Consider the model of searching for ideas introduced in Section 16.5.2. Suppose that the entrepreneur can use any of the techniques he has discovered in the past to produce at any point in time, and he can also stop production at any point and go back to searching.

(a) Formulate the maximization problem of the entrepreneur recursively.

(b) Prove that if the entrepreneur has turned down production with some technique a' at date t, he will never accept technique a' at date t + s, for s > 0 (i.e., he will not accept it for any possible realization of events between dates t and t + s).

(c) Prove that if the entrepreneur accepts technique a' at date t, he will continue to produce with this technique for all datess ≥ t rather than stopping production and going back to searching.

(d) Using parts a and b, show that the maximization problem of the entrepreneur can be formulated as in the text without loss of any generality.

(e) Now suppose that when not producing, the entrepreneur receives income b. Write the recursive formulation for this case and show that as b increases, the cutoff threshold R increases.