Exercise 7.24. (1) In the q-theory of investment, prove that when 00 (i)=0 (for all i), investment

Exercise 7.24. (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 steady-state value k∗ immediately. (2) Prove that as shown in Figure 7.1, the curve for (7.64) is downward sloping in the neighborhood of the steady state. (3) As an alternative to the diagrammatic analysis of Figure 7.1, linearize (7.61) and (7.64), 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.