Evolution of time preference by natural selection

This paper entertains the hypothesis that human time preferences are in evolutionary equilibrium (i.e. that no mutation changing time preferences could be favored by natural selection). This hypothesis implies that the marginal rate of substitution (MRS) holding Darwinian fitness constant must equal the MRS holding utility constant. Furthermore, in a market economy the latter must equal the MRS in exchange. Exploiting these principles, I find that the long-term real interest rate should equal ln(2) per generation (about 2 percent per year) and that young adults should discount the future more rapidly than their elders.

Evolution of Time Preference by Natural Selection By Alan R. Rogers* This paper entertains the hypothesis that human time preferences are in evolu­tionary equilibrium (i.e. that no mutation changing time preferences could be favored by natural selection). This hypothesis implies that the marginal rate of substitution (MRS) holding Darwinian fitness constant must equa! the MRS holding utility constant. Furthermore, in a market economy the latter must equal the MRS in exchange. Exploiting these principles, I find that the long-term real interest rate should equal tn(2) per generation (about 2 percent per year) and that young adults should discount the future more rapidly than their elders. (JEL D91, E43) Preferences are central to economic the­ory, yet little is known about them. Because of (.his ignorance, economists have worked hard to make their assumptions about pref­erences as mild as possible, and economic theory has grown more abstract and general over the years. This has cushioned the the­ory against violations of assumptions but has not increased its power. Economics would gain much from a theory of prefer­ences. The theory of evolutionary genetics may be useful in this regard. If genes affect preferences, then an evolutionary model may succeed in predicting them.1 ‘Department of Anthropology, University of Utah, Sail Lake City, UT 84112. Jack Hirshleifer read several drafts of ihis paper, and his suggestions have improved it enormously. Helpful comments were also provided by James Brannan, Tom Caraco, Elizabeth Casbdan, Brian Charlesworth, Eric Charnov, Colin Clark, Hans Ehrbar, Gordon Getty, Hillard Kaplan, Kristen Hawkes, Elizabeth Manderscheid, John Fezzey, Eric Smith, and Margo Wilson, The paper also benefited from the discussion following presentation of these ideas al the Santa Fe Institute in April 1992, at the meeting of the Western t onomic Associalion in July 1992, and at the Univi sily of Chicago's Rational Choice Seminar in Apri 1993. This work was sup­ported in part hy grants from (he U.S. Department of Health and Human Services (MGN 1 R29 UM39593) and from the National Science Foundation (DBS- 92! 1255). 'This proposal has been made previously by economists including Gary S. Becker (1976), Hirsh­leifer (1977), R. H, Coase, (1978), Paul H. Rubin and Cris W. Paul (1979), Heinrich W. Ursprung (1988). and The proposal that preferences have evolved by natural selection may strike some readers as strange. It is customary to think of selection acting on "biological" charac­ters such as size, shape, and patterns of fertility, but less customary to view prefer­ences in the same light. Yet there seems really to be no basis for this prejudice. Con­sider, for example, preferences concerning mates. Most individuals prefer to mate with opposite-sexed individuals of their own spccies. Presumably, this is because those who preferred otherwise left few offspring: they were selected against. How else- except by natural selection-can such pref­erences be explained? If selection affects mating preference, then why not time pref­erence? Every animal that builds a nest or burrow is deferring immediate consumption Ingemar Hansson and Charles Stuart (1990). In addi­tion, several authors have published evolutionary mod­els of lime preference. Some have used ideas from the theory of optimal foraging in an effort to understand how animals discount rewards over delays measnred in seconds or minutes (John H. Kage) el al., 1986; A, W. Logue, 1988). Others (Gordon Getty, 1989, 1991; Hansson and Stuart, 1990) have tried to explain lime preference over intervals of years and decades. The model 1 present here deals with time preference on this longer lime scale. II differs from Geliy's in empha­sizing preferences ralher than production and in incor- poraiing a more complex model of the life cycle. It differs from that of Hansson and Stuarl in considering lime preference within as well as between generations, and in allowing for sexual reproduction. 460 VOL. #4 ho. .1 for a'delayed reward. Thus, time preference has an ancient evolutionary history. It rnight also be objected that time prefer­ence is shaped by learning and culture, not by natural selection. In response, one can argue that the capacity for culture is itself shaped by natural selection, so that we tend to learn what selection would have favored anyway. I have shown elsewhere (Rogers, 1988) that neither of these positions is en­tirely corrcct. A model of natural selection will predict more poorly where cultural transmission is important. However, the re­duction in predictive power may be small. Thus, a model ignoring culture may prove useful even if patterns of time preference are transmitted culturally. Below, Section 1 defines the marginal rates of substitution and of time preference and sumniiiriy.es previous results tn'm eco­nomies and evolutionary genetics; Section II combines these to derive new results con­cerning the intertemporal utility function and the real interest rate. These results all assume complete certainty about which in­dividual will enjoy the returns from an investment. Section III relaxes this last assumption, allowing the returns to be directed toward an initially unspecified relative of the investor. In this context, the marginal rate of time preference turns out to vary both with the age of the investor and with the delay between investment and return on investment. Finally, the results are summarized and discussed in Sec­tion IV. I. Preliminaries A. Marginal Rates of Substitution in Preference, in Exchange, and in Fitness Immediate benefits are ordinarily pre­ferred over henefits that are delayed. Irving Fisher's (1930) graphical analysis of this matter is shown in Figure 1. The dotted indifference curves there connect points of equal utility. The absolute value of the slope of an indifference curve is called the marginal rate of substitution (MRS) in pref­erences and measures the rate at which pre­Noles: jf(.r) iind /c(jr + 7) denoie consumption al ages .r anti j‘ + r, respectively. The dolled indifference curves show ihe coniour map of a hypothetical u 1 iJity function. The solid "market line" has slope - e'T, where / is the interest rale. The dashed 45° line connects points al which k(x + r) = k(x ). sent and future consumption can be substi­tuted without affecting utility. The solid "market line" in the figure con­nects points of equal wealth. By borrowing, the individual at A can move along the market line toward B. The move from C to B, on the other hand, is accomplished by lending. The absolute slope of the market line is called the MRS in exchange. The optimum B occurs where the MRS in pref­erences is equal to that in exchange. Following Hansson arid Stuart (1990), I define the MRS in fitness as the marginal rate at which two goods can he substituted without changing Darwinian fitness.2 An ‘In models with discrete generations, Darwinian illness is the conditionally expected number oT an individual's offspring, given i(s genotype. In models wilh overlapping generations, limess is measured by R. A. Fisher's (1958) "Malthusian parameter," which measures ihe asymptotic rate of exponential in­crease in the numbers of one's descendants (Brian Charlesworlh, 1980).462 