Socio-Econ. Plann. Sci. Vol. 27, No. 1, pp. 1-8, 1993 0038-0121/93 56.00 + 0.00 Printed in Great Britain. All rights reserved Copyright 0 1993 Pcrgamon FYcss Ltd

Premature Mortality in a General Equilibrium Model of Air Pollution Control

ROBERT E. KOHN Professor Emeritus of Economics, Department of Economics, Southern Illinois University at Edwardsville,

Pdwardsville, IL 62026, U.S.A.

Abstract-A general equilibrium model is developed in which production of a good generates toxic pollution that imposes the risk of premature mortality. This risk is reduced to an optimal level by a Pigouvian tax that causes consumers to demand less of the polluting good. Whereas the value of human life on which courtroom decisions are commonly made is one of lost output, in this model the value of human life includes both a lost output component and a psychological willingness to pay component. Because the dose-response curve for pollution is S shaped and because the disparity between the ex ante statistical value of human life and the ex post certainty value is likely to be large, the model has special implications for Pigouvian taxation.

INTRODUCTION

It is well established [19] that air pollution carries the risk of premature mortality. In a study of 2 million deaths that occurred in the United States in 1970, Mendelsohn and Orcutt [22, p. 851 conclude that “approximately 140,000 deaths a year-9% of all deaths-may be associated with air pollution”. Individuals implicitly associate a dollar value with that risk [25], and there are some good estimates [6, lo] of the monetary benefits of saving lives by reducing air pollutiont. The present paper presents a general equilibrium model, that is, a model incorporating production, consumption, and the determination of all relative prices, in which people derive utility from consuming goods, but disutility from the risk of premature mortality caused by air pollution. In addition to the disutility, there is a physical loss in output because afflicted persons cannot work. The model incorporates the economist’s unique and powerful method of valuing human life. Although there are many general equilibrium models in which the level of pollution adversely affects people’s utility [8, p. 6781, this appears to be the first model in which the risk of death is made explicit. In the context of this general equilibrium model, an optimal level of premature mortality is defined, which is the economist’s analogue of “acceptable risk”.

MARGINAL CONDITIONS FOR AN OPTIMAL LEVEL OF RISK

Consider a simple economy in which there are Ho identical persons, each of whom consumes x units of good x and y units of good y per period. There are n identical firms, each of which produces X units of good x using L, units of labor and K, units of capital, and m identical firms, each of which produces Y units of good y per period using L, and K, units of labor and capital, respectively. The corresponding production functions, X(Lx, K,) and Y(L,, KY), exhibit increasing and then decreasing marginal returns to scale, where the total outputs produced and consumed are:

nX = nX(L,, K,) = H,x and mY = mY(L,, K,) = Z-&y. (1)

tThe fear that air pollution can cause earl_* death is not new. In 1880, a letter to the editor of the Chicago Tribune addressed the “smoke nuisance (that) exists in the busiest portions of the city. ” Its author warned that the “evils resulting from the inhalation of smoke are appalling. . . that, in this city, IO out of every 100 cases of consumption are the result of inhaling smoke.” He called it “terrible to think of’ that 10% of deaths per year are “caused by inhaling smoke into the lungs!” This letter is familiar to me because the author was my grandfather’s brother [9]. My great-uncle was only 17 years old when he wrote it and he had no medical expertise other than that he himself was afflicted by consumption (tuberculosis) from which he died the following year; yet his claim of a 10% increase in mortality because of air pollution, which must have seemed far-fetched a hundred years ago, is astonishingly close to current estimates [22].

2 ROBERT E. KOHN

Producers of good y generate air pollution as a byproduct. In contrast to models in which pollution from one industry undermines productivity in another industry [16, 181, in the present model it is the community’s health that is directly impacted. The air pollution is toxic, and for each unit of good y produced there is a small probability, P, that each person will die prematurely in the period. This probability, which increases with the total output of good y, is represented by the function, P(mY), whose first derivative, P,,,,,, is positive and whose second derivative, P,,,Y,n,b, is positive for low values of m Y and negative for large values of mY. This sequence of signs is consistent with the assumption [22, p. 941 that “biological dose-response curves often assume an S shape”.

