When subsidies for pollution abatement increase total emissions.

Kohn, Robert E.

I. Introduction

In a first-best, deterministic world, it is well-known [2; 28; 301 that a unit tax on emissions, equal to marginal damage, is an efficient mechanism for internalizing the damages caused by polluting firms. A unit subsidy for emissions abated, also equal to marginal pollution damage, has the same desirable property, but only in the short-run. In the long-run such subsidies induce polluting firms to operate at too small a scale and attract an excessive number of new firms to the polluting industry. This can have the perverse result that there are more emissions when abatement is subsidized than when it is not. Nevertheless, economists have been reluctant to give up entirely on the subsidy approach, recognizing with Oates [23, 290] that "Polluters will obviously be far more receptive to measures that assist with the costs of pollution control than to those that place the burden upon themselves." It is therefore not surprising that economists have continued to look for and find new justifications for subsidizing pollution control. Thus Mestelman [20, 187] argues that ". . . the subsidy scheme may be a second-best alternative for externality control . . . when the direct taxation alternative is not politically viable." McHugh [17, 64] shows that ". . . subsidies for pollution abatement expenditures" can be a useful instrument in the case of "cost-increasing technological innovations." Harford [6] demonstrates that subsidies for pollution control inputs may be efficient when enforcement is sufficiently costly. Harford [7] makes still another argument for subsidies in the case in which the day-to-day performance of abatement equipment is uncertain, but the uncertainty can be reduced by maintenance expenditures. Finally, there are schemes for combining subsidies and taxes [14; 26]. Given the continuing interest in subsidies, it is important to examine carefully the disturbing case in which subsidies increase rather than decrease total emissions.

Baumol and Oates [2, 212] were the first to recognize that ". . . 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!" They provide a three-page mathematical proof for this, which they credit to Eytan Sheshinski, but caution their readers [2, 228] that ". . . it is necessary, strictly speaking, to provide consistent examples that go both ways (but to) avoid further lengthening of the argument, we have made no attempt to do so". The reason that the subsidy can cause total emissions to increase is more readily explained in the simple case in which the emission rate is constant per unit of output and there is no technology of abatement. Prior to the subsidy, there is long-run competitive equilibrium with zero profits. When subsidies for abatement are offered, polluting firms reduce their emissions by cutting back their output, moving down their marginal cost curve and moving up their average cost curve.(1) Production now incurs an opportunity cost of foregone subsidies and, as Kneese, [8, 90-92] first observed, the marginal cost curve shifts upward just as it would if there were a Pigouvian tax on emissions. Simultaneously, there is a downward shift in the average cost curve. Firms earn profits under the subsidy until more firms enter the industry and there is a new, zero-profit equilibrium in which the shifted marginal cost curve intersects the downward sloping portion of the average cost curve.

This intersection determines a market price which is necessarily lower than before, for if it were higher, as Mestleman [19, 126] adroitly explains, there would be an incentive for some firms to reject the subsidy, increase their output, and earn a profit at this higher price. In the new long run equilibrium, the market price of the polluting good is lower, and the total output and therefore total emissions are greater than before the subsidy. The dynamics become complicated in a more general context in which the relative prices of inputs change and technological abatement reduces the emission rate per unit of output; it is therefore helpful to have the illustrative examples that Baumol and Oates [2, 288] advocate but do not provide.

The major work on this topic, much of it in response to Baumol and Oates[2], has been done by Mestelman[18; 19; 20]. Using a computable general equilibrium model in which there are two goods, one of whose production is adversely affected by pollution generated during production of the other good, Mestelman simulates the contrary cases in which the subsidy causes total emissions to increase as well as to decrease. In contrast to the simple partial equilibrium case described above, Mestelman[18; 19] obtains a new equilibrium with less total emissions in either of two ways: the first, by allowing the emission rate per unit of output to decline as the firm curtails its output; the second, by having the increase in the number of firms drive up the cost of the scarce managerial input. Mestelman's [21, 523] specification of a managerial input not only adds realism in that ". . . it incorporates active economic agents who have incentives to lead the economy to an optimal equilibrium state," but also enables him to explain [19] the case of decreasing total emissions in this second, novel way.