THE AM ER ICAN ECONOMIC REVIEW JUNE 1994 allele3 that encourages substitution at this rate is neither favored nor disfavored by natural selection: it is selectively neutral. If preferences have been shaped by natural selection, then individuals should be indif­ferent to choices that do not affect fitness. Thus in evolutionary equilibrium the MRS in fitness must equal the MRS in prefer­ences. I exploit this principle below by de­riving an expression for the MRS in fitness and setting it equal first to the MRS in preferences and then to the MRS in ex­change. The analysis will employ two familiar rep­resentations of the MRS in preferences. First, 1 write it in terms of the marginal rate of time preference (MRTP), which measures the extent to which a unit increase in pre­sent consumption is preferred to a unit in­crease in future consumption. If I am indif­ferent between another apple today and e9T apples delayed by r years, then 6 is my MRTP for apples. The MRTP is related to the MRS in preferences by (1) cIk( x + t) dx(x) where k( x) is consumption at age x, U is utility, and the derivative is taken along a line of constant utility. The left side is, by definition, the MRS in preferences. In gen­eral, 0 will be a function of age x, of the time-delay t between investment and re­turn on investment, and of the dated con­sumption levels k(jc) and k(x + r). A second familiar expression for the MRS in preferences is obtained by writing the utility of a consumption stream beginning at time /„ and ending at /, as (2) U= [''e-^'-,a)u[K{t)]di J'o where *■(/) is consumption at time /, (/[*■(/)] ‘The word allele refers to a variant of a gene. Thus, a gene that occurs in two forms such as "altruist" and "nonaltruisl" is a gene with two alleles. is the utility of this dated consumption^ and 17 is the subjective rate of utility discount. The MRS in preferences between k(x + t) and k(jc) is (3) dx( x + r) dx(jf) U constant e' + where hJk] = du[i<)/dK is the marginal utility of dated increments to consumption (Hirshleifer, 1970 p. 127). Finally, I will make use of the fact that the real interest rate i is related to the MRS in exchange by (4) dx(x + t) dx( x) I W const a n( where W is wealth and the derivative is taken along a line of constant wealth, that is, along the solid market line in Figure'!. The left side is, by definition, the MRS in exchange. At simultaneous evolutionary and market equilibrium, each of these marginal rates of substitution must equal the MRS in fitness. In deriving an expression for this latter quantity, I will need results from the evolu­tionary theory of kin selection. B. Results from the Theory of Kin Selection The theory of kin selection (W. D. Hamil­ton, 1964a, b) is about altruism toward rela­tives. It tells when natural selection will favor an "altruist allele," which predisposes its bearer toward some behavior that de­creases its own Darwinian fitness but in­creases that of a relative. Those who pro­vide these benefits are called "donors"; those who receive benefits arc called "re­cipients." Kin selection is relevant to economic problems of investment decision because the returns from an investment may affect the Darwinian fitness of the investor's relativesVOL. N4 NO. 3 ROGERS: EVO LU T IO N OF TIME PREFERENCE 463 Tablf I -How Changes in Fertility and Survival Affect Fi™f..ss tu- Effecl on: liU Probability of act (iii) Additive effect (IV) Reproductive value (v) Discount factor (vl) Relationship to donor Donor Recipient lD(x i) /n( x,) APix,) AP( JT| + t) i-' [)(X | "t- ilx ) rR(jij ^ dx) e~rU] • it) c ,.( r, ■* T+lfiJ 1 r Nmes: The altruist allele is favored (disfavored) by selection if the sum of row products is positive (negative). The notation is defined in the text. To simplify the table, I assume that ihe sex ratio at birth is unjiy, that effects on survival are brief, thal these effects are small enough that second-order terms in AP can be ignored, and that a single recipient is affected by each act of altruism. Source: Rogers (1993 appendix A). rather than that of the investor himself. For example, benefits received by a woman of age 50 cannot increase the number of her own children but may increase that of her daughter. The "investor" of economic par­lance becomes the "donor" in kin-selection theory. The "recipient" in kin-selection the­ory is the individual (or individuals) whose Darwinian fitness is increased by the re­turns from that investment. Often, the re­cipient will be the donor (investor) himself. I will continue to use the term "altruism" in this case, although this usage is nonstan­dard; in ordinary usage it is not altruism unless the recipient is someone else. The fate of an altruist allele depends not only on the delay between investment and return, but also on the ages of the donor and recipient. I assume that the investment is made when the donor's age is and benefits the recipient t years later when the recipient's age is x2. The fate of the altruist allele depends also on the coefficient of relationship,4 r, between donor and recipi­ent. If the donor and recipient are the same individual, then r = 1 and x2 = xt+ t. Subscripts m and f will denote males and females, respectively. In the absence of al­truism, an individual of sex g will survive from age x to x + dx with probability Px(x), He or she will survive from birth to age x with probability /^(jc ). An individual of sex '"Wright's coefficient of relationship (J. F. Crow and M. Kimura, 1970 pp. 69. 137-38) can be interpreted as the fraction of their genes that two individuals can expect to hold in common. It equals 1 if the donor and refipiem are the same individual, j if the recipient is an offspring, J if a grandchild, and so forth. g who reaches age x can expcct to produce mg{x) offspring of sex g before age x + dx. These age-specific rates of survival and fertility are assumed to be constant, which implies, under mild additional as­sumptions. 5 that the population will eventu­ally converge to a stable age distribution. Thereafter, its growth is exponential. I de­note by p the rate of exponential population growth in the absence of altruism. In general, the fate of the altruist allele will depend on how it affects survival and fertility. In this paper, I ignore effects on fertility and consider effects on survival alone. The effect of a given interaction is assumed to be brief, of duration dx. Altru­ism changes survival probabilities at age .r, in the donor and age x2 in the recipient, adding AP(jc,) to the survival of the donor and AP{x] + t) to that of the recipient. In the usual case, the interaction harms the donor and benefits the recipient, so AF(xt) <0 and + t) > 0. For mathematical convenience, I assume that these effects are small enough that second-order terms can be ignored. I have derived elsewhere (Rogers, 1993) the conditions under which the altruist al­lele will increase in frequency, and these are summarized in Table 1. In the table, subscripts D and R indicate the sex of the donor and the recipient. The reproductive 5It is sufficient that there be positive fertility in two adjacent age classes (J. H. Pollard, 1973 pp. 42-44).4(>4 THE AMERICAN ECONOMIC REVIEW JUNE 1994 value (R. A. Fisher, 1958) is defined by and can be interpreted as the expected pre­sent value of an individual's future contri­butions to the gene pool. The equivalence of li U‘) to the expected present value follows from the following observations. First, the probability of surviving from age x to age y is I s(y)/1 g{x). Thus. m (y)ll,(y)/l^ix) is the expectcd number of offspring that an individual of age x will produce at age y. The reproductive value is a sum of these contributions, discounted exponentially at rate p. The rationale for this discounting can be understood as follows. The birth of an altru­ist at time y increases the frequency of the altruist allele by l/[2My)], where N(y) is the size of the population at