Chapter 3, Problem 3CACQ

To determine

(a)

The own price elasticity at price P = $15

Answer to Problem 3CACQ

The own price elasticity at price P = $15 is equal to -1.5

Explanation of Solution

$\begin{array}{c}\ln Q_X^d=7-1.5\ln P_X+2\ln P_Y-0.5\ln M+\ln A\\ \text{Where, }{P}_X=\$15\\ P_Y=\$6\\ M=\$40,000\\ A=\$350\\ {\alpha }_X=-1.5\\ \text{Own price elasticity}:E_{Q_X,P_X}={\alpha }_X\\ =-1.5\end{array}$

Given:Demand function

Here, the demand is elastic because the absolute value of elasticity is greater than 1 (E>1).

Economics Concept Introduction

The price elasticity of demand shows the responsiveness of quantity demanded of good X due to the change in price of good X.

Formula for elasticity when demand function for good X is in log- linear form

$\begin{array}{c}\ln Q_X^d={\alpha }_0+{\alpha }_X\ln P_X+{\alpha }_Y\ln P_Y+{\alpha }_M\ln M+{\alpha }_H\ln H\\ \text{Own price elasticity}:E_{Q_X,P_X}={\alpha }_X\end{array}$

To determine

(b)

The cross-price elasticity of demand between good X and Y.

Answer to Problem 3CACQ

The cross-price elasticity between good X and good Y is equal to 2.

Explanation of Solution

Given:

With the help of demand function with their respective values cross-price elasticity is:

$\begin{array}{c}\ln Q_X^d=7-1.5\ln P_X+2\ln P_Y-0.5\ln M+\ln A\\ \text{Where, }{P}_X=\$15\\ P_Y=\$6\\ M=\$40,000\\ A=\$350\\ {\alpha }_Y=2\\ \text{Cross price elasticity}:E_{Q_X,P_Y}={\alpha }_Y\\ =2\end{array}$

Here, good X and good Y are substitutes (E= +ve)

Economics Concept Introduction

The cross-price elasticity shows the responsiveness of quantity demanded of good X due to the change in price of good Y.

Formula for cross-price elasticity of demand when demand function for good X is in log-linear form.

$\ln Q_X^d={\alpha }_0+{\alpha }_X\ln P_X+{\alpha }_Y\ln P_Y+{\alpha }_M\ln M+{\alpha }_H\ln H$

$\text{cross price elasticity}:E_{Q_X,P_Y}={\alpha }_Y$

To determine

(c)

The income elasticity when income M = $40,000.

Answer to Problem 3CACQ

The income elasticity of demand is equal to -0.5

Explanation of Solution

With the help of the demand function and the respective values of variables, the income elasticity of the consumer can be calculated. The expression for income elasticity derivation is as follows:

$\begin{array}{c}\ln Q_X^d=7-1.5\ln P_X+2\ln P_Y-0.5\ln M+\ln A\\ \text{Where, }{P}_X=\$15\\ P_Y=\$6\\ M=\$40,000\\ A=\$350\\ {\alpha }_M=-0.5\\ \text{Income elasticity}:E_{Q_X,M}={\alpha }_M\\ =-0.5\end{array}$

Here, good X is inferior good (E<0).

Economics Concept Introduction

The income elasticity of a product shows the responsiveness of quantity demanded of the product when the income of consumer changes.

Formula for cross-price elasticity of demand when demand function for good X is in log-linear form:

$\begin{array}{c}\ln Q_X^d={\alpha }_0+{\alpha }_X\ln P_X+{\alpha }_Y\ln P_Y+{\alpha }_M\ln M+{\alpha }_H\ln H\\ \text{Income elasticity}:E_{Q_X,M}={\alpha }_M\end{array}$

To determine

(d)

The own advertising elasticity of demand

Answer to Problem 3CACQ

The advertising elasticity of demand is equal to 1.

Explanation of Solution

With the help of demand function and the respective values of variables, the advertising elasticity of the demand can be calculated. The expression for income elasticity derivation is as follows:

$\begin{array}{c}\ln Q_X^d=7-1.5\ln P_X+2\ln P_Y-0.5\ln M+\ln A\\ \text{Where, }{P}_X=\$15\\ P_Y=\$6\\ M=\$40,000\\ A=\$350\\ {\alpha }_H=1\\ \text{Advertising elasticity}:E_{Q_X,H}={\alpha }_H\\ =1\end{array}$

Economics Concept Introduction

The advertising elasticity of demand shows the responsiveness of quantity demanded of a product due to the change in advertising spent on the product.

Formula for cross-price elasticity of demand when demand function for good X is in log-linear form:

$\begin{array}{l}\ln Q_X^d={\alpha }_0+{\alpha }_X\ln P_X+{\alpha }_Y\ln P_Y+{\alpha }_M\ln M+{\alpha }_H\ln H\\ \text{advertising elasticity of demand}:E_{Q_X,H}={\alpha }_H\end{array}$