Non-paternalistic altruism and welfare economics

Journal of Public Economics 83 (2002) 293–305 www.elsevier.com / locate / econbase

Non-paternalistic altruism and welfare economics Nicholas E. Flores* University of Colorado, Department of Economics, CB 256, Boulder, CO 80309, USA Received 2 February 2000; received in revised form 26 June 2000; accepted 25 October 2000

Abstract Bergstrom showed that a necessary condition for a Pareto optimum with non-paternalistic altruism is classification as a selfish Pareto optimum. This paper shows that Bergstrom’s result does not generalize to the benefit–cost analysis of generic changes in public goods. There may exist good projects that will be rejected by a selfish-benefit cost test, a selfish test error. Selfish test error is linked to preference interdependence between public goods and income distribution, the same condition Musgrave identified as problematic for optimal public goods provision without altruism. Transferable selfish utilities provide freedom from selfish test error.  2002 Elsevier Science B.V. All rights reserved. Keywords: Public Goods; Altruism; Cost–Benefit Analysis JEL classification: H41; D64; D61

1. Introduction Non-paternalistic altruism refers to the situation where a given individual, the altruist, values the welfare of another, the beneficiary. In contrast, paternalistic altruism refers to the situation where the altruist values the beneficiary’s consumption of a particular merit good, irrespective of the beneficiary’s preferences. Non-paternalistic altruism is usually represented analytically by the entry of the beneficiary’s utility function into an aggregation function that represents the altruist’s preferences over own-good consumption and the beneficiary’s utility. *Corresponding author. Tel.: 11-303-492-8145; fax: 11-303-492-8960. E-mail address: [email protected] (N.E. Flores). 0047-2727 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0047-2727( 00 )00162-6

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Bergstrom (1982) showed that a necessary condition for a Pareto optimum with non-paternalistic altruism is classification as a Pareto optimum based only on selfish considerations. Thus optimal provision of public goods ultimately depends upon selfish preferences. As opposed to determining optimal provision, the goal in most benefit–cost analysis is to determine if a particular project could potentially improve welfare. The typical analysis considers the worth of a project independent of who will pay, providing the Hicks–Kaldor compensation principle as justification for avoiding the cost allocation issue. The resulting benchmark of a ‘‘good project’’ is any project for which there exists a cost allocation that would allow for a Pareto improvement. The impact of non-paternalistic altruism has been discussed at length in the benefit–cost literature, including applications in risk analysis 1 and environmental valuation.2 Based on Bergstrom’s result, several studies 3 have further concluded that non-paternalistic altruism can and should also be ignored for the generic, discrete changes encountered in benefit–cost analysis. It has long been recognized that even in the absence of altruism, the optimal provision of public goods cannot always be determined independent of the distribution of income (Musgrave, 1969; Bergstrom and Cornes, 1983). As noted by Musgrave (1969), the optimal provision of public goods may depend upon the optimal distribution of income which in turn will depend upon the choice of social welfare function.4 At the micro level, non-paternalistic altruism can be viewed as a social planner problem where the altruist’s aggregation function is the social welfare function. By applying Musgrave’s reasoning to this social planner problem, one can conversely conclude that the optimal distribution of income may depend upon the provision level of public goods. Similarly for the generic changes in public goods provision considered in benefit–cost analysis, the desired distribution of income between the altruist and beneficiary may change with changes in public goods provision. This paper provides an analysis of public goods provision that explicitly allows for preference interdependence between public goods and income distribution while considering cost allocation as well. Using an expenditure-type analysis it is shown that in some cases, the selfish benefit–cost test will reject a good project which is referred to as selfish test error. The general conclusion from this analysis is that passing a selfish benefit–cost test is only a sufficient condition for good projects. The same selfish transferable utilities (Bergstrom and Cornes, 1983;

1 See Jones-Lee et al. (1985); Viscusi et al. (1988); Jones-Lee (1991); Jones-Lee (1992); Johansson (1994); and Johannesson et al. (1996). 2 See Johansson (1992); Milgrom (1993); Lazo et al. (1997); and McConnell (1997). 3 See Jones-Lee (1991); Johansson (1992); Jones-Lee (1992); Johansson (1994); Johannesson et al. (1996); Lazo et al. (1997); and McConnell (1997). 4 Bergstrom and Cornes (1983) identify necessary and sufficient conditions for the independence of allocative efficiency from the optimal provision of public goods.

