Chapter 8, Problem 15PAA

To determine

To know: The profit-maximizing amounts of electricity, optimal price and company’s profit.

Explanation of Solution

Electricity is produced by two facility which is public utility.

Given is the inverse demand function:

  $\text{P}\text{=}\text{1200-4Q}$

To find the marginal revenue of a public utility, derivation is done for total revenue.

  $\begin{array}{l}P=1200-4Q\\ TR=PQ\\ =1200Q-4Q^2\\ MR=\frac{\partial TR}{\partial Q}\\ =1200-8Q\end{array}$

Cost function of facility 1 for producing electricity:

  $C_1(Q_1)=8000+6Q_1^2$

Marginal cost of facility is given as follows:

  ${\text{MC}}_{\text{1}}{\text{(Q}}_{\text{1}}\text{)}\text{=}{\text{12Q}}_{\text{1}}$

Cost function of facility 2 for producing electricity:

  $C_2(Q_2)=6000+3Q_2^2$

Marginal cost of facility is given as follows:

  $M{C}_2(Q_2)=6Q_2$

Condition for profit maximizing situation is as follows:

  $MR=M{C}_1=M{C}_2$

For facility 1, profit maximizing condition is:

  $MR=M{C}_1$

  $1200-8(Q_1+Q_2)=12Q_1$

  $20Q_1+8Q_2=1200$

  $5Q_1+2Q_2=300$

For facility 2, profit maximizing condition is:

  $MR=M{C}_2$

  $1200-8(Q_1+Q_2)=6Q_2$

  $8Q_1+14Q_2=1200$

  $4Q_1+7Q_2=600$

Solving equations simultaneously,

  $\begin{array}{l}5Q_1+2Q_2=300\\ 10Q_1+4Q_2=600\\ 10Q_1+4Q_2=4Q_1+7Q_2\\ 6Q_1=3Q_2\\ 2Q_1=Q_2\\ Q_1=33.33\\ Q_2=66.66\end{array}$

Thus, the profit maximizing amounts of electricity produced at facility 1 are 33.33 kilowatts hours and that produced in facility 2 is 66.66 kilowatts hours ,thereby totaling the output combined for facility 1 and facility 2 to 100kiilowatt hours.

The optimal price is given as:

  $\begin{array}{l}P=1200-4Q\\ =1200-4(Q_1+Q_2)\\ =1200-4(33.33+66.66)\\ =800\end{array}$

Hence, the optimal price charged by the public utility provider of electricity is $800 per kilowatt hours.

The utility company’s profit is given by:

  $\begin{array}{l}\pi =TR-TC\\ =PQ-(8000+6Q_1^2)-(6000+3Q_2^2)\\ =800\times 100-8000-6000-6{(}^{33.33}-3{(}^{66.66}\\ =80000-14000-20000\\ =46000\end{array}$

Hence, the utility company’s profit is $46000.

Economics Concept Introduction

Introduction:

Profit of a firm is maximized when marginal revenue is equal to marginal cost.

Marginal benefit is the additional benefit to the total for receiving a particular good or service.

Marginal cost is the addition to total cost when one more unit of good is produced.