time y. Thus, one must weight each birth by 1 /[7N{y)\. This is equivalent to weighting by e~py since my assumptions imply that the population is increasing exponentially at rate p. In (jt), births at age y > .v are discounted by \/\lN{x)] In effect, this takes births at age x as a numeraire, or unit of value. The value of a birth at age y is measured by the number of births at age x that would be needed to produce the same effect on allele frequency. The fate of the altruist allele is deter­mined by the sum of the products of the entries in the two rows of Table 1. The allele will increase in frequency when rareh 'The theory holds not only when ihe altruist allele is rare (with frequency near zero), but also when its frequency is near unity. There is no theory for ihe ease of an allele of intermediate frequency, if this sum is positive, and decrease if it is negative. The allele is selectively neutral if the sum is zero. Elsewhere (Rogers, 1993), 1 justify this result in two ways, one rigorous and the other heuristic. Here, I paraphrase the heuristic justification. The altruist allele is expressed only if the donor survives to perform an altruistic act at age Thus, column (ii) shows that the allele's effect is proportional to the proba­bility /D(X[) of this event. It is also propor­tional to the additive effects on the age- specific survival of donor and recipient, as shown in column (iii). Multiplying by repro­ductive value in column (iv) expresses these effects on survival in terms of future births discounted back to age + dx in the case of the donor, and to x2 + dx in the case of the recipient. The additional discount fac­tors in column (v) discount these future births back to the time when the donor's age was zero, thus expressing all births in a common currency. The product of columns (ii)-(v) gives the expected effect of an altru­ist gene on births, appropriately discounted. However, I am not interested in all births, but only in the births of new altruist individ­uals. Thus, column (vi) multiplies by the coefficient of relationship r, which can be interpreted as the probability that the recip­ient has the altruist allele given that the donor does.7 Having justified the entries in Table 1, I will now examine its implications. In this section 1 show how the MRS in fitness can be expressed in terms of demo­graphic parameters, the degree of kinship between donor and recipient, and the over­all rate of population growth. Throughout this section, I assume that the individuals involved in an interaction are known with certainty. 7This interpretation is valid when the inheritance system is diploid and the altruist allele is rare. II. The MRS in Fitness when Recipients are Known with CertaintyVOL. 84 NO. 3 ROGERS: E VO LU T IO N OF TIME PREFERENCE 465 'A. The MRS in Fil ness and Irving Fisher's Indifference Diagram The altruist allele is selectively neutral if the sum of row products in Table 1 is zero, that is, if i'dU, + dx)AP(Xi) + e~prwR( x2 + dx)AP(x} + r) = 0. Rearranging this expression yields the MRS in fitness between increments to the survival of the altruist and survival of the recipient: (6) MRS,, = - A />(*,) t-'nUi + dx) re n'ruR(x1 T dx) re p7t-'R(jr2) where the approximation assumes that the duration dx of effects on survival is small. MRSp specifies the rate at which the sur­vival of the donor can be substituted for the (delayed) survival of the recipient without affecting Darwinian fitness. I now use this result to construct an ana­logue of the indifference diagram in Figure 1. The dotted lines in Figure 2 connect points of equal fitness and may thus be called fitness isograrns. As the graph shows, the fitness isograms are convex to the origin.8 bIn deriving this result, 1 take dx as lhe time uni! so that v(x + dx)= l(x + 1). Since p is constant along a fitness isogram, equation (6) implies that rflnMRS = 1 <■(*, , I) ) fll'(X | - I) I, + 1) 3P(x i) dP(Xn) -MRS,. 4 0 Now substitute &v ( X , -L I) du ( X J +■ 1) Jp(x2) MRSp <M-0 dP(y) f0 = { '•■(v + l)e '/(>■) t v' i ( i > if y < x if >' j To relate this result to economics, I as­sume that P is not only a function of age, but is also a differentiable function of con­sumption. Thus, P- Ptx, k(jc)] where k(jc) is consumption at age x. Further suppose that the mortality trade-off discusscd above is caused by a trade-off in consumption. The donor forgos some consumption at age jc, in return for a delayed increase in the con­sumption of the recipient, who is then of age x2. Specifically, the donor's con­sumption at age jc, changes from k(jc,) to «(*,) + Ak(jc,), while that of the recipient at age x2 changes from k(x2) to x(x2) + Ak(ji:, + t). If these changes in consumption are small, then A/'(*,) = PJjc,,*(*,)] Ak(x,) AP(j:1 + t) = Pk[j:2,k(j:2)] Ak(x, + t) where Plt = dP/dx. Substituting these into equation (6) gives the MRS in fitness be­tween the consumption of the donor and the delayed consumption of the recipient: (7) MRS, a Ak( jc, l t) Ak(jc,) MRS, j ' MRS* is obtained by multiplying MRSp by the ratio of marginal effects of consumption on survival of the donor and recipient. If P lo obtain din MRS, + 1) MRSP SP(x2) f/O," 1) which is always negative. Thus, MRS,. decreases with increasing values of P(xt). and the fitness isogram is convex.466 THE AM E R IC AN ECONOMIC REVIEW JUNE 1994 Survival Consumption ' P{x + r) 1 2 k(x + r) 0 P(x) k(x) Fiourf 2. Indifffrenctl Diagkams on Survival and Consumption Axfs Nates: The dolled lines in each panel arc fitness isograms, which are equivalent lo indifference curves under evolutionary equilibrium. The solid market lines define opportunities for lending and borrowing. The left panel refers to cffects on survival and exhibits Ihe indifference curves implied by equation (6). The image of these curves in (he right panel is even more convex, because of the diminishing marginal effect of consumption on survival. has the properties that are usually at­tributed to utility-increasing with con­sumption, but al a decreasing rate-then the transformation from survival to con­sumption axes will increase the convexity of the fitness isograms (John Hicks and Roy G. D. Allen, 1934), as shown in the right- hand panel of Figure 2. At evolutionary equilibrium, the MRS in fitness must equal that in preferences, and the fitness isograms in Figure 2 become indifference curves. To recover the familiar indifference diagram of economics, we need only introduce a market line, which is drawn as a solid line in both panels of Figure 2. As usual, the market line defines opportunities for lending or borrowing and is straight when drawn on consumption axes. Jn the left panel, however, it is concave to the origin becausc of the nonlinear transforma­tion from consumption axes to survival axes. In adding the market line, one need not assume that credit markets have been im­portant throughout human evolution. The hypothesis of evolutionary equilibrium merely specifies the form of the indifference curves. This equilibrium may have been reached long before the advent of credit markets. Given these preferences, Irving Fisher's (1930) argument (Fig. I) explains how modern humans can maximize utility hy lending and borrowing. The hypothesis of evolutionary equilibrium modifies Fisher's argument only by giving definite form to indifference curves that were arhitrary in Fisher's analysis. Equations (6) and (7) identify several fac­tors that contribute to the preference for immediate rather than delayed consump­tion. First, if the time path of consumption is increasing and the henefits of investment will go to the investor himself, then PJxv i)] > PJx2, k(.v2)], thus inflating MRS„. This is the familiar effect of dimin­ishing marginal utility on indifference curves, which accounts for the increased convexity of indifference curves drawn on consumption axes. But this is not the whole story. MRS„ is further inflated by MRSP, the product of e"7, r~\ and i-'D(jc,)/r(jc2)- The first two of these factors are never less than unity. The third exceeds unity when the donor is adult and is also the recipient. Thus, MRS;J will usually exceed unity for adult donors. This implies that, for such donors, MRS^ will usually exceed unity even along the 45° line, where present and future consumption arc equal. The excep-VOL. H4 NO. 1 ROGERS: EVOLUTION OF TIME PREFERENCE 467 tiijus'will involve transfers from aged adulls with low reproductive value to their off­spring with higher reproductive value. This provides an evolutionary explanation for positive time preference. B. The Marginal Rale of Utility Discount and the Marginal Utility of Consumption The procedure of equating the MRS in Htness with that in preferences also pro­vides a variety of algebraic results. Substi­tuting (6) into (7) and setting the result equal to (3) gives u „.