There is a third way to model the case in which total emissions decline; that is by allowing for technological abatement. Whereas Mestelman [20, 187] ". . . neither treats waste emissions as an input nor allows for direct treatment of potential emissions", the model presented in this paper does provide for technological abatement. Moreover, in the model developed here, the production functions of the firms have the familiar property of increasing, then decreasing returns to scale. Mestleman's creative insight [20, 187], that the subsidy might be a useful second-best instrument, is illustrated quantitatively in the present model and is then shown to have policy implications beyond what Mestelman had originally foreseen.

II. The Numerical Model and the Marginal Conditions for Economic Efficiency

Consider an economy in which the pollution level, measured in, say, micrograms per cubic meter of air, is represented by the fraction, e. The source of pollution is the production of good y, which is produced by a divisible number, m, of identical firms, each using [L.sub.y] units of labor and [K.sub.y] units of capital to make Y units of output. The total output of good y, expressed in general and then specific terms, is

y = mY(Ly, Ky) = m[110[L.sub.y.sup.0.3][K.sub.y.sup.0.9] - [L.sub.y.sup.1.3]

[K.sub.y.sup.1.9]/4]. (1)

The production functions in this model exhibit increasing and then decreasing marginal returns to scale and may be either capital intensive, as in (1), or labor intensive.

In this model the emission rate is a constant E units of pollutant concentration per unit of good y, and the pollution level is

e = mE[1 - B([K.sub.b],Y)]Y = m[1/400,000][1 - [K.sub.b.sup.0.5]/([K.sub.b.sup.0.5] = [alpha]

[Y.sup.0.5])]Y. (2)

The fraction of pollution abated by each firm, B(.), increases with the quantity of abatement capital, [K.sub.b], that it employs but decreases with the quantity of Y that it produces. For simplicity, the format of B(.), which is taken from[10] and is analogous to equation (7) in[5], is numerically specified to exhibit constant returns to scale in abatement[5]. To restrict the number of variables in the numerical model, it is realistically assumed that abatement is accomplished with capital alone. In this model, the parameter, [alpha] in (2) is changed to model the contrasting cases in which the subsidy causes total emissions to either increase or decrease. For the case of increasing emissions, [alpha] is set equal to 0.5, and for the case of decreasing emissions, [alpha] is set equal to 0.01. In effect, abatement is more effective the smaller the [alpha] parameter.

Pollution adversely affects the production of good x, which is produced by a divisible number, n, of identical firms, each employing [L.sub.x] and [K.sub.x] to make X units of output. The total production of good x is

x = nX = nF([L.sub.x],[K.sub.x])G(e) = n[88[L.sub.x.sup.0.9[K.sub.x.sup.0.3] - [L.sub.x.sup.1.9]

[K.sub.x.sup.1.3]/5][1 - [e.sup.2]], (3) in which, for simplicity, pollution damage is represented by a multiplicative factor. The contrasting factor intensities in (1) and (3), together with the following input constraints, have the consequence that both industries are increasing cost industries as in[19]. In this model, the input constraints are

n[L.sub.x] + m[L.sub.y] = 1000, n[K.sub.x] + m[K.sub.y] + [K.sub.b] = 1000,

(4) and preferences are represented by the community utility function,

U = U(x,y) = 10[x.sup.0.5] + [y.sup.0.75] (5)

There are four types of long-run competitive equilibrium allocation that are of interest. The first, [A.sup.*] is the Pareto optimal competitive equilibrium in which a Pigouvian tax, [phi], which equals marginal pollution damage, is imposed on emissions. The second, [A.sub.o], is the laissez-faire competitive equilibrium in which there is no government intervention and hence no inducement for polluting firms to abate. The third allocation, [A.sub.o.sup.s], is a competitive equilibrium in which polluting firms receive a Pigouvian subsidy, also [phi], for reducing emissions below the benchmark level, [EY.sub.o], that are emitted in the laissez-faire equilibrium. To disaggregate the effects of the subsidy, it is assumed that there is no abatement in the allocation [A.sub.o.sup.o], whereas the final allocation, [A.sup.s], is the same as A.sub.o.sup.s] except that there is abatement.

