Microfoundations of Discounting
Alexander T. I. Adamou, Yonatan Berman, Diomides P. Mavroyiannis, Ole B. Peters
MMicrofoundations of Discounting
Alexander T. I. Adamou ∗ Yonatan Berman † Diomides P. Mavroyiannis ‡ Ole B. Peters § ¶
January 9, 2020
Abstract
An important question in economics is how people choose between differentpayments in the future. The classical normative model predicts that a decisionmaker discounts a later payment relative to an earlier one by an exponentialfunction of the time between them. Descriptive models use non-exponentialfunctions to fit observed behavioral phenomena, such as preference reversal.Here we propose a model of discounting, consistent with standard axioms ofchoice, in which decision makers maximize the growth rate of their wealth. Fourspecifications of the model produce four forms of discounting – no discounting,exponential, hyperbolic, and a hybrid of exponential and hyperbolic – two ofwhich predict preference reversal. Our model requires no assumption of behav-ioral bias or payment risk.
Keywords: temporal discounting, growth rates, decision theory ∗ London Mathematical Laboratory, [email protected] † London Mathematical Laboratory, [email protected] ‡ Universit´e Paris-Dauphine, [email protected] § London Mathematical Laboratory and Santa Fe Institute, [email protected] ¶ We thank Ryan Singer for insightful discussions that led to an early outline for this manuscript.We are also grateful to Matthew Gentry, and to seminar participants at the 15th European Meetingon Game Theory, 2019 International Conference on Public Economic Theory, the Danish ResearchCentre for Magnetic Resonance, Royal Holloway, University of London, the University of Cyprus,and Universit´e Paris-Dauphine. We thank Baillie Gifford for sponsoring the Ergodicity Economicsprogram at the London Mathematical Laboratory. a r X i v : . [ ec on . T H ] J a n Introduction
In economics and psychology, temporal discounting – or, simply, discounting – is aparadigm of how decision makers choose between rewards available at different timesin the future. We write here of people and money payments, noting that discountingis also studied in other animals and for other reward types. The basic observationto be explained is this: for two payments of equal size, people prefer typically theearlier payment to the later one. In the discounting paradigm, the later paymentis discounted relative to the earlier payment by multiplying it by a function of thetime period between the payments, called the delay . This operation expresses thelater payment as an equivalent payment at the earlier time, to be compared with theearlier payment actually on offer.Why do people assign lower values to payments further in the future? One plausibleanswer is that a later payment is less likely to be received, because there is more timefor something to go wrong with it. In other words, delay increases risk. Another isthat, for equal payments, the later one corresponds to a lower growth rate which, ifsustained over time, would result in being poorer. Modern treatments of discountingin economics tend to follow risk-based reasoning, while there is a more even splitbetween risk and rate interpretations in psychology.This paper studies the microfoundations of discounting using the rate interpretationin a riskless setting. In our model, a decision maker chooses between two known anddifferent payments to be received at known and different times, such that the growthrate of her wealth is maximized. Our model assumes no behavioral bias and does notviolate standard axioms of choice. It predicts a range of situation-dependent discountfunctions, including those documented in the discounting literature.This literature abounds with models (Cohen et al., 2019). In some, theoretical con-siderations are used to construct the decision maker’s maximand, from which thediscount function is derived. Such models are “normative” in that they say what adecision maker should do if she wants to act optimally in the sense specified. In othermodels, the discount function is chosen to fit empirical data, with theoretical justi-fication sought post hoc or not at all. These models are “descriptive” in that they1redict what decision makers actually do, regardless of whether it is in any senseoptimal.The main normative model in economics is exponential discounting, in which thediscount function decays exponentially with the delay (Samuelson, 1937). For moneypayments, this is derived straightforwardly: either by a no-arbitrage argument, as-suming payments are guaranteed and the earlier payment can be invested during thedelay to earn compound interest at a riskless rate; or by assuming the later paymenthas a constant hazard rate during the delay and the decision maker maximizes theexpected payment (Kacelnik, 1997).Exponential discounting is not descriptive. Experiments on human and non-humananimals suggest that payments can be discounted more for shorter delays and less forlonger delays than is well described by fitting the rate parameter of an exponentialfunction. Furthermore, subjects exhibit a behavior known as preference reversal,where they switch from preferring the later to the earlier payment as time passes.Specifically, the switch happens as the time to the earlier payment – which we call the horizon – gets shorter, while the delay between the payments stays the same (Kerenand Roelofsma, 1995, p. 288). Preference reversal is never predicted by standardexponential discounting (Green and Myerson, 1996, Fig 2). The primary evidenceagainst the main normative model is summarized by Myerson and Green (1995) andreferences therein. In response to its falsification, descriptive models are proposedwith discount functions better able to fit observations.A widely-used descriptive model is hyperbolic discounting, where the discount func-tion is a hyperbola in the delay. The function has one free parameter, known asthe degree of discounting, which determines its steepness. Fitting this parameterto experimental data is more efficient than fitting an exponential function, both atgroup level and for individuals (Myerson and Green, 1995). Furthermore, preferencereversal is compatible with this model (Green and Myerson, 1996, Fig. 2).Kacelnik (1997) remarks on this divergence between normative and descriptive mod-els, noting that the hyperbola “is not strongly explanatory because it did not emergefrom an a priori analysis but purely from its power to describe data efficiently. Incontrast, because of the strong appeal of the a priori argument favoring exponen-tial discounting, several re-elaborations have been made to rescue the rationale that2ed to it.” Most such attempts to adapt the normative model introduce paymentuncertainty, which we discuss in Sec. 1.3. Another approach, favored in behavioraleconomics, is to present non-exponential discounting as a cognitive bias – a deviationfrom optimal behavior – to be documented and quantified in mathematical functionsthat encode human psychology (Loewenstein and Prelec, 1992; Laibson, 1997).
