4. Human Capital Accumulation Consider our basic, two-period model of Human
Capital Accumulation. That is, the individual is born with some level of human capital,
H1, and must decide in each period t = 1, 2 what fraction of time they should invest
in human capital accumulation 0 S I; S 1 for t = 1, 2. Individuals only care about
income earned in each period t, which is a function of the wage m = Ht, and the time
spent working (1 — 1;). They discount future utility by the factor 0 < 5 < 1. Human capital accumulates each period according to the following production technology:
Ht+1 = (1— élHt + H3451: (2)
Wherei,b>0,0<7<1,and0<6<1. o (5 points) Part a) Provide a full statement of the optimization problem that
this individual is solving (listing objective function, choices variables you are
optimizing with respect to, any constraints), and solve for the optimal level of
human capital investment in each period (If, and 1;) as functions of the model
parameters. 0 (5 points) Part b) Suppose that there are differences across people in terms of H1.
That is, H], is a random variable in the population, and some people start life
with higher values of H1, and some people start with lower values. Under what
condition on the parameters does starting with a higher stock of initial human
capital, H1 induce workers to make larger human capital investments in the ﬁrst
period? Under what condition does it induces workers to make smaller human
capital investments? Explain why these conditions make sense. a (5 points) Suppose that, instead of a two-period model, we consider a. model with many periods, T. (e.g. T = 50 for a realistic annual model of human
capital investment behavior). A stande empirical ﬁnding is that the relationship between age and and individuals’ wage exhibits a pattern that looks like this: Figure 1
Lifecycle Wages for Six Cohorts
CPS Duh: 1976‘88 -— $8h1976 "momma—Fawn
4—30311975#~¢omm——5omm Explain how the T period model can rationalize this pattern. That is, what
explanation might this model offer for why wages should follow this pattern?