Soc Choice Welfare (1986) 3:151-160

Social Choice dWelfare

© Springer-Verlag 1988

Intergenerational Preference Orderings L. G. Epstein* Department of Economics, University of Toronto, 150 St. George Street, Toronto, Canada M5S 1A1 Received August 17, 1984/Accepted June 5, 1986

Abstract. This paper investigates preference orderings on infinite horizon intergenerational consumption streams. The trade-offs inherent in the selection of moral precepts for an intergenerational preference ordering are made explicit by establishing the mutual inconsistency of an appealing set of axioms.

I. Introduction This paper is concerned with the ranking of infinite horizon consumption streams. The framework is standard: there is a single aggregate consumption good and a single consumer alive at any instant who is representative of his generation. The problem is to order intergenerational consumption streams in a morally appealing manner. This paper formulates some axioms which are likely to be widely acceptable and then shows that they are mutually inconsistent. The analysis serves to make explicit the trade-offs inherent in the selection of moral precepts for an intergenerational preference ordering. The essence of the analysis can be conveyed by considering the problem of finding maximal consumption paths in sets of the form {(Co,...,c,,...):

ct>O and

+r)kt-ct Vt>0, k o > 0 given} . (1) to capital (kt) in the constant returns to scale

0
r > 0 is the time independent net return technology. The optimal consumption program depends on the intertemporal social welfare function. Discounting is ruled out below by an equity axiom. Thus, if a utilitarian welfare function is adopted, it must embody a zero rate of discount and consumption must increase monotonically and without bound along an optimal path. This policy m a y seem questionable (Rawls 197], p. 287) for it demands "heavy sacrifices of poorer generations for the sake of greater advantages for later * I am grateful to Don Campbell for many valuable discussions and comments. I have also benefitted from the comments of Mike Peters, participants in the Theory Workshop at the Universityof Toronto, and from a referee and editor of this journal.

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ones that are far better off." This objection is met by the maxi-min criterion which implies a bounded, indeed constant, consumption program. But the rigid egalitarianism of the maxi-min criterion rejects any trade-off where one generation loses regardless of the gains to other generations. Thus poor economies are left underdeveloped and the standard of living remains low forever.1 Moreover, these unappealing policy implications of the utilitarian and maxi-min social welfare functions are not restricted to the simple environments in (1). For example, they are valid also in a model of economic growth with an exhaustible resource (Dasgupta and Heal 1979, Chap. 7 and 10) where reproducible capital can be substituted sufficiently for the exhaustible resource to permit unbounded consumption paths to be feasible. It is natural to wonder whether there exist alternative intergenerational preference orderings which combine the attractive features of the utilitarian and maxi-min criteria and which avoid their deficiencies. In particular, is it possible that (a) optimal consumption programs are necessarily bounded so that future generations are not treated "too well" at the expense of earlier ones; and (b) a tradeoff exists between present and future consumption levels, so that some degree of economic growth is conceivably optimal. The existence of such preference orderings is addressed in this paper. Since it is desirable that preference orderings perform well in a broad range of environments, the class of choice sets in (1) is expanded to allow time varying rates of return. When some other, rather compelling axioms are imposed, it is shown (Theorem 1) that (a) and (b) cannot be satisfied simultaneously. Thus, in choosing an intergenerational preference ordering, society must choose which of these precepts it will violate) The difficulty of reconciling seemingly plausible ethical precepts is also demonstrated by Diamond (1965, p. 176) and Campbell (1985). But their results depend crucially on the specification of a topology. For example, one of Diamond's precepts is continuity of the preference ordering in the supremum topology, the importance of which as an ethical principle is far from clear. Svensson (1980) objects to the requirement of continuity and shows that Diamond's impossibility result disappears if an alternative and nontrivial topology is adopted for the set of consumption paths. In contrast, continuity is not required in Theorem 1 below. Rather, the theorem relies on axioms which are more readily interpreted as statements about the nature and implementability of the policy prescriptions of the preference orderings. The paper proceeds as follows: The next section lays out the axioms and proves the impossibility result (Theorem 1). Section 3 presents a final result regarding the performance of equitable preference orderings in environments where a positive

