Economic growth and economic development 382

Introduction to Modern Economic Growth
where k∗ is the golden rule capital-labor ratio given by f 0 (k∗ ) = δ.
(1) Set up the Hamiltonian for this problem with costate variable λ (t).
(2) Characterize the solution to this optimal growth program.
(3) Show that the standard transversality condition that limt→∞ λ (t) k (t) = 0
is not satisfied at the optimal solution. Explain why this is the case.
Exercise 7.19. Consider the infinite-horizon optimal control problem given by the
maximization of (7.28) subject to (7.29) and (7.30). Suppose that the problem has
a quasi-stationary structure, so that
f (t, x, y) ≡ β (t) f (x, y)
g (t, x, y) ≡ g (x, y) ,
where β (t) is the discount factor that applies to returns that are an interval of time
t away from the present.
(1) Set up Hamiltonian and characterize the necessary conditions for this problem.
(2) Prove that the solution to this problem is time consistent (meaning that
the solution chosen at some date s cannot be improved upon at some future
date s0 by changing the continuation plans after this date) if and only if
β (t) = exp (−ρt) for some ρ ≥ 0.

(3) Interpret this result and explain in what way the conclusion is different
from that of Lemma 7.1.
Exercise 7.20. Consider the problem of consuming a non-renewable resource in
Example 7.3. Show that the solution outlined their satisfies the stronger transversality condition (7.55).
Exercise 7.21. Consider the following continuous time discounted infinite horizon
problem:
max

Z

∞

exp (−ρt) u (c (t)) dt

0

subject to

x˙ (t) = g (x (t)) − c (t)
with initial condition x (0) > 0.
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