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The framework of this paper is subjective time perception in the context of intertemporal choice, that is to say, the process of making decisions on dated outcomes (monetary or not) by an individual or a group of individuals. In this setting, the Discounted Utility model and, more specifically, the exponential discounting have been the paradigmatic methodology used to measure the preferences on delayed outcomes. However, this model can only be applied to consistent choices in which individuals do not change their preferences when the involved rewards are delayed the same time interval. Unfortunately, this is not the case of several decision scenarios where time is viewed as a subjective variable. The objective of this paper is to formally analyze the consistency of intertemporal choices governed by a discount function, derived from the exponential, where time has been distorted according to certain psychological traits of the subjects involved in the decision-making. More specifically, the different types of decreasing impatience will be characterized by focusing on the distortion derived from the subjective perspective of time. The findings of this research are very relevant in order to explain the time-related behavior of decision-makers in some noteworthy fields such as finance, psychology, marketing or sociology.

The framework of this paper is behavioral finance and, more specifically, intertemporal choice when individuals exhibit decreasing impatience in their decision-making processes. After characterizing the two main types of decreasing impatience (moderately and strongly decreasing impatience), the main objective of this paper is to generalize these concepts when the criterion of time increase is given by an arbitrary function which describes such increments. In general, the methodology is mathematical calculus but particularly the concept of derivative according to the function which rules the increase of time. The main contribution of this paper is the characterization of this extension of the concept of decreasing impatience by using the aforementioned novel derivative and the well-known Prelec's index.

The framework of this paper is behavioral finance and, more specifically, intertemporal choice when individuals exhibit decreasing impatience in their decision-making processes. After characterizing the two main types of decreasing impatience (moderately and strongly decreasing impatience), the main objective of this paper is to generalize these concepts when the criterion of time increase is given by an arbitrary function which describes such increments. In general, the methodology is mathematical calculus but particularly the concept of derivative according to the function which rules the increase of time. The main contribution of this paper is the characterization of this extension of the concept of decreasing impatience by using the aforementioned novel derivative and the well-known Prelec's index. [ABSTRACT FROM AUTHOR]

Intertemporal choices play a fundamental role in the lives of individuals, and the Discounted Utility model is the essential framework for describing decision makers' attitudes in front of alternatives structured over multiple periods. The classical formulation of the model assumes constant preferences over time, i.e., it assumes that individuals' choices are consistent. Empirical evidence, however, shows that individuals' preferences do not respond to this assumption, generating temporally inconsistent decisions. This paper addresses the problem of temporal inconsistency in order to interpret and describe anomalous choices, i.e., not rationalizable from a theoretical point of view, through the cognitive distortions of the decision-maker. Indeed, even if we assume that the investor is a rational subject, behavioral finance suggests that an anomaly is part of the human being and must be recognized as a systematic condition of the decision-making process. Exploiting the relationship between the rate of impatience and temporal preference, this work aims to demonstrate that the degree of decrease in impatience quantifies the weight of emotional drives in the anomalies of intertemporal choices. An experimental approach based on constructing the hyperbolic factor for each individual in different contexts is presented to test our results. The variability in the collected data highlights that individuals' behavior is very different, suggesting the need to project strategies in personalized finance. [ABSTRACT FROM AUTHOR]

Recent evidence on intertemporal choice suggests that decision-makers may exhibit both increasing and decreasing impatience simultaneously, called inverse-S discounting. This paper introduces trichotomic discounted utility to model inverse-S discounting. Under trichotomic discounted utility the decision-maker distinguishes between short-term delays, medium-term delays, and long-term delays. Exponential discounting holds within each category, but not necessarily across each category. We provide preference foundations for trichotomic discounted utility in the timed outcomes framework and in the consumption streams framework. The key axiom is a weakening of the time consistency axiom that allows for both increasing and decreasing impatience. We also consider a weak, local version of time consistency, and show that this plausible condition is satisfied only by discount functions with exponential "pieces". [ABSTRACT FROM AUTHOR]

