Keywords

JEL Classifications

In the life-cycle model of household behaviour, each household expects a lifetime pattern of rising earnings in youth and middle age followed by retirement. Hence, households plan to save in their first segments of life in order to build resources to dissave, and from which to accrue interest income, during the last (Modigliani 1986). The framework easily incorporates children, with consumption early in a household’s life driven higher and saving for retirement perhaps delayed until middle age (Tobin 1967). In a standard life-cycle model, parents plan for their own life and assume financial responsibility for their children until the latter reach adulthood (say, age 18 or 22) – but not beyond. Elaborations of the framework, on the other hand, extend parental concern, or interest in non-market transactions, to encompass a household’s grown children. Such elaborations expand the scope of the life-cycle model to include bequests.

Conceptually, there are at least three broad categories of models in which bequests play a role. The first, which is often called the ‘altruistic model’, assumes that parents care about the well-being of their grown children. The second, which one might call the ‘joy of giving model’, assumes that parents derive pleasure from making transfers to their adult children’s households but that the pleasure is not specifically dependent upon the children’s utility gain. In the third formulation, parent-to-child emotional and social ties favour and facilitate non-market exchanges that may generate bequests – for example, bequests may emerge as payments to heirs for personal services rendered.

Altruistic Model

A model with ‘altruistic bequests’ (Becker 1974; Barro 1974) extends to grown children parental concerns for minor children typical of standard life-cycle analyses.

Consider a specific example in which each household has one adult, raises one child, and lives two periods. Suppose that a household begun at time t has earnings yt in youth but is retired in old age. It rears its child during its first stage of life; the child initiates its own household thereafter, with the descendant household passing its first stage of life as the parent household lives through its second stage. The time-t parent chooses consumption \( {c}_t^1 \) and \( {c}_t^2 \), respectively, for its two stages of life; derives utility \( u\left({c}_t^1,{c}_t^2\right) \) from this consumption; inherits it in youth; and transfers it+1 in old age to its adult child. Let the interest rate be r. Given it and it+1, the parent household’s lifetime utility is U(.) such that

$$ U\left({i}_t+{i}_{t+1},{y}_t\right)\equiv \underset{c_t^1,{c}_t^2}{\max }u\left({c}_t^1,{c}_t^2\right) $$
$$ \mathrm{subject}\ \mathrm{to}:\; {c}_t^1+\frac{c_t^2}{1+r}+\frac{i_t+1}{1+r}\le {i}_t+{y}_t $$

Let the parent household care δ times as much about its adult child’s lifetime utility as about its own, δ2 times as much about its grandchild’s lifetime utility, and so on. Then the parent household’s dynastic utility is

$$ \sum\limits_{s = 0}^\infty{\delta}^s\cdot U\left({i}_{t+s},{i}_{t+s+1},{y}_{t+s}\right) $$

If yt = y all t, if institutions force bequests to be nonnegative, and if descendant households share the same preference ordering, we can characterize the time-t parent household’s dynastic utility as V(it, y) with

$$ V\left({i}_t,y\right)=\mathop {\max }\limits_{i_t + 1 \ge 0}\left\{U\left({i}_t,{i}_{t+1},y\right)+\delta \cdot V\left({i}_{t+1},y\right)\right\}. $$
(1)

If δ = 0, we have a ‘pure life–cycle model’; if δ > 0, we have an altruistic model in which positive bequests may emerge.

Laitner (1992) studies a second altruistic formulation, one allowing heterogeneous earning abilities. In terms of the framework above, a parent household with earnings yt may know the random variable, say, \( \tilde{y} \), from which the earnings of its descendants will be (independently, in the simplest case) sampled, but the parent cannot observe the sampling outcomes as it makes its bequest plans. Then dynastic utility is

$$ V\left({i}_t,y\right)=\mathop {\max }\limits_{i_{t + 1} \ge 0} \left\{U\left({i}_t,{i}_{t+1},y\right)+\delta \cdot E\left[V\left({i}_{t+1},\tilde{y}\right)\right]\right\}, $$
(2)

where E[.] is the expectations operator.

