Journal of Environmental Management (1991) 32, 221-225

Reflexive Pol lut ion

Robert E. Kohn

Department of Economics, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026, U.S.A.

Received 30 November 1989

Pollution is reflexive when firms are adversely affected by ambient air concentrations of pollutants that they themselves emit. There are two responses to reflexive pollution. In the first, firms are pollution takers and ignore the self-inflicted damage associated with their marginal product. In the second case, firms do account for marginal self-inflicted damage. Firms' perceptions of their marginal costs differ in the two cases as do the corresponding Pigouvian taxes. Nevertheless, the two sets of competitive market prices are both identical.

Keywords: pollution, taxes, market prices, marginal costs.

1. Introduction

Reflexive pollution occurs when an individual firm is adversely affected by a pollutant which the firm itself emits. It is a common occurrence because a firm often suffers damage from the ambient air concentration of a pollutant, such as sulfur dioxide, to which it, as well as other firms, are the contributors. The case of reflexive pollution is remarked upon by Meade (1952) and by Tietenberg (1973) but not formally modelled, either by them or by subsequent investigators. Most recently, Baumol and Oates (1988, p, 42n) look at the case of reflexive pollution but conclude that " . . . the indirect effect on its own output of a firm's emissions . . . operating through . . . the aggregate level of pollution will typically be miniscule, and since it complicates the analysis considerably, we have chosen to ignore it." The topic is however interesting for there are actually two types of responses to reflexive pollution. In the first, firms are assumed to be pollution- takers, much as they are in Burrows (1979), ignoring the self-inflicted damage from their own marginal output. In the second, originally recognized by Tietenberg (1973), the emitter-receptor firms internalize the self-inflicted marginal damage and reduce output accordingly. The first of these two responses falls into the environmental pollution category of Haveman (1973) because the reflexive pollution " . . . fails to alter the private equilibrium of the firm", whereas the second, in which the firm does alter its level of output, falls into Haveman 's congestion category.

Tietenberg (1973) determines the efficient competitive market prices and Pigouvian taxes for the case in which reflexivity is absent, but notes that there is an alternative set of

221 03014797/91/030211+05 $03.00/0 �9 1991 Academic Press Limited

222 Reflexive pollution

taxes that is efficient when pollution is reflexive. This paper derives the competitive market prices and Pigouvian taxes that hold under each of the two types of response to reflexive pollution and mathematically confirms Tietenberg's (1973) insight that both sets of taxes foster an efficient allocation of resources.

2. Marginal products and the marginal rate of transformation

The case of reflexive pollution damage is examined in the larger context in which there is pollution damage across firms and industries as well as within firms and industries. Consider an economy in which there are two goods, x and y, each produced by a large number, n and m respectively, of firms. The numbers of firms are variable and for mathematical convenience, divisible. Production of each good is a function of labour, L x and Ly, of capital, K x and Ky, and of the level of pollution, e, in the economy. The pollution level, e, may be interpreted as the sum of emissions or as a concentration, such as micrograms per cubic meter of ambient air. As in Meade (1952), pollution damage is represented by a multiplicative factor; because outputs are implicit functions of themselves, this greatly simplifies the mathematics. Firms in each industry use the common multiplicative production function for that industry. These are

X = [F,(Lx,Kx) ] [Gx(e)] , Y= [Fy(Ly,Ky)] [Gy(e)] (1)

It is assumed that the functions, F,( .) and Fy(.), exhibit increasing and then decreasing returns to scale. The derivatives, G e and Gy, are negative. (Superscripts in this paper denote partial differentiation with respect to the superscript.)

