Public Choice 51:81-86 (1986). 1986 Martinus Ni jhoff Publishers, Dordrecht. Printed in the Netherlands.

Optimal quantity of a controversial good or service

ROBERT E. KOHN Department of Economics, Southern Illinois University, Edwardsville, IL 62026

1. Introduction

Musgrave and Musgrave (1984: 78) ' . . . define merit goods as goods the pro- vision of which society - as distinct from the preferences of the individual consumer - wishes to encourage, or in the case of demerit goods, to deter.' The categorization of demerit goods ana sevices has become increasingly vague. When substantial numbers of people argue that a particular good or service that is legal (illegal) should be made illegal (legal), it is not clear whether or not such goods or services belong in the demerit category. It would be better to call them simply 'controversial goods and services.'

A controversial good or service (CGS) is one that some people choose to consume, but which other people disdain, often on moral grounds, their utili- ty diminished solely by their awareness that such goods or services are being consumed. Between the two extremes of prohibition and permissibility is the alternative of sumptuary taxes, which as Musgrave and Musgrave (1984:438) note , ' . . , may be imposed to discourage the consumption of demerit goods.' If however, such goods are treated as controversial, the correct purpose of the tax is to foster the Pareto optimal level of consumption.

In this paper, a strong assumption on additivity is made so that the CGS can be treated as a Samuelson public good. A model is constructed in which there are four persons, two of whom consume the CGS and whose marginal rates of substitution in consumption between the CGS and the numeraire good are equal, and two persons whose utility is less because of their aware- ness that the CGS is being consumed and whose marginal rates of substitu- tion are summed rather than equated. The condition for the optimal output and allocation of the CGS is derived, and the sumptuary tax that would foster that allocation in a competitive market economy is determined. The model is then used to illustrate the case in which sumptuary taxation is not efficient and the CGS should be prohibited. In the concluding section of the paper, the model is related to preceding models in the economic literature.

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2. Marginal conditions for economic efficiency

Let (xi, Yi) represent the consumption hundle of the ith person in a two-good, four-person economy. Assume that the first and second persons in the model enjoy both goods, but that the third and fourth persons disparage the con- sumption of good y, which in this model is the controversial good or service (CGS), and deem themselves worse off the larger the sum, yl + y2, consumed by others. The utility functions of the four people are

U1 = UI(x~, Y0 (1)

U 2 = U2(x2, YE) (2)

U 3 = ua(x3, Ya + Y2) (3)

U 4 = U4(x4, yl + y2) (4)

1,2 34 34 where Ux ~' 2 > 0, Uy > 0, Ux' > 0, and Uy' < 0. It is assumed that each consumer's utility function, as he or she perceives it, is the best possible for- mula for that consumer's welfare. It is also assumed that the preferences of consumers are stable, that is, that they are not altered in the process of con- suming their chosen bundle; this may be a strong assumption if consumption of the CGS is traumatic (see Solomon and Corbit, 1973).

The conditions for economic efficiency are derived from a Lagrangian ex- pression that incorporates the above utility functions and a production possibilities function. It is assumed that economic efficiency can be con- sidered apart from distributional equity and that all resources are fully employed, including those used for producing and supplying the CGS. The first three utility levels are given and that of the fourth person is maximized.

L = U4(X4, Yl + YZ) + V I (U1-U I (x1 , Y0) (5) 2

q -V2(Uo-U2(x2 , y2)) + V3(Uo3-U3(x3, Yl + Y2))

+ V4(F(x1 + X2 + X3 + X4, Yl + Yz))

The first order conditions for an internal maximum are

--V1U~ -t- V4Fx = 0 (6) - -VzU 2 + V4Fx = 0 (7)

-V3W3x + V4Fx = 0 (8) U 4 -I- W4Fx = 0 (9)

4 1 3 Oy - V lUy - V3Uy --1- V4Fy = 0 (10)

4 2 3 Uy - V2Uy - V3Uy q- V4Fy = 0 (11)

