Prudence, Intertemporal Substitution, and Savings under Uncertainty

This paper derives the relations between the coefficient of absolute prudence, the equivalent precautionary premium, risk aversion to concentration, and the normality and shift of current consumption under uncertainty, without the time-separable utility assumption. Examples show that Kimball's coefficient of absolute prudence does not fully characterize precautionary saving or saving behavior under uncertainty. It is proved that, whereas a higher rate of intertemporal substitution and a larger coefficient of absolute prudence imply more savings when current consumption is normal, a larger coefficient of intertemporal substitution and a smaller coefficient of absolute prudence imply more savings when current consumption is inferior. Received July 17, 2001; revised version received November 20, 2001     

Precautionary saving in the presence of labor income and interest rate risks

This paper deals with the classic issue of precautionary saving in a standard two-period setting. The literature has identified conditions on the individual??s utility function under which either labor income uncertainty or interest-rate uncertainty can lead to positive precautionary saving. We allow for both sources of uncertainty simultaneously. We extend the Jensen inequality from one risk to two positively quadrant dependent risks. The main result of the paper is that ??positive quadrant dependent?? uncertainty raises saving if and only if ??partial relative prudence?? is larger than 2. We characterize the condition that ??partial relative prudence?? exceeds 2 via preferences over simple binary lotteries, in the style of Eeckhoudt and Schlesinger (Am Econ Rev 96:280?C289, 2006).     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Focusing effect and the poverty trap

I build a dynamic consumption-savings model in which agents׳ choices are distorted by the focusing effect: agents overweight the utility of goods in which their options differ more. I show that the consumption-savings choice depends both on the marginal return on savings and on the total return on savings, implying that the incentive to save may increase with the initial level of wealth. As a consequence, a salience-based poverty trap may exist when the marginal return on savings is sufficiently high and sufficiently flat. I also consider the case of a perfect credit market and show that a poverty trap may emerge when the salience of consumption is bounded above. I discuss policy implications. In particular, imposing upon an agent a punishment for decreasing savings below a threshold leads to a higher level of savings, even when the threshold triggering the punishment is not binding     

Uncertainty and the Voluntary Provision of a Pure Public Good in a Two‐Moment Decision Model

In this paper, we explore the potential benefits of uncertainty that may arise in a two‐moment model of the voluntary provision of a pure public good. We find that an increase in a given contributor i’s risk associated with the aggregate contribution level of the other contributors (i.e., an increase in social uncertainty) induces that contributor to increase his own contribution level if and only if the uncertainty's incremental effect on the expected value of his net marginal utility is negative. Contributor i’s welfare likewise increases when a closely related condition is met, namely that the uncertainty's marginal effect on his expected marginal utility value of the public good exceeds its countervailing effect on the numeraire. Further, the corresponding aggregate contribution to the public good increases in the presence of free‐riding if and only if the incremental effect of contributor i’s contribution on the aggregate expected value of all other contributors’ net marginal utilities is small‐enough positive. We derive similar conditions for the case of private uncertainty, where the increase in contributor i’s risk is associated with his own marginal valuation of the public good. A simple example illustrates these conceptual results. Numerical analysis demonstrates that an increase in private uncertainty can have a nonmonotonic impact on contributor i’s welfare.     

The effect on optimal consumption of increased uncertainty in labor income in the multiperiod case

We consider a multiperiod, additive utility, optimal consumption model with a riskless investment and a stochastic labor income. The main result is that for utility functions belonging to the set F, consumption decreases when we go from any sequence of distribution functions representing labor income to a more risky sequence. A concave utility function belongs to F if its first derivative exists everywhere and is convex.     

Recursive preferences and balanced growth

We study a class of utility functions that are defined recursively by an aggregator W(x,y) where u_t=W(c_t,u_t+1). In single-agent economies it is known that a sufficient condition for the existence of a balanced growth path is that utility should be homogenous of degree γ. In the context of a multi-agent economy we show that this restriction implies that either a balanced growth equilibrium fails to exist or all agents have the same constant discount factor. We suggest a generalization of recursive preferences wherein the intertemporal utility function is time dependent. Within this class we establish that there may exist a balanced growth equilibrium even if agents are different.     