Here we propose a model of temporal discounting compatible with hyperbolic dis-counting, in which neither payment risk nor behavioral bias are assumed. We studythe basic temporal choice problem in which a decision maker must choose betweentwo known, certain, and different payments to be made at known, certain, and differ-ent future times. In our model, she does so by comparing the growth rates of wealthassociated with each option.The temporal choice problem is underspecified. We specify it fully by introducing:the wealth dynamics, treating specifically additive and multiplicative cases; and thetime frame of the decision, meaning the period over which it is appropriate to computegrowth rates.Depending on the specification, our model predicts four different forms of discounting:no discounting; exponential; hyperbolic; and a hybrid of exponential and hyperbolic.This is not an exhaustive list – other dynamics would produce other forms of dis-counting. Two of the discount functions nested in our model, hyperbolic and hybrid,are compatible with preference reversal.The hybrid discount function depends not only on the delay and the horizon, butalso on the decision maker’s wealth and background growth rate. This produces aricher set of predicted behaviors than other specifications. One prediction is thatdecision makers can switch from preferring the earlier to the later payment as theirwealth increases. In other words, richer people discount less steeply, consistent withempirical findings (Green et al., 1996; Epper et al., 2018). Another prediction is that,for small payments, discounting is close to hyperbolic for short delays and close toexponential for long delays. Said succinctly, a normative-only model provides rationale without fit, and a descriptive-onlymodel provides fit without rationale. Ideally, models of discounting, as of other behavioral phenom-ena, would provide both and the distinction would be redundant. payments (not growth rates).In our perspective this reflects a change of circumstances and not of mind. Thefundamental preference – for faster growth – never reverses.Moreover, this paper marks a shift from psychological to circumstantial explanationsof discounting. We predict that changes in the discount function arise from changesin wealth dynamics and time frame, which are properties not of the decision makerbut of her circumstances. When these circumstances are included in the formalism,a single behavioral model – a single maximand – is capable of predicting a rangeof observed behaviors. Since psychological risk preferences, encoded in idiosyncraticutility functions, do not appear in our model, we sidestep recent controversy in theliterature about the suitability of experiments involving money payments to test mod-els of utility-of-consumption flows (Cohen et al., 2019). Such experiments are able tofalsify our model and are planned.The paper is organized as follows. Sec. 2 sets out the temporal choice problem and ourdecision model. In Sec. 3 we present different specifications of the problem and de-scribe how a decision maker discounts future payments in each of them. We concludein Sec. 4.
This paper follows the tradition of adapting the normative discounting model to makeit consistent with observations, i.e. to make it descriptive (Kacelnik, 1997). Ourstrategy is to postulate the growth rate of wealth as the decision maker’s maximand.Computing this requires information about wealth dynamics and time frame, aboutwhich the basic temporal choice problem is silent.Another strategy is to leave the maximand – usually an expected payment or utilitygain – unchanged and introduce uncertainty in the amount or timing of the payment.The uncertainty is chosen so that the effective discount function takes a form known4o fit the data. Adding risk can be viewed as another way of resolving the under-specification of the temporal choice problem. This approach is questionable because,whereas dynamics are unspecified in the problem, uncertainty is explicitly absent –a choice between certain payments at certain times. Payment risk is a sound micro-foundation when the uncertainty absent from the problem formulation is important inreality. Such situations are plausible and likely widespread. For example, a predatordeclining a small but readily-caught prey to search for something more filling riskscatching nothing at all. However, adding payment uncertainty to generate, say, hy-perbolic discounting is not a general prescription. It fails when the real payment riskis negligible, as envisaged in the problem and presented in experiments, e.g.
Myersonand Green (1995).Furthermore, to recover the hyperbola as the discount function, specific forms ofuncertainty or other adaptations are required. Green and Myerson (1996) point outthat an expected payment model (‘risk neutrality’) with a hyperbolic hazard ratepredicts hyperbolic discounting. They note also that the exponential function can bemade consistent with observed behavior, including preference reversal, by allowing itsrate parameter to vary with payment size. Sozou (1998) treats an expected paymentmodel with an uncertain hazard rate, about which the decision maker learns throughBayesian updating. If the prior distribution of the hazard rate is exponential, thenhyperbolic discounting is again obtained. Dasgupta and Maskin (2005) also assumerisk neutrality but keep the hazard rate constant. To recover hyperbolic discountingthey introduce the possibility of payment occurring before the anticipated time.Such adaptations lead to a loss of generality. They make statements of the type: ‘ifthere is uncertainty in a future payment, and if it takes this specific form, then thediscount function is a hyperbola.’ Such ad hoc assumptions reduce the generalityof risk-based models further, since they are useful only when the real risk is of aparticular nature.The strategy we follow leaves payment certainty alone and changes the maximand. Itis long established in biology and psychology that hyperbolic discounting is consistentwith maximizing the rate of change of resources in a model of additive payments. Thisinsight, traced back to the predation model of Holling (1959), is recognized in stud-ies of human discounting (Myerson and Green, 1995; Kacelnik, 1997; Sozou, 1998).Kacelnik (1997, Fig. 2) offers a pictorial representation, similar to ours in Sec. 3, of5ow rate maximization predicts preference reversal. In the rate interpretation, thedegree of discounting is no longer a free psychological parameter. Rather it is con-strained to be the reciprocal of the horizon (Myerson and Green, 1995). This is atestable prediction.This paper extends this strand of the literature by setting the decision maker’s maxi-mand as the growth rate of resources under general dynamics. We generalize furtherby allowing the time frame of the decision – over which growth rates are computed– to be the period from the decision to either the chosen or the later payment. Thiscaptures circumstances akin to opportunity costs, specifically whether receiving theearlier payment frees the decision maker to pursue other payments.Our work contributes to the growing field of ergodicity economics (Peters, 2019) inwhich decision makers maximize the long-time growth rate of resources, rather thanexpectation values of psychologically-transformed resource flows (under, in prospecttheory, psychologically-transformed probability measures). This study joins recentevidence of strong dependence on wealth dynamics of human decisions under uncer-tainty (Meder et al., 2019) and may be used to design similar experiments withoutuncertainty.