[«■( A'V (8) - v r c ■«..[*(-v + 7)1 rp( -vr) ^[ -T 1 J R( T2 ) ; This requires that the marginal utility of consumption be a function of age x as well as consumption k, and that it be propor­tional to the reproductive value r(,v). For example, one might define (9) uk[k] = uji'.x] a PK[ j,*][■( X) (LO) rj = -(In r)/r + p where for simplicity I have suppressed sub­scripts denoting sex.<: Economic models typ­ically assume the function it to be indepen­dent of age. It seems impossible to reconcile this assumption with the hypothesis of evo­lutionary equilibrium. Thus, the evolutionary perspective re­stricts the range of permissible formulations of intertemporal utility. It also broadens their scope to include consumption by rela­9Tim is not ihi: i>r,V .l.-'iniilnr of u ,md rj that is consistent with (N). 1 have included > in I he expression lor 77 because it will vary with the length of delay, provided lhai bene fils ;-jnrivin^L liTilt i\ deiay ol one generation 10 offspring. those arriving after two generations go to grandchildren. and so on. tives, weighted by the coefhcient of relation­ship, This suggests a restriction on the "dy­nastic utility function" of Becker and Robert J. Barro (1988), which assigns arbitrary weights to the consumption of descendant generations. In evolutionary equilibrium those weights are no longer arbitrary. Equation (10) relates the subjective rate 77 of utility discount to r, r, and p. In the special case when recipient and donor are the same individual, r =-- 1 and therefore 77 = p. The relevant value of p is not the current rate of population growth, but that which prevailed on average during some long but unknown period of evolutionary history. Apart from the last couple of cen­turies, p must have been on average very near zero (Colin MeEvcdy and Richard Jones, 1978). Thus, equation (10) implies that rj = 0 when donor and recipient are the same individual. When u satisfies equation (9), there is no additional discounting due to length of delay. Consequently, rj is needed only in models that allow intergen­erational transfers. When intergenerational transfers do oc­cur, equation (10) says that the subjective rate of utility discount is 17 = - (In r)/r. For a given degree of relatedness r between donor and recipient, rj is inversely propor­tional to the delay r between cost (invest­ment) and benefit (return on investment). However, tbis conclusion may not hold in general, since r is likely to vary as a func­tion of 7 [see footnote (9)]. C. The Long-Term Real Interest Rate An expression for the long-term interest rate can be obtained by setting (11) MRS„. = e'T a procedure justified under evolutionary equilibrium by equation (4). To simplify the result, 1 restrict attention to the ease of an investment that benefits the investor's daughter after exactly one generation. By assumption, the mother and daughter are affected at the same age, so that i'u(jf,) = rR(x,), and MRSp = r " lepl, where r \ isTHE AMERICAN ECONOMIC REVIEW JUNE IQQ4 the coefficient of relationship between mother and daughter, and r equals T, the generation length.10 In stationary equilib­rium, the mother and daughter will have equal wealth at this common age, so that MRS,, = MRS,.. I am thus restricting atten­tion to points along the dashed 45° line in Figure 2. Equation (11) becomes 2epT = e‘T, (12) / = (In 2)/ 7' + p. The generation time T is usually a little Jess than 30 years in human populations. For example, T = 28.9 in the 1906 population of Taiwan (Hamilton, 1966). Thus, if p ~ 0, selection should favor long-term interest rates that average (In2)/28.9 = G.024 per year," which is in reasonable agreement with observation. Time-productivity does not appear explic­itly in this analysis. However, in equilibrium the intergenerational MRS,, must also equal the marginal time-productivity of intergen- erationa) investment, for any discrepancy between the two would trigger a capital accumulation or decumulation process eventually restoring the equilibrium (see Hirsh lei fe r, 1970 p. 172). In a Malthusian interpretation, the accumulation process would take the form of a population surge, rather than a capital surge, but would lead in either case to an equilibrium along the 45° line.12 "'There arc several measures of generation length m demography (Ansley J. C'oale. 1972). The inosl ap­propriate here is Ihe mean age of childbearing (i.e.. the average age of mothers of newborn daughters): T = Sl.ac :,'1/r(-f)nj1(.r), "This is similar lo Geliy's (198V, 1991) finding llial i ■ \ , T. The similarity is probably mere coincidence, however, as Paul Romer has pointed out to inc. Getty assumes that i must match the rale of production of aggregate fitness (i.e.. the rate at which siew individuals are recruited into Ihe population). Since the popula­tion replaces ilself once per generation, the recruit­ment rale R must satisfy RT - 1 in stationary equilib­rium. Setting R = i produces Getty's conclusion that i - ! ! T. i:More explicitly, the Malthusian interpretation as­sumes that: (i) A population will grow whenever its members find that marginal time-productivity of inter- generationai investment exceeds MRS„. (The produc- These conclusions require qualification. In equating marginal rates of substitution in preferences and in exchange, I have implic­itly assumed that maximization of utility re­quires some nonzero transfer of consump­tion from donor to recipient. This is not necessarily so: the donor will not dispense benefits to all potential recipients. Further­more, I have assumed that at the time an individual contemplates some altruistic act, he or she somehow knows in advance both the age and the reiatedness of the recipient. I have not allowed for the possibility that there will be no recipient at all. To deaf with these complications, one needs a statis­tical model incorporating uncertainty about which relative, if any. will eventually receive the benefit. I now assume that the identity of the recipient is initially uncertain. Uncertainty might be incorporated in several ways, and I have tried to choose the simplest. I assume that when a benefit arrives it wiil be allo­cated among potential recipients (including the donor herself) so as to maximize its discounted value to the donor. Since the benefits in my mode! are very small, I rule out the possibility of distributing a benefit among several recipients. Bcfow, Subsection III-A will incorporate uncertainty into the theory, Subsection UI-B will show how the parameters of this extended theory can be estimated, and Subsection III-C will use de­mographic data from natural-fertility popu­lations to draw inferences about time pref­erence. livity pays for larger families.) (ii) Marginal time- productivity is. however, a declining function of popu­lation size. (This must be so under any form of density-dependent population regnlation.) Thus, (iii) at stationary equilibrium. MRS,, must equal Ihe margi­nal lime-produclivily of intergenerational investment, (iv) This same equilibrium also requires that She