savings and fertility is largest close to subsistence level and vanishes as income goes to infinity
and C becomes negligible small relative to income.
Adults supply one unit of labor and receive labor income y . In old age they expect to consume
capital income, c2 = (1 + r)sy, where r denotes the expected interest rate and s is the savings
rate. Thus, their budget constraint is
y = c1 + sy + nhy = c1 + c2/(1 + r) + nhy . (4)
Maximization of (3) subject to (1) and (4) provides the following first oder conditions for an
interior solution.
|
∂u |
∂u = (1 + r)- ∂c2 |
⇒ |
βι r 1 ɪ ʌ β2 ----= = (1 + r)- , C1 — C C2 |
(5a) |
|
∂u ∂n π |
∂u = ;■ hy |
⇒ |
β3 β1 h — =----=hy , n C1 — C |
(5b) |
|
∂u n ∂n |
∂π ∂u ∂u ∂h + ∂h = ∂C1ny |
⇒ |
β3 β4 β1 - (1 - π)λ + ɪ = —ny . |
(5c) |
Condition (5a) equates marginal utility from current and future consumption. Condition (5b)
requires that utility from having another child equates child costs in terms of foregone utility from
consumption. According to condition (5c) utility from spending an additional unit of income
on children - derived directly through higher quality children (or the warm glow of giving) and
indirectly through the impact of health expenditure on family size - equates marginal child costs
in terms of foregone utility from consumption. Holding consumption constant we see from (5b)
that higher child expenditure (h ∙ y) is observed together with the desire for a smaller family
(lower n and higher ∂u∕∂n). This is the Beckerian child quantity-quality trade-off.
The presence of mortality and health expenditure generates a second quantity-quality trade-
off visible in (5c). To understand its consequence consider an increase of fundamental survival
probability π. This has a twofold negative effect on the first term on the left hand side: First,
marginal utility from health expenditure driven by the wish for a large family (∂u∕∂n ∙ n = β3∕π)
decreases, because more children survive anyway. Second, marginal returns of health expenditure
(∂π∕∂h = [1 — π]λ) are lower because child expenditure is less effective in preventing death under
the generally improved survival conditions. As a consequence parents reduce their fertility rate
and the right hand side of (5c) decreases.