Hyperbolic discounting and the standard model: Eliciting discount functions

Experiments on static intertemporal choice find evidence of particularly extreme impatience toward immediate rewards. While this is often taken as support for hyperbolic discounting, it could also arise because the most likely participants in experiments may be those with the most immediate need for money. We conduct a calibration exercise and find that the extreme impatience observed in experiments can be accommodated by a standard exponential discounting model with no discounting and expectation of a ‘small’ increase in the base consumption level. The calibration uses existing estimates of curvature of utility.     

Hyperbolic discounting and the time-consistent solution of three canonical environmental problems

In this paper I propose a time-consistent method of discounting hyperbolically and apply it to three canonical environmental problems: (i) optimal renewable resource use, (ii) the tragedy of the commons, and (iii) economic growth and pollution. I show that, irrespective of potentially high initial discount rates, time-consistent hyperbolic discounting leads always to a steady state of maximum yield, or, if the environment enters the utility function, a steady state where the Green Golden Rule applies. While (asymptotic) extinction is a real threat under exponential discounting it is impossible under time-consistent hyperbolic discounting. This result is also confirmed for open-access resources. In a model of economic growth and pollution, hyperbolic discounting establishes the Golden Rule of capital accumulation and the modified Green Golden Rule.     

Empirical and public choice evidence for hyperbolic social discount rates and the implications for intergenerational discounting

The derivation of the correct discount rate for intergenerational projects in Cost Benefit Analysis is particularly contentious. Public choice has resulted in lower discretionary exponential discount rates for many intergenerational projects in Britain and the USA. This is shown to be strong indirect evidence that the true social discount rate may be a hyperbolic (rather than an exponential) function. There is also empirical evidence for this hypothesis. The hyperbolic nature of discounting is also a standard finding in the behavioural sciences. For intergenerational time frames hyperbolic discount rates should be employed together with exponential discount rates in cost-benefit sensitivity analyses.Sincere thanks to Maureen Cropper and Paul Portney for supplying their survey results and to Elaine Barrow and Phillip Judge for graphics assistance. Two anonymous referees also provided valuable comments.     

Social dilemmas,time preferences and technology adoption in a commons problem

Agents interacting on a body of water choose between technologies to catch fish. One is harmless to the resource, as it allows full recovery; the other yields high immediate catches, but low(er) future catches. Strategic interaction in one ‘objective’ resource game may induce several ‘subjective’ games in the class of social dilemmas. Which unique ‘subjective’ game is actually played depends crucially on how the agents discount their future payoffs. We examine equilibrium behavior and its consequences on sustainability of the common-pool resource system under exponential and hyperbolic discounting. A sufficient degree of patience on behalf of the agents may lead to equilibrium behavior averting exhaustion of the resource, though full restraint (both agents choosing the ecologically or environmentally sound technology) is not necessarily achieved. Furthermore, if the degree of patience between agents is sufficiently dissimilar, the more patient is exploited by the less patient one in equilibrium. We demonstrate the generalizability of our approach developed throughout the paper. We provide recommendations to reduce the enormous complexity surrounding the general cases.     

The prevalence of hyperbolic discounting: some European evidence

Experimental matching data are used from the 2000 Bank of Italy Survey of Household Income and Wealth (SHIW) and the 2000 wave of the Center for Economic Research (CentER) Savings Survey at Tilburg University to compare the relative frequencies of hyperbolic and exponential discounters. Among 3200 Italian respondents and 1400 Dutch respondents, less than a quarter exhibited hyperbolic discounting. This finding is both statistically significant and robust with respect to various assumptions regarding utility; moreover, it holds across a wide variety of economic, social and demographic characteristics. The youngest, poorest, most urban and least educated individuals are the most likely to be hyperbolic discounters. In addition, it is found that hyperbolic discounters accumulate less wealth and are somewhat less likely than exponential discounters to utilize commitment devices to constrain their future choices.     

Developments in discounting: With special reference to future health events

Research on time preferences and discounting has two main motivations: to inform decision making by providing a basis for the comparison of future costs and benefits; to explain the influence of the future on current behaviour. This paper introduces the wide range of discount functions which have been advocated in addition to the standard constant rate exponential model. The different approaches taken to estimating the parameters of the discount function are reviewed. The issues raised by discounting in the longer term are discussed. Particular attention is given to time preferences with respect to future health events which have for a number of reasons been a fruitful area for research on time preferences. Specifically the paper examines: the distinctive issues raised when discounting future health events; the methods which have been used to discount health events; the relationship between time preferences and health-affecting behaviour.     

