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BEQUEST MOTIVES, SOCIAL SECURITY, AND ECONOMIC GROWTH.

JUNXI ZHANG [*]

This article conducts a systematic analysis of four bequest motives in a simple model of endogenous growth. It analyzes how bequest motives affect fertility savings, growth, and the effects of pay-as-you-go Social Security. It is found that altruistic and exchange motives give equivalent outcomes if the discount factors are set the same. The outcome under the joy-of-giving motive may involve a higher bequest ratio, higher saving, and better growth rates than that under altruism. If bequests are accidental, the equilibrium values will depend on the probability of survival. Moreover, the results provide testable implications on bequest motives. (JEL D1, D9, H55, J1)

I. INTRODUCTION

Although the importance of bequests in aggregate savings has been established by Kotlikoff and Summers (1981), who reported that 80% of U.S. household wealth is inherited wealth, what bequest motives lead to inheritance remains an unsolved problem. Yet it is well known that bequest motives have important implications for the behavior of financial markets, the macroeconomic impacts of fiscal policies, and the intergenerational transmission of inequality in the distribution of wealth.

At least four hypotheses for the existence of bequests have been discussed in the literature: (1) Bequests may arise from intergenerational altruism, that is, parents obtain utility from their heirs' utility as well as from their own consumption (Barro, 1974); (2) the prospect of bequests is used by parents to induce children to behave as desired by parents (Bernheim et al. 1985); (3) bequests may arise from the "joy of giving," that is, parents leave bequests simply because they obtain utility directly from the bequest (Yaari, 1964); [1] and (4) bequests may be the unintentional by-product of precautionary savings and a stochastic date of death in the absence of an annuity market (Abel, 1985). Following Barro (1974) and Becker (1974), the altruistic bequest motive has been used most frequently.

However, the enormous empirical work using micro data has not led to conclusive evidence. On the one hand, Bernheim et al. (1985), Cox (1987) and Cox and Rank (1992) show that bequests are more consistent with exchange than altruism. Altonji et al. (1992, 1997) also show that altruism is decisively rejected. Moreover, Wilhelm (1996) finds little support for an altruistic theory of bequests. On the other hand, a recent reexamination of the exchange motive by Perozek (1998) shows that the finding of Bernheim et al. is not robust. Laitner and Juster (1996) find some evidence of altruistic bequests. Therefore, a definite conclusion on bequest motives seems far from over. Indeed, though altruism has not been supported by most of those empirical studies, it remains a plausible hypothesis and continues to be used frequently (see, e.g., Coate [1995] and Davies and Zhang [1995]). Abel (1985) argues theoretically that although the simple life cycle model without bequest motives is an inadequate description of the savi ng behavior in the United States as found in Kotlikoff and Summers (1981), accidental bequests by selfish consumers can account for a sizable fraction of aggregate wealth. However, little formal empirical work has been done to test whether bequests are accidental or arise from joy of giving.

Given the lack of consensus on bequest motives, it is interesting to conduct a systematic theoretical analysis of different bequest motives in a particular context. Becker et al. (1990) and Caballe (1995) assume altruism in their models of economic growth, Wiedmer's (1996) analysis of Social Security and growth does not consider bequests. Except Ihori (1994), there seems to be no analysis of other bequest motives in the context of economic growth. The purpose of this article is to analyze how different bequest motives affect fertility, savings, and economic growth. The article will also examine how the effects of pay-as-you-go Social Security depend on bequest motives. We ask two questions: How do bequest motives affect equilibrium results? And, relatedly can the results provide testable implications that differentiate bequest motives? The first question is important in the sense that it may tell us to exercise cautions on drawing hard conclusions from any specific bequest motive. A positive answer to the se cond question becomes crucial particularly in light of inconclusive evidence from empirical studies based on micro data. It is of considerable value if empirical work can make use of aggregate data on fertility, saving, growth, and Social Security in discriminating bequest motives.

Recent contributions in the endogenous growth theory have provided various alternative formulations of endogenous per-capita income growth. To simplify the analysis, this article adopts a simple growth specification--Romer's (1986) type of capital externality. [2] The advantage of this formulation is that the whole analysis is tractable and it allows an intuitive development of the effects of Social Security over a variety of bequest motives. Note that, however, our model below differs from that of Romer (1986) in that we allow for intergenerational transfers and endogenous fertility.

The rest of the article is organized as follows. Section II outlines the basic framework with Romer's style of endogenous growth. This section sets the stage for sections III-V. Section III solves solutions for alternative bequest motives, and section IV compares the solutions. Section V analyzes effects of Social Security under alternative bequest motives. Section VI discusses empirical implications and evidence, and section VII concludes.

II. THE MODEL

The economy produces a single storable commodity. Production for this commodity is organized in competitive markets where firms maximize profit in each period given factor prices. A representative firm i uses capital [[k.sup.i].sub.t] and labor [[l.sup.i].sub.t] to produce output [[y.sup.i].sub.t] through a production function [[y.sup.i].sub.t] = f([[k.sup.i].sub.t], [[l.sup.i].sub.t], [k.sub.t]), where [k.sub.t] represents the economywide capital stock and variables are expressed in per-capita terms. As standard in the literature, for example, Romer (1986), to obtain explicit solutions, production function f is assumed to be homogeneous of degree one in [[k.sup.i].sub.t] and [[l.sup.i].sub.t] when [k.sub.t] is held constant. In addition, to allow for balanced growth, we assume that f exhibits constant returns in [[k.sup.i].sub.t] and [k.sub.t] from a social point of view. Thus, we adopt the Cobb-Douglas form,

(1) [[y.sup.i].sub.t] = f([[k.sup.i].sub.t], [[l.sup.i].sub.t],[k.sub.t]) = A[([[k.sup.i].sub.t]).sup.[beta]][([[l.sup.i].sub.t]).sup.1-[beta]][( [k.sub.t]).sup.1-[beta]],

where A is a scale parameter and [beta] is the capital share. Profit maximization implies that factors are paid their marginal products, expressed at the economywide level:

(2a) [r.sub.t] = [partial][y.sub.t]/[partial][k.sub.t] = [beta]A,

(2b) [w.sub.t] = [partial][y.sub.t]/[partial][l.sub.t] = (1 - [beta])[Ak.sub.t], where [w.sub.t] is the wage rate and [r.sub.t] is the interest rate. In deriving equation (2), we have imposed the equilibrium condition [k.sub.t] = [k.sub.t]. As usual, the infinitesimal influences of each individual capital stock on the aggregate stock and thus on the productivity of labor have been ignored.

At each period t there are two overlapping generations of individuals, the young and the old. A typical parent works, consumes, saves, and rears children in the first period of life (when young) and enjoys the fruits of the savings in the second period (when old). A young parent chooses to have (1+ [n.sub.t]) children. Let v, where 0 [less than] v [less than] 1, denote the units of time needed to rear a child and normalize the parent's time endowment to unity. Then each parent will supply 1 - (1 + [n.sub.t])v units of time to work. Denoting [[c.sup.t].sub.t] ([[c.sup.t].sub.t+1]) be the units of the homogenous final good that a representative member of the generation born in period t consumes when young (old), the lifetime budget constraints are

(3a) [[c.sup.t].sub.t] = [1 - (1 + [n.sub.t]) v] (1 - [s.sub.t]) [w.sub.t] + [b.sub.t],

(3b) [[c.sup.t].sub.t+1] = (1 + [r.sub.t+1])[s.sub.t][1 - (1 + [n.sub.t])v][w.sub.t] - (1 + [n.sub.t])[b.sub.t+1],

where [b.sub.t] is the inheritance received when young, [b.sub.t+1] is her per-child bequest, and [s.sub.t] is her saving rate.

Product market clears when aggregate investment equals to savings (see, for example, Blanchard and Fischer [1989]):

(4) [L.sub.t+1][1 - (1 + [n.sub.t+1])v][k.sub.t+1] = [L.sub.t][s.sub.t][1 - (1 + [n.sub.t])v][w.sub.t].

