The numéraire matters in cost-benefit analysis

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JOURNAL O F Journal of Public Economics 64 (1997) 117-123

ELSEVIER

PUBLIC ECONOMICS

The num6raire matters in cost-benefit analysis Kjell A m e B r e k k e Research Department, Statistics Norway, P.B. 8131 Dep., N-0033 Oslo, Norway

Received ! June 1994; received in revised form I April 1996

Abstract The choice of num6raire is shown to be important in cost-benefit analysis. When a public good is involved, individual consumers" marginal rates of substitution will generally differ. Thus, the less valuable the num&aire is to a person, the higher the number required to express his net benefit, and the more will his interest weigh in the total sum. The choice of money as num6raire is systematically favourable to those who value money the least, relative to alternative num6raires. ©i997 Elsevier Science S.A. Keywords: Num6raire; Cost-benefit analysis; Welfare weights JEL classification: H43

1. I n t r o d u c t i o n The choice of num6raire has been considered a matter of convenience at least since Walras showed how prices expressed in terms of one num6raire could easily be translated into prices in terms of another. Adding the cost and benefits for a person gives the same sign, independent o f the choice o f nun~raire. For example, if one tonne o f reduced emission o f some pollutant is worth $100 to a person, then a p r ~ e c t that reduces emissions by one tonne at a cost of $50 to this person, is to his benefit, In money terms, the cost o f $50 should be subtracted from the benefit of $100. If measured in reduced emissions, the cost is equivalent to 0.5 tonne, while the benefit is 1 tonne. The net benefit is positive independent of unit. While the sign of the net benefit is independent o f the choice of num~raire, the same is not true for the sum o f net benefits, at least not when the willingness to 004%2727/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved Pll S0047-2727(96)016 ! 0-6

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K.A. Brekke I Journal of Public Economics 64 (1997) !17-123

pay for public goods is included. In a cost-benefit analysis it is this sum that is taken as the social net benefit. In this paper I will argue that the choice of num6raire is very important to the sign of the sum of net benefits, and that different num6raires will systematically favour different interest groups. Once it is recognized that the choice of num6raire matters, the question is whether one particular choice is the right one. Should we add net benefits in terms of money rather than, say, tons of reduced emissions? One argument for the choice of money as num6raire could be based on the Hicks-Kaldor criterion that the winners should compensate the losers. While each single project is not to everyone's advantage, it is sometimes argued that the use of the Hicks-Kaldor criterion will be to everyone's advantage in the long run as an effect of many small projects. Each person will be among the winners on one project and among the losers on others. I will argue that there are systematic differences between the different num6raires which makes this argument less convincing. An alternative foundation for cost-benefit analysis is based on the use of Bergson-Samuelsonian welfare functions. Within this framework the choice of money as num6raire may be interpreted as a claim that an additional dollar corresponds to the same amount of cardinal utility independent of wealth, income or other characteristics of the person who receives the dollar. If we cannot in general make this assumption, then it follows that individual net benefit has to be weighted. A shift of num&aire from money to tons of emissions is found to correspond to a shift in weights of factors up to more than one thousand in an empirical example. Unless money is a better measure of cardinal utility than, for example, tons of emissions, the welfare weights are indeterminate to an extent that questions the practical usefulness of cost-benefit analysis. Not only will the value of environmental quality become ambiguous, but it will be shown that the choice of num&aire is a choice of which group to favour.

2. Why the num~raire matters. An example In general, individuals' marginal willingness to pay for public goods differ. Therefore, consumers are faced with differing alternative values, or Lindahl prices, of public goods. This is the intuitive reason why the num6raire matters. The following example may clarify this. Consider two persons, a 'materialist' and an 'environmentalist'. Their preferences are defined over private money ¥, which can be used to buy commodities at given prices, and the environment E. The measurement of E depends on the specific project under consideration and may be in tons of reduced emissions or square kilometres of preserved wilderness. Below, ,one unit of E will be denoted an environmental quality unit. The utility functions are given as

K.A. Brekke I Journal of Public Economics 64 (1997) 117-123

| 19 (1)

UI(Yt,E) = 4 Y s + E

for the materialist, and (2)

U2(Y2,E) = Y2 + 4 E

for the environmentalist. Their initial income is Y~= 1000 and the state of the environment is E = 1000 (according to some measurement rule.) Consider the project AYj = - 1 for j = 1,2 and A E = 1. Individual j ' s equivalent variation, EI~,, is the amount of money that j needs to obtain the welfare change he will derive from the change AE. ~ is defined by

vj(r, + L-~,,E) = V,(~.e + aE). If the compensating variation exceeds the cost -AYj, then the sum EI~j+AYj is positive. In this case j ' s utility will increase as the project is implemented. The welfare c h ~ g e for person j of the project can he expressed in money terms as net benefit, N B j. v, satisfying

v,(r, + Na,.,E) = Uj(r, + a r , , E + aE).

