Chapter 18, Problem 18.1P

(a)

To determine

The expected utility if shareholder offered to share half of the store’s revenue and the lowest share one would accept to manage the firm.

Explanation of Solution

Given the utility function, $\text{Utility= w-100}$

If a shareholder’s share is 50% of the gross profit with C, then expected utility can be derive by inserting the value of the half of gross profit in C’s utility function, we have

$\begin{array}{l}\text{EU=0}\text{.5}\left(\text{500-100}\right)\text{ + 0}\text{.5}\left(\text{200-100}\right)\hfill \\ \text{EU=0}\text{.5}\left(\text{400}\right)\text{ + 0}\text{.5}\left(\text{100}\right)\hfill \\ \text{EU=200+50}\hfill \\ \text{EU=250 }\left(\text{when half reserve is shared with shareholders}\right)\hfill \end{array}$

Yes, C will accept when shareholder offered to share half of the store revenue with her.

$\begin{array}{l}\text{EU }\left(\text{when receiving quarter share}\right)\hfill \\ \text{EU=0}\text{.5}\left(\text{250-100}\right)\text{ + 0}\text{.5}\left(\text{100-100}\right)\hfill \\ \text{EU=0}\text{.5}\left(\text{150}\right)\hfill \\ \text{EU=75}\hfill \end{array}$

$\begin{array}{l}\text{Lowest share that would accept}:EU\ge 0\hfill \\ 0.5\left(1000k-100\right)+0.5\left(400k-100\right)\ge 0\hfill \\ \text{Where k: Clare’s share in profits}\hfill \\ 700k-100\ge 0\hfill \\ K\ge \frac{1}{7}\hfill \end{array}$

The lowest value C would accept is ${\frac{1}{7}}^{th}$ reserve with shareholders.

(b)

To determine

The most C would pay to buy out the store if shareholders decided to sell it off.

Explanation of Solution

$\begin{array}{l}\text{Maximum C is willing to pay to buy out the store= Expected Value of the store}\hfill \\ \text{=}\frac{\text{1}}{2}\text{×1000+}\frac{\text{1}}{2}\text{×400}\hfill \\ \text{=\$700}\hfill \end{array}$

The maximum amount she is willing to pay to buy is $700.

(c)

To determine

The fixed salary she would accept instead of $100 bonus.

Explanation of Solution

Let fixed salary be M

$\begin{array}{l}0.5\left(100+M+100\right)+0.5\left(M-100\right)\ge 0\hfill \\ M-50\ge 0\hfill \\ M\ge 50\hfill \end{array}$

(d)

To determine

The lowest share that would induce her to exert effort.

Explanation of Solution

1. Let the required share be K

One must have

$\begin{array}{l} EU\left(effort\right)>EU\left(noeffort\right)\hfill \\ 0.5\left(1000K-100\right)+0.5\left(400K-100\right)>400K\hfill \\ 700K-100>400K\hfill \\ 300K>100\hfill \\ K>\frac{1}{3}\hfill \end{array}$

Therefore, the lowest share that she would induce her to exert effort is $\frac{1}{3}$ .

1. Maximum expected profit along with fixed salary and bonus.

$\begin{array}{l}\text{M=Fixed salary, B= Bonus}\hfill \\ \text{Maximize profit }\left(\text{π}\right)\text{= Revenues- fixed salary-expected bonus}\hfill \\ \text{π= 0}\text{.5}\left(\text{1000}\right)\text{ + 0}\text{.5}\left(\text{400}\right)\text{-M-0}\text{.5B}\hfill \\ \text{ π=700-M-0}\text{.5B}\hfill \\ \text{ M+0}\text{.5B=700}\hfill \\ \text{2M+B=1400}\hfill \end{array}$