This paper studies a two-agent strategic model of capital accumulation with heterogeneity in preferences and income shares. Preferences are represented by recursive utility functions that satisfy decreasing marginal impatience. The stationary equilibria of this dynamic game are analyzed under two alternative information structures: one in which agents precommit to future actions, and another one where they use Markovian strategies. In both cases, we develop sufficient conditions to show the existence of these equilibria and characterize their stability properties. Under certain regularity conditions, a precommitment equilibrium shows monotone convergence of aggregate variables, but Markovian equilibria may exhibit nonmonotonic paths, even in the long-run. [ABSTRACT FROM AUTHOR]

Many studies show that time preference data from experiments and surveys are related to field behavior. Time preference measures in these studies typically depend simultaneously on utility curvature, the level of impatience, and the change in the level of impatience. Thus, these studies do not allow one to establish which of these three components drive(s) the field behavior of interest. Of these components, the change in the level of impatience is theoretically thought to be the main driver of time inconsistencies and self-control problems. To test this theoretical presumption, one has to measure the change in the level of impatience independently from utilities and levels of impatience. This paper introduces a measure of the degree of decreasing impatience, the DI-index. It measures the change of impatience independently from the level of impatience and independently from utility. It can also be used to test various discounting models. An experiment finds no correlation between the degree of decreasing impatience and self-reported self-control problems in daily life, suggesting that changing impatience is not the sole driver of self-control problems. Data and the supplementary material are available at https://doi.org/10.1287/mnsc.2017.3015. This paper was accepted by Manel Baucells, decision analysis. [ABSTRACT FROM AUTHOR]

The framework of this paper is the concept of derivative from the point of view of abstract algebra and differential calculus. The objective of this paper is to introduce a novel concept of derivative which arises in certain economic problems, specifically in intertemporal choice when trying to characterize moderately and strongly decreasing impatience. To do this, we have employed the usual tools and magnitudes of financial mathematics with an algebraic nomenclature. The main contribution of this paper is twofold. On the one hand, we have proposed a novel framework and a different approach to the concept of relative derivation which satisfies the so-called generalized Leibniz’s rule. On the other hand, in spite of the fact that this peculiar approach can be applied to other disciplines, we have presented the mathematical characterization of the two main types of decreasing impatience in the ambit of behavioral finance, based on a previous characterization involving the proportional increasing of the variable “time”. Finally, this paper points out other patterns of variation which could be applied in economics and other scientific disciplines.

The framework of this paper is intertemporal choice and, more specifically, the so-called delay effect. Traditionally, this anomaly, also known as decreasing impatience, has been revealed when individuals reverse their preferences over monetary or non-monetary rewards. In this manuscript, we will analyze the delay effect by using preference relations and discount functions. The treatment of the delay effect with discount functions exhibits several scenarios for this paradox. Thus, the objective of this paper is to deduce the different expressions of the delay effect and their mathematical characterizations by using discount functions in stationary and dynamic settings. In this context, subadditivity will be derived as a particular case of decreasing impatience. Finally, we will introduce a new discount function, the so-called asymmetric exponential discount function, able to describe decreasing impatience.

There is a growing amount of literature suggesting people tend to behave inconsistently over time, which is driven by decreasing impatience. In addition, many studies have found relations between discounting estimates from experiments and field behavior, such as smoking cessation and dieting. However, these studies often did not separate time inconsistency from other factors such as utility curvature or the level of discounting. In order to establish the relation between field behavior and the degree of time inconsistency, it is therefore necessary to obtain a pure measure of the latter that is not distorted by these other factors. The present study implements a recently introduced measure of deviations from constant impatience, called the “Decreasing Impatience (DI)-index,” to estimate the degree to which people deviate from constant impatience. We provide the first extension of DI to health outcomes, both for individual and societal discounting using three different starting points. Moreover, we include a survey gathering information about several health-related behaviors, in order to test for the relationship between the amount of decreasing impatience and healthy behavior. We observe that decreasing impatience is the modal preference, although constant and increasing impatience are no exceptions, and, hence, these types of discounters should not be neglected. Furthermore, the DI-index is higher for individual health outcomes than for societal health outcomes, but is not distributed differently among the three classes of discounters. The DI-index decreases with starting period for individual health outcomes, but not for societal health outcomes. Very few significant relations between time inconsistency and self-reported health-related behavior were found.