Wealth effects and agency costs

We analyze how the agent's initial wealth affects the principal's expected profits in the standard principal–agent model with moral hazard.We show that if the principal prefers a poorer agent for all specifications of action sets, probability distributions, and disutility of effort, then the agent's utility of income must exhibit a coefficient of absolute prudence less than three times the coefficient of absolute risk aversion for all levels of income, thus strengthening the sufficiency result of Thiele and Wambach (1999). Also, we prove that there is no condition on the agent's utility of income alone that will make the principal prefer richer agents. Moreover, we show that, for an interesting class of problems, the principal prefers a relatively poorer agent if agent's wealth is sufficiently large. Finally, we discuss how alternative ways of modeling the agent's outside option affects the principal's preferences for agent's wealth.     

Time Horizon and the Discount Rate

We consider an economy à la Lucas (1978, Econometrica46 , 1429-1446) with a risk-averse representative agent. The exogenous growth rate of the economy follows a random walk. We characterize the set of utility functions for which it is efficient to discount more distant cash flows at a lower rate. The benchmark result is that, when the growth rate is almost surely nonnegative, the yield curve is decreasing if and only if relative risk aversion is decreasing with wealth. Relaxing the assumption on the absence of recession requires more restrictions on preferences, such as increasing relative prudence. Journal of Economic Literature Classification Numbers: D81, D91, Q25, Q28.     

Taxation of Labor Income and the Demand for Risky Assets

This article analyzes the effect of labor income risk on the joint saving/portfolio-composition problem. Given decreasing absolute prudence, we find that even when labor income risk increases overall saving, it tends to lower investment in a risky asset. Applying the theory to public finance, we argue that realistic increases in marginal tax rates on labor can cause large enough reductions in after-tax labor income risk to cause significant increases in risky investment.     

Choice under uncertainty with the best and worst in mind: Neo-additive capacities

We develop the simplest generalization of subjective expected utility that can accommodate both optimistic and pessimistic attitudes towards uncertainty—Choquet expected utility with non-extreme-outcome-additive (neo-additive) capacities. A neo-additive capacity can be expressed as the convex combination of a probability and a special capacity, we refer to as a Hurwicz capacity, that only distinguishes between whether an event is impossible, possible or certain. We show that neo-additive capacities can be readily applied in economic problems, and we provide an axiomatization in a framework of purely subjective uncertainty.     

Dynamic stability and reform of political institutions

This paper examines endogenous institutional change in a class of dynamic political games. The political aggregation rules used at date t+1 are instrumental choices under rules at date t. Effectively, rules are “players” who can strategically delegate future policy-making authority to different rules. A political rule is stable if it selects itself. A reform occurs when an alternative rule is selected. The stability of a political rule is shown to depend on whether its choices are dynamically consistent. For instance, simple majority rules can be shown to be dynamically consistent in many common environments where wealth-weighted voting rules are not. The result extends to political rules that incorporate private activities such as extra-legal protests, threats, or private investment. The approach is one way of understanding various explanations of institutional change proposed in the literature. A parametric model of public goods provision gives an illustration.     

An essay on the Life Cycle: Characterizing intertemporal behavior with uncertainty,human capital,taxes, durables,imperfect capital markets,and non-separable preferences

This paper formulates a general characterization of a household's portfolio choice and savings behavior in an environment with uncertain future interest rates, prices, wages, and factors influencing tastes. Savings may be invested in three types of assets: financial assets; human capital, which is non-tradable; and consumer durables, in which investment may be partially irreversible. Risk-return relations determine the optimal allocation of resources across assets at a point in time. The optimal intertemporal allocation of resources is determined by a restriction on the planned growth rate for the marginal utility of after-tax wealth, where growth rates depend on rates of time preference and measures of long-term riskless rates of interest. Given special assumptions, this marginal utility follows a martingale process as a consequence of optimizing behavior. Pricing formulae are developed for evaluating shifts in uncertain future income, wage, and price profiles. The relations characterizing portfolio and savings behavior presented here do not rely on particular distributional assumptions; they account for all forms of uncertainty including wage uncertainty induced by human capital investment; they allow for the non-marketability of assets; and the main results apply for very general functional form assumptions for preferences. In later sections, results are extended to incorporate income taxes and to account for a wide variety of imperfections in asset markets.     