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Bergstrom, 1989) that circumvent Musgrave’s problem of preference interdependence for the optimal provision of public goods without altruism are shown to avoid the problem identified in this paper as well. The implications for applied benefit–cost analysis are discussed.

2. A model of non-paternalistic altruism The model involves a single altruist and a group of beneficiaries. Altruism operates in one direction only.5 The altruist’s preferences are defined over expenditures for the personal consumption of market goods, xA , the levels of the B 1 2 I public goods, Q, and the utility levels of the I beneficiaries, U 5fu , u , . . . , u g. The beneficiaries’ preferences are defined over expenditures for their personal consumption of market goods, x i , i [ h1, . . . , Ij and the levels of the public goods. It is assumed that preferences can be represented by twice differentiable utility functions, yielding the following functional representations. u A 5 G sxA , Q, U Bd u i 5 u isx i , Q d, i [ h1, . . . , Ij

(1)

The objective of the altruist is to maximize G when allocating income between own consumption and the I beneficiaries. The allocation process will depend upon the various functional forms in (1), the altruist’s income, the levels of the public goods, and the incomes of the beneficiaries. The functional form of G also handles paternalistic altruism when Q is the merit good. The direct benefit of Q to the altruist could be the result of personal enjoyment, the merit aspect of Q, or both. From an analytical perspective, there is no distinction between these motivations. The direct benefit of Q will be treated as a selfish benefit. Altruism is revealed by income transfers from the altruist to the beneficiaries. The altruist maximizes u A by allocating income between personal consumption and income transfers to the beneficiaries, conditional upon Q. The necessary conditions for the optimal allocation of income between the altruist and the beneficiaries is as follows. GU B u ix l Gi 5 ]] # 1 ;i [ h1, . . . , Ij Gx

(2)

l Gi is the altruist’s marginal value for transferring income to individual i. It may be the case that some beneficiaries receive transfers while others do not. For those receiving transfers, Eq. (2) holds with equality. For the remainder of the analysis, 5

The results in this paper could also obtained in a model of two-way altruism under the conditions discussed by Lindbeck and Weibull (1988) where ‘‘if the difference in wealth is sufficiently large and the degree of altruism is sufficiently large, then a transfer is given in equilibrium, from the wealthier to the poorer individual.’’

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it is assumed that each beneficiary receives an income transfer and therefore (2) initially holds with equality for each beneficiary.6 With regard to Bergstrom’s (1982) result, it is important to recognize that most benefit–cost analysis focuses (a) on discrete changes and (b) attempts to establish whether or not a project is good. The applied goodness criterion is whether the change, supported by a cost allocation, could lead to a Pareto improvement, i.e. the Kaldor–Hicks compensation test (Hicks, 1939; Kaldor, 1939). Clearly if a project passes the selfish benefit–cost test, then it is a good project under altruistic preferences as well. The next section establishes that the converse is not always true. There may exist good projects that fail the selfish benefit–cost test, a selfish test error.