The marginal conditions for a Pareto optimum and their reformulation in the context of a competitive economy in which the prices of the goods are px and py, are adapted from[10]. These are useful here because all but one of these marginal conditions are satisfied in the subsidy equilibrium, [A.sup.s]. First there is the condition for equal marginal rates of technical substitution in production, which are represented by ratios of corresponding marginal products. As a consequence of cost-minimization in a market economy, these in turn are equal to the wage rate, w, divided by the price of capital, r; that is,

[F.sub.l]/[F.sub.k] = [Y.sub.l]/[Y.sub.k] = w/r (6)

Condition (6) is satisfied in all four types of competitive allocation examined in this paper.

The marginal condition for the efficient scale of an individual firm in industry x, and by inference, the efficient number of firms, n, in that industry, is

[[F.sub.l]G][L.sub.x] + [[F.sub.k]G][K.sub.x] = FG = X. (7)

This condition, that firms operate at the point of locally constant returns to scale, which is satisfied when free entry in a competitive market economy drives total profit, [pi.sub.x], to zero,

[pi.sub.x] = [p.sub.x]X - w[L.sub.x] - r[K.sub.x] = 0, (8) holds in all four types of competitive equilibrium. The particular class of production functions used in the numerical examples has the convenient properties, given (7), that [K.sub.x] equals (40/[L.sub.x]) and, given (6), that [L.sub.x] equals the square root of (160[Y.sub.k]/[Y.sub.l]); this makes two of the variables dependent and simplifies the computer search for optimal and equilibrium solutions.

The marginal condition for the efficient scale of an individual firm in industry y, and by inference, the efficient number of firms, m, in that industry, is

[Y.sub.l][L.sub.y] + [Y.sub.k][[K.sub.y] + [K.sub.b] = Y[1 - [B.sub.y][Y.sub.k]/[B.sub.k]. (9)

This condition, which is satisfied in a market economy when free exit restores total profit, [pi.sub.y], of each polluting firm to zero, that is

[pi.sub.y] = [p.sub.y]Y - w[L.sub.y] - r[K.sub.y] + [K.sub.b] - [phi]E[1 - B]Y = 0, (10) holds only when firms pay a Pigouvian tax, and it is therefore satisfied in [A.sup.*] alone.

The condition that the marginal product, measured in units of good x, of capital allocated to abatement should equal the marginal product of capital in direct production in industry x, is

n[FG.sub.e][EB.sub.k]Y = [F.sub.k]G. (11)

This translates into the familiar equality of the Pigouvian tax, [phi] and the marginal cost of abatement, [Mathematical Expression Omitted] or -r[[EB.sub.k]Y, which is satisfied because of cost-minimization by pollution-abating firms:

[phi] = r/[[EB.sub.k]Y, (12) where the tax equals marginal pollution damage measured in dollars,

[phi] = -n[FG.sub.e][r/([F.sub.k]G)] = n[FG.sub.e][w/([F.sub.l]G)]. (13)

Condition (12) is satisfied in [A.sup.*] but also in [A.sup.s] because the Pigouvian subsidy, [phi], has the same definition as in (13). Finally, the condition that the marginal rate of substitution in consumption equals the marginal rate of transformation,

[U.sub.y]/[U.sub.x] = [[F.sub.l]G - n[FG.sub.e]E(1 - B - [B.sub.y]Y)[Y.sub.l]]/[Y.sub.l], (14) is satisfied by consumer optimization, in which

[U.sub.y]/[U.sub.x = [p.sub.y]/[p.sub.x], (15) because

[p.sub.x] = w/[F.sub.l]G] (16) and

[p.sub.y] = w/[Y.sub.l] + [phi]E[ 1 - B - [B.sub.y]Y]. (17)

These conditions hold in [A.sup.*] but also in [A.sup.s], in which the Pigouvian subsidy is [phi].

