Utility functionals with nonpaternalistic intergenerational altruism

IOURNAL OF ECONOMIC THEORY 49, 241-265 (1989)

Utility Functionals with ~o~~ater~alistic Intergenerational Altruis

HAJIME How

Faculiy of Economics, Tohoku lJniversit.v, Sendai 980, Japan

AND

SADAO KANAYA

Faculty of Economics, Tokyo Metropolitan Uhmsity, 1-I Yakumo 1-chome, Meguro-ku, Tokyo iS2, Japan

Received July 18, 1987; revised December 20, 19gg

The purpose of the paper is twofold. The tirst purpose is to investigate the conditions for the existence and uniqueness of stationary utility functionals which embody a two-way non-paternalistic intergenerational altruism. The specific forms of the utility functionals are also derived in the linear case. The second purpose is to investigate whether these functionals can generate consistently optimum distribution programs. Using a simple overlapping generations framework, it is shown that the answer to this question is negative as long as the younger generation’s utility is’ given any weight in each period’s objective functional. Journal of Economic Literature Classification Numbers: 022, 024, 113. I? 1989 Academic Press, k~c.

I. INT-R~DLJcT~~N

In recent years it has been increasingly realized that intertemporaI resource allocation is also an integernerational resource allocation and must be analyzed in terms of utility functions which explicitly incorporate intergenerational altruism. Notable examples are the analyses of bequest behavior pioneered by Strotz [30] and developed by Whelps and F’~Ilak [25-j, Kohlberg [18], Goldman [12], Ray and Rernheim [27], Leininger [21], EIarris [14], and Ray [26], of intergenerational justice by Arrow [2] and asgupta [lo], of the neutrality of government bonds and of the

* The authors thank the participants of various seminars, particularly K. fwata, H. Ryder, and D. Fudenberg, for useful comments. They also thank two anonymous referees for many useful comments and suggestions. The first author’s research was partially tinanced by a grant from the Japan Foundation for the Promotion of Economic Research.

241 @O22-0531/89 $3.00

Copyright :c 1989 by Academic Press, Inc. All rights of reproduction in any fern resewed.

242 HORI AND KANAYA

steady-state interest rate by Barro [3], Carmichael [9], Burbidge [7, 81, Buiter and Carmichael [6], Tsukamoto [31], Abel [ 11, and Bernheim and Bagwell [.5], and of the perpetuation of inequality by Laitner [20] and Loury [22].

Two different approaches have been adopted by these authors in specifying intergenerational altruism. The first is to directly assume that each generation’s utility depends on the consumptions of its own and other relevant generations. The second is to assume that the altruism is non-paternalistic so that each generation’s utility depends on its own consumption and its neighboring generations’ (i.e., children’s and parents’) utility levels through what Koopmans l-191 called an aggregator.

Although the second approach has a strong intuitive appeal, an aggregator itself is not a utility function. Rather, what it specifies is a (non-autonomous) difference equation which the sequence of the utilities of successive generations has to satisfy. Therefore, in order to apply the aggregator approach to intergenerational altruism, we have to resolve the issue of the existence and (possibly) uniqueness of a solution for such a difference equation.

The purpose of the present paper is twofold. The first purpose is to consider the existence and uniqueness of a solution for the difference equation defined by an aggregator. For the case where altruism extends only from parents to children, this issue was considered by Lucas and Stokey [23] and Streufert [29], For the more general case where altruism extends both ways between parents and children, partial analyses were given by Hori [15], Kanaya [16-j, and Kimball [17], under the additional assumption that the aggregator is linear. The present paper follows the latter line of investigation and considers both linear and nonlinear aggregators.

Roughly speaking, a solution of the above difference equation is a utility functional which relates the consumption stream of past, present, and future generations to the utility of the present generation. Thus one can define each generation’s optimum intergenerational income distribution program in terms of this utility functional. The second purpose of the paper is to consider if different generations’ optimum programs are dynamically consistent in the sense of Strotz [30]. Streufert [29] gave an affirmative answer to this issue for the case where altruism extends only from parents to children. It will be shown below, however, that this affirmative result does not generalize to the case where altruism extends also from children to parents.

The paper is organized as follows. Sections II through V deal with the existence and uniqueness of solution functionals to difference equations delined by aggregators. Section II clarifies two solution concepts. Sections III and IV consider linear and non-linear aggregators, respectively, and Section V makes some remarks concerning the nature of solutions.

INTERGENERATIONAL ALTXuI§M 243

Sections VI through VIII consider the dynamic consistency of optimum programs. Section VI defines the concept of consistency. After a lemma in Section VII, Section VIII gives the consistency an inconsistency res

II. Two SOLUTION CON~EFTS

The starting point of the following analysis is the postulate that each generation’s utility depends positively on its own consumption an neighboring generations’ utility levels and that this dependence is stationary. For tractability, we also postulate that each generation’s own consumption affects its utility separably.

I,et C~ be the fth generation’s own consumption vector, let I’ be a stationary function that summarizes the separable effects of own consump tion on the utility, let P’Z = P’(c~), and let .CJr be the ftb generation’s utility level. Then the above postulates can be represented by an increasing function G: R3 3 R such that

We will call V (and V!) consumption utility and G an aggregator. As was noted in the Introduction, (1) defines a second-order non

autonomous difference equation that regulates the behavior of the U*‘s. Since the equation is non-autonomous, this behavior will depend in the infinite sequence of the Vf’s. Thus, roughly speaking, solutions of (1) are functionals which are defined on a certain set of infinite sequences an specify the dependence of the Uis on the vt’s in such a way that (1) E identically met. But apparently such a definition is still too broads Two issues are involved.