We begin by formalizing a basic riskless temporal choice problem, where discountingis used to express a later payment as an equivalent payment at an earlier time. Wedefine a
Riskless Intertemporal Payment Problem (RIPP):
Definition 1 (Riskless Intertemporal Payment Problem) . A Riskless IntertemporalPayment Problem (RIPP) is a comparison between two vectors, a ≡ ( t a , ∆ x a ) , b ≡ ( t b , ∆ x b ) . A decision maker at time t with wealth x ( t ) must choose between twofuture cash payments, whose amounts and payment times are known with certainty.The two options are: a . an earlier payment of ∆ x a at time t a > t ; and b . a later payment of ∆ x b at time t b > t a . a or b is required. Here we explore what happens if thatcriterion is maximization of the growth rate of wealth, i.e. if a is chosen when itcorresponds to a higher growth rate of the decision maker’s wealth than b , and viceversa . Growth rates
A growth rate is defined as the scale parameter of time in the growth function ofwealth subject to dynamics. So, if wealth grows as x ( t − t ) = f ( g ( t − t )), where f denotes the growth function, then the growth rate is g . Different dynamics correspondto different growth functions and, therefore, to different forms of growth rate. Wetreat explicitly multiplicative and additive dynamics (Peters and Gell-Mann, 2016),noting that more general dynamics can be treated similarly (Peters and Adamou,2018). Multiplicative dynamics
Ignoring, for the moment, possible payments ∆ x a and ∆ x b , a common assumptionis that wealth grows exponentially in time. We label this dynamic as multiplicative.It corresponds to investing wealth in income-generating assets, where the income isproportional to the amount invested. Wealth grows as x ( t ) = x ( t ) e g ( t − t ) , (2.1)and the scale parameter of time in the exponential function is g . This growth rate, g , resembles an interest rate or a rate of return on investment. Its dimension is thereciprocal of time, e.g.
5% per year.
Additive dynamics
Another possibility is additive dynamics, where wealth grows linearly in time. Thiscorresponds to situations where investment income is negligible and wealth changesby net flows that do not depend on wealth itself. In this case wealth grows as x ( t ) = x ( t ) + g ( t − t ) , (2.2)and the scale parameter of time in the linear function is g . This growth rate, g ,7esembles a net savings rate from additive income, such as from labor earnings andwelfare payments less consumption. It is measured in units of currency per unit oftime, e.g. $5000 per year.The functional form of the growth rate differs between the dynamics. The growthrate between time t and t + ∆ t can be extracted from the expression for the evolutionof wealth over that period. Under multiplicative dynamics it is g = ln x ( t + ∆ t ) − ln x ( t )∆ t , (2.3)and under additive dynamics it is g = x ( t + ∆ t ) − x ( t )∆ t . (2.4)The matching of growth rate with dynamics is crucial. An additive growth ratecomputed for wealth following a multiplicative process would vary with time, as woulda multiplicative growth rate computed for additively-growing wealth. The correctgrowth rate extracts a stable parameter from the dynamics, allowing processes withthe same type of dynamics to be compared.Given the wealth dynamics, a RIPP implies two growth rates: g a , associated withoption a ; and g b , associated with option b . This permits defining growth-optimalbehavior: Definition 2 (Growth-Optimal Preferences) . The preference relation (cid:37) is growth-optimal if given the wealth dynamics, a decision time t , an initial wealth x ( t ) , andpayments a ≡ ( t a , ∆ x a ) and b ≡ ( t b , ∆ x b ) :1. a (cid:31) b [‘ a is preferred to b ’] if and only if g a > g b a ∼ b [‘indifference between a and b ’] if and only if g a = g b a ≺ b [‘ b is preferred to a ’] if and only if g a < g b In words, Definition 2 states that a growth-optimal decision maker prefers option a if her wealth grows faster under this choice than under option b , and vice versa . Sheis indifferent if the growth rates are equal. Definition 2 is consistent with the von8eumann-Morgenstern axioms (von Neumann and Morgenstern, 1944): completenessis satisfied by design. It also satisfies transitivity (see proof in Appendix A). Inde-pendence and continuity are irrelevant since in this setup all the payments and timesare certain. Figure 1 illustrates a RIPP, corresponding to the basic question that arises in temporaldiscounting, e.g. ‘would you prefer to receive $100 tomorrow or $200 in a month’stime?’ Growth rates depend on time increments, not times themselves, so it is usefulto define the two fundamental time increments in the problem: the period from thedecision to the earlier payment, called the horizon , H ≡ t a − t ; (2.5)and the period between the payments, called the delay , D ≡ t b − t a . (2.6)The discount function is a function of the delay. It is the multiplicative factor which,when applied to the later payment, renders the decision maker indifferent, i.e. δ ( D ) ≡ ∆ x a ∆ x b (cid:12)(cid:12)(cid:12)(cid:12) a ∼ b . (2.7)Depending on the model specification, the discount function can also vary with othervariables in the problem, such as the horizon, payment sizes, initial wealth, and anybackground growth rate.Despite its apparent simplicity, solving the temporal choice problem requires addi-tional assumptions. In our model, two assumptions are needed. The first concerns thedynamics under which the decision maker’s wealth grows, as discussed in Sec. 2.1.This determines the appropriate form of the growth rate. The second assumptionconcerns what we call the time frame of the decision, specifically whether a decisionmaker receiving the earlier payment at t a is free immediately to make her next deci-sion, or whether she must wait until the later time t b . This determines the appropriate9 imeW ealth t t a t b x ( t ) x ( t ) + ∆ x a x ( t ) + ∆ x b Horizon, H Delay, D Figure 1: The basic setup of the model. A decision maker faces a choice at time t between option a , which guarantees a payment of ∆ x a at time t a , and option b , whichguarantees a payment of ∆ x b > ∆ x a at time t b > t a . We define the time between thedecision and the earlier payment as the horizon , H ≡ t a − t ; and the time betweenthe two payments as the delay , D ≡ t b − t a .time period for computing the growth rate under each option. Full specification al-lows the decision maker’s maximand – the growth rate of her wealth – to be evaluatedand her options compared. For concreteness we confine our attention to ∆ x b > ∆ x a ,which is the most commonly considered dilemma.We describe four different specifications of this basic setup. In each we calculate thegrowth rates of wealth, g a and g b , associated with options a and b . From this analysiswe infer the discount function as δ ( D ) ≡ ∆ x a ∆ x b (cid:12)(cid:12)(cid:12)(cid:12) g a = g b , (2.8) i.e. the ratio of earlier to later payment under the constraint that the growth ratescorresponding to each are equal. The four specifications predict four forms of dis-counting – no discounting, exponential, hyperbolic, and a hybrid of exponential andhyperbolic – and, in some cases, preference reversal.10 Results
We begin by describing the four different model specifications. Each specifies twoaspects necessary to quantify the growth rate of wealth: the time frame of the decision;and the dynamics under which wealth evolves.
Time frame
A key aspect, often left unspecified or implicit in the literature, is whether receivingthe earlier payment frees the decision maker to pursue the next payment. We treatthis by specifying the time frame of the decision, which we illustrate with the followingscenarios:1. Denise is a day laborer. Every evening she looks at a job market forum andchooses a job for the next day. Jobs pay different wages and take differentamounts of time, although always less than a day. Denise is paid as soon as shecompletes the job and goes home. She cannot do more than one job each day.2. Fiona is a freelancer. She works on projects ranging from a few days to manymonths and she can only work on one project at a time. As soon as she finishesa project, she gets paid and can move on to the next project.In the first scenario, the important element to note is that no matter which choice ismade, it does not affect the timing of future choices. Denise’s next decision is alwaysmade the next evening. The time frame is independent of the choice, so we say it is fixed .In the second scenario, the time frame depends on the choice made. The timing ofFiona’s next choice is determined by her current decision, e.g. choosing a shorterproject frees her sooner for the next opportunity. We call this the adaptive timeframe.In our model, we must compute the growth rate of wealth using the time periodover which the growth rate is effective. This is the time period between successivedecisions. With a fixed time frame it is the period from decision to later payment, i.e. t b − t = H + D . In Denise’s case, this is one day. With an adaptive time frame,11t is the period from decision to chosen payment, i.e. t a − t = H for option a and t b − t = H + D for option b . This specification is appropriate to Fiona’s situation. Dynamics
As described in Sec. 2, the wealth dynamics can also take different forms. We addresstwo common cases: additive and multiplicative wealth dynamics. We note that underthe multiplicative dynamics it is assumed that the payment itself is reinvested at abackground exponential growth rate, r . For additive dynamics there is essentially noreinvestment of the payment. Income in this dynamic is independent of wealth.We discuss the four specifications, as illustrated in Fig. 2. In each case we: computethe growth rates g a and g b associated with each option; compare them to determinethe conditions under which each option is preferred; determine whether preferencereversal is predicted; and, finally, elicit the form of temporal discounting equivalentto our decision model. F ixed Adaptive
TIME FRAME
Additive M ultiplicativeAdditive M ultiplicative
DYNAMICS A B C DFigure 2: The four model specifications, determined by specifying a time frame andwealth dynamics. The labels A, B, C, and D, are used for the different cases.
