Fiscal Policy With Impure Intergenerational Altruism

NBER WORXING PAPER. SERIES FISCAL POLICY WITH IMPURE INTERGENERATIONAL ALTRUISM Andrew B. Abel B. Douglas Beroheim Working Paper No. 2613 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 June 1988 This work is part of the NBER'S program in Taxation and was supported by the National Science Foundation, through Grants No. SES8408618 and SES860763O. We would like to thank seminar participants at the various presentations of this paper for helpful comments. Any views expressed here are those of the authors, and should not be attributed to any other individual or organization. NBER Working Paper #2613 June 1988 FISCAL POLICY WITH IMPURE INTERGENERATIONAL ALTRUI SM ABSTRACT Recent work demonstrates that dynastic assumptions gtarantee the irrelevance of all redistributional policies, distortionary taxes, and prices——the neutrality of fiscal policy (Ricardian equivalence) is only the "tip of the iceburg.' In this paper, we investigate the possibility of reinstating approximate Ricardian equivalence by introducing a small of friction in intergenerational links. If Ricardian equivalence depends upon significantly snorter chains of links than do these amount stronger neutrality results, then friction may dissipate the effects that generate strong neutrality, without significantly affecting the Ricardian result. Although this intuition turns out to be essentially correct, we show that models with small amounts of friction have other untenable implications. We conclude that the theoretical case for Ricardlan equivalence remains tenuous. Professor Andrew B. Abel The Jharton School University of Pennsylvania Philadelphia, PA 19104 Professor B. Douglas Bernheim Department of Economics Stanford University Stanford, CA 94305 1. Introduction In recent years, Robert Barro's [1974] version of "Ricardian equivalence" has stimulated much controversy concerning the effects of government budget deficits and social security programs. known paper, Barro supplented In his well— the traditional overlapping generations model with intergenerational altruism, and argued, in essence, that voluntary transfers between parents and children cause the representative family to behave as though it is a single, infinite—lived individual——a "dynastic" unit. From the point of view of the family, neither debt nor social security alters available alternatives; both are therefore neutral. Barro's analysis identifies the strength of Thus, intergenerational altruism as a key factor in determining tne effects of government bond issues and public pension programs. Recently, Bernheim and Bagwell [1988] have argued against the applicability of Ricardian equivalence by demonstrating that Barro's assumptions guarantee distortionary the irrelevance of all redistributional policies, taxes, and prices——the neutrality of fiscal policy is only the "tip of the iceburg." Their results rely on the existence of intrafamily linkages, which arise whenever two unrelated individuals produce a common child, Bernheim and Eagwell concluded that, since these other propositions do not hold even approxirrately, assert that the world is approximately dynastic. one cannot Accordingly, all conclusions following from the dynastic framework (including Ricardian equivalence) are suspect. Bernheim and Bagwell also noted that it might be possible to reinstate approximate Ricardian equivalence without generating untenable consequences by introducing a small amount of friction would long chains. cumulate friction'. Intuitively, with each link and would become substantial for Since Ricardian equivalence (for debt redeemed within a few generations) presumably depends on short chains while the Bernheim— Bagwell results presurrably depand on long ones (we note that these presumptions may be erroneous——see section 7), the introduction of friction do mit just the trick, The purpose of this paper formally is to evaluate the preceding argument by introduciing various forms of friction into a model with altruistically motivated intergenerational transfers. frictions arising from three sources: We focus on the derivation of pleasure directly from the act of giving; incomplete information about others' preferences; and egalitarian social norms that constrain parents to divide transfers evenly between children. The first two sources of friction turn out to be quite similar analytically, and give rise to qualitatively similar results, approxite amount of In particular, one can obtain Ricardian equivalence by introducing a sufficiently small friction. Furthermore, for any given amount of friction, one can reinstate the relevance of other redistributional policies by taking the population to be sufficiently large (it follows from this that taxes will distort prices will play an important allocational role). However, there is a hitch: by simultaneously taking friction to be snll and population to be large, one drives each individual's behavior, marginal propensity and to consume out of wealth to zero. In resolving —3-. several paradoxes posed by Bernheim and gwell, one therefore merely encoiters another. introduction of The elitarian constraints generates some intruiging results. Most importantly, one obtains exact Ricardian equivalence in a world where other redistributional policies have Since there is no need to assume that significant allocative effects. this source of friction is "small," one does not encounter the own— wealth effect puzzle noted above. We are troubled however by the rather ad hoc nature of this constraint. generates a new paradox: In addition, its imposition we show that an exogenous increase in the wealth of any given individual is never Pareto improving. we conclude that the theoretical case tenuous Consequently, for Ricardian equivalence remeins even when one explicitly recognizes sources of economic friction. We ornize our discussion as follows. Section 2 lays out the basic model, cescribes an appropriate notion of equilibrium, presents and some technical results which facilitate the analysis of subsequent sections. In sections 3 through 6 we consider, respectively, specialized cases in which a) there is no friction, b) altruists derive utility in part directly from the act of giving, c) agents have incomplete information about each others' preferences, and d) parents are constrained to divide transfers equally between their children. Section 7 contains some concluding remarks. We defer all technical to the appendices. Appendix A contains a complete treatment of comparative statics for cases b and c above, while manipulations and proofs —4— Appendix B treats case d. We present proofs of specific results in Appendix C. 2. The Model ccnsider an economy comprised of 2N households. We Despite the fact that we treat each household as if it consists of a single individual, one should for the purpose of interpretation think of households as nErried couples. The population is evenly divided between of households, henceforth referred to as "parents" and two groups children," Thus, there are N parents (labelled p., i = 1,...,N), and N children (labelled k., i = 1,... ,N), Eve parent has two children, spouses and every child has two parents (reflecting the fact that originally come from different households). particular that p's convention, = kNl kj. children are k and k1 We assume in (where, by It is therefore appropriate to think of intrafamily relations as a kind of circle (pictured in figure 1), consisting of an outer layer (parents) and an inner layer (children). This representation of intrafamily relations is unquestionably highly stylized, and does not reflect the full complexity of family networks, particularly in cases where these networks span more than two generations (see Bernheim and Bagwell [1986]). On the other hand, this framework has the advantage of rendering our current analytic objectives tractable, while in all likelihood doing very little violence to the underlying economic issues. argue We return to this point in section 7, and that more realistic modelling of family networks would only tend to strengthen our conclusions. —5— Parent endowed with (Ci, 1 is endowed with wealth, w. a transfer to Parent P. child k 1 t. > — 1 convention, We t.1 (C. p = w. + T. 1 1 + t.i1 u. = well being is so that the well- is given by k. probability > 0, T. > 0, from suppose that children are completely selfish, being of child consumption parents p and p1 p1), and consumes all available resources C.1 With T.1 — of course, to non—netivity constraints by is (T), and a transfer to child k (tj, 1 1+1 1 0). Child i receives transfers (where, similarly, child divides his wealth between C.1 = subject, W It, parent u(c.) is also completely selfish, so that his given by = u(Ci felicity function for parents is identical to that for children——this restriction is inessential). With probability (note that the (1 — it), parent p. is altruistic; this entails