Note that the term, [L.sub.t+1][1 - (1 + [n.sub.t+1])v], on the left-hand side represents the effective labor supply of the next generation. Making use of equations (2b), (4), and [L.sub.t+1] = ([1 + [n.sub.t])[L.sub.t], per-capita capital stock can be shown to grow at the rate

(5) [g.sub.t] = [gamma](1- [beta])[As.sub.t]/(1 + [n.sub.t]) - 1,

where the parameter is defined as [gamma] [equivalent] [1 - (1 + [n.sub.t])v]/[1 - (1 + [n.sub.t+1])v], which equals one along the balanced growth path, as shown in the next section. It is clear from equation (5) that the growth rate depends positively on the gross saving rate but negatively on fertility. Because different bequest motives affect the saving and fertility rates, they will eventually affect the rate of economic growth. The subsequent section will conduct a balanced growth analysis on the effect of alternative bequest motives on this rate.

III. ALTERNATIVE BEQUEST MOTIVES

We consider four different bequest motives in this article: altruistic, exchange, joy of giving, and accidental. To facilitate the comparison, we will adopt a general specification of the utility function by modeling the number of children in the parent's utility. This will allow us to focus on the different effects of alternative bequest motives.

Altruistic Model

In this subsection, we examine a popular form of bequest motive--the parent is altruistic in that she cares about the well-being of the child. Let the utility attained by each child be [V.sub.t+1], the utility of each adult is

(6) [V.sub.t]([b.sub.t]) = [U.sub.t]([[c.sup.t].sub.t], [[c.sup.t].sub.t+1], 1 + [n.sub.t]) + [rho][V.sub.t+1] ([b.sub.t+1]),

where [rho] [epsilon] (0, 1) is the degree of pure altruism. To maintain tractability and obtain analytical solutions, we adopt a logarithmic instantaneous utility function throughout the article:

(7) [U.sub.t]([[c.sup.t].sub.t], [[c.sup.t].sub.t+1], 1 + [n.sub.t]) = [[alpha].sub.1] ln [[c.sup.t].sub.t] + [[alpha].sub.2] ln [[c.sup.t].sub.t+1] + [[alpha].sub.3] ln (1 + [n.sub.t]).

A typical agent's decision problem is then given as follows: She seeks to maximize (6), by choosing the consumption profile ([[c.sup.t].sub.t], [[c.sup.t].sub.t+1]), the saving rate [s.sub.t], the number of children 1 + [n.sub.t], and bequest [b.sub.t+1], subject to constraints (3a)-(3b). For an interior solution, utility maximization yields the following first-order conditions (FOCs):

(8a) [[alpha].sub.1][1 - (1 + [n.sub.t])v][w.sub.t]/[[c.sup.t].sub.t] = (1 + [r.sub.t+1])[[alpha].sub.2] x [1 - (1 + [n.sub.t])v][w.sub.t]/[[c.sup.t].sub.t+1],

(8b) v([[[alpha].sub.1] (1 - [s.sub.t])[w.sub.t]/[[c.sup.t].sub.t] + [[[alpha].sub.2] (1 + [r.sub.t+1])[s.sub.t][w.sub.t]/[[c.sup.t].sub.t+1]]) + [[alpha].sub.2][b.sub.t+1]/[[c.sup.t].sub.t+1] = [[alpha].sub.3]/(1 + [n.sub.t]),

(8c) [[alpha].sub.2](1 + [n.sub.t])/[[c.sup.t].sub.t+1] = [rho]([dV.sub.t+1]/[db.sub.t+1].

Using the envelope theorem gives

(9) [dV.sub.t+1]/[db.sub.t+1] = [[alpha].sub.1]/[[c.sup.t+1].sub.t+1].

Accordingly, the FOC (8c) becomes

(8c') [[alpha].sub.2](1 + [n.sub.t])/[[c.sup.t].sub.t+1] = [rho][[alpha].sub.1]/[[c.sup.t+1].sub.t+1].

In deriving the FOCs, the standard transversality condition has been omitted. Moreover, it can be easily verified that the second-order conditions are satisfied.

Throughout the article, we focus on a balanced growth path (BGP) equilibrium, which is a collection of variables {[[c.sup.t].sub.t], [[c.sup.t].sub.t+1], [k.sub.t]} over time such that these extensive variables grow at constant rates, and intensive variables such as the saving rate, [s.sub.t], and the fertility rate, 1 + [n.sub.t], are constant.

It is well known that there are no transitional dynamics in endogenous growth models with Romer's type of capital externality, because it can be eventually reduced to the simple "AK" model. Thus, our discussion in what follows will be focused on the BGP. We first study the relationship between fertility and growth. Notice that the interest rate, r, is constant as given by equation (2a), so we define a constant parameter z [equivalent] (1 + [beta])A/(1 + r). Then equation (5) implies that

(10) (1 + g)(1 + n)/(1 + r) = zs.

Defining the bequest-income ratio, [[delta].sub.t], as

[[delta].sub.t] = (1 + [n.sub.t])[b.sub.t+1]/{(1 + [r.sub.t+1])

x [1 - (1 + [n.sub.t])v][w.sub.t]},

then it is straightforward to show that balanced growth requires that the bequest-income ratio to be constant. Also, it proves convenient to express variables as functions of the saving rate, for comparison in the next section. The following proposition discusses the effects of the saving rate on the equilibrium values of the bequest-income ratio, [delta], the fertility rate, 1 + n, and the growth rate, 1 + g.

PROPOSITION 1. In the altruistic model, the bequest-income ratio and the growth rate are increasing functions of the saving rate, while the relationship between the fertility rate and the saving rate is ambiguous, depending on the model's parameter values.

Proof. Algebraic manipulation of equations (8a) and (8b) with the use of the budget constraints (3a)-(3b) and (10) yields [3]

(11) [delta] = [([[alpha].sub.1] + [[alpha].sub.2])s - [[alpha].sub.2]]/[[[alpha].sub.1] + [[alpha].sub.2]/(zs)],

(12) 1 + n = (s/v)[{s + ([[alpha].sub.1]zs + [[alpha].sub.2])

/[([[alpha].sub.2] + [[alpha].sub.3])(1 + z) - [[alpha].sub.2]

-([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3])zs]}.sup.-1].

Equation (11) suggests that [delta] is an increasing function of s. However, an inspection of equation (12) indicates an ambiguous functional relation between 1 + n and s. The result for the growth rate, 1 + g, follows immediately from equations (5) and (12)

(13) 1 + g = (1 - [beta])Av{s + ([[alpha].sub.1] zs + [[alpha].sub.2])

/[([[alpha].sub.2] + [[alpha].sub.3])(1 + z) - [[alpha].sub.2]

-([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3])zs]},

which increases as s rises.

The result that high bequest is associated with high saving is simply the reflection of the arbitrage condition across generations of (8c'). Higher saving raises [[c.sup.t].sub.t+1], which can only be matched by raising [[c.sup.t+1].sub.t+1] via bequest (holding [n.sub.1] constant). For the growth rate, note that in this model, apart from being a factor of production, capital also generates external effects that will be translated into persistent growth. Finally, the ambiguous result on fertility arises from two opposing income and substitution effects. On the one hand, high saving reduces the parent's current income, thereby reducing her demand for children. On the other hand, if the current consumption reduces more than the increase in saving, fertility tends to go up.

Given the importance of saving in Proposition 1, it is essential to find the determinants of the saving rate in the altruistic model. From equations (8a), (8c'), and (10), we obtain

(14) s = [rho]/z.

Note that because the saving rate is always smaller than one, we impose model parameters to satisfy z [greater than] [rho]. It is clear from equation (14) that the main determinant of the saving rate is the degree of altruism. The more the parent cares about the child's utility, the more she will save to have enough resources in the second period so that more bequests will be available for her child.

It is worth mentioning that substituting (14) into (10) immediately reveals the following relationship: (1 + g)(1 + n)/(1 + r) = [rho]. This is the standard "modified golden rule" in an infinite-horizon model. It has an important implication that the economy is dynamically efficient, suggesting that there is no overaccumulation of capital over time in this model once an altruistic individual takes her children's utility into consideration. Abel et al. (1987) derive a sufficient condition for dynamic efficiency and conclude that this condition is almost surely satisfied for the U.S. economy.

