Economic growth and economic development 381

Introduction to Modern Economic Growth
x (0) = 0 and x (1) = 1, where y (t) ∈ R and f is an arbitrary continuously differen-

tiable function. Show that the unique solution to this maximization problem does
not satisfy the necessary conditions in Theorem 7.2. Explain why this is.
Exercise 7.16. * Consider the following maximization problem:
Z 1
x (t)2 dt
max −
x(t),y(t)

0

subject to

x˙ (t) = y (t)2
x (0) = 0 and x (1) = 1, where y (t) ∈ R. Show that there does not exist a continu-

ously differentiable solution to this problem.

Exercise 7.17. Consider the following discounted infinite-horizon maximization
problem
max

Z

0

subject to

∞

¸
∙
1
1/2
2
exp (−ρt) 2y (t) + x (t) dt
2
x˙ (t) = −ρx (t) y (t)

and x (0) = 1.
(1) Show that this problem satisfies all the assumptions of Theorem 7.14.
(2) Set up at the current-value Hamiltonian and derive the necessary conditions, with the costate variable µ (t).
(3) Show that the following is an optimal solution y (t) = 1, x (t) = exp (−ρt),
and µ (t) = exp (ρt) for all t.
(4) Show that this optimal solution violates the condition that limt→∞ exp (−ρt) µ (t),
but satisfies (7.55).
Exercise 7.18. Consider the following optimal growth model without discounting:
Z ∞
[u (c (t)) − u (c∗ )] dt
max
0

subject to

k˙ (t) = f (k (t)) − c (t) − δk (t)
with initial condition k (0) > 0, and c∗ defined as the golden rule consumption level
c∗ = f (k∗ ) − δk∗
367