A manager (M) and a worker (W) interact as follows: First, the players make a joint decision, in which they select a bonus parameter p and a salary t. The salary can be any number (positive or negative). The bonus parameter p must be between 0 and 1; it is the proportion of the firm’s revenue that the worker gets. The default decision is “no employment,” which yields a payoff of 0 to both players. If the players make an agreement on p and t, then, simultaneously and independently, the worker chooses an effort level x and the manager chooses an effort level y. Assume that x Ú 0 and y Ú 0. The revenue of the firm is then r = 20x + 10y. The worker’s effort cost is x 2 , whereas the manager’s effort cost is y 2 . Each player gets his share of the revenue and his transfer, minus his cost of effort. The players have equal bargaining weights (1>2 and 1>2). The game is depicted in the following illustration:
Compute the negotiation equilibrium of this game by answering the following questions:
(a) Given p and t, calculate the players’ best-response functions and the Nash equilibrium of the effort-selection subgame.
(b) Finish the calculation of the negotiation equilibrium by calculating the maximized joint value of the relationship (call it v*), the surplus, and the players’ equilibrium payoffs. What are the equilibrium values of p, t, x, and y?