A firm has two divisions, each of which has its own manager. Managers of these divisions are paid according to their effort in promoting productivity in their divisions. The payment scheme is based on a comparison of the two outcomes. If both managers have expended “high effort,” each earns $150,000 a year. If both have expended “low effort,” each earns “only” $100,000 a year. But if one of the two managers shows “high effort” whereas the other shows “low effort,” the “high effort” manager is paid $150,000 plus a $50,000 bonus, but the second (“low effort”) manager gets a reduced salary (for subpar performance in comparison with her competition) of $80,000. Managers make their effort decisions independently and without knowledge of the other manager’s choice.
(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 effort is costly to the managers (such as a costly signal of quality). In particular, suppose that “high effort” costs an equivalent of $60,000 a year to a manager who chooses this effort level. Draw the game table for this new version of the game and find 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 effort 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 parts (a) and (b).