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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Theor Popul Biol. Author manuscript; available in PMC Mar 1, 2007.
Published in final edited form as:
PMCID: PMC1513193

The Co-evolution of Intergenerational Transfers and Longevity: An Optimal Life History Approach1


How would resources be allocated among fertility, survival, and growth in an optimal life history? The budget constraint assumed by past treatments limits the energy used by each individual at each instant to what it produces at that instant. We consider under what conditions energy transfers from adults, which relax the rigid constraint by permitting energetic dependency and faster growth for the offspring, would be advantageous. In a sense, such transfers permit borrowing and lending across the life history. Higher survival and greater efficiency in energy production at older ages than younger both favor the evolution of transfers. We show that if such transfers are advantageous, then increased survival up to the age of making the transfers must co-evolve with the transfers themselves.

Keywords: evolution, longevity, mortality, intergenerational transfers, life history, optimal energy allocation

1 Introduction

A growing literature seeks the optimal solution to the “general life history problem”, how to allocate resources among fertility, mortality and growth from birth to death. Most optimal life history studies of which we are aware assume that the individual can use only the energy that it produces (forages) in each period, and the life history is optimized subject to this strict budget constraint (Cichon [1997], Cichon and Kozlowski [2000], Vaupel et al [2004], Abrams and Ludwig [1995], Taylor et al. [1974], Goodman [1982], Schaffer [1983], Stearns [1992], Clark and Mangel [2000]; the important exceptions are Kaplan and Robson [2002] and Robson and Kaplan [2003]). But what if individuals were permitted to borrow and lend over their life cycles? Markets for loans do not, of course, exist in nature, but intergenerational transfers from adults to juveniles are common and serve a similar function. Transfers permit a stage of nutritional/energetic dependence early in life with rapid growth and development, followed by a corresponding adult stage of “repayment” in which transfers are made to the young. The strict period-by-period energy constraint is then replaced by a looser version which, in a steady state, requires that the survival-weighted and discounted sum of transfers received minus transfers made over the life cycle must be zero, similar to a life cycle budget constraint with borrowing and lending at an interest rate equal to the population growth rate.3

Many species, including all mammals, most birds, many insects, and some fish and reptiles, make various forms of intergenerational transfers (see Clutton-Brock [1991]). The duration and magnitude of such transfers are extraordinary in the case of human beings and some dolphins and whales, and the longevity of these species (see Carey and Gruenfelder [1997]) motivates our exploration of the possible correlation between intergenerational transfers and the optimal life history strategies. We will consider how the life history changes shape when intergenerational transfers are permitted and confer a selective advantage. Lee (2003) took the existence of transfers as given, and did not consider physiological tradeoffs. In this paper we will examine the conditions under which transfer behavior (parental care) evolves, and consider how mortality co-evolves, when tradeoffs are explicitly modeled through the energy budget constraint.

The analysis we present is formally applicable to cooperative breeders, that is, groups of individuals in which some members across the age spectrum potentially provide food and care to young that are not necessarily their own offspring.4 In such cases, we can imagine a lineage carrying a mutation forming a stable population in aggregate, and living in small cooperatively breeding groups. Each group can be viewed as a microcosm of the lineage, with random departures from its stable age distribution. To obtain analytic results, we need the stable age distribution to write the balancing constraint on transfers.5 Within each group, all members share the same genotype which might include a gene promoting longevity, transfer behavior, or punishment of freeriding, for example. Transfers take place within these groups. Humans are cooperative breeders, and it has been argued that their longevity, particularly in postreproductive years, is related to their transfer behavior.6 There is also evidence (Brown [1987]) that cooperatively breeding bird species live longer than others.

The evolution of altruistic behavior raises difficult questions addressed by a large literature. We acknowledge these difficulties, but here we simply assume that some genotype can solve these difficulties and support transfer behavior. Although humans and other species we have in mind do not reproduce clonally, we believe that our analysis captures the central forces at play.