Exchange Model

This subsection considers a different exchange motive for private transfers from the donor (the parent) to the recipient (the child). Similar to the existing models in the literature with strategic bequest motives, for example, Bernheim et al. (1985) and Cox (1987), we assume that the parent bequeaths each child in exchange for services provided. Let [a.sub.t+1] be the services the offspring provides for her parent; alternatively, [a.sub.t+1] can be taken as the services that the parent expects from each child. Then, in this case, the motive for the parent to leave bequest to each child, [b.sub.t+1], is to exchange for services from them, [a.sub.t+1]. A simple reduced form of the parent's utility function is given by

(15) [V.sub.t]([b.sub.t], [a.sub.t]) = [U.sub.t] + [[alpha].sub.4] ln [a.sub.t+1] - [[alpha].sub.5] ln [a.sub.t],

where [U.sub.t] is given in equation (7) and [[alpha].sub.4] and [[alpha].sub.5] are constant parameters. [4] [[alpha].sub.4] represents the utility of service from each child to the parent, and [[alpha].sub.5] represents the disutility to the parent associated with providing services to her parent one period earlier. In general, we assume that [[alpha].sub.4] [not equal to] [[alpha].sub.5].

In addition to the parent's budget constraints (3a) and (3b), a behavioral constraint should also be considered in the parent's maximization problem. The child will offer services if and only if the change in utility from entering into the transfer-services relationship with the parent is nonnegative. The nonnegative self-selection constraint is

(16) [V.sub.t+1]([b.sub.t+1], [a.sub.t+1]) [greater than or equal to] [V.sub.t+1],

where [V.sub.t+1] is the child's "threat" utility level, under which minimum or possibly no transfer-service exchange occurs between the parent and the child. [5]

Let [[lambda].sub.t] be the Lagrange multiplier for the self-selection constraint (16). Each working adult maximizes (15) subject to constraints (3a)-(3b) and (16) by choosing the vector ([s.sub.t], [n.sub.t], [[delta].sub.t], [a.sub.t+1]). The resulting FOCs for this model are

(17a) [[alpha].sub.1][1 - (1 + [n.sub.t])v][w.sub.t]/[[c.sup.t].sub.t]

= (1 + [r.sub.t+1])[[alpha].sub.2][1 - (1 + [n.sub.t])v]

X [w.sub.t]/[[c.sup.t].sub.t+1],

(17b) v[[[alpha].sub.1](1 - [s.sub.t])[w.sub.t]/[[c.sup.t].sub.t] + [[alpha].sub.2]

X (1 + [r.sub.t+1])[s.sub.t][w.sub.t]/[[c.sup.t].sub.t+1]]

+ [[alpha].sub.2][b.sub.t+1]/[[c.sup.t].sub.t+1] = [[alpha].sub.3]/(1 + [n.sub.t]),

(17c) [[alpha].sub.2](1 + [n.sub.t])/[[c.sup.t].sub.t+1] = [[lambda].sub.t][[alpha].sub.1]/[[c.sup.t+1].sub.t+1],

(17d) [[alpha].sub.4]/[a.sub.t+1] = [[lambda].sub.t]([[alpha].sub.5]/[a.sub.t+1]).

Note that equations (17a) and (17b) are the same as equations (8a) and (8b) in the altruistic model, and equation (17c) differs from (8c') only in that the discount rate [rho] in the right-hand side of (8c') is replaced by [[lambda].sub.t]. From equation (17d), [[lambda].sub.t] = [[alpha].sub.4]/[[alpha].sub.5], which is the intergenerational ratio of taste for attention (hereafter, the attention taste ratio). This ratio can also be viewed as a discount factor similar to that in the altruistic model. Therefore, similar equilibrium solutions can also be obtained for the exchange model, and the results are given in Table 1.

Joy-of-Giving Model

In the joy-of-giving model, we assume that the parent cares about the amount of bequests rather than the welfare of her children. The parent's utility function is given by

(18) [V.sub.t]([b.sub.t]) = [U.sub.t] + [[alpha].sub.6] ln [b.sub.t+1].

The FOCs for this model are

(19a) [[alpha].sub.1][1 - (1 + [n.sub.t])v][w.sub.t]/[[c.sup.t].sub.t]

= (1 + [r.sub.t+1])[[alpha].sub.2][1 - (1 + [n.sub.t])v]

X [w.sub.t]/[[c.sup.t].sub.t+1],

(19b) v[[[alpha].sub.1] (1 - [s.sub.t])[w.sub.t]/[[c.sup.t].sub.t] + [[alpha].sub.2](1 + [r.sub.t+1])

X [s.sub.t][w.sub.t]/[[c.sup.t].sub.t+1]] + [[alpha].sub.2][b.sub.t+1]/[[c.sup.t].sub.t+1]

= [[alpha].sub.3]/(1 + [n.sub.t]),

(19c) [[alpha].sub.2] (1 + [n.sub.t])/[[c.sup.t].sub.t+1] = [[alpha].sub.6]/[b.sub.t+1].

Notice that equations (19a) and (19b) are the same as (8a) and (8b) in the altruistic model, and equation (19c) differs from (8c') in that the child's first-period consumption [[c.sup.t+1].sub.t+1] in the right-hand side of (8c') is replaced by the amount of the parent's bequests [b.sub.t+1]. Once again, we can obtain the equilibrium solutions in this model similarly to the previous two models.

Accidental Model

As a final variant of introducing bequest motives into the basic model, we model bequests as an unintentional by-product of precautionary savings due to uncertainty in the date of death. Following Abel (1985), we assume that there is a probability p that the parent dies at the end of her period of life after having children. If she survives to the second period of life, she does not work and consumes all her savings with nothing left. But if the parent dies at the end of the first period, her unconsumed wealth solely constituted by savings (including the accrued interest) will be equally divided among her children.

To account for the uncertain life, we adopt a utility function similar to (7):

(20) [U.sub.t]([[c.sup.t].sub.t], [[c.sup.t].sub.t+1], 1 + [n.sub.t]) = [[alpha].sub.1] ln [[c.sup.t].sub.t] + (1 - p)[[alpha].sub.2] ln [[c.sup.t].sub.t+1] + [[alpha].sub.3] ln(1 + [n.sub.t]),

which differs from (7) only by the presence of the probability of being alive in the second period, 1 -- p. According to the specification in (20), the parent derives no utility from leaving (accidental) bequests. As an extension to Abel (1985), our analysis below allows for both endogenous fertility and growth.

The parent maximizes the utility function in (20) subject to her lifetime budget constraints:

(21a) [[c.sup.t].sub.t] = [1 - (1 + [n.sub.t])v](1 - [S.sub.t])[w.sub.t] + [b.sub.t],

(21b) [[c.sup.t].sub.t+1] = (1 + [r.sub.t+1])[S.sub.t][1 - (1 + [n.sub.t])v][w.sub.t],

where [b.sub.t] is the bequest that the individual may receive from her parent when she is born. Because different individuals receive bequests of different sizes depending on the mortality history of the earlier generations of their families, this model differs from the earlier models in this section by featuring heterogenous agents. Hence, modifications are needed to convert variables into per-capita ones (i.e., per person in the young generation).

As shown in the appendix, the (aggregate) saving, bequest, fertility and growth rates are given by

(22a) S = (1 - p)[[alpha].sub.2][1 + p(1 - p)/z]/[[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]],

(22b) [delta] = p(1 - p)[[alpha].sub.2][1 + p(1 - p)/z]/[[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]],

(22c) 1 + n = (1/v)[[alpha].sub.3][1 + p(1 - p)/z] / {[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3] X [1 + p(1 - p)/z]},

(22d) 1 + g = (1 - [beta])AS/(1 - p)(1 + n)

= {(1 - [beta])Av[[alpha].sub.2]/[[alpha].sub.3]} X {[[alpha].sub.1] + (1 - p) [[alpha].sub.2] + [[alpha].sub.3] X [1 + p(1 - p)/z]}/[[[alpha].sub.1] + (1 - p)[[alpha].sub.2]].