First, the standard approach to picking one among many of the possible solutions of (1) is to specify initial, boundary, or transversality conditions which the sequence of the Ur’s has to satisfy. In the present context? however, what these conditions mean economically is not clear. Since, by assuming the stationarity of G and V, we are essentially assuming that preferences do not vary among different generations “)Y a better approach would be to seek stationary functionals that satisfy (1) identically. adopt this latter approach.

i This is of course a strong assumption. However, for the practical purpose of economy- wide intergenerational income redistribution, through social security for example, such an assumption seems almost inevitable. If the social welfare function to be adopted in such a context is to be based on the actual preferences, to foresee future generations’ preferences correctly will be impossible. It seems that all one can do is to hope that intergenerational altruism is so intrinsic in human nature that it does not change much over time, and when a substantial change occurs, modify the plan accordingly.

244 HORI AND KANAYA

Second, there is the issue of how to treat past generations, One natural way would be to regard past generations’ utility levels as givens. If the framework is that of non-overlapping generations where each generation lives only one period, then CJ- i is given when the tth generation makes the choice. In this case, since all the consumption utilities from the lth period on affect Ut while the utility levels of all the past generations except the immediate one are irrelevant by (l), the tth generation’s utility functional will look like

The usefulness of this expression is not restricted to the case of non- overlapping generations; as long as each generation lives only finite periods, a successive application of (2) yields the relevant utility functional. Suppose each generation lives two periods and let et = (cj, cf), where et is the tth generation’s consumption in its ith period. Then substituting Ut-l=4(Ut-l, T1)=4(Ut-*, V(C~-~,C~-~), Win (2), we obtain

where, for the tth generation in its lirst period, Utpz and ciP i are givens while cf- i and V are variables.

But this is not the only way to treat past generations. Each individual may possibly be so obedient that he may care how his parents would respond to his and his descendants’ future consumption plans if they were alive. In this case, parents’ bodies die but their souls live eternally, and it is no longer possible to regard past generations’ utility levels as givens. Since past generations’ utilities also depend on V( ~ i , I’r - z, and so on, the tth generation’s utility functional will look like

where t-lv= {vs}:Lm, and (‘~ ’ v Vl ; V + ‘) is the inlinite sequence IVslFLm viewed from the tth generation’s standpoint.

Therefore, in the following, we look for stationary solutions of (1) which are either of the form (2) or of the form (4). Moreover, it will be natural to impose two additional restrictions on CJ~ and $: (i) 4 and $ are well- delined for all bounded sequences, and (ii) C$ and $ are increasing in each argument. For convenience, we shall call a solution of the form (2) a mortality solution, and a solution of the form (4) an eternity solution.

INTERGENERATIONAL ALTRUISM 245

III LINEAR AGGREGATORS

In this section, we consider the problem under the linearity assumption for aggregators:

Al. G(U, V3 U’)=aU+bV+cU’, a>O, b>Q c>O.

rJnder this assumption, we can obtain explicit forms of the solutions of (I ). We ln-st consider the mortality solution of (1). The following results are

basically due to Hori [15] and Kanaya [ 161.

THEQREM 1. Suppose G satisfies A.1. Then a linear mortality Jolution q‘ (1) exists if ancl Only if

(a,c)eAuB

and is given by

where A={(a,c):a>O,c>O,a+c<l}

u{(a,~):a>O,c>O,a+c=l~c<$j,

and

l+Jl-4ac 2a

ProoJ Suppoe a solution of (1) is linear so that

where y5 6, and the fs are constant coefficients to be determined. Substitute u ~+1=yU~+~~Xo/?~V~+1+~+8forU~+1in(l)andsolvefor U(toobtain

In order for the right-hand sides of (7) and (8) TV be identical, it

246 HORI AND KANAYA

to hold that y=a/(l-cy), 8=&j/(1-cy), /$,=b/(l-cy), and p3= (c/(l-cy))j3e1, ~=1,2 ,.... Thus

l3= 0 if c# 1 -cy arbitrary if c= 1 -cy (9)

Furthermore, in order for the right-hand side of (7) to be well-defined for all the bounded sequences ( lJ- r, { Vt+ S}FE 0) and increasing in each argument, it is necessary and sufficient that

O<L< 1. 1 -cy (10)

Combining (9) and (lo), we obtain the theorem. 1

This theorem implies that a linear mortality solution of (1) is unique if u + c < 1. Although some strengthening of this uniquenes result is possible, using the notion of differentiable functionals,(2) it is an open question whether one can obtain a uniqueness result without any additional requirement on the admissible class of solutions.

We now turn to the eternity solution of (1). Part of the results prese,nted below were also obtained by Kimball [17].(3) As in the Introduction, the

* Let 1 a denote the space of bounded sequences of real numbers endowed with the supremum norm and let ~~.~~ denote the norm. Then a functional 4 dellned on 1 m is said to be dqferentiable (in the sense of Frechet) if, at each x61m, there is a continuous linear functional 4’(x) (continuity being defmed in terms of the supremum norm on lm) such that

lim l~(x+~)-~(x)-~‘(x~.~l=~ IIM -II IVII

where d’(x). h is the value of the linear functional 4’(x) evaluated at h 6 lw. (See Dieudonne [ll], p. 149, for example.) With this notion of differentiability, Theorem 1 can be strengthened to the following: Suppose G satisfies A.1. Then a d$ferentiable mortality solution of (1) exists zf and only zf (5) holds. Moreover, such a solution is linear and thus given by (6) lj” (a, c) E ,4. The proof of this proposition is available from the authors upon request.

