in Section 4 of this chapter. 9 For an illuminating discussion of the subject of this section, see Kneese and Bower, Managing Waler QualilY, pp. 98-109. See also A. P. Lerner, "Pollution Abatement Subsidies," American Economic Review LXI(December, 1972), 1009-10. 10 In an unpublished note, Yakov Amihud has argued that in the presence of risk the lumpsum payment, vs', may reduce the marginal risk of the subsidized firm and may therefore induce it to maintain an output level larger than that of the taxed firm. On this see, for example, A. Sandmo, "On the Theory of the Competitive Firm Under Price Uncertainty," American Economic Review LXI (March, 1971), 65-73.
~.v
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"
,
subsidy given by the constant US.. Because, by definition, a lump-sum subsidy does not affect behavior, it should hardly come as a surprise that the choice between a tax and a subsidy policy does not influence any of the firm's decisions. This, then, is the basic argument rationalizing the intuitive notion suggested at the beginning of this chapter that a tax and a subsidy, like the carrot and the stick, should be able to achieve the same result. II Strictly speaking, this conclusion is, however, incorrect. For suppose that, in the absence of taxes and subsidies, our firm's maximum profits are zero. Then the imposition of a tax would ultimately force it to close its doors, but the subsidy program could end the precariousness of its existence. Put another way, it is not quite legitimate to describe the component US.in the subsidy (1) as a lump-sum payment, for it may influence the firm's decision between continuation and cessation of operations. This suggests immediately the provision that is required for the subsidy program to establish a set of incentives identical to those of the tax: The lump-sum payment (us.) must not be contingent upon the firm's decision to stay in business.12In principle, this payment must be made to the polluter, whether potential or actual, so that it has no direct influence on any choice that confronts him.13Note that once this stipulation is introduced, the choice of the benchmark level of emissions becomes wholly arbitrary in terms of any implications for optimal resource use; the selection of a value for s. affects only the magnitude of the subsidy payment. The administrative infeasibility of such a system of payments is evident. The lump-sum subsidy must be paid not only to those who continue polluting activities, but also to any potential polluters. For example, a firm that chooses to cease its operations altogether must continue to receive the subsidy payment indefinitely (otherwise the subsidy program might have induced the firm to remain in business). Similarly, potential entrants into the polluting activity must be eligible for the subsidy to
11
prevent them from initiating waste generation simply to qualify for the lump-sum payment. The difficulty of identifying these economic units (along with the obvious political obstacles to such a system of payments) imply that we must restrict our consideration of this form of subsidy program to the conceptual realm, as it does not represent a real policy alternative. Throughout this chapter, it is therefore assumed that subsidy payments in any period t are limited to firms that are actively in business during this period. The equivalence of the incentives under the tax and subsidy programs then vanishes, and we conclude Proposition One. For the individual firm, the choice between a tax and a subsidy program to induce a decrease in pollution emissions may determine whether or not the firm continues its operations. However, other things being equal, no other decision of the profit-maximizing firm will be influenced by the choice between the two fiscal measures provided the marginal tax and subsidy rates are equal. Note that Proposition One does not enable us to reach any unambiguous conclusions about the relative desirability of taxes and subsidies in practice. If the firm stays in business, its level of output will be identical under the two fiscal programs. However, we know (ignoring any external effects) that monopoly outputs are normally less than optimal.'4 It is thus conceivable that a subsidy, if it permits a monopoly to continue its operations, may be a second-best solution superior to a tax that leads to the cessation of production. However, when we turn next to the case of pure competition, the conclusions are unambiguous. As we have already shown in Chapter 4, the appropriate taxes imposed on detrimental externalities are indeed capable of yielding a Pareto optimum. In the next section, we will see, however, that, for the competitive industry, subsidies may be expected to produce pollution levels very different from those corresponding to a Pigouvian tax program. We find that subsidies must unavoidably violate the necessary conditions for Pareto optimality (Table 1 of Chapter 4).15.16
14 As we noted in Section 1 of Chapter 6, Buchanan and others have pointed out that the imposition of efftuent charges on monopoly firms may actually reduce welfare, because they will induce a fall in the level of an output that is, perhaps, already less than optimal. See his "External Diseconomies, Corrective Taxes, and Market Structure," American Economic Review LlX (March, 1969), 174-77. IS In most of this chapter, we will take the utilization of resources achieved by the Pigouvian tax as the standard of optimality against which to measure the subsidy program. It is easy to argue the propriety of this procedure intuitively. After all, the tax merely makes the individual pay all of the social costs of his activity. The optimality of a system of pure. competition in the absence of externalities follows in part from this characteristic
I 1
There is a different argument whose invalidity is shown in Section 5. Suppose there are two industries, A and B, and that a tax rate of t on A's output will achieve the desired reallocation of resources from A to B. Then surely the same thing can be accomplished by an r dollar subsidy to A if a sufficiently greater subsidy is provided to B and (fiscal and monetary) policy keeps the levels of employment of resources from changing. As will be shown later, this argument is, in fact, incorrect so long as the relative prices corresponding to a given optimum for the economy are unique, and the absolute prices are fixed by some normalization rule or otherwise.