non—paternalistic altruism for his child (as in Frro [1974]), and possibly some concern for the magnitude of his bequests (as in Andreoni [1986]): = u(C.) + [u(c,) + u(c.1)} + m[v(T.) + v(tj] —6— (a, For simplicity, we assume that > 0). the random events that determine parental preferences are distributed independently over parents. Throughout, we also assuma that u() v() and are twice continuously differentiable and strictly concave. The final allocation of resources is determined through a simultaneous move game, in which each parent chooses his own consumption, as well preferences are private information; while parent not he himself is limited to as intergenerational transfers. Each parent's p. knows whether or altruistic, his fnformation concerning others owledge is of the distribution of preferences described It is therefore necessary to employ a solution concept that above. The natural choice is to focus allows for incomplete information. attention on Eayesian Nash Eauilibria (see Harsanyi [1967—68]). In a Bayesian Nash Equilibrium (hencefort BNE), we assign parent a function mapping his preferences into decisions. to each These decisions m.st maximize his expected utility given associated preferences, and given the distribution of induced by their assigned fLmctions. a particularly simple form. obviously we need set T1, = I all (T,t) being 0, rerdless altruistic. solves In the current context, a parent i only describe the choices, parent i, t. When other parents' decisions is selfish, of what other (Tt1) rents which are * *N he Accordingly, (T.,t.)i1 BNE has will do. contingent Thus, upon is a BNE if for —7— (3) max T ti T.1 - u(W. 1 t,) 1 + (1 - .)[u(w.1 + T. + 1 tx—1 + u(w. + i+1 ) Ti+1 + t) i' + u(w + i+1 + Iu(w. + T,) 1 1 (subject to non—negativity equals either 3 or 1 t)]} 1 We note constants). information (so that definition reduces to the sore standard + a[v(T in ) i w it is complete), this notion of initially distributed evenly within generations. 1 I )j passing that when a Nash equilibrium. Throughout much of our analysis, we will assume that W, + v(t resources are That is, =W = w 1 fcr all i. When we assume symmetric endowments, we will also focus the attention on symmetric equilibria, which have the property that magnitudes of all transfers (conditional upon the parent being * * * altruistic) are identical (i.e., T. = 1 = T for all i 1 t. We now present three technical results which justify the comparative these statics performed in subseqnt sections. The first of establishes existence. Theorem 1: For all endowment profiles (W,w.) 1 1 i1 , a BNE exists. Next, we show that symmetric equilibria do indeed exist when endowments are symmetric. Theorem 2: If endowments are distributed symmetrically, there exists a symmetric BNE. Furthermore, the associated then * transfer, T is independent of N. The second portion of this result establishes that is in resources some the allocation of important sense independent of population size. This conolusion wIll feature prominently in the ensuing analysis. Finally, establish a iqueness result. we If Theorem 3: a = 0 and a = When altruism is In particular, a > 0 or then there is a unique it > 0, BNE. 0, imperfect or (a it positive), equilibrium is unique. we lose nothing at all by focusing on symmetric equilibria for the case of symmetric endowments. In a frictionless world, there mey indeed be a multiplicity of equilibria (more on later), but all such equilibria are equivalent, so once analysis If ain this our involves no loss of generality. Throughout the following equilibria sections, we will focus on interior (i.e., parents meke positive transfers to their children). Since we will be priuarily concerned with environments that are "almost" symmetric and frictionless, it is sufficient to assume that u'(W) < u'(w) As a final preliminary step, we describe two types policies of particular interest. use of governnt follows: debt. of "fiscal" The first of these corresponds to the The level of debt, 8, affects endowments as -.9— dw. 1 for all That is, the government redistributes resources from the i. younger generation to the older generation, presunbly by deferring into taxes the future. Note that this experiment is a pure case of intergenerational redistributIon, since all members of the same are affected identically. The second type of fiscal policy considered here amounts to a pure redistributions within the parents' generation. In particular, generation represents a transfer to parent dW. d.1 I p, if . financed out of "general revenues' ji —1/(N — 1) otheraise It would also be natural to analyze a third type of OliC consisting of redistributions within the children's generation. Analytically, such policies are extremely similar to redistributions within the parent's generation, so we do not consider them explicitly. Note that, taken together, these three sets of instruments are comprehensive, in the sense that they allow the governrint to achieve any conceivable distribution of resources. Throughout the rest of this paper, we focus on the extent to which fiscal instruments redistribute consumption in equilibrium. policy p transfer, (where p tj, is either public debt, we define For each 5, or an intragenerational a distributional index: —10— N = j =1 dC (Ii + dc. The logic of this index is straightforward. If the policy effect on the consumption of any individual, then 0 R = 0. has no p Thus, corresponds to the Ricardian equivalence hypothesis, and R 1 = for all i corresponds to the Rernheim—Bagwell neutrality Note also that, proposition. in the absence of operative intergenerational linkages, for each of the policies described above, R =- 1 redistribution tells 1 3, us how - (redistributing endowments leads to a one—for—one of consumption). Thus, a iue of R between and 0 closely behavior conforms to each of the polar cases. Perfect Aitni We begin by considering a frictionless world, in which altruism is nonpaternalistic (a = received much prior attention (see perfectly [1988]), this section contains no = 0). Since such environrnts have rro [1974] and Bernheim new and results as such. Rather, gwell we restate known neutrality results within the context of our current model in order to provide a "base case" with which to compare the results of subsequent sections. Under the assumptions specified in section 2, an interior equilibrium nust satisfy (4A) u(C) = u(c) (4B) u(C.) = u(c) —11— where * C1 =L1 * * —T.1 —ti and * T,1 + * c.1 = for all w.1 + * t.i—I Given our concavity assumptions, these conditions are also i. Note that (4A), (4B), (5A), and sufficient to establish an equilibrium. (58) form a system of 4N equations in 4N unknowns. Ordinarily, one would think that the system would be fully determined. However, brief inspection reveals that one of the equations given in (4A) and (48) is redundant (recall that S 0N1 Thus, the 01). system is under— determined. does not, however, This allocation. To see this, reflect real * * Note that (4A), and (58) over we sum (5A) (C.1 + c.1 (4B), and (6) — w. — 1 (omitting system of 2N equations in 2N unknoms. that indeterrniriancy of resource consumption is fully determined. i to obtain w.1 ) = o the redundant equation) form a Accordingly, In fact, it seems likely we have already established that there is a a-uique solution to this system of equations (Theorem 3). contrast, transfers are indeterminant. refer to figure 1. Suppose that an In point, If every ain parent simply increases T. by $1 To understand this equilibrium prevails. and decreases t by $1, —12— the allocation of real resources retrains unchanged. profile of transfers is also an equilibrium. Thus, the new Equilibrium transfers are therefore defined only up to an additive constant, with the sole restriction that all Accordingly, we transfers sist be positive. ny ignore transfers completely, and describe the equilibrium consumption profile directly through equations (4A), (4B), and Simple inspection of (6), these equations reveals that the of resources depends allocation only upon total wealth, N ] (W. + i=1 Changes in the distribution of wealth have no effect on the consumption of any individual. Several neutrality results follow immediately from this observation. We begin with rro's [1974] well—iciown version of Ricardian equivalence: Propositioni: If a=it=O, then 5 =0. The proof simply consists of noting that + wj) = 0 and invoking the preceding observations. Bernheis and Bagwell [1988] have criticized two grounds. First, they argue linkages, that, rro's analysis in a world with intrafamily rro' s assumptions (perfect non—çaternalistic altri.iism on —13— coupled with policies operative transfers) imply that all redistributional are neutral. In the current context, Proposition 2: if 5 = = 0, then R we obtain 0 for all i This result follows directly from the observation that 3 analogously (ii1 (. + )) = 0 to Proposition 1. indicates that policies that redistribute resources beten apparently unrelated members of the same generation have no Proposition effects 2 on resource allocation. Using this result, one can also show elaborate environments, apparently distortionary taxes have no effects on behavior, and that prices are not only indeterminate, but also play no role in the resource allocation process that, see in somewhat more Bernheim and gwell [1988] and Bernheim [1986]). Bernheim and Bagwell also offered, but did not emphasize, a second criticism of the dynastic framework: as the population size increases, each individual's merginal propensity to consume out of his own wealth falls to zero. As we shall see, this observation turns out to be particularly important in models that incorporate small friction, amounts of in the current context, we have * Proposition To (along 3: If a = = establish Proposition 3, 0, we dC. = 0 lim N- for all I. i argue as follows. 