III. Numerical Example in Which Total Emissions Increase under the Subsidy

The competitive allocation, [A.sup.*], is derived with a computerized iterative search routine in which [L.sub.y] [K.sub.y], and [K.sub.b], which as explained earlier determine [L.sub.x], [K.sub.x] and with (4), m and n as well, are systematically varied until a maximum value of U is found. The wage rate is assumed to be $10.00, and from the optimal solution and equations (6), (12), (16), and (17), the dollar values of r, [phi], [p.sub.x] and [p.sub.y] are derived. The solution, which satisfies all of the above marginal conditions for efficiency, is contained in the [A.sup.*] column of Table 1. In this case, the marginal cost of good y is the sum of the direct marginal cost of production, w/[Y.sub.l], plus the Pigouvian tax per unit of output, [phi]E[1 - B - [B.sub.y]Y].

[TABULAR DATA OMITTED]

The competitive allocation, [A.SUB.o], is derived with a search routine in which [L.sub.y] alone is varied. In this particular allocation, [K.sub.y] equals (40/[L.sub.y]) and [L.sub.x], [K.sub.x], m and n are endogenous as before. The value of [L.sub.y] is found at which the sum,

[psi] = [[U.sub.y]/[U.sub.x] - [p.sub.y]/[p.sub.x].sup.2] = [pi.sub.y.sup.2],

(18) is driven to zero. The objective function, (18), incorporates the two competitive market equilibrium conditions that are not made endogenous in the computer program. In all of these models, [pi.sub.x] is zero by virtue of the endogenous equation (7). The total profit, [pi.sub.y], of each individual firm in industry y has a different formula in each of the four competitive equilibrium allocations. For purposes of comparison, the respective specifications of total profit are as follows:

[A.sup.*] : [pi.sub.y] = [p.sub.y]Y - w[L.sub.y] - r[[K.sub.y] + [K.sub.b]] - [phi]E[1 - B]Y

(19)

 [A.sub.o] : [pi.sub.y] = [p.sub.y]Y - w[L.sub.y] - r[K.sub.y] (20)
 [A.sub.o.sup.s] : [pi.sub.y] = [p.sub.y]Y - w[L.sub.y] - r[K.sub.y + [phi]E[[Y
.sub.o] - Y] (21)

The formula for the market price, [p.sub.y], which is identical in [A.sup.*] and [A.sup.s], but different in [A.sub.o] and different again in [A.sub.o.sup.s], is derived by differentiating [pi.sub.y] with respect to, say, [L.sub.y], then setting the derivative equal to zero and solving for [p.sub.y]. In the case of [A.sup.*], and again of [A.sup.s], the formula for [p.sub.y] is that previously given in (17). Note also that condition (6) can be derived for industry y by taking the derivative of [pi.sub.y] in (19) through (22) with respect to [K.sub.y] and then dividing each [Mathematical Expression Omitted]. The rounded numerical solution for [A.sub.o] is shown in Table I. This is the competitive equilibrium allocation in which the total profit as defined in (20) is zero, [p.sub.y] is w[Y.sub.l], and [phi] is zero. The value of Y in the fourth row of the [A.sub.o] column becomes the benchmark [Y.sub.o] in the remaining models in Table I.