It follows that

x 1 2 2 3 3 U~y/U 4 (12) Uy/Ux = Uy/U~ = Fy/Fx - Uy/Ux -

For persons 1 and 2, who enjoy the CGS, the marginal rates of substitution in consumption must be the same and must also equal the marginal rate of transformation in production (taken as positive) augmented by the sum of

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y ,+y~ - . . . . 1 ,~ - ~-< - - L - / - / - - - -~ , . / / ~+b 4-c

~,// / / / I I I

I I

! / /

/ / I '111 '1 , I I I

X 1 X 2

U~ / / / F 'l 1

Figure 1.

the marginal rates of substitution in consumption by the disapproving per- sons, 3 and 4, who are willing to give up some of good x for less consumption of y. (Note that the signs of Uay ' 4/Ux3' 4 are the reverse of U~' E/UI' z.)

The optimal solution is graphically illustrated in Figure 1. Persons 1 and 2 consume (Xl, yl) and (x2, y2), respectively, on indifference curves U 1 and U 2 . The common marginal rate of substitution is (a + b + c) units of good x for 1 unit of good y. This is the inverse of the slope of the line through the joint consumption bundle, (xl + x2, yl + y2). The remaining persons, 3 and 4, consume (x3, Yl + Y2) and (x4, yl + y2) on indifference curves U 3 and U 4 , which have the conventional curvature for the case of a wanted and an un- wanted good. Their marginal rates of substitution of good x for good y are - a and - b. The marginal rate of transformation of good x for good y is c. For a numerical example, see the appendix to this paper.

3. An efficient sumptuary tax and the case in which consumption of the CGS should be prohibited

The appropriate tax on the CGS is the sum of the amounts of money that of- fended persons are able and willing to pay at the margin to restrict consump- tion of the CGS by one unit. This is analogous to the Pigouvian tax on a pollu- tant (see Kohn, 1975: 15-28). In the case of the simple model in Section 2 of this paper, setting the marginal cost (and price) of the numeraire good, x, at unity, the price of good y, is its marginal private cost, Fv/Fx, plus the tax,

3 3 4 which is - ( - Uy/Ux - Uy/U~ per unit. The case in which the CGS should be banned is the one in which the

necessary conditions for economic efficiency are not sufficient. The deriva-

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U 4 U4*

\ \ l u 2 \ _ _ _

"+'~I - - - \~- - - ? : . . . . . . I \ , ' ;"" 1~/"~ / i u, ~,',' /i [ /

I I

,,', , ' , ,~,/ / , , / , , I_ '_,V/ ! \ ' '

i i I I,'% ; 'V I ' ,~ I / t x\ ~/ '"' /X 11/ I /X r I I/' ~ ./ : \ ]

F

-i!

I n, ~ n,3+', m, m 2 - -ml - -m 2

n`3

Figure 2.

tion of the second order conditions for (5) is complicated. However, the violation of these conditions can be graphically illustrated. In Figure 2, the indifference curves of persons 1 and 2 are relatively steep and intersect the x-axis, and the indifference curves of persons 3 and 4 are convex upward rather than to the right. (Such a reversal of convexity would hold if persons were primarily offended that the CGS was being consumed in even the smallest amount and their disutility then increased at a decreasing rate.) In Figure 2, the necessary conditions are satisfied when y~ + y2 is consumed. However, it is possible to increase person 4's utility from U 4 to U 4. , without changing the remaining utility levels, by reducing consumption of the CGS to zero. Person 1 must receive m~ units more of good x and person 2, m2 units more. However, person 3 is satisfied with m3 units less and production increases by m4 units. Although, person 4 has m3 + n~ - ml - m2 units less of good x, he or she is on a higher indifference curve. This case is included in the appendix.