1 Thisis not necessarilythe case if the maxi-min criterion is applied to utility paths rather than to consumption paths and if generations are altruistic. Moreover,in such models, the maxi-mincriterion may be intertemporallyinconsistent. See Dasgupta (1974),Calvo (1978), Phelps and Riley (1978). z The incompatibilityof (a) and (b) is not established for all planning problems, such as the one consideredby Heal and Dasgupta. The conflictbetween(a) and (b) is establishedonlyif it is requiredthat the preferenceorderingperform well also for the environmentsconsideredbelow. Interest in the latter stems from the simplicityand importancein the economicsliterature of the Knightian constant returns specification.Of course, small open economiesface exogenousreal interest rate profilessuch as in (2).

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constant level of consumption is not permanently sustainable. Proofs are relegated to an appendix. The following notation and terminology are adopted:

l~={C=(co,...,c,,...):

sup A< oo} ,

li++= P = ( p o , . . . p , . . . ) : p t > O V t a n d

l T = { C e l °° :c,>0Vt} , pt
•

Consumption paths C generally lie in lT while price paths lie in l~++. For each scalar c > 0 and for each C e l t and T>O, rC denotes the path (Cr,Cr+i . . . . ), (c,C) represents (c, Co, cl . . . . ) and (c) represents the constant path (c, c . . . . ). Similar oo

notation applies to elements of l 1 ÷. PC denotes the inner product ~ptct. When o there is no ambiguity, the limits of summation are suppressed. A preference ordering on l~ is a reflexive binary relation ):. The related indifference ( ~ ) and strict preference (>-) relations are defined by

C~C'.~C)~C'

and

C'~C

C ~ C ' <=>C~C'

and not

,

C'~C

.

Note that a preference ordering need not be complete or transitive.

2. Preference Orderings Often social welfare analysis is based on the utilities of the persons involved and only indirectly on their consumption levels. Sen (1977) calls this approach "welfarism" and presents some arguments against it. It is not adopted here. Rather, the precepts described and the subsequent analysis relate directly to preference orderings on the consumption space I~°. In the introduction, it was argued that preference orderings should perform well on all choice sets of the form

{ C e l T " for all t>=O, O<=kt+l =(1 +rt)k~-e~,

ko given} .

(2)

r, is the net marginal product of capital in period t. (Each rt could be restricted to be positive though that restriction is not imposed in order to simplify the notation and exposition.) It is convenient to adopt the following equivalent expression for (2):

{Cel~:PC<=ko},

where

P=(Po

. . . . .

Pt ....

),

p t = ( l +ro)-i. . .(l + r t ) - l >O ,

(3) t>0 .

Refer to P as a "price profile" though it represents technological and not market trade-offs. Since the paper establishes the incompatibility of a set of axioms, the analysis is strengthened by considering a subclass of the constraint sets in (3). Moreover, one expects that equitable orderings will perform better in environments in which a positive constant level of consumption is feasible. Thus P is restricted to lie in ll++. (In that case the constant consumption level c is feasible if c
154

L.G. Epstein Two examples of elements of l~++ are given by: p , = ( l + r ) -~ ,

Vt>0,

for some

r>0,

(4)

which is the c o m m o n time autonomous specification corresponding to (1); and pt=(1 +r0)-l...(1

+rt) -1

,

where

1 -brt=o~ -t

,

t = 0 , 1. . . . .

(5)

is a fixed number in (0, 1) . In the case of (5), the net marginal product rt grows without bound as t ~ 0% possibly reflecting technical change. K o o p m a n s (1967, p. 125), in discussing the issue of existence of maximal elements, argues that ethical principles should be screened to determine whether in given circumstances they are capable of implementation. Thus we first impose the requirement that the preference ordering permit the choice of maximal elements. We require that maximal elements exist for all price profiles in 11++.