We begin by considering what life history for a lineage-founding individual would produce the greatest number of living descendants at a specified future date, optimizing subject to the usual budget constraint that does not allow transfers (section 2). We show that the appropriate measure of fitness to be maximized for this individual is the Malthusian parameter. This sets the stage for considering the conditions under which intergenerational transfers would be selected. We investigate when such transfers increase fitness (section 3), and if they do, how low mortality coevolves with them (section 4). The last two sections contain extensions and conclusions.

2 A Model of Optimal Life History

We first consider the case in which transfers are not an evolutionary option. The analysis could be carried out for a life history of potentially unlimited length, but we will instead consider the more realistic case of an individual who is not fertile past age y.7 To avoid the complications of mating and sexual reproduction, we will consider a population of females reproducing asexually. To unify the terminology and notation, we call the age interval [a, a+1) age a+1, and assume that all decisions affecting age a+1 are made at time a. The probability that a person survives from a to age just below a + 1 is denoted pa+1. Fertility at age (a + 1) takes place just before a + 1, conditional on survival, and is denoted ma+1.

At age a, a typical individual expects to have energy or resources which, following Abram and Ludwig (1995), Cichon (1997) and Vaupel et al. (2004), she allocates to fertility (ma), maintenance (pa) and growth (za). We can think of growth as an increase in body size, but we could also think of it as other kinds of physical investment such as development of the brain, as in Kaplan and Robson (2002) and Robson and Kaplan (2003). Because the individual can potentially reproduce in all periods before y, there is a tradeoff between energies devoted to reproduction, growth and maintenance: Having more children early in life comes at the expense of her growth and survival probability, which in turn affects her later fertility.

2.1 The Maximization Problem

The disposable resource or energy of an individual aged a depends on her body size, denoted wa. Specifically, her age-a budget (energy) constraint is written as


where ba, ca, da are constant coefficients, which express the rate at which energy can be used to achieve various levels of survival, fertility or growth. ζa is a production coefficient linking body-size with the net production, or acquisition through foraging, of disposable energy.8 It is easy to see that one of the four coefficients (ba, ca, da, ζa) in (1) is redundant, and so at each age a, we normalize ζa to be 1. This simplifies the expressions in what follows, but note that whenever we need to combine units of energy from different ages, we will have to convert the units appropriately. When this happens we will alert the reader.

The body size of an individual grows according to the following rule: wa+1 [equivalent] wa + za. The initial body size w1 is itself an important intergenerational transfer from the mother. In our analysis, w1 is given, while the adult size is part of the optimization problem through allocation of energy to growth. Thus the ratio of birth size to adult size is endogenous in our analysis. Given our linear homogeneous budget constraint, scale is irrelevant, so only this ratio matters. In this paper our emphasis is on intergenerational transfers occurring after birth.9

We expect that natural selection will maximize reproductive fitness, measured as the representation of an individual’s genes at some future date τ. Since we are assuming clonal reproduction, this is equivalent to maximizing the number of living descendants at some date τ, which may be far beyond the individual’s finite lifespan.10 Consider an individual age a at time t. Let Va,t(.) be her contribution to the number of descendants at time τ. Here t will measure the remaining length of time until τ, when fitness is assessed, so for individuals closer to τ, t will be smaller. Bellman’s (1957) principle of optimality can be used to maximize the expected number of future descendants at τ. According to this principle, energy is allocated at age a and period t so as to maximize the contribution to fitness assessed at τ, assuming that the energy in all future ages and periods is also allocated optimally.

2.2 The Solution

For any a [set membership] {1, 2,…,y}, let the age-a strategy be θa [equivalent] (pa, ma, za) and its feasible set be Ω(wa). For any t, the Bellman equations can be written as follows, for which the interpretation is given in Appendix A.11


We denote the optimum in (2) by θa* = (pa*, ma*, za*). Now, we try to write (2) in terms of V1,t for different t’s. Let [var phi]a [equivalent] (p1*… pa*ma*) be the net maternity function. Starting from the age-y equation, lagging each equation by one period, substituting it into the equation one line above, and iterating the process, we obtain


Manipulating (3) can give us a steady-state optimal solution for θa as well as the corresponding Va(wa). Instead of proceeding in this direction, we shall conform with the literature and apply the results derived by McNamara (1991).

