Chapter 8, Problem 8.5P

A

To determine

In order to maximize the total produce it sells, subject to that the firm does not operate at loss, find out the number of units that it produces and also the price that the firm should charge.

Answer to Problem 8.5P

Given the conditions, it can be derived the total produce as $\frac{a-c}{b}$ units, and also charge a price of ‘c’ dollars.

Explanation of Solution

Subject to the given constraint, it can be derived that the firm operates until where the TR=TC.

Implies,

  $\begin{array}{l}TR=TC\\ Pq=cq\\ (a-bq)q=cq\\ a-bq=c\\ bq=a-c\\ q=\frac{a-c}{b}\end{array}$

Thus, it can be found that the firm is producing $\frac{a-c}{b}$ units, and charge a price of ‘c’ dollars.

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.

B

To determine

On the condition that the total revenue collected is maximized, find out the produce and the price it should charge.

Answer to Problem 8.5P

After necessary derivations, it can be concluded that the total revenue is maximized at a/2 dollars, and a/2b units. Further, it should be observed that when the price elasticity of demand becomes -1, the TR is maximized.

Explanation of Solution

The Total Revenue equation is as follows:

  $\begin{array}{l}TR=Pq\\ TR=aq-b{q}^2\end{array}$

Now, it shall be derived that the Marginal Revenue equation as below;

  $\begin{array}{l}MR=\frac{d}{dq}TR\\ MR=\frac{d}{dq}(aq-b{q}^2)\\ MR=a-2bq\end{array}$

Now, in order to maximize TR, it shall be equated that MR to 0;

  $\begin{array}{l}MR=0\\ a-2bq=0\\ 2bq=a\\ q=\frac{a}{2b}\end{array}$

Now, substitute this value of q in the above equation;

  $\begin{array}{l}P=a-b(\frac{a}{2b})\\ P=a-\frac{a}{2}\\ P=\frac{a}{2}\end{array}$

Thus, it can be concluded that the total revenue of the firm is maximized at a price rate of a/2 dollars, and at a quantity of a/2b units. Further, it should be observed that when the price elasticity of demand becomes -1, the TR is maximized.

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.

C

To determine

Supposing that the firm wishes to maximize its sales subject to a profit making of 1% on its sales, find out the number of units that the firm should produce and also the price that it should charge.

Answer to Problem 8.5P

After necessary calculations, the firm should produce a quantity of $q=\frac{(a-1.01c)}{b}$ , and at a price of P=1.01c, so as to earn a profit of 1% on sales.

Explanation of Solution

  $\begin{array}{l}P=1.01MC\\ P=1.01c\end{array}$

Now, substituting this value in the equation,

  $\begin{array}{l}1.01c=a-bq\\ bq=a-1.01c\\ q=\frac{(a-1.01c)}{b}\end{array}$

  $\begin{array}{l}TR=\frac{(1.01c).(a-1.01c)}{b}\\ TC=\frac{c(a-1.01c)}{b}\\ \mathrm{Pr}ofit=\frac{\left\{\left(1.01c\right).\left(a-1.01c\right)-c(a-1.01c)\right\}}{b}\\ \mathrm{Pr}ofit=\frac{\left\{(a-1.01c)(1.01c-c)\right\}}{b}\\ \mathrm{Pr}ofit=(0.01c)q\\ \mathrm{Pr}ofit=0.01cq\\ \mathrm{Pr}ofit=0.01TC\end{array}$

Thus, in order to earn a profit of 1% on the sales, the firm should produce a quantity of $q=\frac{(a-1.01c)}{b}$ , and at a price of P=1.01c.

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.

D

To determine

Supposing that the firm wishes to maximize its profits per unit, find out the number of units that the firm should produce and also the price that it should charge.

Answer to Problem 8.5P

The firm should produce a quantity of $q=\frac{(a-c)}{2b}$ , at a price of $P=\frac{(a+c)}{2}$ .

Explanation of Solution

In order to maximize the profits per unit, the firm should equate MR with MC;

  $\begin{array}{l}MR=MC\\ a-2bq=c\\ 2bq=a-c\\ q=\frac{a-c}{2b}\end{array}$ ,

Substituting this value of q, in the demand equation;

  $\begin{array}{l}P=\frac{\left(a-b\right)\left(a-c\right)}{2b}\\ P=\frac{a-(a-c)}{2}\\ P=\frac{(2a-a+c)}{2}\\ P=\frac{a+c}{2}\end{array}$

Average Profit per unit will be=AR-AC

  $AP=\frac{a+c}{2-c}=\frac{a+c-2c}{2}=\frac{a-c}{2}$

Thus, in order to earn profits per unit, the firm should produce a quantity of $q=\frac{(a-c)}{2b}$ , and at a price of $P=\frac{(a+c)}{2}$ .

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.

E

To determine

Contrast the solutions derived in parts A through D to the output that would be preferred by the firm.

Answer to Problem 8.5P

We can find that the solution from part A was that of a Perfectly Competitive market, part B is of a sales maximizing firm, and of part D is of a monopoly firm.

Explanation of Solution

On comparing, it can be found that the solution derived in part A was that of a Perfectly Competitive market, part B is related to a sales maximizing firm, and solution of part D was related to a monopoly firm.

Moreover, the quantity produced in parts A and B is more than in the parts C and D.

Further, the price charged in parts A, B and C is lower than the price determined in part D.

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.