3. The selfish benefit–cost test and test error The most prominent feature of non-paternalistic altruism models is the altruist’s desire or need to balance personal consumption and the utility levels of the beneficiaries. Restricting the welfare analysis to selfish values implies that for the altruist, beneficiaries’ utilities are a given and no longer part of optimization problem. Similarly for the purpose of the welfare analysis, the income transfer link is severed for the beneficiaries. De facto eliminating the potential for income transfers facilitates the construction of a set of expenditure functions that provides considerable insight into the implications of focusing on selfish values. For each beneficiary, one can derive an expenditure function of personal utility level and the levels of the public good, e i (Q, u). The expenditure function is the level of market good expenditure needed to maintain a utility level u with public goods at levels Q. These expenditure functions have the same analytical and conceptual properties as expenditure functions when altruism is not operable. Letting the 0 superscript denote the status quo, the expenditure function exactly equals expenditures on market goods before any changes, e i (Q 0 , u 0 ) 5 x i . For a project that increases public goods to Q 1 , the beneficiary’s willingness to pay can be written using the expenditure function. WTP i 5 e i (Q 0 , u 0 ) 2 e i (Q 1 , u 0 )

(3)

While beneficiaries’ motives are completely selfish, the transfers they receive result from altruism are not. From the perspective of the beneficiary, selfish 6 While it is assumed that the altruist acts upon the altruistic component of preferences by transfer, there may be social barriers to transfer. For example a beneficiary may feel insulted by receiving an income transfer. In these cases, the opportunity to provide public goods that increase the beneficiary’s utility may be the only way in which altruism is revealed. Relative to the analysis in the next section, initial efficient income transfers could not be assumed.

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willingness to pay with transfers may differ from (3) since the beneficiary must grapple both with the benefit of receiving more of the public good and subsequent changes in the transfer that result from changes in the public good and / or changes in income from paying for the change. Defining a selfish value, even for a beneficiary is not that straightforward. The representation in (3) is consistent with selfish values presented in the literature. This definition is pure, and defensible, in the sense that it is defined as if no altruism were operating. Now consider providing the change in Q at no charge to beneficiary i. Then the new, higher level of utility will be defined by u 1 5 u i (x i , Q 1 ). Expenditures before and after the change in Q are identical, e i (Q 0 , u 0 ) 5 e i (Q 1 , u 1 ). A result of this identity is that (3) can be rewritten as the change in expenditures for the utility change with Q fixed at Q 1 . The identity provided in (4) will prove useful in the analysis of the altruist’s value for the same change in beneficiary i’s utility. WTP i 5 e i (Q 0 , u 0 ) 2 e i (Q 1 , u 0 ) 5 e i (Q 1 , u 1 ) 2 e i (Q 1 , u 0 )

(4)

Similarly, one can construct an expenditure function for the altruist that is a function of Q, the level of total utility as measured by G, and the vector of beneficiaries’ utility levels, e A (Q, U B , G). Prior to any changes, the expenditure function equals the altruist’s expenditures on market goods, e A (Q 0 , U 0 , G 0 ) 5 xA . For the increase in Q, the altruist’s selfish willingness to pay can be written as the change in expenditures, maintaining U B 5 U 0 . WTP AS 5 e A (Q 0 , U 0 , G 0 ) 2 e A (Q 1 , U 0 , G 0 )

(5)

As was the case for the beneficiaries’ selfish values, the altruist’s definition of selfish willingness to pay derives from the elimination of transfers and holding beneficiaries’ utilities fixed. The representation of selfish willingness to pay for the altruist as provided in (5) is the isolated benefit for personal consumption of the change in public goods. The benefit–cost analyst is concerned with identifying good projects. If the sum of the selfish values exceeds the cost of providing the additional Q, then the project obviously deserves a good rating. When beneficiaries’ utilities are held at the status quo level, altruism plays no role. However if beneficiaries are allowed a utility gain, then the situation becomes more interesting. Consider giving beneficiary i the increase in Q at no cost while maintaining the status quo expenditure level on market goods. While one can no longer consider the beneficiaries’ willingness to pay, one can consider the altruist’s willingness to pay for the increase in Q and the increase in beneficiary i’s utility. Letting UI / i denote the vector of the other I21 beneficiaries’ utility levels, the altruist’s willingness to pay for the combined increase in Q and u i can be written using the expenditure function. The small script u entry in the modified expenditure function is the utility level for beneficiary i.

