Preference for Flexibility and Dynamic Consistency

Dekel, Lipman, and Rustichini [3] characterize preferences over menus of lotteries that can be represented by the use of a unique subjective state space and a prior. We investigate what would be the appropriate version of Dynamic Consistency in such a setup. The condition we find, which we call Flexibility Consistency, is linked to a comparative theory of preference for flexibility. When the subjective state space is finite, we show that Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency and that it implies that the decision maker is a subjective state space Bayesian updater. Later we characterize when a collection of signals can be interpreted as a partition of the subjective state space of the decision maker.     

Subjective states: A more robust model

We study the demand for flexibility and what it reveals about subjective uncertainty. As in Kreps [D. Kreps, 1979. A representation theorem for ‘preference for flexibility’. Econometrica 47, 565–577], Nehring [K. Nehring, 1996. Preference for flexibility and freedom of choice in a Savage framework. UC Davis Working Paper; K. Nehring, 1999. Preference for flexibility in a Savage framework. Econometrica 67, 101–119] and Dekel et al. [E. Dekel, B. Lipman, A. Rustichini, 2001. Representing preferences with a unique subjective state space. Econometrica 69, 891–934], the latter is represented by a subjective state space consisting of possible future preferences over actions to be chosen ex post. One contribution is to provide axiomatic foundations for a range of alternative hypotheses about the nature of these ex post preferences. Secondly, we establish a sense in which the subjective state space is uniquely pinned down by the ex ante ranking of (random) menus. Finally, we demonstrate the tractability of our representation by showing that it can model the two comparative notions “2 desires more flexibility than 1” and “2 is more averse to flexibility-risk than is 1.”     

Subjective random discounting and intertemporal choice

This paper provides an axiomatic foundation for a particular type of preference shock model called the random discounting representation where a decision maker believes that her discount factors change randomly over time. For this purpose, we formulate an infinite horizon extension of [E. Dekel, B. Lipman, A. Rustichini, Representing preferences with a unique subjective state space, Econometrica 69 (2001) 891-934], and identify the behavior that reduces all subjective uncertainties to those about future discount factors. We also show uniqueness of subjective belief about discount factors. Moreover, a behavioral comparison about preference for flexibility characterizes the condition that one's subjective belief second-order stochastically dominates the other. Finally, the resulting model is applied to a consumption-savings problem.     

Subjective probability over a subjective decision tree

A state space has been assumed as a primitive for modeling uncertainty, which presumes that the analyst knows all the uncertainties a decision maker (DM) perceives. This is problematic because states are private information of the DM, and hence are not directly observable to the analyst. Dekel et al. [Representing preferences with a unique subjective state space, Econometrica 69 (2001) 891-934] derive, rather than assume, the subjective state space from preference over suitable choice objects.In a dynamic setting, a decision tree, that is, a pair consisting of a state space and a filtration, has been taken as a primitive. This assumption is also problematic—a decision tree should be derived rather than assumed as a primitive. We formulate a three-stage extension of the above literature in order to model a DM who anticipates subjective uncertainty to be resolved gradually over time. We identify also subjective beliefs on the subjective state space.     

Preference for flexibility in infinite horizon problems

Summary. We consider the extension of the classical problem of preference for flexibility to many periods. Preferences are defined over sets of infinite paths of choices. The main result provides a set of axioms on preferences that yield an additive representation over a subjective state space. This space is the set of preferences over choice today and feasible set tomorrow. The main new axiom, stochastic dominance, is a stronger form of the assumption of monotonicity. Received: September 11 2000; revised version: December 18, 2001     

Ambiguity,data and preferences for information – A case-based approach

We model decision making under ambiguity based on available data. Decision makers express preferences over actions and data sets. We derive an α-max–min representation of preferences, in which beliefs combine objective characteristics of the data (number and frequency of observations) with subjective features of the decision maker (similarity of observations and perceived ambiguity). We identify the subjectively perceived ambiguity and separate it into ambiguity due to a limited number of observations and ambiguity due to data heterogeneity. The special case of no ambiguity provides a behavioral foundation for beliefs as similarity-weighted frequencies as in Billot et al. (2005) [3].     

On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L_∞(M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's (J. Econ. Theory4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.     