The competitive allocation, [A.sub.o.sup.s], is derived by varying [L.sub.y] and [K.sub.y], with [L.sub.x], [K.sub.x], m and n determined endogenously, until (18) is again driven to zero. In this case, [p.sub.y equals [w/Y.sub.l] + [phi]E], where [phi] is defined in (13) above. This is the perverse case in which the subsidy for abatement causes total emissions to increase from 0.317958 to 0.319574. To qualify for the equilibrium amount of subsidies, each competitive polluting firm reduces its output from [Y.sub.o] = 1046.61 to Y = 932.249. As a result, the direct marginal cost of production, w/[Y.sub.l], falls to $0.225519. This is illustrated in Figure 1, where the w/Y.sub.l] curve shifts upward because the firrn is now employing less capital.(2) At the new equilibrium level of output, it can be confirmed from the numbers in Table I that marginal cost equals average cost,

w/[Y.sub.l] + [phi]E = $0.233642 = (w/[L.sub.y] + r[K.sub.y] - [phi]E[[Y.sub.o] - Y])/Y =

$0.233642, (23) which also equals the market price, [p.sub.y]. Although firms are operating in the range of increasing returns, they are earning zero profits. The market price of good y has declined absolutely and relative to both the price of good x and to the decline in nominal income, so that consumption of the polluting good increases from 127183 to 127830 units per period. This larger output is made possible by an increase in the number of firms to m = 127830/932.249 = 137.120. That the community is unambiguously worse off under the subsidy than under the laissez-faire equilibrium is confirmed by the decrease in total utility from 8323.16 to 8310.19.

The competitive allocation, [A.sup.s], is derived by varying [L.sub.y], [K.sub.y] and [K.sub.b], with [L.sub.x], [K.sub.x], m and n determined endogenously, until the sum,

[eta] = [[U.sub.y/[U.sub.x] - [p.sub.y]/[p.sub.x].sup.2] = [pi.sub.y.sup.2] +

[r - [phi][EB.sub.k]Y].sup.2], (24) is driven to zero. This objective function is identical to (18) except that marginal condition (12) for abatement efficiency, the same that holds for Pigouvian taxation, is incorporated. This condition, which firms satisfy when they maximize profits, can be obtained by setting the derivative of (22), with respect to [K.sub.b] , equal to zero. Although [pi.sub.y] is different in [A.sup.s] than in [A.sup.*], the formula for [p.sub.y] is the same in both. In the equilibrium allocation, [A.sup.s], the pollution level, e = 0.319355, is higher than the pollution level in the original competitive allocation prior to the subsidy, [A.sub.o], and thus illustrates the perverse case identified in the title of this paper. The emission level is only slightly lower than in [A.sub.o.sup.s] because the fraction of pollution abated, B(.), is only 0.0007, but total utility is nevertheless higher. A simple explanation for the perverse increase in total emissions in the shift from [A.sub.o] to [A.sup.s] is that the proportional increase in polluting output, 643/127183 = 0.0051, exceeds the proportional decrease, B, in the emission rate.

IV. Numerical Examples in Which Total Emissions Decrease under the Subsidy

In the absence of an abatement technology, the case in which the subsidy causes total emissions to decrease could not be simulated with the present numerical model in which the emission rate is a constant. This supports the view that, in the absence of abatement, the case of increasing emissions is indeed robust. Although the case of decreasing total emissions was simulated by letting the emission rate decline with output,

E = Y/400,000,000, (25) this case is not particularly interesting and the results are not reported here. Such a case in which the emission rate varies with output is familiar to economists from the work of Carlton and Loury[3; 4] and although Mestehuan[20] makes use of this kind of nonlinearity with his parameter [delta], the commonly made assumption, as in [9, 38-65], is that emission factors are constant with respect to output. The more realistic explanation for the case in which the subsidy causes total emissions to decrease rather than to increase, one that Mestelman[20] cannot simulate with his model, is that the subsidy induces substantial technological abatement. For most polluting industries, this is the more relevant case [9, 38-65].