Specification: the period for computing the growth rate is that between the decisionand the later payment, t b − t = H + D ; and wealth dynamics are additive. Here,irrespective of the initial wealth and in addition to the chosen payment, wealth grows12t a background additive growth rate, k . In other words, the decision maker alwaysreceives an amount k ( t b − t ) over the period, on top of what else she chooses.In the economic context, we might interpret this sinecure-like income as coming from ajob or welfare scheme which does not interact with the payments under consideration.In biological contexts, we might interpret negative k as a metabolic rate of energyexpenditure. Provided it takes the same value under the two choices, k appears inneither preference criterion nor discount function. We include it only for completeness.We begin by writing down the final wealth under each of the two options, evaluatedat t b : x a ( t b ) = x ( t ) + k ( t b − t ) + ∆ x a ; (3.1) x b ( t b ) = x ( t ) + k ( t b − t ) + ∆ x b . (3.2)The growth rates are: g a = x a ( t b ) − x ( t ) t b − t = ∆ x a H + D + k ; (3.3) g b = x b ( t b ) − x ( t ) t b − t = ∆ x b H + D + k . (3.4)Since ∆ x b > ∆ x a , option b is always preferred to option a . This is a trivial case:under additive wealth dynamics and comparing growth rates over the same timeperiod, only payment size matters. There is no discounting and the discount function δ is undefined, because the indifference condition is never satisfied. Specification: the period for computing the growth rate is that between the decisionand the later payment, t b − t = H + D ; and wealth dynamics are multiplicative.This specification corresponds to the classical temporal discounting, where wealthcompounds continuously at the background multiplicative growth rate, r , and pay-ments are reinvested at this rate.We note that in this case the earlier payment, ∆ x a , if chosen, is treated as growingexponentially from its receipt at t a to t b . Therefore, the wealths evolve from t to t b
13s follows: x a ( t b ) = x ( t ) e r ( t b − t ) + ∆ x a e r ( t b − t a ) ; (3.5) x b ( t b ) = x ( t ) e r ( t b − t ) + ∆ x b . (3.6)The corresponding growth rates are: g a = ln x a ( t b ) − ln x ( t ) t b − t = 1 H + D ln (cid:18) x a e rD x ( t ) e r ( H + D ) (cid:19) + r ; (3.7) g b = ln x b ( t b ) − ln x ( t ) t b − t = 1 H + D ln (cid:18) x b x ( t ) e r ( H + D ) (cid:19) + r . (3.8)The criterion g a > g b is simple. Only the numerator in the second term of thelogarithm is different, so only this must be compared. Thus, g a > g b if∆ x a e rD > ∆ x b . (3.9)We see that the decision criterion depends on a single time period, the delay D , andon the background growth rate of wealth, r . The discount function is obtained bysetting the growth rates to be equal, which happens when ∆ x a e rD = ∆ x b . This yields δ ( D ; r ) = ∆ x a ∆ x b (cid:12)(cid:12)(cid:12)(cid:12) g a = g b = e − rD , (3.10)which is the classical exponential discounting result (Samuelson, 1937). The inter-pretation is straightforward: if it is possible to reinvest the earlier payment such thatit will exceed the later payment amount at the later payment time, then option a ispreferred to option b . Note that the payments, ∆ x a and ∆ x b , appear in the growth rates, g a and g b , only as theirratios to the initial wealth, x ( t ). An equivalent choice problem to a decision maker with differentwealth has payments scaled up or down, such that these ratios are unchanged. The same commentapplies to case D. Indeed, this result corresponds to the historical use of the term “rate of discount” to describea riskless interest rate in the money market, e.g.
Jevons (1863). .4 Case C – Adaptive time frame with additive dynamics Specification: the period for computing the growth rate is that between the decisionand the chosen payment, either t a − t = H or t b − t = H + D ; and wealth dynamicsare additive. Like in case A, regardless of the initial wealth and in addition to thechosen payment, wealth grows at the background additive rate, k . Unlike case A,options a and b are evaluated at t a and t b , respectively: x a ( t a ) = x ( t ) + k ( t a − t ) + ∆ x a ; (3.11) x b ( t b ) = x ( t ) + k ( t b − t ) + ∆ x b . (3.12)The growth rates are: g a = x a ( t a ) − x ( t ) t a − t = ∆ x a H + k ; (3.13) g b = x b ( t b ) − x ( t ) t b − t = ∆ x b H + D + k . (3.14)It follows that the criterion g a > g b is∆ x a H > ∆ x b H + D . (3.15)So, in this specification, the decision maker cares about the linear payment rate undereach option.Preference reversals are observed changes in decisions as time passes, i.e. as thehorizon gets shorter. We can test whether they are predicted in our model by varying H while holding other variables constant. In the present specification, indifferenceoccurs at the horizon for which Eq. (3.15) becomes an equality, i.e. at H = H PR where H PR ≡ D ∆ x a ∆ x b − ∆ x a . (3.16)Since t = t a − H , this can be expressed as a critical decision time at which thedecision maker is indifferent: t PR0 ≡ t a − H PR = ∆ x b t a − ∆ x a t b ∆ x b − ∆ x a . (3.17)15or H < H PR ( t > t PR0 ), the payment rate under option a exceeds that under option b and the earlier payment is preferred. The converse is true for H > H PR ( t < t PR0 ).Fig. 3 illustrates how the dependence of payment rate on horizon leads to preferencereversal under additive dynamics with an adaptive time frame, c.f. (Kacelnik, 1997,Fig. 2).
