per capita (y).
π = π(y) , — > 0, lim — = 0, lim π = a < 1 . (2)
∂y y→∞ ∂y y→∞
In accordance with the empirical evidence the marginal income effect on child survival vanishes
as income goes to infinity. For later reference we introduce geographic location by the following
definition.
Definition 1. Consider two regions, 1 and 2, for which (2) holds. We say that region 1 is a
geographically unfavorable location if π1 (y) < π2 (y) for all income levels. If the opposite holds,
region 1 constitutes a favorable location.
3. Households
Consider the decision problem of young adults who derive utility from consumption now (c1)
and in old age (c2), from having a family (of size n), and from child expenditure (h). Abstracting
from gender differences any adult is allowed to reproduce without matching. Actual family size,
n ≡ n ∙ π, differs from fertility, n, because child survival is uncertain. To keep the analysis
tractable we consider n as continuous variable. Thus, the parent under investigation can be
regarded as an economy’s average adult who bears n children, spends a fraction h of his income
on each child, and observes a fraction π of them surviving childhood. Survival during adulthood
is assumed to be certain. Summarizing, his or her utility is given by
u(c1,c2, n, h) = βι log(cι - c) + β2 log(c2) + вз log(nπ) + β4 log(h) (3)
for y > c, and c1 = y ≤ c otherwise.
In order to obtain a closed form solution we consider a logarithmic formulation of the utility
function. Yet, because of the incidence of subsistence consumption, c, this assumption is less
restrictive than usually . The intertemporal elasticity of substitution is - in deviation to standard
models - not constant during the process of economic development. It assumes a value of
zero at subsistence level and converges towards one as income and consumption go to infinity.
See, among others, Atkeson and Ogaki (1996) for evidence that the elasticity of intertemporal
substitution rises with increasing income and wealth accumulation.
The incidence of subsistence consumption generates a hierarchy of needs. If income raises from
subsistence level, current consumption becomes less important and young adults increasingly
care for future consumption and about the size of their family. This positive income effect on