12 Kneese and Bower note this condition (Managing Water Quality, p. 104). 13 We must say no "direct" influence here, because a set of lump-sum taxes or subsidies will have income effects leading, in general, to a new general equilibrium set of relative prices. The point is that such a program has no direct price effects in the sense of altering the terms of choice in the initial equilibrium situation.
---~e--
--
,
Before turning to the behavior of the industry in the next section, the reader should note that Proposition One refers explicitly to the individual firm and applies only with "other things being equal." This means that, if the tax or subsidy has no effect on the price of the firm's output, then the firm (if it stays in business) will operate at the same level of output with the same level of waste emissions under both fiscal programs. However, as we shall see in the next section, a system of taxes in a competitive industry will generate a different industry supply curve (and hence a different price) than a subsidy program. As a result, the new equilibrium output and emissions level for the competitive firm will differ under the two sets of fiscal incentives. 3 The case of the competitive industry
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c
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c
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Matters turn out quite differently in the competitive industry, because exit and entry are an integral element in the determination of total output. Here we can expect the choice between a tax and a subsidy to have a significant effect on total output. In fact, the results of a subsidy may well prove surprisingly unsatisfactory, as we will now show. In this section the argument will proceed on the simplifying premise that emissions are a single-valued function of industry output, and in the next section it will be generalized to take account of the possibility of changing emissions independently of output (abatement). It may be helpful to consider the argument first in diagrammatic terms. In Figures 14.1a and 14.1b, we depict the equilibrium positions of a rep()
I
I
0
0
<T
I
Footnote
15 (conI.)
I
I
:
of its operation. The tax program, in effect, internalizes all externalities and makes a competitive system operate as if no externalities were present. That is why the tax system always yields optimal results and why, if a subsidy program leads to a different pattern of resources utilization, it is likely not to be optimal. However, we must be careful in using this argument. Because a Pareto optimum is normally not unique, one cannot be certain from the observation that the allocation of resources under the subsidy program differs from that under taxes that the former is not itself Pareto-optimal. This point will be examined further in Section 5. 16 Note that Table I of Chapter 4 shows that Pigouvian taxes will sustain Pareto-optimal exit and entry decisions by all the firms in a competitive economy and not just optimal decisions on nonzero activity levels. The exit-entry decisions relating to emissions of pollution are represented by conditions 5 and 5c in Chapter 4, which show that the equilibrium emissions of the firm will be zero under a Pigouvian tax regime if, and only if, that is true in the corresponding Pareto optimum. However, we recall that these results depend on competitive behavior and on each polluter being a "small" source of emissions. As we saw in Chapter 4, if the marginal damages from the firm's emissions are not (approximately) constant over the range of its discharges, then the firm's Pigouvian tax bill will not equal the total damages that its emissions impose on society. In such cases, the Pigouvian tax will not provide t,he correct incentive for the firm's entry-exit decision.
