Chapter 14, Problem 3E

A.

To determine

To calculate: The total profit function of the firm.

Answer to Problem 3E

The detailed solution with explanation is provided in the next section.

Explanation of Solution

The demand function for two classes of goods is given as;

  ${\text{P}}_1=100-2Q_1$

Total Revenue for the manufactured item is,

  $\begin{array}{l}T{R}_1=\text{Price}\times \text{Quantity}\\ =(100-2Q_1)Q_1\\ =100Q_1-2Q_1{}^2\end{array}$

Total Revenue for semi-manufactured item is,

  $\begin{array}{l}{\text{P}}_2=80-Q_2\\ T{R}_2=\text{Price}\times \text{Quantity}\\ =(80-1Q_2)Q_2\\ =80Q_2-Q_2{}^2\end{array}$

Total Cost function is,

  $\begin{array}{l}TC=20+4(Q_1+Q_2)\\ =20+4Q_1+4Q_2\end{array}$

Total Profit function is,

  $\begin{array}{l}\text{=Total Revenue}-\text{Total Cost}\\ =100Q_1-2Q_1{}^2+80Q_2-Q_2{}^2-20-4Q_1-4Q_2\\ =-20+96Q_1+76Q_2-2Q_1{}^2-Q_2{}^2=-20+96Q_1+76Q_2-2Q_1{}^2-Q_2{}^2\end{array}$

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

B.

To determine

To calculate: The profit maximizing level of price and output for each of the two freight categories.

Answer to Problem 3E

The detailed solution with explanation is given in the next section.

Explanation of Solution

First, differentiate the total profit function with respect to the Q₁

  $\begin{array}{l}M{\pi }_1=\frac{d}{d{Q}_1}(-20+96Q_1+76Q_2-2Q_1{}^2-Q_2{}^2)\\ {\text{=96-4Q}}_1\text{=0}\\ {\text{4Q}}_1\text{=96}\\ {\text{Q}}_1\text{=}\frac{96}{4}\\ \text{=24}\\ {\text{P=80-Q}}_2\\ =80-24\\ =\$56\end{array}$

Now, we shall differentiate the total profit function with respect to the Q₂

  $\begin{array}{l}M{\pi }_2=\frac{d}{d{Q}_2}(-20+96Q_1+76Q_2-2Q_1{}^2-Q_2{}^2)\\ {\text{=76-2Q}}_2\text{=0}\\ {\text{2Q}}_2\text{=76}\\ {\text{Q}}_2\text{=}\frac{76}{2}\\ \text{=38}\\ {\text{P}}_2=80-Q_2\\ =80-38\\ =\$42\end{array}$

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

C.

To determine

To calculate: The marginal revenue for each market at the given levels of output.

Answer to Problem 3E

The detailed solution with explanation is given in the next section.

Explanation of Solution

Marginal Revenue in each market

  $\begin{array}{l}\text{Total Revenue}=100Q_1-2Q_1{}^2{\text{+80Q}}_2-Q_2{}^2\\ {\text{MR}}_1\text{=}\frac{d}{d{Q}_1}{\text{(100Q}}_1-2Q_1{}^2{\text{+80Q}}_2-Q_2{}^2)\\ =-4Q_1\text{+100}\end{array}$

Now, we shall substitute the values of output

First, we shall differentiate the total profit function with respect to the Q₁

  $\begin{array}{l}=-4Q_1\text{+100}\\ =-4(24)+100\\ =-96+100\\ =\$4\end{array}$

Marginal Cost for manufactured goods is $4

  $\begin{array}{l}\text{Total Revenue}=100Q_1-2Q_1{}^2+80Q_2-Q_2{}^2\\ M{R}_2=\frac{d}{d{Q}_2}(100Q_1-2Q_1{}^2+80Q_2-Q_2{}^2)\\ \text{=}-{\text{2Q}}_2+80\end{array}$

Now, we shall substitute the values of output

  $\begin{array}{l}\text{=}-2(38)+80\\ =-76+100\\ \text{=\$4}\end{array}$

Thus, the marginal cost for semi-manufactured goods is $4

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

D.