8y Proposition 2 with a similar result for children), equalizing the distribution —14- of resources within generations has no effect on consumption. can invoke Theorem 3, to conclude that the Thus, we distribution of consumption is symmetric both before and after the incremental infusion of wealth. It is trivial to check that all i) dC./dW. C (s C i) for all 1 and c (a c 1 for are both increasing in aggregate resources. Thus, < i/N, from which the result follows immediately. Empirically speaking, Proposition 2 (along with its corollaries) Proposition 3 are both untenable, Indeed, since not hold even as an approximation in the raal. world, and critical sense not even approximately like the these properties do eality is in model described dome here. Accordingly, rnheim and gwell conclude that it is inapropriate to take the Ricardian equivalence result even as a "rule of thumb" guide to policy, without first specifying the nature of the approximation in great detail. 4. Joy undertake this task in subsequent sections, We of Giving in this section we analyze the case in altruistic and, in addition, the transfers T1 and (7E) u'(C1) parents are and m = 0. Parent i chooses to satisfy t u'(C.) = (7A) all care directly about the size of the Formally, a > 0 transfers they aeke. which = u'(c.) + u'(c.1) av'(T) + av'(t) In deciding on the optimal transfers, parent i considers reducing his owr consumption, C, by one unit. If he transfers this unit to child —15— 1, the parent's utility is increased by transfers this unit of consumption good to child parent's utility is increased by u'(c11) conditions (7A) and (7B) show that that the in utility comparative static To obtain differentiates the entire system i), a + mv'(Tj; i + along consumer chooses if he then the The first—order T1 and T so t consumption from increasing either or is t.. results for this model, one formed by equations (7A) and (7B) (for constraints. with the budget 1, av'(t.). rrarginal utility loss from decreasing his own equal to the nBrginal each + u'(c) The following result is extremely helpful for evaluating the effects of specific policy exercises. Theorem 4: If it = 0, > 0, and the initial distribution of endowments is symmetric, then = j-k - + + where X —1 = 2[1 + Since the formula for Given the nature inverse of the of the other. right ) v(T) u"(c) solves X solutions. - X) hand side v"(T) u"(c) + v"(T) u"(C) + [av"(Tfl2 u"(C)u"(c) X is quadratic, there are, of course, two of this formula, one root is simply the If a ) 0, then the expression on the strictly exceeds 2, so that one solution exceeds unity, —16— while the other lies value of and C does not depend dC/dW k the It is easy to check that 1, upon whether one uses the larger or For convenience, we henceforth adopt the convention that smaller root. 0 beeen < X < 1. Now consider the effects of a Ricardian redistribution in which is increased by one unit i. and w It is of course feasible for all is decreased by one unit C and c this inriance in the face of this experirint, is not, in general, optimal as argued below. all Suppose that and increase mchanged. to in and T1 parents t However, intain their each by 1/2, for to remain unchanged of consumption - own In this consumption unchanged case all c. will However, the first—order conditions (7A) and (7B) will be satisfied because the increase in av'(T.) and av'(t). all be fail transfers leads to a reduction Therefore, the narginal utility of parent i's consumption, u'(Ci), would exceed the right—hand sides of (7A) and (7B), which represent the narginal utility associated with an additional transfer. To re—establish optimality, parent i would increase his own his transfers. Therefore, the Ricardian experiment increases the consumption of parents and reduces the consumption and decrease consumption of children. not remain unchanged in the face of a Ricardian experinnt ss based on the fact that increased transfers would reduce av'(T1) and av'(t.) and therefore violate the The argument first—order that consumption would conditions (7A) and (7B). However, if a is srrll then this effect will be small and the impact on consumption will be minimal. —17— Thus, we would expect the effect of deficits on consumption to be continuous in Likewise, one would expect to obtain a. a similar continuity property with respect to the effect of transfer policies on Formelly, we have consumption. If Proposition 4: is endowments as 0 and the initial distribution of = symmetric, then urn R 6 = urn R i 0. = a+0 tails us that by taking friction to be small we can Proposition 4 obtain both m Ricardian equivalence and the stronger neutrality properties arbitrarily good approximations. By itself, this result does not that for ¶. for & it bolster the Ricardian position. However, the key point is quality of the approximation depends upon N, whereas does not. Indeed, since public debt does not alter the symmetry of the then by Theorem 2 endowments, contrast, R In ' varies with R a is completely independent of systematically with We is wealth, j i. becomes very on the Note that dC/d is (1) the direct effect of the increase in dC/dW; and consumption of the reduction for all N effect of t.1 consumption in large economies. sum of tao components: parent N. as R therefore consider in detail the distribution of the In keeping with the intuition given in the introduction to this paper, we wish to explore the behavior of large. N. (2) the effect on parent in parent j's wealth by (N l's — 1)1 units, To evaluate these components in a large economy, we take the limit of the formula given in Theorem 4 (recalling that, since endowments are symmetric, C, , and T do not depend on N): —18— dC* X)1 = Thus, an increase in Wk (2 + has a positive effect on magnitude of the effect declines geometrically as but the C. j—k increases—— friction dissipates the effect on more distant relatives. We depict this pattern graphically in figure 2. Consider now the tao component effects of For a large .. economy, the effect on parent j's wealth is negligible. follows from Theorem 4 that even summing over all effect on arent distantly related to to i there is no I, j i's consumption (i.e. effect (ii) above is eaual to Intuitively, in large economies almost all zero). Indeed, it j are only i, so that the friction in any chain linking j almost completely dissipates the effects of changes in j's wealth. Thus, In a large economy, the effect on parent i's consumption of the redistributive transfer 'r• is the same as the effect on parent i's consumption of an increase In parent l's wealth (effect (i) above). Inspection of Theorem 4 reveals that, even in a large economy, dC/dW1 is positive. This follows from the fact that if parent i received an additional ixit of wealth and did not increase his own consumption, then he would increase his transfers c1 and c.÷1. In this case, T1 and t1, thereby increasing the rrrginal utility of his own consumption, u'(C1), would exceed the right—hand sides of the first— order conditions (7A) and (7B). conditions, parent i In order to satisfy the first—order would increase his own consumption. —19— Finally, since the effects of wealth injections are localized, in to redistribute consumption from large economies we would expect the general population to the close relatives of limit 1 R + 1. t = C, a If of endowments is symmetric, then — N —I I lim . N-- dc.i dW. > 0, and the initial distribution lim = 1. Furthermore, > 0. 