Effectiveness: For each P ~ l~++, there exists C* that is maximal in { C ~ l ~ :PC < 1 ), i.e., C* ~ l~, P C * <_1 and P C > 1 for all CE I~ such that C>-C*. Note that there is no less of generality in normalizing so that k0 = 1. Also, there may be m a n y maximal elements for a given P. C* denotes a typical maximal element and not the set of all maximal elements. 3 It is important to note that there is ethical content in the requirement that the maximal path lie in l~. A preference ordering on l~° m a y have an extension to a larger set of paths A3 l~, and there may exist a maximal path in {C ~ A : P C < 1 } which does not lie in l~. Such a path is unbounded and reflects a willingness to sacrifice the consumption of early generations for the benefit of later ones whose consumption is far larger and indeed unbounded. Such an "excessively generous" attitude towards future generations is ruled out by the effectiveness axiom. 4 Though effectiveness guarantees the existence of a maximal element C*, for any Psl~++, there remains the question of whether the path C* will actually be implemented. Presumably, the T-th generation determines consumption in period T. It does so given the capital inherited from the past and given its ranking of consumption profiles that extend into its future. Thus generation 0 can consume c~, but generation T can deviate from r C * if it is not maximal given the T-th period constraint set and preference ordering. A critical question is: what intergenerational preference ordering is used by succeeding generations ? Here it is assumed that each generation adopts the notion of intergenerational justice that is embedded in ~ and applies it to rank 3 At first glanceit might seemthat someform of continuity of ~ is beingsmuggledinto the analysisvia the effectivenessaxiom, But that is not the case. Compactness of the constraint set and closed upper contour setsfor ~ in sometopology on l~ are sufficientfor the existenceof maximalpaths, at least given completeness and transitivity. But they are not necessary for the existence of a maximal path. 4 Thereis an alternative interpretation of effectivenessin which the constraint set {C :C ~ l~, PC <=1} reflects a technological constraint. In that case there is no ethical content in the axiom and the ensuing analysis must be reinterpreted. But such technological constraints seem unnatural and contrived. Moreover, such a reinterpretationwould not be faithful to an essentialfeature of the growth model noted in the introduction, that feasibility does not rule out unbounded consumption paths.

Intergenerational Preference Orderings

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consumption paths that extend into its future. This is a natural assumption for the simple reason that the precepts embedded in ~: that are appealing or even compelling at t = 0 should be no less so when viewed from the perspective of a subsequent generation at t > 0 . It is also consistent with the social contract conception of distributive justice which is based on choice in an original position of primordial equality. Rawls (1972, p. 139) offers the following relevant comment: " . . . it is important that the original position be interpreted so that one can at any time adopt its perspective. It must make no difference when one takes this viewpoint or who does so : the restrictions must be such that the same principles are always chosen." Dasgupta (1974, p. 412) supports this view and argues that whereas claims of future generations must be taken into account in determining a just savings policy, the claims of past generations are not relevant to our sense of justice. Thus the problem facing generation T is to find maximal paths in

f C e lT • ~oo Pt + r ct < 1 - T-1 t ~ p,c* , where maximality is defined with respect to ~ . t=O T-1 0 ) Of course, 1 - ~ ptC* is the capital inherited by generation T assuming .that o previous generations have undertaken consumption levels c* . . . . , c*_ 1With this background, the next axiom m a y be stated.

Consistency: For each P e l~++, there exists C* which is m a x i m a l in { C ~ l ~ : P C <=1 } and such that f o r infinitely many values o f T it is the case that TC* is m a x i m a l in C~I~ :~p~+rct
ptc* •

0

Consistency requires that there exists at least one path C* that is maximal for generation 0 and which remains maximal for infinitely m a n y succeeding generations. A stronger axiom is obtained if "infinitely m a n y " is replaced by "all". 5 If the stronger consistency axiom is satisfied, then C* is implemented if for each T, the T-th generation picks TC* from among all maximal paths. (C* is certainly implemented if maximal paths are unique in the problems solved by each generation.) If there does not exist such a path C*, it is ambiguous which consumption program is followed through time. The preference ordering ~ induces a collection of inconsistent criteria that are applied in different generations. This inconsistency could possibly be resolved as in Phelps and Pollak (1968), Peleg and Yaari (1973), for example, but it is not clear how faithfully that resolution would reflect the ethical norms adopted for ~ . And it is the ethical properties of the final 5 Koopmans (1960)considersthe functional structure of continuous utility functions which implythe strong form of consistency. Here, the existence of a utility function is not assumed. Indeed ~ is not necessarilycomplete,transitive or continuous. Koopmans imposesthe stationarity postulate that for all c, C and C', C ~ C ' ~ (c, C)~-(c, C'). Roughly speaking, the strong form of the consistency axiom formulatedhere imposesan analogous requirementonly if (c, C') is maximalfor someprice profile. Note that not every path in 12~is optimal for some Pc l~+,even if ~ is convex and representable by a utility function. (See Kurz and Majumdar (1972) regarding the difficulties in finding supporting prices in infinite dimensional spaces.)