(3)

where N B i . v ~ E I ~ j + A Y i , with equality in the linear case considered in this example. Similarly, the net benefit may be expressed in environmental quality terms as NBj. E, satisfying

uj(~.E + Na 9

= u,(r, + a r , , e + aE),

(4)

e.g., the net benefit in environmental units, for person 1 above, wil' be NBI.E = -3, as is easily verified by inspection. Individual and total net benefit in this example are reported in Table 1. We see from the table that the average N B is negative in environment units, and the project looks bad. This will be the case even in a society with almost 80% environmentalists (utility U2). At the same time N B is positive if it is computed in money terms. The average N B will be positive even if only 20% in society are environmentalists. The choice of scale is thus very important in this example. Note that if the average net benefit is reported only in money terms, then there is no way to recover the fact that the average net benefit is negative when the average is computed in environmental quality units. To see this, note that the Table I Net benefit in different units Enironmentalunits Income units

I

2

Total

-3 - 0.75

+ 0.75 +3

- 2.25 + 2.25

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average net benefit in monetary terms is also consistent with the case U~ = U2 = Y~+ ~ E . In this case the average net benefit would be pogitive also when computed in environmental units. If a decision-maker finds environmental quality units to be the appropriate unit of aggregation, then he will be unable to detect that the project is bad according to his judgement. Why do money and not environmental units as num&aire favour the environmentalist? The net benefits of the project is positive for the environmentalist. If this net benefit is expressed in money terms, then it becomes a large number because money is of low value to the environmentalist. However, if the net benefit is expressed in environmental quality units, then the net benefit would be a ,.mall number, since environmental quality is important to the environmentalist. Similarly, the net benefit is negative for the materialist, but because he values money high, the net benefit is only a tiny number if expressed in money terms, but large in environmental quality units. More generally, we would expect it to be favourable for a group if the good chosen as num6raire is one that they value less than other groups. Where does the standard argument for the independence of the num6raire break down? In equilibrium, the marginal rate of substitution between two private goods will be equal for all consumers. In an economy with only private goods, the num6raire would not matter. In the current example, however, one of the goods is public and the marginal rate of substitution between a public and a private good may be different for different individuals. If two consumers have equal marginal utility of the private good, then the marginal utility of the public good will be unequal, and vice versa. Implicitly, all individuals are assumed to have equal marginal utility of the numeraire good. The choice of num&aire is thus not only a normalization of prices but also a normalization of marginal utility as well. This double normalization explains the dependence of the choice of num6raire.~ This explanation in terms of normalizing marginal utility shows that the importance of the num6raire is based on an interpretation of cost-benefit analysis as a measure of changes in total utility. If side payments were allowed, then the environmentalists surely could compensate the materialists in the above example. The environment would not be a reasonable num6raire because side payments in environmental quality units are not feasible. However, lump-sum ~ansfers are in general not feasible either. It is sometimes argued that lump-sum transfers are not required, since the winners and losers will not be the same individuals in all projects. In the long run, almost all wi!l be winners. But, since the winners and losers in a project are independent of the choice of num6raire, the argument would not support any particular choice of num6raire. A similar argument was used by Edgeworth (1888) ~A similar case of double normalizationthat makes the num~raire important is pointed out in Atkinson and Stem (1974), where the num&airenormalizesboth consumerand producedprices, thus implicitly assumingthat the nura6raire good is untaxed.

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to support utilitarianism, then framed in terms of utility units and not monetary units. Finally, it may he argued that side payments • .~ are not required because the tax system can be used to correct general biases 7 Since taxes are in monetary units, this would support the choice of money as num6raire. Using the tax system to correct general biases requires that we can tax Gr subsidize a commodity that is correlated to the required compensation. Note, however, that in the example above, both the environmentalist and the materialist have the same income, and compensation would require a redistribution between consumers with different preferences, not different income.

