Exercise 6.12. Let us return to the problem discussed in Example 6.5. (1) Show that even when

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Exercise 6.12. Let us return to the problem discussed in Example 6.5. (1) Show that even when the sequence of labor earnings {w (t)}∞ t=0 is not constant over time, the result in Exercise 6.11 can be applied to obtain exactly the same result as in Example 6.5. In particular, show that the optimal consumption profile is still given by (6.31). (2) Using the transversality condition together with a (0) and {w (t)}∞ t=0, find an expression implicitly determining the initial level of consumption, c (0). What happens to this level of consumption when a (0) increases?

(3) Consider the special case where u

(c) = ln

c. Provide a closed-form solution for c (0). (4) Next, returning to the general utility function u (·), consider a change in the earnings profile to a new sequence {w˜ (t)}∞ t=0 such that for some T < ∞, w (t) < w˜ (t) for all t

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