’ Kimball [17] asserts that (I) has linear solutions if and only if ac c l/4, and that there are two degrees of freedom in choosing a solution. See his Proposition 2. But, except the one given in our Theorem 2, these solutions are well-defined only for sequences { V!} y= ~ such that VC converges to zero exponentially as t + oo or t + - co. In Proposition 3, he obtains a uniqueness result using the additional assumption of “bounded” altruism, while we obtain a uniqueness result by requiring that a solution be well-detined for all bounded sequences { V,} y= ~~. Moreover, while he implicitly assumes the linearity of a solution throughout, it is a consequence in our approach.


infinite sequence { V3j;= ~ * viewed by the tth generation will be denote by (I-’ V; Vr; V+ I).

THEOREM 2. Suppose G satisfies A.1. Thea an eternity solutiorz of ( exists ij and mly $

aad is uniquely given by

where

Ct= l-&zG 1-%JGiz 2c

and P= 2a .

Prooj We will iirst prove that (12) is a solution of (I) if a + c c 1. IY will also prove the “if” statement of the theorem. Using the easily ve~i~ab~~ relations

m2-x+a=O and afi2-/3+c=O, (13)

one can check that (12) satisfies (1) identically. Moreover, a + c c 1 im that

l-4ac>O, O<lX<l, and O<j?<l. (141

Therefore the $ given in (12) is increasing in each argument and we& defined for any bounded sequence { Vr} y! - ~ ~ Thus it is a solution.

Next, in order to prove the “only if” statement of t (I) has an eternity solution $. Let {V} denote t with VT = V for all t and let U(V) = $( { V}). Since $(rPIV; Vl; V+‘)=tJ({V})=U(V) for all f, we obtain

(l-a-c)U(V)=bK (15)

But U(V) is increasing in V because so is $ in each argument. Therefore (15) implies that l>a+c.

Finally, uniqueness results from Theorem 4, to be proved later.

Two remarks seem in order at this point. First, a comparison between (6) and (12) shows that the distinction

between a mortality and an eternity solution is not superficial. In fact, the margina rate of substitution between the consumption utihties of its own

248 HORIANDKANAYA

( vt) and its parent generation ( vrmr) is c! in the eternity solution (12), while it is u/?/(uf12 + c) in the mortality solution (6). But E = u/?/c and thus

4 CC>7 c@ +c.

Namely, the eternity solution places more weight on the parent generation’s consumption than the mortality solution does.

Second, the difference of the existence conditions for a mortality solution (Theorem 1) and an eternity solution (Theorem 2) can be explained as follows. As was argued by Becker [4], there is an infinite regress of utility interactions among generations: P’t affects Ut which affects U*+r which in turn affects Ut and so on. In order for this inlinite regress to converge, some restriction on the degrees of altruism, a and c, is necessary. In the case of a mortality solution, only the existing and future generations inter- act, while in the case of an eternity solution, past generations also enter the interactions and therefore the restriction on u and c becomes more stringent.

IV. NON-LINEAR AGGREGATORS

We now give an extension of our existence and uniqueness results to non-linear aggregators. For this purpose we assume that:

A.2. G is increasing in each argument,

G(0, 0,O) = 0,

and moreover, there exist u > 0, b > 0, and c > 0 such that

lG(U, V, U’)-G(& 7, t?‘)[ <u /U- 01 +b IV- vi +c lU’- i??

and a+c<l.

The results concerning the mortality and eternity solutions are given in Theorems 3 and 4. The tool for the proof is Banach’s Contraction Mapping Theorem, which was applied by Lucas and Stokey [23] to a similar problem for the case where altruism extends only from parents to children. For the Contraction Mapping Theorem and its proof, see, e.g., Smart [28].

We first consider the mortality solution. Note that we want a solution 4 to be well-defined for all the bounded sequences of the form (llJ+ l, V). We can identify the set of all these sequences with I *, which was defined in footnote 2.

INTERGENERATIONAL ALTRUISiM 249

THEOREM 3. g G satisfies A.2, then a mortality sohAm 4 oi (I) satisfying

exists and is unique. Moreover, such a solutioti is continuous with respect to the I m-nornl.

- - and let B( U3 V) be the set of functionals $ defined on X(U, V) and satisfying (16) and

Then with the norm

- - B(U, V) is a closed subset of the Banach space consisting of ah the - - bounded functionais detined on X( U, V), and is therefore a complete metric - - space. Moreover, B( U, V) is nonempty because the functional & defined - - - - &U, I”)=0 for all (U, V’)EX(U, V) is in B(U, V). - - - -

For each 6 E B( U, V), define a functional T$ on .X( U, V) by

In the right-hand side of (18), 4(&U, VI)? V*) is well-defined for aI1 - - ( U, V’ ) E X( U, V) because

and because, in view of (17) and Da bV/( 1 - c - a),

- - for all (U, V’) E X( U, V). Therefore TJ is also well-defined.