Convex Time Budgets and Individual Discount Rates in the Long Run

The purpose of this study is to use experimental data to estimate individual discount rates and test for hyperbolic discounting over a long time horizon. To do this, we employ the convex time budget experimental approach with cash payments over a 20 year time horizon. To date, there are few experimental studies that explore discount rates beyond a 1 year time horizon. Previous experimental studies that focus on short time horizons find high discount rates, which may not be applicable to decisions that affect outcomes in the distant future. Our findings are quite similar to the average rate of 4.9% found by Grijalva et al. (Environ Resour Econ 59:39–63, 2014), who similarly use a 20 year time horizon, but use the multiple price list elicitation method along with payment via government savings bonds. We find annual discounts rates in the range of 1.9–5.5%, depending on the specific model used. We also find evidence for declining discount rates, and that this hyperbolic pattern of behavior is related to the confidence subjects have in receiving distant-future payments.     

Quasi-hyperbolic discounting and social security systems

Hyperbolic discounting has become a common assumption for modeling bounded rationality with respect to individual savings decisions. We examine the effects of hyperbolic discounting on the comparison of alternative social security systems. We show that this form of bounded rationality breaks the equivalence between funded and pay-as-you-go (PAYG) systems established in Sheshinski and Weiss [Sheshinski, E., Weiss, Y., 1981. Uncertainty and optimal social security. Quarterly Journal of Economics 95, 189-206]. Intergenerational transfers within a PAYG economy are usually secured by the social security system and independent of longevity, whereas this is not the case for the funded economy. The savings level under hyperbolic discounting is lower than under exponential discounting [Laibson et al., 1998], but the ratio between the savings level under hyperbolic discounting within a funded economy and a PAYG economy depends on the effectiveness of the commitment devices. It is shown that if individuals are hyperbolic discounters, then in a PAYG economy any change in the mandated level of intergenerational transfers is neutralized by individuals’ voluntary bequests. This does not apply to a funded system.     

Understanding intertemporal choices

Previous experimental results show clearly that many subjects do not optimize when solving a life-cycle consumption problem. What do they do? This paper attempts to resolve this question, looking at the discounting, hyperbolic and rolling models as possible explanations. Data from two experiments (one an experiment with a typical subject pool and the second an experiment with subjects from the CentER panel) is used, and the advantage of having experimental data is exploited, which means that one can actually estimate the hyperbolic model. It is shown that the (exponential) discounting model appears to give the best explanation – suggesting that subjects do look ahead (as they should) but increasingly less as time passes (as they should not in the context of these experiments).     

Exponential discounting bias

We address intertemporal utility maximization under a general discount function that nests the exponential discounting and the quasi-hyperbolic discounting cases as particular specifications. Under the suggested framework, the representative agent adopts, at some initial date, an optimal behavior that shapes her consumption trajectory over time. This agent desires to take a constant discount rate to approach the optimization problem, but bounded rationality, under the form of a present bias, deviates the individual from the intended goal. As a result, decreasing impatience will end up dominating the agent’s behavior. The individual will not be aware of her own time inconsistency and, therefore, she will not revise her plans as time elapses, what makes the problem relatively simple to address from a computational point of view. The general discounting framework is used to approach 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 utility maximization model to the case of habit persistence is also considered.     

Discounting and altruism to future decision-makers

Is discounting of future decision-makers’ consumption utilities consistent with “pure” altruism toward those decision-makers, that is, a concern that they are better off according to their own, likewise forward-looking, preferences? It turns out that the answer is positive for many but not all discount functions used in the economics literature. In particular, “hyperbolic” discounting of the form used by Phelps and Pollak (Rev. Econ. Studies 35 (1968) 201) and Laibson (Quart. J. Econ. 112 (1997) 443) is consistent with exponential altruism towards future generations. More generally, we establish a one-to-one relationship between discount functions and altruism weight systems, and provide sufficient, as well as necessary, conditions for discount functions to be consistent with pure altruism.     

Decreasing Impatience: A Criterion for Non‐stationary Time Preference and "Hyperbolic" Discounting

Despite recent interest in hyperbolic discounting, there has been little discussion of exactly what property of time preferences is instantiated by hyperbolic or quasi‐hyperbolic functional forms. The paper revives an earlier proposal in Prelec (1989) that the key property is Pratt–Arrow convexity of the log of the discount function, which corresponds to decreasing impatience(DI) at the level of preferences. DI provides a natural criterion for assessing the severity of departure from stationarity in that greater DI is equivalent to more choices of dominated options in two‐stage decision problems, as well as greater convexity of the log of the discount function. Inefficient choices may arise as intentional precommitments, or as unintended reversals of preference by “naïve” agents.     

