Economic growth and economic development 383

Introduction to Modern Economic Growth
Suppose that u (·) is strictly increasing and strictly concave, with limc→∞ u0 (c) =
0 and limc→0 u0 (c) = ∞, and g (·) is increasing and strictly concave with limx→∞ g 0 (x) =

0 and limx→0 g0 (x) = ∞.

(1) Set up the current value Hamiltonian and derive the Euler equations for an
optimal path.
(2) Show that the standard transversality condition and the Euler equations
are necessary and sufficient for a solution.
(3) Characterize the optimal path of solutions and their limiting behavior.
Exercise 7.22.

(1) In the q-theory of investment, prove that when φ00 (i) = 0

(for all i), investment jumps so that the capital stock reaches its steadystate value k∗ immediately.
(2) Prove that as shown in Figure ??, the curve for (7.62) is downward sloping
in the neighborhood of the steady state.
(3) As an alternative to the diagrammatic analysis of Figure ??, linearize (7.59)
and (7.62), and show that in the neighborhood of the steady state this system has one positive and one negative eigenvalue. Explain why this implies
that optimal investment plans will tend towards the stationary solution
(steady state).
(4) Prove that when k (0) < k∗ , i (0) > i∗ and i (t) ↓ i∗ .

(5) Derive the equations for the q-theory of investment when the adjustment
cost takes the form φ (i/k). How does this affect the steady-state marginal
product of capital?

(6) Derive the optimal equation path when investment is irreversible, in the
sense that we have the additional constraint i˙ ≥ 0.

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