The probability, z, that each individual person will die in the period, which corresponds to 2; in [21], is the probability per unit of output times the total output of good _Y:

z = [P(m Y)]m Y. (2)

Given the exposure of H, persons, it follows that the total expected mortality per period in this economy is zH,,. To keep the model simple, the possibility of pollution abatement by firms and self-protection or risk reduction by potential victims is excluded [I 1, 16-18, 261. The identical utility function of each person is:

C’ = U(x, )?, z) (3)

where U, and U, are positive and U,, negative. That the sign of U, is the opposite of the sign of U, and Uy implies that each individual is willing to forego some consumption of goods in exchange for a small reduction in his or her own risk of dying within the period. Alternatively, this same individual, though placing a high, even infinitely high, value on his or her own life, is willing to accept a slightly greater risk of dying in exchange for more goods?. Following Bergstrom [4, pp. 16-181 rather than Jones-Lee [13, p. 5581, it is assumed that each person’s utility is affected by his own probability of a premature death and not by that of others. In the current model, it is assumed that the relationship between each individual’s risk of premature mortality, z, and the level of polluting production, mY, is deterministic and that z is known with certainty.

In the absence of any deaths, the total quantity of labor in this economy is simply the number of persons, H,,. It is assumed that the zH,, persons that die from the toxic pollution sicken at the beginning of the period and, although they continue to consume until the end of the period, when they die, they are unable to work. Hence, the total quantity of productive labor is:

H,[l - z] = nL, + mL,. (4)

Regardless of the number of premature deaths caused by toxic pollution, it is assumed that the steady state population reverts to H, at the beginning of each new period. This strong assumption and the highly restrictive eqn (4) which are in the spirit of the underlying one-period models of premature mortality [3, 4, 201, confine all of the benefits and costs of reducing pollution to the same period in which the pollution is emitted. To incorporate the loss of output in future periods because of current deaths [l], or the deaths in future periods because of current pollution [12], would unnecessarily complicate this model. Finally, it is assumed that the total quantity of capital is fixed at K,,. The fixed constraint on capital:

&=nK,+mK, (5)

is typical of models of efficient resource allocation. The marginal conditions for economic efficiency are obtained from the Lagrangian expression:

2 = &[U(X,Y, ~11 + A[fGx - nX(L,, Kx)l + Y[&Y - mY(L,, KY)1

+ p[z - fWY(L, K,))mY(L,, KY)1

+cc{H[l -z]-nL,-mL,} +j3[&--nK,--mK,]. (6)

tThe insight that people are willing to give up (accept more) consumption in exchange for a very small decrease (increase) in the risk of death is elucidated by Bergstrom [3, 41. In practice, Knetch and Sinden [15] find that individuals require a greater amount of compensation for a given increase in risk than they are willing to pay for the same decrease in risk. Smith and Desvousges [27] also find such an asymmetry, but in the reverse direction.

Premature mortality and air pollution 3

Equating to zero the derivatives of (6) with respect to x, y, z, L,, K,, L,, K,, n and m yields the following values for the multipliers:

and

L = -U,= - U,[X,L,+X,K,]/X, y = -U,, a = U,X,, /I = U,X, (7)

P = &KJ,XL - Uzl = WY YL - UxXLIIIP + mYP,,)Y,l

= WY YK - UJJKP + m YP,,) YKl

= [UyY- U,X,L,- U,X,K,]/[(P+mYP,,)Y]. (8)

Setting the second and third expressions in (8) equal to each other gives the condition that the marginal rate of technical substitution of capital for labor must be the same in both industries; that is:

XL/X, = YJYP (9)

The remaining condition for efficiency in production is on the scale of individual firms and, by implication, on the number of firms in each industry:

X= X,L,+X,K, and Y= Y,L,+ Y,K,. (10)

Derived from (7) and (8), respectively, (10) implies that all firms produce at the point of locally constant returns to scale.

The above conditions for efficient production are conventional and hold in the absence of pollution. However, the marginal condition for efficient consumption, which follows from the first two equalities in (8) is unique:

U,/U,=X,/Y,+[-U,/U,+X,]H,[P+mYP,,I. (11)

Expression (11) states that U,/U,, the marginal rate of substitution in consumption of good x for good y, should equal the marginal rate of transformation in the absence of pollution, which is X,/Y,, plus the loss of [- U,/U,+ X,] units of good x to all H, persons as a result of the pollution risk, [P + m YP,,,,,], associated with the marginal unit of good y. As in Conley’s model [7j, there are the following two components of this loss:

The first component of the loss is the psychological impact, which is preceded by a minus sign since Uz is negative. The marginal rate of substitution of good x for z, which is - U,/U,, represents the quantity of good x that each person would be willing to forego to reduce the probability of his or her own premature death from pollution during the period. Such a tradeoff, extrapolated to ab units of good x for one whole unit of z, at the person’s optimal combination, (z *, x *), holding consumption of y constant at optimal y*, is characterized in Fig. 1. These ab units of good x correspond to economists’ [3-5, 131 definition of the statistical value of life. It is an ex ante value that contrasts with the much larger quantity, x’ -x *, that a person would have to receive posthumously to accept certain death from pollutiont.