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resentative competitive firm (firm i) and the corresponding competitive industry under three different sets of circumstances: the equilibrium point, (yC, pC), when there is no public environmental program; point (y', pi) with a unit tax on pollution emissions; and point (yV, pV) when there is a unit subsidy, v (equal to I), for reductions of emissions below some benchmark level. Starting from the no-program solution, we note that the unit pollution tax produces an upward shift in the firm's marginal and average cost curves (to MC"V and AC,). If, instead of having no environmental program, a system of subsidies is instituted (under which we assume there are no negative subsidy payments), the firm's marginal cost shifts up to MC"v' but its average cost is now reduced to ACv' From our earlier results, we know that the tax and subsidy programs have identical effects on the firm's marginal costs. Consequently, in Figure 14.la, the sole difference in the firm's cost relationships under the two programs is that its average cost under the system of subsidies (ACv) will be less than its average costs (AC,) under the pollution tax or in its absence (ACc)' However, entry and exit can be depended upon to drive price down to the firm's minimum level of average cost. The result may actually be no change or even an increase in the equilibrium emissions of the individual firm under an emissions tax. For example, if emissions are strictly proportionate to output, the equilibrium output of the representative competitive firm must be exactly the same with and without the tax, for a fixed tax per unit will then shift its average cost curve directly upward by a uniform vertical distance (it will not be increased by full amount of the unit tax because rent will also be affected by the accompanying change in industry output) and so the firm's cost minimizing output and emissions levels will remain completely unaffected by the tax.17 However, a subsidy program will generally decrease the equilibrium emissions of the competitive firm. Geometrically, we see this by noting that the new marginal cost curve, MC"v' must now cut the original (noprogram) cost curve, ACc, at a point that lies to the left of the old equilibrium point, J. But, ACv' the average cost curve with subsidy, must lie
17 Robert Kohn has demonstrated recently that although this result holds in a partialequilibrium competitive model, it does not, in general, hold in a general-equilibrium analysis. In a full general-equilibrium selling, Kohn demonstrates that this proposition is valid only if firms' production functions are homothetic. See his, "A General Equilibrium Analysis of the Optimal Number of Firms in a Polluting Industry," Canadian Journal 01 Economics XVIII (May, 1985),347-54. Note, however, that the result must hold for a perfectly competitive equilibrium, where at the equilibrium point production functions must always be linearly homogeneous locally.
below ACc' and so, given a positive slope of the marginal cost curve, the new equilibrium point, L, must lie still further to the left of 1.\8 Turning now to the emissions of the industry, which are, of course, the primary concern of policy, we note that the tax program, because it raises every firm's average and marginal costs, must result in a leftward shift of the industry supply curve, from Sc to Sf; price rises from pCto p' and industry output falls from yCto y' with a consequent decline in the industry's emission of pollutants. This happens though each firm that continues to operate produces the same output in both cases, because the tax will drive some firms from the industry. Similarly, the subsidy will induce the entry of firms (producing the rightward shift of the industry supply curve from Sc to Sv); the result is a fall in price(to pV) and an increasein industry output (to yV) and in industry emissions. Note that, although the individual firm produces less under the subsidy than it would under either the tax or in the absence of any program, the industry output under the subsidy (yV) exceeds both y' and yC; thus, the entry of new firms more than offsets the reduction in emissions by the individual firm. More specifically, if waste emissions are a fixed and rising function of the volume of industry output (no abatement technology available), Figures 14.la and 14.1b suggest the disturbing conclusion that, although a subsidy program may reduce the emissions of each firm by itself, the subsidies, far from yielding a reduction in total industry emissions like a pollution tax, may, in fact, increase emissions from their unregulated level! It is easy to show that this paradox must result if emissions increase with output, and if the slopes of the industry supply and demand curves are respectively positive and negative, as we normally assume. For, on the premise that the subsidy program as described by (I) never involves a negative subsidy payment (that is, a payment by the firm to the government), some reduction in average cost to the industry must result. Hence, with a subsidy, the long-run competitive supply curve must shift downward and so, with a negatively sloping demand curve, equilibrium output and pollution must be increased above the levels they would have reached in the absence of government intervention. In sum:
18
.11
I I
If emissions are strictly proportionate to output, so that we may .write s = by, this result
is trivial if the average curve has a single minimum and a continuous first derivative. For if g(y) represents the firm's average cost in the absence of a tax or subsidy, with minimum point given by dg(y)/dy
= 0, at
that
point
the slope
of the average
cost curve
with
subsidy is d[g(y)ub(y'y)/yl/dy= dg(y)dy-d[uby'/yl/dy= uby'/y2 > 0,
so that the average cost minimizing output in the absence of subsidy must be greater than that under subsidy.
