Chapter 11, Problem 7E

a)

To determine

To Evaluate: The effect of the price cut on the (i) total revenue (ii) Contribution margin of model A based on this analysis, should the firm lower the price of model A.

Answer to Problem 7E

1. Total Revenue: Increase in the total revenue by $77,715 has been detected due to fall in price.
2. Contribution model of Model A: The contribution margin has increased by $9,540

Explanation of Solution

Given:

The following table has current information on the sales, profitability, and costs of the three models:

MODEL	AVERAGE QUANTITY SOLD(UNITS/MONTH)	CURRENT PRICE	TOTAL REVENUE	VARIABLE COST PER UNIT	CONTRIBUTION MARGIN PER UNIT	CONTRIBUTION MARGIN
A	15,000	$30	$450,000	$15.00	$15	$225,000
B	5,000	35	175,000	18.00	17	85,000
C	10,000	45	450,000	20.00	25	250,000
Total			$1,075,000			$560,000

Lowering the price of Model A to $27

Elasticity of demand to be $-2.5$

Explain:

The price elasticity of demand $\left(E_D\right)$ measures the responsiveness of change in quantity demanded to change in price.

The arc price elasticity of demand measures the price elasticity between two prices of the good. It is calculated as

  $E_D=\frac{\frac{Q_2-Q_1}{\left(Q_1+Q_2\right)/2}}{\frac{P_2-P_1}{\left(P_1+P_2\right)/2}}\mathrm{.......}\left(1\right)$

  $=\frac{Q_2-Q_1}{Q_1+Q_2}\times \frac{P_1+P_2}{P_2-P_1}$

The arc price elasticity of demand for model A of tennis rackets is $E_D=-2.5$

For the quantity demanded $Q_{A1}=15,0000$ , the price is $P_{A1}=\$30$ .The Company is considering to lower the price to $P_{A2}=\$27$ .

Substitute appropriate values in (1) and rearrange to find $Q_2$ .

(Prices of other models are constant)

  $-2.5=\frac{Q_{A2}-15,000}{15,000+Q_{A2}}\times \frac{30+27}{27-30}$

  $-2.5=\frac{Q_{A2}-15,000}{15,000+Q_{A2}}\times \frac{57}{-3}$

  $-2.5\left(15,000+Q_{A2}\right)=-19\left(Q_{A2}-15,000\right)$

  $-2.5Q_{A2}-37,500=-19Q_{A2}+285,000$

  $-2.5Q_{A2}+19Q_{A2}=285,000+37,500$

  $16.5Q_{A2}=322,500$

  $Q_{A2}=19,545$

(i) Total Revenue (TR) is the total receipt of the company from the sales of tennis rackets.

The change in the total revenue is calculated as

  $\Delta T{R}_A=T{R}_{A2}-T{R}_{A1}$

  $=P_{A2}{Q}_{A2}-P_{A1}{Q}_{A1}$

  $=\left(\$27\right)\left(19,545\right)-\left(\$30\right)\left(15,000\right)$

  $=\$77,715$

There has been an increase in the total revenue by $\$77,715$ due to fall in price.

(ii) Contribution margin per unit(C) is the difference between the price per unit and variable cost per unit.

The new contribution margin per unit is calculated as

  $C_{A2}=P_{A2}-V$

  $=\$27-\$15$

  $=\$12$

Contribution margin is the marginal profit per unit sale. It is calculated as

Contribution Margin (A) $=C_{A2}{Q}_{A2}$

  $=\left(\$12\right)\left(19,545\right)$

  $=\$234,540$

The contribution margin with $P=\$30$ was $\$225,000$ .The new contribution margin is $\$234,540$ .

Thus, the change is

  $\Delta \text{ in contribution margin(A) =}\$234,540-\$225,000$

  $=\$9,540$

The contribution margin has increased by $9,540.

The change in the contribution margin is positive. Thus, lowering the price appears to be worthwhile.

Economics Concept Introduction

Introduction: Total Revenue refers to the total receipts a vendor can yield from the selling goods and services to the buyers. Cost received by sellers from buyers by the selling number of goods is known as Total revenue.

b)

To determine

To Evaluate: The effect of the price cut on the (i) total revenue (ii) Contribution margin for the entire line of tennis rackets. Through this analysis, should the firm lower the price of model A.

Answer to Problem 7E

1. Total Revenue: There has been an effect in the total revenue of the entire line of tennis rackets by $\$59,395$ due to fall in price of model A.
2. Contribution margin of Model A: There has been a decline in contribution margin by the entire line of tennis rackets by $12 due to fall in price of model A.The total change in the contribution margin is negative. Thus, lowering the price of model A should not take place.