The case of a strong abatement technology, which is modeled by letting [alpha] equal 0.01, is illustrated in Table II. The [A.sub.o] and [A.sub.o.sup.s] solutions are the same as before; only [A.sup.*] and [A.sup.s] are different. The fraction of pollution abated, B, is 0.3331 in the optimal allocation, [A.sup.*], in Table II, as compared to 0.0007 in the previous table.(3) Here, the simple explanation for the decrease in total emissions in the shift from [A.sub.o] to [A.sup.s] is that the proportional increase in polluting output is less than the proportional decrease in the emission rate.

[TABULAR DATA OMITTED]

Although the result in Table II, that the subsidy induces abatement that significantly reduces total emissions, is almost trivial, it does provide a useful numerical affirmation of Mestelman's[20] insight that the subsidy might be a useful second-best instrument if Pigouvian taxation is not politically viable. It is clear from the final row of Table II that although the allocation [A.sup.s] is inferior to [A.sup.*], it is superior to the laissez-faire alternative, [A.sub.o]. This is a robust result for the 0.3288 value of B in the allocation, [A.sup.s], in Table II is well below the average for the six pollutants examined in [9, 681.

V. Conclusion

When polluting firms are paid a subsidy to abate, total emissions in the economy may either increase or decrease. This paper is a response to the call of Baumol and Oates [2, 228] for ". . . consistent examples that go both ways."(4) In his presidential address to the Eastern Economic Association, Oates [23, 290] again warned that ". . . in a competitive setting, subsidies will lead to an excessively large number of firms and industry output . . . it is even conceivable that aggregate industry emissions could go up!" Based on the general equilibrium model presented here, in which the emission rate is a constant and there is no technological abatement, the perverse case that continues to concern Oates[23] is the only case that could be successfully simulated. However, this perverse case is likely to be realistic only for industries in which there is very little opportunity for technological abatement.

The sequence depicted in Tables I and II, in which the subsidy for abatement fosters a long-run competitive equilibrium, holds only when the output of good y is large relative to that of good x. This occurs when the exponent of y in (5) is higher than 0.71 +. At values of 0. 71 and below, the initial allocation, [A.sub.o], includes larger quantities of good x (more than 40,000), smaller quantities of good y (less than 115,000), but greater pollution damage and higher unit subsidies, [phi]. In this lower portion of the production possibility frontier, there are always optimal solutions, [A.sup.*], obtainable by taxation, but there are no long-run equilibrium allocations, [A.sup.s]. Mestleman [19, 126] attributes a similar finding in his partial equilibrium model to relatively high rates of subsidy causing the new price to exceed the minimum average cost of firms that choose to decline the subsidy.

The failure to achieve long-run general equilibrium when the subsidy is relatively high is an interesting subject for further research. Some preliminary analysis suggests that, as the subsidy increases, the long-run equilibrium scale of polluting firms decreases toward the scale at which long-run marginal cost is a minimum.(5) Test runs, in which the exponent of y in (5) is varied across the critical level from 0.71 to 0.72, indicate that a long-run equilibrium is achieved only when the subsidy per unit of output, which is [phi]E[Y.sub.o] - Y]/Y, exceeds the decrease in the average cost of labor and capital, which is [w[L.sub.y] + r[K.sub.y]]/Y, in the shift from [A.sub.o] to A.sub.o.sup.s]. This condition, for which the intuition is not yet clear but which is satisfied by the examples in Tables I and II, differs from Mestelman's [19, 1261 partial equilibrium condition for the maximum subsidy. It is reassuring to note that whenever, in the present general equilibrium model, the subsidy does yield a new long-run competitive equilibrium such as [A.sub.o.sup.s] or [A.sup.s], the price of good y is always lower than the initial laissez-faire price in [A.sub.o]. Therefore, polluting firms never have the Mestelman [19, 126] option of rejecting the subsidy to enjoy a competitive advantage.(6)

Of the two tables, Table II is the more robust and therefore the more policy relevant. The observation in the second table that total utility in [A.sup.s], though less than that in [A.sup.*], exceeds total utility in [A.sub.o] dramatically reinforces Mestelman's[20] argument for subsidizing abatement on second-best grounds. However, the payment of subsidies would be costly for the government and it is unrealistic to assume (to avoid compounding the inefficiency) that the funds could be raised by lump-sum taxes. There is also [23, 296] an "administratively contentious matter" that might even be a source of nonconvexity[24], which is that of determining the benchmark level of emissions on which the subsidy is based. Finally, there is the already-discussed problem[19] that a subsidy for abatement, equal to marginal pollution damage, may exceed the level consistent with long-run competitive equilibrium.

