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Uncertainty and Hyperbolic Discounting Uncertainty and Hyperbolic Discounting By PARTHA DASGUPTA AND ERIC MASKIN* Empirical studies in economics and behav- they have become more impatient than they ioral ecology suggest that, ceteris paribus, ani- were in January. mals and humans appear to place less weight on Both the O’Donoghue-Rabin and Strotz ex- the future than on the present, i.e., they act as amples accord with “hyperbolic discounting,”2 though they discount future payoffs. Further- which has attracted considerable interest among more (and more interestingly), they do so with economists because it appears to shed light on discount rates that increase as the time before important economic phenomena such as house- those payoffs are realized grows shorter.1 In hold saving behavior (see David Laibson, 1997; other words, subjects act as though they become B. Douglas Bernheim et al., 2001; Christopher less patient when payoffs are more imminent. Harris and Laibson, 2003). An anecdotal (human) example is offered by One might ask whether there is some reason Ted O’Donoghue and Matthew Rabin (1999): for such behavior. The point of view that we when offered the choice in February between a take is that, to a considerable extent, animal and painful seven-hour task (e.g., preparing a tax human behavior is shaped by preferences that return) on April 1 and a painful eight-hour task are the outcome of evolutionary forces. That is, on April 15, most of us, they suggest, will opt cravings, urges, or instincts—the operational for the earlier date. But as April 1 approaches, manifestations of preferences—induce the ani- we are apt to change our minds, if we can, and mal or human to make the right choice in the postpone the pain to April 15, even though it “average” situation that it, he, or she is likely to will then be greater. Thus, we behave as though face, where the “right” choice means the one we discount the later pain more as time grows that maximizes survival (and, therefore, the op- short. Robert H. Strotz (1956) discusses a sim- portunity to reproduce).3 We show that if the ilar phenomenon involving positive payoffs. “average” situation entails some uncertainty Early in the calendar year, he notes, many peo- about when payoffs are realized, the corre- ple attempt to lay money aside for Christmas. sponding preferences may well entail hyper- As time goes by, however, they may find them- bolic discounting, giving rise to preference selves spending the money on summer vaca- reversals. tions or back-to-school clothes. It is as though To get a rough understanding of this result, imagine that a decision maker (DM) is offered the choice between a small (positive) payoff relatively soon (prospect P) or a big payoff * Dasgupta: Faculty of Economics, University of Cam- relatively late (prospect PЈ). Furthermore, sup- bridge, Sidgwick Avenue, Cambridge CB3 9DD, United pose that, for either P or PЈ, there is a small but Kingdom (e-mail: [email protected]); Maskin: Institute of Advanced Study, Einstein Drive, Princeton, NJ 08540 (e-mail: [email protected]). We are grateful to the Beijer International Institute of Ecological Economics, Stockholm, 2 Strictly speaking, hyperbolic discounting requires that and the National Science Foundation (Grant SES-0318103) for the discount rate vary inversely with the time to payoffs. But research support. For comments on earlier versions of this the term has come to be applied more generally to any paper, we thank Roland Be´nabou, Douglas Bernheim, David manifestation of increasing impatience as time horizons Colander, Nick Davies, Eddie Dekel, Avinash Dixit, Ernst shrink. See Ariel Rubinstein (2001), who cautions that Fehr, Christopher Harris, Alex Kacelnik, John Kagel, Adi many observed behaviors are consistent with a variety Livnat, Franc¸ois Maniquet, Ariel Rubinstein, Larry Samuel- of preferences in addition to those involving hyperbolic son, Peter Sozou, and three referees. This paper was previ- discounting. ously entitled “Uncertainty, Waiting Costs, and Hyperbolic 3 In this respect, our approach is similar to that of Larry Discounting.” Samuelson and Jeroen Swinkels (2004), who—although 1 The empirical literature on birds (particularly pigeons their approach is otherwise quite different from ours—share and starlings) is summarized in James E. Mazur (1987) and our view that an animal’s urges are evolutionarily deter- Leonard Green and Joel Myerson (1996); that on humans in mined substitutes for full information about the choices it George Ainslie (1992). faces. 