Labor supply and efficient contracts.

Consider a labor market with one firm and one worker. The firm is risk neutral. It operates under constant returns to labor and is subject to multiplicative technological shocks s1. Thus, if it uses It hours of labor, its output is s1I1. The worker has utility E(In(Itwt) It) where I is hours of labor, w is the hourly wage, and b is larger than one. I1 can take any nonnegative value.
(a) Characterize the competitive spot market equilibrium graphically and algebraically.
(b) Suppose that an econometrician can observe ?i and Ii in that market. Can he estimate b? Why or why not?
(c) Assume that Si can be observed by both sides so that contracts can be made contingent on Si' Characterize the equilibrium under a symmetric information contract.
(d) Suppose that an econometrician can observe ?i and Ii in that market. Can he estimate b? Why or why not? If you were the econometrician, what additional information would you try to obtain in order to estimate b?