IV. COMPARISON OF MODELS WITH ALTERNATIVE BEQUEST MOTIVES

In this section we compare the saving, bequest, fertility, and growth rates under the four outcomes--altruistic (al), exchange (ex), joy-of-giving (jg), and accidental bequests (ac), where subscripts are used to distinguish alternative models. We choose the altruistic model as the benchmark to conduct the analysis. For ease of comparison, equilibrium solutions under each case are summarized in Table 1 and comparisons of these results are given in Table 2.

The following proposition presents the results on the comparison between the altruistic model and the exchange model.

PROPOSITION 2. Suppose the preference parameters are given such that [[alpha].sub.4]/[[alpha].sub.5] = [rho], then the solutions are identical for the altruistic and the exchange models; that is, [S.sub.al] = [S.sub.ex], [[delta].sub.al] = [[delta].sub.ex], [n.sub.al] = [n.sub.ex], and [g.sub.al] = [g.sub.ex]. However, if [[alpha].sub.4]/[[alpha].sub.5] [greater than] ([less than])[rho], the exchange model yields higher (lower) saving, growth rates and bequest ratio, and the fertility rates under the two models are indeterminate; that is, [S.sub.ex] [greater than] ([less than]) [S.sub.al], [[delta].sub.ex] [greater than] ([less than])[[delta].sub.al], [g.sub.ex] [greater than] ([less than])[g.sub.al], but [n.sub.ex] may be higher or lower than [n.sub.al].

Proof. Substituting t = [[alpha].sub.4]/[[alpha].sub.5] from equation (17d) into (17c) and making use of the assumption that [[alpha].sub.4]/[[alpha].sub.5] = [rho] yield an equation identical to (8c'). Because equations (17a) and (17b) are equivalent to (8a) and (8b), respectively, the solution set (17) for the exchange model is the same as the set (8) for the altruistic model. Of course, direct comparisons of the two models in Table 1, in conjunction with Proposition 1 yield the same conclusions.

Proposition 2 bears strong intuition. Recall that [[alpha].sub.4]/[[alpha].sub.5] represents the attention taste ratio or the discount factor in the exchange model, and [rho] represents the discount factor in the altruistic model. This proposition implies that when these two ratios are the same in logarithmic utility functions, the two models generate identical solutions. If [[alpha].sub.4] [greater than] [rho][[alpha].sub.5], people discount less in the exchange model and so would demand more services. As a result, higher bequest must be left, which in turn induces higher saving and growth.

Next, we compare the solutions for the joy-of-giving model with those of the altruistic model.

PROPOSITION 3. When the preference parameter are given such that [[alpha].sub.6] [greater than or equal to] [rho][[alpha].sub.1], compared to the solutions of the altruistic model, the equilibrium outcome in the joy-of-giving model involves a higher saving rate, growth rate, and bequest ratio, but indeterminate fertility; that is, [S.sub.jg] [greater than] [S.sub.al], [[delta].sub.jg] [greater than] [[delta].sub.al], [g.sub.jg] [greater than] [g.sub.al], but [n.sub.jg] may be higher or lower than [n.sub.al]. However when [[alpha].sub.6] is sufficiently small such that [[alpha].sub.6] [much less than] [[alpha].sub.1], the above results will be reversed except that the fertility rate remains indeterminate.

Proof. We prove this proposition only for the case of [[alpha].sub.6] [greater than or equal to] [[rho][alpha].sub.1], as similar proofs can be made for other cases. There are two alternative ways. First, as the FOCs (19a) and (19b) in the joy-of-giving model are identical to the FOCs (8a) and (8b) in the altruistic model, it remains to compare (19c) with (8c'). If [[alpha].sub.6] = [[rho][alpha].sub.1], the marginal utility in the righthand side of (19c) becomes [[rho][alpha].sub.1]/[b.sub.t+1], whereas the marginal utility in the right-hand side of (8c') becomes [[rho][alpha].sub.1]/[[c.sup.t+1].sub.t+1]. Because [b.sub.1+1] [less than] [[c.sup.t+1].sup.t+1] in the budget constraint (3a) updated for one period, the comparison yields that s = [rho]/z in the altruistic model as shown in equation (14) and s [greater than] [rho]/z in the joy-of-giving model. Hence, other results follow by Proposition 1.

Second, we substitute equations (10) and (12) into (19c) under the assumption that [[alpha].sub.6] = [[alpha].sub.1] to have an expression for the bequest ratio [[delta].sub.jg] as in Table 1. Equating it to equation (11), we yield an expression for the saving rate [S.sub.jg] also shown in Table 1. Given our usual condition on parameter restrictions [rho]/z [less than] 1, we obtain that [S.sub.jg] [greater than] [S.sub.al]. Proposition 1 then gives all other results. [6]

Similar to the discussion after Proposition 2, if we equate the two discount rates [[alpha].sub.6]/[[alpha].sub.1] and [rho], the marginal utility that she obtains from giving bequests in the joy-of-giving model is higher than that in the altruistic model, as implied by equations (19c) and (8c'). Consequently, more bequests will be given in the former, which is then associated with more saving (see again [8c']). Higher saving leads to higher growth. Moreover, because the growth rate depends positively on the saving rate but negatively on fertility, the higher growth rate in this joy-of-giving model largely comes from higher saving. It should be noted that Abel and Warshawsky (1988) suggest that [[alpha].sub.6] may be much larger than [[rho][alpha].sub.1]. Therefore, the first part of Proposition 3 is practically more relevant.

Finally, it is less clear-cut when the solutions of the accidental model are compared with those of the altruistic model. We have the following.

PROPOSITION 4. When the probability that the parent dies at the end of her first period of life is sufficiently high that it is close to one, the accidental model involves lower saving and bequest rates than the altruistic model, and the fertility and growth rates are indeterminate; that is, [S.sub.ac] [less than] [S.sub.al] and [[delta].sub.ac] [less than] [[delta].sub.al] but [n.sub.ac] and [g.sub.ac] may be higher or lower than [n.sub.al] and [g.sub.al], respectively. When the probability is sufficiently small such that it is close to zero, the bequest ratio in the accidental model is lower than that in the altruistic model, [[delta].sub.ac] [less than] [[delta].sub.al], and other rates are indeterminate. Otherwise, no explicit relationships can be drawn.

Proof. Under the assumption that p is close to one, (22a) and (22b) imply that [S.sub.ac] and [[delta].sub.ac] are close to zero. However, comparisons of the fertility and growth rates (22c) and (22d) to their counterparts in Table 1 for the altruistic model yield no clear conclusions, depending on other parameter values. When p is not close to one, direct comparisons are in general impossible except that [[delta].sub.ac] is close to zero when p is close to zero. For example, if p = 0, [S.sub.ac] can be higher than [S.sub.al] for sufficiently small value of [rho].

V. THE PRESENCE OF SOCIAL SECURITY

This section introduces unfunded or pay-as-you-go Social Security system into our basic framework. At each date a long-lived government taxes the labor income of the young at a flax rate [tau] and transfers the tax revenue to the old. In the presence of Social Security, the individual's budget constraints (3a) and (3b) now become (for the first three models only)

(23a) [[c.sup.t].sub.t] = [1 - (1 + [n.sub.t])[nu]](1 - [tau])

X [w.sub.t] + [b.sub.t],

(23b) [[c.sup.t].sub.t+1] = (1 + [r.sub.t+1])[S.sub.t][1 - [n.sub.t])v]

X [w.sub.t] + (1 + [n.sub.t])[tau]

x [1 - (1 + [n.sub.t+1])v][w.sub.t+1]

- (1 + [n.sub.t])[b.sub.t+1],

where [n.sub.t] and [w.sub.t+1] are the average levels of fertility and wage. In equilibrium, [n.sub.t+1] = [n.sub.t] = [n.sub.t] and [w.sub.t+1] = [w.sub.t+1]. We assume that individuals will take them as given when making private decisions.