con­sumption of a daughter al age 25 equals that which her mother enjoyed at age 25: consequently MRS„ = MRSr. I thank Jack Hirshleifer for this Malthusian interpretation. III. Uncertainty Concerning RecipientsVOL. &) VO. 3 ROGERS: EVOLUTION O h TIME PREFERENCE V69 A. The Evolutionary Discount Function As before, I assume that an altruist allele adds AP(xt) to the donor's survival at age x}, and AP(xt + t) to the recipient's when the donor's age is jr, + t. Now however, the recipient's age, reproductive value, and co­efficient of relationship are all random vari­ables. This requires that the row-products in Table 1 be replaced by their expectations. The altruist allele is selectively neutral when the sum of these expected row-producls is zero. If the duration dx of effects is small, this sum is approximately L:u(x<) AjU(*i) + e"''TE(rt;R(jr2)}A/5(jrl + r) = 0 where E denotes the expectation. In taking this expectation, I define rR(A'2) = 0 when there is no recipient at all. Rearranging gives (13) MRS, AP(xl + t) AP{xt) f'u( -O E{re-"TL,K(jc2)} To facilitate presentation of numerical re­sults, I define an evolutionary discount func­tion A, which satisfies (14) MRSP = exp ,JT + T A( x, iv) dw This formulation is general enough to ac­commodate almost any pattern of discount­ing. For example, when A is a constant, future survival benefits are discounted expo­nentially at a constant rate. On the other hand, hyperbolic discounting (George Ainslie and Nick Haslam, 1992) implies that A is a hyperbolic function of r.13 For example, suppose future benefus are valued at (l+ftr)-1 times their nominal value. Equating this lo expl - f* + 7A(,\. y)dy 1 shows ihal A(j,.v + r) = a /(I + ar). In evolutionary equilibrium, the MRS in fitness must equal that in preferences. Com­bining equations (f) and (14) gives 1 .,r + t (15) 0 = 0( x, x + t) -= - 1 X(x,w)dw. r Jx The average value of A over any interval predicts 6, the MRTP over that interval. Note however, that this rate refers to effects on survival rather than on consumption. Consequently, A is more closely related to the marginal rate of utility discount [rj in equation (2)] than to the MRTP for con­sumption. To define A, note that liquations (13) and (14) imply that (16) E(Z(jc,j£ + t)} ; exp| - j A( x, w) dw where Z(x, x + t) = e~',rn:R(x2)/ uD(x Z measures the beneficial effect of a unit increase in the recipient's survival relative to a unit increase in the donor's. To calcu­late E'(Z}, one must make some assumption about the process by which the recipient is selected. There are several possibilities worth investigating, but I will consider only one: I assume that the benefit is awarded to the singie individual whose value of Z is largest. The problem thus is to take the expectation of Z under this assumption. This expectation depends on the joint dis­tribution of the numbers of relatives of each age and degree of relatedness who are alive at particular times. 1 assume that the num­ber of same-sexed offspring produced at age x by a parent of sex g is a Poisson random variable with mean /«„(*), and that off- a spring of sex g survive independently to age a with probability l/.a). The Appendix de­scribes an algorithm for calculating E{Z} and A(a_, y) under these assumptions. When costs and benefits affect consump­tion, Z is replaced by V = 470 THE AM E R IC A N ECONOMIC R EV IEW JUNE 19Q4 Age Figurf 3. A( '-Specific Fkrtii.itv (Ffmales) Notes: These schedules are normalized so lhal all have ihe same maximum value. Sources: 1906, Taiwan (Hamilton, 196(1)'. standard natural fertility (Coale anil T. .1. Trussell. 1474); 1973 Libya and 19th-ceruury Utah (Mahjoub A. El-Faedy and Lee L. Bean. I9N7). The rest of the analysis proceeds as before, except that it is necessary to know the form of the function Pk[.*,k] and to take the expectation with respect to the joint distri­bution of wealth, age, and relatedness among potential recipients. This task is not attempted here. B. Methods for Estimating the Evolutionary Discount Function The evolutionary discount function de­pends on the population's growth rate p, on male and female fertility schedules [mm(.v) and mf(A-)] and survivorship functions [/m(,v) and /r(;r)]. The relevant values of these parameters are not those of contemporary populations, but those which prevailed dur­ing recent evolutionary history. These past values are, of course, unknown. Fortunately, age-specific rates of human birth and death obey certain regularities that allow strong inferences about A. The first of these regularities concerns the shape of the age-specific (female) fertil­ity schedule in so-called "natural-fertility populations.''14 Although such populations A natural-fertility population is one iri which birlh conlrol is either absent or else is applied independently vary greatly in overall rates of fertility, there is little variation in the shape of the fertility schedule: the ratio of fertilities at any two ages is relatively constant from population to population, as is illustrated by Figure 3. Thus, one can be fairly confident that age- specific fertilities in ancient human popula­tions were roughly proportional to those in contemporary natural-fertility populations. Less is known about male paternity schedules. 1 have found appropriate data for only three natural-fertility populations: those shown in Figure 4. The paternity schedules illustrated there are broadly simi­lar, but they vary more than do fertility schedules of females. Among 19th-century Mormons, paternity peaked when men were in their mid-twenties. In contrast, the peak comes at least a decade later among Turk­men and Libyans. Human survival schedules are also broadly similar, notwithstanding the emphasis that is often placed on variation among them. Coale and Paul Demeny (1983) define four families of model life tables, of which the of the number of a woman's existing children. In natural-ftrlility populations, women may use birlh con­trol lo space hirihs, but they do not use it to achieve a largel family size (Elizabeth A. Cashdan, 19S5').VOL. 84 NO. 3 ROGERS: E VOLUT ION OF TIME PREFERENCE 471 19th lurk Turkmei r'-----' I '---------i------------!------------1-----------1 ---------1------------1--------i 0 10 20 30 40 50 60 70 80 Age Figure 4, Age-Specific Paternity (Mai.es) Notes: These schedules are normalized so lhal all have the same maximum value. Sources: Libya and frontier Utah (El-Faedv and Bean, 1987); Turkmen af Persia (William Irons, 1979). "West" family seems most reasonable as a model for prehistoric human mortality. Within this family, one can choose survivor­ship schedules for a great variety of levels of overall mortality, as measured by e£, the expectation of' life at birth. Finally, 1 must make an assumption about the rate p of population growth during pre­history. Although the human population has been growing for several centuries, its rate of growth prior to this must on average have been very low (McEvedy and Jones, 1978). Thus, I assume that p = 0.15 C. Evidence on Discounting: Inference from Natural-FerliUty Populations Figure 5 presents the evolutionary dis­count function implied by the fertility and paternity data of 19th-century Utah (El-Faedy and Bean, 1987) and the Model '"This implies both that 1 = L^/m(jr)nim(jr) and that I - Er/f( j)mf(.v). To satisfy these requirements, fertil­ity and paternity schedules are rescaled as follows. Let m*(x) denote the schedule of female fertility as given by the data. Prior to entering this schedule into the analysis, I rcscalc it as tn((x)= An analogous rescaling is used for male paternity schedules. West life table with mortality level 12 (Coale and Demeny, 1983 p. 47). This mortality level implies that the expectation of life at birth, eJJ, is approximately 45 years. In the figure, "age at investment" refers to the age at which a decision is made between an immediate and a delayed benefit. Ages be­yond the