Writing Va,t as Vt(a), McNamara (p.235) transformed the problem of life history in (2) into the following recursive process:


where the operator T connects the state space across different ages. In this age-specific stochastic process, the transformation operator constitutes a Leslie matrix. Then the well-known Perron-Frobenius theorem can be applied to show the existence of a steady state. In that steady state, the value function Vt(a) of all ages grows at a constant rate, which is the dominant eigen value of the Leslie matrix. In particular, we can write V1,t as V1,t = A(λ*)tt. From now on, we shall normalize A to be 1 to simplify the notation.12

Let θ [equivalent]1, …, θy). We rewrite the maximization problem in (2) as


Let la* = p1* … pa* be the probability of survival from birth to age a. In view of the definition of [var phi]a, we can rewrite (4) as


This is the Euler-Lotka equation. Thus, we have

Proposition 1

The solution to the value function in (2) has the form V1,t = (λ*)t, where λ* is the Malthusian parameter solved from (5).

In the analysis above, we derive what a selfish agent that maximizes its own clonal replication would do, and show that the objective to be optimized turns out to be the Malthusian parameter. Explicit characterization of the generic optimization problem in (2) helps us specify and analyze the intergenerational transfers. The setting in (2) also provides us with a decision framework for the comparative static analysis of the next section.

2.3 A Corner Solution Pattern

In reality, many species first grow and then become fertile once they have reached their adult size. They cease growth or grow very slowly once they start bearing offspring. Our main interest is in species that make intergenerational transfers, such as mammals or birds, and these exhibit determinate growth of the sort described. The following proposition, proved in Appendix B, shows that the determinate growth pattern emerges in our model:13

Proposition 2

Values of ma and za cannot both be interior solutions at the same time.

Given the linear energy constraint in (1), proposition 2 is intuitive and also consistent with that found in Taylor et al. (1974), and Vaupel et al. (2004) and is convenient for our later analysis. Biological interepretations and reasons behind the choice of determinate and indeterminate growth can be found in Heino and Kaitala [1999]). Simple differentiation of (2) tells us that any growth in size at age a has the benefit of increasing the number of future offspring at various ages by a constant factor: pa+1/ca+1 at age a + 1, pa+1pa+2/ca+2 at age a + 2, …. Furthermore, in a steady state the value of a new-born at time t is proportional to λt. Thus the steady-state tradeoff between increasing size and bearing offspring is a constant, which depends on the parametric value of ca’s and da’s. Therefore, a corner solution of either ma or za must arise.

Substituting (1) into (2), one sees that the objective function at any age is a concave function of pa. Thus, by restricting the range of the parametric values of (ba, ca, da), an interior solution of pa can often be obtained (see also footnote 7). In what follows, we shall concentrate on analyzing the case in which an organism first grows for r periods, and then stops growing and reproduces. In our notation, the organism would have ma = 0 in the first ar periods, and would have za = 0 when ar + 1.14

3 Optimal Life History and Transfers

In some species, parents invest in their offspring after birth by making transfers of food, guarding against predators, warming or ventilating them, and so on. We shall focus on the most prevalent form of transfers by mammals, when adult individuals aged jr + 1 transfer something to offspring aged ir. We ask when such a transfer would raise the intrinsic growth rate λ, and therefore be selected.

Recall that equation (1) was normalized at each age a by dividing through by ζa Because of this, when we consider transfers between ages i and j, we must use a conversion factor ηij (= ζij). Let the transfer given by an individual at age-j be Tj and the amount received by an age-i be Ri. The demography imposes a feasibility condition on these transfers in a steady state:


where g(.) characterizes the technology for receiving transfers and converting them into the equivalent of energy directly produced by the age-i child recipient,15 with g(0) = 0, and g′(.) > 0. If there is no cost of converting the transferred energy, then g(R) simply equals R. We introduce g(.) to reflect the likelihood that the transfer process becomes less efficient at very high rates. This nonlinearity makes it possible to consider interior optimal transfers.

