u'(1 - Z ) = (1 -τ) y'( Z) r
α-β((1 -τ)(1 + σ2 ) y ( Z ) + τy )
Calculation of the partial
derivates is straightforward. Regarding the redistribution parameter τ, we have
dZ__J_
dτ D
u(1-τZ)+β(1 -τ) y‘( Z ) r ( y-(1+σ2 ) y ( Z ))
where D = β(1 -τ)2r(1 + σ2)(y'(Z))2 -u"(1 -Z)-y''(Z)u'(1 -Z)/y'(Z)> 0. The first term
in the square bracket of equation (4) reflects that the incentive to invest in education is
reduced when τ rises and the return to education declines. The sign of the second term
depends on the relative income position. For individuals with income below the mean y,
more redistribution increases income and decreases the marginal utility of consumption,
which partially lowers investment in education. For rich people, however, income decreases
and thus the marginal utility of consumption increases, partially working in the direction of
higher investment. In principle, this indirect effect of income redistribution may be so strong
that its total effect on education investment is positive. However, for a representative
individual with income close to mean income,5 the effect of redistribution on her optimal
effort level is negative. Similarly, in a two-country model with two differing social security
systems and migration of labor, Poutvaara (2007) finds a less redistributive, earnings-based
pension system of the target country to increase investment in human capital of prospective
migrants compared to flat-rate pension benefits.
Regarding uncertainty, it follows that
dZ _ β(1 -τ)2ry'( z) y ( z) < 0
(5)
dσ2 D
Increased uncertainty in the return to education, σ2 , decreases investment in education. The
model of, for example, Charles and Luoh (2003) predicts a similar relationship, by showing
that individuals prefer less risky investments, all else equal.
The result that volatility in the return to education reduces education investment is, however,
not universally true, but depends on theoretical assumptions. While our model considers the
investment in effort at school as an asset, education investment may also have similarities
with real options. In a model where education investment is the time devoted to non-
compulsory education, and assuming that, after having left education for the labor market, the
individual cannot return to education, Hogan and Walker (2007) show that investment is
5 Since mean income seems to be higher than median income in all income distributions, it is more reasonable to
assume that the representative individual has income below the mean than above the mean.