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WTP AS 1i 5 e A (Q 0 , u 0 , U 0I / i , G 0 ) 2 e A (Q 1 , u 1 , U 0I / i , G 0 )

(6)

By adding and subtracting terms, (6) can be expressed in terms of selfish willingness to pay plus willingness to pay for the increase in beneficiary i’s utility at the new level of Q. WTP AS 1i

5

e A (Q 0 , U 0 , G 0 ) 2 e A (Q 1 , U 0 , G 0 ) 1 e A (Q 1 , u 0 , U 0I / i , G 0 ) 2 e A (Q 1 , u 1 , U 0I / i , G 0 )

(7)

If there exists a beneficiary i such that WTP SA1i . WTP SA 1 WTP i , then the selfish benefit–cost test can fail to identify a good project. Proposition 1 formally states this result. Proposition 1. Define WTP i , WTP SA , WTP SA1i respectively as in Eqs. (4), (5), and (7). Suppose there exists some beneficiary i for which WTP SA1i . WTP SA 1 WTP i and the total costs of the project that provides the increase in Q from Q 0 to Q 1 are such that WTP AS 1i 1 o j ±i WTP j . TC . WTP AS 1 o i WTP i . Then the project will fail a selfish benefit–cost test despite being a good project. Proof. Suppose that for beneficiary i,WTP AS 1i . WTP AS 1 WTP i and WTP AS 1i 1 o j ±i WTP j . TC . WTP AS 1 o i WTP i . Since the sum of selfish willingness to pay is less than total cost, the project will fail the selfish benefit–cost test. However since WTP AS 1i 1 o j ±i WTP j . TC, there exists a cost allocation in which beneficiary i pays nothing for the project that supports a Pareto improvement and thus the project is good. Proposition 1 identifies when selfish test error, the erroneous rejection of a good project by the selfish benefit cost test, will occur. The antecedent conditions can be restated as at the new, higher level of Q, the altruist has a higher value for the change in beneficiary i’s utility than beneficiary i. The existence of a single beneficiary i with the antecedent conditions is sufficient for the erroneous rejection of a good project. The altruist may also have a relatively higher value for the utility change for other beneficiaries as well. The analysis of a second or third utility change requires a sequence of utility changes which entails considerably more notation and does not provide additional insight. From the definitions, equations, and identities presented above, it is clear that selfish test error occurs when e A (Q 1 , u 0 , U I0/ i , G 0 ) 2 e A (Q 1 , u 1 , U I0/ i , G 0 ) . e i (Q 1 , u 1 ) 2 e i (Q 1 , u 0 ). Using the fundamental theorem of calculus, the expenditure differences can be written in integral form. u0

≠e A (Q 1 , s, U 0I / i G 0 ) A 1 0 0 0 A 1 1 0 0 e (Q , u , U I / i , G ) 2 e (Q , u , U I / i G ) 5 ]]]]]] ds ≠u

E

u1

(8)

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u1

E

≠e i (Q 1 , s) e i (Q 1 , u 1 ) 2 e i (Q 1 , u 0 ) 5 ]]] ds ≠u

(9)

u0 A

1

0

0

0

A

1

1

0

0

i

1

Using (8) and (9), e (Q , u , U I / i , G ) 2 e (Q , u , U I / i , G ) . e (Q , u 1 ) 2 e i (Q 1 , u 0 ) can be written in integral form. u1

u1

u0

u0

≠e A (Q 1 , s, U I0/ i G 0 ) ≠e 1 (Q 1 , s) ]]]]]] ]]] 2 ds . ds ≠u ≠u

E

E

(10)