1 - Taken together, 4 and 5 may well appear to resolve Propositions the difficulties raised by Bercheim and Bagwell. taking in the dC* dC* obtain so that We sumrxarize these conclusions in Proposition 5. Proposition 5: lim 1, Ricardian equivalence a Specifically, one can to an arbitrarily good approximation by sufficiently small. If for a given a the population is sufficiently large then, as in a model with no altruistic linkages, a one dollar intragenerational transfer will redistribute one dollar of I 1 P 1). The recipient of such a consumption in equilibrium (i.e. transfer will act that is, he will these resources as though he has received an injection of new wealth—— completely ignore the fact that the governrrent acquired by levying taxes on individuals to whom the recipient is operatively linked. Taking the population to be large does not, affect the approximate validity of Ricardian equivalence, Thus, with a small and N large relative to a1, deficits are approximately however, neutral, but intragenerational redistributions are not. Formally, have we —20— Eroposition 6: for any sequence sequences (ak,Nk) k- (o,). for the argument to 0 and = k, lim R Note that one does not ha with <ak,N>kl for all Ok) Nk - decreasing function There exists a obtain = (ak,Nk) = Urn R + 0 and (0,) and 1. R' + I More generally, (a,,N,) + work. for all must be sufficiently large Nk that such N for each (o,) ¶ consistent with any limiting value for cannot justify Ricardian equivalence small and the population is large. simply by arguing that However, for the moment gwehl appears Bernheim and that one must turn to the actual values of including 0. Thus, R ', one friction the logical puzzle posed to be mitigated. It is by seems empirical evidence in order to determine whether a and N are consistent with approximate Ricardian equivalence, but inconsistent with the collateral neutrality results. Yet this resolution is unsatisfactory. a takes small if one simultaneously (so that Ricardian equivalence is approxirrtely true) large (so that intragenerational transfers remain relevant), then in the limit each individual's consumption is necessarily unrelated to his own wealth. More precisely, and N Proposition 7: Suppose a = of endowments is symmetric. Let lim (ak,Nk) = (o,). Then 0, < and that 0k' Nk > the initial distribution be such that —21— urn k+ Thus, * dC. = 1 by introducing friction through resolve the difficulties if one raised 5, one cannot 2 and 3 by Propositions takes friction to be small without simultaneously of section 3: letting the population get very large, then in the limit everything is neutral; if one takes friction to be small while letting the population grow, then in the limit each individuals marginal propensity to consume out of wealth falls to zero. Propositions Suppose we take each k. 6 and 7 may at some sequence first appear -* (ak,Nk) to be inconsistent. (O,) with N. > N(z) for Sy proposition 7, we know that in the limit consumption does not depend upon an individual's own alth. This seems to imply that consumption depends upon aggregate wealth, from which it would follow that all redistributive policies are neutral. proposition 6 tells us that R 1. Quite to the contrary, The key to this puzzle is the fact that, in the limit, consumption is a function of local aggregates, rather than global aggregates. i That is, the consumption of individual depends only upon the wealth holdings of the limit, i is close" relatives. In has an infinite number of close realtives (even though these relatives form a negligible subset of the entire population), and so i's own wealth is irrelevant. dollar from i to j (where I related) will transfer one dollar relatives to j and his However, a and j redistribution of are only very of consumption from close relatives. one distantly i and his close —22— In summary, we find that (a small -* equivalence one can simultaneously take friction 0) and population large (N - ) such holds arbitrarily well, and such that that Ricardian redistrjbutions real effects (changes in wealth only affect consumption locally). However, in doing so one necessarily produces an untenable result: individuals' consumption is unrelated to his own wealth. 5. Incowplete have each Inforation Now consider an economy in which a fraction selfish and the retraining fraction 1 — it of it of the parents are the parents are altruistic. parent knows whether he is altruistic or selfish, and knows the fraction it of selfish parents, but does not know whether any other Each particular parent is selfish or altruistic. that there is no joy of giving motive (a For simplicity, we assume 0). Rather than treat this case in detail, we will simply indicate its formal similarity to the joy of giving model. and (6) it = 0, Specifically, if the utility of each parent is given by u(C.) + [u(y.) + u(y.1)J + a [v(T.) + where C.1 = W. 1 — T i tI — and y.1 = w. + T. ÷ t. 1 1 —1 v(tj] a > 0 —23— Alternatively, when a = 0 and it > 0, parent is expected (given that he is altruistic) reduces to (9) u(C.) + (recall that, (i - m)[u(y.) j + m[u(wj + T) transfers j being First, it). Clearly, (1 — differences disappear u() it as rather t and Finally, in (9) (likewise w.i+1 + comparative as quantitative to 0. Second, in (9) the third term . Yet both it and goes than in (9) the a are riasures Merely changing the index is inconsequential. Third, in possibility that u(s) of it appears in place of irrelevant. t, this difference in scale can have no qualitative consequences, and even (9) and nothing if he is completely selfish). second term is multiplied by is multiplied by t)i + u(w1÷1 altruistic, with the understanding We note four differences between (8) and (9). of friction. ÷ with incomplete information, we interpret T, choices conditional upon that u(y.1) + utility t.1 v(). are identical, this too is v() w Since we never ruled out the + in place of T. appears in place of t.). 1 statics for the instruments T Clearly, this cannot affect t., since 1 w. in independent it should not the strong similarities between (8) and (9), be surprising that formal analysis of the two models is virtually Given therefore treat these models simultaneously in Appendix both by analyzing a slightly sore general formulation that subsumas specifications. Since Appendix A gives a complete characterization of identical. We comparative statics for the general formulation, it is possible to A —24- obtain direct analogs of Propositions a = 0, it 0 > 4 through 7 for the case by mimicking the proofs in Appendix C. of We leave details to the interested reader. 6. Despite its apparent promise, the introduction of friction does not appear to resolve successfully all of the puzzles posed and gwell's analysis. isotivated which 15 in Bernheim We now turn to a less obvious alternative, by the empirical obsertion that testators often - choose to divide bequests equally among their heirs (see Menchik [1980]). This phenornon has puzzled previous analysts, in that it appears to contradict the implications of all widely subscribed theories concerning bequest motives (see the discussion in Summers [198]). to an in section Shleifer and We offer no new explanation of equal division here, but rather simply assume that altruistic subject rnheim, elithrian constraint. rents Like rraximize utility the introduction of friction 4, the constraint itself is somewhat ad hoc, but, as we shall see, its introduction generates some intniiging implications. We leave the task of justifying the equal division assumption for future work. Accordingly, we set assuming that parent p. a = it = 0, and modify our basic model by maximizes utility subject to the constraint that t. 1 Formally, Theorems I = T. 1 through 3 do not apply to this case. We therefore —25— provide the following result: Theorem 5: constraints. Suppose a = it = 0, and that parents face egalitarian there exists a For every endowment profile Furthermore, if endowments are distributed unique equilibrium. symmetrically, then the equilibrium is symmetric, and the associated * irideendent of N. equilibrium transfer, T , is As in section 4, it is useful to derive some preliminary comparative static results that allow us to compute the effects of We therefore provide the following theorem: various policy experiments. Therem 6: constraints. dC (-kl X a = it= 0, and that parents face egalitarian Let initial endowments be distributed symmetrically. —2\(1 + k where Suppose XN)_1(l + XN — N)_1(1 3I)(1 + — X)1 )(i — for N)_1(1 — )_1 j Then = k otherwise solves +X_1240) Once again, the formula for X is quadratic. Since the right hand side is strictly less than —2, one solution is less than while the other lies of the other). begin 0 and —1 (one is simply the reciprocal For convenience, we choose the second root (both yield the same value of We beten —1, dC/dWk) and adopt the convention that 0 > X > —1. our analysis of egalitarian altruism by noting that —26— exactly (i.e., not approximately, as in the establish this property, we need not assume Ricaran equivalence holds preceding sections). To that endowments when are distributed symmetrically——the result obtains even the financial status of children differs within families, Proposition 8: Suppose litarian constraints. Then a = it = 0, harents face Rd = it is important to qualify Proposition The 8 in the following previous models yielded Ticardia equivalence way. (or approxirnat equivalence) for all transfers involving a parent and his children, Here, that is not the case, Policies that entail differential treatment of children within