The impossibility of compromise: some uniqueness properties of expected utility preferences

Summary. We focus on the following uniqueness property of expected utility preferences: Agreement of two preferences on one interior indifference class implies their equality. We show that, besides expected utility preferences under (objective) risk, this uniqueness property holds for subjective expected utility preferences in Anscombe-Aumann's (partially subjective) and Savage's (fully subjective) settings, while it does not hold for subjective expected utility preferences in settings without rich state spaces. Indeed, when it holds the uniqueness property is even stronger than described above, as it needs only agreement on binary acts. The extension of the uniqueness property to the subjective case is possible because beliefs in the mentioned settings are shown to satisfy an analogous property: If two decision makers agree on a likelihood indifference class, they must have identical beliefs. Received: November 15, 1999; revised version: December 29, 1999     

On preferences with infinitely many subjective states

Models with subjective state spaces have been extremely useful in capturing novel psychological phenomena that consist of both a preference for flexibility and for commitment. Interpreting the utility representations of preferences as capturing these phenomena requires one to use the notion of a sign of a state. For linear preferences, we completely characterise the sign of a state in terms of its analytic representation as an integral with respect to a signed measure. In models with finitely many states, a state is either positive or negative, but never both. We show that in models with infinitely many states, a state can be both positive and negative. Thus, models with finitely many states may not capture all the behavioural features of an infinite model. Our methods are also useful in constructing utility functionals over menus with desired local properties.     

Choice and individual welfare

We propose an abstract method of systematically assigning a “rational” ranking to non-rationalizable choice data. Our main idea is that any method of ascribing welfare to an individual as a function of choice is subjective, and depends on the economist undertaking the analysis. We provide a simple example of the type of exercise we propose. Namely, we define an individual welfare functional as a mapping from stochastic choice functions into weak orders. A stochastic choice function (or choice distribution) gives the empirical frequency of choices for any possible opportunity set (framing factors may also be incorporated into the model). We require that for any two alternatives x and y, if our individual welfare functional recommends x over y given two distinct choice distributions, then it also recommends x over y for any mixture of the two choice distributions. Together with some mild technical requirements, such an individual welfare functional must weight every opportunity set and assign a utility to each alternative x which is the sum across all opportunity sets of the weighted probability of x being chosen from the set. It therefore requires us to have a “prior view” about how important or representative a choice of x at a given situation is.     

The α-MEU model: A comment

In Ghirardato et al. (2004) [7], Ghirardato, Macheroni and Marinacci propose a method for distinguishing between perceived ambiguity and the decision-maker?s reaction to it. They study a general class of preferences which they refer to as invariant biseparable. This class includes CEU and MEU. They axiomatize a subclass of α-MEU preferences. If attention is restricted to finite state spaces, we show that any α-MEU preference relation, satisfies GMM?s axioms if and only ifα=0 or 1, that is, the preferences must be either maxmin or maxmax. We show by example that these axioms may be satisfied when the state space is [0,1].     

Status Quo Bias in Bargaining: An Extension of the Myerson-Satterthwaite Theorem with an Application to the Coase Theorem

We generalize the Myerson-Satterthwaite theorem to study inefficiencies in bilateral bargaining over a divisible good, with two-sided private information on the valuations. For concave quasi-linear preferences, the ex ante most efficient Bayes equilibrium of any mechanism always exhibits a bias toward the status quo. If utility functions are quadratic every Bayes equilibrium is ex post inefficient, with the expected amount of trade biased toward the disagreement point. In other words, for the class of preferences we study, there is a strategic advantage to property rights in the Coase bargaining setup in the presence of incomplete information. Journal of Economic Literature Classification Numbers: C78, D23, D62, D82.     

On a Class of Stable Random Dynamical Systems: Theory and Applications

We consider a random dynamical system in which the state space is an interval, and possible laws of motion are monotone functions. It is shown that if the Markov process generated by this system satisfies a splitting condition, it converges to a unique invariant distribution exponentially fast in the Kolmogorov distance. A central limit theorem on the time-averages of observed values of the states is also proved. As an application we consider a system that captures an interaction of growth and cyclical forces: of two possible laws, one is monotone, but the other is unimodal with two periodic points. Journal of Economic Literature Classification Numbers: C6, D9.     

What is an ‘endogenous state space’?