1

1

Proposition 2. If 2 euu0 ≠e A (Q 1 , s, U I0/ i G 0 ) / ≠u ds . euu0 ≠e 1 (Q 1 , s) / ≠u ds, then there exists some u [ [u 0 , u 1 ] such that 2 ≠e A (Q 1 , u, U 0I / i G 0 ) / ≠u . ≠e 1 (Q 1 , u) / ≠u. Proof. (contra positive) Suppose that for all u [ [u 0 , u 1 ], 2 ≠e A (Q 1 , u, U I0/ i G 0 ) / ≠u # ≠e 1 (Q 1 , u) / ≠u. By the properties of the Riemann integral, 1 1 2 euu0 ≠e A (Q 1 , s, U 0I / i G 0 ) / ≠u ds # euu0 ≠e 1 (Q 1 , s) / ≠u ds which proves the proposition. Corollary 1. Suppose that for all u [ [u 0 , u 1 ], 2 ≠e A (Q 1 , u, U 0I / i G 0 ) / ≠u # 1 1 ≠e 1 (Q 1 , u) / ≠u, then 2 euu0 ≠e A (Q 1 , s, U I0/ i G 0 ) / ≠u ds # euu0 ≠e 1 (Q 1 , s) / ≠u ds. Whenever WTP AS 1i . WTP AS 1 WTP i , there exists the potential for erroneous rejection of a good project by the selfish benefit–cost test. A necessary condition for WTP AS 1i . WTP AS 1 WTP i is the existence of some u [ [u 0 , u 1 ] such that the integrand on the left hand side of (10) is strictly greater than the integrand on the right. While the contra positive of Proposition 2 facilitates a very simple proof, the contra positive has additional value in that it identifies a sufficient condition for the absence of potential selfish test error. The contra positive is stated as Corollary 1. Proposition 2 facilitates the establishment of the link between selfish test error and preference interdependence between Q and the distribution of income. Through total differentiation, the marginal value of beneficiary utility for the altruist and the beneficiary can be expressed in terms of the utility functions. ≠e A (Q 1 , u, U 0I / i , G 0 ) Gu 2 ]]]]]] 5 ] ≠u Gx A

1

0

0

1

1

1

≠e (Q , s) 1 and ]]] 5 ]1 ≠u ux

(14)

1

≠e (Q , u, U I / i G ) ≠e (Q , u) Gu 1 i 2 ]]]]]] . ]]] ⇔ ] . ]1 ⇔ l G . 1 ≠u ≠u Gx u x

(15)

The altruist having a relatively higher value for the increase in beneficiary utility than the beneficiary implies that additional income transfer is desirable for the altruist at the new, higher level of Q. Proposition 2 and Eq. (15) emphasize this

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point. A sufficient condition for WTP AS 1i . WTP AS 1 WTP i is that for every u [ [u 0 , u 1 ], 2 ≠e A (Q 1 , u, U 0I / i G 0 ) / ≠u . ≠e 1 (Q 1 , u) / ≠u. This result is stated as a proposition without proof. Proposition 3. If for every u [ [u 0 , u 1 ], 2 ≠e A (Q 1 , u, U 0I / i G 0 ) / ≠u . ≠e 1 (Q 1 , 1 1 u) / ≠u, then 2 euu0 ≠e A (Q 1 , s, U 0I / i G 0 ) / ≠u ds . euu0 ≠e 1 (Q 1 , s) / ≠u ds which further A A i implies that WTP S 1i . WTP S 1 WTP . Before proceeding with simple examples of selfish test error, a discussion of the analytic techniques used above is in order. Selfish values for both the beneficiaries and the altruist were defined to mock the situation that for practical purposes, altruism does not exist at least for considering the change in public goods. The definitions of selfish value are consistent with discussions of selfish values in the existing literature.7 In examining the willingness to pay of the altruist, these definitions are analytically convenient since they facilitate the comparison of the beneficiary’s value with the altruist’s value of the change in beneficiary utility. As the propositions above show and the examples below demonstrate, there exist situations when the altruist values beneficiary’s utility change more highly than the beneficiary, resulting in selfish test error.