the same family y well have real effects, since the egalitarian constraint prevents parents from offsetting such redistri— butions, This observation leads naturally into our next result. Just as the equal division requirement prevents parents from offsetting redistributions within the family, offsetting more government taxes private individuals from complex transfer policies. Suppose for example that the parent p., and distributes the absence of elitarianism, p. his incremental tax, and subsidy. it precludes p1+1 In the presence of an the procedes to will decrease will raise T1 t P. by the amount of by his incremental elitar1an constraint, these are proscribed. Instead, the actual responses of p11 will offset the policy only partially. alternatives Accordingly, one might well suspect that In egalitarianism p and introduces —27— a kind of friction, which attenuates the effects of a perturbation as further from its source. In large populations, one might once find that policies of iritragenerational redistribution lead to one moves ain sensible consequences. Taking limits of the formulas in theorem 6 (and recalling that, with symmetric endowments, k and the equilibrium allocation are independent of N), we obtain dC* 1 —— dW / — an , 3 dC l3kI( = dW1 + As expected, the effect of p.'s geometrically as - X)(1 )_1 wealth on becomes "distant" j for p's from k j consumption declines However, the most i. striking feature of these formulas follows from the negative. Accordingly, a windfall for parent consumption of i — j is A p. (i when j) i — j is p. fact that is X raises the odd, and lowers it when even (see figure 3). moment's reflection suggests that this pattern is quite natural. In response to an infusion of wealth, parent p1 increases both his consumption and his transfers. Upon seeing that one child (k.,k+1) off, transfer less. is better fall. Parents parents and result, the As a p2 and transfer more. p,1 and p.2 p÷ choose to consume more resources of children k. i—i and and k. i+2 respond by choosing to consume less, The pattern then repeats. From these results it is easy to establish the relevance of —28— intragenerational redistributions in large econornies In fact, parent p 1 will respend to a transfer funded from general revenues (rn ) just as he would to an injection of new wealth; affect does not anish 2Etion9: as the population grows - This result has initial Let if if and that parents face endowments be distributed ho -- N- rn > 0, and a serve > 1. an larger redistributive effect on there are egalitarian intergenerational transfers, than there are no private transfers at all (i.e. contrary to Urn 8 rather peculiar implication, which is that intragenerational transfer has consurrtion 0, dC dC N 'i lim one = a = Suppose tarian constraints, symmetrically. Then furthermore, the pure wealth R' the implications of previous analyses, to magnify rather than dampen the 1). That is, private transfers > redistributive effects of government policies, Even so, it might appear that egalitarianism provides the ideal resolution to the paradoxes raised by Bernheim and Eagwehl. After all, one obtains exact Ricardian equivalence without assuming that source of friction is small. not pass to o this In contrast to previous sections, one need limits simultaneously, thereby producing a paradoxical wealth effect. Yet this conclusion is premature, for the imposition of egalitarianism produces a paradox of its own. Specifically, consider the welfare effects of an exogenous increase in the wealth of some consumer. Ordinarily, we would think of this occurrence as -29- unambiguously desirable. rnodelr Indeed, Not so within the context of the current roughly speaking, only one half of the population would benefit, while the other half would lose. Proposition 10: = Suppose egalitarian constraints. Let initial Formally, 0, = and that parents face endowments be distributed symmetrically. a) If N is even and b) If N is odd and < 0 dU/dW1 Thus, 111 1, then 3 I 3 + is 2) consume more (and, resources of js if family have declined, and odd. parents 3 merely 7. In this — 1 and 3 ÷ 1 feel that the implications of Proposition note that one most simple guides We do 10, not deny cannot accept the egalitarian framework reexamining the validity Closing j crust be worse off even while surprising, are not obviously counterfactual. abandoning - own behavior optimally. The reader may well without odd. accordingly, give less to their children), then the adjusting his We is The intuition for this result follows directly from our discussion of figure 3; this. 3 an exogenous increase in the wealth of any given consumer is never a Pareto improvement. after 1ff < 0 N — 3 ÷ 2) < N/2, then niin(j, min(j, N — dtj/dW1 of some very basic premises, and to welfare analysis. Reirks closing, it is important to emphasize that we have conducted analysis in a way that is likely to significantly overstate the plausibility of approximately Ricardiari worlds. More generally, the —30— case for Ricardian equivalence is even less compelling for two reasons. First, our model spans only two generations. While it is therefore adequate for analyzing the effects of deferring taxes to the next generation, it is unsuited for drawing inferences about the impact of longer term debt, Just as friction compounds through successive linkages between families, it will also compound as intergenerational chains lengthen. to Accordingly, in a more general model, we would expect find that relatively while temporary deficits are approxirratelyneutral, relatively permanent ones are - not. intrafamily linkages are actually much more complicated than the network modelled here. As we extend consideration to a larger number of generations, we generate a proliferation of paths linking Second, different members of the same generation (see Bernheim and gwell, detailed discussion). Linkages actually fora a "web", rather than the circle illustrated in figure 1 As a result, the "distance" between two arbitrarily selected individuals may be quite section 4, for a small on average, even when the population example, that we add one more is quite large. generation, meintaining our assumption every child two parents. Then, that every parent has two children, and ioring redundancies (i.e., sibling don't have the same grandparent in—laws), each is directly linked through his grandchildren to 10 other grandparents, who are in turn linked to 10 others, and so forth. suggests that each 2L This household is connected through chains involving or fewer links to on the order of only for Suppose, L other households, rather than to households, as in the current model. Formal analysis of —31— random graphs indicates that Bollabas this intuition is essentially correct (see [1981]). These observation suggest that, in a more realistic model, the Bernheim—Bagwell puzzles would be much more If most individuals roixist. are connected through relatively few links, then it may be very difficult to eliminate the approximate neutrality of intragenerational transfers without assuming a or it very large. Similarly, each individual would in such a world have a tremendous number of "close" relatives so that, once agedn, the marginal propensity to consume out of own wealth might be extremely small in the absence of large friction. Overall, it is very difficult to see how one could introduce just enough friction in a model with a realistic pattern of interfamily to produce approximate Ricardian equivalence without also generating untenable results as in Bernheim and Sagwell. Jhile one can, perhaps, avoid these problems by invoking an egalitarian constraint, linkages in addition generates some disturbing welfare results. Conseqntly, the theoretical case for Ricardian equivalence remains tenuous at best. this alternative seems very ad hoc, and xA Complete Comparative Statics for Joy of Giving and Incomplete InfornEtion Models This appendix presents the comparative statics analysis of a model that nests the joy of giving model in Section 4 and the incomplete Recall inforn'tion model in Section 5. C. = consurrtion of that adult i = consumption of child i Also = wealth of adult i = wealth of child i = transfer from adult i to child i = transfer from adult i to child i + 1. recall that C. = W. — 1 1 (Al) Let - y adults T. 1 t. 1 denote the consumption of child i if he receives transfers from i and i — 1, = (A2) + T. + t1_1 Let (A3) (A4) where = Z. z1 is a dummy variable. the transfer from adult i w1 = + T. + t. In particular, if to child i and z = 0, then Z is is the transfer from —33— adult to child I i + consumption of child I 1; — z. 1. i Alternatively, if i + u(C.) + L[u(y.) The equations (Al — i , ii = a, = 0 chooses A) contain both and = w() + to rrexiniize w(zjj the joy of giving model and the v(). p = t and T1 the To obtain private information model, set (i joy of giving model, set Alternatively, to obtain — r it), = = 1, sit, the and u(). = first—order conditions are obtained by substituting (Al The into is the if he does not receive a 1 u(y.1)J + w(Z) incomplete inforrration model. p I + 1. Now suppose that adult (A5) Z if he does not receive a transfer from adult is the consumption of child transfer from adult 1, then (A5) and differentiating with respect (A6a) (T.) —u'(C.) + u'(y.) (A6b) (t.) _ut(C) ÷ pu'(y ÷ to nw'(Z.) + and T1 = A4) — t1: 0 T1w'(z.) = 0 Now totally differentiate the first—order conditions with respect to T., 1 t., 1 (A7a) W i and w. 1 to obtain —u"(C.)