The owner of a rm needs to decide how much to invest to improve its production process.More investment means that the workers are more ecient in generating revenue but is costly. The following game models this investment decision.

The timing of the game is as follows:The game begins with the owner, player 1, choosing an investment level > 0. Two employers, player 2 and 3, who work for the owner observe the chosen investment , and subsequently decide simultaneously on eort levels, e2, e3 > 0.

After the investment and eorts have been decided, revenue for the rm is generated according to the following function:

(, e2, e3) = e2 + e3 + e2e3 .

Notice that the higher is, the more ecient the workers are at converting eort into revenue. Revenue is shared among the three parties in the following way: e [0, 1] proportion of the revenue accrues to the owner while the remaining revenue is split among the workers equally so that each worker obtains (1) proportion of the revenue.

Both investment and eort are costly. The owner incurs a cost of 3 for investment while each worker i incurs a cost ofe by exerting eort ei. Therefore, the utility functions of the players of this game are as follows:

u1(, e2, e3) = (e2 + e3 + e2e3)3 ,

u2(, e2, e3) =(e2 + e3 + e2e3)e ,

u3(, e2, e3) = 1 2 (e2 + e3 + e2e3)e .

Part a: [5 pts] Describe the strategies of each of the players in this extensive form game.

Part b:[5 pts] Suppose that = 1/2.Solve for the NE of the simultaneous move in the proper subgame after the manager has already chosen the investment level to be . (Your solution should be written in terms of )

Part c:[10 pts] Suppose that = 1/2.Using your answer to part b, solve for the SPNE of this game. What is the SPNE strategy prole? What is the SPNE outcome? What are the payos to each of the players in the SPNE?

Part d:[Extra Credit + 10 pts] Solve for the unique SPNE for any > 0.What happens to the manager's prots as increases? Explain intuitively.