Because fertility is zero in the first r periods of life, from (1) we have zs = (wsbsps)/ds, sr, si. For the age-i, zi = (wi+Ribipi)/di because the age-i agent receives transfer Ri. And because there is no body growth in periods sr+1, we know from (1) that ms = (wsbsps)/cssr+1, sj. For the age-j, mj = (wjTjbjpj)/cj, because the age-j agent gives a transfer Tj. Finally, since body size does not grow after age r + 1, we have ws = wr+1sr + 1.

With this background information and assuming steady state, equation (4) can be rewritten as


where all ps and ms are evaluated at their optimal values. Note that the value of Ri is implicit in this equation, and is here taken as given and fixed. Note also that the form in which (7) is written assumes that r is given, whereas it is in fact endogenous, and varies as ps, ms and zs vary. For small variations in the neighborhood of the optimum, however, r will not change. To see this, imagine that we carried out many otpimizations of the form of (7), sequentially taking r equal to a for every possible discrete age group a. If we now choose the value of r associated with the greatest maximum value of λ, that will be the optimal r which occurs in (7). Note that this value of r is a function of the level of transfers Ri. However, due to the discreteness of the age groups, for small variations of Ri the value of the optimal r will not change. For this reason, we can differentiate (7) with respect to transfers Ri to determine the effect on λ of a marginal increase in transfers.

Starting from a scenario with no transfers (Ri = 0), we shall evaluate how the steady state selection criterion λ will be affected by the introduction of a marginal transfer. The case of optimal transfers will be discussed briefly later. We now differentiate (4), and use the steady state condition V1,t = λt to obtain16




is the conversion factor between the transfer Tj and the effective value of the transfer received, and


which is the compound factor of accumulating size from age i to maturity (the end of age r). That is, an increment to growth at age i will result in larger body size and increased foraging productivity at age i + 1, which in turn raise body size at age i + 2, and so on. In (8) the coefficient of dλ is negative by the stability condition of λ. In the case when Ri = 0, (8) only needs some minor revision: Gij has to be evaluated at Ri = 0 and g(Ri) = 0.

Substituting the formula for Gij into (8), we have

Proposition 3

The sign of dλ/dRi, which is the selection impact of a marginal transfer from age-j to age-i, is identical to the sign of


On the right hand side of (9), the first term (Ki[.]) is the lifetime expected sum of fertility increase, from age r + 1 to age y, due to the increased body size. Transfers to young individuals lead to larger adult body sizes, which in turn generate more energy for growth and other purposes. The envelope theorem tells us that the net marginal benefit of a change in Ri can be evaluated by the net increase in reproduction. Because ms is weighted by λs, we obtain the first term as shown. The second term captures the lost fertility at age j due to the out-transfer. Term Bij must be positive for the transfer to be selected.

Examination reveals that Bij > 0 is more likely to be met under the following conditions: 1) When there are more age-j adults relative to age-i offspring to share the costs of the transfer (either larger pipj−1, or lower fertility, or both). Higher background mortality (larger coefficients ba) would work against the evolution of transfers. It also follows from (9) that transfers are less likely to evolve in the context of rapid population growth (larger λ), for example for an opportunistic species or under favorable climatic conditions, and more likely to evolve when carrying capacity is saturated. 2) When the adults are relatively more efficient than the child at generating energy per unit body size (smaller ηij [equivalent] ζij). Perhaps this is more likely for carnivores than herbivores, since catching prey requires more skill, speed, strength, and weaponry (teeth, claws). 3) When there is a lower cost to augmenting body size between age i and age j (smaller ds, s = i, …, r in Ki), which makes the investment from adults more rewarding. 4) When survival from age r + 1 to j is high, so that low adult mortality is a predisposing factor for the evolution of transfers, at least up to the age of transferring.17

Transfers might be concentrated on younger offspring or older offspring. In general, the compounded returns to early transfers (larger Ki for smaller i) favor transfers to the young, and the more so when the young convert energy more efficiently into body size (smaller da for small a’s). Transfers to older juveniles might still evolve, if their survival were sufficiently high. Transfers to infants will be more likely if infants are relatively helpless, and unable to forage effectively (low ζi and hence low ηij). This would be more likely true for carnivores but less so for herbivores. If older juvenile productivity relative to body weight increased, then transfers to that age would be less likely to evolve. Finally, a context of more rapid population growth favors transfers to older juveniles (as revealed by the λyi term).