[dW. — dT. — dtij + u'(y.)[dw1 + T)W"(Z4)[dW7 + dT1J (A7b) —u"(C.)[dW. — dT1 + — dtj + w"(z. )[dw. u"(y. + = + + dt.1J 0 1)[dw. dt1J dT = 0 1 + dT.1 + dtj —34— e assume that initially W. = W = and restrict our attention to syametric equilibria. a 5 u"(C,) u"(y.) + nw"(z.) + W W1 for all i, and we Let < 0 b a u"(C.) < 0 e xu(y.) 0 f 5 w"(z.) 0 and f e a = b + and observe that + f Using the definitions of < 0. a dT. + 1 b dt, + a dt. = b dU. — (e + 1 (A8b) b dT. + a dt. + 1 e dT. = b dW, — 1 x• 1 b, e we can rewrite (A7a, b) as (ABa) Let a 1 be a 2 difference 1 i—I i+1 column vector such that x1 = (e f)dw.1 ÷ f)dw. i+1 dt]. [dT,, 1 1 The linear equation system in (A8) can be written as ra bi Le 0]1[x] (A.9) 1 r = o 1 L—b —e1 1 —a] [x.] 1 [b [: (e + f)dw1 + b Now observe that (Alo) dW1 — :r=[; J dW.1 — (e + f)dw. —35— and then pre—multiply both sides of (A9) by the rratrix on the right—hand side of (MO) to obtain +g,3 x.=Mx. 3 3—1 where e _ae a e e b (i +—)dw. e b a 2 + be and PdW. e j—l g = LdW. The behavior of equation - x. = 3 - [dT., 3 - i)(e dW.1 + f) is governed by the linear difference dt.J 3 in (All) and the boundary condition that = MxN The boundary condition in (A12) exploits the fact that the N adults are located around a circle and adult N + 1. I is formally the same as adult For the purposes of our analysis, it is sufficient to allow and g2 to be nonzero and to restrict 0 8. for In this case, it follows from (All) and (A12) that 2 = M + Mg1 + g2 j = 3,4,5,..., N. —36— and (A14) = xN MN2X2 Substituting (A14) into (A13) yields an expression for the exogenous changes and g1 = (A15) (I — MN) 1{Mg1 ÷ g2] Using the boundary condition in (A12), the expression for x. = N3 and the fact that solution for x1 (A16b) = x. = < for j = 2,,.. ,N x2 in(A15) we have a complete x1,. ...,XN. (A16a) Let in terms of N—i M N—i (I — N ) [Mg1 + g2] + M2(I be the two - MN) + 1[Mg1 characteristic g2] 2,.. .,N j ; roots of the rtrix N. Observe that + (A17a) (A17b) It follows from (A17b) that other. that Let both It (A18a) X = tr M = = det M = - a - e2 1 and denote the smaller root are reciprocals of each X1. It follows from (A17a) positive. can be directly verified that the ntrix roots, X and > are N = PAP1 N can be written as —37— where 1 1 p = I (A18b) (b÷eX) L k 1 (b÷eX1) 0 A= (A18c) — 1 and - + = (Alad) — ? ) (b = PA j—2 eX) -a + eX) a Now observe that N j—2 (k19) (I N —1 N ) — (I N —1 A ) — —1 Substituting (AlSb) and (AlBc) into (A19) yields (A2o) M2(I - We of parent 1. I — -2 '(b+eX)X2 (b+eX_1)X(2) comparative statics First, we examine the effects of an increase in the wealth In particular, we let > 0 dW1 =. . . =dWn =dw1 =. (A21a) r 1 = are now prepared to analyze two exercises. dW 2 MN)I g1 [] . and .=dw N =0. Inthiscasewehave dW1 —38— r g2=1 a I e JdW1 L and b-a r e I (A22) + Mg1 g2 = 2 12 a —e J be L Using dW1 —ab (A17a) we can rewrite (A22) as rb—a e + Mg1 g2 = b —a La New cbserve dW1 + A + ] that (A24) F{Ng1+g2] r 1 = b + a) (A25) Substituting (A25) (b — a)C (b—a)X+a(X+C1) a that (b + a)(b - a)/a into (A24) = —bR + A —1 ) - a yields b-a A+ (A26) + a(X + dW1 b-a e(A_X)L(b+) Observe frcm (A17a) b-a — e 2[Mg1 + g2 a 1 = A — A1 [ b—a —1 a & 1 —1 J dW1 I —39— (A26) by (A20) and use (A16b) to obtain Pre—multiply x .=1—? N — b—a (x j—l ÷ xN_÷l) e j—2 + xN—+2 —_1)_1 ) r (A27) 1 I L(b + eX)(1 [ e (b + + characteristic equation beX (A28) 2 = I — I tr N + I 2 (a — b 2 —e b-a XJ + —1 + )(1 b—a—1 det 2 i N e = X and To simplify (A27) recall that the roots — satisfy = 0 whicn can be the written as — be Now observe that (A29)(b + eX )(i + I Substituting (A30) (b b_ai = 1{( — e e 2 2 2 be - ae)X, + (e + b —ab)X 1 + be} (A28) into the right—hand side of (A29) yields eX1 )(i + a i — a _aX.L+X, e j 1 i=1,2 Substituting (A30) into (A27) yields (1 — XN)l(x — k 3-2 k1)1 +XN—j+2 (A31) [ + XN_3 b—a + + b - a e I = 2,... e + dW (X1 N_+1) To calculate Formally, xi from (A12) and (A14) x1, observe can be written as XN1 ttat where + = x1 = 1N_l + g. (O,dW1). Therefore = N—i N_1 — + b—a e N (X +1)1 (A32) +xN-i + b — e a dW1 l)J (XN + Now we consider the alternative exercise of increasing the wealth of child 1- by > 0.- In this case dw1 1 = (A33) + ) dw1 [a and g2 = = 0. = In this case, x. = Equation (A34) can be M2(I PA2(I — x. = (A34) (A3 5) e1 — MN) iMg1 which can be written as AN)P_1FAP_1g1 j = 2,., .,N rearranged to yield x.=PA N—i—i (I—A) P It follows from (Aied) and (A33) that (A36) — P1g 1 — r 1+— e e( - h_i) b+e + —a 2 +ae b a2 —ae b I I dw1 —41— -a Now observe that eX. + b + 2 + ae = ek b I 2-b2-ae a — so that b in light of (A17a), Simplifying (A'7) 2 —a + eX, + (A37) ae + b = e{X — I — eX. + b Recalling tnat j X = e 2 — set be j X a — b ae = e{-X. + H, and X2 can use we a—e (A39) — X ) + yields 2 A38) ( — X e = Pg1 (A38) to rewrite (A36) as -1 s—e j Now use (A19), + A2O), A35) and —)(i e — X N—i ) (k — (A39) to obtain —i—iT—(X X r ) - 1at (Mo) j j + (X a—e eX)1 - 4 (b L To —i + X + —1 + X';_÷ii ) lj_1 eX)[ - X1 =2,...,N. simplify (MO) observe that (A41) (b + ek — I )(a e x.) — = {ab — be # (me — e2 b2 Substituting (A28) into (Ml) yields (A42) i + eaX) (b+eX.)(a_e_X)=a(l b i b 1 — beX —42— Substituting (A42) into (A4O) yields = - — + + dw (A43) + j = 2,... To calculate x1, note that (P33) we obtain = (i forrrelly x1 + g1. Using (A43) and - - - i + = ,N. 1)1 (x +xNi) N)-l( (1 + (A44) dw1 L1 Appendix + XN) + (x + B Egalitarieni am This economy in appendix whioh presents the oompantive statics analysis of all parents divide their estates equally among the their ohildren. Recall that ci = consumption of adult i = consumption of child i = wealth of adult = wealth of = child i i from parent i to child Observe that (31) i transfer from parent c.1 = w. — 2'T. 1 1 to child i + 1. i, which equals transfer -43- and csw +T1 +Ti_i (32) Parent i chooses Ti to .mximize (33) + u(ci; $u(o) + and the first-orGer condition for tots razisization problem is (34) _2u'(Ci) Totally and + u'(o) differentiating this + a B&(ci+i) 0 first—order condition with respect Ti. yields Wj _2u*(Ci)(dwi — 2dTi] + uM(ci)(dvi d''i t Pu"(ci+i)(dwi+i (35) that initially '¼ restrict our attention to syrntrio We assume a 4u(Ci) b ! 2u"(Ci) < o e s $u(oi) < and observe we oem write to that (35) + 2Pu"(oi) a — 2(b + e). — W and Vi + — + dTii] dTi " + for all i, and we Let equilibria. < Using the definitions of a, b, and e, as e dT,1 + a dTi + e dTi+i — b Si — e — e (86) —44- The linear difference equation in (BE) can be written as by defining the 2 X 1 column vector second—order in companion form [dT., dT.1] x. = Therefore, x. = 1 Mx. i—i I = + h. 1 2,...,N where o —1 e L1 h. The behavior of = [ dW.1 (dw1 — + dwj1 is governed by (B7) and the boundary condition = The boundary condition in (Be) may be represented as adult Mx + h1 reflects N + the fact that formally adult 1. For the purposes of our analysis, it is sufficient to allow and h2 = h to be nonzero and to restrict 0 for this case, it follows from (B7) and (Be) that =M x = 2 .M 1 XN N—2 + x + i = 3,4,5,... ,N. h1 ifl —45— Using the fact that x. = M2x2 for j = 2,... ,N, B1) — we can use to obtain a complete solution for N—i (I—N) i—2 (: ='4 (Blia) (Bitt) x. Let = ) A2 be the 1r N — M L1+h2i4h1 N—i / [Nn + h2j two ccaracteristic roots i of = 2, . . . the catrjx N '4. Observe that + (Bt2a) (312b) It i = 2 bet '4 — < —2 = follows from (Bi2bj that the roots are reciprocals of each other. Let k be the larger root (Bi2a) that < -i < It can be directly (Bi tr K = '2 therefore X follows from < 0. verified that '4 = a) the PAPi where ii (B13b) (Bi 3c) It = A = rx Lo 0 riatrix '4 can be written as —46— X 1 = (213d) — X — —1 Lx i- Now observe that (B14) N 1—2 N —1 (I—N) =PA 1—2 N —1 —1 (i—A) Substituting (213b, c, d) into (214) and performing the trix multiplication yields i—2 N N—i - (I—N ) N = (i—h ) —1. —1—1 Ei—i (—X ) I N—i÷i i-2 N-i+2 (215) I We are now prepared to analyze two .i—2 N—i+2 —(h. comparative ÷X ) I —(.1-3÷N-i-3 ) statics exercises. First, we examine the effects of an increase in the wealth of adult 1. In particular, let In this case = h1 [ > 0 dW1 0 j = and and dW2 h2 •.. = rb/e)dw, = L o dW = = dw1 ... } It follows + i+1) dw, from = (Sub) and (215) that x. = (1 — xN)( — T1) (816) (i_1 1 I b XN j dW N—i+2 1—2 Therefore, (217) 1 = b (1 — XN)_i(x - x1)1['1 + ?N_i+i] I = 2,... -47— i= Now consider the alternative exercise of increasing the wealtn of child 1. In particular, let dw1 .