Botl~ industries pollute and the level of pollution in the airshed is

e = [Ex]X + [Ey]y = nExX+ mEy Y (2)

where E, and Ey represent constant rates of emission per unit of good, x and y, respectively. The assumption of constant emission factors and the additivity of emissions are simplifications that are commonly used in empirical as well as theoretical models (see Kohn, 1978, pp. 38-65). It is further assumed that the ratios, (Ex)/e and (Ey)/e, are very small. Taking the derivatives of (1) and (2) with respect to L x and Ly yields four marginal products of labour for the two industries,

XLx= F~xGx[1- mFyG~Ey]A, YLx= nFL, GxExFyGeyA (3)

where A, a pure positive fraction, equals 1/(1 -nFxGeEx-mFyGyEy), and

yLy = FLyGy[1 _ nFxG~xEx]A, XLy = mFLyyGyEyFxG~A (4)

The partial derivatives with respect to K X and Ky are analogous. The interpretation of the derivatives with respect to L x are as follows. The marginal product, XLx, is the direct increase in the output of each firm in industry x, when each firm employs the marginal unit of labour, minus the loss in output of each firm that occurs because of the increase in pollution due to the increase in output of each of these n firms, and plus the increase in output because the m firms in industry y emit less when pollution causes each of their output levels to decline by YLx.

Given that the total quantities of labour and capital in this economy are fixed, the

R. E. Kohn 223

marginal rate of transformation is a simple combination of the marginal products derived above and the technically efficient number of firms in the two industries,

ay/ax = ( E L y - - (m/n) YCx)/(XCx - (n/m)XLy ) (5)

In the next two sections of the paper, it is shown how efficient market prices are based on marginal products that differ from those defined above but that their ratio is nevertheless equal to (5).

3. Efficient taxation when firms are pollution takers

Assume a market economy in which firms are price-takers and, following Burrows (1979), pollution-takers as well. In the present context, the latter implies that the individual firm takes the pollution level as given and ignores the small increment, (Ex)/e or (Ey)/e, that a unit increase in its own output would contribute to the pollutant concentration in the airshed. The marginal pollution damage, D, is the summation of the value of the foregone marginal output in the two industries. The value of output is based on direct marginal costs alone, w/(FLxxGx) a n d w/(F~yGy) where w is the wage rate; in the case of reflexive damage, it would introduce double-counting to value the foregone output at the market prices as it is in Kohn (1988, p. 628). This definition of marginal pollution damage, which holds regardless of whether or not there is a tax on pollution, is

D = - nFxGexw/(FLxxGx) - mFyGeyw/(~yGy) (6)

The efficient Pigouvian tax rate, Tper unit of emissions, which is the same for all firms in both industries, is equal to D.

Given that firms are price and pollution takers, the market prices, when emissions are taxed, are

Px = w/FL~G~) + (Ex)T, Py = w/(FLYGy) -~- ( E y ) T (7)

Observe that the marginal product of the pollution taking firm can be derived from (3) or (4) by letting m = n = 0. Dividing Px by Py and substituting for T yields

P x / e y = [FLyay(1 -- n F x G e E x ) - - mFLxxGxExFyGy]/[F~xGx( l - mFyGyEy) - - F l ~ y y y a y E y F x a e x ] . (8)

Multiplying through by A and substituting (3) and (4) yields (5); it follows that the ratio of prices in the case of pollution-taking equals the marginal rate of transformation and is therefore efficient.

The optimal solution can be illustrated by the following numerical example. There are n firms in industry x, each using an identical production function to produce the quantity, X. The total output of this industry is x = n X = n(48L~/3K 4/3- 5/3 /3 Lx K~x ) ( I - e ) , where the pollution level is e = n(1/2,0OO,OOO)X+ m(1/1,OOO,OOO)Y. Similarly, the total production of a second industry is y-m(48Ly- 4/3Ky2/3 Ly7/3KyS/3) ( l - e ) . To derive a definitive numerical solution, assume a community utility function, U = Ln(x)+ Ln(y) and fixed quantities of inputs equal to 800 units of labour, the price of which is $1-00 per unit, and 800 units of capital. The efficient Pigouvian tax is $1792.68. The efficient quantities of inputs are L x = 1.826144, K x = 8.761630, Ly = 9.130721, Ky-- 1-752326 and the direct marginal products of labour are F~xG x = 71.82408 and F~yG v = 73.82728. Other

224 Reflexive pollution

elements of the optimal solution are n = 76.85828, m=72.24465, FxG~x = - 8 6 3 . 5 4 8 2 , FyGey = - 8 8 7 " 6 3 2 8 , Px = $0"0148192 and Py = $0-0153378. The optimal level of pollution in the airshed is e = 0.08868227.