Explanation of Solution

Given:

The following table contains recent information on the sales, costs, and profitability of the three models:

MODEL	AVERAGE QUANTITY SOLD(UNITS/MONTH)	CURRENT PRICE	TOTAL REVENUE	VARIABLE COST PER UNIT	CONTRIBUTION MARGIN PER UNIT	CONTRIBUTION MARGIN
A	15,000	$30	$450,000	$15.00	$15	$225,000
B	5,000	35	175,000	18.00	17	85,000
C	10,000	45	450,000	20.00	25	250,000
Total			$1,075,000			$560,000

Lowering the price of Model A to $27

Elasticity of demand to be $-2.5$

Explain:

The arc-cross price elasticity of demand measures the cross price elasticity between two prices of the related good. It is calculated as

  $E_{AB}=\frac{\frac{Q_{B2}-Q_{B1}}{\left(Q_{B1}+Q_{B2}\right)/2}}{\frac{P_{A2}-P_{A1}}{\left(P_{A1}+P_{A2}\right)/2}}\mathrm{.......}\left(2\right)$

  $=\frac{Q_{B2}-Q_{B1}}{Q_{B1}+Q_{B2}}\times \frac{P_{A2}-P_{A1}}{P_{A1}+P_{A2}}$

The arc cross- price elasticity of demand between model A and model B of tennis rackets is $E_{AB}=0.5$

The original and new price of model A rackets are $P_{A1}=\$30$ and $P_{A2}=\$27$ , respectively.

The original sales of model B are $Q_{B1}=5,000$

Substitute appropriate values in (2)

  $0.5=\frac{Q_{B2}-5,000}{5,000+Q_{B2}}\times \frac{30+27}{27-30}$

  $0.5=\frac{Q_{B2}-5,000}{5,000+Q_{B2}}\times \frac{57}{-3}$

  $0.5\left(5,000+Q_{B2}\right)=-19\left(Q_{B2}-5,000\right)$

  $2,500+0.5Q_{B2}=-19Q_{A2}+95,000$

  $0.5Q_{A2}+19Q_{B2}=95,000-2,500$

  $19.5Q_{B2}=92,500$

  $Q_{B2}=4,744$

The arc cross-price elasticity of demand between model A and model C of tennis rackets is $E_{AC}=0.2$ .

The original and new price of model A rackets are $P_{A1}=\$30$ and $P_{A2}=\$27$ , respectively.

The original sales of model C are $Q_{C1}=10,000$

Substitute appropriate values in (2)

  $0.2=\frac{Q_{C2}-10,000}{10,000+Q_{C2}}\times \frac{30+27}{27-30}$

  $0.2=\frac{Q_{C2}-10,000}{10,000+Q_{C2}}\times \frac{57}{-3}$

  $0.2\left(10,000+Q_{C2}\right)=-19\left(Q_{C2}-10,000\right)$

  $2,000+0.2Q_{C2}=-19Q_{C2}+190,000$

  $0.2Q_{C2}+19Q_{C2}=190,000-2,000$

  $19.2Q_{C2}=188,000$

  $Q_{C2}=9,792$

The change in the total revenue is calculated as

  $\Delta T{R}_C=T{R}_{C2}-T{R}_{C1}$

  $=P_{C1}{Q}_{C2}-P_{C1}{Q}_{C1}$

  $=\left(\$45\right)\left(10,000\right)-\left(\$45\right)\left(9,792\right)$

  $=-\$9,360$

The change in the total revenue of the entire line of tennis rackets is

  $\Delta TR=\Delta T{R}_A+\Delta T{R}_B+\Delta T{R}_C$

  $=\$77,715+\left(-\$8,960\right)+(-\$9,360)$

  $=\$59,395$

There has been an increase in the total revenue of the entire line of tennis rackets by $59,395 due to fall in price of model A.

Contribution margin for model B is

Contribution margin (B) $=C_B{Q}_{B2}$

  $=\left(\$17\right)\left(4,744\right)$

  $=\$80,648$

The contribution margin with $Q_{B1}=5,000$ was $\$85,000$ .The new contribution margin is $80,648. Thus, the change is

  $\Delta \text{ in contribution margin}\left(B\right)=\$80,648-\$85,000$

  $=-\$4,352$

Contribution margin for model C is

  $\text{Contribution margin(C)}=C_C{Q}_{C2}$

  $=\left(\$25\right)\left(9,792\right)$

  $=\$244,800$

The contribution margin with $Q_{C1}=10,000$ was $250,000. The new contribution margin with $244, 800.Thus, the change is

  $\Delta \text{ in contribution margin(C) =}\$244,800-\$250,000$

  $=\$5,200$

The change in the contribution margin by the entire line of tennis rackets is

  $\Delta \text{ in contribution margin(Total) =}\$9,540+\left(-\$4,352\right)+\left(-\$5,200\right)$

  $=\$12$

There has been a decline in contribution margin by the entire line of tennis rackets by $12 due to fall in price of model A.

Economics Concept Introduction

Introduction: Contribution Margin refers to the difference with the trading price and variable cost. It is the measurement of the ability of the company to enclose the variable cost with revenue. Contribution margin includes fixed costs and profit it is the amount that contributes to the payment for making a profit.