=dWN=O. Inthiscase,n1Ll h2 oP = [—1 dw1, and dw2 0J dw1 > 0 = = dw, dW1 = and so that a-c Mh1 It follows from x. = (1—X + (bllbP N, —1 ) (k—. h2 = j L dw1 (015) and (810) a - —1 —1 e ) e a i = that i—2 (. i-i N—i÷1 (X12 1Oi+2) 2, , Therefore, ,i + (X + N - r. 1 a — e e i—i N—i+2 +kN—i+1 )+,,? i—2 +X ) 1 - kN)(X - _1) 1=1,., (Xi_3 + + .N-i+2,' ) —48— Appendix C Theorem p's strategy space; let is compact and convex, a = continuous in in {(T,t)jT = S.1 Let 1: t + denote an S1 1 T, element of t is S. > o}. = S Note that S4. is by assumption and (it is easy to verify) quasi—concave (sl,...,sN), Thus, by Debreu's {1952] Social Equilibrium Existence Theorem, s.. (s,... ,s) there exists a profile of strategies c definition * (T ,T j must sax u(W T,t Q.E.D. In a symrrtric equilibrium with transfer level — T — t t) + {(i + m{u(w + T) — m)[u(w + t + we know that the solution always entails T > 0, > 0, * T + + u(w t)j} + a[v(T) < W. + t + 5y concavity of T = t, so we T)) + v(t)] u simply v, and require solves sax u(W — 2T) + T subject to y(T) = and T) T + u(w + to T T satisfy subject * which satisfies our equilibrium. Theorem 2: that — utility p's Further, and K W. U < T K W/2. max u(W — O<T<W/2 arg — m)u(w + T + 2{(1 Zr) Let y: + 2[(1 T) [0,W/2] - — lt)u(w + mu(w 0,W/2] + T + T) + T)} + 2v(T) be defined as + mu(w -4- T)} + 2av(T) Since this objective function is continuous and strictly concave, y is a continuous fumction. By the intermediate value theorem, there —49— * exists T * such that = y(T T * ), as required. equilibrium condition is independent of equilibrium independent of Theorem c01 1 3 C is and is KS 1 1 1 = w. ÷ T. 1 1 That is, C. 1 p's that the recains a symmetric T N. (T,t.)1, For any BNE =W so N, Finally, note * let —t.1 ÷ t.i—i p. consumption contingent upon being altruistic, and p. consumption contingent JDOn p. 1 i—I being 0 oN We will first establish trat (C.,c.). must be identical 1 1 i=1 altruistic. in all SNE. Suppose this claim is false. N and (C,c.). L,c.). 1 1 i=1 1 1 i=1 rise to distinct profiles of generality, we some tray suppose that either . Without or > > loss c for j. Take first the case of we see that either T, K T., or Now we use induction. t. > C. t. K Through t, p's budget constraint, Without loss of generality, t. K t,. we assume and Then there are two SNE which give —o —o N • K t. .. 3+1. 3+1 Since Suppose that for some it must then be the case i > 0, > Ci, that t. . > 0, we 3+1 have u'(C° ) —< j+i (inequality (1 — t)u'(C mey occur if • ) + C.= 0). mu'(w. 3+1+1 Now we + t. are .) 3+1 that * av'(t. •) 3+1 ) —50— For suppose not. Using strict concavity of C..> 3+1 t. and C. 3+1 u'(?.) But — (i < < t j+i ., j+l more to Ic. 3+1+1 Next, Ic. 3+1+1 since + u'(w.1 t. . 3+1 < t. and 3+1 must then be the case that T. ) < — / (1 Now we argue that of concavity u \ —0 lt)u'(c. ) 3+1+1 > o? c? 3+1+1 3+1+1 Since it - C? > C? . 3+1+1 3+1+1 For suppose . . v, along with c?3+1+1 and from - + . su'(w,3++1 . then, T. > T. . 3+2+1 3+1+1 - + — , > C, we have 3+1+1 - + + is a contradiction. 'a budget have constraint, we must - —0 u'(C. 3+1+1 v, along with could increase his utility by transfering which , and wewouidhave s)u'G?1) this implies that u — T, . ) + 3+1+1 not. > c? . 3+1+1 / cv'T. Using strict T.3+1+1 . and > T - * 3+1+1 we would have u'(C? . 3+1+1 But more ) < (1 — a)u'(c?3+1+1 ) . this implies +i+1 could to Ic. . 3+1+1 is ; which if — . > budget constraint, t. . < j+i+1 3+1+1 . + T.3+1+1 . ) + av'(T.3+1÷1 increase his utility by transferring —o > C. . T. . and C. . , then by P. . 3+1+1 j+i+1 3+1+1 3+1+1 This completes the induction step. t. . 3+1+1 Note that induction implies This violates the au'(w.3+1+1 a contradiction. T. Finally, + aggrete budget c? > C and constraint. ?>c for all i. Accordingly, we have a contradiction for the first case. Now turn to the second case (c? > c? 3 3 for some j). By k's 3 —51— budget constraint, either T.3 T. assume T.. > T, 3 > or t. > 'J— 3— . Witnout loss of that have already demonstrated above We C I,3 implies C.3 > 3 This returns us to the first case, which yields a contradiction. The preceding argument suffices to establish that if m 3 and c.3 it c.3 and > = 0, T. 3 > allocation of consumption. all BNE yield the same or 0 > that consumption it > 0. Let 1 i1 1 Now suppose denote the unique BNE profile. first that Suppose 0° > for some j. 0 Then either (i) > 0 T and T, = or (ii) 0 hold; is - (i 3—> strict concavity of furthermore, (i) u We obtain budget Next, suppose that + t.1 = W., either 1 (1 — (i) cu'(w,) + av'(T.) + sv'(O) of these conditions can most one value of from t. Suppose we know T Knowing m+1 , 3 budget constraint. p's t. Thus, T. T.. 3 Then we obtain we can T1 calculate t m+1 Applying induction, we conclude that constraint. all transfers are uniquely T.1 for at 's budget constraint. p1ts ÷ v, only one can hold Now procede by induction. from Bitu'(w. + T.) it)u'(c) 3 and uniquely determined. from km+1 + and u'(C) By it)u'(c) — u'(C) determined. C 0 < 0 = T.1 for all < W. 1 + T.) +ltu'(wi+1 +W lt)u'(c?) + mu'(w. + I Consider any j. i. Since and sv'(T.) —T = (1 - lt)u'(c?1) + I —TI ) — I ) ÷v'(W —52— or 0 and T. = (ii) (1 + av'(O) < (1 — + u'(w.) — + T. = W. 1 1 or (iii) (1 — 5)u'(c) + u'(w. By strict concavity of hold; Thus, + w.) u and furthermore, (i) can -v, + av(W.) av'(W) 1 + > (1 — u'(w.1)- + v'(0) only one of these three conditions hold for at most one value of is unicuely determined for each T.1 + and + can w) 1 + Ttu'(w. + i, as is t. 1 T. 1 (t.1 = T). 1 Q.E.D. Proofs of Propositions Theorem 4: 1, 2, and The formula for 3 were given in the text. X + X follows directly from Without loss of generality, take substitution into (A17a). k = 1. Then from (A32), dC 1 dW dT dt, dW dW = 1 = 1 - 2(1 = 1 — 2(1 — - = 1 — 2(1 — xN1( - + (1 + XN)lx — + xN_1 (1 + XN)(x + (i - (X + x1)/2)(1 + + X)1(k — — + ;)(1 + XN\ —53— = - 1 2(1 + (av'(Ti + — N)_1(x + 1 as desired. dC. dW = )1 - N. / + 1)_1[(j kN)( - — ÷) M+.av"Tj u"(c) (i N I - N—1 A ) ( —1 mv"(T) —1 — ÷ + ______ From (A31). we have i dT dt. dU1 dW1 N —1 — ) A = (1 = —1 — X) —1 + -,A j + ( v"(T) u"(c) + - XN)_l(kl - (x note - (A — always f cv'(T) — [vu(T)1 2u(C)u(c)' ---- --- Finally, u'(C) —2 j—1 N— 4- +2 + + -A ÷ 2(1 + * )j(k3_1 ÷ 2 4-1)(1 - )1(x1 - that the N— labelling of parents is arbitrary, relabel to make any given parent with whom he shares his children P2 p1, + so av\ that we can and either of the parents Relabelling produces the desired formula. Proposition 4: levels w therefore and By Theorems 2 and 3, W, the unique equilibrium for all symmetric endowment is symzretric. be characterized by the first order condition u'(C) — u'(w ÷ - C) — avl(W_—_C) = o It can —54— Since u and v are concave, the equilibrium value of is also the C solution to u(w ÷ u(C1 sax + 2av( W-C ) — C) + W 0<c<W (note that this yields (Ci) as the first order condition). Since this problem satisfies all the hypotheses of the maximum theorem, its solution is continuous in a =0 = 0 (by assumption (w a. Let C° < C°< W); let ÷W w — C° and it follows that (Ci), v"(T) dC = a and 0 u"(C) + u"(c) noting that C + c0, T° > C, we immediately C°, C° - c°)/2. From Letting denote the equilibrium value for we also have = him aO 0. Thus, 0 c - c and 0 C , , ha a0 have v(T) + = = him R a0 o T + T 0, Since with c = w + W — C, 0. from Theorem 4 we have Next, dC him a+0 = I urn a+0 [(1 — XN)_ i—i jx We + x_l — xN—i÷i XY1a] v"(T) u"(C) 2 + have written this as the limit of the product of two Since Note urn a+0 ) = 1, the limit that this limit of the second expression is does not depend upon i. expressions. 4v'(T)/u'(C). Furthermore, the first —63— As in the proof of Theorem 4, relabelling produces the desired formula for the derivatives with respect to Equilibrium is characterized by the set of first Proposition 8: order conditions 1 Implicitly + T. i+1 + [u"(ci) dT. 4u'(C) There are verify & (w.i+1 + T. i+1 = TV)] 1 2u(.1 differentiating these conditions with respect to 2u"(C.) + = ÷ I — 1 C + + 1 6 yields u"(c.1)J dTi—i ÷ (cj( d6 such equations in N — N dT. is dT. + u unknowns c.1;( (the 1 i+1 dT. dT./do). dô One can by inspection that dT. dó for all i — satisfies these equations. 1 2 From this, it is trivial to verify the desired result. Proposition Q.E.D. 9 dC. dC. Part 1: urn = urn N- 1 > 0 I Recall that dO. 1 1 dC — L __i. dW. 1 dW —56-- Since the right hand side is positive and finite, we have the desired conclusion. Q.E.D. ition5 Fart 1 * ha R N÷ 1: = = 1, Without loss of generality, take him F N- 1 + > 1, For any sequence positive integers 1 k =-{i K k = {i 1 1 choose another sequence of Nk wi±h + Mk First, we prove that + 0. < i < N = = k or N—N k k < = < N = k or N + !'