Consider another version of the game between Carina and Wendy, where Carina selects any effort level e on the job. Assume e = 0. The revenue of the firm is equal to $800e. Carina's disutility of effort is e2 (measured in dollars). Carina and Wendy interact as before; first, they jointly determine Carina's compensation package, and then (if they agree) Carina selects her level of effort.
(a) Suppose Carina's effort is not verifiable, so Carina and Wendy can write a contract specifying only a salary t for Carina. Assume Carina and Wendy have equal bargaining weights. Draw the extensive-form game and compute the negotiation equilibrium. Does Carina expend effort?
compensation package states that Carina gets a fraction x of the revenue of the firm [leaving the fraction (1 - x) to Wendy]. Calculate Carina's optimal effort level as a function of x. Then, under the assumption that Wendy has all of the bargaining power, calculate the value of x that maximizes Wendy's payoff. (Do not graph the bargaining set; this setting does not fit very well into our bargaining theory. Technically, this is an application of the generalized Nash bargaining solution [see Appendix D].)
(c) Now suppose that the contract is of the form w(p) = xp + t, where w is the amount paid to Carina and p is the revenue of the firm. That is, the contract specifies that Carina receive some base salary t and, in addition, a fraction x of the firm's revenue. Assume the players have equal bargaining weights. Calculate the negotiation equilibrium of this game. (Start by finding Carina's optimal effort decision, given t and x. Then, holding t fixed, determine the number x that maximizes the players' joint value. Finally, determine the players' negotiation values and find the salary t that achieves this split of the surplus.)