L.LL,
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POliCY
:
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I
Proposition Two. In a competitive industry, where pollutiii'g are a fixed and rising function of the level of industry outpij1;'eq and subsidy rates will normally not lead to the same outpllt levellio . the same reductions in total industry emissions. Other thingsoeingeqUt. the subsidy will yield an output and emission level not only greater.than those that would occur under the tax, but greatereven than they WoUld: be in the absence of either tax or subsidy.J9 As already noted, the explanation of our paradox is straightforward. The subsidy does indeed reduce the level of emissions per firm. BuUt, necessarily attracts into the industry enough additional firms to offset tl1is reduction and more. Thus, we can hardly expect the of effect the subsidy on the decision of the firm to continue or discontinue o~erations to bean insignificant matter. A further examination of the case in which emissions depend exclusively on output can sharpen these results and offer some additional insights. Let us simplify still further by assuming emissions to be strictly proportionate to output. Then we may write s = by (b some constant) as the emissions-output relation and let s. = the base pollution level for calculation of the subsidy, y. = the corresponding output level where s. = by., sV = the emissions of the representative firm after imposition of subsidy rate, v, per unit of emissions, and y v = the output of the representative firm under a subsidy program with a pollution benchmark, s., and a subsidy rate v= t per unit of reductions in emissions, SV= byv. With subsidy rate v, the total subsidy payment to the representative firm must be v(by.-byV) so that the subsidy per unit of output will be vb(y.-yV)/yv=vb[(y./yV)-I]. (5) This will be positive if, and only if, y. > yV (that is, so long as the benchmark emission level at which zero subsidy is paid is set higher than the
19 We note that our proof of this proposition employs a partial-equilibrium framework. Mestelman has shown that in a general-equilibrium setting, it is possible that a subsidy will not result in an increase in industry output and emissions. This result, although presumably unlikely, could occur in response to relative factor price adjustments within a general-equilibrium system. See S. Mestelman, "Production Externalities and Corrective Subsidies: A General Equilibrium Analysis," and Management IX (June, 1982), 186-93. Journal of Environmental Economics
I's)e'l~lof emissions under the subsid)1\1rogram). "Thus, so long as ..>.yu,the subsid)1\1rogram must \1roduce a unilorm downward shilt in
.U\~
~,\~
11,1
I
industr)1 su\1\11)1 curve b)1 the amount indicated b)1 (5), though
one
I ..1 I
'.thansnot generall)1equal to the u\1ward shilt that results hom a tax \1rogram. This, incidentall)1, \1oints u\1 the im\1ortance 01 the value 01 the 'Second\1arameter in a subsid)1 \1rogram: the benchmark \1ollution level (s,.). The larger s., the more the industr)1 supply curve shifts down and the larger win be the industr)1's output (and emissions). This is in contrast to our earlier conceptual subsidy that was made equivalent to a tax by paying the subsrdy to all "potential" polluters; there, the benchmark pollution level had no direct effect on the industry supply curve. Now from (5), we can immediately derive a second paradoxical conclusion: Proposition Three. If emissions rise monotonically with industry output, the more effective the subsidy program is in inducing the individual firm to reduce its emissions, the larger is the increase in total industry emissions that can be expected to result from the subsidy. This follows at once, for the smaller the value of yV relative to y., the larger will be the unit subsidy payment (5) and so the larger will be the resulting downward shift in the industry supply curve. In other words, the more effective the subsidy program is in inducing the desired behavior on the part of the individual firm, the worse for society the corresponding subsidy program will be! 20 To summarize, we see that in a competitive industry the consequences of a given tax and subsidy rate are far from similar; a subsidy intended to curb pollution may produce exactly the opposite outcome by inducing increases in total emissions. Note also that the problem need not be limited to competitive industries. Under oligopoly, for example, a subsidy
20
I
I
I
I
A moment's thought shows that this proposition must hold where the output-emissions function for the industry takes the more general form g = G(y) where G' > O. The result
is in no way dependent on. our simplifying proportionality premise s
= by.