Conceptually, a model with altruistic bequests provides an extension of the life-cycle model’s parental concern for minor children’s well-being to a more or less symmetric concern for grown children. Empirically, bequests and inter vivos transfers to adult children certainly occur in practice (Modigliani 1986; Kotlikoff 1988). The formulation with heterogeneous earnings predicts that bequests need not be universal but are most likely in the case of very prosperous parents. Social commentators frequently criticize bequests as a source of inequality, and the second point in the preceding sentence shows how bequests can contribute to cross-sectional dispersion of private wealth holdings. Bequests may have played a larger role in national wealth accumulation in the past, when long retirement spells were perhaps less common (Darby 1979), and a model with both life-cycle saving and altruistic bequests can provide a framework for analysing the change (Laitner 2001).

Loans for education fail to generate collateral for creditors; hence, parental and/or public support may be important for ensuring efficient educational investment. Since benefits of education last long into adulthood, the model with altruistic bequests provides a logical framework for studying parental contributions (for example, Tomes 1981). For instance, suppose that a child’s earnings are an increasing, concave function f (.) of ability, a, and parental support for education, e, in the child’s youth: yt+1 = f(at, et). With homogeneous agents, at = a all t, and (1) becomes

$$ V\left({i}_t,y\right)= \mathop {\max }\limits_{i_{t + 1} \ge 0,e_t \ge 0}\left\{U\left({i}_t,{i}_{t+1}+{e}_t\cdot \left(1+r\right),y\right)+\delta \cdot V\left({i}_{t+1},f\left(a,{e}_t\right)\right)\right\}. $$
(3)

Then it+1 > 0 ensures efficient provision of education et regardless of the degree of parental concern for the child, δ. If, on the other hand, the tangible bequest is zero, investment in education can be inefficiently low.

A second prominent application of the altruistic model relates to fiscal policy. In a standard life-cycle model, when government turns from tax to deficit finance, national consumption may rise for a time, and the economy’s long-run capital intensity may decline. Reformulating the life-cycle model to include altruistic bequests can overturn this result (for example, Barro 1974). Debt service and repayment for current government borrowing may extend far beyond the life span of existing households, but not beyond the time horizon of dynasties. Maximization in (1) may yield an outcome in which the non-negativity constraint never binds, and Barro (1974) shows that in that case tax and deficit finance may have identical implications for aggregate consumption, capital accumulation, and interest rates. The latter equivalence is often referred to as ‘Ricardian neutrality’. (With heterogeneity of agents, as in formulation (2), non-negativity constraints will, on the other hand, tend to bind for some households – Laitner 1992 – and then outcomes resembling Ricardian neutrality, while still possible, may be more in doubt – for example, Bernheim 1987.)

Recent dynamic general equilibrium analyses of long-run growth and business cycles frequently employ the so-called ‘representative agent’ paradigm. Utility maximization over an infinite time horizon for a set of identical agents determines desired private consumption, saving, and labour supply. It seems fair to say that the life-cycle model with altruistic bequests, as in Barro (1974) and related papers, provides the most basic motivation for this approach.

Turning to empirical findings, the widespread existence of bequests (and inter vivos gifts) within family lines is well established (Modigliani 1986; Kotlikoff 1988). The pure life-cycle model does not seem able to explain as much national wealth as we see, and estate building seems a plausible explanation for the remainder (Kotlikoff 1988). However, despite some consistency with the altruistic model, empirical evidence often seems to fail to support the implications of pervasive Ricardian neutrality (for example, Altonji et al. 1992, 1997). Long-standing evidence that households with multiple children tend in practice to divide their bequests equally (for example, Menchik 1988) also seems contrary to implications of the simplest versions of the altruistic model. Perhaps altruistic bequest behaviour is, in practice, concentrated among the highest-income households (as might be implied by formulation (2)).