Redistributive effects of minimal equal sacrifice taxation

We examine the circumstances under which minimal equal sacrifice taxation reduces income inequality. Focusing on absolute sacrifice and relative inequality, we show that after tax incomes are more equally distributed than before tax incomes and that after tax inequality decreases with the revenue to be raised if and only if the elasticity of the marginal utility of income is decreasing. However, other things equal, after tax inequality is shown to decline as the utility function becomes more concave. We investigate next how these conclusions extend when one takes the absolute inequality view and considers that it is the relative loss in utility that has to be equalized across individuals.     

Stochastic growth: a duality approach

We re-examine the representative agent's optimal consumption and savings under uncertainty in the presence of investment constraints using martingale representation and convex analysis techniques. This framework allows us to explicitly quantify precautionary savings which induces a higher average growth rate than in a certainty setup. We provide a closed form solution for a Cobb-Douglas economy. The effect of uncertainty on portfolio selection is analyzed. Consumption growth rate and risk free interest rate exhibit a U-shaped relationship. Uncertainty negatively affects expected consumption growth rate; such a result seems to be supported by empirical evidence.     

ON THE CONCAVITY AND QUASICONCAVITY PROPERTIES OF UTILITY FUNCTIONS

Concavity and quasiconcavity have always been important properties in financial economics particularly in decision problems when an objective function has to be maximized over a convex set. Both properties have mainly been used as purely technical assumptions. In this paper, we link concavity and quasiconcavity of a utility function to the basic concepts of risk aversion, prudence, risk vulnerability and temperance. We show that concavity means the agent is more risk vulnerable than prudent. In particular, we can see when a function is both concave and quasiconcave and when it is only quasiconcave.     

Risks, Financing Constraints, and High Savings Ratio in the Rural Economy of China: A Model Incorporating Precautionary Savings and Liquidity Constraints

This paper makes an in-depth investigation on the phenomenon of high savings rate in the rural economy of China between 1978 and 2003. On the basis of precautionary savings theory, we construct a model incorporating the risks, liquidity constrains, and aging population to explain the existence of high savings rate in the rural economy of China. We measure risks with Gini coefficient and marginal propensity to save. We find that these risk indices are positively associated with the higher savings rate and the higher degree of prudence of rural households. Our findings pose an urgency of the reform of rural financial system and the improvement in social security system in the rural economy of China. Translated from Economic Science, 2005, (1) (in Chinese)     

An income fluctuation problem

A consumer has a t period planning horizon problem, at each period, he gets additionals incomes that are i.i.d. random variables, and he must decide how much of his income will be spent in consumption, yielding some utility, and how much will be saved to maximize the total expected utility. No borrowing is allowed, and for the amounts saved no interest is paid. Using the concepts of competitive prices it is shown that as t → ∞ the corresponding limit of the consumption policy is strictly bounded above by the expected value of the random income.     

Moral Hazard,Income Taxation and Prospect Theory*

The standard theory of optimal income taxation under uncertainty has been developed under the assumption that individuals maximise expected utility. However, prospect theory has now been established as an alternative model of individual behaviour, with empirical support. This paper explores the theory of optimal income taxation under uncertainty when individuals behave according to the tenets of prospect theory. It is seen that many of the standard results are modified in interesting ways. The first‐order approach for solving the optimisation problem is not valid over the domain of losses, and the marginal tax schedule offers full insurance around the reference consumption level. The implications of non‐welfarist objectives under income uncertainty are also examined.     

On the shape of optimal tax schedules

Take consumers to be described by a parameter h (skill, needs, etc.) with utilities defined on N commodities, including factor supplies. Our main result is that if in the optimum each component of this consumption vector is bounded and bounded away from zero over the population, each marginal tax must be zero at both ends of the corresponding tax schedule. For the income tax case, it is also shown that if the marginal tax rate at the top of the scale is positive, one can construct another tax schedule which is strongly Pareto superior (dominating the first one at all or most income levels), requirements of information being low.