Uncertain Lifetime Research Articles

Demand Changes and Real Exchange Rate Dynamics in a Finite‐Horizon Model with Sectoral Adjustment Costs

In this paper, we use the Blanchard‐type uncertain lifetime overlapping generations framework to develop a three‐good open economy to examine the effects of demand changes on the economic performance of small open economies with sectoral adjustment costs. Our simulation results reveal a discernible Harberger‐Laursen‐Metzler effect on both temporary and permanent terms‐of‐trade shocks. The inclusion of sectoral adjustment costs generates persistent deviations of the real exchange rate from its long‐run equilibrium in response to terms‐of‐trade shocks, with the degree of sectoral adjustment costs having an effect on the subsequent dynamic transition of the economy. We also find that an increase in the mortality rate leads to a lowering of the current account.

Persistent inequality with endogenous rate of time preference

Purpose – The purpose of this paper is to provide a theory that can explain the persistence of inequality in an economy where household agents are identical in terms of their preferences and have access to the same production technology.Design/methodology/approach – An overlapping generations model is developed where agents are imperfectly altruistic and face uncertain lifetimes. The rate of time preference of an agent depends on her probability of survival which is increasing in her level of consumption. An agent's initial endowment of human capital jointly determines her patience and willingness to accumulate human capital and other productive assets.Findings – It was found that inequality may persist in the economy as a result of endogenous rate of time preference. Agents with low initial endowment of human capital are impatient and choose not to invest in human capital. Agents with initial endowment of human capital above a certain threshold choose to invest in human capital as they expect to survive ...

Discounting and Consumption Over an Uncertain Horizon: Draw-Down Plans for Family Trusts

Individuals, endowments and trusts face uncertain lifetimes. When the planning horizon of an entity is stochastic and Pareto distributed, hyperbolic discounting and time-varying consumption rates are optimal. We derive expressions for the optimal rate of consumption (draw-down) from wealth for family trusts facing positive probabilities of extinction at each generation. Using birth statistics for the UK, we compute family extinction probabilities and show that they are well-approximated by a Pareto distribution, hence family trusts will discount hyperbolically. Numerically optimised consumption paths for family trusts with CRRA preferences are decreasing but always higher than for infinitely-lived trusts.

Life-cycle asset allocation with annuity markets

This paper derives the optimal consumption and portfolio choice pattern over the life-cycle for households facing uninsurable labor income risk, ruin risk, stochastic capital markets, and uncertain lifetime. Our model posits a dynamic utility maximizer with CRRA and Epstein/Zin preferences who has access to liquid stocks, bonds, and illiquid life annuities. We show that a considerable fraction of wealth is gradually annuitized until retirement and beyond to skim the mortality credit. The remaining liquid wealth is mainly invested in stocks to compensate for riskless investment in life annuities. The model allows us to assess the economic importance of common explanations for the empirically low annuity demand by analyzing participations rates, annuitization fractions, and welfare effects.

Wealth transfers and the role of collateral when lifetimes are uncertain

We develop a general equilibrium model of wealth transfers in the presence of uncertain lifetimes and default. Without introducing exogenous debt constraints, agents are allowed to make collateral-backed promises at any state of their life span.

The HIV Anticaptory Saving Motive: An Empirical Analysis in South Africa

This paper studies the effect of the HIV/AIDS epidemic on saving behaviour. Two important characteristics of HIV result in opposing forces on savings: mortality increases, which reduces savings, and long-term illness risk increases, which enhances savings. We use a two period life-cycle model with uncertain lifetime including perceived HIV contamination risk to illustrate both the opposing effects of the HIV epidemic on individual savings and test the predictions of our model with data obtained from an economic experiment with real monetary incentives performed in South Africa. The empirical results show that increased mortality decreases the amount of savings and that having a high perception of HIV contamination risk increases savings. The latter effect confirms the HIV anticipatory saving hypothesis.

Following the rules: Integrating asset allocation and annuitization in retirement portfolios

Financial advisers have developed standardized payout strategies to help Baby Boomers manage their money in their golden years. Prominent among these are phased withdrawal plans offered by mutual funds including the “self-annuitization” or default rules encouraged under US tax law, and fixed payout annuities offered by insurers. Using a utility-based framework, and taking account of stochastic capital markets and uncertain lifetimes, we first evaluate these rules on a stand-alone basis for a wide range of risk aversion. Next, we permit the consumer to integrate these standardized payout strategies at retirement and compare the results. We show that integrated strategies can enhance retirees’ well-being by 25%–50% for low/moderate levels of risk aversion when compared to full annuitization at retirement. Finally, we examine how welfare changes if the consumer is permitted to switch to a fixed annuity at an optimal point after retirement. This affords the retiree the chance to benefit from the equity premium when younger, and exploit the mortality credit in later life. For moderately risk-averse retirees, the optimal switching age lies between 80 and 85.