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consumption profiles generated by the resolution process, rather than properties of or ~-maximal paths, that would presumably be of concern to generations in the original position or to an impartial analyst. Thus in the absence of consistency a substantially different analysis seems to be required. But the principal reason for imposing consistency is simply that the purpose of the entire exercise is to find a preference ordering that will resolve intergenerationaI conflicts. If the ordering itself generates such conflicts, it is unacceptable. Indeed, on this basis the strong form of consistency described above should be adopted, but the weaker form suffices for our purposes. The following is a common formulation of equity: Equity: C ~ C' if C' can be obtained from C by interchanging the consumption levels of a finite number of generations. The next axiom imposes a weak form of monotonicity.

Monotonicity: C' >- C whenever inf (c; - G) > 0 and C' ~ C whenever inf (c; - ct) >=O. Diamond imposes the much stronger requirement that C' )> C whenever ct,> = ct for all t with at least one strict inequality.

Partial Transitivity: (i) C >- C' and inf (c~ - c't') >= 0 ~ C >- C". (ii) I f i n f (ct -c~) > 0 and if C' can be obtained from C" by a finite permutation of consumption levels, then C>-C". (iii) I f C'>-C" and if C can be derived from C' by a finite permutation of consumption levels, then C>-C". Partial transitivity is implied if C ) : C ' and C ' ~ C " , with at least one strict preference, implies C>- C". But the axiom is much weaker. It requires that C>- C" only if one ore more of the hypothesized rankings of the pairs, C, C' and C', C" is due to monotonicity (parts (i) and (ii) or equity (parts (ii) and (iii)). Moreover, part (iii) is used in the discussion below and in the proof of the Corollary, but not in the proof of Theorem 1. In the introduction, an argument was suggested in support of the next axiom.

Substitution: For every c > 0, there exists C ~ l~ with Co< c and C>-(c). It must be possible to improve upon any positive path by reducing t--0 consumption and somehow varying the consumption of later generations. If partial transitivity is maintained, then violation of the substitution property has clear implications for the policy prescriptions generated by ~ . Let substitution be violated at (c) and let P ~ 11++ be any price vector such that (~Pt)- 1 = c. Then (c) is maximal for that P. That is, as long as (c) is the largest consumption level that it is feasible to enjoy forever along a constant program, then it is maximal, regardless of returns to investment implicit in P. (If (c) is not maximal then there exists C>-(c) such that P C < 1. Sincep~ > 0 Vt, ct < c for some t or else PC> c(~p,) = 1. Let C' be obtained from C by switching consumption in periods 0 and t. By partial