It can be easily shown that in the presence of Social Security, the FOCS of the first three models are similar to those given in the preceding section except that the two-period consumption levels [[c.sup.t].sub.t] and [[c.sup.t].sub.t+1] are given by the budget constraints (23a) and (24b). As shown below, the solutions depend on whether the saving rate is independent of the tax rate. So our strategy is to obtain the solutions for the first two models, because the saving rates in these models are independent of tax, and then make modifications for the other two models. For the altruistic and exchange models, the FOCs yield the same equation determining the saving rate as equation (11), [S.sub.al] = [rho]/Z and [S.sub.ex] = ([[alpha].sub.4]/[[alpha].sub.5])/z, where z [equivalent](1 - [beta])A/(1 + r) is a constant as defined in section III, suggesting that both are independent of [tau], analogous to the well-known neutrality result in Barro (1974). The equations for the bequest-income ratio and the fertility rate are (j = al and ex) [7]

(24) [[delta].sub.j] = [([[alpha].sub.1] + [[alpha].sub.2])[S.sub.j] - [[alpha].sub.2]/[[alpha].sub.1] + [[alpha].sub.2]/([ZS.sub.j])] + [ZS.sub.j][tau],

(25) 1 + [n.sub.j] = ([S.sub.j]/v)[{[S.sub.j] + ([[alpha].sub.1][ZS.sub.j][tau] + [[alpha].sub.2])

/[([[alpha].sub.2]+ [[alpha].sub.3])(1 + Z) - [[alpha].sub.2]

- ([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3])[ZS.sub.j]

- ([[alpha].sub.1][ZS.sub.j] + [[alpha].sub.2])Z[tau]]}.sup.-1].

From equations (24) and (25), we find that in both models [delta] increases with [tau] but (1 + n) decreases with [tau]. Moreover, equation (5) then implies that (1 + g) is an increasing function of [tau] for each model.

For the joy-of-giving model, however, we find that the saving rate is no longer independent of the Social Security tax. Specifically, the saving rate in the presence of social security is shown to be a decreasing function of [tau]:

(26) [S.sub.jg] = [[alpha].sub.6] + z([[alpha].sub.6] + [[alpha].sub.2]) - [[alpha].sub.2]Z[tau]]

/ {Z[([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.6] + [[alpha].sub.1]Z[tau]]}.

Evidently, equation (26) corresponds to its counterpart in Table 1 only when [tau] = 0. Thus, we conclude that the presence of Social Security ([tau] [greater than] 0) lowers private saving in this model. As for other equations, we give their expressions in Table 3, which presents all results of different models.

For the accidental model, the saving rate is also shown to decrease as the Social Security tax is raised (see the appendix for derivation)

(27) [S.sub.ac] = {(1 - p)[[alpha].sub.2]/[[[alpha].sub.1]+(1 -p)[[alpha].sub.2]]}

X [1 + p(1 - p)/Z - [tau]]/[1 + [[phi].sub.1][tau]],

where [[phi].sub.1] = [[alpha].sub.1]Z/{[(1-p).sup.2][[[alpha].sub.1] + (1-p)[[alpha].sub.2]} [greater than] 0 is a constant. Moreover, the bequest ratio is shown to be negatively related to the tax rate. Other equations are given in Table 3.

We summarize the results on the responses of the choice variables to changes in Social Security taxes in the next proposition with full details given in Table 4.

PROPOSITION 5. Depending on the underlying bequest motives, the Social Security tax has different effects on the choice variables:

(1) For the altruistic and the exchange models, it induces higher bequest and growth but lower fertility. In addition, the saving rate is found to be independent of the Social Security tax rate, suggesting that the higher growth rate is the result of a pure fertility effect;

(2) For the joy-of-giving model, higher taxes induce lower saving and growth, whereas the effects on bequest and fertility are ambiguous;

(3) For the accidental model, an increase in the tax rate lowers saving, bequest, and growth rates, while the fertility rate can be higher or lower.

Proof. Direct examinations of the equations in Table 3 yield the results.

This proposition suggests that Ricardian equivalence holds for the saving rate in the altruistic and exchange models, as it is independent of the Social Security tax rate. However, the bequest ratio and fertility and growth rates are not independent of the Social Security tax rate, suggesting that Social Security is not neutral overall even under altruistic bequest motive.

Under the altruistic or exchange motive, Social Security reduces fertility and increases bequests and the growth of per-capita income. Notice that the Social Security tax reduces the effective after-tax wage when young and therefore reduces the cost of raising children. On the other hand, the increase in the bequest raises the cost of children. The negative effect on fertility indicates that the latter dominates the former in our model. If bequests are accidental or motivated from joy-of-giving, however, Social Security depresses saving and economic growth (see, e.g., Feldstein [1974], Wiedmer [1996]).

VI. EMPIRICAL IMPLICATIONS AND EVIDENCE

Our analysis offers several very useful testable implications on bequest motives. First, although the saving rate is independent of the Social Security tax in altruistic and exchange models, it is not so in joy-of-giving and accidental models. Thus, an empirical analysis of the effect of Social Security on saving would differentiate between altruistic/exchange models and joy-of-giving/accidental models.

Choosing between altruistic and exchange models is more difficult but not impossible. From Table 3, we observe that the effect of the Social Security tax on the bequest ratio must be less than 1 in the altruistic model, but it may be less or larger than 1 in the exchange model. Hence, if the estimate of the effect is larger than 1, the altruistic model would be rejected in favor of the exchange model.

Choosing between joy-of-giving and accidental models can be based on regressions of bequest or fertility on Social Security. According to Proposition 5, the effect of Social Security on bequest or fertility can be positive or negative in the joy-of-giving model, but the effect must be negative in the accidental model. Hence, if the estimate of the effect is positive, the joy-of-giving model would be rejected in favor of the accidental model.

What is the empirical evidence on the relations mentioned above? Unfortunately, the empirical literature has been inconclusive on the effect of Social Security on saving rates, [8] which is essential in choosing between bequest motives in terms of our analysis. On the one hand, Feldstein (1974) and other studies (e.g., Rossi and Visco [1995] and Feldstein [1996]) offer time-series evidence that Social Security has reduced saving. On the other hand, Leimer and Lesnoy (1982) find that their time-series evidence does not support Feldstein's hypothesis that Social Security has substantially reduced personal saving in the United States. In fact, they report that, if anything, the postwar evidence suggests that Social Security may have increased saving. Barro and MacDonald (1979) conclude that the cross-country evidence provides neither empirical support for the hypothesis that Social Security depresses saving nor an empirical refutation of that hypothesis. In a recent study of cross-country data, Zhang and Zhang (1997) find that Social Security does not affect saving. There is much evidence of a negative effect of Social Security on fertility, but this can be consistent with any motives. There seems little empirical study of Social Security on bequests. Hence, a further study of the impact of Social Security on saving and bequest would be especially useful in testing bequest motives.

Finally, it has been observed that in some developed countries, fertility rates are below replacement levels. Our analysis can also shed light on this important empirical fact. As discussed earlier, most of the recent evidence from micro studies rejects altruism and supports exchange. We thus take the exchange model to illustrate. [9] Note that in our unitary-sex model economy, the number of children is expressed as 1 + n. Hence below-replacement fertility means n [less than] 0. It can be shown that, to obtain n [greater than] 0, the model parameters must satisfy

[Z/([[alpha].sub.4]/[[alpha].sub.5])][[[alpha].sub.1]([[alpha].sub.4] /[[alpha].sub.5])+[[alpha].sub.2]]/{Z[[alpha].sub.2]+(1+Z)[[alpha].su b.3]-([[alpha].sub.4]/[[alpha].sub.5])([[alpha].sub.1]+[[alpha].sub.2 ]+[[alpha].sub.3])-[[[alpha].sub.1]([[alpha].sub.4]/[[alpha].sub.5])+ [[alpha].sub.2]]Z[tau]} [less than] 1/v - 1.