H D H D H D
Figure 3: Preference reversal in case C. From left to right panel, t increases and H decreases – i.e. both payments get closer – while all other parameters are heldconstant. Initially, option b is preferred, having the higher payment rate (slope ofdashed line). At the critical time, t = t PR0 , given by Eq. (3.17), both options implythe same payment rate. At later times, option a has the higher payment rate and ispreferred.Finally, we compute the discount function under this specification. When g a = g b ,we have δ ( D ; H ) = ∆ x a ∆ x b (cid:12)(cid:12)(cid:12)(cid:12) g a = g b = HH + D = 11 + D/H . (3.18)Thus we recover the widely-used descriptive model of discounting in which the dis-count function, δ , is a hyperbola of the delay, D . We note that δ also depends onthe horizon, H . Indeed, the degree of discounting parameter – usually treated as apsychological parameter – appears in our model as 1 /H , the reciprocal of the horizon(Myerson and Green, 1995). As the horizon gets shorter, 1 /H becomes larger, δ getssmaller, and the later payment becomes less favorable. No knowledge of the decisionmaker’s psychology is required in this setup, only the postulate that she prefers herwealth to grow faster rather than slower.16 .5 Case D – Adaptive time frame with multiplicative dy-namics Specification: the period for computing the growth rate is that between the decisionand the chosen payment, either t a − t = H or t b − t = H + D ; and wealth dynamicsare multiplicative.We follow the same steps as in the previous cases. Wealth evolves to: x a ( t a ) = x ( t ) e r ( t a − t ) + ∆ x a ; (3.19) x b ( t b ) = x ( t ) e r ( t b − t ) + ∆ x b . (3.20)The corresponding growth rates are: g a = ln x a ( t a ) − ln x ( t ) t a − t = 1 H ln (cid:18) x a x ( t ) e rH (cid:19) + r ; (3.21) g b = ln x b ( t b ) − ln x ( t ) t b − t = 1 H + D ln (cid:18) x b x ( t ) e r ( H + D ) (cid:19) + r . (3.22) Preference reversal
When the later payment is sufficiently large, ∆ x b > ∆ x a e rD , preference reversal ispredicted, and a threshold horizon, H PR , exists. For shorter horizons than H PR ,the earlier payment is preferred ( g a > g b ) and vice versa . The discount function andthreshold horizon are not expressible in closed form for general parameter values.They become tractable in the limit of small payments, which we present below. If thelater payment is too small, ∆ x b < ∆ x a e rD , the earlier payment is always preferred inthis specification. Wealth effect
Our model predicts another type of preference reversal here, elicited by varying theinitial wealth, x ( t ), rather than the horizon. As x ( t ) →
0, the earlier payment ispreferred regardless of the size of the later payment. If the later payment is large This can be shown by comparing the H → H → ∞ limits of g a and g b in Eq. (3.21) andEq. (3.22). x b > ∆ x a e rD (cid:18) H + DH (cid:19) , (3.23)then it becomes preferable to the earlier payment as x ( t ) → ∞ . Thus, the decisionmaker switches from preferring earlier to later payments as her wealth increases. Wecall this the wealth effect . It is illustrated pictorially in Fig. 4, which shows thevariation of growth rates, g a and g b , for each option as initial wealth, x ( t ), increasesfrom left to right. Other parameters are held constant. H D H D H D
Figure 4: Wealth effect in case D, with logarithmic vertical scales. Initial wealth x ( t )increases from left panel to right panel ($500, $2277, $5500) with all other parametersheld fixed ( t = today, t a = 1 year from today, t b = 2 years from today, ∆ x a = $1000,∆ x b = $2500, r = 0 .
03 per year). At small wealth, option a is preferred, havingthe higher growth rate according to Eq. (3.21) and Eq. (3.22). At a larger wealth, x ( t ) P R ≈ $2277, both options imply equal growth, with reversal occurring as wealthincreases further.Figure 5 shows the difference in growth rates, g a − g b , as a function of initial wealth, x ( t ), for the same parameters as in Fig. 4. The earlier payment is preferred whenthis difference is positive, which happens for wealth below some threshold. For largerwealth, the growth rate difference is negative and the later payment is chosen.We interpret this as follows. Assuming multiplicative dynamics and an adaptive timeframe, it is growth-optimal for people of lower wealth to choose an earlier, smallerpayment; and growth-optimal for wealthier individuals to hold out for the later, largerpayment. This is consistent with the findings of Epper et al. (2018), that “individualswith relatively low time discounting are consistently positioned higher in the wealthdistribution.” It is likely consistent with (Green et al., 1996), in which people with18igure 5: The difference in growth rates, g a − g b , as a function of initial wealth, x ( t ),in case D. For small initial wealth the earlier, smaller payment is preferred, whereasfor large initial wealth the later, larger payment is preferred. Parameters as used inFig. 4.higher incomes were observed to discount less steeply.We exemplify the wealth effect by presenting a calculation using the same parametersas in Fig. 4. Suppose a decision maker faces a choice between receiving $1000 afterone year (option a ) or $2500 after two years (option b ), and that she has access toa riskless interest rate of 0.03 per year. If she has $500 initially, she evaluates thegrowth rate corresponding to option a as g a = 11 ln (cid:18) e . × (cid:19) + 0 . ≈ . , (3.24)and to option b as g b = 12 ln (cid:18) e . × (cid:19) + 0 . ≈ . . (3.25)Thus, the decision maker would prefer the earlier, smaller payment, as 1 . > . i.e. , 11 times more than19n the previous setting, a similar calculation yields g a ≈ .
19 per year and g b ≈ .
21 per year, so the later, larger payment is preferred.