250 HORI AND KANAYA

Now, by A.2 and using (17) twice,

Since u + c < 1 implies that c + (1 - &%k)/2 < 1, it follows that T is a contraction. Therefore, by the Contraction Mapping Theorem, T has a - - unique lixed point in B( U, V). Let it be denoted by dcD, vj. By the construc- tion of T, $= dcD, vj satisties

- - for all (U, V')EX(U, V).

In order to see that dcD,vj can be uniquely extended to the whole space of bounded sequences, let 0’ and p’ be such that p’> v and n’a max( D, b V’/( 1 - c - a)). For such a (D’, P’), we can again obtain a unique fixed point 4cD,,v,j E B(D’, V’). If we restrict the domain of dcO, Pj to - - X(U, I’), then such a restriction is clearly a member of B(U, I’) and - - satisfies (19) for all (U, V’) EX(U, I’). Since a fixed point of - - - - T: B( U, I’) + B( U, V) is unique, this restriction coincides with dcD, vj; in other words, dcD,,Pj is a unique extension of #cD,vj to X(D’, P’). Since (D’, P’) is arbitrary, we can extend $cD, vj uniquely to Ia this way. Let 4 denote the unique extension. It is clear that 4 satisfies (16), (17), and (19) for all (U, V1)elm.

INTERGENERATIONAL ALTRLXW 251

To show that the C$ thus obtained is a solution of (1 ), it remains to show that it is increasing in each argument. For this purpose, recursively defme - - a sequence of functionals { @} z= O on X( UY V)

where T@-’ is defined by (18). This sequence converges to the unique fixed point $CD,V,. (See Remark 1.2.3. of Smart [28].)

Now note that each 4’ is non-decreasing in each argument. In fact, this is trivial for n = 0. Suppose $n is non-decreasing for some n. Then sime @+‘(U, V’)=G(U, VI @(d’(U, VI), V2)) and since G is mcreasmg, @+I is non-decreasing~

This clearly implies that $CD, PJ is also non-decreasing. Let dCO, Vj = $ for notational convenience. Since $ satisfies (19) and G is mcreasing> &U, VI ) is increasing in U and VI. Suppose $ is increasing in Vt, t = 1, 2, .~.? 3, and let (ill5 VI) and (U, VI) be such that Vf= vi for t#,r+ I, Vs+l> ps+l~ Since Vs+ l is the sth term of V2 and since $(U, VI) 2 &U, VI) because 7 is non-decreasing, we obtain

by the inductive hypothesis. Therefore, since G is mcreasmg m eat argument,

- - Thus $ = dCU Vj is increasing. Since this is true for an arbitrary ( UY V)? t# is increasing. This shows that a mortahty solution of (I ) satisfying (16) exists.

In order to show that a solution of (I) satisfymg (16) is umque, it wih suffice to show that such a solution also satisfies (l7), because we know from the existence proof that a solution of (I) satisfying (14) and (17) is unique. For this in turn it will suffice to show that if a solution 4 of (I) satisfies (16), then

252 HORI AND KANAYA

for any U, U’, V1={Vz}~=i, and V”={V;}y=r, where /I= (1 - dG)/2~. In fact (20) and (21) imply that

which implies (17). As a byproduct, (21) also implies the continuiuty of 4. To prove (20), note that by A.2 and (16)

km 011 G c ldcwk Oh 0) - w4 ON + c Ma 011 < ((I- 4=3/2 + cl IqqO, O)l.

Since c + (1 - Ji?G)/2 < 1 if u + c < 1, (20) follows. In order to prove (21) we will first prove that, if V’ and Vf differ only

in their fth terms, denoted by Vt and Vi, then

The proof of (22) will be by induction on t. For t = 1, by A.2 and (16),

from which (22) follows. Suppose (22) holds for some t > 1. Let V2 ad V: be the sequences that are obtained by eliminating the lirst terms from V’ and Vi. Thus V2 and Vf+ l differ only in their tth terms, which are Vr + l and V;+l. By A.2, (16), and the inductive hypothesis,

l&V w-N4 C+J

from which (22) follows for t + I also. Finally, given V1 = { V3}FC, and VI’= { V~}~cl, let

INTERGENERATIONAL ALTXUISM 253

Then Vi’ and Vi’+I differ only in their (? + 1 )st terms. Letting VA’ = VI for convenience, use (16) and (22) in

to obtain (21). In Theorem 3, we had to impose an additional requirement, inequality

(16), m order to obtain uniqueness. Also, a comparison of Theorems I and 3 suggests that the existence condition in Theorem 3 (namely Q + c < I) may be more restrictive than is actually necessary. The fatter restriction was necessitated basically because we resorted to the Contraction ~a~~i~g Theorem. Whether uniqueness can be obtained without an additional requirement such as (16) and whether the existence condition of Theorem 3 can be relaxed are open questions.

We next turn to the eternity solution of (I) under assumption A~2. will identify the set of all the bounded sequences of the form { Vllyz ~~ with lm.

THEOREM 4. If G satisfies A.2, ( 1) has a unique eternity solution and the soiution is continuous with respect to the I m-nor~~.

Prooj Let Y(P)= {{Vt}yzpm : IVJ < Vfor all t] and let set of all the bounded functionals on Y(P) endowed with the supremum norm. Then B(p) is a nonempty Banach space. For each $ E B(V), define T$: Y(Q+I? by

(T$)(-‘V; Vo; V1)=G($(-2V; Vpi; V’), Vo> $(‘V; V,; If’)).

