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Many studies specify human mortality patterns parametrically, with a parameter change affecting mortality rates at different ages simultaneously. Motivated by the stylized fact that a mortality decline affects primarily younger people in the early phase of mortality transition but mainly older people in the later phase, we study how a mortality change at an arbitrary age affects optimal retirement age. Using the Volterra derivative for a functional, we show that mortality reductions at older ages delay retirement unambiguously, but that mortality reductions at younger ages may lead to earlier retirement due to a substantial increase in the individualʼs expected lifetime human wealth.

In the twentieth century, people tended to retire earlier in most developed countries, despite the fact that there was a steady and significant increase in life expectancy. In the US, for example, the life expectancy at birth for men who were born in 1900 was about 50 years. This figure increased substantially to over 70 years for those who were born in the middle of the century, and increased further to almost 80 years for those born in the 1990s.1 Despite increasing life expectancy, however, labor force participation rates of men aged 65 and over steadily declined from over 60% in 1900 to around 20% at the last decade of the century, while the rates of men aged 55 to 64 declined from around 90% to below 70% for the same period (Costa [8, Fig. 2.1]).2 Various explanations have been proposed for this seeming paradox. For example, the important role of the generous benefits provided by the social security system has been analyzed by Gruber and Wise [10] and [11], and the wealth effect associated with sustained economic growth has been examined by Costa [8]. To complement these well-known explanations, several researchers (such as Bloom et al. [3], Kalemli-Ozcan and Weil [17]) examine the relatively neglected question of how mortality decline affects retirement age. In particular, Kalemli-Ozcan and Weil [17] show that people may retire earlier if the decrease in the variability of age at death associated with mortality decline is very significant. In this article, we continue this line of inquiry by studying the differential incentives for early retirement provided by mortality changes at different stages of demographic transition. A salient feature of mortality changes in many countries is that, while life expectancy has increased steadily over the last two centuries, mortality decline does not occur uniformly across age groups (Lee [18], Wilmoth and Horiuchi [23], Cutler et al. [9]).3 In the earlier stage of mortality transition, a decline in mortality pertains mainly to younger people, particularly infants and children, whereas in the later stage, an “aging of mortality decline” has occurred, characterized by “successively larger reductions in mortality rates at older ages, and by smaller reductions at younger ages” (Wilmoth and Horiuchi [23, pp. 484–485]).4 An illustration of this pattern is given in Fig. 1, which is based on the survival data, starting from age 20, of different cohorts of men in the US. The upper panel shows the survival curves of men born in four different decades: the 1900s and 1910s at the beginning of the century, and the 1980s and 1990s towards the end of the century. It is observed that the improvement in survival probabilities at various ages is more substantial for the earlier period. Moreover, unlike the later period, the mortality decline at the beginning of the century pertained mainly to younger people. This is seen more clearly in the lower panel of Fig. 1, in which we plot the mortality rates (in log) at different ages.5 The mortality decline occurred in greater concentration among people aged 20 to 50 from the 1900s to 1910s.Motivated by the stylized facts regarding the aging of mortality decline, we study the effect on optimal retirement age of a change in mortality at an arbitrary age. To do so, we use a method different from that of most existing studies, which usually address the effect of mortality decline by modeling the human mortality pattern using a parametric approach and considering the derivative of retirement age with respect to a change in the survival parameter. One characteristic of this approach is that a change in the survival parameter usually affects mortality rates at different ages simultaneously. For example, in an article examining whether an individualʼs saving level or retirement age adjusts more substantially when there is a mortality decline, Bloom et al. [3] show in a model with age-invariant mortality rates (as in Blanchard [2]) that the optimal response to a mortality decline is to delay retirement, with zero (or possibly negative) effect on saving rates. However, a change in the parameter of an age-invariant mortality process affects mortality rates at all ages to the same extent.6 Thus, although this widely used specification is helpful for analyzing some economic phenomena (such as government debt policies), it is not appropriate if oneʼs objective is to isolate the impact of mortality change at a particular age. In other parametric specifications of the survival curve (such as in Boucekkine et al. [6]), a mortality parameter change does not go so far as to affect mortality rates at all ages to the same extent, but it still affects mortality rates at different ages simultaneously. Thus, if there are systematic differences in how changes