age at investment are "future ages." Thus, the line marked by open circles shows the discount function pertaining to some investment that might be undertaken by newborn infants,16 whereas the line marked by stars pertains to investments by young adults. To understand what these curves mean, consider a hypothetical 20-year-old woman who has been offered some survival benefit that will not arrive until she is 40. Since she is female and is now age 20, the starred curve in the upper panel of Figure 5 ap­plies. It indicates that the average discount l6The notion of investment by newborns is not quite as preposterous as it sounds. An infant who demanded less of its parents, for example, by crying less or eating less, would allow resources to be diverted from itself to other siblings. To an evolutionist, this constitutes an investment in these siblings. Nonetheless, 1 do noi mean to argue that such investments are of any great importance.472 THE AM E R IC AN ECONOMIC REVIEW JUNE 1Q94 y, future age Figure 5. Evolutionary Discount Function Notes: A(.v, y) is the average evolutionary discount rate wilbin a five-year age interval. "Age al investment." rv, refers to the age al which a decision is made between an immediate and n delayed benefit. "Future age," y, refers to ages beyond the age al investment. The dolled lines show the rate of interest predicted by equation (12). where the generation iime is 7} = 27.98 for females and T„, = .10.45 for males. The figure is based on male and female fertility of 19th-century Utah Mormons (El-Faedy and Bean 1987) and on the Model West life table (mortality level 12. f',',=-47.5 for females and 44.5 lor males) (Coale and Demeny, 1983 p. 47). rates within the four five-year intervals spanning ages 20-40 are 0.059, 0.050, 0.012, and 0.007, respectively. The average of these is 0.032, and this implies that the future benefit should be discounted by a factor of exp[-20x0.032] = 0.529.17 The 20-year-old, therefore, should value this delayed benefit at only about half of its nominal value. In genera!, one applies a MRTP that is an average of A over the relevant interval. ''Equation 05) says that this average is equal to 8 = 0.032. and equation (1) implies that ihe future benefit is discounted by a factor of e~e', where r = 20 is the time delav. This same female should discount a shorter delay at a higher rate. For example, she should discount a delay of five years at an average rate of 0.059, nearly twice the average rate required for a 20-year delay. Figure 5 predicts strong age differences in the MRTP applied to delays of 5-10 years. Such benefits should be discounted only slowly by preteen-age children, very rapidly by young adults, and at intermediate rates by older adults. Figure 5 also suggests that individuals of all ages will apply roughly the same dis­count rate (about 2 percent per year) to benefits that are delayed more than 50 years. This is consistent with equation (12) andVOL. 84 NO. 3 ROGERS: E VOLUTION OF TIME PREFERENCE 473 A(20,;v' 0.0b 0.06 O.n-i -| 0.02 u.oc Bv izi !ji >1 M 11 >lI i : v 0 20 30 40 SO 60 y, futuj-i; ii£<: 70 80 T 90 100 Fn;i id 6. En i r of MoRTAnn'Li vdi on 1Ht Even l-nonary DisrnuNT Func i ion Notes: Shown are evolutionary discount functions for females calculatcd using the 19lh-eenturv Utah fertility and paternity schedules and Ihe Model West mortality schedule, with two extreme levels of mortality. The eurve indicated hy "a" symbols has extremely high mortality: the expectation of life at birlh (rJI) is only 20 years for females and 18 for males. The curve indicated by " v " symbols has exlreinely low moriality (cfj = 80 for females, 76.6 for males). shows that the approximations underlying that equation introduce no serious error. It is interesting that the curves for various ages of altruism lie very nearly atop one another, so that A(x, y) = A*( y). The pat­tern of time preference predicted here is well approximated by a discount function that depends on a single argument. This has an important implication: it precludes re­versals of preference. Reversals of prefer­ence occur when an individual changes his mind about the relative value of two alter­native rewards as the time of their arrival approaches. For example, 1 may plan on Monday to diet on Tuesday, but when Tues­day arrives I may succumb instead to the temptation of a chocolate cake. Empirical work on this issue has relied on theoretical work showing that preference reversals must occur unless future benefits are discounted exponentially at a constant rate (Robert Strotz, 1956; Ainslie, 1975). However, an important limitation of this analysis seems to have gone unnoticed: it assumes that the discount function depends on delay, but not on age. The present model provides a counterexample to the conclusion of Strotz and of Ainslie. To see why, consider two benefits that will arrive at ages y and y + r, respectively. At time x < v the two bene­fits will be discounted by factors of e xp[ - //A*iz)dz] and exp[ - // + 7 A*(z)dz], respectively. The ratio of these discount fac­tors is always exp r A *{z)dz regardless of the value of x. Thus, there can be no reversal of prefere nce even when A* is far from constant. This implies that evi­dence for nonconstant rates of discount cannot by itself account for reversals of preference. To apply the Strotz-Ainslie analysis it is necessary also to show that the pattern of time preference is age-indepen­dent. The value of the results in Figure 5 de­pends on their sensitivity to the underlying demographic parameters. If each set of de­mographic data implies a markedly different evolutionary discount function, then the particular results shown in Figure 5 are of little interest. In Figure 5 the curves for different ages at investment are quite simi­lar. Thus, it is probably sufficient to com­pare populations using a single age at in­vestment, say, 20. This is done in Figure 6, which uses the 19th-century Utah fertility474 THE AME RICAN ECONOMIC REVIEW JUNE 1994 A(20,y) 0,0(5 0.06­0.04 J 0.02­0.00- &lintl*| l-'i'TriliLy 0 inih-s'^'TiiciTy Vtflh jr 3 1 l'< v:i l MM) LrtiV. ill J Sr.niulard NnHiinil ftrtlRy i 111 -i------7----------;----- r ■ 20 30 [0 .j" T r t>U ;/, future (0 8U 90 Id) Fiiii.'ki 7. Eitkt op Fi.krii.iTY Schfuuix on mil Evolutionary DlSf cJ' v I Fl \'< ! 1II- Fi MAI FS NoH'i: f acli curvc uses ;i different schedule of female fertility. All curves use ihe Model Wcsl mortality schedule (level 12) and Ihe I Olh-cenlury Utah Mormon paler- nily schedule. and paternity schedules together with two variants of the Mode] West mortality sched­ule. One schedule assumes extremely high mortality (e'f, = 20 for females, 18 for males), while the other assumes extremely low mor­tality (£•," = 80 for females, 76.6 for males). As the figure shows, mortality has only a minor effect on the evolutionary discount function. The similarity of these curves is remarkable in view of the extremely differ­ent conditions that they represent. The mortality experience of all real populations can be expected to fall within the extremes represented. The small effect of differences in mortal­ity rates is surprising. Intuitively, one might expect discounting to be rapid when mortal­ity is high, since the chances of living to enjoy a delayed benefit arc then decreased. However, the assumption of zero popula­tion grow'th implies that populations with high mortality must also have high fertility. This elevated fertility increases the delayed benefit's effect on fitness. Apparently, these opposing effects balance almost exactly. Figure 7 shows the effect of various fertil­ity schedules on the female evolutionary discount function. The three curves there use identical schedules of survival and of paternity. All three are similar in implying a high MRTP among young adults followed by damped oscillations that approach a long-term rate of about 2 percent. The main differences are in