The Dynamic Cost of the Draft

0 < β < 1, implying diminishing marginal productivity of human capital in the
supply of labor services. Marginal and average labor productivity is independent
of working time and equal to human capital raised to the power of β.8

The individual maximization problem is based on an explicit representation
of the utility function:
where c is consumption of goods, v is demand for leisure, p is the rate of time
preference, 7 is the inverse of the intertemporal elasticity of substitution (σ), a
measures the weight assigned to leisure in the instantaneous utility function, and
the coe∏icienl μ reflects the rate at which the marginal utility of leisure decreases
as the amount of leisure is increased.

The specification of the instantaneous utility function implies that individuals
smoothe consumption over the life cycle, whereas leisure is not necessarily con-
sumed in every period. Marginal utility from leisure is a ∙ (1 — 2μv_ijt),i 2 fc, ng,
where marginal utility is constant if μ is equal to zero. A natural upper bound for
μ is 1∕2e, which implies that the marginal utility of leisure goes to zero if time is
spent entirely on leisure. Contrary to popular constant elasticity of substitution
formulations of the utility function, the additively separable utility function al-
lows the point in time at which the individual begins to retire to be endogenous.
Retirement begins when the individual starts to demand a positive amount of
leisure, which means that less time than the normal work week is devoted to
work and human capital formation.

The quadratic utility function with respect to leisure allows us to capture
two labor market features. First, the demand for leisure is concentrated at the
end of the life cycle. Active labor market participation is phased out at old age
since labor productivity falls. The fall in productivity at old age follows from
intertemporal maximization of lifetime utility. Since the return to investment
in human capital depends on remaining lifetime, human capital accumulation is
concentrated in the beginning of the life cycle and phased out towards the end
of life. Human capital depreciates at a constant rate, and the opportunity cost
to leisure thus declines when learning ends. Second, the model captures the idea
that people typically require a higher marginal wage compensation when leisure
time is scarce.

Individuals are born without financial wealth, and they can save and borrow
without liquidity constraints at the market interest rate, r. The lifetime budget
constraint states that the present value of lifetime expenditures on consumption

⁸ Marginal and average labor productivity will also depend on working time if we use ho-
mothetic Cobb-Douglas or CES specifications. If one of these specifications is applied, labor
productivity may increase at the end of the life cycle when working time decreases.

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