Abstract: Recent evidence on intertemporal choice suggests that decision makers may exhibit both increasing and decreasing impatience simultaneously, called inverse-S discounting. This paper studies inverse-S discounting behaviour. A formal connection between inverse-S discounting and inverse-S probability weighting is established. A tractable, yet flexible, discounted utility model of such behaviour is introduced. A preference foundation for the simple inverse-S discounting model is provided, both in the context of timed outcomes and the context of consumption streams. The key axiom is a weakening of the time consistency axiom that allows for both increasing and decreasing impatience.
Abstract: Experimental studies suggest that individuals exhibit more risk tolerance when prospects are delayed, leading to preference reversals that cannot be attributed to discounting. When data suggests that utility is time independent, probability weighting functions, such as those used to model prospect theory preferences, can account for such risk tolerance. We propose a descriptive model with a two-parameter probability weighting function where one of these parameters depends on the time at which a prospect is resolved. The time-dependent parameter is responsible for the curvature of the probability weighting function and can be seen as an index of insensitivity towards changes in probabilities. We provide conditions that characterize increased sensitivity towards more distant probabilities; this can account for the observed risk tolerance towards delayed prospects. In our framework, the discount function can be quite general and independent of utility for outcomes or probability weighting, allowing the model to be compatible with latest empirical findings of nonconstant discounting. In a simple application to bargaining we show that it can be advantageous for an individual to advance or delay the bargaining time if an opponent displays increased sensitivity to delay.
Abstract: Variational preferences (Maccheroni, Marinacci and Rustichini, 2006) are an important class of ambiguity averse preferences, compatible with Ellsberg-type phenomena. In this paper, a new foundation for variational preferences is derived in a framework of two-stages of purely subjective uncertainty. A similar foundation is obtained for purely subjective maxmin expected utility (Gilboa and Schmeidler, 1989). By establishing their axiomatic foundations without the use of extraneous probabilities, the conceptual appeal and applicability of these ambiguity models is enhanced.
Abstract: Choice under risk is modelled using a piecewise linear version of rank-dependent utility. This model can be considered a continuous version of NEO-expected utility (Chateauneuf, Eichberger and Grant, 2007). In a framework of objective probabilities a preference foundation is given, without requiring a rich structure on the outcome set. The key axiom is called complementary additivity.
Abstract: This paper studies intertemporal choice in a dynamic framework with continuous time. A model called continuous quasi-hyperbolic discounting is considered, extending the popular quasi-hyperbolic discounting to an integral form. Dynamic preference axioms, time consistency and time invariance, are formulated and used to provide a foundation for an integral form of exponential discounting; the central model of dynamic, intertemporal choice in economics. A relaxion of the time consistency axiom, complementary time consistency, is formulated to provide a dynamic preference foundation for continuous quasi-hyperbolic discounting. A preference condition for present bias is also characterised in the context of the model.
Abstract: A model of choice under purely subjective uncertainty, Piecewise Additive Choquet Expected (PACE) utility, is introduced. PACE utility allows for optimism and pessimism simultaneously, but represents a minimal departure from expected utility. It can be seen as a continuous version of NEO-expected utility (Chateauneuf et al, 2007) and, as such, is especially suited for applications with rich state spaces. The main theorem provides a preference foundation for PACE utility in the Savage framework of purely subjective uncertainty with an arbitrary outcome set.
Abstract: Two-Stage Exponential (TSE) discounting, the model developed here, generalises exponential discounting in a parsimonious way. It can be seen as an extension of Quasi-Hyperbolic discounting to continuous time. A TSE discounter has a constant rate of time preference before and after some threshold time; the switch point. If the switch point is expressed in calendar time, TSE discounting captures time consistent behaviour. If it is expressed in waiting time, TSE discounting captures time invariant behaviour. We provide preference foundations for all cases, showing how the switch point is derived endogenously from behaviour. We apply each case to Rubinstein's infinite-horizon, alternating-offers bargaining model.
Abstract: We provide a preference foundation for decision under risk resulting in a model where probability weighting is linear as long as the corresponding probabilities are not extreme (i.e., 0 or 1). This way, most of the elegance and mathematical tractability of expected utility is maintained and also much of its normative foundation. Yet, the new model can accommodate the extreme sensitivity towards changes from 0 to almost impossible and from almost certain to 1 that has widely been documented in the experimental literature. The model can be viewed as ‘‘expected utility with the best and worst in mind’’ as suggested by Chateauneuf, Eichberger and Grant (2007) or, following our preference foundation, interpreted as ‘‘expected utility with consistent optimism and pessimism’’.
Abstract: This paper reconsiders the Bargaining Problem of Nash (Econometrica 28:155–162, 1950). I develop a new approach, Conditional Bargaining Problems, as a framework for measuring cardinal utility. A Conditional Bargaining Problem is the conjoint extension of a Bargaining Problem, conditional on the fact that the individuals have agreed on a “measurement event”. Within this context, Subjective Mixture methods are especially powerful. These techniques are used to characterise versions of the Nash and the Kalai–Smorodinsky solutions. This approach identifies solutions based only on the individuals’ tastes for the outcomes. It is therefore possible to do Bargaining theory in almost complete generality. The results apply to Biseparable preferences, so are valid for almost all non-expected utility models currently used in economics.