Characterization of stationary preferences in a continuous time framework

We characterize preference relations on continuous time consumption paths which admit an exponential discounting representation. We provide two theorems as such, one in the cardinal framework and another in the ordinal framework. Our characterizations parallel the known characterizations in discrete time framework. In the cardinal framework, we adopt the axioms of Epstein (1983), which characterize a stationary preference relation in discrete time, and obtain the exponential discounting model as a special case of the discounting model proposed by Uzawa (1968). In the ordinal framework, we adopt the axioms of Bleichrodt et al. (2008) which were proposed to generalize Koopmans’ classical characterization of stationary preferences.     

Optimal money supply in models with endogenous discount factor

This paper studies the Friedman rule for the optimal quantity of money in money in the utility (MIU) and cash–credit models while considering two specifications for the endogenous discount factor. In the first specification, the discount factor depends directly on the utility level. In the second, the discount factor depends on every component of the utility function. We show that under the former specification the Friedman rule is the optimal policy. Under the latter, however, while the Friedman rule is optimal for the MIU model, it is not optimal for the cash–credit model.     

On time-consistent policy rules for heterogeneous discounting programs

This article considers a new concept of social optimum for an economy populated by agents with heterogeneous discount factors. It is based upon an approach that constrains decision rules to be temporally consistent: these are stationary and unequivocally ruled by the state variable. For agents who differ only in their discount factors and have equal weights in the planner’s objective, the temporally-consistent optimal solution produces identical consumption for the agents at all time periods. In the long run, the capital stock is determined by a modified golden rule that corresponds to an average-like summation of all discount factors. The general argument is illustrated by various two-agent examples that allow for an explicit determination of the temporally consistent decision rules. Interestingly, this temporally consistent solution can be simply recovered from the characterization of a social planner’s problem with variable discounting and can also be decentralized as a competitive equilibrium through the use of various instruments.     

Management of a capital stock by Strotz's naive planner

The capital management problem posed by R.H. Strotz is analyzed for the case of the ‘naive’ planner who fails to anticipate changes in his own preferences. By imposing progressively stronger restrictions on the primitives of the problem – namely, the discounting function, the utility index function, and the investment technology – the planner's behavior is characterized first as the solution to an ordinary differential equation and then via explicit formulae. Inasmuch as these characterizations leave the discounting function essentially unrestricted, the theory can accommodate, in particular, decision makers who discount time according to the hyperbolic and ‘quasi-hyperbolic’ curves used in applied work and said to be supported by psychological studies. Comparative statics of the model are discussed, as are extensions of the analysis to allow for credit constraints, limited foresight, and partial commitment.     

Learning by doing,endogenous discounting and economic development

In this paper, we focus on a growth model where the discount rate is decreasing in capital accumulation and endogenous growth is made possible through learning by doing, knowledge accumulation being a by-product of gross investment. In such a model, the utility function has to be restricted to take positive values implying that the elasticity of marginal utility is lower than one. The presence of endogenous discounting generates a steady-state of stagnation which can be saddle-path stable or unstable depending on the marginal productivity of knowledge. In the case of long run growth, the fact that the elasticity of marginal utility is lower than one implies the existence of two asymptotic balanced growth paths: the one with the higher growth rate being a saddle point while the one with the lower growth rate not being a saddle point. We also study the optimal solution which is characterized by a unique balanced growth path. The policy consists as usual in subsidizing investment in order to internalize the externality.     

证券投资的风险偏好与期望效用决策模型

本文把期望效用理论用于投资决策分析，按实际风险偏好的系统分类构造模型，据此证明决策人效用函数必为线性函数和指数函数；前者对应风险中立，后者对应风险厌恶和风险追求；函数参数的不同取值，唯一确定了各类风险偏好的性质和程度。由此产生的分析方法和模型，可同时适用于正态或非正态收益分布的证券组合，为投资价值的评估提供有效而实用的程序。     

Non-stationary additive utility and time consistency

Within a continuous time life cycle model of consumption and savings, I study the properties of the most general class of additive intertemporal utility functionals. They are not necessarily stationary, and do not necessarily multiplicatively separate a discount factor from “per-period utility”. I prove rigorously that time consistency holds if and only if the per-period felicity function is multiplicatively separable in $t$, the date of decision and in $s$, the date of consumption, or equivalently, if the Fisherian instantaneous subjective discount rate does not depend on $t$. The model allows to explain “anomalies in intertemporal choice” even when the agents are time consistent and various empirical regularities. On the other hand, the model allows to characterize mathematically the “effective consumption profile” of naive, time-inconsistent agents.     

