No vem ber 21, 2008Larry KarpARE 263Not e s o n hy per bolic d is co u ntin g11DiscountratesIt is importan t to distin gu ish between the discount rate for consum ptio n an dthe discoun t rate for utility; the latter is also known as the p ur e rate of timepreference.Let θ (t, τ) equa l th e nu mber of units of utilit y that you would be willingto give up at calendar time τ in exc h an ge for one additiona l unit of utility attime t+τ,witht>0. θ (t, τ) is th e discount factor for utility. (Equivalently,θ (t, τ) isthepresentvalueattimeτ of an additional unit of utility at timeτ + t.)Holdin g the current calen da r time (τ) fixed,thepurerateoftimeprefer-ence, ρ (t, τ ),isdefined as the rate of decrease of the utility discount factor:ρ (t, τ) ≡−∂θ(t,τ)∂tθ (t, τ).The fam iliar case arises when the pure rate of tim e preference is a con-stan t, i.e under exponen tial discounting:θ =exp(−ρt)−dθdtθ (t)= ρ,aconstant.These notes consider the case where ρ is a function of time. It is im-portant to distinguish between the tw o reasons why ρ mightbeafunctionoftime: it could depend on calendar time (τ) or on the amount of time betw eenthe periods when the transfer of utility occurs (t) In the former case, the dis-count rate is said to be non-stationary. This form of discountin g leads to a1These notes are based on a series of five papers, two of which are co-authored withYacov Tsur, and one of which is co-authored with Tomoki Fujii.1non-stationary optimal control problem, but aside from the non-stationarit y,the contr ol problem is standard.In con trast, if the discount rate depends on the amount of time betweenthe periods when the transfer occurs, decisions that are optimal at the cur-rent time τ are (t y pically) not optimal at subsequen t times. That is, theoptim al (from the standpoin t of the decisionmaker at time τ ) plans are time-inconsistent. In order to obtain a time consisten t rational expectationsequilibriu m (mor e precisely, a subgam e perfect equilibrium ) we need to solv ethe problem as a dynamic game amongst different generations of decision-makers, agents who mov e sequen t ially. This period’s decisionmaker actstoda y, next period’s decisionm aker acts in the next period, and so on.Hereafter, I consider the case where pure rate of time preference is sta-tionary,butdependsontheamountoftimebetweenthepointsatwhichthetransfe r in utility occurs, t. The case of most interest arises when ρ (t) isnon-increa sing and strictly decreasing over some in terval; this case is oftenreferred to as h yperbolic (or quasi-hyperbolic) discoun ting.The consumption discount factor equals the present value at time τ of anadditional unit of consumption t units of time in the future:θ (t) U0(cτ+t) .The consumption discoun t rate (also know n as the social discount rate) equalsthe rate of decrease of the consumption discoun t factord(θ(t)U0(cτ+t))dtθ(t)U0(cτ+t)= ρ (t)+d(U0(cτ+t))dtU0(cτ +t)=ρ (t)+η (t) g (t)whereη (t) ≡−U00U0c (elastiticy of marginal utility, or measure of risk a version)g (t)=dcdtc(growth rate).The social discoun t factor migh t be nonstationary because ρ, η or g de-pend on calendar time. Mere time dependence (non-stationarit y ) does notcause time inconsistency. In this setting, time-in consisten cy arises only ifthepurerateoftimepreferencedependsontheamountoftimebetweenthet wo periods when the transfer of consum ption occurs.22 Time incon siste n cyTo illustrate the time incosistency of optimal plans under non-constant dis-coun ting, consider the three period cak e-eating model in w hic h future utilityis discounted with discoun t factor β,butdifferen t periods in the future areindistinguish able today. Here I am considering a discrete time setting; ithas the follo wing relation to the con tin uous time setting above. Supposethatthecontinuouspurerateoftimepreferenceisr>0 during the first unitof time and is 0 thereafter. Then, at time 0, the discount factor applied toperiod 1 and to period 2 consumption is β ≡ e−r.Thus, in period 1 the ma ximizat ion pro blem ismax [U (c1)+β (U (c2)+U (c3))]subject toc1+ c1+ c1≤ 1Thesolutiontothisproblemrequiresc2= c3(1)U0(c1)=βU0(c2)c1+ c1+ c1=1.Now consider how the decisionmaker in period 2 beha ves. In period 2,when c1has already been ch o sen, the objective ismax [U (c2)+βU (c3)]subject toc2+ c3≤ 1 − c1Recall that each of the succession of decisiomakers (three, in this setting)discounts the future using the discount factor β. The necessary condition isU0(c2)+βU0(1 − c1− c2) , (2)which does not satisfy equation (1). Th u s, the decisionmaker in period 2does n ot want to carry out the plan chosen by the decisionma ker in period 1:theplanthatisoptimalfromthestandpointofthedecisionmakeratperiod1istimeinconsistent.In order to obtain the subgame perfect equilibrium , we solve this modelbackwards. Solve equation 2 to obtain the function c2= C (c1).Substitute3this function in to the maxim ization problem for the agent at period 1 towrite her problem asmaxc1U (c1)+β [(U (C (c1)) + U (1 − c1− C (c1)))] .The necessary condition to this problem isU0(c1)=β½[(U0(1 − c1− C (c1))) − U0(C (c1))]dCdc1− (U0(1 − c1− C (c1)))¾.3 A rationale for hyperbolic discountingMo st applications of h yperbolic discoun tin g in volve fairly short decision pe-riods, e.g. a few yea r s or perhaps the life of an individual. Hypeberbolicdiscounting is also im portant for very long-lived problems, e.g. those involv-ing climate c han ge. One rationale for hy perbolic discoun ting is based on timeperspectiv e - the tendency to foreshorten time periods as w e peer further intothe future.A function s(n) captures time perspective by assigning a perceived lengthtoayearthatbeginsn yearsfromnow. Thisfunctionsatisfies s(0) = 1,s0(·) ≤ 0 and s(∞) ≥ 0; undistorted time corresponds to s(·) ≡ 1.Therelation betw een real time (t) and perceiv ed (foreshortened) time isS(t)=Zt0s(ζ)dζ.From the standpoint of toda y, the time period from now un til t “looks like”aperiodfromnowuntilS(t).Theconstantpurerateρ0represents impatience as applied to the per-ceiv ed time S. From today’s perspectiv e, the presen t value of a utilit y streamU(c(S)),S≥ 0,isZ∞0U(c(S))e−ρ0SdS.Making a c hange of variables from S to t (i.e. from foreshortened time toreal time), the pa yoff expressed in real time isZ∞0expµ−ρ0Zt0s(ζ)dζ¶U(c(t))s (t) dt.4The utility discoun t factor is