where « is consumption at age x. Furthermore, I assume that the fertility and mortality effects in the table were produced by changes in consumption. Specifically, the donor’s consumption at age a changed from « to « + Ax, while that of the recipient changed from 5) to 6) +An, If these changes are small, then the fertility and mortality effects are
AP Am
AkP,(2) (6) Akm, (2) (7)
Q
Q
where P,, = OP(z, &)/Ok is the marginal effect of consumption on survival, and m, = Om(z, k)/OK the marginal effect on fertility. Substituting these into equation 5 and rearranging gives the MRS
al (2) (1) 4 POay() Age pr 1) 4 PMylt
MRSp = —— ~~ = (¢ ) Me Tin (8) Ak@) tT) \m? + PP
This generalizes Eqn. 7 of my earlier paper [15], which excluded the marginal effect of consump- tion on fertility.
2.2.3 The long-term real rate of interest
The long-term interest rate is found by setting setting MRSp = e'* (9)
where i is the interest rate over delay 7. This procedure equates the MRS in fitness (the left-hand side of the equation) with that in exchange (the right-hand side), and is justified as follows. The argument in figure 3 shows that, in evolutionary equilibrium, the MRS in fitness must equal that in preferences. Furthermore, in market equilibrium the MRS in preferences must equal that in exchange. In studying equation 9, we are examining the implications of the hypothesis that both equilibrium assumptions hold true.
As in my previous paper on time preference, I concentrate on intergenerational investments in which the investment benefits the investor’s daughter after exactly one generation. By assumption, the mother and daughter are affected at the same age, so that the two reproductive values in MRS; are equal. In stationary equilibrium, the mother and daughter will also have equal wealth at this common age, so that the marginal effects of consumption on their fertility and survival are equal as well. Consequently, the right-most fraction in equation 8 equals unity, and MRS = e’ /r, where r = 1/2 (since the two individuals are mother and daughter), and 7 equals the generation length, T.. Equation 9 becomes 2e°" = e*”, or
i= (In2)/T +p (10)
The relevant rate of population growth is not the current one, but some sort of average rate over re- cent evolutionary history. Since evolutionary changes are usually slow, the last couple of centuries of rapid growth have probably had no large effect. Prior to that, o must on average have been near zero. Thus, equation 10 suggests that 1 ~ (In2)/T. The generation time T’ is usually a little less
HOUSE_OVERSIGHT_011161