Interview design for revealing preferences of policy makers

A policy maker is asked a few simple questions about his preference. Then the model represents it by a quadratic utility function, which can be made monotonic and quasi-concave (= to provide the convexity of the preference). The design of the interview with a policy maker is aimed at attaining the following goals: (a) no ambiguous output (= degeneration of the model), (b) ordinal approach to preferences (= asking questions about ordinal preferences and providing the uniqueness of the ordinal preference at the model output, regardless of its representation by a quadratic utility function), (c) stability of the model (= the model's input–output transformation is continuous). We also describe briefly the implementation of our model in a user-friendly interface to a corresponding computer program.     

Intertemporal choice with different short-term and long-term discount factors

This paper proposes a new axiomatic model of intertemporal choice that allows for dynamic inconsistency. We weaken the classical assumption of stationarity into two related axioms: stationarity in the short-term and stationarity in the long-term. We obtain a model with two independent discount factors, which is flexible enough to capture different time preferences, including a greater impatience for more immediate outcomes (when a long-term discount factor exceeds a compounded short-term discount factor). Our proposed model can accommodate some experimental results that cannot be rationalized by other existing models of dynamic inconsistency (such as quasi-hyperbolic discounting and generalized hyperbolic discounting).     

Competitive equilibrium cycles with endogenous labor

In this paper, we study a two-sector optimal growth model with elastic labor supply. We show that the modified golden rule is saddle-point stable when the investment good is capital intensive. To characterize stability with a capital intensive consumption good, we focus on either additively separable or homothetic preferences. In the first specification, we show that optimal oscillations require the elasticity of intertemporal substitution in consumption to be high enough while the elasticity of labor needs to be low enough. At the same time, we prove that with a linear utility in leisure the modified golden rule is always saddle-point stable. In the second specification for preferences, we show that the local dynamic properties of the optimal path depend instead on the shares of consumption and leisure into total utility. We prove that endogenous fluctuations are even more likely with homothetic preferences.     

Patient Decision Making: Exponential versus Hyperbolic Discounting

Evidence strongly indicates that human decision makers discount the future using a hyperbolic function instead of following an exponential, which has been the long‐standing assumption. We compare the exponential and hyperbolic models in evaluation of a simple research and development project selection problem, finding that the exponential function favors short‐term choices, while the hyperbolic function favors the long‐term. We demonstrate applications of the hyperbolic function, concluding that the traditional exponential—which does not faithfully represent typical human preferences—leads toward myopic decision making when compared to the hyperbolic. Copyright © 2012 John Wiley & Sons, Ltd.     

A von Neumann–Morgenstern representation result without weak continuity assumption

In the paradigm of von Neumann and Morgenstern (1947), a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible. In this work, we replace the assumption of weak continuity by monotonicity. More precisely, on the space of lotteries on an interval of the real line, it is shown that any affine preference order which is monotone with respect to the first stochastic order admits a representation in terms of an expected utility for some nondecreasing utility function. As a consequence, any affine preference order on the subset of lotteries with compact support, which is monotone with respect to the second stochastic order, can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations for affine preference orders on the subset of those lotteries which fulfill some integrability conditions. The subtleties of the weak topology are illustrated by some examples.     

Simple axioms for countably additive subjective probability

This paper refines Savage’s theory of subjective probability for the case of countably additive beliefs. First, I replace his continuity axioms P6 and P7 with a simple modification of Arrow’s (1970) Monotone Continuity. Second, I relax Savage’s primitives: in my framework, the class of events need not be a σ-algebra, and acts need not have finite or bounded range. By varying the domains of acts and events, I obtain a unique extension of preference that parallels Caratheodory’s unique extension of probability measures. Aside from subjective expected utility, I characterize exponential time discounting in a setting with continuous time and an arbitrary consumption set.     

Preference for equivalent random variables: A price for unbounded utilities

Savage's expected utility theory orders acts by the expectation of the utility function for outcomes over states. Therefore, preference between acts depends only on the utilities for outcomes and the probability distribution of states. When acts have more than finitely many possible outcomes, then utility is bounded in Savage's theory. This paper explores consequences of allowing preferences over acts with unbounded utility. Under certain regularity assumptions about indifference, and in order to respect (uniform) strict dominance between acts, there will be a strict preference between some pairs of acts that have the same distribution of outcomes. Consequently in these cases, preference is not a function of utility and probability alone.     