3. H o w important is the choice o f num6raire?

In the previous section we only calculated the sign of bib E. To simplify comparison with aggregation in monetary units, we transform the result into monetary units. T o achieve this, I assume that the cost o f the project is uniformly distributed to all individuals, i.e. zlY~ = (1/I)AY = AYl~ Then we can find the per capita cost that make.s the project socially indifferent to the status quo, when aggregated in environmental units. That is, AYm is chosen so that N 8 E = 0 . To examine the emoirical importance of the choice o f num6raire, we have made calculations based on a study by Strand (1985). In that study the average net benefit o f lowering emissions into the air from automobiles was estimated. The analysis is based on a survey of 2059 individuals. Of these, 1852 responded, which is about 90%. The preject under analysis was to lower emissions into the air from automobiles by 50%. The respondents' net benefit varied from zero (277 respondents) to over 12 000 kroner (five respondents). If the project is to improve the environment, AY~ will become zero if willingness to pay (WTP) is zero for at least one individual. The reason for this is that for this individual, any monetary cost will correspond to infinite negative net benefit in environmental quality units. In what follows I assume that W T P = 0 is reported when WTP is in fact positive but small. The lowest interval of positive net benefit was 10-90 kroner, and in the following calculations I have replaced the response ¢ f r p = o by W T P = 5 . Alternatively, the zero bidders could have been removed from the data. This would have a minor effect on the resuh.3 To aggregate in environmental units, the distribution aspects of the project have :See, for example, p. 160 in Frank (1992) who argues: "'Once we focus on finding a general policy for making large number of decisions and recognize that compensationsfor general biases is possible through the tax system, it becomesclear that we have had a perfectlygood .socialchoice mechanism all along, namely the cost-benefit criterion.'" ~ZeroWTP are much discussed in the valuation literature. A hypothesisis that a zero WTP may not reflect low valuation,but rather are protest bids from persons who do not accept the idea of expressing envionmental quality in monetary terms. For a discussion, see Mitchell and Carson (1989).

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to be known. An analysis of the income distribution effect of a carbon tax in Alfsen et al. (1992) shows that it is reasonable in this case to assume that the cost is evenly distributed. Thus the procedure above seems justified in this case. With these assumption I found A¥, =31 kroner. Thus with a cost of 31 kroner to each individual, the average environmental quality improvement demanded is equal to AE, while individual costs above 31 kroner would make total net benefits in environmental terms, negative. This can be compared with the average willingness to pay in money terms of 685 kroner. Thus, with average per capita cost less than 685 kroner, the total net benefit in monetary terms is positive. Note that with the per capita cost at 500 kroner, a person with WTP= 12 000 kroner has a net benefit equal to 11,500 kroner, and 0.96 environmental unit. Similarly, a person with WTP= 10 kroner has a net benefit of - 4 9 0 kroner or - 4 9 environmental units. The •se of environmental units in aggregation is equivalent to giving the monetary net benefit of the materialist (WTP= 10) a weight 1252 times the weight given to the monetary net benefit from the environmentalist (WTP= 12 000).

4. Conclusion In this paper I have demonstrated that the value of environmental quality will depend cruciaUy on the choice of num6raire, and that different choices favour different groups. The empirical example discussed shows that a shift in num6raire corresponds to huge shifts in welfare weights. The social cost of reduced emissions from automobiles was reduced from 685 kroner to 31 kroner by a change in aggregation units. Even though efficiency and distribution cannot be separated in theory, it may be thought that is it wise for practical purposes to keep them apart. In the example discussed in this paper, distributional problems remairl even though both consumers had the same income. Individuals with different tastes will prefer different social states, and one particular choice of valuation principle will favour groups with particular types of preferences.

Acknowledgments Thanks to the Norwegian Resource Council for financial support through the Economy and Ecology program. Helpful comments from Rolf Aaberge, Karine Nyborg, Amartya Sen, the Editor, Nicholas Stem, and an anonymous referee are gratefully acknowledged.

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References AIf~n, K., A. Brendemoen and S. GlomsrOd, 1992, Benefit of climate policies: Some tentative conclusions, Discussion Paper 69, Central Bureau of Statistics, Oslo. Atkinson, A.B. and N.H. Stem, 1974. Pigou, taxation and public goods. Review of Economic Studies 41, 119-128. Edgeworth, F.Y., 1888, Mathematical ~ychics (London). Frank, R.H., 1992. Melding sociology and economics: James Coleman's foundation of social theo~'y, Journal of Economic Literature 30, 147-170. Mitchell, R.C. and R.T. Carson, 1989, Using surveys to value public goods: The contingent valuation method (Re~urces for the Future, Washington, DC). Strand, J., 1985, Verd~tting av redu~ne luftforurensinger fra biler i Norge (Valuation of reduced air pollution from automobiles in Norway), Memorandum no. I, Department of Economics. University of Oslo (in Norwegian).