Chapter B, Problem 9E

a.

To determine

To evaluate the if the price, output, total revenue, total cost and total profit level will have maximizing franchise experience.

Explanation of Solution

Given information:

Given demand function is

  $P=\$50-0.001Q$

Marginal Cost (MC) = $6

Fixed cost = $300,000

  $\begin{array}{l}TR=P\times Q\\ =\left(\$50-0.001Q\right)\times Q\\ =\boxed{50Q-0.001Q^2}\end{array}$

Therefore, $TR=\boxed{50Q-0.001Q^2}$

  $\begin{array}{c}MR=\frac{d\left(TR\right)}{d\left(Q\right)}\\ =\frac{d\left(50Q-0.001Q^2\right)}{d\left(Q\right)}\\ =\frac{d\left(50Q\right)}{d\left(Q\right)}-\frac{d\left(0.001Q^2\right)}{d\left(Q\right)}\\ =\boxed{50-0.002Q}\end{array}$

Therefore, $MR=\boxed{50-0.002Q}$

  $\begin{array}{c}MR=MC\\ 50-0.002Q=6\\ -0.002Q=6-50\\ -0.002Q=-44\end{array}$

  $\begin{array}{c}Q=\frac{-44}{-0.002}\\ =\boxed{22,000}\end{array}$

Therefore, the output is 22,000 pound of cookies

  $\begin{array}{c}P=\$50-0.001Q\\ =50-0.001\left(22,000\right)\\ =50-22\\ =\boxed{\$28}\end{array}$

Therefore, the price per pound of cookie is P = $28

  $\begin{array}{c}TR=P\times Q\\ =28\times 22,000\\ =\boxed{\$616,000}\end{array}$

  $\begin{array}{c}TC=VC+FC\\ TC=\left(Q\times MC\right)+FC\\ =\left(22000\times 6\right)+300000\\ =\$132000+\$300000\\ =\boxed{\$432000}\end{array}$

  $\begin{array}{c}\mathrm{Pr}ofit\left(\pi \right)=TR-TC\\ =\$616000-\$432000\\ =\boxed{\$184000}\end{array}$

Economics Concept Introduction

Introduction:In economics, total cost is the amount of all the costs incurred by a business to achieve a certain level of production. It is expressed usually as a sum of all fixed costs.

b.

To determine

To evaluate the if the parent company charges each franchisee a fee equal to 5% of total revenues along with computation of the values of price, output, total revenue, total cost and total profit.

Explanation of Solution

  $\begin{array}{c}T{R}_1=P\times Q\times \left(1-5\%\right)\\ =\left(\$50-0.001Q\right)\times Q\times 95\%\\ =\left(\$50Q-0.001Q^2\right)\times 0.95\\ =47.5Q-0.00095Q^2\end{array}$

  $T{R}_1=\boxed{47.5Q-0.00095Q^2}$

  $\begin{array}{c}M{R}_1=\frac{d\left(T{R}_1\right)}{d\left(Q\right)}\\ =\frac{d\left(47.5Q-0.00095Q^2\right)}{d\left(Q\right)}\\ =\frac{d\left(47.5Q\right)}{d\left(Q\right)}-\frac{d\left(0.00095Q^2\right)}{d\left(Q\right)}\\ =47.5-0.0019Q\end{array}$

Therefore, $M{R}_1=\boxed{47.5-0.0019Q}$

  $\begin{array}{l}MR=MC\\ 47.5-0.001Q=6\\ -0.0019Q=6-47.5\\ -0.0019Q=-41.5\end{array}$

  $\begin{array}{c}Q=\frac{-41.5}{-0.0019}\\ =\boxed{21842}\end{array}$

So the output is 21842 pound of cookies.