Then T$ E IZ( P) and T maps B( r) into itself. If $I~, $* E

Thus T is a contraction and has a unique fixed point r/C V, E That I/I(~) can be extended to a unique $J: /m + .R and that $ is

increasing in each argument can be proved analogously to Theorem 3. Next, in order to show that $ is continuous, let C(V) be the set of con-

tinuous and bounded functionals on Y(v) endowed with the su~rem~m norm. Then C(P) is a closed subset of B( V) and is a complete metric space. We can regard the T constructed above as an operator mapping C(p) into itself. Thus T has a fixed point in C(V). Since C(V) cz l?(p) and s&e a fixed point in B(V) is unique, the fixed point of T in C(P) coincides with $cVj and therefore $cVJ is continuous. Continuity of $ follows from this.

254 HORIANDKANAYA

Finally, to show that an eternity solution is unique, let $ be an eternity solution. Then, since $ is well-defined and non-decreasing on I a by definition of a solution, it is bounded on Y(v) for any ra 0. Thus its restriction to Y( P), denoted by g(F), belongs to B(P). By the uniqueness of a fixed point in Z?(P), it follows that I,&( rJ = +( vj. Thus $ = $. fl

V, SOME REMARKS

Before proceeding to the next issue, some remarks may be in order concerning the nature of the solutions obtained in the previous sections.

The first remark concerns the domain of definition of the solution functionals. For convenience, we have made the requirement that solutions be defined for al the bounded sequences of vr. But this requirement may be too strong; for example, the consumption utility v may be a bounded function, or the technologies may be such that there is a known bound on the feasible streams of future consumptions. In such a case, it is possible to replace the domains of solutions by smaller sets without changing the results. The simplest is the following. Let Z be an arbitrary interval in the real line and say that { Vt}yc l (or { Vt}yf em) is bounded in Z if there exist y and P in Z such that JJ< vt < v for all t. Let X(Z) and Y(Z) be the sets of all the { vt}yz 1 and { vr} y= -~ that are bounded in Z. Then we can replace the domains of 4 and ti by Rx X(Z) and Y(Z).

In fact, all the suficient conditions for the existence of a solution are obviously still valid after this replacement. The necessary condition for existence and the uniqueness result in Theorem 1 are based on the linearity of solution functionals and are valid even if 4’s domain is replaced by Rx X(Z). The necessary condition for existence in Theorem 2, which was derived from (14), is based on the requirement that tj be increasing and its domain contain multiple constant sequences. But this latter requirement is satislied by Y(Z). The uniqueness result of Theorem 2 is based on the increasingness requirement on $, which implies that the sequence {Uz}yzPa delined by Uf=i,QPiv; vt; v’+i) is bounded if {vz}Fz-a is bounded by & vz < v and if the constant sequences defined by y and F are both in the domain of $. This implication is still valid if I,& domain is replaced by Y(Z). Finally, the uniqueness results of Theorems 3 and 4 are still valid if ~~5’s domain is replaced by R xX(Z) and I,!?s domain by Y(Z) because we obtained the uniqueness by first considering bounded domains and then enlarging them. These bounded domains can be replaced by bounded subsets of R xX(Z) and Y(Z).

The second remark concerns an ordinal extension of our results. In the text, we started from some given G and P’, and supposed that these given


G and P’ satisfy Al or A.2. But this supposition is not essential an ordmal extension is possible.

To see this, first note the following. Let G* and P’* be given, let I* be the smallest interval such that v*(c) E I* for ah conceivable C’S, Let X(1*) and Y(I*) be as above, and let U* denote the utility that satisfies (I ) with respect to G* and P’*. Let f and I? be arbitrary continuous and increasi transformations of U* and k’* and let U and P’ be such that U* =j( and v* = /I( P’). Define G by

Then, with z* = arbitrary interval, qS*: Z* x X(1*) + a mortahty (an eternity) solution of G* if and on (*: Y(I) +R) is a mortality (an eternity) solution 0 Z=f-l(Z*), I=k’(Z*), and, with the notation Ir((v)= and NV= {~~~s~}3~t~

From this we can deduce the following. Let G* and Y* be given suppose that there exist continuous and increasing transformations j a such that the function G defined above satisfies A-1 or A.2~ Then the results for the solutions of G given in Theorems I through 4 have straigbtforwar~ imphcations for the solutions of G*.

VL THE NOTION OF CONSISTENCY

We now turn to our second issue: If each generation chooses an mter- generational distribution program which is optimum in terms of a utihty functional derived above, will different generations’ optimum programs consistent with each other?

The notion of consistency we adopt is due to Strotz [30].4*s In order to define it precisely, we need some more specifications of the model.

4 A related notion of consistency of preferences was proposed by Hammond [ 131. Str~tz’s notion of consistency is detined in terms of a particular program and is therefore weaker than Hammond’s, in the sense that the existence of a consistently optimum program does not imply the consistency of preferences in Hammond’s sense.

‘It was shown in Streufert [29] that, as long as each generation cares only about its descendant generations, dynamic consistency holds if and only if (i) the aitruism is of a nonpaternalistic nature and (ii) each generation cares only about its direct descendant generation. The following analysis considers how the introduction of the parent generation affects dynamic consistency.

256 HORI AND KANAYA

In order for the two-way altruism between the parent and child generations to be meaningful, the model has to allow the possibility of transfers in both directions, which in turn requires that at least two generations coexist in each period. Hence in the following we adopt the simplest overlapping generations framework in which each generation lives two periods.