It does,
how-
ever, depend upon our assumption that there is no pollution-abatement technology so that the firm is unable to reduce emissions per unit of output. We will relax this condition in the next section. One way to get around the difficulty of Proposition Three is to reduce the unit subsidy with the number of firms so that, with n firms in operation, the individual firm will
receive(v.jn)(s.-s')
instead of v(s.-s')
wherev. is a constant.Therewouldappear
to be serious practical difficulties to such a variable-subsidy arrangement. In any event, it does not help in the more fundamental difficulty described in Proposition Two, for any positive subsidy payment must, in the conditions we are discussing, than decrease the pollution emissions of a competitive industry. increase rather
...AA"'~ ,"'. ~...~ ~
- r--
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program may induce the entry of new firms or the opening of additional
plants that can produce preciselythe same sort of result.21
4 Industry equilibrium with abatement technology
As was shown in Chapter 4, appropriate taxes wiIIalways lead to optimal industry outputs even when the emissions of the firm depend not only on its outputs but also on the resources it devotes to their abatement. However, we have seen in Section 6 of Chapter 7 that, where emissions depend on the levels of both of these types of activities by the firm, the level of the polluting output may very well be increased by the imposition of a tax simply because the corresponding Pareto-optimal level of that output is greater than it would be in a competitive market equilibrium. Indeed, if several industries produce the pollutant or if the community has several different pollutants to contend with, the optimal tax may conceivably result in an increase in the industry's emissions of the pollutant. However, such anomalies generally arise only in cases that violate the concavity-convexity conditions that are usually assumed to hold. The issue before us here, rather, is the effect in the "normal case" of a tax or a subsidy upon outputs and emissions of the industry in long-run equilibrium, when abatement techniques (whose effectiveness can be increased by increased abatement expenditures) are available. We will prove the following result: 22 Proposition Four. Under "normal" concavity-convexity conditions, where the competitive industry adjusts both outputs and abatement outlays and all inputs are purchased on competitive markets so that cost functions are fixed, then a marginal addition to a tax on emissions wiIIalways reduce the total industry output of the pollution-generating commodity and reduce total industry emissions. On the other hand, a marginal addition to a subsidy for reduction of emissions wiII increase output of the commodity in question, but its effect on total industry emissions cannot in general be predicted from the maximization conditions, and may go in either direction. To derive this result, we use the same basic notation as before in writing our s!Jbsidy-profit relationships. Abatement outlays wiII be dealt with
21
only implicitly, taking them to be a function, a = G(y, s), of output, y, and emissions, s, with Gy> 0 and Gs< 0 (i.e., if emissions are to be held constant, more money must be spent on abatement as output increases, etc.). We use the same expression, u(s*-s), to represent both the subsidy and the tax payment, where u is the subsidy (tax) rate and s* is the base from which reduced emissions are calculated. Thus, if s* > s, the expression represents a subsidy payment, but if s* = 0, it becomes an emissions tax payment of us. We must begin the proof of the proposition by deriving some results about the behavior of the representative firm. Then we will turn in succession to an examination of ay/av and as/av,where y and s are the output and emissions rate of the representative firm. Finally, we will examine the effect of a rise in v on industry output and emissions, any/av and ans/av, where n is the number of firms in the industry. We may now express the firm's cost function as the result of its selection of an emissions level that minimizes its total cost, given the level of its output. Thus, letting C and c, respectively, be the firm's cost function with and without the subsidy (tax), we have C(y, v) = Min[c(y, s) - u(s*-s)].
I I I
I I
\
This immediately yields the maximum conditions
(6)
cs+u=O,
css>O.
\ I
Using the usual comparative statics approach to determine the effects of a change in v and the interrelations of the other variables in equilibrium, we differentiate the equation in (6) totally with respect to y and s, and then, in turn, with respect to s and v and set the total differentials equal to zero to obtain (7) as cys as 1
_=__ ay css'
_
_=--<0
av css
and, by (6), (7), and the second-order conditions
as
cy=cy+(Cs+u) Cyy- Cyy+Cys-y a which imply, by (7), that D= CyyCSS-c;s>O. ay =cy (8) Css
We are grateful to Lionel Robbins for this observation.
as - CyyCss- > 0 _ c;s
22 We are deeply indebted to Eytan Sheshinski, who provided the following proofs and to Peter Coughlin for his helpful comments. We must also. thank Michael Braulke and Alfred Endres, "On the 'Economics of Effluent Charges," Canadian Journal of Economics vm (November 1985), 891-4, for pointing out some errors in our earlier formulation in the first edition of this book..
(9)
We turn next to long-run competitive equilibrium with the zero profit condition for the firm
yp(ny)
Then we obtain the effect upon industry output, ny, from (15) and (16) as dny dy -=n-+ydv dv
*
- C(y, v) = o.
