40 - oO 30 7 [o} oO Std. Dev. 20- ce) [e} oO 10 - [o} Qo | | 0 20 40 60 80
Mean
Figure 4: Mean and Standard Deviation of Age-specific Fertility Based on the following sets of fertility data: 1906 Taiwan [11], Standard Natural Fertility [4], 1973 Libya and 19th century Utah [6].
3 Uncertainty about recipients
Thus far, I have assumed that the recipient is known with certainty at the time the investment is made. No allowance has yet been made for the possibility that the benefit may eventually go to someone other than the intended recipient. As in my previous paper on time preference, I will incorporate uncertainty by assuming that when the benefit arrives, it will be allocated among potential recipients (including the donor herself) so as to maximize its discounted value to the donor. As before, I rule out the possibility of distributing the benefit among several recipients. The development below differs from that of my previous paper in two ways. First, it allows the interaction to affect fertility as well as survival. Second, it will incorporate diminishing marginal returns to consumption.
3.1 Model
We begin as before, with table 1. The difference is that, under uncertainty it is not the row-sum itself that must equal zero, but the expected value of this sum. I assume changes in fertility and survival are caused by changes in consumption, as discussed above in section 2.2.2. In addition, I use the model of diminishing marginal returns defined above in section 2.3. Thus, equations 6-7 and 11-12 allow equation 5 to be re-expressed as
0= An (am + BPYY) [KO + Ane?" Bf (am) + BPA) /n (14)
where £ denotes the expectation. In taking this expectation, I define v@) = 0 when there is no recipient at all.
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