To determine

To Calculate: The marginal revenue for each market at the given levels of output.

Answer to Problem 3E

The detailed solution with explanation is given in the next section.

Explanation of Solution

Marginal Revenue in each market

  $\begin{array}{l}\text{Total Revenue}=100Q_1-2Q_1{}^2{\text{+80Q}}_2-Q_2{}^2\\ {\text{MR}}_1\text{=}\frac{d}{d{Q}_1}{\text{(100Q}}_1-2Q_1{}^2{\text{+80Q}}_2-Q_2{}^2)\\ =-4Q_1\text{+100}\end{array}$

Now, we shall substitute the values of output

First, we shall differentiate the total profit function with respect to the Q₁

  $\begin{array}{l}=-4Q_1\text{+100}\\ =-4(24)+100\\ =-96+100\\ =\$4\end{array}$

Marginal Cost for manufactured goods is $4

  $\begin{array}{l}\text{Total Revenue}=100Q_1-2Q_1{}^2+80Q_2-Q_2{}^2\\ M{R}_2=\frac{d}{d{Q}_2}(100Q_1-2Q_1{}^2+80Q_2-Q_2{}^2)\\ \text{=}-{\text{2Q}}_2+80\end{array}$

Now, we shall substitute the values of output

  $\begin{array}{l}\text{=}-2(38)+80\\ =-76+100\\ \text{=\$4}\end{array}$

Thus, the marginal cost for semi-manufactured goods is $4

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

E

To determine

To calculate: The profits.

Answer to Problem 3E

The detailed solution with explanation is provided in the next section.

Explanation of Solution

Total Profit function is

  $\begin{array}{l}\text{Total Revenue}-\text{Total Cost}\\ =100Q_1-2Q_1{}^2{\text{+80Q}}_2-Q_2{}^2-20-4Q_1-4Q_2\\ \text{=}-20+96Q_1{\text{+76Q}}_2-2Q_1{}^2-{\text{Q}}_2{}^2\\ $ \text{=}-\text{20+96(24)+76(38)}-2{(24)}^2-{(38)}^2$=-20+2,304+2,888-1,152-1,444\\ =\$2,576\end{array}$

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.

F

To determine

To calculate: The difference in profit levels between differential pricing and uniform pricing cases.

Answer to Problem 3E

The detailed solution with explanation is provided in the next section.

Explanation of Solution

Elasticity of demand manufactured good (E₁)

  $\begin{array}{l}\text{=}\frac{\partial Q_1}{\partial P_1}\times \frac{P_1}{Q_1}\\ {\text{P}}_1{\text{=100-2Q}}_1\\ {\text{Q}}_1\text{=50-}\frac{1}{2}{\text{P}}_1\end{array}$

Now, differentiate with respect to the price

  $\begin{array}{l}\frac{\partial Q_1}{\partial P_1}=-\frac{1}{2}\\ {\text{E}}_1\text{=}\frac{\partial Q_1}{\partial P_1}\times \frac{P_1}{Q_1}\\ =-\frac{1}{2}\times \frac{45.4}{27.3}\\ \text{=-0}\text{.83}\\ \end{array}$

Elasticity of demand manufactured good (E₂)

  $\begin{array}{l}\text{=}\frac{\partial Q_2}{\partial P_2}\times \frac{P_2}{Q_2}\\ {\text{P}}_2{\text{=80-Q}}_2\\ {\text{Q}}_2{\text{=50-P}}_2\end{array}$

Now, differentiate with respect to the price.

  $\begin{array}{l}\frac{\partial Q_2}{\partial P_2}=-1\\ E_2\text{=}\frac{\partial Q_2}{\partial P_2}\times \frac{P_2}{Q_2}\\ \text{=-1}\times \frac{45.4}{34.6}\\ \text{=-1}\text{.3}\end{array}$

Economics Concept Introduction

Introduction: A mathematical relationship between the total profit and output of a firm is called profit function.