4. Concluding remarks

The problem of the CGS requires no new tools of analysis. The precedent for including quantities of specific goods consumed by one person in the utility function of another person was established by Duesenberry (1949). Daly and Giertz (1972: 2) formalized the case in which simple awareness of

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what others are consuming can foster disutility as well as utility. This is a departure from the earlier view of Mill (1859: 78), who denied economic significance to the disutility of persons ' . . . who consider as an injury to themselves any conduct which they have a distaste for, and resent . . , as an outrage to their feelings.' In these studies and that of Danielson (1975), the disutility associated with the consumption of a particular good is reduced by redistributing that good.

The prospect of reducing disutility, not by redistributing the offending good but by reducing its total consumption is posed by Mishan (1969: 339), but there the problem is some physical by-product of consumption rather than pure awareness that the good is being consumed. In the present paper, it is the awareness that a particular good or service is being consumed that causes disutility to people who do not wish to consume that good

themselves. The basic model here is that of the Samuelson public good, except that

it is a 'bad' , and the quantity of that bad is the total quantity consumed. There is thus a double summation: a vertical summation of marginal rates of substitution by nonconsumers and a horizontal summation of quantities consumed. The latter summation has no precedent as an argument in utility functions, and its significance may be questionable. Giertz and Sullivan (1977) develop a model in which consumption of a particular good by mul- tiple recipients gives utility to multiple donors. But they are careful to assume that each of the n recipients consumes the same quantity. In the case of a CGS such an assumption would be unrealistic. However, it appears that horizontal summation may be crucial to the derivation of an efficient sump- tuary tax for the CGS.

5. Appendix

The following is a numerical example. If the utility functions and corre- sponding constraints are U 1 = xl + 36y 2/3 = 606, U z = xz + 72y~/2 = 452, U 3 = 200x~ - 3(ya + y2) z = 140,000 and U 4 = 100x4 - (yl + Y2) 2 , and the production possibilities constraint is (Xx + xz + x3 + x4) 2 + 15(y~ + y2) 2 = 2,400,000, the optimal allocation is y~ = 64, Yz = 36, Xl = 30, x2 = 20, x3 = 850, x4 = 600, and U 4 = 50,000. The common marginal rate of substitution for the consumers of both goods is 6 units of x per unit of y. This equals the marginal rate of transformation, which is 1 unit of x per unit of y, minus the sum of the marginal rates of substitution of the disapproving persons, which are - 3 for person 3 and - 2 for person 4. In a competitive economy, the marginal cost of good y is 1, the efficient sumptuary tax is 5, and the total tax revenue is 500.

The case in which the CGS should be banned can be simulated by re-

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placing the utility functions of persons 3 and 4 with U 3 = x3 - 60(yl + y2) 1/2

= 250 and U 4 = x4 - 40(y~ + y2) 1/2. The marginal conditions are conve-

niently satisfied with the same numerical values and marginal rates of

substitution as above except that U 4 = 200. If, however, there is no produc-

tion of the CGS, so that y~ = Yz = 0, then the same utility levels, 606,452,

250, can be attained by the first three persons, but the utility level of person 4 will be 241 instead of 200.

REFERENCES

Daly, G., and Giertz, J.F. (1972). Benevolence, malevolence and economic theory. Public Choice 13: 1-19.

Danielson, A.L. (1975). A theory of exchange, philanthropy and appropriation. Public Choice 24: 13-26.

Duesenberry, J. (1949). Income, saving and the theory of consumer behavior. Cambridge: Harvard University Press.

Giertz, J.F., and Sullivan, D.H. (1977). Donor optimization and the food stamp program. Public Choice 29: 19-35.

Kohn, R.E. (1975). Air pollution control." A welfare economic interpretation. Lexington: Heath.

Mill, J.S. (1859). On liberty. New York: Norton, 1975. Mishan, E.J. (1969). The relationship between joint-products, collective goods, and external

effects. Journal of Political Economy 77: 329-348. Musgrave, R.A., and Musgrave, P.B. (1984). Public finance in theory and practice. New York:

McGraw-Hill. Solomon, R.L., and Corbit, J.D. (1973). An opponent-process theory of motivation. Journal

o f Abnormal Psychology. 81: 158-171. Reprinted 1978 in the The American Economic Review 68: 12-24.