1290 VOL. 95 NO. 4 DASGUPTA AND MASKIN: UNCERTAINTY AND HYPERBOLIC DISCOUNTING 1291 positive probability that at any time t before the discounting that does explain these reversals anticipated payoff dates, the payoff will be re- (Propositions 1 and 1*). In Section III, we show alized early at t rather than when expected. that our analysis extends to forms of uncertainty Assume that the DM initially opts for prospect considerably more general than allowed for in PЈ; the reason for making this choice, of course, Section II (Proposition 2). Finally, in Section is PЈ’s bigger payoff, enhanced by the chance of IV, we turn to atypical decision problems and early realization.4 But, as time passes, if the dynamic inconsistency. payoff from PЈ does not materialize, the likeli- hood of early realization declines. Of course, for I. Discounting and Hazard Rates the same reason, the likelihood of early realiza- tion wanes for P too—but this does not matter Before turning to hyperbolic discounting, we so much because P’s payoff is anticipated must first ask the question, “Why should a DM sooner anyway. Hence, with time, the more discount at all?” A conventional answer, pro- immediate prospect P becomes increasingly at- vided by both the economics and zoology liter- tractive relative to PЈ and eventually the DM atures, is that, in a typical situation that an may switch to P. animal or human may face, future payoffs run Although, as this example illustrates, our some risk of disappearing or depreciating model predicts preference reversals that accord (Menahem E. Yaari, 1965). Suppose, for exam- with hyperbolic discounting, these reversals ple, that a blackbird tends to hang around a will be entirely dynamically consistent as long particular raspberry bush waiting for the fruit to as the DM confronts typical decision problems ripen.5 Before this happens, however, a flock of (i.e., the DM will actually profit from reversing crows (which don’t care about ripeness) may herself). However, the DM’s urges may lead her descend on the fruit, devouring it all. Hence, the astray—resulting in dynamic inconsistency—if blackbird should discount the payoff of getting she faces decisions for which the payoff- the fruit, where the discount rate is the hazard realization times depart from the norm. Thus, rate of the “crow-arrival” process (the hazard evolution also enables her to learn how to over- rate at time t is 1/⌬t times the probability that come or neutralize those urges when an atypical the crows will arrive between times t and t ϩ situation recurs. For example, in experimental ⌬t, conditional on their not having arrived al- settings, pigeons discover how to commit them- ready). In particular, if this process is Poisson— selves not to switch from “patient” to “impa- i.e., the hazard rate is independent of t—the tient” choices (Howard Rachlin and Leonard implied discount rate is constant. Thus, if a Green, 1972; for more on this see Section IV). prospect (e.g., eating raspberries) has a payoff V And people susceptible to the impulse to spend (the calories in the berries) at time T (the time at their holiday savings prematurely find that they which the berries ripen) and the hazard rate is a can thwart that inclination by putting their constant r, the prospect should be evaluated as money in illiquid Christmas accounts (Strotz, though the payoff is eϪrTV.6 1956). In Section I, we examine the standard ratio- nale for discounting (viz., to take account of the 5 We are indebted to the ornithologist Nick Davies for risk that future payoffs may disappear or depre- our bird and fruit examples. 6 Another explanation for discounting—treated in an ear- ciate) and discuss why some previous explana- lier version of this paper (Dasgupta and Maskin, 2002)— tions for hyperbolic discounting turn out not to turns on the idea that waiting for a payoff to materialize is be consistent with the O’Donoghue-Rabin and often costly, either because the blackbird may have to use Strotz phenomena. In Section II, we introduce up energy while waiting (a physiological cost), or because uncertainty about when payoffs are realized and by hanging around the raspberry bush it may lose other opportunities for food (an opportunity cost). This alternative show that this leads to a version of hyperbolic explanation for discounting has the advantage of immedi- ately explaining the empirical finding (see Green and My- erson, 1996) that larger payoffs are discounted less than 4 There is a connection here with the theory of sequential smaller ones. Specifically, suppose that a bird is offered the search among risky prospects (Martin L. Weitzman, 1979), choice between a prospect with reward ␣V at time T and one which shows that, ceteris paribus, choosing prospects with with reward ␣VЈ at time TЈ, where ␣ is a scalar and VЈϾ higher dispersions before those with lower dispersions is the V Ͼ 0 and TЈϾT. Experiments suggest that as ␣ rises, the optimal search strategy. bird is more likely to favor the latter prospect. This effect is 1292 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 This model suggests an immediate potential This model cannot explain, however, the sort explanation for hyperbolic discounting.
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