Discounting in the small payment limit
In many applications of discounting, it is plausible to assume that the payments aresmall relative to wealth: ∆ x a (cid:28) x ( t ) e rH ; (3.26)∆ x b (cid:28) x ( t ) e r ( H + D ) . (3.27)We can express the threshold horizon and the discount function in closed form in thislimit. Setting g a = g b and using the first-order approximation ln(1 + (cid:15) ) ≈ (cid:15) for (cid:15) (cid:28) H PR ≡ D ∆ x a e rD ∆ x b − ∆ x a e rD , (3.28)and δ ( D ; H ; r ) = ∆ x a ∆ x b (cid:12)(cid:12)(cid:12)(cid:12) g a = g b ≈ He rH ( H + D ) e r ( H + D ) = e − rD D/H , (3.29)which is a product of hyperbolic and exponential discount functions. This hybridcase has interesting behavior in the long- and short-delay limits. As shown in Fig. 6,discounting is close to hyperbolic for short delays and to exponential for long delays.Thus, for the same dynamic and the same time frame, and assuming small paymentsrelative to initial wealth, our choice criterion predicts both approximately hyperbolicand approximately exponential discounting, depending on the delay.
This paper explores temporal discounting under the postulate that decision mak-ers maximize the growth rate of their wealth. We consider a basic temporal choiceproblem between two known, certain, and different payments at known, certain, anddifferent future times. To compute growth rates, the problem must be further speci-fied. We add information about wealth dynamics, treating additive and multiplicativecases, and the time frame of the decision, meaning the period over which growth isevaluated. 20igure 6: The discount function in the small payment limit in case D. The solid blackcurve is the hybrid δ = e − rD / (1 + D/H ), for r = 0 . H = 0 .
65 years.This is close to the hyperbolic discount function 1 / (1 + D/H ) (black dotted) for shortdelays and to the exponential discount function e − rD (grey dashed) for long delays.Preference reversal is an observed behavior in which decision makers switch from pre-ferring later to earlier payments as time passes. It is incompatible with the classicalnormative model of exponential discounting. Our model generates four different formsof discounting, depending on the decision maker’s circumstances – no discounting, ex-ponential, hyperbolic, and a hybrid of exponential and hyperbolic. The hyperbolicand hybrid forms predict preference reversal without, as is commonly needed, as-sumptions of behavioral bias or payment risk.The hybrid case – corresponding to multiplicative dynamics and an adaptive timeframe – suggests another type of preference reversal, called the wealth effect. Here adecision maker switches from an earlier to a later payment as her wealth increases.In other words, richer people discount less steeply than poorer people, in line withempirical findings (Green et al., 1996; Epper et al., 2018).Our main contribution is the prediction of non-exponential discounting and preferencereversal in a model that does not violate standard axioms of choice (von Neumann21nd Morgenstern, 1944). Changes in the discount function arise only from changes inwealth dynamics and time frame. This marks a shift from psychological to circum-stantial explanations of discounting. Our model assumes no dynamic inconsistency,in that the decision maker prefers at all times the option with the highest growthrate. If corroborated empirically, it would be both a normative and a descriptivemodel. Experimental tests are feasible because the model works directly with moneypayments rather than utility-of-consumption flows (Cohen et al., 2019).The temporal choice problem we study is riskless. A planned extension of this workis to explore the consequences for our model of payment uncertainty. Uncertaintydecreases the long-time growth rate associated with a payment (Peters and Gell-Mann, 2016). This would make a risky payment less desirable in our model, withoutreference to the risk preferences of the decision maker.The dynamics discussed here do not cover the entire range of wealth dynamics. Al-though multiplicative and additive wealth dynamics are common and intuitive, otherwealth dynamics are possible, which would lead to other forms of discounting in ourmodel. Our decision criterion can be adapted to general dynamics using the growthrates described in (Peters and Adamou, 2018).We end with a caveat. The temporal choice problem involves two future payments.In the small horizon limit, the growth rate corresponding to the earlier payment inthe adaptive time frame diverges. This indicates a loss of model realism. We link thisto the breakdown of an implicit assumption: that the growth rates we compute aresustained over sufficiently long periods to be meaningful to the decision maker. Theyare, in effect, the growth rates of wealth achieved under repetition of the choice. Weshare the view of Kacelnik (1997, p. 60) that “the discounting process used for the one-off events seems to obey a law that evolved as an adaptation to cope with repetitiveevents.” As the horizon shrinks, this imagined repetition occurs at a frequency sohigh that the choice problem no longer resembles a real situation. References
Cohen, J. D., K. M. Marzilli Ericson, D. Laibson, and J. M. White (2019):“Measuring Time Preferences,” Journal of Economic Literature,
Forthcoming .22 asgupta, P. and E. Maskin (2005): “Uncertainty and Hyperbolic Discounting,”American Economic Review, 95, 1290–1299.
Epper, T., E. Fehr, H. Fehr-Duda, C. T. Kreiner, D. D. Lassen, S. Leth-Petersen, and G. N. Rasmussen (2018): “Time Discounting and Wealth In-equality,” .
Green, L. and J. Myerson (1996): “Exponential versus Hyperbolic Discountingof Delayed Outcomes: Risk and Waiting Time,” American Zoologist, 36, 496–505.
Green, L., J. Myerson, D. Lichtman, S. Rosen, and A. Fry (1996): “Tempo-ral Discounting in Choice between Delayed Rewards: The Role of Age and Income,”Psychology and Aging, 11, 79–84.
Holling, C. S. (1959): “Some Characteristics of Simple Types of Predation andParasitism,” The Canadian Entomologist, 91, 385–398.
Jevons, W. S. (1863): A Serious Fall in the Value of Gold Ascertained, and ItsSocial Effects Set Forth, E. Stanford.
Kacelnik, A. (1997): “Normative and Descriptive Models of Decision Making:Time Discounting and Risk Sensitivity,” in Characterizing Human PsychologicalAdaptations, ed. by G. R. Bock and G. Cardew, John Wiley & Sons Ltd., 51–70.