This inequality suggests that three parameters are important: the child-rearing cost v, the attention taste ratio ([[alpha].sub.4/[[alpha].sub.5]), and the factor productivity A (via parameter z). Moreover, an increase in the ratio of [[alpha].sub.4]/[[alpha].sub.5], or a decrease in the values of v or A tends to cause an increase in fertility. Thus, the above inequality tends to hold for high ratio of [[alpha].sub.4]/[[alpha].sub.5] and low values of [nu] and A.

The occurrence of below-replacement fertility implies that the above inequality may be violated, due to one or any combination of the three possibilities: (1) sufficiently low ratio of [[alpha].sub.4]/[[alpha].sub.5]; (2) high [nu]; and (3) high A. The literature on fertility determinants has identified high cost of children (i.e., high v here) as the major reason for fertility decline in developed countries in recent decades.

To see whether the incidence of below-replacement fertility may eventually cause difficulty in implementing the pay-as-you-go system, we analyze how this case affects the present values of Social Security benefits and contributions by each individual. For the exchange model in the presence of Social Security, one can easily show that the following relationship holds: (1 + g)(1 + n)/ (1 + r) = [[alpha].sub.4]/[[alpha].sub.5]. Notice the left-hand side represents the present value of Social Security benefits received by an old individual over the Social Security tax under the pay-as-you-go system. [10] Since the ratio of [[alpha].sub.4]/[[alpha].sub.5] can take any positive value of larger or smaller than one, the present value of each individual's Social Security benefits may be higher or lower than that person's Social Security taxes. If the former is higher than the latter, below-replacement fertility should not cause significant problems for the pay-as-you-system. The reason is that output growth is fast e nough to offset the burden cause by the fertility decline.

However, when each individual's Social Security taxes exceed the present value of her Social Security benefits, below-replacement fertility may have adverse impact on the Social Security system. More specifically, if below-replacement fertility is caused by a decline in the ratio of [[alpha].sub.4]/[[alpha].sub.5], rather than by such factors as a rise in the cost of raising children and/or a drastic improvement in productivity, the gap between the present value of an individual's Social Security benefits and that of her contributions will widen. When such a gap grows, meaning that each individual will receive less of her initial contribution, it would become increasingly difficult for the policy makers to maintain the pay-as-you-go system, and hence its sustainability would be in doubt.

VII. CONCLUSIONS

We have conducted a systematic theoretical analysis of different bequest motives in a simple endogenous growth model with and without Social Security. To isolate the effect of the variation in bequest motives, all other aspects of the models are set the same. To keep the analysis trackable (in particular to obtain analytical solutions and facilitate comparisons), a logarithmic utility function has been used.

Under the model's assumptions, it is found that altruistic and exchange motives give equivalent outcomes if the discount factors are set the same

between the two motives. This is a surprising result but is unlikely to be generalized to other utility functions. However, the result is important because it tell us that it is sometimes impossible to differentiate between altruism- and exchanged-motivated motives.

Under a reasonable preference restriction, the outcome under joy-of-giving motive involves a higher bequest ratio and higher saving and growth rates than those under altruistic motive. When the probability that the parent cannot survive to the second period of life is sufficiently high, the accidental model involves a lower saving rate and bequest ratio than the altruistic model. When the probability is sufficiently small, the bequest ratio in the accidental model is also lower than that in the altruistic model.

In both altruistic and exchange models, the saving rate is independent of Social Security taxes. Under the two bequest motives, Social Security increases the bequest ratio and growth rate, but reduces fertility. Under the joy-of-giving motive, Social Security depresses saving and growth. If bequests are purely accidental, Social Security reduces bequest, saving, and growth. These results show that bequests motives are important in understanding the effects of pay-as-you-go Social Security systems. Given data on fertility, savings, bequest, Social Security, and economic growth, the results are also useful in testing which bequest motive is more prevalent among the different bequest motives. Our hope is that this study will stimulate more (empirical) research on bequest motives from analyzing aggregate relationships between Social Security, on one hand, and fertility, savings, bequests, and growth on the other.

Junsen Zhang: Professor, Department of Economics, Chinese University of Hong Kong, Shatin, N. T., Hong Kong. Phone 852-2609-7043, Fax 852-2603-5805, E-mail jszhang@cuhk.edu.hk

Junxi Zhang: Associate Professor, School of Economics and Finance, University of Hong Kong, Hong Kong. Phone 852-2857-8502, Fax 852-2548-1152, E-mail jjzhang@econ.hku.hk

(*.) We are grateful to two referees and the editors for very helpful comments and suggestions. Any remaining omissions or errors are our own.

(1.) The idea is that the act of giving itself generates utility (a "warm glow") for parents.

(2.) The analysis would become more complicated by also including human capital accumulation. In such a more general model, Social Security would affect per-capita income growth through fertility, savings, and human capital investment. Romer's (1986) model is also used in an interesting analysis of Social Security and growth by Wiedmer (1996), who finds that unfunded Social Security reduces growth. We will see that this result does not always hold in our models (see also Zhang and Zhang [1995]). Our model and results also differ from Ihori (1994).

(3.) We assume hereafter that bequests are operative. It can be shown that a necessary and sufficient condition is [rho] [greater than] (1 + [g.sup.0])(1 + [n.sup.0])/(1 + [r.sup.0]), where a variable with the superscript 0 denotes its value for the economy without bequest motive. This condition resembles the one studied in a model without endogenous growth by Weil (1987, 385) and Abel (1987, 1043). It is straightforward to verify that under the condition bequests will be operative. To see this, notice that in such an economy, it can be shown that the saving rate is given by [s.sup.0] = [[alpha].sub.2]---/([[alpha].sub.1] + [[alpha].sub.2]). Also notice that the condition [rho] [greater than] (1 + [g.sup.0])(1 + [n.sup.0])/(1 + [r.sub.0]) is equivalent to s [greater than] [s.sup.0]. The latter implies that s [greater than] [[alpha].sub.2]/([[alpha].sub.1] + [[alpha].sub.2]). Hence, [delta] [greater than] 0 according to equation (11).

(4.) Note that [[alpha].sub.4] ln [a.sub.t+1] is the utility associated with receiving services [a.sub.t+1] from each child and [[alpha].sub.5] ln [a.sub.t] is the disutility for the parent who gives [a.sub.t] to her parent. Here th e formulation seems to suggest that the parent only values the utility from services of each child but not aggregate. Given the logarithmic functional form, however, it also encompasses the case of aggregate services, because part of [[alpha].sub.4] ln[(1 + [n.sub.t])[a.sub.t+1]], [[alpha].sub.4] ln(1 + [n.sub.t]), can be grouped into the instantaneous utility function [U.sub.t]([[c.sup.t].sub.t], [[c.sup.t].sub.t+1], 1 + [n.sub.t]).

(5.) The self-selection constraint is not any indication of parental altruism toward children. It simply means that there is a minimum level of utility under which the child will not engage in any exchange with the parent. It also means that any exchange must be acceptable to the child as well to the parent. See Cox (1987) for further discussion.

(6.) Note that for positive fertility in the joy-of-giving model, we require [[alpha].sub.3] [greater than] [[alpha].sub.6].

(7.) Nonnegativity of fertility yields an upper bound for the tax rate. Therefore, our analysis in what follows will be focused on the tax rate in the range between zero and the upper bound.

(8.) See Seater (1993) for a survey of the literature.

(9.) The issue can be analyzed similarly for the other three motives.

(10.) It can be seen as each young person paying one unit of the Social Security tax and each old person receiving (1 + g)(1 + n) units of Social Security benefits. This can be easily verified from examining (23a) and (23b). The second term on the right of (23b) is the benefits, and part of the first whole term on the right of (23a) is the contribution. In a steady state, the ratio becomes (1 + n)[tau][1 - (1 + n)v][w.sub.t+1]/(1 + r) over [tau][1 - (1 + n)v][w.sub.t], which is (1 + n)([w.sub.t+1]/wt)/(1 + r), or simply (1 + n)(1 + g)/(1 + r).