The uncertain lifetime and the timing of human capital investment

I examine the effects of mortality decline on fertility and human capital investment decision of parents taking into account the uncertainty about child survival. I propose a model, where parents decide on their fertility before the uncertainty is realized, but they choose to invest only in human capital of their surviving children. The model implies a positive relationship between mortality and fertility and a negative one between mortality and educational investment. It has been argued elsewhere that as, in reality, most of the mortality decline occurred in infancy, it should not affect the human capital investment decision, which comes later in life. Thus, increased survival chances should not promote growth by raising the human capital investment. This paper argues the contrary and proposes a mechanism where mortality decline at any age before the teen years can promote growth by raising human capital investment regardless of the timing of the educational investment.

Life-Cycle Asset Allocation and Optimal Consumption Using Stochastic Linear Programming

We consider optimal consumption and (strategic) asset allocation of an investor with uncertain lifetime. The problem is solved using a multi-stage stochastic linear programming (SLP) model to be able to generalize the closed-form solution obtained by Richard (1975). We account for aspects of the application of the SLP approach which arise in the context of life-cycle asset allocation, but are also relevant for other problems of similar structure. The objective is to maximize the expected utility of consumption over the lifetime and of bequest at the time of death of the investor. Since we maximize utility (rather than other objectives which can be implemented more easily) we provide a new approach to optimize the breakpoints required for the linearization of the utility function. The uncertainty of the problem is described by discrete scenario trees. The model solves for the rebalancing decisions in the first few years of the investor's lifetime, accounting for anticipated cash flows in the near future, and applies Richard's closed-form solution for the long, subsequent steady-state period. In our numerical examples we first show that available closed-form solutions can be accurately replicated with the SLP-based approach. Second, we add elements to the problem specification which are usually beyond the scope of closed-form solutions. We find that the SLP approach is well suited to account for these extensions of the classical Merton setting.

Optimal life insurance purchase and consumption/investment under uncertain lifetime

In this paper, we consider optimal insurance and consumption rules for a wage earner whose lifetime is random. The wage earner is endowed with an initial wealth, and he also receives an income continuously, but this may be terminated by the wage earner’s premature death. We use dynamic programming to analyze this problem and derive the optimal insurance and consumption rules. Explicit solutions are found for the family of CRRA utilities, and the demand for life insurance is studied by examining our solutions and doing numerical experiments.

Are Americans Saving “Optimally” for Retirement?

We solve each household’s optimal saving decisions using a life cycle model that incorporates uncertain lifetimes, uninsurable earnings and medical expenses, progressive taxation, government transfers, and pension and social security benefits. With optimal decision rules, we compare, household by household, wealth predictions from the life cycle model using a nationally representative sample. We find, making use of household‐specific earnings histories, that the model accounts for more than 80 percent of the 1992 cross‐sectional variation in wealth. Fewer than 20 percent of households have less wealth than their optimal targets, and the wealth deficit of those who are undersaving is generally small.

Optimal long-run fiscal policy: Constraints, preferences and the resolution of uncertainty

We construct a computational dynamic stochastic overlapping generations general equilibrium model with uncertain lifetimes and explore the impact of policy stickiness (specifically, a major reform will preclude future reforms for a generation) on optimal long-run fiscal policy. Under such circumstances, entitlement reforms exhaust a valuable option to move in the future. We explore the conditions under which the gain to waiting is large enough to induce optimizing policymakers to delay reforming a suboptimal system. We also allow for the uncertainty to have ARCH characteristics and explore the impact of time-varying uncertainty on the optimality of delayed policy action.

The existence, uniqueness, and optimality of the terminal wealth depletion time in life-cycle models of saving under uncertain lifetime and borrowing constraint

In life-cycle models of saving under uncertain lifetime and borrowing constraint, the consumer's wealth must be depleted before the maximum lifetime. This paper investigates the existence, uniqueness, and optimality of the terminal wealth depletion time. It is proved that the optimal terminal wealth depletion time, if such exists, must be unique. If the equation that determines the optimal terminal wealth depletion has multiple solutions, then the location of the optimal solution will depend on the configuration of the solutions. An optimality test is developed to verify whether a candidate solution for the terminal wealth depletion time is indeed optimal. The paper introduces a method new to economics, the Dubovitskii–Milyutin adjoint equation, to analyze the properties of the optimal control problem.

Current Account, Capital Formation and Terms of Trade Shocks: a Revisit of the Harberger-Laursen-Metzler Effect

This paper extends the Blanchard model of a closed-economy to a three-good (exportable, importable and non-tradable goods) open-economy model with capital accumulation and uncertain lifetimes to study the impacts of terms of trade shocks on the current account. The simulation results show that a model with uncertain lifetimes is more appropriate to describe a small open economy like Taiwan at the steady-state equilibrium than a model with infinite lifetimes. We find that the Harberger-Laursen-Metzler effect is discernible for temporary or permanent terms of trade shocks. Furthermore, the steady state is a saddle point and the speed of convergence of capital and consumption is quite low.