4 Coevolution of Transfers and Longevity

Natural selection should move the life history toward the optimal θs [equivalent] (ps, ms, zs) to maximize the intrinsic growth rate λ. We now switch back to treating transfers, Ri, as given, and consider how the optimal levels of pk depend on the level of transfers, for variations that are small enough such that the optimal age of sexual maturity r, corresponding to the given level of transfers Ri, does not change. We will develop one result (Proposition 4) that holds in the neighborhood of the optimal level of transfers, Ri*, and another result (Proposition 5) that holds when transfers are below this optimal level.

>From the corner-solution pattern presented in section 2.3, it follows that the problem of finding the optimal life history reduces to searching for the optimal pa’s that maximize λ in equation (7). Given that the transfer in question is from age j to age i, it is natural to consider separately the first order conditions for pk when kr and when kr + 1. We shall discuss these cases separately below.

The immature age range corresponds to kr. Differentiating the right hand side of (7) with respect to pk and using the envelope theorem, we see that its first order condition is proportional to the following expression:


where I(k) = 1 if rki, and I(k) = 0 otherwise. The term associated with I(k) is from the differentiation of (6) (dTj/dRi), which is nonzero only if k is in the range between i and j. The Kk factor enters (10) because it is the relevant compound growth factor up to age-k.

The sexually mature range corresponds to kr + 1. Differentiating (7) and using the envelope theorem yields the following first order condition:


where I(k) = 1 only if r + 1 ≤ kj − 1, and I(k) = 0 otherwise. In what follows, we shall ask the following comparative static question: how does the optimal pk change when Ri increases toward its optimum Ri*?

We note that in the ongoing process of evolution, the cumulation of marginal changes should eventually maximize a species’ fitness and hence exhaust the selection advantage of increasing transfers by choosing the optimal Ri* such that


Suppose the optimal transfer from age j to age i, denoted Ri*, has been attained. Totally differentiating (10) we have


If we evaluate the derivative around the optimal Ri*, we know from (12) that dλ = 0. The coefficient of dpk is negative by the second order condition. Thus, we know that as Ri increases towards the optimum Ri*, whether pk moves in the same or opposite direction hinges on the sign of [partial differential]Δpk/[partial differential]Ri.

Partially differentiating (10) with respect to Ri, we get


For rki, I(k) = 1, the last two terms of the above expression cancel, and hence the we see that [partial differential]Δpk/[partial differential]Ri is indeed positive. This means that if Ri increases towards its optimum Ri*, then pk is also increasing for any rki.

Following similar steps (see Appendix C) we see from (11) that for kr + 1, pk moves in the same direction as Ri around the optimum Ri* if and only if [partial differential]Δpk/[partial differential]Ri is positive for r + 1 ≤ kj − 1. Summarizing the above discussion, we have

Proposition 4

Consider a transfer Ri from age j to age i. As Ri increases towards the optimum within the neighborhood of the optimal Ri* that maximizes the fitness index, survival from age i to age j must increase.

What about the evolution of survival before age i? We summarize the result in the following proposition, and the proof is given in Appendix D.

Proposition 5

If an increased transfer from age j to age i improves fitness, the survival probability up to age i must also increase.18

Why is it that Proposition 4 applies only to survival improvements between the age of receiving and the age of giving the transfer, while in Proposition 5 survival also improves at ages from birth to i? Improved survival from age i to j always imparts an efficiency gain when transfers are increased, so it is selected. Increased survival from birth to age i does nothing to conserve the investment in transfers, and in this sense does not impart any efficiency gains when transfers increase. However, it raises the number of births surviving to age i, and thus raises λ, other things equal. In the neighborhood of the optimal transfer, the effect of this increase in survivors to age i is exactly offset by a reduction in transfers per offspring age i, so λ is unaffected, and the survival improvement to i is not selected.

Once an adult is both past the age of providing transfers and no longer fertile, her continuing survival makes no contribution, positive or negative, to reproductive fitness. Mathematically, with respect to a transfer from age j to age i, we can say nothing about the comparative statics with respect to ps for sj.