Intergenerational Preference Orderings

157

transitivity, C' b-(c). By construction, c~ < c. But this contradicts the hypothesis that the substitution axiom is violated at (c).) If also ~ is monotonic, complete and if indifference is transitive, then (c) is uniquely maximal. (If C :~ (c) is also maximal, then C ~ (c) by completeness. Argue as above and apply equity and the transitivity of indifference to obtain C', such that C'~(c) and c'- (c) and yet c ~ + e < c, so that the substitution axiom is satisfied at (c), which is a contradiction.) The major result of this paper is that the axioms stated thus far are mutually incompatible. Theorem 1: There does not exist a preference ordering on l~ that satisfies equity, effectiveness, consistency, monotonicity, partial transitivity and substitution. Moreover, if ~ satisfies thefirst five axioms, then the substitution axiom is violated at the constant path (c) for each c e S, where S is an unbounded subset of (0, oo). The violation of substitution described in the Theorem is widespread in that it occurs at infinitely many constant consumption paths. Dasgupta and Heal (1979, p. 311) express the view that a refusal to trade off present consumption for greater future consumption is more appropriate for richer societies where equality considerations should dominate. It is interesting to note, therefore, that the violation of substitution is established on a set S which is unbounded above. That is, a preoccupation with equality along many, though not all, high level constant consumption programs, is implied by the remaining axioms. The proof, provided below, actually establishes a stronger version of the Theorem than is stated. For example, it assumes only that maximal elements exist and that consistency is exhibited, for all scalar multiples of a single price path P c l~++ for which Pt converges to zero "sufficiently" quickly as t ~ oo. The equity axiom could also be weakened since it is used only to establish the existence of a maximal path C*, for a suitable P, along which c* < c*+1 for all t. Moreover, it is difficult to conceive of a widely acceptable preference ordering which would not imply (weakly) increasing consumption through time for some price profile P with Pt declining sufficiently quickly. The interpretation of the impossibility result in Theorem i is open, depending on how compelling one finds the axioms. It has just been noted that both the consistency and equity axioms can be weakened without invalidating the Theorem, and there remains little with which to quarrel in these axioms. Monotonicity and partial transitivity both appear to be mild restrictions. Effectiveness has two aspects. The less contentious is the existence of maximal elements, and it has just been noted that this existence is required only for a small set of price profiles. That leaves the other aspect of effectiveness, namely that C*e l~, and the substitution axiom. The interpretation of Theorem 1 offered here is thus that society must choose between these two precepts. Theorem I is "tight" in the sense that if any one of the axioms is deleted, then the remaining precepts can be satisfied. (This is demonstrated in an earlier working paper version of this paper that is available from the author upon request. 6) If the There is one exception. An example has not been constructed to show that all but the partial transitivity axiom can be satisfied.

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L.G. Epstein

substitution axiom is deleted, the remaining axioms are satisfied by the maxi-min criterion, 7 defined by C ~ C' ¢*- infer_-> inf c~ . Indeed, the maxi-min ordering is the only ordering which satisfies both the first five axioms in the statement of Theorem 1 and appropriate homotheticity and continuity requirements, as proven in the cited working paper.

3. Concluding Remarks This paper has described some trade-offs between ethical precepts that are unavoidable in environments where a positive constant level of consumption is feasible. To conclude, consider decision problems in which such paths are not feasible. Not surprisingly, equitable preference orderings perform even less satisfactorily in such environments, as the final theorem shows.

Theorem 2: Let ~ be a preference ordering on l~ that satisfies equity, monotonicity and partial transitivity. Let P = (Po . . . . . Pt . . . . ) be such that ~ P t = oe and Pt > Pt +1 >O for all t. I f C* is maximal in { C e l ~ : P C < l } , then C * = 0 . Rates of return to investment implicit in P are positive since prices are falling though time, but they are not so large as to permit a positive constant level of consumption to be sustained (since ~p~ = oe). For such a price profile, only 0 can be a maximal path. In particular, no maximal path exists if monotonicity is strengthened to the form adopted by Diamond and stated in Sect. 2.

Appendix Proof of Theorem 1 : Pick PeP+ + such that ~ pt/PT*O as T ~ . (For example, T+I adopt (5).) By effectiveness and consistency, there exists a maximal and consistent path C*. It is straightforward to show that c*
Vt~0 .

(6)

(Suppose to the contrary that c* > c*+1 for some t. Let C be the path obtained from C* by switching consumption in periods t and (t+ 1). By equity, C ~ C * . Moreover, since Pt >Pt+l, PC* - P C = (Pt+l -p~)(e*+l - c * ) > 0. Thus 3e > 0 such that P [C + (e)] < PC* = 1. C + (~) >- C by monotonicity. Therefore, C + (e) >- C* by partial transitivity, which contradicts the maximality of C*.) Since C* e l~, sup c* < oo and so g - l i m e * exists. By monotonicity C* + 0 and so ? > 0. In short, c* converges monotonically to the positive limit ?. The substitution axiom is violated at the constant path (~). To prove this, suppose to the contrary that 3 C such that C>-(e--) and Co< ~. By monotonicity, (e--)?eTC*VT. Thus by partial transitivity it follows that C ~ T C * V T . C* was 7

See footnote 1.

