s small business has grown to the point where he plans to hire a full-time manag

Question

s small business has grown to the point where he plans to hire a full-time manager Pere, an architect, has ittle nclination and abity to manage a medium-sized, fast-growing business himself He plans to sem-retire, devoting his working hours to consulting on issues project management Pierre's account ng system is quite simple. There is no of design and or other recognition lags. Consequentl, the fim's payoff and its net income for the year are equal Pierre is negotiating with Yvonne as a possible manager. He wants the manager to work hard, since his past experience is that hard work generates a net income (before any manager compensaton) of $2.000 90% of the time and S900 10% of the time, Pierre's recent experience, when he has not worked hard, is that the $2,000 net income is gener- red only 10% of the time; otherwise, net income OS900. During the negotiations, Plerre ascertains that Yvonne is both risk and effort averse er utility for money is equal to the square root of the amount of money received. Her sutlity for effort is 4 if she works hard and 1.1 if she shirks. Her reservation utility is 11. Plerre decides that Yvonne is ideal for the job. He quickly offers her a series of one- ear contracts, with annual cash compensation of a $100 salary plus 10% of net income fore manager compensation. Yvonne immediately accepts equired ote: Take calculations to two decimals Show calculations to demonstrate why Yvonne accepts the position. Which act will she take? After two years, Pierre is worried because net income has been $900 each year. He decides to change Yvonne's next year's compensation contract. After consulting a compensation specialist, he offers her a salary of $52.30 plus a profit share of 9.21% af net income before manager compensation. Yvonne hesitates, but decides to accept Show calculations to demonstrate why she hesitates but accepts. Plerme is risk neutral, with utility equal to the amount of profit received after manager compensation is Pierre's expected utility higher or lower under the new contract in part b, compared to the original contract in part a? Show calculations and explain why there s a difference.

Explanation / Answer

Answer :

a . Calculation to demonstrate whu yvonne accepts the position

Lets donate hard work by "a1" and shirking by "a2" :

Thus Expected utility of Yuvonne each activity will be as under -

EU (a1) = 0.9 * {100+(.10*2000)}.5 + 0.1 * {100+(.10*900)}.5 - 4

= 0.9 * (300).5 + 0.1 (190).5  - 4

= 0.9 * 17.32 + 0.1 * 13.78 - 4

= 15.59 + 1.38 - 4

= 12.97

There for expected utility of Yuvone is 12.97.

Shrink utiity of Yvone :

EU (a2) = 0.1 * 17.32 + 0.9 * 13.78 - 1.1

= 1.73 + 12.40 - 1.1

= 13.03

Thus, Yvonne accepts the postition because her expected utility is greater than as compared to her reservation utility that is 11.

Part B

Lets donate hard work by "a1" and shirking by "a2" :

Thus Expected utility of Yuvonne each activity will be as under -

EU (a1) = 0.9 * {52.30+(.0921*2000)}.5 + 0.1 * {52.30+(.0921*900)}.5 - 4

= 0.9 * (236.5).5 + 0.1 (135.19).5  - 4

= 0.9 * 15.38 + 0.1 * 11.63 - 4

= 13.84 + 1.16 - 4

= 11

There for expected utility of Yuvone is 11.

Shrink utiity of Yvone :

EU (a2) = 0.1 * 15.38 + 0.9 * 11.63 - 1.1

= 1.54 + 10.47 - 1.1

= 10.91

The reason Yvonne is hesitating is because her new utility is lower than the older compensationcontract. She eventually accepts the offer because if she works hard her utility would be 11 thatequals to her reservation utility, so it is acceptable to her.

Part C

Pierre's utility under both contract (past exp)

Old contract: EU= 10%*(2000-100-200)+ 90%*(900-100-90)=809

New contract: EU= 90%*(900-52.3-82.89)+ 10%*(2000-52.3-184.2)= 864.73

Pierre's utility under both contract (recent exp)

Old contract: EU= 90%*(2000-100-200)+ 10%*(900-100-90)= 1601

New contract: EU= 10%*(900-52.3-82.89)+ 90%*(2000-52.3-184.2)= 1663.63

The difference in the utility is because of change in compensation structure of Yvonne.

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