Infinitely repeated games in the laboratory: four perspectives on discounting and random termination

This paper compares behavior under four different implementations of infinitely repeated games in the laboratory: the standard random termination method [proposed by Roth and Murnighan (J Math Psychol 17:189–198, 1978)] and three other methods that de-couple the expected number of rounds and the discount factor. Two of these methods involve a fixed number of repetitions with payoff discounting, followed by random termination [proposed by Sabater-Grande and Georgantzis (J Econ Behav Organ 48:37–50, 2002)] or followed by a coordination game [proposed in (Andersson and Wengström in J Econ Behav Organ 81:207–219, 2012; Cooper and Kuhn in Am Econ J Microecon 6:247–278, 2014a)]. We also propose a new method—block random termination—in which subjects receive feedback about termination in blocks of rounds. We find that behavior is consistent with the presence of dynamic incentives only with methods using random termination, with the standard method generating the highest level of cooperation. Subject behavior in the other two methods display two features: a higher level of stability in cooperation rates and less dependence on past experience. Estimates of the strategies used by subjects reveal that across implementations, even when the discount rate is the same, if interactions are expected to be longer defection increases and the use of the Grim strategy decreases.     

Discounting the Distant Future: An Experimental Investigation

We use a laboratory experiment to elicit discount rates over a 20-year time horizon using government savings bonds as a payment vehicle. When using a constant (exponential) discount rate function, we find an implied average discount rate of 4.9 %, which is much lower than has been found in previous experimental studies that used time horizons of days or months. However, we also find strong support for non-constant, declining discount rates for longer time horizons, with an extrapolated implied annual discount rate approaching 0.5 % in 100 years. There is heterogeneity in discount rates and risk preferences in that people with more optimistic beliefs about technological progress have higher discount rates. These findings contribute to the debate over the appropriate discount rate to use in comparing the long-term benefits of climate change mitigation to the more immediate costs.     

Time preference and savings behaviour

A number of studies have examined various determinants of savings rate. This article contributes to this literature by empirically testing whether the time preference (discounting behaviour) is another important determinant of savings rate. To this end, we estimate the hyperbolic Euler equation using the generalized method of moments (GMM) to examine whether the short-run discount factor can account for savings behaviour. The empirical results show that people exhibit short-run patience (impatience) when savings rate increases (decreases), which is in line with the theoretical prediction. This result implies that the time preference also plays an important role in determining savings behaviour. Various sets of instruments and different sample periods do not reverse the main finding.     

Behavior in second-price auctions by highly experienced eBay buyers and sellers

We report on sealed-bid second-price auctions that we conducted on the Internet using subjects with substantial prior experience: they were highly experienced participants in eBay auctions. Unlike the novice bidders in previous (laboratory) experiments, the experienced bidders exhibited no greater tendency to overbid than to underbid. However, even subjects with substantial prior experience tended not to bid their values, suggesting that the non-optimal bidding of novice subjects is robust to substantial experience in non-experimental auctions. We found that auction revenue was not significantly different from the expected revenue the auction would generate if bidders bid their values. Auction efficiency, as measured by the percentage of surplus captured, was substantially lower in our SPAs than in previous laboratory experiments.     

Spatial discounting of ecosystem services

The impact of conservation efforts targeted at preserving ecosystem services largely depend on the welfare implications associated with spatial variations in the provision of ecosystem services. While there is ample empirical evidence of spatial discounting or decay of the valuation of ecosystem services, there are still few underpinnings based on welfare economic theory. We establish a theory of spatial discounting that closely follows the concept of time discounting pertaining to climate change, and show spatial discount rates in the consumption, ecosystem service, and willingness to pay (WTP) numeraires. We consider the role of key parameters such as pure rate of spatial preference, consumption change, ecosystem services change, population density, and elasticity of marginal utility. We find that the spatial discount rate of WTP for ecosystem services that frequently appears in the empirical literature is the difference between the ecosystem service discount rate and consumption discount rate, where the ecosystem service discount rate includes both physical distance decay and welfare effects. Finally, we use numerical simulations to illustrate how the three different spatial discount rates vary with the spatial distance from the source of ecosystem services and with consumption patterns, implying many more possible spatial variations of WTP.     

Bid Functions in Auctions and Fair Division Games: Experimental Evidence

In auctions a seller offers a commodity for sale and collects the revenue. In fair division games the object is collectively owned by the group of bidders who equally share the revenue. We run an experiment in which the participants face four types of allocation games (auctions and fair division game under two price rules, first– versus second–price rule). We collect entire bid functions rather than bids for single values and investigate price and efficiency of the different trading institutions. We find that the first–price auction is more efficient than the second–price auction, whereas economic rationality assuming heterogeneous bidders suggests the opposite. Furthermore, we study the structure of individual bid functions.