Hyperbolic Discounting and Savings BehaviorJohn Du¤yEcon 2713 Notes Week #10John Du¤y () H yperbolic Disco unting and Savings Behavior Econ 2713 Notes W eek #10 1 / 26Hyperbolic DiscountingEvidence of multiple (split) personalties has been around for sometime, most severely in the condition known as schizophrenia.However, lesser forms of split personality seem commonplace.In particular, there is evidence that people evaluate immediate choicesdi¤erently from future choices: delayed events are devalued heavilyrelative to immediate ones. This can often lead to self-defeatingbehavior.As Ainslee and Haslan (1992) argue, this split personality “may notbe a freak of nature but the condition of nature itself.”One way to capture this kind of behavior, of course, is withdiscounting. The standard, discounted utility approach involvesexponential discounting. But behavioralists have argued in favor of adi¤erent version known as hyperbolic discounting.John Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 2 / 26Exponential DiscountingStandard discounted utility model of intertemporal choice isintroduced by Samuelson (1937).Consider two sequences for consumption, fc0,c1, ..., cTg andfc00, c01, ..., c0Tg. The …rst sequence will be strictly preferred to thesecond if and only if:∞∑t=0δtu(ct) >∞∑t=0δtu(c0t)where u() is a concave utility function.Until recently, the discounted utility model of intertemporal choicehad not been subjected to the same degree of scrutiny as the expectedutility model of choice under uncertainty. But this is now changing.Several patterns of intertemporal choice have been shown to con‡ictwith the standard exponential discounting model.John Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 3 / 26Stationarity PropertyConsider an individual who is indi¤erent between an additional x unitsof consumption at time t and y > x units at a later time τ > t, givena constant baseline consumption in amount c in all periods.Under the discounted utility model, we should have thatδt[u(c + x)  u(c)] = δτ[u(c + y)  u(c)] oru(c + x)  u(c) = δτtu(c + y)  u(c)The stationarity property is that preference between the twoconsumption adjustments depends only on the absolute time intervalseparating them, τ  t. However, this appears to be at odds withsurvey evidence.Thaler (1981) reports that when individuals are given a choicebetween one apple today and two tomorrow, most choose one appletoday. However, when individuals are given a choice between oneapple in 50 days versus two apples in 51 days, the same group ofdecision-makers choose two apples in violation of the stationarityproperty.Behavioralists argue that this calls for non-stationary discounting(decreasing functions of time).John Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 4 / 26Modelling Time-Dependent DiscountingLoewenstein a nd Prelec (1992) formalize the problem as follows. Letv() denote a value function, and φ(t) a discount function. Assumeφ(0) = 1 (no discounting in the present).Suppose that a person is indi¤erent between receiving x now andy > x s periods later. However, consistent with the apple example, ifboth outcomes are postponed by a common delay in time t, then thedecision maker strictly prefers the large amount y to the smalleramount x.Formally,v (x) = φ(s)v (y ) ) φ(t)v (x ) < φ(t + s)v (y )To maintain the …rst indi¤erence would require a time-dependentdiscount factor:Ainslie (1975) proposes φ(t) =1t.John Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 5 / 26Modelling Time-Dependent Discounting, cont’d.Loewenstein a nd Prelec (1992) provide an axiomatic derivation of themore general form:φ(t) = (1 + αt)β/αα, β > 0The α coe¢ cient determines the extent to which the discountfunction departs from constant (continuous form exponential)discounting: limα!0φ(t) = eβt.With the addition of a positive α, it is possible to get a time pro…lefor φ(t) that looks like a “step function”:Hyperbolic Versus ExponentialDiscounting00.20.40.60.811.20 1 2 3 4 5TimePhi(t)exponential hyperbolicJohn Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 6 / 26Quasi-Hyperbolic DiscountingOriginally due to Phelps and Pollak (1968) this version of hyperbolicdiscounting was resurrected by Laibson (1997) and O’Donoghue andRabin (1999).The quasi-hyperbolic form is a dicrete-time approximation to thecontinuous hyperbolic discounting function: Letting utdenote theinstantaneous utility obtained in period t, an individual’sintertemporal preferences at time t may be represented as follows:Ut(ut, ut+1, ..., uT) = ut+ βT t∑t=1δτut+τwhere β, δ  1.For β = 1, these preferences correspond to the standard exponentialdiscounting case. However, for β < 1 this representation captures thetime-inconsistent preference for immediate grati…cation.The result is a declining rate of time preference and a violation oftime consistency.John Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 7 / 26Quasi-Hyperbolic Discounting IllustratedWe can write the quasi-hyperbolic discount function as follows;φQH( τ) =1 if τ = 0βδτif τ = 1, 2, 3,...Alternatively as the sequence φ(τ) = f1, βδ , βδ2, βδ3....}Example: β=12, δ=1, φQH( τ) = f1,12,12,12, ...g Essentially alldiscounting occurs between the current and immediate future period.Compare with the discrete form exponential discount function:φE( τ) =1 if τ = 0δτif τ = 1, 2, 3,...John Du¤y () H yperbolic Discounting and Savings Behavior Econ 2713 Not es W eek #10 8 / 26Laibson (1997)Laibson cites Strotz (1956), an early observer of time inconsistentbehavior, on the many commitment devices in place to correct againstself-defeating behavior, e.g., personal advisors, joining the army,costly to liquidate investments, which Laibson labels “golden eggs.”Picking up on the latter, Laibson imagines that consumers withdynamically inconsistent preferencesu(ct) + βT t∑t=1δτu(ct+τ),can invest in two instruments, a liquid asset xtand an illiquid assetzt; the illiquid asset (e.g., a house) takes one period of model time toliquidate or borrow against.With dynamically inconsistent preferences, it is standard practice tomodel the decision-ma ker, a consumer in this case, as playing a gamewith a sequence of temporal