To specify the technical constraints, let ci be the lth generation’s consumption vector in its ith period, i = 1,2, C* the total consumption goods vector in the tth period, and kr the capital goods vector in the lth period. Let Pz(k) be the set of consumption goods vectors and capital goods vectors that are producible from a capital goods vector k in the fth period. We say that a sequence of consumption goods vectors {CS}Tct is (k, t)-&zdde if there is a sequence of capital goods vectors {ks}Trl with kt = k such that (CS, ks+ i) E Ps(k3) for all s > t. We also say that a distribution program (eye i , { (ci , c:)}F= z) is (k, t)-@asibZe if there is a (k, t)-feasible sequence of consumption goods vectors { CS} p! ~ such that

c:- 1 + c; = cs, s= t, t+ 1, . . . .

Although this feasibility constraint on intergenerational distribution programs is couched in terms of the production possibility set of the whole economy, it can also be interpreted as a budget constraint facing each price-taking infmite line of family.6 Thus let uz E R be the asset carried over from the (t - 1)st to the tth periods, We, rr, and pf be the wage rate, interest rate, and price vector expected (correctly) to prevail in the tth period, and let

Then Pf(k) = {(C, k’) 1 k’ = (k -ptC)(l + rz)}, and a (k, t)-feasible distribution program is an intra-family distribution program (cf- r , { (ci , cz) } F! 0 satisfying the budget constraint

Let ~ttcf-l, {C c:, CT:)}:! ,) denote the objective functional in the tth period. We say that a distribution program (z;- r, {(F:, I?:)};=~) is (k, t)-

‘However, a “family line” is not a well-detined concept under the currently prevalent inheritance system. For example, if a family has more than one child, which child’s utility is to be represented by fit + t in U? = G( UC - t , Vt, fir + r )? Also to be noted is the public goods property of parents’ and children’s utilities, pointed out by Nerlove, Razin, and Sadka [24]. One institutional setting which is free of these difticulties is the system of premogeniture.


optimum with respect to Hz (or simply (k, t)-optimum) 8 it is (k> t)-feasible and if for any (k9 f)-feasible program (c;- i, {(c: ~ c:) 1 F= 1),

Let the m-esent period by denoted by t = 1, let kl = k be given, and let (Ci, {(Ci, Fz)]FE1) be (k, 1)-optimum with respect to H1. (Fi, {(Fi, E~)~~zl) is consistently optimum with respect to a sequence Qf objectice functionals { HS 1 SmC i ( or simply coizsistently optimum) if there is an associated sequence of capital stock vectors {EJjb?= 1 with El = Jz such that the subprogram (Cf-i, {FJ, Ft}FSt) is (Et, t)-optimum wit for all t > I.

It remains to relate the objective functional Ht to the utility functionals we have derived in the first part of the paper. Since two generations coexist in each period, in most of the rest of the paper we will take either the older generation’s utility functional (Theorem 5) or the younger generation’s utility functional (Theorem 6) as each period’s objective functional. In each of these two cases, we will consider both a mortality solution and an eternity solution.

Although this assignment of objective functionals to periods is rather artificial, it sheds light on where different generations’ interests agree, where they disagree, and when the disagreement gives rise ‘to dynamic inconsistency. In fact, the intuition given by Theorems 5 an 6 leads to a more general inconsistency result in Theorem 7+ where a ergsonian welfare function for the two coexisting generations is taken to be the objective functional.

VII. A LEMMA

This section presents and proves a lemma which we need in treating the eternity utility functional.

LEMMA. Suppose A.2 holds and Iet $ denote the eternity solution of (I ). Then tkere is a functional 6: ia -+ R wkich is increasing in each argument and satisfies, for all { Vt} y= ~ ~ E I =,

I)-lv; vo; vl)=e(ov;$(ov; v,; P)).

Proo$ We first assert that there is a functional Q: lm x increasing in each argument and satisfies, for all (‘Y, U’EK,

19(‘c U) = G(Q( -‘K d(‘t U)), Vo, U)

258

and

HORIAND KANAYA

p(ov; U)-tqOV; U’)l < l-41 -4ac

2u IU- U’i. (25)

The proof of this assertion is analogous to the proof of Theorem 3. Let X(Q={{IJ(}y=- m:supi ~~~~~},z(~)={u:~u~~~~/(l-c-u)}, and let B(P) be the set of functionals g: .X(v) x Z(p) -P Z(v) satisfying (25). Define an operator r B( p) -+ B( P) by (7’@(‘K U) = G(& ~ ’ v; g( ‘I’; U)), vo, U). Then T has a fixed point 13~~), which can be extended to an increasing functional (3: Z”’ -+ R. A detailed proof is omitted.

We next show that there is a functional $: Za + R such that

(G-v; vo; V)= e(Ov; lJi(Olq VI; P)). (261

The proof of (26) also uses the Contraction Mapping Theorem. Let Y(F) and B(F) be as in the proof of Theorem 4, and define an operator TdI(~)+B(~) by

(T+')(-'V; Vo; V1)=6(oP'p,Y(oP'; VI; V2)),

By (25) for any rj’, ij* E B( v),

II W1 - W*ll < 1-Jl-4U

2~ IW -be

Since (1 - dG)/2a < 1 if u + c < 1, T is a contraction and has a unique fixed point which satislies (26). Since v>O is arbitrary, this fixed point can be extended to lm, and this extension gives the desired $.

