8. Two young partners at a consulting firm, Bob and Paul, decide to leave the firm and set up their own partnership. They agree to maintain the partnership for one year and to split the profits 50/50. If both work hard, the new firm will make profits of $22 million. If one of the consultants works hard while the other shirks, the firm's profits will be $18 million. If both shirk, the firm's profits will be $14 million. The disutility of working hard is equivalent to $2 million in a year. In this "partnership game", there is (are) Nash equilibrium (equilibria). 9. Consider the following simple principal/agent problem. A firm hires an employee to perform a sales function. Let e denote employee's effort level. Suppose e is not observable by the prin- cipal. Assume that every additional unit of effort increases sales by $200, and the employee's marginal cost of effort satisfies the following equation: mc(e ) = 0 if e 10 The employee's salary is determined as: $600 base salary plus 20 percent commission on sales. Under this salary scheme, the employee will choose the effort level e at , and the firm's profit would be $ 10. Consider the following three-stage duopoly game. There are two firms in the market: Firm I and Firm 2. In the first stage, a first-price auction is conducted to determine the order of moves. The firm with the highest bid (first mover) pays a cost equivalent to its bid and chooses its action in the second stage. After observing this action, the other firm (second mover) chooses its action in the third stage. When there is a tie in the first stage (both firms submit the same bid), a coin is tossed to determine the winner, and only the winner pays its bid. Each firm has the following three possible actions: small (expansion), medium (expansion) and large (expansion), and the payoffs they obtain in the market (not the final payoffs yet) are shown below: The second mover Small Medium Large Small 22, 16 33, 20 17, 29 The first mover Medium 25, 17 18, 13 21, 16 Large 12, 17 10, 15 15, 21 The final payoff is equivalent to the payoff obtained in the market minus the cost (if any) paid in auction. In the unique subgame perfect Nash equilibrium (SPNE) of this three-stage duopoly game, both firms will bid $ in the first-stage auction