5 Extension and Discussion

5.1 General Optimal Transfers

So far we have discussed the impact of a transfer from one age j to another age i, but of course transfers may be provided by adults of various ages, and received by children of various ages. The feasibility constraint in (6) need not hold for each (i, j) pair, but rather resources must be balanced over the lifecycle. Specifically, let g(Ri) indicate the energy cost of all transfers received by an individual at age i. The life cycle feasibility constraint is:


similar to that in Lee (2003).

In general it will be optimal for adults of many ages to make transfers, and then the marginal benefit of transferring from each age must be equalized. Likewise, the marginal benefit at each age of receiving must be equalized. As long as we have interior solutions, we should have a system of equations to solve for such optimal transfers. Details will not be provided here, but one should note that the co-evolution result of transfers and longevity we derived in the previous section would not be affected by such complications.

5.2 Selection and Population Density

For a given set of the parameters ba, ca, da and ζa, for all a, there will be some optimal growth rate associated with the optimal life history, and only by chance will it be zero. If the growth rate is positive, then nothing in our model prevents population density from increasing without limit. It is beyond the scope of this paper to consider the dynamic trajectory as density changes. However, we will sketch the way density could be introduced into the model and provided that density is at an unchanging equilibrium level, no change in the analysis would be required.

Density is measured as the total body mass of the population per environmental resource. The main effect of greater density would be to make foraging more difficult and thereby to reduce the energy yield for a given body weight, that is to reduce the coefficients ζa. The conversion of energy into body weight, fertility, or survival as expressed by the other parameters would not be affected to a first approximation, although a more elaborate analysis might permit density to affect mortality (through contagion) and fertility (through limited breeding sites) directly, in addition to the indirect effect through energy production that is now included. So long as the relationship is monotonic, its precise functional form need not concern us. We can simply multiply ζa at every age by some factor that makes the corresponding optimal growth rate λ* equal unity.

When transfer behavior evolves, the founder of the mutant line actually experiences a lower NRR than otherwise, because she foregoes some adult fertility and survival in order to divert energy to caring for her existing offspring, although she received no such care in her youth. In this way she gets the lineage started, and subsequent members of the lineage realize a higher NRR as a result of her initial sacrifice and the improved life history it made possible. Of course, the inclusive reproductive fitness of the first individual is also raised thereby, even though her own NRR is reduced.

5.3 The Linear Technology

We assumed that the budget constraint is linear homogeneous, reflecting a linear tradeoff technology. We could, instead, give equation (1) a nonlinear form, for example by allowing the cost of fertility at age a to be a nonlinear function ca(.) of ma. In this case, second order derivatives would appear in the comparative statics formulas. It would be possible to derive comparative static results if we are willing to assume the sign of these second derivatives,

6 Conclusion

The optimal life history approach seems well suited for exploring the positive selection of life history characteristics. In this paper, we are able to connect formally the optimization problem for an individual life history and the aggregate criterion of the growth rate. This enables us to carry out a comparative static analysis of the effects of these parameters, in contrast to the previous literature which has explored optimal life histories through numerical solutions.

Previous applications of the optimal life history approach have assumed that the individual’s energy budget must balance at every age. Intergenerational transfers replace this instantaneous budget constraint with one that holds over the life cycle. Here we model intergenerational transfers, and ask under what conditions they would improve reproductive fitness and be selected. With transfers, a period of juvenile dependency with more rapid growth and development can be funded by contributions from adults. We consider what features of an initial life history without transfers would make it more likely that intergenerational transfers would confer a selective advantage. Factors favoring the selection of transfers include the ratio of adults to juveniles, greater efficiency of the old relative to the young in producing energy per unit of body size, and the efficiency of juveniles in converting energy into body size. We also discussed the factors favoring transfers to younger versus older offspring, and favoring transfers from older versus younger adults. Because lower mortality favors the selection of transfers from old to young, and because lower mortality coevolves with increased transfers, we find the longevity and transfers should increase in a mutually reenforcing way, as argued in Carey and Judge (2001) and Lee (2003).