Conceptually, as one considers couples instead of single parents, dynasties will interact through marriage. Assortative mating can preserve the logic of the analysis of the parthenogenetic theoretical construct (Laitner 1991). Mating patterns that are random theoretically could, in contrast, expand to an overwhelming degree the scope of interpersonal connections that ‘neutralize’ incentives for self-interested behaviour (Bernheim and Bagwell 1988).

The preceding formulations assume that a parent cares about his child but that the reverse is not true. A number of papers analyse two-sided altruism. Implicitly, in fact, all formulations with altruistic transfers are two sided – in model (1), for example, the parent cares about his child’s utility relative to his own with a ratio of weights δ:1, while the child cares about his parent’s utility relative to his own with weights in a ratio of 0:1. Unless parents and children agree on each other’s relative importance, strategic behaviour may arise if agents have sufficient latitude in their set of feasible actions. In Laitner (1988), for instance, though parents and children care about each other, each may care less about the other than about itself – in which case a parent with low earnings may intentionally limit his life-cycle saving in youth in order to induce a larger transfer from his child during his retirement.

In the simplest life cycle model, a household saves before retirement in order to preserve an even level of consumption for the remainder of its life. An altruistic model extends the time frame of such behaviour: a household may use bequests (and inter vivos gifts) to promote evenness of consumption for its entire family line.

Joy of Giving Model

A joy-of-giving model provides a donor with pleasure that is independent of recipient utility and outside resources. For example, our two-period household above might solve

$$ \mathop {\max }\limits_{i_{t + 1} \ge 0} \left\{U\left({i}_t,{i}_{t+1},{y}_t\right)+W\left({i}_{t+1}\right)\right\}, $$
(4)

with the new function W(.) being unrelated to lifetime utility U(.) or to recipient earnings yt+1. In this approach, the parent household has preferences over its own lifetime consumption and the size of the bequest that it provides to its offspring, rather than over the descendant’s consumption or utility. An example is Blinder (1974).

A possible advantage of this framework is that it does not require as great an ability on the part of donors to manifest empathy and rationality as the altruistic model. Another advantage is its analytic simplicity. In applications, authors may seek to specify the utility function W(.) in a manner that can mimic, at least to some degree, the model with altruistic bequests (for example, Modigliani 1986).

Exchange

The emotional ties of parents and their children may lead parents to prefer attentions from their grown children over services purchased in markets. Similarly, emotional bonds, tradition, or social norms may give trades between relatives lower transaction costs than those based on market contracts. Relatives may also have more complete information about one another than anonymous market participants do. Such factors may lead parents to make transaction and insurance arrangements with their grown children, and parental payments may take the form of bequests or inter vivos gifts.

In traditional societies, a household’s eldest son might labour on his parents’ farm, supporting his parents in their old age. In return, the son might expect to inherit the farm at his parents’ death. One can view such a bequest as a payment for services, and neither altruistic nor joy-of-giving impulses on the part of parents (or their son) need be determinants of the transfer’s size.

Bernheim et al. (1985) provide a model in which elderly parents desire attention from their adult children, and the parents can be thought of as paying for the services through their bequest.

Many economists note the relative infrequency with which households purchase annuities. Transactions costs and adverse selection, due to private information about one’s likely longevity, may be the underlying reason. In practice, parents may circumvent annuity markets by making implicit contracts with their grown children: in return for care and support in old age, the parents agree to bequeath their assets to their children. The children take the place of an insurance company: if their parents die young, the children’s efforts receive generous remuneration; if the parents live a long time, their bequest may be small or non-existent, and the children’s reward per hour of effort will be low. Kotlikoff and Spivak (1981) show that such arrangements can be surprisingly efficient. Friedman and Warshawsky (1990) illustrate a related point: they show that parents who have some inclination (either joy of giving or altruistic) to bequeath to their children may eschew market annuities with even modest transactions costs, preferring self-insurance, under which their children can inherit unspent parental resources.

See Also