In the last term, S is “rescaled” (for notational
convenience) to be the fraction of time spent in
school by the child in time period 1.
The remaining
time during that period, 1 - S, is spent working.
For simplicity, equations (3) and (4) assume no
borrowing or saving.
The only way to transfer
income between periods 1 and 2 is to invest in
children's education.
The ability to borrow or save
would reduce parents’ need to invest in children's
education but would not eliminate it because it is
prudent for parents to diversify their investments
among several different risky alternatives.
Finally, equation (5) relates the child's cognitive
skills to his or her income:
Y
c
=
π
A,
(5)
where
π
is the productivity of cognitive skills in the
labor market.
8

Substitution of (2) into (5), of (5) into (3) and (4), and
finally of (2) - (4) into (1) gives:
U = Y
1
- pS +
δ
Y
2
+ ((1-S+
δ
)k
π
+
σ
)
α
f(Q)g(S)
(6)
A. Exogenous School Quality
At first, assume that school quality is exogenous, so
that S is the only choice variable.
Differentiating (6)
with respect to S, setting the derivative equal to zero,
and totally differentiating this condition gives the
impacts of changes in the model's various parameters
on the optimal (utility-maximizing) value of years of
schooling.
Optimal years of schooling increases (and thus the
cognitive skills acquired by the child increase) when
the following increase:
•
the child's learning efficiency (
α
)
•
school quality (Q)
•
the weight (
δ
) parents give to future
consumption relative to current consumption
•
parental tastes for schooling (
σ
)
9