Economics 202A Midterm Exam
October 22, 2013
Instructions: You have 1 hour 20 minutes for this exam. Take the …rst few minutes
to look it over, and pace yourself. Don’t panic; the problems are easier than they
look. (Really.) If you get stuck, move on to something else and come back to the
di¢ cult bit later if you have time. Each question is worth 50 points.
1. A Ramsey model with human capital. Assume a constant-returns pro-
duction function of the form Y=F(K; H; L)where Hdenotes the stock
of human capital. The labor force grows according to _
L=nL; and letting
lower case letters denote ratios to Las usual, we shall write y=f(k; h)
F(K=L; H=L; 1). Technology does not change over time.
(a) A fraction sof saving goes into accumulating Kand a fraction 1s
goes into accumulating H: If both types of capital depreciate at the same
rate , then:
_
K=s[F(K; H; L)C]K;
_
H= (1 s) [F(K; H; L)C]H:
Show how to write these two equations in terms of k; h; per capita consump-
tion c=C=L; and the "intensive" production function f(k; h):
(b) We would like to endogenize the share sabove (in principle, it need
not be constant over time without further assumptions). To that end, set up
the optimal control problem of maximizing
Z1
0
u[c(t)]e(n)tdt
(where > n), subject to your equations from part (a) for _
kand _
h, as well
as initial conditions for physical and human capital. What are the control
variables and what are the state variables?
(c) Write down the Hamiltonian for the optimal control problem, letting
kand hdenote the costate variables for the two accumulation constraints.
What are the …rst-order conditions for the controls, and what do you learn
from them? Explain your answer intuitively. (To economize on writing e¤ort,
denote @F=@K =@f=@k =fk, and likewise for fh. Also, recall I pointed
out in class that everything I said about the Maximum Principle applies to
vectors of control and state variables.)
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