4. Examples This section provides several examples for which WTP AS 1i . WTP AS 1 WTP i . The class of G in the examples consists of those functions which are linear combinations of Cobb Douglas utility functions. For simplicity, the examples deal with a single public good and a single beneficiary. Eq. (16) presents the utility functional forms while Eqs. (17) and (18) present the respective expenditure functions. G(xA , q, u 1 ) 5 g x aaA q bA 1 (1 2 g ) u 1

S D

u e 1 (q, u) 5 ] q b1

S

1 / a1

G 2 (1 2 g ) u 1 e A (q, u 1 , G) 5 ]]]]] g q bA

u 1 (x 1 , q) 5 x 1a 1 q b 1

(16) (17)

D

1 / aA

(18)

Four different, but similar, sets of parameter values are used in the examples. The initial distribution of income for each of the examples is efficient from the 7

This definition is consistent with the marginal conditions outlined in Bergstrom (1982); Johansson (1992); Johansson (1994); Jones-Lee (1991); and Jones-Lee (1992).

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Table 1 Cobb Douglas examples

aA 5 0.7 bA 5 0 g 5 0.5 a1 5 0.7 b1 5 0.3 q 0 5 1 q 1 5 2 x A0 5 1000 x 1 5 1000 1 WTP 5 257 WTP AS 5 0 WTP AS 11 5 313.06 WTP AS 11 2 WTP AS 5 313.06 . WTP 1 aA 5 0.7 bA 5 0.1 g 5 0.5 a1 5 0.7 b1 5 0.3 q 0 5 1 q 1 5 2 x 0A 5 1000 x 1 5 1000 1 WTP 5 257 WTP AS 5 94.28 WTP AS 11 5 377.82 WTP AS 11 2 WTP AS 5 283.54 . WTP 1 aA 5 0.7 bA 5 0.2 g 5 0.5 a1 5 0.7 b1 5 0.3 q 0 5 1 q 1 5 2 x 0A 5 1000 x 1 5 1000 WTP 1 5 257 WTP AS 5 179.67 WTP AS 11 5 436.48 A A WTP S 11 2 WTP S 5 256.81 , WTP 1 aA 5 0.7 bA 5 0.2 g 5 0.5 a1 5 0.7 b1 5 0.3 q 0 5 1 q 1 5 1.25 x A0 5 1000 x 1 5 1000 WTP 1 5 91.20 WTP AS 5 61.77 WTP AS 11 5 153.17 A A WTP S 11 2 WTP S 5 91.40 . WTP 1

altruist’s perspective. The parameter values and willingness to pay values are provided in Table 1. In these examples, the largest difference between the altruist’s value for the beneficiary’s utility gain and the beneficiary’s value for the change is when the altruist has no selfish value for q. In the second example, the altruist has a positive selfish value for the increase in q and the altruist’s value for the utility gain is still greater than the beneficiary’s value for the change. In the third example, the altruist has a lower value for the utility change than the beneficiary. The fourth example uses identical utility parameters as in the third example, but the increase in q is 0.25 rather than 1. For this smaller increase, the altruist has a relatively higher value for the utility change. The four examples demonstrate that for this class of preferences, utility parameters as well as the size of the change can matter.

5. Transferable utility and freedom from selfish test error Bergstrom and Cornes (1983) provided a class of preferences that are not subject to Musgrave’s (1969) interdependence problem for the optimal provision of public goods without altruism. Bergstrom (1989) also applied this class of preferences to a problem associated with the rotten kid theorem (Becker, 1974, 1981). The class of preferences, transferable utility, consists of those preferences for which selfish preferences can be represented by functions of the form u i (x i , Q) 5 A(Q) x i 1 B i (Q) for beneficiaries and g(xA , Q) 5 A(Q) xA 1 B A (Q) for the

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altruist. Further suppose that G consists of quasi-concave aggregation functions H that are functions of the beneficiaries’ utilities and the altruist’s selfish utility. G(xA , Q, U B ) 5 H f g(x a , Q), U Bg

