Exercise 16.10. In the stochastic permanent income hypothesis model studied in Section 16.5, suppose that c (t) ≥ 0, u (·) is twice continuously differentiable, everywhere strictly concave and strictly increasing, and u00 (·) is increasing. Suppose also that w (t) has a nondegenerate probability distribution. (1) Show that consumption can never converge to a constant level.

(2) * Prove that if u (·) takes the CRRA form and β < (1 + r) −1 , then there exists some a <¯ ∞ such that a (t) ∈ (0, a¯) for all t. (3) * Prove that when β ≤ (1 + r) −1 , there exists no a <¯ ∞ such that a (t) ∈ (0, a¯) for all t. [Hint: consider the case where β = (1 + r) −1 and take the stochastic sequence where w (t) = wN for an arbitrarily large number of periods, which is a positive probability sequence. Then generalize this argument to the case where β ≤ (1 + r) −1 ]. (4) * Prove that when β ≤ (1 + r) −1 , marginal utility of consumption follows a (nondegenerate) subermartingale and therefore consumption must converge to infinity. [Hint: note that in this case (16.24) implies u0 (c (t)) ≥ Eu0 (c (t + 1)) and use this equation to argue that consumption must be increasing “on average”].