To the psychological component of the loss is added the expected decline in output, XL, associated with the marginal victim’s inability to work during the period because of premature illness and impending death. (Although XL is in units of good x per person and - U,/U, is in units of good x per unit of probability, the two are additive because the marginal unit of probability, AZ, is equivalent to one whole life.) The willingness to pay for survival throughout the period is summed over all H,, persons, as is the lost output. Finally, the total loss for AZ = 1, which is [ - U, U, + X,] [H,,], is multiplied by the fractional probability of premature death associated with the marginal unit of good y. This is i?z/iT(mY), which is the final square bracketed term in (11). The units of H,[P + mYI',,,,] are in persons per unit of good y, so that the right-hand side of (1 I), like the left-hand side, cancels out to units of good x per unit of good y. The efficient solution includes an optimal or acceptable level of risk, z *. In the next section of this paper, a dollar value is associated with that risk.

TEkrgstrom [3, p. 51 characterizes the case in which an individual places an infinite posthumous value on his or her own life by drawing the indifference curve in Fig. 1 asymtotic to the vertical line at z = 1.0.

ROBERT E. KOHN

Fig.

I I I

0 Z' 1.0 z

I. Tradeoff of consumption for risk of premature mortality

COMPETITIVE MARKET ECONOMY WITH A PIGOUVIAN TAX

In general, a competitive but polluting economy can be made efficient by Pigouvian taxation [3 I]. In the context of the economy described in the preceding section, let w dollars be the given wage rate. It follows that r, the rental price of capital is wX,/X, dollars and that px, the price of good x, is [w/X,] dollars. By taxing producers of good y, the government fosters the optima1 level of risk for the population. The correct Pigouvian tax per unit of good y is:

4 =Px[- UZ/U, + ~Llm~ + mY~m,I (12)

which sums the subjective cost, predicated on marginal willingness to pay to reduce the risk, plus the tangible component of lost output. The marginal rate of substitution, - UzjU,, times px now measures the dollar value of a statistical human life. This is a number that economists are able to estimate from a variety of market choices that individuals commonly make [14, pp. 36-101; 291. The other elements of the Pigouvian tax, the population H,, at risk, the probability, [P + m UP,,], of premature death associated with the marginal unit of polluting output, and the value of output per capita, pxX,, also correspond to numbers that economists estimate [6, lo]. In our deterministic model, the Pigouvian tax on the polluting good is equal to the known marginal damage. If the risk of premature mortality were treated as a stochastic variable, whose expected value is z, then the right-hand side of (12) would denote expected marginal damage. In that case, the Pigouvian tax is correctly equated to expected marginal damage only if consumers are risk-neutral. If consumers are risk-averse, it is shown in [ 171 that the Pigouvian tax should exceed expected marginal damage. To keep the present mode1 simple, it has been assumed that z is known with certainty.

The price of good y in this competitive market economy is

py = M’/YL + 4. (13)

It can be demonstrated that these market prices and the above Pigouvian tax foster the marginal conditions for efficiency derived in the preceding section of this paper. Substituting w/X, for pA and (12) for 4, it is readily seen, given that consumers maximize utility causing U,/U, to equal pu/px, that efficiency condition (11) is satisfied. Likewise, it is easy to show that cost minimization in the employment of inputs causes (9) to be satisfied while free entry and exit of firms and the resulting zero profit condition cause (10) to be satisfied. However, because the second derivative, P mr,mr7 is positive for low values of mY and negative for large values, which is characteristic of

Premature mortality and air pollution

Fig. 2. The S shaped dose-response curve.

the S shaped dose-response curve, there is a troublesome potential for multiple equilibria in this hypothetical economy.