Finally we show that 0 satisfies (23) by showing that $ = *. In fact, using (26) and (24)

iJ( -Iv; vo; VI) = e(v; tJ(V; vl; V))

= G(O(-lb’; 6(‘P’; $(‘V; Vl; if*))), Vo, $(‘I,‘; VI; V*))

= G(d( -‘K $( -‘V; Vo; V)), Vo, I&OK Vl; V*))

=G($(-*Y El; V’), Vo, $(‘V; Vl; V*)).

Therefore $ is an eternity solution of (1). The uniqueness of such a solution implies IJ = $. 1

VIII. CONSISTENCY AND INCONSISTENCY RESULTS

We are now ready to state and prove the results concerning the consistency of optimum programs. We lirst take either the older generation’s


utility functional (Theorem 5) or the younger generation’s utility functional (Theorem 6) as the objective functional in each period. After reviewing the intuition provided by these analyses, we consider the case where a

sonian social welfare function for the two coexisting genei-atons is t ctive functional in each period (Theorem 7). In the rest of the paper

we let 4 denote the mortality solution obtained in Theorem I or 3 an the eternity solution obtained in Theorem 2 or 4.

THEQREM 5. Suppose the older generation’s utility functional is the objective functional in each period. Then a (k, 1 )-optimum program is ~o~siste~t~?l optimum ty

(i) the existence conditions in Theorem 1 or 3 are satisfied and q5 is the utility jiunctional~ or $

(ii) assumption A.2 is satisfied and @ is the utility fu~ct~o~~~.

ProojC Let (Ci, {(- ci, ,?i)]F= r) be a (k, I)-optimum program and jet {kS>;= I be an associated sequence of capital stock vectors. It will suffice to show that if (FfP I, {(Fi, Ez)]F= t) is (Et, r)-optimum, then e> UC WZ+J is (iQz+r, t-t- I)-optimum. Let (cf, {(ci9 c~)];~~+~) be an arbitrary (Et + I, t + 1 )-feasible program. Then ($ r, (2: ~ cf)> {(c:~ c~)~~=~+r) is clearly (I%~, t)-feasible. Let VS= V(Ft, Cz), sat- I9 V( = V(?l, cf), and VS = V(ct , cz), s > t + 1, where Ff- I is a given in the tth eriod. We first consider the program’s consistency with respect to 4. Let

u*-1 =fj(Dr-2; P-‘) and U~+I=@(~~+z;V~+,, P’*), where DIPz is a given in the lth period. Then by the inductive hypothesis,

Moreover, since 4 is a solution given in Theorem I or 3, it satisfies

$q~t-I> ~r~-4~~t-l, W~D~~z-l- ut-11,

where b = (I + 4=)/2c or (1 - dG)/2c, an therefore, if (a, C) satisfies the existence conditions in Theorem 1 or 3.

260 HORIANDKANAYA

It follows from (27), (28), (29), and (30) that

qqr7-,, F)-qqDz-l, ~~)~(l-c~)(~~~~-u~~~)/c~o. (311

fhx fft+lCcf, {(4, cZl}2t+lJ = 4tot-l; v(zi, $1, {VG, 4}Zl+ll is the objective functional in the (t + 1)st period and since (cf, {(ci, c~)}~!~+i) is an arbitrary (kf+i, t-t- l)-feasible program, (31) shows that (Ff, {(Fi, Ff)}FZt+l) is (k(+i, t+ 1)-optimum.

We next turn to the program’s consistency with respect to $. In this case the givens in the tth period are I?- i and ‘+21? By the same reasoning that led to (27), the inductive hypothesis implies that

$(f-2F Prp l; rq21+y2~ Ptpl; V).

By the lemma in the preceding section, this can be rewritten

t9(z-1v; lj(f-lF; rt; vt+l))>q-l~ l+byF vt; v+l)).

Since 0 is increasing, it follows that

l)(t-lq Fr; vt+1)2$(t-1E vt; v+l).

Therefore (c?:, {(Fi, ?~)}~=~+r) is (kt+l, t+ 1)-optimum. 1

We next turn to the case where only the younger generation’s utility counts. In order to obtain a sharp result, we now adopt the following differentiability assumptions, which enable us to characterize optimum programs by the Euler equations:

A.3. V has positive partials with respect to all the arguments. A.4. 4 has positive partials with respect to all the arguments. AS. $ has positive partials with respect to all the arguments .

Note that under A.4 and A.5, the aggregator G and the functional 0 of the lemma also have positive partials with respect to all the arguments.

Let cij be the jth element of ci and call a program (ci, {(ci, cz)}FZr) interior if, for some s > 1 and somej,

cfj#O and c~+~,~#O.

THEOREM 6. Suppose the younger generation’s utility functional is the objective functional in each period. Then an interior distribution program which is (k, l)-optimum is not consistently optimum $

(i) assumptions A.3 and A.4 are satisfied and q5 is the utility functional, or lf

(ii) assumptions A.2, A.3, and A.5 are satis$ed and $ is the utility functional.


Proof Let (F& { (?i, Pi)}:= 1) be an interior (k, l)-optimum program. Without loss of generality, we can assume that

and f& # 0.