-l)/2 k - Define — N k +2Ci<N = = + 3 < = ic < N = k Note that = dC, do, -j/2 iP + i€Ek aeK, K 1 dC• = c' L 1 —+ \' L * 1 1 -÷ dO, do, 1 1 — do. 1 d'r 1 IcKic (where the final equality follows from the fact that aggregate consumption). dC. 1 d'v dC. 1 — =dW 1 1 = (N — Noting that N v / \1 L" =2 1)'(N dO. 1 ) N dO, ' 1 dO, — 31 3 ' does not alter -57— along with a we dC. —l I — i dC1 L similar expression for and using symmetry around have — = N - - K dC - > 2 1)N--2k — 2 + 2dW dc. + — k + 2 dT. i=21 dt. dT. + k- 1)_i = - 1)_i dt, (1 - - 2) d- = 1 M 1)1NL - i=21 dW dC (2M, dC. !Nk:! N1 + 2 j - (2Mk - 2) - 2 - - 2 2) - - dt dT where the last equality follows from the fact that + T1 = + Recall that the symmetric equilibrium allocation is independent of Mk Further and N — (Nk 1 W1. Now we take limits. + and = 1 C1 p, k + N 1)_i + 0, From N. as k + N(N We are therefore left with lj.m , it is obvious that since dtM • 0 (given then + 1, and - R (A1), > 1, as (Nk desired. 111(Mk 0 K — < i). 1) + 0. —58— Now we argue that N t 2 1 = dC v L ( 1=1 I < 1. urn 2 N' do + I 1 1 = (N 1Y(N <(N - i1 {N( I I) - — ÷ IN + it From Theorem 4, — + (II + I )}/ - dC, is clear that > 0 for all i, It follows that 1 do, > 0 for all the first order condition for some i, othen,'ise would be violated. Furthermore, from the derivation in the proof of proposition 4, it is clear that '>0 < (N — 1r1N( = (N — 1)N and ÷1) From this, it is immediately clear that dO, Part 2: lim —'i- = N- dm. 1 > 0. + lim F Thus, N( ÷)}/2 1 < 1. dO. lim ''s' > 0. N+o dW.1 As before, without loss of generality, take dO — — N dO — 1\1rN L __.i. dW1 i = 1. We know that 4c.i dW Since the symmetric equilibrium allocation is independent of second term disappears in the limit. dO1 as desired. rent N, the The first term converges to Finally, using Theorem 4, —59-. (1 )1 dC, - = Proposition 6: exists such that N Fix any a. We for all N > Na N(s) = Na . Consider 300e sequence N N(s k ) for all a. 0, R + k=> I. SVU(TL ow that — F sa ,Na / Then, for each Further F0 ) (2+ Q.E.D. li 1 < a. For each ÷ a, > as I - F is independent of K a N, so = 1, so there k , let wftn . Râ Since 0 follows from proposition 4. Proposition 7: dO. = [(1 - By Theorem 4, xNl(1 - F)lv(T (2 )(i + From this formula, it is possible to deduce toe following three properties: dO. (i) decreasing in (iU (111) lim N dO, sO < 4 i Property (i) IS straightforward to check. the fact that (1 — XN)_1(l + XN) is Propeerty (ii) follows from decreasing in property (iii) as follows. In the proof of proposition 4, we showed that N. We establish —60.. °-i ['cj[4N u'(c°) - urn [(i - aO u(C°) u'(C°) + Thus, hr = dW. u'(C°) ÷ = Now suppose there rnust ftuq°) [ N1[ that u'(c exist a sequence a k. all > 0 ut(C°) < ] ) propositionis false Then by property (ii, converging to <akNk>kl 1 for !iL]—1 [ v"(T°) ut(C°) u"(c°) ) + u(C the [4N u"(c°) (0,) such that > i > o By Proposition 3, we can without loss of generality take for all k. Choose k* such that — a for some all p. a > C. k Consider a subsequence such that p > > WNk,ak dWlNk*,ak N so him p- dC1 dW 1 1 N, ,ak But this contradicts property (iii). N.÷ K p Then, by property (ii), > > N N, ÷ a * ÷ a k for —61— Theorem 5: Each player's strategy set is the interval (he chooses a transfer belonging to this interval). compact and convex. T.. i without if we have this as Obviously, this is utility is cor.tinuous, and concave in As in Theorem 1, we immediately have Next, note that all p's Further, [O,W/21 existence. an equilibrium where = t. 1 for T,1 constraint, this configuration the constraint is imposed, since the effect imposing a relains an equilibrium when of this is only to limit deviations. the existence of a symmetric Thus, equilibrium when endownnts are symritric is giaranteed by Theorem 2. Finally, we come to uniqueness. Throughout our argument, we will first order refer to parents' solution, can written as be 2u'(C.) there and (T.) 1 i=1 exists some C. < C.. 1 1 = (u'(c) + contrary to tne theorem, that Now suppose, equilibria, conditions, which, for an interior . (TY1 1 i=1 for which i Inspection of < T I'. i+1 Now we procede T i+m C. i+m k. < T i+m , > T. C. . i÷rn arid p.1 '5 first order condition (recalling that By 's budget T.1 — is1 , and by induction. c i÷m two distinct P.'S budget constraint, By either c. < c., or loss of generality, assume the latter. constraint, are Then, without loss of generality, T. > is strictly concave) reveals that Without tnere < c 's p. 1+nl . i-s-rn T.1 Then by Then, by first p i÷m < 01+1. budget k11's T.i+i — < T. i+i Suppose 0.1 that is odd, m 'S budget constraint, first order condition, c.i÷m+1 > c.i÷÷1 . and constraint, T.i+m+i > T. i+m+1 , T. 1+m+1 — u T.i+m+i By > —62-- T, z+m I'. i+m that Suppose < T. is even, I. 1+m i+m a p.14-rn's budget constraint > i+m+-1 '5 budget i+rn+1 Then, by first order condition, i+m 'S p By 1+::: > c. c. i÷m constraint, T.i.+m=1 < T. i+rn+1' -T.i+m+1 >T.11-rn -T.i+m T. i÷rn+1 Applying induction, we see that for all rn. a taking The Theorem 6: substitution into dC1 +m 1÷m+1 i÷m±1 = N - . >> i+m 1 1 yields a contradiction. X + X formula for follows directly from From (B17), we have (B12a). dT1 = 1 — 2 = I 4 — = 1 + (X + = 1 — (1 - + 2)(1 (1 + X)(1 = —2X(1 as j.+rn k By i÷rn+1 ) C. C. and , + XN_1)(l + — — XN)(l - X)1(1 N)_1/x x)1(l Xl)(1 - x)1(l — + + — — desired. Also from (517), dC, - — dT. -2 dW1 dW1 = - = (x = x1(x - (1 _1 + ? —(1 + + 2)(1 X)(1 - — N ?) XN)1(l —1 — + XN_i+1) (x — x—1 ) —1 )_1(_1 (x—1 + + xN— XN_i+I) +1 and —64— Since the distribution of endowments is symmetric before and after change in W, the equal division constraint is not binding; thus, the formula for a = a is exactly as in the proof of Proposition 4 (taking dC/dW 0): dC Since < 0 < u"(c)[u'(C) c and since 1 (N — 1) limit of = is 0. and = him (N — N+ From Theorem C are independent of N, the Thus, dC. ha + - lr'N dC i i 6, this last term equals dC = hIm N÷ —2X(1 — 1 X)1, which is strictiy positive as desired. Part im 2: R 1 Nk proof of Proposition 5. R a. 1 N (I! dC. = 1=1 1 as + For any > k + and k as in the dc. 1 dO. C k' Nkt Note that I]/2 dc. { Consider some define + 1 1 lCPk , icKk satisfying dC. + I j—'- 1 0 < a < —2X(1 1 - I/2 iKk do. + X)1. I By the argument in dC Part 1, there exists argument N1 similar to that such that for all N > N1, —i > a. By an given-in Part 1, it is easy to show that —65— = Consider then some exists n for N > all + 0 > satisfying sucn that N2 k(1 = X)(1 > X) - + —X(1 . N2, (0 X)(i — X)1. There Thus, for N > R 1 1 > — 2k do. / + dC, + i iKk 1 L i 'k + 1 do. 1 L Proceding exactly as in the proof of Proposition 5, for N > k From (B17), 1)_i Nk[l = First, 1 -(i — Nk 2 — + that R 1 > Note XN)_l(k assume that — N 1 + Note that p.'s well—being (2Mk -2) as k + that > - + . Thus, m1n{c,} 1. Q.E.D. that 1)1(x + x_I is even. dT.1/dW1 is given by C min{s,n} is positive if and only if odd. — dTM it is clear Proposition 10: * it then follows dT — us J rran{,'1} naxN1,N2}, R1 dT./dW1 + I ijK, ,j + 2)( + Then it is easy to check that is odd. < 0, and Now consider dT.1/dW1 < 0. p, with Recall that -66- I This u(V is strictly must declins as — 2T) + (u(w + increasing in WI tins. Nezt,assussthat N We + T1_1 T3_1) N—(j—1)<j—I, then dT1/dW1>O it I * j < N/2 thsn (j + 1) I < N — and dT31/d%ç- implies that P1 + Thus, Tj+e isodd. Thsnitissasytocbsokthatif i—I <N—(i—I),dT1/dW1>O iff dT1_1/dW1 u(w + p1s utility analogously for j even. and reason + Further, 3 isodd. if itt — N—i isodd. Accordingly, (Ci + I) — 1], and so ann.stivs itt- 1 isodd. is wons offr ccnvsrsely it 3 As is--svsn. above; If, this - en the otherhand,N—j+2<N/2,then N—((j—I)—1]<(j—I)—I,and and are nsptive itt N is odd. Lain, this dT3_1/dW1 4?1 1/dW is worse ottj ocnvensly it N — j is nen. Q.E.D. implies — p1 j —67— References Andreoni, James R. [1986], impure Altruism and Donations to Public Goods: A Theor of 'Wa-Glow' Giving," mirneo, University of Wisconsin. Barro, Robert S. 1974, 'Are Government Bonds Net ealtn?" Journal of Political Economy 81, 1095—1117. Bernheim, B. Douglas z1986] "On the 7oluntary end Involuntary Provision of Public Goods," American Economic Review 76, 789—793. Berheim, B. Doug.as and Kyle Bagwell L1988], "05 Everything Neutral?" Journal of Political Economy 96, 308—336. Bernheim, B. Douglas, Andrei Shleifer, and wrence Summers [1985], "The Strategic Bequest Notice," Journal of Political Economy 93, 1045— 1076. [i91], 'The Diameter of Random Graphs," Transactions Bollabas, B. the American Nathemetical Society 267, No. 1. of Debreu, Gerard [1952], "A Social Equilibrium Existence Theorem," Proceedings of the National Academy of Sciences of the U.S.A. 38, 886—893. John [1967—68], "Games with incomplete Information Played by Bayesian Players," nagement Science 14, 159—82, 320—34, 486—502. Harsanyi, Menchik, Paul 0. [1980], "Primogeniture, Equal Sharing, and the U.S. Distribution of Wealth," Qrterly Journal of Economic 94, 299—316. The Structure of Intrafamily Linkaes p1 / / parents N 7- p2 I \ children I \ p5 N 7 p4 / / / pative Figure 2: Statics with Frictcon / I' \ / / / r / / -4 A / / N N N --.j-3 j-2 Figure 3: 2ative Statics with Egaitanarianism * urn dC. N- dwk / I I / f / / / / / __________________________ j-3 . j-2 \ / j-i / \ \ \ \ j j+1 /-_ ' ' j+2 / / \/ j+3