  $\begin{array}{c}P=\$50-0.001Q\\ =50-0.001\left(21842\right)\\ =50-21.842\\ =\boxed{\$28.158}\end{array}$

Therefore, the price per pound of cookie is P = $28.16

  $\begin{array}{c}T{R}_1=P\times Q\times 95\%\\ =28.16\times 21842\times 0.95\\ =\boxed{\$584,317.184}\end{array}$

  $\begin{array}{c}T{C}_1=VC+FC\\ T{C}_1=\left(Q\times MC\right)+FC\\ =\left(21842\times 6\right)+300000\\ =\$131052+\$300000\\ =\boxed{\$431052}\end{array}$

  $\begin{array}{c}\pi =T{R}_1-T{C}_1\\ =\$584317.184-\$431052\\ =\boxed{\$153265184}\end{array}$

Economics Concept Introduction

Introduction: In economics, total cost is the amount of all the costs incurred by a business to achieve a certain level of production. It is expressed usually as a sum of all fixed costs.

c.

To determine

To evaluate the if the parent company considers a fixed franchise fee structure along with computation of the values of price, output, total revenue, total cost and total profit.

Explanation of Solution

Initial fixed price is $300,000. Now adding $25,000 to the fixed price is $325,000

  $\begin{array}{c}TC=VC+FC\\ T{C}_2=\left(Q\times MC\right)+FC\\ =\boxed{6Q+325000}\end{array}$

  $\begin{array}{c}T{R}_2=P\times Q\times \left(1-1\%\right)\\ \left(\$50-0.001Q\right)\times Q\times 99\%\\ =\left(\$50Q-0.001Q^2\right)\times 0.99\\ =\boxed{49.5Q-0.00099Q^2}\end{array}$

  $\begin{array}{c}M{R}_2=\frac{d\left(T{R}_2\right)}{d\left(Q\right)}\\ =\frac{d\left(49.5Q-0.00099Q^2\right)}{d\left(Q\right)}\\ =\frac{d\left(49.5Q\right)}{d\left(Q\right)}-\frac{d\left(0.00099Q^2\right)}{d\left(Q\right)}\\ =\boxed{49.5-0.00198Q}\end{array}$

  $\begin{array}{c}MR=MC\\ 45.5-0.00198Q=6\\ -0.00198Q=6-49.5\\ -0.00198Q=-43.5\\ Q=\frac{-43.5}{-0.00198}\\ =\boxed{21969}\end{array}$

Therefore, the output is $\boxed{21969}$ pound of cookies.

  $\begin{array}{c}P=\$50-0.001Q\\ =50-0.001\left(21969\right)\\ =50-21.969\\ =\boxed{\$28.04}\end{array}$

Therefore, the price per pound of cookie is $\boxed{\$28.04}$ .

  $\begin{array}{c}T{R}_2=P\times Q\times 99\%\\ =28.04\times 21969\times 0.99\\ =\boxed{\$609850}\end{array}$

  $\begin{array}{l}T{C}_2=VC+FC\\ T{C}_2=\left(Q\times MC\right)+FC\\ =\left(21969\times 6\right)+325000\\ =\$131814+\$325000\\ =\boxed{\$456814}\end{array}$

  $\begin{array}{c}\pi =T{R}_2-T{C}_2\\ =\$609850-\$456814\\ =\boxed{\$153036}\end{array}$

Economics Concept Introduction

Introduction: In economics, total cost is the amount of all the costs incurred by a business to achieve a certain level of production. It is expressed usually as a sum of all fixed costs.

d.

To determine

To evaluate the fee arrangement recommended to be adopted by the parent company.

Explanation of Solution

With 5% fee arrangement, overall profit is $153,265. It can receive per unit of $30,753 $\left(\$28.16\times 21,842\times 0.05\right)$ . With fixes fee arrangement, overall profit is $153,036. It can receive per unit fee of $31,160 $\left(25000+\left(\$28.04\times 21969\times 0.01\right)\right)$ . Therefore, it is suggested that fixed fee structure gives better fee per unit.

Economics Concept Introduction

Introduction: In economics, total cost is the amount of all the costs incurred by a business to achieve a certain level of production. It is expressed usually as a sum of all fixed costs