Summary. In the literature on choice under unforeseen contingencies, the decision maker behaves as if she aggregates possible instances of future rankings indexed by a set S. The set S is interpreted as a subjective state space even though subsequent rankings need not conform to any one of the aggregated utilities. This paper proposes a definition for a subjective state space under unforeseen contingencies that is topologically unique, derives its existence from preference primitives as opposed to the representation of preferences, and does not commit to an interpretation in which states correspond to future realized rankings. The definition topologically concurs with and extends the identification of the essentially unique subjective state space due to Dekel, Lipman and Rustichini [4].Received: 28 October 2003, Revised: 13 October 2004, JEL Classification Numbers: D11, D81, D91.I thank Eddie Dekel, Alan Kraus, Bart Lipman, Chris Shannon, and the referee for some helpful remarks.     

A theory of (relative) discounting

We derive a representation theorem for time preferences (on the prize-time space) which identifies a novel notion of relative discounting as the key ingredient. This representation covers a variety of time preference models, including the standard exponential and hyperbolic discounting models and certain non-transitive time preferences, such as the similarity-based and subadditive discounting models. Our axiomatic work thus unifies a number of seemingly disparate time preference structures, thereby providing a tractable mathematical format that allows for investigating certain economic environments without subscribing to a particular time preference model. This point is illustrated by means of an application to sequential bargaining theory.     

Cobb-Douglas preferences under uncertainty

This paper axiomatizes Cobb-Douglas preferences under uncertainty. First, we extend the original Trockel (Econ Lett 30:7–10, 1989)’s axiomatic foundation to a general state space framework based on the Strong Homotheticity Axiom, obtaining also the incomplete case a la Bewley (Decis Econ Financ 25:79–110, 2002). We show that this key axiom for the Cobb-Douglas expected utility specification is refuted by Ellsberg’s uncertainty aversion behavioral pattern. Our main result provides a set of meaningful axioms characterizing Cobb-Douglas min-expected utility preferences, an important class of uncertainty averse preferences for studying the consequences of ambiguity in finance and other fields. Finally, we present briefly how to obtain more general representations like the variational case.     

Ambiguity Without a State Space

Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyses a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of objective ambiguity without explicit reference to any state space. We characterize a representation that integrates a non-linear transformation of first-order expected utility with respect to a second-order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion.     

Lexicographic expected utility with a subjective state space

This paper provides a model that allows for a criterion of admissibility based on a subjective state space. For this purpose, we build a non-Archimedean model of preference with subjective states, generalizing Blume et al. (Econometrica 59:61–79, 1991), who present a non-Archimedean model with exogenous states; and Dekel et al. (Econometrica 69:891–934, 2001), who present an Archimedean model with an endogenous state space. We interpret the representation as modeling an agent who has several “hypotheses” about her state space, and who views some as “infinitely less relevant” than others.     

Nonseparable preferences and optimal social security systems

In this paper, we consider economies in which agents are privately informed about their skills, which evolve stochastically over time. We require agents' preferences to be weakly separable between the lifetime paths of consumption and labor. However, we allow for intertemporal nonseparabilities in preferences like habit formation. In this environment, we derive a generalized version of the Inverse Euler Equation and use it to show that intertemporal wedges characterizing optimal allocations of consumption can be strictly negative. We also show that preference nonseparabilities imply that optimal differentiable asset income taxes are necessarily retrospective in nature. We show that under weak conditions, it is possible to implement a socially optimal allocation using a social security system in which taxes on wealth are linear, and taxes/transfers are history-dependent only at retirement. The average asset income tax in this system is zero.     

A dynamic two country Heckscher–Ohlin model with non-homothetic preferences

We examine the properties of a two-country dynamic Heckscher–Ohlin model that allows for preferences to be non-homothetic. We show that the model has a continuum of steady state equilibria under free trade, with the initial conditions determining which equilibrium will be attained. We establish conditions under which a static Heckscher–Ohlin theorem will hold in the steady state, and also conditions for a dynamic Heckscher–Ohlin theorem to hold. If both goods are normal, each country will have a unique autarkic steady state, and all steady state equilibria are saddle points. We also consider the case in which one good is inferior, and show that this can lead to multiple autarkic steady states, violations of the static Heckscher–Ohlin theorem in the steady state. Furthermore, there may exist steady state equilibria that Pareto dominate other steady states. These steady states will be unstable if discount factors are the same in each country, although they may exhibit dynamic indeterminacy if discount factors differ.