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TABLE 1

Results of Different Models with Alternative Bequest Motives (No Social Security)

Altruistic model

[S.sub.al] = [rho]/z, [[delta].sub.al] = [([[alpha].sub.1] + [[alpha].sub.2])([rho]/z) - [[alpha].sub.2]]/[[alpha].sub.1] + [[alpha].sub.2]/[rho]],

1 + [n.sub.al] = ([S.sub.al]/v)[{([rho]/z) + ([rho][[alpha].sub.1] + [[alpha].sub.2])/[([[alpha].sub.2] + [[alpha].sub.3])(1 + z) - [[alpha].sub.2] - [rho]([[alpha].sub.1] + [[alpha].sub .2] + [[alpha].sub .3])]}.sup.-1],

1 + [g.sub.al] = (1 - [beta])A[S.sub.al]/(1 + [n.sub.al]).

Exchange model

[S.sub.ex] = ([[alpha].sub.4]/[[alpha].sub.5])/z, [[delta].sub.ex] = [([[alpha].sub.1] + [[alpha].sub.2])([[alpha].sub.4]/[[alpha].sub.5])/z - [[alpha].sub.2]]/[[alpha].sub.1] + [[alpha].sub.2]([[alpha].sub.5]/[[alpha].sub.4])],

1 + [n.sub.ex] = ([S.sub.ex]/v)[{[([[alpha].sub.4]/[[alpha].sub.5])/z] + [[[alpha].sub.1]([[alpha].sub.4]/[[alpha].sub.5]) + [[alpha].sub.2]/[([[alpha].sub.2] + [[alpha].sub.3])(1 + z) - [[alpha].sb.2] - ([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3])([[alpha].sub.4]/[[alpha].sub.5])]}.sup.-1],

1 + [g.sub.ex] = (1 - [beta])A[S.sub.ex]/(1 + [n.sub.ex]).

Joy-of-giving model

[S.sub.jg] = [[alpha].sub.6] + z([[alpha].sub.6] + [[alpha].sub.2])]/[z([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.6])], [[delta].sub.jg] = [[[alpha].sub.6]/([[alpha].sub.6] + [[alpha].sub.2])][[alpha].sub.6] + [[alpha].sub.2])][z([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.6])],

1 + [n.sub.jg] = ([S.sub.jg]/v)[{[[[alpha].sub.6] + z([[alpha].sub.6] + [[alpha].sub.2])]/[z([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.6])] + ([[alpha].sub.6] + [[alpha].sub.2])/([[alpha].sub.3] - [[alpha].sub.6])}.sup.-1],

1 + [g.sub.jg] = (1 - [beta])A[S.sub.jg]/(1 + [n.sub.jg]).

Accidental model

[S.sub.ac] = (1 - p)[[alpha].sub.2][1 + p(1 - p)/z]/[[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]], [[delta].sub.ac] = p(1 - p)[[alpha].sub.2][1 + p(1 - p)/z]/[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]],

1 + [n.sub.ac] = (1/v)[[alpha].sub.3][1 + p(1 - p)/z]/{[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3][1 + p(1 - p)/z]},

1 + [g.sub.ac] = (1 - [beta])A[S.sub.ac]/[(1 - p)(1 + [n.sub.ac])].

Note: z = (1 - [beta])A/(1 + r) above is a constant.
TABLE 2
Comparison of the Equilibrium Solutions for Models Given in Table 1
(Benchmark: The Altruistic Model)
 Parameter
 Restrictions
Exchange model [[alpha].sub.4] = [[rho][alpha].sub.5]
 [[alpha].sub.4] [greater than]
 [[rho][alpha].sub.5]
 [alpha].sub.4] [less than]
 [[rho][alpha].sub.5]
Joy-of-giving model [[alpha].sub.6] [greater or equal to]
 [[rho][alpha].sub.1]
 [[alpha].sub.6] [less than less than]
 [[rho][alpha].sub.1]
Accidental model If p is sufficiently
 large (close to 1)
 Saving Rate Bequest Ratio Fertility Rate
 (s) ([delta]) (1 + n)
Exchange model same same same
 higher higher indeterminate
 lower lower indeterminate
Joy-of-giving model higher higher indeterminate
 lower lower indeterminate
Accidental model
 lower lower indeterminate
 Growth Rate
 (1 + g)
Exchange model same
 higher
 lower
Joy-of-giving model higher
 lower
Accidental model
 indeterminate


TABLE 3

Results of Different Models with Alternative Bequest Motives (With Social Security)

Altruistic model

[s.sub.al] = [rho]/z, [[delta].sub.al] = [([[alpha].sub.1] + [[alpha].sub.2])([rho]/z) + [[alpha].sub.2]]/[[[alpha].sub.1] + [[alpha].sub.2]/[rho]] + [rho][tau], 1 + [n.sub.al] = ([S.sub.al]/v)[[([rho]/z) + {([rho][[alpha].sub.1] + [[alpha].sub.2])/[([[alpha].sub.2] + [[alpha].sub.3])(1 + z) - [[alpha].sub.2] - [rho]([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3]) - ([rho][[alpha].sub.1] + [[alpha].sub.2)z[tau]]}].sup.-1], 1 + [g.sub.al] = (1 - [beta])[AS.sub.al]/(1 + [n.sub.al]).

Exchange model

[S.sub.ex] = ([[alpha].sub.4]/[[alpha].sub.5])/z, [[delta].sub.ex] = {[([[alpha].sub.1] + [[alpha].sub.2])([[alpha].sub.4]/[[alpha].sub.5])/z - [[alpha].sub.2]]/[[[alpha].sub.1] + [[alpha].sub.2]([[alpha].sub.5]/[[alpha].sub.4])]} + ([[alpha].sub.4]/[[alpha].sub.5])[tau], 1 + [n.sub.ex] = ([s.sub.ex]/[nu])[([([[alpha].sub.4]/[[alpha].sub.5)/z] + [[[alpha].sub.1]([[alpha].sub.4]/[[alpha].sub.5]) + [[alpha].sub.2]]/{([[alpha].sub.2] + [[alpha].sub.3])(1 + z) - [[alpha].sub.2] - ([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3])([[alpha].sub.4]/[[alpha].sub.5]) - [[[alpha].sub.1]([[alpha].sub.4]/[[alpha].sub.5]) + [[alpha].sub.2]]z[tau]}).sup.-1], 1 + [g.sub.ex] = (1 - [beta])[AS.sub.ex]/(1 + [n.sub.ex]).

Joy-of-giving model

[S.sub.jg] = [[[alpha].sub.6] + z([[alpha].sub.6] + [[alpha].sub.2] - [[alpha].sub.2]z[tau]]/{z[([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.6]) + [[alpha].sub.1]z[tau]]}, [[delta].sub.jg] = [[[alpha].sub.6](1 + z[tau])/([[alpha].sub.6] + [[alpha].sub.2])][[[alpha].sub.6] + z([[alpha].sub.6] + [[alpha].sub.2]) - [[alpha].sub.2]z[tau]]/[z([[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.6]) + [[alpha].sub.1]z[tau]], 1 + [n.sub.jg] - ([s.sub.jg]/[nu])[([s.sub.jg] + {([[alpha].sub.6] + [[alpha].sub.2])/[([[alpha].sub.3] - [[alpha].sub.6])(1 + z[tau])]}).sup.-1], 1 + [g.sub.jg] = (1 - [beta])[AS.sub.jg]/(1 + [n.sub.jg]).

Accidental model [*]

[s.sub.ac] = (1 - p)[[alpha].sub.2][1 + p(1 - p)/z - [tau]]/{[[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]](1 + [[phi].sub.1][tau])}, [[delta].sub.ac] = p(1 - p)[[alpha].sub.2][1 + p)(1 - p)/z - [tau]]/{[[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]](1 + [[phi].sub.1][tau])}, 1 + [n.sub.ac] = (1/v)[[alpha].sub.3][1 + p(1 - p)/z - [tau]](1 + [[phi].sub.2][tau])/{[[[alpha].sub.1] + (1 - p)[[alpha].sub.2]](1 + [[phi].sub.1][tau]) + [[alpha].sub.3][1 + p(1 - p)/z - [tau]](1 + [[phi].sub.2][tau])}, 1 + [g.sub.ac] = (1 - [beta])[AS.sub.ac]/[(1 - p)(1 + [n.sub.ac])].