It is well known that when marginal damage declines as the quantity of pollution increases, which would be the case along the upper portion of the S shaped dose-response curve characterized in Fig. 2, an iterative sequence of successively larger Pigouvian taxes can bring the economy to a local optimum that is inferior to the global optimum [2, 28, 311. This could occur if the initial, laissez-faire competitive equilibrium were at a point such as A in Fig. 2, and the succession of Pigouvian taxes led to an equilibrium at a local, but not global, optimum at point B. Fortunately, Mendelsohn and Orcutt [22, p. 941 find that current pollution levels in the United States are “too low to trace out an S shape”. (They find, in effect, that a 50% probability of dying from air pollution would be associated with concentrations a thousand times larger than those observed!) It therefore appears that current risk levels are characterized by a point such as C in Fig. 2; so that an iterative sequence of Pigouvian taxes would bring the economy safely down the dose-response curve, from C to D, where the efficient slope, az/a(mY), is achieved. It follows that the nonconvexity of the S shaped dose-response curve is not a realistic problem for implementing the present model.

A NUMERICAL EXAMPLE AND SOME POLICY IMPLICATIONS

In contrast to the competitive optimum, which is fostered by the Pigouvian tax, perfect competition in the absence of a tax results in a laissez-fair equilibrium in which there is too much of the polluting good and too great a risk of premature mortality. The shift from the laissez-faire equilibrium to the optimal equilibrium is usefully simulated with the following numerical example in which the production functions are:

H,x = nX = n[l 10L;6K;6 - Lk6Kk6/6] (14)

and H,,y = mY = m[108L$‘K0;’ - L$‘K$‘/2]. (19

Assuming, realistically, that current pollution risks are on the lower portion of the S shaped dose-response curve, let the probability of each person’s premature death from toxic pollution be simply:

z = m Y [P] = m Y [(m Y/1OO,OOO,OOO)2]. (16)

ROBERT E. KOHN

Table I. Competitive market equilibrium without and with a Pigouvian tax”

Parameterh Without the tax With the tax

.r = &Y/H, 112.4018 132.8605

, =mY,H,, 45.85005 21.61125

P= 2.102227 x IO ’ 4.670459 x IO ’

0.009638721 0.001009344

H,,[I ~ ;]= 990.3613 998.9907

.Y, - 75.66383 75 33632 ,., _ 92.59256 92 19179

PI = $132.1636 $132.7381

-P,lL’,:L,l- %2,X72,976 %l.l57,223

pl[lJ’,,‘Lf,]=p,- $108 0000 $272.0135

[P + mYP,,]- I401 I38 x IO ’

d1 = $163.5440

i- 2.704 X.228

‘The equations and the correspondmg values of the key de&Ion variables.

/., , l,, and m. are given in the text. The corresponding quantities of capital

are K, = 60/L, and K,. = 36/L,. The wage rate is fixed at $10.000

“All parameters are defined in the text

The common utility function is:

1’ = .\.‘)7’!J)?(/( 10 + [500:]?) (17)

and the constants are H, = K, = 1000. Using (4) (5) and (lo), the computer search for a maximum Ho U reduces to three independent variables whose values in the long-run, laissez-faire equilibrium are approximately L, = 7.708546, L, = 5.971014, and m = 41.46537. In the laissez-faire solution, U,/U, equals X,/Y,_ alone, which violates the crucial marginal condition (11). The key variables in the efficient long-run competitive solution are L.y = 7.742057. L., = 5.996971, and m = 19.54455. (Although it is unrealistic to assume that a fraction of a firm can operate, this abstraction is necessary for the marginal conditions to hold exactly. If the numbers of firms are constrained to integers, the equalities at the margins are only approximated.)

In this numerical example, the wage, IV, is fixed at $10,000. The data in Table 1 illustrate that the imposition of the Pigouvian tax, 4, raises the relative market price of good y, so that consumers purchase more of good x and less of good 1’. Consequently, there is less pollution, and the probability of each person’s premature mortality per unit of good y, that is, P, and the probability of each person’s premature mortality per period, that is, z, decrease. Premature deaths accordingly decline from approximately 10 per period, without the tax, to 1 per period, with the tax. The value of a statistical life, correspondingly decreases from $2,872,976 to $1,157,223, reflecting, in part. the

lessened risk of death [26, p. 501 and, in part, the greater availability of goods?. The numbers derived from this example are for purposes of illustration only. The statistical value

of a human life in the optimal solution is less than the typical mean values, ranging from $1.6 to $4.0 million, that economists have been estimating [8, p. 7131. The acceptable risk of approximately 1 death per 1000 per period in the final column of Table 1 is much less stringent than the U.S. Environmental Protection Agency’s maximum acceptable cancer risk, which is 1 additional cancer case in 10,000 lifetimes. The optimal cancer risk per polluting firm per period in this model, Hoz/m,

is much greater than the California minimum under Proposition 65 which limits a firm’s emissions to levels such that the risk of cancer in the surrounding area is less than 1 case per 100,000 persons [24, p. 181.