the timing of the peak. The fertility data for Libya and Utah imply that discounting should peak in the late teens or early twenties, while the Taiwan and standard natural-fertility schedules gen­erate a peak in the late twenties. These differences mirror those in the fertility schedules (Fig. 3). Figure 8 shows the cffect of various pater­nity schedules on the evolutionary discount functions of females and of males. All the curves employ identical schedules of sur­vival and of female fertility. They show that, whereas the evolutionary discount functions of females arc insensitive to the paternity schedule, those of males are somewhat more sensitive. The Utah Mormon paternity schedule implies that discount rates should peak when males are in their twenties, while the other schedules imply a somewhat later peak. Without knowing more about the pa­ternity schedules that characterized human evolutionary history, it is difficult to choose between these alternatives. In spite of their differences, the predicted male discount functions arc broadly similar. All imply high rales of time preference by young adults and lower rates among their elders. All con­verge toward rates of roughly 2 percent in old age. In each figure, rates oscillatc up and down with a period of roughly a generation. To understand these oscillations, consider aHr,. Future Age Figure 9. Valuf.s to Ego or Sci.f and or Descendants Nolei: This shows the log of contributions to Ego's reproductive success of hersi lf, a child, a grandchild, and a great-grandchild, assuming that individuals of each genera­tion are horn 25 years apart (calculaled from demographic data for Taiwan, 1906). In the special case when the marginal effect of consumption on survival is the same for the donor and all potential recipients, this is equivalent to the MRS in fitness for con­sumption. Equating the marginal rates of substitution in fitness and in exchange [equation (4)] shows that the real interest rate should be near 2 percent. As before, the result is in good agreement with reality. IV, Discussion and Summary This paper has explored the potential of a new method of economic analysis, that of setting the MRS in fitness equal to those in preferences and in exchange, as first sug­gested by Hansson and Stuart (1990). This procedure is justified by the hypothesis that human preferences are at evolutionary equi­librium: that no mutation changing prefer­ences could be favored by natural selection. The quantitative results derived from this hypothesis all depend on demographic pa­rameters. The relevant parameter values are those which prevailed on average during humans' recent evolutionary history. Since these are unknown, I have instead relied on demographic statistics from modern natural- fertility populations, whose vital rates are thought to resemble those of preindus­trial populations. It would be unwise, how­ever, to take any single modern population as the examplar of our unknown ancestors. We do not know whether prehistoric human demography was more similar to that of 19th-century Taiwan, or that of 19th-cen­tury Utah, to name just two possibilities. Nonetheless, it seems likely that species- wide mean demographic parameters have for a very long time fallen within the range spanned by modern natural-fcrtility popula­tions. And as I have shown, the results are hardly changed regardless of which natural-fertility population one examines. Thus, the results are probably not compro­mised by ignorance about prehistoric de­mography. The results may be summarized as fol­lows: (i) The hypothesis of evolutionary equilib­rium gives definite form to indifference curves that are arbitrary in the classical Fisherian analysis. In addition to the classical indifference diagram, which is drawn on consumption axes, I intro­duce a complementary diagram (Fig. 2) that displays the same information on survival axes. A market line that is lin­ear on consumption axes becomesVOL. H4 NO. 3 ROGERS: EVOLUTION OF TIME PREFERENCE 477 concave'on survival axes. In evolution­ary equilibrium, indifference curves are more convex on consumption axes than on survival axes, but they are convex in either case. (ii) Algebraic results arc obtained by equating two expressions for the MRS in preferences: one derived from the hypothesis of evolutionary equilibrium, and the other from a familiar additively separable intertemporal utility func­tion. The result implies that the marginal utility of dated consumption depends on age as well as on level of consumption. I suggest a formulation in which the subjective rate 17 of utility discount is zero when an investment benefits the investor himself. In the general case, 17 ~(ln/-)/T where 7 is the delay between investment and re­turn on investment and r is the coef­ficient of relationship between the in­" vestor and the recipient. Thus, for a fked degree of relatedness r < 1, the subjective rate of utility discount is in­" versely proportional to 7. (iii) The analysis confirms some classical ideas about the causes of time prefer­ence, but it also suggests revisions. It identifies three factors that lead hu­mans to prefer immediate over delayed benefits: (a) an expectation of rising consumption, (b) declining reproduc­tive value, and (c) the possibility that delayed benefits may accrue to children or other descendants rather than to the investor. Factor (a) is well known to economists (Fisher, 1930 pp. 73-75). Factor (h) combines a familiar and a novel effect. After reproductive matu­rity, reproductive value declines both because of a steady decrease in the expectation of future life and because of a decline with age in fertility. The first of these effects is well known (Fisher, 1930 pp. 84-85), but the effect of declining fertility has apparently not been discussed. Its implications are im­portant, however, for it underlies all of the age effects discussed below. Finally, - factor (c) makes definite what bas heretofore been arbitrary; the strength of the bequest motive. Survival benefits to descendants are discounted by a fac­tor of 2~'\ where d is 1 for offspring, 2 for grandchildren, and so forth. (iv) The hypothesis of evolutionary equilib­rium implies that the marginal rate of time preference (MRTP) will vary both with age and with the length of delay. Young adults should discount at a higher rate than their elders, especially over short delays. The predicted age effect receives support from the com­mon observation that young adults seem to live as if there were no tomorrow, and from the fact that crime rates are high among this age group18 (T. Hirschi and M. Gottfredson, 1983; Walter R. Gove, 1985; Martin Daly and Margo Wilson, 1990). It may also be relevant that empirical studies of college stu­dents have concluded that humans dis­count rapidly (Richard Thaler, 1981; John K. Horowitz, 1991; Gordon C. Winston and Richard G. Woodbury, 1991), while studies of individuals near­ing retirement have reached the oppo­site conclusion (L. J. Kotlikoff et al., 1982; Alan Blinder et al., 1983). (v) The predicted discount functions imply that young adults should discount longer delays at progressively lower rates. This pattern is often reported in empirical studies (Thaler, 1981; Win­ston and Woodbury, 1991) and has been used to explain the phenomenon of preference reversal. Yet no reversals of preference are implied under the pre­sent model. As shown above, noncon­stant discount rates need not imply preference reversals unless the pattern of time preference is age-independent. (vi) The hypothesis of evolutionary equilib­rium implies that the interest rate islt is plausible lo suppose that a high MRTP does, in fact. encourage crime: Ihe benefits of crime are often immediate, and its costs are delayed. This point of view is often expressed in the criminological litera­ture (James Q. Wilson and Richard J. Herrnstein, 1985 p. 205), and is supported by at least one empirical