Q. Suppose that a Professor P has recruited a teaching assistant, Mr. A. Professor P needs to increase her payoff function, x-s, where x is the number of hours taught by Mr. A and s is 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) Elucidate how much will Professor P have to pay Mr. A also 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 no. of hours which he needs to work. What values of a and b must Professor P prefer so as to maximize her payoff? What are her and Mr. A's payoffs?