4. Efficient taxation when reflexive damage is internalized

Tietenberg (1973, p. 250) reasons that:

" . . . for firms that are adversely affected by their own pollution, the efficient tax rate is equal to total marginal social damage minus the marginal social damage already internalized."

His conclusion can be verified mathematically. The perfectly competitive firm in industry x, fully aware of reflexive damage, calculates its marginal product as

XLx = FLxGx + FxGexEx XLx = FLxxGx/(1 - FxGeEO (9)

This is equivalent to (3) for the case in which m = 0 and n = 1. To maximize its total profit, the firm adjusts its output until marginal cost equals the market price. It follows that the price, ~x, is

7~• w[1 - FG~E]/[FLx•215 + [E• x (lO)

where zx, the Tietenberg tax for firms in that industry is derived by substituting (n - 1) for n in (6) to net out the damage consciously internalized by the taxed firm. This tax is

~• = T + WFxG~x/(F~xG~), T,y : T-I- wFya~/(F~yGy) (11)

Because G~ is negative, ~ is less than T. A comparison of (7) and (10) indicates that in the absence of a Pigouvian tax, perceived marginal cost is higher and output lower when the firm internalizes its self-inflicted damage than when it does not. It is this characteristic that moves the second response to reflexive pollution out of Haveman's (1973) env i ronmenta l pol lut ion category and into the congest ion category.

Assuming that all firms respond to reflexive pollution as Tietenberg predicts and the Pigouvian tax is appropriately adjusted, it is easy to prove by substituting (11) into (10) that

7"Cx = 71;y , Px=Py (12)

5. Concluding remarks

This paper develops the mathematics for the case of reflexive pollution, considered by Meade (1952), Tietenberg 0973), and Baumol and Oates (1988), in which a firm's emissions ultimately inflict damage upon itself. If firms are pollution takers as well as price takers and ignore the reflexive pollution associated with their own output, the market prices of the goods are the same as if each firm takes self-inflicted damage into account in calculating its marginal cost.

For an industry with many firms, the amount of pollution inflicted by an individual firm on itself is, as Baumol and Oates surmise, probably very small so that the differences in actual prices and Pigouvian taxes in the two cases are also likely to be very small. The mathematical analysis is of theoretical interest, however, because it depends

R. E. Kohn 225

u p o n two different in terpreta t ions of marg ina l input products , because the P igouvian tax is based upon a counter- in tu i t ive va lua t ion of foregone output , and because Tietenberg 's (1973) insight may be also viewed as counter - in tu i t ive and therefore requir ing proof.

References

Baumo~, W.J. and Oates, W. E. (1988). The Theory o f Environmental Policy: Second Edition. Cambridge: Cambridge University Press.

Burrows, P. (1979). Pigouvian taxes, polluter subsidies, regulation, and the size of a polluting industry. Canadian Journal of Economics 12, 494-501.

Haveman, R. H. (1973). Common property, congestion, and environmental pollution. The Quarterly Journal of Economics 87, 278 287.

Kohn, R. E. (1978). A Linear Programming Model for Air Pollution Control. Cambridge: MIT Press. - - ( 1 9 8 8 ) . The limitations of Pigouvian taxes as a long-run remedy for externalities: Comment. Quarterly

Journal of Ecomomics August, 625 630. Meade, J. E. (1952). External economies and diseconomies in a competitive situation. The Economic Journal

62, 54-67. Tietenberg, T.H. (1973). Specific taxes and the control of pollution: a general equilibrium analysis. The

Quarterly Journal of Economics 87, 503-522.