(10)
Because in competitive equilibrium the firm selects its output to maximize its total profit at the (fixed) equilibrium price, we obtain, differentiating 00) with respect to y,
p(ny)
dn dv > 0 if s*-s> <0 0, so that v(s*-s) is a subsidy; -vs, an emissions tax.
= s-s yp'
- Cy(Y, v) = O.
(11)
if s*=O, so that v(s*-s)=
Again taking total differentials of the first-order conditions in equilibrium conditions (10) and (II) in the standard comparative statics procedures, we obtain nyp' [ np' y2p' dY
Cv
To determine the effect upon emissions, we get, by (7), ds as dy +------as dv - ay dv av which, by (15), cys dy css dv 1 css'
yp' ][ dn ] = [ Cyv ] dv where the determinant of the system, A, satisfies
-
(12)
Cyy
=-
~
css [ 1
CYS-
A= y2p'Cyy< 0 by the second-order conditions. Next, solving (12) as a pair of linear equations in dy, dn, and dv, we get
yD
I (s-s*)css+
yCys)+ 1 , ]
which, by the definition of D in (9),
v A Ycyy From the definitionof C(. ) and (6), we obtain as C,,= -(s*-s)+(cs+v) av = -(s*-s)<O and, by (14)and (7),
C
yv
dy - = -1 [yp ,Cvd
y p Cyv] = -
2,
1
=[Cv- yCyv]'
(13)
yD (s-s*)cys+
ycyy).
Finally, we determine the effect of a change in v on the industry's total emissions, ns:
(14)
-=n-+sdv dv = - -n
_ - -
dns
ds
dn
dv
(s-s*)Cys+ * YCyy)2 (CysY+ (s-s*)c"s)+ Y D *
ns
s(s-s*)
2' yp
= as= _
ay
1
YD
cYS
css
.
* Cys
Substituting the two preceding expressions into (13) gives dy - _dv
n 2 [y(s-s y D
2 )Cys+ Y Cyy+ yscys+s(s-s
s(s-s*) )css] + ~ yp
Yc yy [
s-s +y-
css ]
_ -
n
(s-s yD
1
*
)css+ YCys) ,
(15)
= - y2D [2y(s-s*)cys+ y2cyy+ (S-S*)2css] sOn - Y 2D (YCys+(s-s*)css)+ s(s-s*) 2 yp
where by (8) and (9) D = Cyycss' From (12), we also obtain
dn - _ np' d v - V2 D'c
.
.
(17)
CvCyy
[Cyvy-CvJ+
CyS -y+s-s* y2Cyy [ .css n = -~(cysy+(s-s*)css)+~' yD
=--
n
Y 2P 'c yy s-s* +y2p' ] s - s* yp
(16)
Now it is readily shown thai the first term in (17) is negative, because the bracketed quadratic form y2cyy+ 2y(s-s*)cys+
=Css (s-s*)+-Ey
(S-S*)2css
C 2
(
css
) (
+
c C c2 ss yy- ys y2>0 css
)
by (8).
We see at once that dns . . . f h dv < 0 I s*= 0, t e emISSIOns case, tax
but that its sign is indeterminate if s* > 0 and s* - s > 0, the subsidy case, because then the last term in (17) must be positive.23 Q.E.D. 5 Uniqueness of the tax solution for Pareto optimality
Similarly, if the victims of a detrimental externality continue to suffer from it when one shifts from one optimum to another, but in both cases generate no externalities themselves, they will be required in both cases to receive zero compensation for the damage they suffer (neglecting lumpsum payments). Thus, a shift between Pareto optima cannot introduce compensation of the victims of externalities. In sum, we have Proposition Five. If every activity that yields detrimental externalities in one Pareto optimal solution also does so in some other solution, both solutions will call for Pigouvian taxation of these activities. Moreover, there will always be zero compensation and zero taxation of the victims of the externality. The analogous proposition (with unit subsidies instead of taxes) applies to external benefits. However, as we will see now, for any particular Pareto optimum, there is a formal sense in which a complex system of subsidies can generally be substituted for a simple Pigouvian tax. For the choice of unit of account does offer a degree of freedom in the selection of the price, tax, and subsidy values called for by the solution in Chapter 4. It is easy to show that the solution summarized in Table 1 of Chapter 4 is unique except for the factor of proportionality permitted by our price normalization convention.24 This is, of course, what we would expect: For a particular competitive equilibrium, relative prices will be determined uniquely, with taxes serving as prices for the generation of externalities. Thus, we can multiply all prices, taxes, and subsidies by the same constant, call it (1- k), without violating the optimality requirements. Now it is true that in a formal sense, by using some appropriate value of k, we do get a system in which taxes and subsidies replace one another. As an illustration, assume for simplicity that any increase in taxes produces an equal increase in price and that, as in Chapter 4, only commodity 1 produces externalities, requiring the imposition on that good alone of a tax at rate t. Then the price of that item is changed from PI to (PI + tl)' Now, if all prices and taxes are reduced by the factor (1- k) this becomes
(1-k)(PI + tl)
.
