Professor P has hired a teaching assistant, Mr. A. Professor P wants to maximize her payoff function, x-s, where x is the number of hours taught by Mr. A and s is the total wages she pays him. If Mr. A teaches for x hours and is paid s, his utility is s-c(x) where c(x) =X^2/2. Mr. A's reservation utility is zero, meaning that if he does not take the job, then his payoff will be zero. .
(a) If Professor P chooses x and s to maximize her utility subject to the constraint that Mr. A is willing to work for her, how much teaching will Mr. A be doing?
(b) How much will Professor P have to pay Mr. A? What are their payoffs?
(c) Suppose that Professor P uses a scheme of the following kind . Professor P sets a wage schedule of the form s(x) = ax + b and lets Mr. A choose the number of hours that he wants to work. What values of a and b should Professor P choose so as to maximize her payoff? What are her and Mr. A's payoffs?