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Problem 3: |30 pts + 10 extra] The owner of a firm needs to decide how much to invest o improve its production process. More

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Problem 3: |30 pts + 10 extra] The owner of a firm needs to decide how much to invest o 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 nvestment level o 2 0. Two employers, player 2 and 3, who work for the owner observe or a subsequently decide simultaneously on effort levels, ez, e3 2 0. After the investment and efforts have been decided, revenue for the firm is generated accordi o the following function: 30 TT (@, e2, (3) = de2 + are3 + eze3. Notice that the higher o is, the more efficient the workers are at converting effort into revenue. Revenue is shared among the three parties in the following way: BE [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 - 8) proportion of the revenue. Both investment and effort are costly. The owner incurs a cost of a for investment a while each worker i incurs a cost of te? by exerting effort ex. Therefore, the utility functions of the players of his game are as follows: u1 (0, ez, e3) = (aez + aes + eze3) - 03, 12(0, ez, e3) 2 - -P(aez + aeg + eze3) - 4' u3(@, ez, e3) ) = -P(aez + aes + eze3) - 73. Part a: /10 pts/ Describe the strategies of each of the players in this game. Part b: /10 pts/ Suppose that B = 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 a. Part c: /10 pts/ Suppose that B = 1/2. Solve for the SPNE of this game. What is the SPNE strategy profile? Part d: /Extra Credit + 10 pts/ Solve for the unique SPNE for any S > 0. What happens to the manager's profits as # increases? Explain intuitively

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