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Consider a life-time utility maximizing individual living in two periods. In period one, she
invests in her human capital. The investment is modeled as the time devoted to education, Z,
instead of leisure L, where Z + L is normalized to unity. Effort at school has opportunity costs
in terms of foregone leisure, but is an investment in future income. In period two, she
consumes her return to education. Assuming separability over time, the life-time utility in
expectational terms is

E{V} =u(1-Z) +rE{U(C)}

where r is the discount rate, C is consumption, and utility functions u and U are concave.

The welfare state is an institution that, in period two, transfers a fraction τ of the income from
high income earners to individuals with low income, and thus reduces the consumption
inequality in society. The uncertainty of the representative agent’s future income is captured
by the stochastic return of her education investment (ε), where E(ε) = 0 and Var (ε)=σ² . We
write consumption in period two as

C = (^{1 -τ}) y ( Z)(^{1 + ε}) ^{+ τ}y

where the right hand side is the representative individual’s expected income after
redistribution. The deterministic part of the income is the productivity that depends on effort
in school, y(Z), with diminishing returns (y'(Z) > 0, y*(Z) < 0). Because of income
redistribution only a fraction (1-τ) of the consumption is related to own productivity, y, while
the fraction τ is related to the average productivity in society, y. This formulation implies that
transfer-related income component can be written as τ(y - y (Z)(1 + ε)), and that individuals
with the stochastic productivity y(1+ε) below (above) the societal mean productivity y will
have a positive (negative) transfer. Thus, the redistribution factor τ is an indicator of the
generosity of the welfare state. For simplicity, the implicit taxation and transfer rules are not
written down in the model.

The individual maximizes equation (1) with respect to effort Z subject to constraint (2). The
first order condition is

u'(1 - Z ) = (1 -τ) y'( Z) rE {u'( c)(1 + ε)}.                                                    (3)

In optimum, the marginal cost of effort is equal to the expected marginal return to effort. To
keep the analysis simple, we continue with the quadratic utility function
U₂ ( C ) = αC -βC²∕2. Then the first order condition (3) can be written as

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