The ordinal Nash social welfare function

A social welfare function entitled ‘ordinal Nash’ is proposed. It is based on risk preferences and assumes a common, worst social state (origin) for all individuals. The crucial axiom in the characterization of the function is a weak version of independence of irrelevant alternatives. This axiom considers relative risk positions with respect to the origin. Thus, the resulting social preference takes into account non-expected utility risk preference intensity by directly comparing certainty equivalent probabilities. The function provides an interpretation of the Nash-utility-product preference aggregation rule. Necessary and sufficient conditions for the function to produce complete and transitive binary relations are characterized.     

Nonrecursive separation of risk and time preferences

Recursive utility disentangles preferences with respect to time and risk by recursively building up a value function of local increments. This involves certainty equivalents of indirect utility. Instead we disentangle preferences with respect to time and risk by building up a value function as a non-linear aggregation of certainty equivalents of direct utility of consumption. This entails time-consistency issues which are dealt with by looking for an equilibrium control and an equilibrium value function rather than a classical optimal control and a classical optimal value function. We characterize the solution in a general diffusive incomplete market model and find that, in certain special cases of utmost interest, the characterization coincides with what would arise from a recursive utility approach. But also importantly, in other cases, it does not: The two approaches are fundamentally different but match, exclusively but importantly, in the mathematically special case of homogeneity of the value function.     

Dynamic theory of preferences: Habit formation and taste for variety

We analyze individual preferences over infinite horizon consumption choices. Our axioms provide the foundation for a recursive representation of the utility function that contains as particular cases the classical Koopmans representation (Koopmans (1960)) as well as the habit formation specification.We examine some of the consequences of our axiomatization by considering a standard consumer choice problem, and show that typically in the space of concave utility functions satisfying our axioms the consumer displays a taste for variety. The latter means that such a consumer selects optimally time variant consumption programs for any given time invariant sequence of commodities’ relative prices and for all possible sequences of market discount factors. In contrast, if a concave utility function satisfies Koopmans’ axioms the consumer does not display a taste for variety.     

Diversification preferences in the theory of choice

Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von Neumann–Morgenstern expected utility model, but the equivalence fails to hold in other models. This paper analyzes axiomatizations of the concept of diversification and their relationship to the related notions of risk aversion and convex preferences within different choice theoretic models. Implications of these notions on portfolio choice are discussed. We cover model-independent diversification preferences, preferences within models of choice under risk, including expected utility theory and the more general rank-dependent expected utility theory, as well as models of choice under uncertainty axiomatized via Choquet expected utility theory. Remarks on interpretations of diversification preferences within models of behavioral choice are given in the conclusion.     

On time preference,rational addiction and utility satiation

A basic consumer problem with a unique good is considered, current consumption of this good influencing in a positive manner consumer intertemporal utility, while past consumption exerts a negative influence. Moreover, in the line of Fisher, a specification of preferences is retained so that the rate of time preference, assumes a long-run value – this means for a stationary consumption-path – that is non-monotonic as a function of consumption: impatience increases for low level of consumptions but decreases for higher ones. Such a framework allows for an integrated appraisal of addiction, satiation and the rate of time preference. It is shown that the emergence of an addiction phenomenon in the neighbourhood of an unsatiated long-run position exactly corresponds to letting the rate of time preference be an increasing function of past consumption habits. When addiction becomes sufficiently strong, the unsatiated stationary state becomes unstable and the satiated steady state becomes the only admissible stationary position.     

Land of addicts? an empirical investigation of habit‐based asset pricing models

This paper studies the ability of a general class of habit‐based asset pricing models to match the conditional moment restrictions implied by asset pricing theory. We treat the functional form of the habit as unknown, and estimate it along with the rest of the model's finite dimensional parameters. Using quarterly data on consumption growth, assets returns and instruments, our empirical results indicate that the estimated habit function is nonlinear, that habit formation is better described as internal rather than external, and the estimated time‐preference parameter and the power utility parameter are sensible. In addition, the estimated habit function generates a positive stochastic discount factor (SDF) proxy and performs well in explaining cross‐sectional stock return data. We find that an internal habit SDF proxy can explain a cross‐section of size and book‐market sorted portfolio equity returns better than (i) the Fama and French ( 1993 ) three‐factor model, (ii) the Lettau and Ludvigson ( 2001b ) scaled consumption CAPM model, (iii) an external habit SDF proxy, (iv) the classic CAPM, and (v) the classic consumption CAPM. Copyright © 2009 John Wiley & Sons, Ltd.