than 30 years in human populations. For example, 7 = 28.9 in the 1906 population of Taiwan [11]. Thus, if p ~ 0, selection should favor long-term interest rates that average (In 2)/28.9 = 0.024 per year, in reasonable agreement with observation.
These results are identical to those of my earlier paper on time preference [15, Eqn. 12], and extend those results to the more general context in which selection acts via fertility as well as mortality.
2.3. Diminishing marginal returns to consumption
I now introduce the standard assumptions of economic analysis: that consumption helps in some sense and that each successive unit of consumption helps less than the last. In the present context, this will mean both that m(x) and P(a) each increase with consumption, and also that marginal effects decline as consumption increases.
Although these assumptions are unremarkable in economics, they may seem problematic here. Eating too much can be bad for you, and animals on restricted diets often seem to live longer than those with unrestricted access to food [7, Sec. 10.3.1]. Yet this is no real cause for skepticism: food is just one of many consumer goods, and wealthy people do live longer than poor ones.
To capture the diminishing marginal effect of consumption, I will assume that
m(z,k) = m*(x)K°
P(a,k) = Px (a)KP
where 0 < a, @ < 1, and attention must be restricted to to parameter values such that P stays within the interval [0,1]. Here m*(x) and P +‘) are, respectively, the fertility and survival probability of a “standard” individual of age c—one who consumes a single unit of resource.
To justify this particular formulation, I appeal to the data in figure 4. There, the vertical axis measures the variation of age-specific fertility across populations, and the horizontal axis measures mean age-specific fertility. The graph shows that fertility is most variable at age classes where fertility is high. At least some of this variation must reflect variation in consumption. Thus, it is sensible to build a model in which the effect of consumption is greatest on age classes with high fertility.
Marginal fertility and survival become
Ti, = —m(c, K) (11) ~ Powe. Pe = —P(e,k) (12) and the MRS in fitness is , pr (4) 1 —v)POy® (2) MRS» = (< ) ye +(1— 7)PYv K (13) r ym + (1 = y)PQy@ ] \ 6
where ‘y = a/(a + 3) measures the importance of marginal fertility relative to marginal survival.
3I need to repeat this exercise with survival data.
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