The framework of this paper is subjective time perception in the context of intertemporal choice, that is to say, the process of making decisions on dated outcomes (monetary or not) by an individual or a group of individuals. In this setting, the Discounted Utility model and, more specifically, the exponential discounting have been the paradigmatic methodology used to measure the preferences on delayed outcomes. However, this model can only be applied to consistent choices in which individuals do not change their preferences when the involved rewards are delayed the same time interval. Unfortunately, this is not the case of several decision scenarios where time is viewed as a subjective variable. The objective of this paper is to formally analyze the consistency of intertemporal choices governed by a discount function, derived from the exponential, where time has been distorted according to certain psychological traits of the subjects involved in the decision-making. More specifically, the different types of decreasing impatience will be characterized by focusing on the distortion derived from the subjective perspective of time. The findings of this research are very relevant in order to explain the time-related behavior of decision-makers in some noteworthy fields such as finance, psychology, marketing or sociology., Competing Interests: The authors declare that they have no conflict of interest., (© 2023 The Authors.)

Under non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides 'agent-specific' results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model. [ABSTRACT FROM AUTHOR]

Researchers have advocated that the acquisition of user preferences is important to the successful adoption of electronic negotiation systems. In this paper, we focus on one such preference, namely time preference, wherein the price of a good/service varies according to the delivery/consumption time. Time preference is a behavioral aspect that varies across buyers. We discuss how different types of preferences can be elicited, represented and integrated with electronic negotiations. We discuss three experiments to study the effect of time preferences on negotiation. The first is a preference elicitation experiment involving 36 subjects. The next two are agent-to-agent negotiation experiments, one based on the individual preferences obtained earlier and the other based on an expanded dataset on both individual preferences as well as negotiation parameters. The agent-based experiment compares outcomes and efficiencies between the standard exponential discounting model and two behavioral models of time preference. Our results bring out the preferences of subjects, as well as the extent to which negotiation is affected and enhanced by the incorporation of time-preferences. [ABSTRACT FROM AUTHOR]

This paper measures deviations from constant discounting and compares these deviations for health and money. Our measurements make no assumptions about utility and do not require separability of preferences over time. In an experiment, most subjects were decreasingly impatient, but a substantial minority was increasingly impatient. The deviations from constant discounting were more pronounced for health than for money suggesting that time preferences are domain-specific. Hyperbolic discounting (Loewenstein and Prelec, Quarterly Journal of Economics, 107, 573-597, 1992) and proportional discounting (Mazur, Quantitative Analyses of Behavior, 5, 55-73, 1987) best described time preferences. Quasi-hyperbolic discounting, the most popular model to accommodate deviations from constant discounting, was rejected for both health and money. [ABSTRACT FROM AUTHOR]

Many empirical studies on intertemporal choice report preference reversals in the sense that a preference between a small reward to be received soon and a larger reward to be received later reverses as both rewards are equally delayed. Such preference reversals are commonly interpreted as contradicting constant discounting. This interpretation is correct only if baseline consumption to which the outcomes are added, remains constant over time. The difficulty with measuring discounting when baseline consumption changes over time, is that delaying an outcome has two effects: (1) due to the change in baseline consumption, it changes the utility from receiving the outcome, and (2) it changes the factor by which this utility is discounted. In this paper we propose a way to disentangle the two effects, which allows us to draw conclusions about discounting even when baseline consumption changes over time. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

Abstract: This paper reports the elicited time preference of human subjects in a laboratory setting. The model allows for non-linear utility functions, non-separability between delay and reward, and time inconsistency including future bias in addition to present bias. In particular, the experiment (1) runs a non-parametric test of time consistency and (2) estimates the form of time discount function independently of instantaneous utility functions, and then (3) the result suggests that many subjects exhibiting future bias, indicating an inverse S-curve time discount function. [Copyright &y& Elsevier]

