Chapter 15, Problem 15.2P

a

To determine

Expected utility if half of the revenue goes to shareholders. Whether contract will be accepted and lowest share of revenue to be accepted to be given to shareholders.

Answer to Problem 15.2P

C will accept the contract because $EU=75$ is greater than the utility from the next best alternative, $U=0$

Explanation of Solution

Given:

C’s utility function: $U=w-100$ and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is

  $U=100$ .

If the shareholders share 50% of the gross profit with C, then expected utility can be derived by inserting the value of half of gross profit in C utility function,

  $\begin{array}{l}EU=0.5\left[500-100\right]+0.5\left[200-100\right]\\ =0.5\left[400\right]+0.5\left[100\right]\\ =200+50\\ =250\end{array}$

C will accept the contract if the expected utility from the contract is greater than the utility derived from the next best alternative which is equal to zero. Since $EU=250$ , therefore contract will be accepted.

If the shareholders share 25% of the gross profit with C, then expected utility will be

  $\begin{array}{l}EU=0.5\left[250-100\right]+0.5\left[100-100\right]\\ =0.5\left[150\right]+0.5\left[0\right]\\ =75+0\\ =75\end{array}$

In this scenario also, contract will be accepted because $EU=75$ is greater than the utility from the next best alternative, $U=0$

The lowest share which C will accept to manage the firm would depend upon whether the provided share yields positive expected utility. The lowest share cab be solved as follows:

  $\begin{array}{l}0.5\left(1,0000s\right)+0.5\left(400s\right)-100\ge 0\\ 500s+200s-100\ge 0\\ 700s-100\ge 0\\ 700s\ge 100\\ s\ge \frac{100}{700}\\ s\ge \frac{1}{7}\end{array}$

Hence, the share which needs to be given in order to make C accept the contract will be

  $s\ge \frac{1}{7}$

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

b)

To determine

Maximum amount paid to shareholders.

Answer to Problem 15.2P

The maximum amount which C will pay to buy out the store is $700.

Explanation of Solution

Given:

C’s utility function: $U=w-100$ and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is $U=100$ .

The highest amount which C will be ready to pay to buy out the store will be the expected value $\left(EV\right)$ of the gross profit which will maximize C’s expected utility, given the probabilities of earning the gross profits of $1,000 and $400 and her utility function.

Expected value can be calculated as follows:

  $\begin{array}{l}EV=0.5\left(1,000\right)+0.5\left(400\right)\\ =500+200\\ =700\\ \end{array}$

Now to see whether $EV=700$ maximizes the expected utility or not, should calculate the expected utility by assuming $EV=700=w$

  $\begin{array}{l}EU=700-100\\ =600\end{array}$

Thus, the expected utility of $EV=700$ is maximum and hence the maximum amount which C will pay to buy out the store is $700.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility.

c)

To determine

Fixed salary to be paid to C.

Answer to Problem 15.2P

C needs to be paid a $fixedsalaryof\$100$ in addition to $100 bonus in order to accept the contract.

Explanation of Solution

Given C’s utility function: $U=w-100$ and there are even chances of earning a gross profit of $1000 and $400. Also, the next best alternative utility function of C is $U=100$ .

Now if C is given a bonus of $100, if the store earns a profit of $1,000,then to calculate the fixed salary of $sx$ which needs to be paid by C so as to induce to accept the contract

  $\begin{array}{l}0.5\left(b+SX\right)-100\ge 0\\ 0.5\left(100+X\right)-100\ge 0\\ 50+0.5X-100\ge 0\\ -50+0.5X\ge 0\\ X\ge \frac{50}{0.5}\\ X\ge 100\\ \end{array}$

Hence, C needs to be paid a $fixedsalaryof\$100$ in addition to $100 bonus in order accept the contract.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction which will be achieved after consumption of certain goods and services. It is estimated utility