A classic bargaining problem involves a union and management in contract negotiations. If management hires w ≥ 0 workers, the company produces f (w) revenue units, where f is a continuous, increasing function. The maximum number of workers who are represented by the union is W. A person who is not employed by the company gets a payoff p0 ≥ 0, which is either unemployment benefits or the pay at another job. In negotiations with the union, the firm agrees to the pay level p and to employ 0 ≤ w ≤ W workers. We may consider the payoff functions as
Assume the safety security point is u∗ = 0 for the company and v ∗ = Wp0 for the union.
(a) What is the nonlinear program to find the Nash bargaining solution?
(b) Assuming an interior solution (which means you can find the solution by taking derivatives), show that the solution ( p∗, w∗) of the Nash bargaining solution satisfies
(c) Find the Nash bargaining solution for (w) = ln(w + a) + b, a > 0, 1 a > p0, b > − ln a.