بیمه عمر، پس انداز احتیاطی و میراث احتمالی

We present a model of life-insurance purchase that takes into account the age of the beneficiary. The beneficiaries considered herein are young children with no resources whose consumption needs are protected by purchasing life insurance if the breadwinner dies. We show that income transfer grows as the child ages; however, the size of contingent bequest shrinks because the need for protection diminishes. Consequently, among the beneficiaries, the younger one would receive a larger bequest. The aggregate demand for life insurance is positively related to the number of children, their consumption needs, and the length of time to independence.

The standard model of demand for life insurance, e.g., Fischer (1973), assumes that the breadwinner maximizes his expected utility over an uncertain lifetime by choosing the optimal level of consumption and life insurance.1 The model assumes that beneficiaries will receive bequests if the breadwinner dies and nothing if the breadwinner survives. The demand for life insurance thus obtained is derived from the bequest function alone. Shorrocks (1979, p. 415) points out that such a model is unsatisfactory because, among other things, “the bequest function is independent of the number or circumstances of the recipients, the utility associated with a transfer arises purely from the act of donation.” Subsequently, Lewis (1989) presents a model of the demand for life insurance in which the breadwinner maximizes the beneficiaries' utility, not own utility. He argues that the results of his model are appealing because they simulate the actual calculation of insurance purchase the household makes. Many questions arise from Lewis' findings. First, computing the needs of beneficiaries is not the only decision that the breadwinner makes. An altruistic breadwinner chooses his own consumption, decides on the size of bequests, and allocates resources to his heirs while he is alive. Can we incorporate these decisions in a single model? Second, the role of child's age in the breadwinner's decision making has not been thoroughly studied in the literature. This is an important issue because the breadwinner purchases life insurance to protect his beneficiaries by providing them financial assistance in the event of his death. Since the need for protection changes with a child's age, so does the decision on contingent bequest. What, then, is the bequest–age relation? Third, we must inquire into the relationship between the age of each child and the distribution of bequests when there are multiple children. Is it based on an equity principle, on primogeniture, or on something else? Last, we have to address Shorrocks' criticism. What role does the number of children play in deciding income transfer and insurance purchase? This paper provides some answers to these questions. The approach is to extend Lewis' model such that life insurance purchase is jointly determined with the breadwinner's own consumption and with inter vivos income transfer to heirs. To accomplish this, we include in the objective function of the breadwinner's optimization problem each recipient's utility function and a bequest function for each child. In other words, we assume the breadwinner is altruistic toward his dependents while he is alive, not just after his death. We also assume that each child will become independent at some point. In the model, each child's utility function enters the breadwinner's objective function at birth and exits it after the age of independence; bequests to adult children are an act of donation. Our proposed theory of intergenerational transfer is different from other models in at least two aspects. First, in other models that emphasize the interaction between generations, beneficiaries have income and engage in strategic actions. Those models apply mainly to adult offspring, and usually exclude young children from the analysis. For example, in testing the altruistic theory,2Wilhelm (1996) excluded children under the age of 25. In contrast, the beneficiaries in our model are financially constrained young children who have no significant strategic options. Second, we contend that the tender age of the dependent plays an important role in breadwinner's decision making. Other models simply overlook this issue. The exception is Laitner and Juster (1996, p. 895) who recognized the point and presented a three-period model in which the middle period is a time of giving to young offspring. Our findings are the following. First, the inclusion of beneficiaries' utility functions in the breadwinner's objective function enables us to draw implications for income transfer to young children. We show that if the actuarial rate of interest is greater than the subjective discount rate, then the income transfer to each child rises as the child ages and that the total expenditure on children is significantly influenced by the number of children and peaks just before the oldest child reaches independence. These results support Espenshade's (1984) findings, in which older children are more expensive because, among other things, they are physically larger and have more activities. Our results are attributed to the breadwinner's lifetime uncertainty and the unfairness of the insurance market, and thus provide an additional explanation for these findings. Second, our theory implies that the age profile of bequests to each child falls over time. The intuition is this: Since the purpose of life insurance purchase is to maintain the beneficiary's standard of living up to the age of independence in the event of the breadwinner's untimely death, such a need for protection declines as the child nears the age of independence. Third, we show that birth order matters. Our birth-order result is based on equal protection for all children until independence. Since the younger one has further to go before reaching independence, more protection is needed. Therefore, when there are multiple children, the younger child would inherit a larger bequest. This result is not contradictory to primogeniture because primogeniture applies mainly to adult heirs, whereas our model applies to younger children. Fourth, we address Shorrocks' point that life insurance purchase must reflect the needs and circumstances of the recipients. We show that, in addition to the standard loading factor effect, the demand for life insurance is positively related to the number, the time to independence, the age differentials, and the standard of living of the beneficiaries. Finally, our proposed model adds new results to life cycle theory. We show that precautionary saving arises from uncertainty about the breadwinner's lifetime when there are dependent children in the family. This precautionary saving rises with the breadwinner's hazard rate and the loading factor of the insurance market. We work out a numerical example using data from a CDC life table to illustrate the significance of this delayed consumption effect.

In this paper, we presented a theory of life insurance purchase of an altruistic breadwinner supporting his liquidity-constrained children. The model takes into account the age and the consumption need of the dependent children and derives several results. The key feature is that the breadwinner's own consumption, the gifts inter vivos, and the purchase of life insurance are jointly determined in our model. We show that, as each child ages, the size of contingent bequest shrinks because the need for protection diminishes. Among beneficiaries, the younger one would receive a larger bequest. The aggregate demand for life insurance is positively related to the number, age differentials, the living standards, and the time needed for each child to reach adulthood. While Lewis' objective is similar to ours, there are major differences between the two papers. By including each dependent's utility function in the breadwinner's optimization problem, we have a dynamic model from which emerge several interesting implications that go beyond Lewis' findings. First, our theory provides new insights into the life cycle theory that there is precautionary saving in a world of unfair insurance. Second, our theory sheds light on the role of the dependent's age in the breadwinner's decision making. Because age determines the need for protection it gives rise to birth order effects on the distribution of bequests. Third, we show that the number of children matters in determining expenditure on children. Fourth, we obtain a more detailed demand for life insurance formula that incorporates the number and circumstances of the recipients. Finally, there is a difference in modeling. Lewis assumes that the breadwinner chooses the optimal level of life insurance to maximize the beneficiary's utility, while assuming his own consumption and the gifts to the heirs are exogenously given. In contrast, in our model all three variables are jointly determined. In this approach the breadwinner still maximizes the child's expected utility from birth to independence. To make the model tractable, we have made some simplifying assumptions. Specifically, we have assumed away child mortality, ignored the aspects of human capital investment on children, minimized the role played by the surviving spouse in the purchase of life insurance, assumed only interior solutions to the optimization problems, and discussed only term life insurance. What would happen if annuities and other types of life insurance are available? Relaxing some or all of these assumptions would test the robustness of the theory developed in this paper. These are for future research.