Although the numbers in Table 1 are not entirely realistic, it is clear that they have important analogues in the policy-making arena. These data can be substituted into (12) to derive the hypothetical Pigouvian tax, 4, on good y. That the subjective component of 4 is much larger than lost output is consistent with a recent finding [23, p. 3861 that the subjective value of a single year of lost life “greatly exceeds annual earnings”.

Kropper and Oates [S. p. 7131 define the value of a statistical life in terms of the willingness to pay for a small reduction in risk by the entire community. It is instructive to paraphrase their definition, substituting only the optimal pX[UL/U,] from the present numerical example, as follows. “If reducing exposure to some substance reduces current probability of death by 10-s for each of 200,000 persons in a population, it will save two statistical lives. If each person is willing to pay $11 S7223 for the 10m5 risk reduction, then the value of a statistical life is the sum of these willingnesses to pay ($11.57223 x 200,000), divided by the number of statistical lives saved, or $1,157,223.” The two approaches are identical and the value that the entire community places on a statistical life is the same as that held by each identical member of the community!

Premature mortality and air pollution 7

In the optimal solution in Table 1, in which the probability of dying from pollution is approximately O.Od’l, each person’s marginal rate of substitution of money for survival is $1,157,223. If any one of the identical persons knew that the probability of his or her own death were 1 .O, in theory he or she would posthumously require twelve and a half billion dollars to remain at the attained level of utility?. Although it is usually assumed that the government revenue from Pigouvian taxes should be redistributed in lump-sums to consumers, it is sometimes proposed that the tax revenue be used to compensate victims of pollution. In the context of the present model, there are q%mY dollars of Pigouvian tax revenue that must be transferred in lump-sums to consumers so that gross disposable income in the economy equals gross national product. Using the substitutions, mYP = z and w =pXXL, it can be seen from (12) that the tax revenue, 4mY, exceeds by (p,[- Uz/U, + X,]H,,[mY]*P,,) dollars the sum of the ex unre compensation of (pX[ - Uz/U,]H,,z) dollars to all H, households for their statistical risk of premature mortality plus the loss of (w&z) dollars in wages to the Z&z disabled persons. This discrepancy holds because the Pigouvian tax is based on marginal damage, which is higher than average damage. However, compensation is usually thought of as an ex post award [30]. Although the tax revenue is more than enough to pay ex ante compensation plus lost wages, it may not be sufficient to pay true, ex post compensation.

CONCLUDING REMARKS

This paper focuses exclusively on the risk of death from air pollution. In a more general model, the damages would include, in addition to the monetized risk of premature mortality, the medical and human costs of nonfatal illness, damage to crops, vegetation and materials and the costs associated with soiling and decreased visibility. It is significant, however, that in Freeman’s [lo, p. 1281 comprehensive point estimates of total air pollution damage, the costs of premature mortality alone account for more than two thirds of the total damage. Because premature mortality is the major component of air pollution damage, it is useful to have a general equilibrium model in which the risk of death is explicit. In the model developed here, the marginal damage per unit of the polluting good is the sum that all households are willing to pay to eliminate the risk of premature death associated with the marginal unit of the polluting good plus the marginal value of output lost because persons who will die at the end of the period are too ill to work during the period. To foster economic efficiency, a Pigouvian tax, equal to the marginal damage, should be imposed on the polluting good. The model presented here is a simple one-period model in which all of the costs and benefits of pollution control occur in the same period that the pollution is emitted. If preferences, technology, and resources remain constant, the same equilibrium allocation will prevail period after period.

In legal contests, the value of a human life is often based on the discounted value of the future net output of the deceased rather than on any measure of psychological willingness to pay. It is significant that in our model, psychological and tangible costs must be added. In addition to monetizing the impact of air pollution on mortality, the model reveals a potential for nonconvexity in the S shaped dose-response curve, although at present levels of pollution, this does not seem to pose a practical problem for Pigouvian taxation. The numerical example suggests that Pigouvian tax revenue based on ex anfe mortality costs is likely to be insufficient for paying ex post

compensation to victims. Finally, the model demonstrates that, because a dollar value can be associated with a statistical life, there is an optimal level of risk of premature mortality.

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8 ROBERT E. KOHN

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