in mortality at different ages affect the retirement age, the differences are not clearly illuminated through these methods. In this article, we consider a life-cycle model of consumption and retirement choice with lifetime uncertainty, and characterize the optimal retirement age and consumption path. We then investigate the effect of a change in age-specific mortality rate on the optimal retirement age. Since the optimal retirement age is a function of mortality rates that are themselves dependent on age, the appropriate method to use is the derivative of a functional. We use the Volterra derivative (Volterra [22], Ryder and Heal [20], Bommier [5]) to show that, while a mortality decline among older people unambiguously leads to a delay in retirement, a mortality decline among young people may lead to earlier retirement. We provide economic intuition of this systematic difference by decomposing the effect of a mortality change on retirement age into the “years-to-consume” and “lifetime human wealth” effects. (More detailed discussion of these effects will be provided in Section 3.) When a mortality decline occurs among an older age group, as in the later phase of mortality transition, the increase in individualsʼ expected lifetime human wealth is usually negligible, and thus the years-to-consume effect dominates. On the other hand, when a mortality decline occurs among a younger age group, as in the early phase of mortality transition, it turns out that the increase in individualsʼ expected lifetime human wealth can be quite substantial, dominating the years-to-consume effect. The remainder of this article is organized as follows. Because the effect of mortality changes on retirement age is most clearly seen in a life-cycle model in which perfect annuities are present and disutility of labor does not depend on life expectancy, we begin by analyzing this basic model. In Section 2, we use the model to examine consumption and retirement decisions, and in Section 3, we consider the effect of a mortality decline at an arbitrary age on optimal retirement age. Section 4 extends the model to allow for the dependence of disutility of labor on life expectancy to capture the “compression of morbidity” hypothesis (as in Bloom et al. [3]) and to allow for imperfect annuities. We demonstrate the robustness of the main results in the basic model vis-à-vis these different specifications. Concluding remarks are presented in Section 5.

An important feature of mortality transition is that a mortality decline affects mainly younger people in its early phase but pertains to older people in the later phase. Motivated by these stylized facts, we examine the effect on optimal retirement age of a change in mortality at an arbitrary age. This article makes two significant contributions. First, we show that a mortality decline at older ages (above the optimal retirement age) will always result in a delay in retirement, while a mortality decline at younger ages may or may not lead to early retirement. The intuition of this result is as follows. When a mortality decline occurs among older people, the lifetime human wealth effect is absent and thus the years-to-consume effect always leads to a delay in retirement. For mortality decline at younger ages, we find that, under some general conditions concerning the wage profile, there is a threshold age before which the lifetime human wealth effect dominates and the optimal retirement age decreases when there is a mortality decline. On the other hand, optimal retirement age increases with a mortality decline after the threshold age. We show that this systematic difference is also present when the model is extended to account for compression of morbidity and annuity market imperfection. Our analysis of the effect on retirement age of mortality decline at an arbitrary age is complementary to the study by Kalemli-Ozcan and Weil [17], who focus on another important feature of mortality transition: the decrease in the uncertainty about the age at death. Interestingly, both articles show that a mortality decline may lead to early retirement under some conditions (the uncertainty effect dominates the horizon effect in their model, and the lifetime human wealth effect dominates the years-to-consume effect in our model). The counterintuitive result emphasized in these two articles, which is obtained under different modeling assumptions corresponding to the two salient aspects of mortality transition, should be given more serious attention.22 The second contribution of this article is methodological. Existing studies analyzing the relation between mortality change and retirement age often use indexes or parameters to represent mortality changes, and consider the derivative(s) of the optimal retirement age with respect to the mortality parameter(s). However, such a change in the mortality parameter usually affects the instantaneous mortality rates (and thus the survival probabilities) at different ages. We show that the more fundamental question about the effect on retirement age of a mortality decline at an arbitrary age can be addressed by using the Volterra derivative. As mortality changes at different ages may also have systematically different effects on other important variables such as growth rate of output per capita (e.g., Boucekkine et al. [6], Zhang et al. [25]), the analysis in this article leads us to believe that the use of the Volterra derivative will be helpful in other economic–demographic studies in the future.