The argument that a subsidy is not usually an adequate substitute for a tax, because the former will generally not satisfy the conditions for a Pareto optimum, may at first leave the reader uncomfortable because of the non uniqueness of the Paretian solution that is inherent in the concept. We know that there will usually be a substantial set of Pareto optima with each optimum corresponding to a different distribution of benefits among the affected parties. Must it not be true then that one can get from one such solution to another with a suitable redistribution (that is, with different combinations of unit taxes and subsidies of the activities of the affected parties)? Will not all such tax and subsidy programs be Pareto-optimal? There is an element of validity to this argument. Either by changes in the initial income distribution or through lump-sum taxes or subsidies, one can get from one of the optimal, solutions to any other. But any Pareto optimum achieved in this manner must always end up satisfying the necessary optimality conditions derived in Chapter 4. If those necessary conditions call for a tax per unit of output, then a per-unit subsidy on just that item simply will not do; it will generally prevent the attainment of optimality. That is, of course, the nature of a necessary condition. That the move from one Pareto optimum to another will not change the Pigouvian taxes into subsidies follows immediately from one highly plausible assumption: that the change from one optimum to another does not transform any activity from a generator of external benefits into one that yields detrimental externalities, or vice versa. The product whose manufacture yields noxious fumes does not begin to emit Arpege. Our result follows at once, for we have demonstrated in Chapter 4 that optimality always requires taxation of activities that produce detrimental externalities and subsidization of those that yield external benefits in accord with the standard Pigouvian formula.
23
= PI+
[tl-
k(pI+
tt)]
= PI +t~
(18)
To show that the sign is indeterminate, it is necessary. strictly speaking~ to provide consistent examples that go both ways. To avoid further lengthening of the argument, we have made no attempt to do so.
24 Recall the condition Pi' wi' (9) of Chapter 4, where Wi'is the Lagrange multiplier corresponding to the labor constraint. This condition may be interpreted as setting the price of labor (leisure) equal to the marginal utility derivable by consumers from a unit addition to society's labor supply. Comparison of optimality relationships (3)-(5) of Chapter 4 with market equilibrium conditions (3<)_(5C) indicates immediately that all optimal
=
prices and taxes can simply be multiplied by the same factor, q.
which is tantamount to the original price plus a subsidy t~, if t~< 0, that is, if
t~=t1-k(Pl+t1)<0
or l>k>tJ!(PI+tlO.
(19)
SITE A' POLLUTING FIRM
However, any other good, i;: I, that was previously untaxed will now have its price changed from Pi to Pi(l-k)=Pi-kPi=Pi+t7, t7= -kPi<O. (20)
Thus, for k sufficiently large to satisfy (19) (that is, to permit a subsidy to the production of commodity I), (20) must represent a set of universal subsidies that together with (18) will yield exactly the same Pareto optimum as the simple Pigouvian tax, t" on commodity I alone. Of course, the subsidy option is extremely cumbersome because it requires one subsidy value to be determined for each activity in the economy in place of the one tax on the externality-generating output. Nevertheless, it is true that Proposition Six. If the necessary conditions for any specific Pareto optimum can be satisfied by a set of Pigouvian taxes, it is generally possible to satisfy those conditions also with a subsidy to the externality-generating activity, counterbalanced by subsidies to other activities. However, this substitution, in effect, amounts only to a variation in the unit of account that leaves all relative prices and taxes unchanged.25 Of course, this sort of substitution can hardly be considered a practical proposal, and it certainly is not what the advocates of subsidy proposals have in mind. Yet it is perhaps useful to recognize to what (very limited) extent there is, theoretically, a choice in the matter. 6 Subsidies to polluters...