An alternative approach that combines the political feasibility of the subsidy, the economic efficiency of the Pigouvian tax, and requires no cash payments by the government, is the assignment to existing polluters of an efficient quantity of transferable discharge permits. This policy approach, which is not only well-studied [16; 22; 29], but is already in the early stages of implementation[25], is a quasi-subsidy because the freely given permits can be sold by their recipients.(7) It appears that economists' continuing interest in subsidizing pollution abatement and their interest in transferable discharge permits may usefully coalesce.

(1.) Inspired by a lecture given by Baumol, Amihud [1, 116] constructs a case of uncertainty in which ". . . a subsidy may even cause the (risk-averse) firm to increase, rather than decrease, its emissions." (2.) Figure 1, which is a partial equilibrium characterization, is strictly valid only at the two levels of Y shown. Basically, this is a general equilibrium analysis in which [L.sub.y], [K.sub.y] and r change with Y. (3.) One would expect the ratio, y/x, in the [A.sup.*] allocations to be higher in Table II because marginal damage is lower, but in this general equilibrium model the reduced damage causes the price of good x to decline by a larger percent than does the price of good y. (4.) A useful feature of[2], which has become something of an unofficial handbook of environmental economics, are these calls for further research. Elsewhere in the book, Baumol and Oates [2, 42] call attention to their omission of a particular term that ". . . reflects the indirect effect On its own output of a firm's emissions ... operating through ... the aggregate level of pollution". In response, it is shown[11] that if the firm does take this "indirect effect" into account, and the Pigouvian tax is correspondingly reduced, as Tietenberg[28, 120] argues that it should, the effects cancel out and the market price is unaffected. In the present model, such a complication, in which industry y would itself be adversely affected by the pollution level, is assumed away. (5.) The fact that solutions exist only when y is large relative to x suggests the possibility that this may be occurring only in the convex-to-the-origin range of the production possibility frontier, as depicted in[15]. Although nonconvexity is common in externality problems[12; 30], a simple test confirms that the frontier at [A.sup.*] is properly concave at optimal (x, y) in both tables. (6.) The present model cannot be used to simulate the Mestelman[19] case in which an increasing number of firms in the polluting industry pushes up the cost of a scarce managerial input, raising the price of the polluting good, and lowering total emissions. However, the polluting industry in the present model is capital-intensive and, based on[27], the expansion of that industry from [A.sub.o] to [A.sub.o.sup.s] might be expected to drive up the price of capital and thereby duplicate the Mestelman effect. In fact the opposite occurs, for r declines from 25.5212 to 24.5582, but because this is not a shift from one production-efficient allocation to another, Stolper-Samuelson theory does not apply. In reducing their output, firms in the polluting industry become less capital-intensive. For firms in the labor-intensive industry to employ the released capital, its price must decline. In the context of this simple numerical example in which [L.sub.x] equals 4([r.sup.1/2) and [K.sub.x] equals 10([r.sup.-1/2), it is easy to confirm from Table I or II that [K.sub.x]/[L.sub.x] does rise in the shift from [A.sub.o] to [A.sub.o.sup.s]. Because r declines under the subsidy, the U-shaped average cost curve of polluting firms shifts downward and to the right, rather than to the left as depicted in Figure 1. (7.) The disadvantage of marketable discharge permits is that, unlike Pigouvian taxes, they do not generate government revenue[13, 30]. For an excellent discussion of the relative merits of these two policy instruments, see Oates[23].

References

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