Abstract: This paper argues that observations of non-stationary choice behavior need not necessarily imply specific properties of the individual’s discount function. As we show, the observed preference reversals in intertemporal choice are consistent with constant discounting and can alternatively be explained by decreasing absolute risk aversion together with the individual’s risk perception. This risk may concern the size of the actual outcome or the endowment consumption stream to which the outcome is added. Both types of uncertainty naturally appear in the context of intertemporal choice. We show how relative degrees of changes in risk over time can determine choices. [ABSTRACT FROM AUTHOR]

Abstract: The commonly used hyperbolic and quasi-hyperbolic discount functions have been developed to accommodate decreasing impatience, which is the prevailing empirical finding in intertemporal choice, in particular for aggregate behavior. However, these discount functions do not have the flexibility to accommodate increasing impatience or strongly decreasing impatience. This lack of flexibility is particularly disconcerting for fitting data at the individual level, where various patterns of increasing impatience and strongly decreasing impatience will occur for a significant fraction of subjects. This paper presents discount functions with constant absolute (CADI) or constant relative (CRDI) decreasing impatience that can accommodate any degree of decreasing or increasing impatience. In particular, they are sufficiently flexible for analyses at the individual level. The CADI and CRDI discount functions are the analogs of the well-known CARA and CRRA utility functions for decision under risk. [Copyright &y& Elsevier]

This paper analyzes a model of discounted utility under habit formation. Habit formation means that utility in each period is determined by the difference between the received outcome and the customary outcome at that point in time. Preferences are rational, in the sense that the decision maker correctly anticipates the habit formation process and behaves in a dynamically consistent manner, i.e., plans are truly carried out. The purpose is to demonstrate that this generalization of the discounted utility model accounts, jointly and in a meaningful way, for the most striking anomalies with respect to classical discounted utility theory. The common difference effect is explained first. This effect, also called decreasing impatience effect, has usually been viewed as evidence against exponential discounting. Then, the frequently observed preference for increasing sequences of outcomes is examined. Such preference, under the traditional discounted utility model, would be regarded as a paradoxical case of negative discounting. A discussion of the descriptive, prescriptive, and normative values of the model is included. [ABSTRACT FROM AUTHOR]

The framework of this paper is the concept of derivative from the point of view of abstract algebra and differential calculus. The objective of this paper is to introduce a novel concept of derivative which arises in certain economic problems, specifically in intertemporal choice when trying to characterize moderately and strongly decreasing impatience. To do this, we have employed the usual tools and magnitudes of financial mathematics with an algebraic nomenclature. The main contribution of this paper is twofold. On the one hand, we have proposed a novel framework and a different approach to the concept of relative derivation which satisfies the so-called generalized Leibniz's rule. On the other hand, in spite of the fact that this peculiar approach can be applied to other disciplines, we have presented the mathematical characterization of the two main types of decreasing impatience in the ambit of behavioral finance, based on a previous characterization involving the proportional increasing of the variable "time". Finally, this paper points out other patterns of variation which could be applied in economics and other scientific disciplines. [ABSTRACT FROM AUTHOR]

The framework of this paper is intertemporal choice and, more specifically, the so-called delay effect. Traditionally, this anomaly, also known as decreasing impatience, has been revealed when individuals reverse their preferences over monetary or non-monetary rewards. In this manuscript, we will analyze the delay effect by using preference relations and discount functions. The treatment of the delay effect with discount functions exhibits several scenarios for this paradox. Thus, the objective of this paper is to deduce the different expressions of the delay effect and their mathematical characterizations by using discount functions in stationary and dynamic settings. In this context, subadditivity will be derived as a particular case of decreasing impatience. Finally, we will introduce a new discount function, the so-called asymmetric exponential discount function, able to describe decreasing impatience. [ABSTRACT FROM AUTHOR]