study (F. Buss, 1964).478 THE AMERICAN ECONOMIC REVIEW JUNE 1994 should be close to (17) : = (In 2)/ T + p where T is the generation time and p is the rate of population growth [see equation (12)]. The idea that the real interest rate should depend on the rate of population growth goes back at least to Paul Samuelson (1958), who found an equilibrium at i = p. However, this equilibrium is dynamically unstable in Samuelson's model until money is in­troduced. Hansson and Stuart (1990) find that this equilibrium is stable after all, when preferences evolve by natural selection. Their result differs from equation (17) because they assume (un- realistically) that parents and offspring are genetically identical. The ln(2) in my result is really - In r, where r = \ is the genetic correlation between parent and offspring. When offspring are ge­netically identical to parents, r = 1 and the first term in (17) disappears. This term measures the discounting that oc­curs because human offspring are not genetic replicas of their parents. The two terms on the right side of (17) change on vastly different time scales. The correlation between parents and offspring does not change at all, and large changes in p produce only modest changes in T.19 A large change in T would require evolution­ary change in some parameter such as age at maturity or the rate of reproductive senescence. Thus, important changes in T happen only on an evolutionary time scale. On the other hand, p can change dramati­cally from one decade to the next. Thus, one might expect the interest rate to track the rate of population growth over short time scales. However, I would argue otherwise. Popu­lation growth enters into the present analy­sis onlv via its effect on the evolution of preferences. If preferences respond via ge­netic evolution, then this response will also be slow. This led me to argue above that'the relevant value of p is a long-term average, which will be near zero if human popula­tions grew only slowly prior to the last cou­ple of centuries. The real interest rate should therefore remain relatively constant, near ln(2) per generation or roughly 2 per­cent per year, regardless of the current rate of population growth. If this view is correct, then models relating the interest rate to population growth will not be very helpful. The more important effect is that arising from the fact that offspring are imperfect genetic replicas of their parents. Appendix: An Algorithm for Calculating A(jc,y) I assume that the distribution of the num­ber of offspring of sex g born to an individ­ual of sex £ at age .t (given survival to that age) is Poisson, with mean ms{x), and that each offspring of sex g survives to age a independently with probability lg{a). I also assume a sex ratio of unity, which implies that m((.t) is also the expected number of opposite-sex births. These assumptions im­ply that the numbers of descendants of each later generation born when the ancestor's age is y are also Poisson random variables. Let hj(y \ x) denote the expected number of female descendants of generation i born when the ancestor's age is >\ given that the ancestor lives at least to age x. Since the sex ratio is unity, /!,(y|.t) is also the condition­ally expected number of male descendants of generation i born at this time. Here, children are in generation 1, grandchildren in generation 2, and so forth. Let D denote the sex of the donor, where De(m,f), The conditionally expected num­ber of daughters born when the donor is of age y is ™d(>') ^d( y)/i d( x) if y < x if y > x igSee, for example, Nathan Keyfitz's (1968 table 7.4) where kD(x) ll-1{x)m^(x). An equal num- tabuiation of the mean ;ige of childbearing. ber of sons is also expected at age y.VOL. 84 NO. 3 ROGERS: EVOLUTION OF TIME PREFERENCE 479 Genbrations 2,3,... refer to grandchil­dren, great-grandchildren, and so forth. Fe­male descendants of generation i + 1 are produced by both males and females of generation /. Thus, -/MzUXU.v z)+ kf(y- z)]dz Consider first the expected numbers of descendants of generations 1,2,... (i.e., of genuine descendants excluding the ancestor herself). The number of descendants of sex g, age z, and generation i who are alive when the ancestor's age is y is also Poisson with mean uj = h;{ y - z UysO) '>0 where the subscript / indexes the various feasible values of g, i, and z, given and y. There are only two feasible values of g: m for male descendants and f for females. Since descendants cannot be older than their ancestor, 0 < z < y. I consider only genera­tions ; = 1,2.......5 since I am calculating the discount function only up to an ancestral age of about 100. This is sufficient since children begin to appear only when the an­cestor is about 20, grandchildren when she is about 40, and so forth. None of this applies to the donor herself. The probability that she will be alive to receive the benefit at age y is not Poisson, but Bernoulli, with mean lu(,y)/iu(x). The coefficient of relationship between an ancestor and a descendant in generation i is 2"'. Thus, equation (16) implies that a benefit accruing to a descendant of sex g, generation /, and age z when the donor is of age y would be discounted by a factor of 2; = •'V U) y. Let Z denote the vector whose /th entry is Zj, and u the vector whose /th entry is u,. The expectation of is calculated by the following algorithm: (i) Insert into Z the discount factor for the donor herself: v{y)/ i:(x), and insert a zero in the corresponding position of u. (ii) Sort Z and u by order of decreasing values of Zy. Let j* represent the posi­tion of the donor in the sorted vectors. (iii) Zj is the largest available value with probability exp P. ; exp j -1 -E«. J -r I t* "I - i = I r ) i exp - 52 (I - e "‘) H:)/H x) if j < j* if ; ■ j* Hz)\ t if ./>./* Kx) J where /, as above, indexes all the feasible values of i and 2, for given values of x and The expression for the case where / < j* follows because the distribution of the num­ber of individuals of type / (where / < /*) is Poisson with mean u,. Consequently, the probability of zero individuals of type i is the probability of zero individuals of types 1,2,...,/ - 1 is exp[ -22/and the probability that at least one individual of type / is present is 1 - e~u\ The other cases follow from analogous arguments. Taking the expectation over this distribution gives the discount function, E{Z{xty)) = ZPiZr This completes the algorithm. This algorithm calculates the discount function favored hy natural selection, as­suming that deaths and births are the only factors influencing discounting. Once this discount function has been calculated, it remains to calculate the discount-rate func­tion, A. Typically, demographic data are grouped in five-year age classcs, and it is only possible to calculate the mean value of A within these intervals. The mean discount480 IH E AMERICAN ECONOMIC REVIEW JUNE 1994 rate within a five-year age class is (Al) A( x,y) = - j* +5A(r,w) dw y = - ln{ £{Z( j, y + 5)}/£{Z(.r,.v)}]/5. REFERENCES Ainslie, George. "Spccious Reward: A Behav­ioral Theory of Impulsiveness." Psycho­logical Bulletin, July J975, 52(4), pp, 463-96. Ainslie, George and Haslam, Nick. "Hyperbolic Discounting," in George Loewenstein and Jon Elster, eds., Choice over time. New York: Russell Sage Foundation, 1992, pp. 57-92. Becker, Gary S. "Altruism, Egoism, and Ge- nctic Fitness: Economics and Sociobiol­ogy." Journal of Economic Literature, Septemher 1976, 14(3), pp. 817-26. Becker, Gary S. and Barro, Robert J. "A Refor­mulation of the Economic Theory of Fer­tility." Quarterly Journal of Economics, February 1988, 103(1), pp. [-25. Blinder, Alan; Gordon, R, A, and Wise, D. 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Greenwich, CT: JAI Press, 1991, pp. 325-42.Copyright of /American Economic Review is the property of /American Economic /Association and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.