Exercise 6.7. Let us return to Example 6.4. (1) Prove that the law of motion of capital stock given by 6.28 monotonically converges to a unique steady state value of k∗ starting with any k0 > 0. What happens to the level of consumption along the transition path? (2) Now suppose that instead of (6.28), you hypothesize that π (x) = axα + bx +

c. Verify that the same steps will lead to the conclusion that b = c = 0 and a = βa. (3) Now let us characterize the explicit solution by guessing and verifying the form of the value function. In particular, make the following guess: V (x) = A ln x, and using this together with the first-order conditions derive the explicit form solution.