(*.) Parameters are defined as: [[phi].sub.1] = [[alpha].sub.1]z/{[(1 - p).sup.2][[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]]} and [[phi].sub.2] = z/[(1 - p).sup.2].
TABLE 4
The Responses to Social Security Taxes for Models Given in Table 3
(Benchmark: The Altruistic Model)
 Response of Saving
 to Tax ([partial]s/[partial][tau])
Altruistic model zero
 (benchmark)
Exchange model zero
Joy-of-giving model negative
Accidental model negative
 Response of Bequest
 to Tax ([partial][delta]/[partial][tau])
Altruistic model positive
 (benchmark)
Exchange model positive
Joy-of-giving model either sign
Accidental model negative
 Response of Fertility
 to Tax ([partial](1 + n)/[partial][tau])
Altruistic model negative
 (benchmark)
Exchange model negative
Joy-of-giving model either sign
Accidental model either sign
 Response of Growth
 to Tax ([partial](1 + g)/[partial][tau])
Altruistic model positive
 (benchmark)
Exchange model positive
Joy-of-giving model negative
Accidental model negative


ABBREVIATIONS

BGP: Balanced Growth Path

FOC: First-Order Condition

APPENDIX

In this appendix, following Abel (1985), we derive the equilibrium solutions without and with Social Security for the accidental model. In the absence of Social Security, we first solve the representative individual's problem conditional on the bequest [b.sub.1] received at birth (in the budget constraint (21a), and then calculate the bequests received by each individual as they grow up from different families with divergent mortality history of the earlier generations.

Given [b.sub.1], the parent chooses [S.sub.1] and [n.sub.t] to maximize (20) subject to (21a) and (21b). The FOCs for an interior solution are

(A1a) [[alpha].sub.1][1 - (1 + [n.sub.t])v][w.sub.t]/[[c.sup.t].sub.t]

= (1 - p)(1 + [r.sub.t+1])[[alpha].sub.2]

x [1 - (1 + [n.sub.t])v][w.sub.t]/[[c.sup.t].sub.t+1],

(A1b) v[[[alpha].sub.1](1 - [S.sub.t])[w.sub.t]/[[c.sup.t].sub.t] + (1 - p)[alpha.sub.2]

x [S.sub.t][w.sub.t]/[[c.sup.t].sub.t+1]] = [[alpha].sub.3]/(1 + [n.sub.t]).

Substituting the budget constraints (21a) and (21b) into the FOCs (A1a) and (A1b) and rearranging, we obtain the optimal values of the first-period consumption and savings (wealth) as linear functions of the present value of life-time gross income if no children are raised, [w.sub.t]+[b.sub.t],

(A2) [[c.sup.t].sub.t] = {[[alpha].sub.1]/[[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]]}

x ([w.sub.t] + [b.sub.t]) [equivalent] [d.sub.1]([w.sub.t] + [b.sub.t]),

(A3) [S.sub.t] = {(1 - p)[[alpha].sub.2]/[[alpha].sub.1] + (1 - p)[[alpha].sub.2] + [[alpha].sub.3]]}

x ([w.sub.t] + [b.sub.t]) [equivalent] [d.sub.2]([w.sub.t] + [b.sub.t]),

where [S.sub.t] [equivalent] [1 - (1 + [n.sub.t])v][s.sub.t][w.sub.t] is the amount of (net) savings, and [d.sub.1] and [d.sub.2] are constants. The second-period consumption can be easily calculated from the budget constraint (21b).

As in Abel (1985), we use j to denote the number of consecutive previous generations in an individual's family that died at age 1, or did not live to the second period of their life. According to this notation, j = 0 refers to an individual whose parent lived two periods and therefore left no bequest to her, and j = 1 refers to an individual whose parent died at age 1 leaving a bequest but her grandparent lived two periods leaving no bequest. From (A2) and (A3), it follows that for a type-0 individual,

(A4) [[c.sup.t].sub.t](0) = [d.sub.t][w.sub.t],

(A5) [S.sub.t](0) = [d.sub.2][w.sub.t].

When j [greater than or equal to] 1, (A2) and (A4) suggest that

(A6) [[c.sup.t].sub.t](j) = [d.sub.1][b.sub.t](j) + [[c.sup.t].sub.t](0).

Similarly, from (A3) and (A5), we have

(A7) [S.sub.t](j) = [d.sub.2][b.sub.t](j) + [S.sub.t](0).

As noted in the text, if an individual who is a j - 1 type dies after one period, she leaves a bequest (including the accrued interest) of the size

(A8) [b.sub.t](j) = {(1 + [r.sub.t])/[1 + [n.sub.t-1](j - 1)]}[S.sub.t-1](j - 1),

to each of her children who are j-type individuals. It should be noted that in our context of endogenous fertility, the number of children that a parent has is not the same across generations, because it depends on her first-period consumption as implied by (A1b). By definition [S.sub.t-1) = [1 - (1 + [n.sub.t-1](v][S.sub.t-1][w.sub.t-1], so (A8) can be simplified as

(A8') [1 + [n.sub.t-1](j - 1)][b.sub.t](j)

/ ((1 + [r.sub.t]){1 - [1 + [n.sub.t-1](j - 1)]v}[w.sub.t-1])

= [s.sub.t-1](j - 1).

To proceed, we must impose the BGP conditions; that is, along the BGP, the saving and fertility rates are not functions of time. Then, (A8') implies that along the BGP the bequest-income ratio is constant, which is denoted as [delta], From the above equations for each type-j individual, j = 1,2,3,..., we can obtain respective aggregate ones (per person in the young generation). Let an asterisk, *, represent a variable expressed at the aggregate level. From (A7) with the use of the definition of [S.sub.t], we obtain an expression linking the saving rate [s.sup.*] to s(0) [the saving rate for an individual whose parent lived two periods and left no bequest to her; because there is a fraction (1 - p)[p.sup.j] of the population of type j, [s.sup.*] is defined as [[[sigma].sup.[infinity]].sub.j=0] (1 - p)[p.sup.j] s(j)]:

(A9) [s.sup.*] = [d.sub.2](1 + r)/[(1 + [n.sup.*])(1 + g)][[delta].sup.*] + s(0).

Because a fraction p of the population dies early leaving bequests, the aggregate bequest ratio can be obtained (see [A8']):

(A10) [[delta].sup.*] = [ps.sup.*].

Because only a fraction (1 - p) of young individuals survives to the second period of life, the product market-clearing condition requires that (see equation (10) in the text)

(A11) (1 - p)(1 + [n.sup.*])(1 + g)/1 + r = [zs.sup.*].

Substituting (A10) and (A11) into (A9) and rearranging yield equation (22a) into the text where the asterisk, *, is dropped for comparative purposes with the other three models in the paper. Then substituting it into (A9) have (22c). Finally, (22d) comes directly from (A11).

In the presence of Social Security, the life-time budget constraints for an individual become

(Al2a) [[C.sup.t].sub.t] = [1-(1 + [n.sub.t])v](1 - [s.sub.t] - [tau])[w.sub.t] + [b.sub.t],

(A12b) [C.sup.t].sub.t+1] = (1 + [r.sub.t+1])[S.sub.t][1 - (1 + [n.sub.t])v][w.sub.t]

+ [(1 + [n.sup.*])/(1 - p)][tau]

x [1 - (1 + [n.sub.t+1])v][w.sub.t+1],

where the second term in the right-hand side of (Al2b) represents the Social Security benefits received by the individual when old. Then, the equilibrium equations can be derived analogously as in the above, where the results are given in Table 3.
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Author:ZHANG, JUNSEN; ZHANG, JUNXI
Publication:Economic Inquiry
Date:Jul 1, 2001
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