This is an extension of the previous exercise. Consider the following stage game between a manager (also called the "Principal") and a worker (the "Agent").
Let the manager be player 1 and the worker be Player 2. Simultaneously, the manager chooses a bonus payment p E [0, x) and the worker chooses an effort level a E [0, co). The stage-game payoffs are
u1(p, a) = 4a - p and u2(p, a) = p - a2.
(a) Determine the efficient effort level for the worker.
(b) Find the Nash equilibrium of the stage game.
(c) Suppose the stage game is to be played twice (a two-period repeated game) and there is no discounting. Find all of the subgame perfect equilibria.
(d) Suppose the stage game is to be played infinitely many times in succession (an infinitely repeated game) and assume that the players share the discount factor 6 < 1. Find conditions on the discount factor under which there is a subgame perfect equilibrium featuring selection of the efficient effort level in each period (on the equilibrium path).