A Simple Separation Theorem

Let us start with the partial equilibrium schooling decisions and establish a simple general result, sometimes referred to as a “separation theorem” for human capital investments.

Consider the schooling decision of a single individual facing exogenously given prices for human capital. Throughout, we assume that there are perfect capital markets. The separation theorem referred to in the title of this section will show that, with perfect capital markets, schooling decisions will maximize the net present discounted value of the individual (we return to human capital investments with imperfect capital markets in Chapter 21). In particular, consider an individual with an instantaneous utility function u (c) that satisfies Assumption 3 above. Suppose that the individual has a planning horizon of T (where T = ∞ is allowed), discounts the future at the rate p > 0 and faces a constant flow rate of death equal to ν ≥ 0 (as in the perpetual youth model studied in the previous chapter). Standard arguments imply that the objective function of this individual at time t = 0 is

Now suppose that this individual is born with some human capital h (0) ≥ 0. Suppose that his human capital evolves over time according to the differential equation

where s (t) ∈ [0,1] is the fraction of time that the individual spends for investments in schooling, anddetermines how human capital evolves as a function of

time, the individual’s stock of human capital and schooling decisions. In addition, we can impose a further restriction on schooling decisions, for example,

whereand captures the fact that all schooling may have to be full-time, i.e.,

s (t) ∈ {0,1}, or that there may exist thisother restrictions on schooling decisions.

The individual is assumed to face an exogenous sequence of wage per unit of human capital given by [w (t)]t=θ, so that his labor earnings at time t are

where 1 — s (t) is the fraction of time spent supplying labor to the market and ω (t) is non­human capital labor that the individual may be supplying to the market at time t.

Finally, let us assume that the individual faces a constant (flow) interest rate equal to r on his savings (potentially including annuity payments as discussed in the previous chapter). Using the equation for labor earnings, the lifetime budget constraint of the individual can be written as

The Separation Theorem, which is the subject of this section, can be stated as follows:

Theorem 10.1. (Separation Theorem) Suppose that the instantaneous utility func-

The intuition for this theorem is straightforward: in the presence of perfect capital mar­kets, the best human capital accumulation decisions are those that maximize the lifetime budget set of the individual. Exercise 10.2 shows that this theorem does not hold when there are imperfect capital markets and also does not generalize to the case where leisure is also an argument of the utility function.

10.2.