(19)

The expenditure function for beneficiary i takes the following form. u 2 B 1 (Q) e 1 (Q, u) 5 ]]]] A(Q)

(20)

Beneficiary i’s willingness to pay for the utility change takes a very simple form. u1 2 u0 WTP i 5 e i (Q 1 , u 1 ) 2 e i (Q 1 , u 0 ) 5 ]]] A(Q 1 )

(21)

Suppose the altruist pays WTP i while beneficiary i receives the full utility gain from u 0 to u 1 . The simple form of WTP i and the fact that the altruist has transferable selfish utility imply that the reduction in selfish utility for the altruist exactly equals the beneficiary’s gain in utility, the essence of transferable utility. With regard to the potential for selfish test error, the question then is whether H( g 0 2 (u 1 2 u 0 ), u 1 , U I0/ i ) . H( g 0 , u 0 , U I0/ i )? For transferable selfish utilities, the verity of this inequality is equivalent to WTP SA1i . WTP SA 1 WTP i . The following proposition establishes that for quasi-concave H, the answer to the question is no, H( g 0 2 (u 1 2 u 0 ), u 1 , U I0/ i ) # H( g 0 , u 0 , U I0/ i ). Proposition 4. Suppose that u i (x i , Q) 5 A(Q) x i 1 B i (Q) for all i [ h1, . . . , Ij and B B A G(xA , Q, U ) 5 H f g(x a , Q), U g where g(xA , Q) 5 A(Q) xA 1 B (Q) and H is a A 1 0 0 0 A 1 1 0 0 quasi-concave function. Then e (Q , u , U I / i , G ) 2 e (Q , u , U I / i , G ) # i 1 1 i 1 0 e (Q , u ) 2 e (Q , u ). Proof. Define g 0 5 g(xA , Q 0 ) and note that g(xA 2 WTP SA , Q 1 ) 5 g(xA , Q 0 ). Suppose the altruist pays WTP i 5 (u 1 2 u 0 ) /A(Q 1 ) for the beneficiary utility gain. Then g(xA 2 WTP AS 2 WTP i , Q 1 ) 5 g 0 2 (u 1 2 u 0 ). From the necessary conditions of the altruist’s utility maximization problem, at ( g 0 , u 0 ), Hg /Hu 5 1 which implies that the equation of the hyperplane tangent to H at ( g 0 , u 0 ) is of the form a 1 u 5 g. Note that ( g 0 2 (u 1 2 u 0 ), u 1 ) lies along the tangent hyperplane. By the quasi-concavity of H, H( g 0 2 (u 1 2 u 0 ), u 1 ) # H( g 0 , u 0 ) and so e A (Q 1 , u 0 , U 0I / i , G 0 ) 2 e A (Q 1 , u 1 , U 0I / i , G 0 ) 5 e i (Q 1 , u 1 ) 2 e i (Q 1 , u 0 ). Proposition 3 provides a class of preferences that are free from selfish test error, regardless of the size of the change in Q or particular utility parameters. The result does depend upon income transfers initially being efficient. As noted earlier, one way that altruism is revealed is through income transfers. The analytic method used here does not allow transfers. It is therefore fair game to ask do problems

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disappear when transfers are adjusted in step with changes in the public goods? 8 An earlier version of this paper provided an example where G consisted of a linear combination of Cobb-Douglass utilities, the same preferences used for the examples of Section 4. In that example, the beneficiary automatically accounted for transfers and provided a willingness to pay that exceeded selfish willingness to pay as defined here. The net result for the altruist was that maintaining optimal transfers actually resulted in a lower G utility level for the altruist after the specified change. As pointed out by an anonymous referee, this example ‘‘can be interpreted as an instance of the point made in Bergstrom’s ‘Fresh Look at the Rotten Kid Theorem.’’’ 9 The main point of that paper is there may be instances when a self-interested beneficiary may not work for the best interests of the collective unit. That is, rotten kids can really be rotten. The example demonstrates that altruism can still have an impact, even when transfers automatically adjust. However when directly applying the results from Bergstrom (1989), the problem of altruism under automatic transfers will also disappear if selfish preferences are represented by transferable utility functions.

