Consider the dynamic programming problem that leads to Figure  8.4.

This problem asks you to solve the problem numerically with one change: preferences are logarithmic, so that u(C) = ln C. Specifically, it asks you to approximate the value function by value-function iteration, along the lines of equation (8.73), with V0(X) assumed to equal zero for all X.

(a) As a preliminary, explain why V1(X) = lnX.

(b) Since it is not literally possible to find Vn (X) for every X from 0 to infinity, proceed by discretizing the problem. Choose an N, and define e ≡100/N.

Now, assume that Y can take on only the values

e, 3e, 5e, ... , 200 −

e, each with probability 1/N. Likewise, assume that C can only take on the values

e, 3e, 5e, ... , and find the Vn (X)’s only for X equal to

e, 3e, 5e, ..., up to some upper bound B that you choose (and assume that Vn (X) = Vn (B) for X > B).
Finally, only do some finite number of iterations. (Use whatever programming language or software you wish; MATLAB is a natural candidate.) Plot or sketch the resulting V(•).

(c) Comment briefly on the process of solving the problem numerically. For example, explain why you chose the values of N, B, and the number of iterations that you did. Did you encounter anything unexpected?

(d) Using the value function you found, find C(•), and plot or sketch that.

(e) Compare the C(•) you found with that in Figure 8.4.

What are the main similarities? The main differences?