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chosen so that for infinitely m a n y values o f T, TC* is maximal in the set oo. C' ~ l+. ~Pt +rct! <=1 -

ptc* By taking subsequences, it can be assumed that 0 the maximality holds for all values o f T. Since C is preferred to rC* it c a n n o t lie in co

the above feasible set; that is, Simple algebraic manipulations yield

T-1

oo

Zp +T>I- Y p,c* 0

0

Yr. 0

~3

(Co - c*) + ~,Pt +r (ct - c*+r)/Pr > 0

VT.

(7)

1

The first term in (7) is negative for large T since Co -c*--->Co - ~ < 0. The second term co

is b o u n d e d in absolute value by (sup c~+ ~). ~,p~ +r/Pr, which converges to zero as 1 T ~ oe by the choice o f P. Thus (7) is impossible and the substitution axiom c a n n o t be satisfied at (0. It remains only to prove that the substitution axiom is violated at (e) for any c e S where S is u n b o u n d e d . Argue as follows : Let P be the above price profile and for any w > 0, let P/w be the indicated scalar multiple o f P. Denote by C* (P/w) a maximal consistent path for the price profile P/w. The above arguments apply virtually unaltered. Thus maximal c o n s u m p t i o n profiles converge m o n o t o n i c a l l y to limiting values ((P/w), and the substitution axiom is violated at the constant path (6(P/w)). Define the desired set S as S - { ~ ( P / w ) : w > 0 } . If S is b o u n d e d then S C [0, fl] for some fl > 0. Pick w > 2ft. ~ P t . Then the constant path (2fl) is feasible given the price vector P/w. Also, c* (P/w) < (( P/w) < fl < 2 fl V t. Thus (2 fl) >- C* (P/w) by monotonicity, which contradicts the maximality o f C*(P/w). Hence S is unbounded.

Proof of Theorem 2." Precisely as in the p r o o f o f Eq. (6) show that c* 0 for some T_->0. Then c* > c~ > 0 V t > T. Thus ~ptc* = oo which contradicts the feasibility and hence maximality o f C*.

References Calvo G (1978) Some notes on time inconsistency and Rawls' maximin criterion. Rev Econ Studies 45: 97-I 02 Campbell DE (1985) Impossibility theorems and infinite horizon planning. Soc Choice Welfare 2: 283-293 Dasgupta P (1974) On some alternative criteria for justice between generation. J Public Econ 3:405-423 Dasgupta P (1974) On some alternative criteria for justice between generations. J. Public Econ 3:405-423 Dasgupta P, Heal G (1979) Economic theory and exhaustible resources. University Press, Cambridge Diamond P (1965) The evaluation of infinite utility streams. Econometrica 33:170-177 Koopmans TC (1960) Stationary ordinaI utility and impatience. Econometrica 28:287-309 Koopmans TC (1967) Intertemporal distribution and optimal aggregate economic growth. In: Ten economic studies in the tradition of Irving Fisher. John Wiley and Sons, New York Kurz M, Majumdar M (1972) Efficiency prices in infinite dimensional spaces: A synthesis. Rev Econ Studies 39 : 147-158 Peleg B, Yaari M (1973) On the existence of a consistent course of action when tastes are changing. Rev Econ Studies 40:391-401

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Phelps E, Pollak R (1968) On second-best national savings and game equilibrium growth. Rev Econ Studies 35:185-199 Phelps E, Riley J (1978) Rawlsian growth: Dynamic programming of capital and wealth for intergeneration 'maximin' justice. Rev Econ Studies 45:103-120 Rawls J (1971) A theory of justice. Harvard University Press, Cambridge, Mass Sen A (1977) On weights and measures: Informational constraints in social welfare analysis. Econometrica 45:1539-1572 Svensson LG (1980) Equity among generations. Econometrica 48:1251-1256