For notational simplicity, we will write I$ and ~7; for Cfj an We lirst consider the program’s consistency with respect to 4. The

following proof needs an expression for the marginal effect of a change in CL on U?. But this marginal effect varies according to which ancestor generation is regarded dead. In order to distinguish between these marginal effects, we write (JUJ&i) 1 U7 for the marginal effect of a change in C: in Ut if the last generation to be regarded as dead is the z th generation and its utility level is UT. Thus, for s 2 t,

and

where in the right-hand side of the second equation

Now, since (Ei7 {(E:, Z:)jF!i) is an optimum program in the first perio from the first period’s viewpoint, (CT, Ei) is an optimum division plan of C2 = 5: + ?i between the iirst and second generations. Since the first generation is the younger in the Iirst period, the last generation it regards dead is f = - 1. Thus.

262 HORI AND KANAYA

(32) and (33) imply that

(34)

Next consider if (CT, Fi) is an optimum division of C2 for the second generation in the second period. Since the 0th generation is dead now, (Cf, C;) is optimum only if (i?U.J&$) 1 U0 - (8UJi3c~) 1 U0 = 0 when evaluated along (ET, {(Ci, Zz)}sZZ). But

Since the expression within the parentheses vanishes due to (34), and since the second in the right-hand side term is non-zero by assumption, (35) does not vanish. Thus (F:, Fi) is not optimum and therefore G {(C F:l):J . 1s not optimum from the second period’s point of view.

We next turn to consistency with respect to $. As above, in order foi (c?& {(Fi, Fz)}FZ1) to be consistently optimum, (FT, Ci) must be an optimum division of CZ = ?f + ?i for both the first and second generations. If we let UZ = $(fPIV; Vf; V+‘), t = 1,2, then this requires that (8U&) - (aU&3ci) = 0 for t = 1,2 when evaluated along (F& {(Zi, Fz)];= 1). But by the lemma, U1 = f3(’ K U2). Thus

where W/WI = aO(lK U&3Vl and i3@/i3Uz = ZqlK UJXJ~. Since (&)‘/a Vl)(d VI /de:) > 0, the requirement is not satislied. 1

An intuition for these consistency and inconsistency results is as follows.(‘) Take any two adjacent generations, say the tth and the (t + 1)st generations. Once the distribution of consumption goods between them is fixed, the future stream of consumptions (ci+ , , { (ci , c:)} F! ~ + Z) cannot cause disagreement between them because, due to the non-paternalistic nature of altruism expressed by (l), it affects Ut only through Uf+ l .@) Thus the distribution of consumption goods between themselves, namely Cc?, ct+ 11, is th e only possible source of disagreement for them. And in fact

‘The following analysis was prompted by the comments of one of the referees for this journal.

*This also shows why, as was proved by Streufert [29] and Kanaya [16], the dynamic consistency does not carry over to the case where a generation’s nonpaternalistic altruism extends to the grandchild as well as the child generations.


they disagree over (c:, ci+r) b ecause, as can, be seen from (I )> it affects CJz indirectly through U* + 1 and also directly through V,.

IIowever, the existence of a disagreement itself does not generate a dynamic inconsistency. In fact, if only the older generation’s utility counts9 then, since (cf, cj+ i ) is a consumption plan for the (t + I )st period, it has to pass the optimality test of the tth generation alone before it is actually carried out, and therefore a dynamic inconsistency does not arise. e other hand, if the younger generation’s utility counts, then ((cf, c:+ ,) has to pass the optimality tests of both the tth and (t + I )st generations. Thus the above-mentioned disagreement between these generations leads to a dynamic inconsistency.

This reasoning suggests that the inconsistency result will still hold if the younger generation’s utility is given any weight in each period’s objective functional. Thus we arrive at

THEOREM ‘7.9 Suppose that the tth period’s objective functional is given bY

where W, is dlyferentiable and satisfies 8 W,/C~U,- l 2 0, 8 Wt/3Ciz > 0~ Then a~ interior distribution program which-6 (k, 1 )-optimum is not consistently optimum iy

(i) assumptions A-3 and A.4 are satisfied and the CJr’s are given by c$$ or if

(ii) assumptions A.2, A.3, and A.5 are satisfied and the Ur’s are given bY $.

Z+oo$ Since the proof is similar to the proof of Theorem 6, we will give only the outline.

Let (.$, I(?:, C:)}p= r) be an interior (k, l)-optimum for Wl(Uo, Ui), assume that ~7;~ # 0 and Ei, # O9 and write 2; and Ci for rFT1 and &

’ Using the notion of consistency proposed by Aloe1 [l], Kimball [17] asserts that the linear eternity functional (12) does not produce any inconsistency. This difference of Kimball% result and ours can be traced to the difference of the consistency notions. Strotz’s notion of consistency, which we are using, is related to the issue: Does the interaction of different generations lead to a game-theoretic situation? On the other hand, although there is some ambiguity, Abel and Kimball’s notion seems to be related to the issue: Given a game-theoretic situation, does there exist a consistent course of actions? It may be worth mentioning that the latter notion of consistency does not require the coeffkients in the objective functional to be geometric.

264 HORI AND KANAYA

By the optimality of the program with respect to HI, we have

(36)

where we are using the same notation for the partials as in the proof of Theorem 6. As was noted above, the 0th and the 1st generations agree on CC:, {F:, C:l)f&). -I- ogether with (36), this implies

which in turn implies

But the first and second generations disagree on (cf , ci) and therefore (37) implies

(37), (38), and the assumption that aWJaU* >O imply that

(36) and (39) constitute a dynamic inconsistency. 1

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Utility functionals with nonpaternalistic intergenerational altruism