When transfer behavior evolves, the founder of the mutant line actually experiences a lower NRR than otherwise, because she foregoes some adult fertility and survival in order to divert energy to caring for her existing offspring, although she received no such care in her youth. In this way she gets the lineage started, and subsequent members of the lineage realize a higher NRR as a result of her initial sacrifice and the improved life history it made possible. Of course, the inclusive reproductive fitness of the first individual is also raised thereby, even though her own NRR is reduced.

A central finding is that on the one hand, lower mortality makes the evolution of transfers more likely, and on the other hand, if increased transfers do evolve then longevity should coevolve. With transfers from adults to juveniles, costly resources are diverted from immediate reproduction to care for existing offspring, and concurrent life history investments in reducing mortality serve to protect these investments.

Mathematical Appendix

Part A

The interpretation of equation (2) is as follows: p1m1 in the first term on the right hand side of (2) characterizes the event that an age-1 individual survives (with probability p1) and bears m1 offspring. Since each of these offspring is valued V1,t−1 in period t − 1 (because the offspring is one period closer to τ), V1,t−1 should be multiplied by p1m1 to obtain the expected value. The V2,t−1 in the second term of (2) is the value function of this individual at age-2. With probability p1 the individual will survive to face this state, and so V2,t−1 should be multiplied by probability p1. The age-2 body size should be w2 = w1 + z1 instead of w1. The interpretations of other expressions are similar, so we move forward to the last equation. For an individual aged y in period t, py and my are chosen to maximize the expected value of the last birth. At age y, there is no gain from further growth. This generates the age-y expected value pymyV1,t−1. Since y is the last fertile age by assumption, there is no second term for the last equation.

Part B

Let g(α) [equivalent] maxx f (x, α). The envelope theorem (see Simon [1976]) says that when x has an interior solution, dg/dα = [partial differential]g/[partial differential]α around the neighborhood of the maximum, for the indirect effect through x is absorbed by the first order condition of x. Applying the envelope theorem to (2), we see that


For the age-a problem, concerning the trade-off between ma and za, we have the following first order condition (in terms of economics, MRS equals price ratio) for an interior solution:


In the steady state, V1,t is a constant of power λ for all t, as shown in the text, and so the above expression can be further rewritten as


Canceling pa* in the numerator and the denominator of [A2], we see that both sides of [A2] are not dependent on any age-a choice variable. Thus, expression [A2] could hold only by accident in a steady state, which in turn implies that ma and za cannot be interior solutions at the same time.

Part C

Partially differentiating Δpk in (11) with respect to Ri, we have


For r+1 ≤ kj −1, I(k) = 1, the last two terms in the above expression cancel, and hence [partial differential]Δ pk/[partial differential]Ri s positive. As such, we know that pk and Ri also move in the same direction for r + 1 ≤ kj − 1 as Ri moves toward the optimum, Ri*

Part D

For ki − 1, the first order condition for pk is similar to that in (10), except that I(k) = 0 ∀ki −1:


Partially differentiating the above expression with respect to Ri yields


where Bij is given in (9). We know that dλ/dRi = 0, and hence Bij = 0 around the optimum Ri*. Thus, for ki − 1, dpk/dRi is close to zero around the optimum transfer. But we are able to say more about the change of pk in the process when Ri increases toward Ri*

Equation (13) says that for any dRi, the sign of dpk/dRi is the same as that of [[partial differential]Δpk/[partial differential]Ri] + [[partial differential]Δpk/[partial differential]λ] · [dλ/dRi]. Partially differentiating [partial differential]Δpk with respect to λ, using (10) to simplify the result, and substituting in the dλ/dRi formula, we have






It is easy to see that terms in the square brackets of (A3) are positive. We showed in section 3 that whenever the increase of Ri improves fitness, Bij must be positive. Thus, as Ri moves toward the optimum Ri* to improve fitness, Bij > 0 must hold in the process, which means that dpk/dRi > 0 ∀ki − 1. Thus, we have proved proposition 5 in the text.


2Lee’s research for this paper was funded by NIA grants R37-AG11761 and P01 AG022500-01. Marc Mangel, David Steinsaltz, conference participants at the Max Planck Insitute for Demographic Research at Rostock, and two anonymous referees provided useful comments on an earlier draft.

