The owner of a firm needs to decide how much to invest to improve its production process. More investment means that the workers are more efficient 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 effort levels, e₂, e₃ 0.

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

(,e₂,e₃) =e₂+e₃+e₂e₃

Notice that the higher is, the more efficient the workers are at converting effort into revenue. Revenue is shared among the three parties in the following way: [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 1/2(1) proportion of the revenue.

Both investment and effort are costly. The owner incurs a cost of ³ for investmentwhile each worker i incurs a cost of 1/4e²_i by exerting effort e_i. Therefore, the utility functions of the players of this game are as follows:

u1(,e₂,e₃) =(e₂+e₃+e₂e₃)³

u2(,e₂,e₃) =(1)/2 (e₂+e₃+e₂e₃)1/4e²₂

u3(,e₂,e3) =(1)/2 (e₂+e₃+e₂e₃)1/4e²₃

Part a: Describe the strategies of each of the players in this extensive form game.

Part b: 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: Suppose that = 1/2. Using your answer to part b, solve for the SPNE of this game. What is the SPNE strategy profile? What is the SPNE outcome? What are the payoffs to each of the players in the SPNE?

Part d: Solve for the unique SPNE for any >0. What happens to the manager's profits asincreases? Explain intuitively.