An extension of quasi-hyperbolic discounting to continuous time

Philosophy of Science

How should one discount utility across time? The conventional wisdom in social science is that one should use an exponential discount function. Such a function is a representation of the axioms that provide a well-defined utility function plus a condition known as stationarity. Yet stationarity doesn’t really have much intuitive normative pull on its own. Here I try to cast it in a normative glow by deriving stationarity from two explicitly normative premises, both suggested by the philosophical thesis of temporal neutralism. Putting the argument in this form helps us better understand exponential discounting and challenges to it.

Journal of Economics, 2014

We address intertemporal utility maximization under a general discount function that nests the exponential discounting and the quasi-hyperbolic discounting cases as particular speci…cations. The suggested framework intends to capture one important anomaly typically found when addressing the way agents discount the future, namely the evidence pointing to the prevalence of decreasing impatience. The referred anomaly can be perceived as a bias relatively to what would be a benchmark exponential discounting setting, and is modeled as such. The general discounting framework is used to address a standard optimal growth model in discrete time. Transitional dynamics and stability properties of the corresponding dynamic setup are studied. An extension of the standard growth model to the case of habit persistence is also considered.

2012

Empirical evidence shows that the exponential discount function employed in standard macroeconomic models falls short on two counts: it implies time-consistent preferences and assumes that the discount factor is a constant, exogenous parameter. As discounting behavior is crucial in the determination of individual intertemporal choice, it seems reasonable to expect that deviating from the standard exponential discounting behavior would significantly affect aggregate outcomes. Thus so, the main question of this dissertation is, "What are the macroeconomic and welfare consequences of time-inconsistent and endogenous discounting?" In this endeavor, this research focuses on three important applications: 1) on resource allocation and welfare in a social planner's economy; 2) on raising the retirement age in an economy where time-consistent and time-inconsistent discounters co-exist; and 3) on monetary policy effectiveness in a standard neoclassical growth model. Indeed, the results show that moving beyond exponential discounting affects both macroeconomic and welfare outcomes to the extent that policy recommendations based on exponential discounting must at the very least be taken with a grain of salt.

Journal of Economic Theory, 2007

This note derives the dynamic programming equation (DPE) to a differentiable Markov Perfect equilibrium in a problem with non-constant discounting and general functional forms. We begin with a discrete stage model and take the limit as the length of the stage goes to 0 to obtain the DPE corresponding to the continuous time problem. We characterize the multiplicity of equilibria under non-constant discounting and discuss the relation between a given equilibrium of that model and the unique equilibrium of a related problem with constant discounting. We calculate the bounds of the set of candidate steady states and we Pareto rank the equilibria.

Management Science, 2006

A ccording to most models of intertemporal choice, an agent's discount rate is a function of how far the outcomes are removed from the present, and nothing else. This view has been challenged by recent studies, which show that discount rates tend to be higher the closer the outcomes are to one another (subadditive discounting) and that this can give rise to intransitive intertemporal choice. We develop and test a generalized model of intertemporal choice, the Discounting By Intervals (DBI) model, according to which the discount rate is a function of both how far outcomes are removed from the present and how far the outcomes are removed from one another. The model addresses past challenges to other models, most of which it includes as special cases, as well as the new challenges presented in this paper: Our studies show that when the interval between outcomes is very short, discount rate tends to increase with interval length (superadditive discounting). In the discussion we place our model and evidence in a broader theoretical context.

Theoretical Economics Letters

The decision about how much to save for retirement is likely to be dependent on when an individual plans to be retired, and vice versa. Yet, the established literature on hyperbolic discounting and life-cycle saving behavior has for the most part abstracted from choice over retirement. Two notable exceptions are Diamond and Kőszegi [1] and an important follow-up study by Holmes [2], which demonstrates that time-inconsistent retirement timing is impossible when saving behavior is explicitly modeled in a stylized three-period setting. In this paper, we build upon the framework of Diamond and Kőszegi [1] and Holmes [2] by generalizing the assumptions about initial income and assets. We show analytically and via simple numerical examples that time-inconsistent retirement can exist in a three-period life-cycle model of consumption and saving.

2007

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 nontransitive 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. (E.A. Ok), yusufcan@umich.edu (Y. Masatlioglu). 1 The exponential discounting model was first formulated by Samuelson [31]. The axiomatic foundations of this model is explored by the definitive works of [15], and [16] in the case of time preferences over consumption streams, and of [8] in the case of time preferences over the prize-time space.

Games and Economic Behavior, 2010

In this paper we elicit preferences for money-time pairs via experimental techniques. We estimate a general speci…cation of discounting that nests exponential and hyperbolic discounting, as well as various forms of present bias, including quasi-hyperbolic discounting. We …nd that discount rates are high and decline with both delay and amount, as most of the previous literature. We also …nd clear evidence for present bias. When identifying the form of the present bias, little evidence for quasi-hyperbolic discounting is found. The data strongly favor instead a speci…cation with a small present bias in the form of a …xed cost, of the order of $4 on average across subjects. With such a …xed cost the curvature of discounting is imprecisely estimated and both exponential and hyperbolic discounting cannot be rejected for several subjects.

2016

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2013

Liminal discounting, the model proposed here, generalises exponential discounting in a parsimonious way. It allows for well-known departures, whilst maintaining its elegance and tractability. A liminal discounter has a constant rate of time preference before and after some threshold time; the liminal point. If the liminal point is an absolute point in time, liminal discounting captures time consistent behaviour. If it is expressed in relative time, liminal discounting captures time invariant behaviour. We provide preference foundations for all cases, showing how the liminal point is derived endogenously from behaviour. We give applications to Rubinstein bargaining games, showcasing the model?s use in microeconomic theory.