3Some transfers take the form of stored output, for example paralyzed prey, and therefore occur with a delay. In steady state, this constraint will still hold: the survival-weighted and discounted transfers made and received over the life cycle must be equal. Out of steady state, the budget constraint would be more complicated to accommodate storage.

4Whether or not cooperative breeding evolved to facilitate intergenerational transfers, the demography of cooperative breeding groups provides an analytic setting in which stable population methods can be appropriately used.

5The analysis for transfers within parent-offspring sets is more difficult, because their age distributions cannot plausibly be viewed as microcosms of the lineage. However, the technical difficulties in analyzing the parent-offspring case do not seem to point to substantive differences in the explanation of transfer behavior between this and the cooperative breeding contexts.

6See Clutton-Brock (1991), Kaplan and Robson (2002), Lee (2003) and Lahdenpera et al. (2004).

7It is not necessary to assume that fertility is 0 past some age y. However, absent this assumption, fertility and survival would never reach zero in our optimization setup. As long as fertility and survival are not infinitely costly, death will never be optimal in our model setup. This is because fertility ma occurs only after survival pa, so it can never be optimal to spend all energy on fertility at some age and none on survival. For this reason, our analysis focuses on survival rather than on life span. We could define the end of life as that age at which the probability of survival to the next period drops below some specified level, such as .001. Perhaps for similar reasons, Cichon and Kozlowski (2000) adopted this approach.

8Note that allowing the coefficients (b, c, d) to be age-dependent is just a general way to write down the energy constraints. The comparative statics results we show later in Propositions 3 and 4 do not depend on the relative size of such coefficients, although the age-specific life-history trajectories do. See Chu and Lee (2005) for more discussion of the shape of optimal age-specific mortality.

9The linear form of the budget constraint, as in Taylor et al. (1974) and Vaupel et al. (2004), is to some extent restrictive. Appropriate nonlinear effects would include an upper bound of unity for pa, with increasing costs as this limit is approached; a dependence of pa on body weight wa; and a dependence of fertility on body weight.

10As we shall see, once the population reaches steady state, the fitness measure is also stable. For this reason, τ should exceed the approximate number of periods from the time in question until the steady state is reached.

11See Ross (1983) for more details of the technical background for (2).

12Note that this result could also be derived from the stationary difference equation in (3). There is, however, a subtle difference between our setup and that of McNamara’s (1991). In the latter paper, body growth was not a choice variable, whereas in our setting it is. Therefore, the state-space in our model contains both age and age-specific weight, instead of age alone. However, convergence toward a steady state can still be obtained in our setting; details are skipped here.

13Suppose with effort za there is probability q(za) of achieving size wa + g1, and probability 1 − q(za) of achieving size wa + g2. In this case, our corner-solution argument will not hold. This may correspond to the case of indeterminate growth. An interior solution could also arise if the rates of converting energy into fertility and into body size were not constant, but rather varied with the amount of energy so converted. See Stearns (1992) and Taylor et al. (1974).

14For instance, when y = 3 (three periods of life) and r = 1 (the first period being childhood), the condition is d1λ2 < c1p2/c2 + p3/c3) and d2λ > c2p3/c3, according to Appendix B.

15We characterize the transfer by its energy cost to the individual making the transfer. The function g(.) should have a subscript i to indicate that this is a function specific to age-i. However, since our analysis applies to any unspecified i, for the time being we drop this subscript i for simplicity of notation.

16Note that the derivatives of the right hand side of (4) with respect to ps (s = 1,…, y) are zeros due to the first order conditions of maximizing over ps, hence these terms do not appear in the differentiation. This is again an application of the envelope theorem.

17This can be seen by canceling the (prpj−1) term in the numerator and denomenator of the second term of (9), and inspecting the remaining terms in Bij.

18As Ri approaches Ri*, [partial differential]pk/[partial differential]Ri → 0, ∀ki − 1.

Contributor Information

C. Y. Cyrus Chu, Institute of Economics, Academia Sinica, email: wt.ude.acinis.etag@uhcsuryc.

Ronald D. Lee, Department of Demography, University of California, Berkeley, email: ude.yelekreb.gomed@eelr.


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