2. Consulting: A University employs a Professor whose base output is a constant K. The Professor chooses to spend effort e consulting, giving her income pe at personal cost C(e) e/2. Unfortunately, consulting lowers the productivity of the Professor; her output at the University is y = K - ye, where y y. Suppose the University cannot contract on consulting hours and simply pays the Professor a constant wage w. The University's profit is then II = y-w while the Professor's utility is U = w+pe-C(e). The Professor has outside option U. = (a) Given wage w, how much consulting does the Professor choose to do? (b) The University chooses the Professor's wage w to maximize profits. In the optimal contract, what is the University's profit II and the Professor's wage w if consulting is allowed? What is the University's profit II and the Professor's wage w consulting is banned? (c) For which values of y, y should the University allow the Professor to do consult- ing? Intuitively, why does the University allow consulting at all if it harms the University? Why does the decision depend on the consulting rate ? (d) We now suppose that the University can contract on the amount of consulting, e, and offers a contract (w, e) consisting of a fixed wage w and consulting effort e. What is the optimal choice of e?