6. Discussion and implications Economic studies, public opinion polls, and personal experiences indicate that altruism is alive and well in economic decisions involving both public and private goods. The approach recommended in the literature for analyzing the provision of public goods has been to concentrate on measuring ‘‘selfish values’’ to avoid measurement error. Justification for this approach has been based on Bergstrom’s result that a necessary condition for the optimal provision of public goods is optimal provision based upon selfish preferences. However as shown above in Proposition 1 and demonstrated through simple examples, there are cases where potentially welfare-improving changes in public goods may be rejected by the selfish benefit–cost test. Selfish test error is due to the issue that Musgrave (1969) identified as a problem for determining the optimal allocation of public goods without altruism: preference interdependence between public goods and the distribution of income. Based on the results from above, we can conclude that Bergstrom’s (1982) result does not fully generalize to the benefit–cost analysis of generic changes in public goods. Passing the selfish

8

Johansson (1992) and Johansson (1994) analyze changes around a social optimum and in both cases notes that a necessary condition for a social optimum is the optimal distribution of income. Based on this condition, these two analyses assume adjustment of income to maintain transfer efficiency. 9 The citation is Bergstrom (1989).

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benefit–cost test is not a necessary condition for a good project, although sufficient. It is important to recognize that the analysis presented above does not undermine Bergstrom’s (1982) original result. The problem lies in the fact that benefit–cost analysis deals with a more modest problem, determining whether a specific project can lead to a Pareto improvement when starting from an inefficient level of public goods. So while Bergstrom’s result stands, the extrapolation of Bergstrom’s result to general cases of benefit–cost analysis does not. At the conceptual level, the paper shows there can be legitimate altruistic values resulting from increases in public goods. Therefore studies that potentially encompass altruistic values need not be discredited on this account as suggested by Milgrom (1993) and echoed by others. But what about practical implications? Let us consider the case for focusing on selfish values. Some valuation techniques, such as contingent valuation and stated-preference conjoint studies, provide direct evidence that altruism is operating. People express concern for others, either in their desire to provide public goods or worry over tax burden on others. The same could be said for public discourse over actual choices at the ballot box. While we usually do not have evidence of altruism through revealed preference data, this is not to say that altruism is not present. For example, environmental quality can increase demand for a recreational activity while also increasing income transfers. Although the transfer may go unmeasured, altruism is still operating and being incorporated into the data nonetheless. Should we change the way we analyze revealed preference data? Probably not, but we should not assume that revealed preference data is free from the reach of altruism. Johansson (1994) suggested that purely selfish values can be elicited through stated preference questions by asking individuals to answer willingness to pay questions conditional upon on others paying their maximum willingness to pay, i.e. when no utility gains are accruing to others. However in the presence of income transfers, beneficiaries will not be providing a purely selfish benefit. They will incorporate income transfers into their decision and so the concept of selfish value as presented in the literature is undermined under the best of circumstances. These observations lead to the question do selfish values practically exist in a world with altruism and income transfers? The analysis presented above makes clear that benefit–cost analysis with altruism cannot simply be conducted independent of who pays. This result is particularly important for studies using stated-preference techniques. If researchers are eliciting values for public goods, they need to make clear the costs to others in order to minimize measurement error. Stated-preference conjoint analysis is seeing widespread application in public goods valuation. In a random utility context, this technique may facilitate that measurement of the altruistic component of public goods provision, including testing for transferable utilities. Analyses that specifically focus on measuring the impact of altruism will lead to a better understanding of the relative importance of altruism in benefit–cost analysis.

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Acknowledgements The author thanks John Loomis, Don Fullerton and Jack Robles for helpful comments.

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