Keren, G. and P. Roelofsma (1995): “Immediacy and Certainty in IntertemporalChoice,” Organizational Behavior and Human Decision Processes, 63, 287–297.
Laibson, D. (1997): “Golden Eggs and Hyperbolic Discounting,” Quarterly Journalof Economics, 112, 443–478.
Loewenstein, G. and D. Prelec (1992): “Anomalies in Intertemporal Choice:Evidence and an Interpretation,” Quarterly Journal of Economics, 107, 573–597.
Meder, D., F. Rabe, T. Morville, K. H. Madsen, M. T. Koudahl, R. J.Dolan, H. R. Siebner, and O. J. Hulme (2019): “Ergodicity-Breaking RevealsTime Optimal Economic Behavior in Humans,” arXiv:1906.04652.
Myerson, J. and L. Green (1995): “Discounting of Delayed Rewards: Models ofIndividual Choice,” Journal of the Experimental Analysis of Behavior, 64, 263–276.23 eters, O. (2019): “The Ergodicity Problem in Economics,” Nature Physics, 15,1216–1221.
Peters, O. and A. Adamou (2018): “The Time Interpretation of Expected UtilityTheory,” arXiv:1801.03680.
Peters, O. and M. Gell-Mann (2016): “Evaluating Gambles using Dynamics,”Chaos, 26, 23103.
Samuelson, P. A. (1937): “A Note on Measurement of Utility,” Review of EconomicStudies, 4, 155–161.
Sozou, P. D. (1998): “On Hyperbolic Discounting and Uncertain Hazard Rates,”Proceedings of the Royal Society B: Biological Sciences, 265, 2015–2020. von Neumann, J. and O. Morgenstern (1944): Theory of Games and EconomicBehavior, Princeton University Press. 24
The Transitivity of Growth-Optimal Preferences
In this appendix we show that growth-optimal preferences satisfy transitivity for allfour cases described in the paper. To prove transitivity we assume three payments, a ≡ ( t a , ∆ x a ), b ≡ ( t b , ∆ x b ) and c ≡ ( t c , ∆ x c ), where t a < t b < t c . We also assume adecision time t < t a and an initial wealth x ( t ). These payments define three RIPPs:a comparison between a to b , between b to c , and between a to c .In each of the four cases we will show that if a ≺ b and b ≺ c , then a ≺ c . We willalso show that if a ∼ b and b ∼ c , then a ∼ c . Case A
In case A (see Sec. 3.2), we show that growth rate maximization is achieved bychoosing the larger payment. Therefore, a ≺ b iff ∆ x a < ∆ x b and b ≺ c iff ∆ x b < ∆ x c .It follows that a ≺ c because ∆ x a < ∆ x c . If a ∼ b and b ∼ c then ∆ x a = ∆ x b and∆ x b = ∆ x c , so ∆ x a = ∆ x c and therefore a ∼ c . Case B
In case B (see Sec. 3.3), we show that growth rate maximization is achieved bycomparing the earlier payment to the later payment discounted by an exponentialfunction, so a ≺ b ⇐⇒ ∆ x a < ∆ x b e − r ( t b − t a ) ; (A.1) b ≺ c ⇐⇒ ∆ x b < ∆ x c e − r ( t c − t b ) . (A.2)It follows that ∆ x b e − r ( t b − t a ) < ∆ x c e − r ( t c − t b ) e − r ( t b − t a ) = ∆ x c e − r ( t c − t a ) , so∆ x a < ∆ x c e − r ( t c − t a ) = ⇒ a ≺ c . (A.3)Similarly, a ∼ b ⇐⇒ ∆ x a = ∆ x b e − r ( t b − t a ) ; (A.4) b ∼ c ⇐⇒ ∆ x b = ∆ x c e − r ( t c − t b ) . (A.5)25t follows that ∆ x b e − r ( t b − t a ) = ∆ x c e − r ( t c − t a ) , so∆ x a = ∆ x c e − r ( t c − t a ) = ⇒ a ∼ c . (A.6) Case C
In case C (see Sec. 3.4) only the linear payment rate of each option matters to thedecision maker, so a ≺ b ⇐⇒ ∆ x a t a − t < ∆ x b t b − t ; (A.7) b ≺ c ⇐⇒ ∆ x b t b − t < ∆ x c t c − t . (A.8)It follows that ∆ x a t a − t < ∆ x c t c − t , and a ≺ c . Similarly, a = b ⇐⇒ ∆ x a t a − t = ∆ x b t b − t ; (A.9) b = c ⇐⇒ ∆ x b t b − t = ∆ x c t c − t , (A.10)so ∆ x a t a − t = ∆ x c t c − t , and a ∼ c . Case D
Like in case C, the time frame in case D (see Sec. 3.5) is adaptive. For this reasonthe growth rate associated with each payment depends only on the payment timeand the decision time. In other words, under both RIPPs comparing a to b , and a to c , the growth rate associated with payment a , g a , is the same. Similarly, g b is thesame in both RIPPs comparing a to b , and b to c , and g c is the same in both RIPPscomparing a to c , and b to c .It follows that a ≺ b ⇐⇒ g a < g b ; (A.11) b ≺ c ⇐⇒ g b < g c , (A.12)26o g a < g c , and a ≺ c . Similarly, a ∼ b ⇐⇒ g a = g b ; (A.13) b ∼ c ⇐⇒ g b = g c , (A.14)so g a = g c , and a ∼ cc