SAVINGS, INTERGENERATIONAL TRANSFERS, AND THE DISTRIBUTION OF WEALTH

This paper investigates the quantitative importance of different savings motives on the distributions of wealth and consumption and aggregate capital accumulation by solving an overlapping generations model with intragenerational heterogeneity. Agents differ in age, ability, earnings shocks, and inherited bequests. In the baseline economy, there are uninsurable idiosyncratic risks associated with uncertain lifetime and earnings shocks. The model is calibrated to the U.S. economy and solved numerically. Then the allocations of the baseline economy are compared with those of an economy with complete annuity markets, one without earnings uncertainty, and one without altruism.

Optimal Life Insurance Purchase, Consumption and Portfolio Under an Uncertain Life

In this dissertation, we set up a new quantitative model to investigate the problem of optimal life insurance purchase, consumption and portfolio investment strategies for a wage earner under an uncertain lifetime. We use two approaches, namely, the dynamic programming technique and the martingale technique, to analyze our model. First, by using the dynamic programming technique, we derive HJB equations for our problem and obtain explicit solutions for CRRA utility functions with or without subsistence requirements. Then we study the demand of life insurance by examining our explicit solutions and doing numerical experiments. Since explicit solutions to HJB equations are very rare, we develop a numerical method which is Makov Chain Approximation combining with logarithmic transformations. This numerical method is stable and easy-to-implement for our problem. Second, by using the martingale technique, we set up the mathematical framework for our model in a very general setting. A series of fundamental results are found, for example, the bankruptcy condition, the characteristic of admissible consumption, insurance premium and portfolio strategies, existence of optimality, etc.

The Political Economy of Environmental Taxes with an Aging Population

This paper develops a two-period overlapping-generations model with environmental externalities and uncertain lifetimes, and studies how two sources of population aging, greater longevity and a lower rate of population growth, affect the politically determined environmental tax and the quality of the environment. It is shown that greater longevity and a lower rate of population growth have entirely different effects on these factors; greater longevity has no effect on the environmental tax and has a non-positive effect on the environmental quality, whereas a lower rate of population growth has a negative effect on the tax and has a positive effect on the quality. In addition, the political decision-making on the tax leads to an inefficient allocation of capital and environmental quality.

Diminishing Impatience: Disentangling Time Preference from Uncertain Lifetime

A decision maker with time consistent preferences may exhibit diminishing impatience, when uncertain lifetime is accounted for. Uncertain lifetime captures not only the risk of mortality, but also the possibility that a promise for a delayed reward might be breached, or a postponed consumption might not be realized. The restrictions that time consistency imposes on additive intertemporal preferences are characterized. It is shown that if the hazard rate of mortality is diminishing, then a time consistent agent will exhibit diminishing impatience. A demographic model that allows for unobservable heterogeneity in frailty (risk of mortality) accommodates diminishing impatience, even in the presence of stationarity and time consistency.

Diminishing Impatience and Non-Expected Utility: A Unified Framework

This work suggests that diminishing impatience originates in the distinction between the certain present and the risky future. It models this difference using the metaphor of uncertain lifetime. This metaphor captures not only the real risk of mortality, which can never be assumed away, but also the implicit risk that a promise of future benefit may not be kept. Uncertain lifetime transforms any planned consumption path into a lottery. A simple functional representation of preferences over stochastic consumption paths is derived. It explicitly allows for time inconsistency when the future is risky, and can be used to study present bias, commitment devices and sophisticated behavior. This representation allows to relate diminishing impatience to well known behavioral regularities in the field of choice under risk and uncertainty. A simple parametric condition can account for the modal choice pattern observed in diminishing impatience, the Common Ratio Effect and the Ellsberg Paradox. Furthermore, it is shown that existing experimental evidence supports this work's hypothesis that diminishing impatience may be traced to the certainty of the present contrasted with the risk associated with any future path. The representation developed is applied to analyze the optimal rate of resource depletion when there exists uninsurable uncertainty that the unconsumed stock may disappear.

The Distribution of Gains from Access to Stocks

Recent stock market and economy-wide developments have raised questions as to the continuation of the dramatic upward trend in household stock market participation observed in the 1990s. We study, computationally and econometrically, determinants of net gains from access to stocks that govern participation decisions. We first solve a calibrated intertemporal model with income risk, participation costs, uncertain lifetime, and retirement to shed light on often conflicting effects of various demographic and market factors on net gains. We then apply modern binary quantile regression techniques to data from the 1995 and 1998 US Surveys of Consumer Finances to uncover the changing roles of demographics as we move from infra-marginal to marginal stockholders and to most likely potential entrants; and to estimate the demographic and other characteristics of these groups. We contrast our findings to standard probit and logit. Findings indicate that gains from stock market access have multiple determinants whose roles are often conflicting and not the same at different points in the gains distribution. Economic downturns can have significant effects on participation decisions of investors around the participation margin through their influence on household incomes, wealth, and employment. The role of education in participation is found to be more limited than usually estimated in participation probits, and confined to the low end of the gains distribution. Continuation of growth in stock market participation is likely to pose challenges, in view of more limited finances, younger age, more limited education, and above all significantly less self-declared willingness to assume financial risk by those most likely to consider entry compared to those marginal investors who have already passed the participation threshold.