A firm has two divisions, each of which has its own manager. Managers of these divisions are paid according to their eort in promoting productivity in their divisions, which is judged by comparison with other managers and other divisions. If both managers are judged as having expended high eort," each earns $150,000/year. If both are judged to have expended low eort," each earns $100,000/year. But if one of the two managers shows high eort" while the other shows low eort," the high eort" manager is paid $150,000 plus a $50,000 bonus, while the second (low effort") manager gets a reduced salary of $80,000. Managers make their eort decisions independently and without knowledge of the other manager's choices.
(a) Assume that expending effort is costless to the managers and draw the payoff table for this game. Find the nash equilibrium of the game and explain whether the game is a prisoners' dilemma.
(b) Now suppose that expending high eort is costly to the mangers (such as a costly signal of quality). In particular, suppose that high eort" costs an equivalent of $60,000/year to a manager that chooses this eort level. Draw the game table for this new version of the game and nd the Nash equilibrium. Explain whether the game is a prisoners' dilemma and how it has changed from the game in part a.
(c) If the cost of high eort is equivalent to $80,000/year, how does the game change from that described in part b? What is the new equilibrium? Explain whether the game is a prisoners' dilemma and how it has changed from the games in part a and b.
(d) Assume now that the owner of the rm can't actually observe manager eort, but instead determines eort based on relative output of the two divisions, meaning that the manager with the division that produces the most is assumed to have provided the highest eort. If the payos are the same in part c, does this change the outcome of the game? If the managers play this game repeatedly, does it matter whether eort is directly observable?