Chapter 6, Problem 7P

Lorena likes to play golf. The number of times per year that she plays depends on both the price of playing a round of golf as well as Lorena’s income and the cost of other types of entertainment—in particular, how much it costs to go see a movie instead of playing golf. The three demand schedules in the following table show how many rounds of golf per year Lorena will demand at each price under three different scenarios. In scenario D₁ Lorena’s income is $50,000 per year and movies cost $9 each, hi scenario D₂, Lorena’s income is also $50,000 per year, but the price of seeing a movie rises to $11. And in scenario D₃, Lorena’s income goes up to $70,000 per year, while movies cost $11. LO6.5

a. Using the data under D₁ and D₂, calculate the cross elasticity of Lorena’s demand for golf at all three prices. (To do tins, apply the midpoints approach to the cross elasticity of demand.) Is the cross elasticity the same at all three prices? Are movies and golf substitute goods, complementary goods, or independent goods?

b. Using the data under D₂ and D₃, calculate the income elasticity of Lorena's demand for golf at all three prices. (To do this, apply the midpoint approach to the income elasticity of demand.) Is the income elasticity the same at all three prices? Is golf an inferior good?

To determine

Cross price elasticity of demand and income elasticity of demand.

Explanation of Solution

In scenario D₁, Lorena’s income is $50,000 per year and movies cost $9 each.  In scenario D₂, Lorena’s income is also $50,000 per year, but the price to watch a movie rises to $11. And in scenario D₃, Lorena’s income rises up to $70,000 per year while the movies cost $11.

Cross price elasticity of demand can be calculated by using the midpoint formula:

$\text{Price elasticity of demand}=\frac{\frac{\text{New quantity}-\text{Old quantity}}{\frac{\text{New quantity}+\text{Old quantity}}{2}}}{\frac{\text{New price}-\text{Old price}}{\frac{\text{New price}+\text{Old price}}{2}}}$ (1)

Table -1 shows the demand schedule for the three different goods.

Table -1

Price	Demand 1	Demand 2	Demand 3
50	15	10	15
35	25	15	30
20	40	20	50

Substitute the respective values in Equation (1), to calculate the cross elasticity when the demand changes from 15 units to 10 units and the price changes from $9 to $11:

$\begin{array}{c}{\text{Cross price elasticity of demand (E}}_1)=\frac{\frac{10-15}{\frac{10+15}{2}}}{\frac{11-9}{\frac{11+9}{2}}}\\ =\frac{\frac{-5}{\frac{25}{2}}}{\frac{2}{\frac{20}{2}}}\\ =\frac{\frac{-5}{12.5}}{\frac{2}{10}}\end{array}$

$\begin{array}{c}=\frac{-0.4}{0.2}\\ =-2\end{array}$

Substitute the respective values in Equation (1), to calculate the cross elasticity When the demand changes from 25 units to 15 units and the price changes from $9 to $11.

$\begin{array}{c}{\text{Cross price elasticity of demand (E}}_2)=\frac{\frac{15-25}{\frac{15+25}{2}}}{\frac{11-9}{\frac{11+9}{2}}}\\ =\frac{\frac{-10}{\frac{40}{2}}}{\frac{2}{\frac{20}{2}}}\\ =\frac{\frac{-10}{20}}{\frac{2}{10}}\end{array}$

$\begin{array}{c}=\frac{-0.5}{0.2}\\ =-2.5\end{array}$

Price elasticity is 2.5 (Ignore the sign).

Substitute the respective values in Equation (1), to calculate the cross elasticity when the demand changes from 40 units to 20 units and the price changes from $9 to $11:

$\begin{array}{c}{\text{Cross price elasticity of demand (E}}_3)=\frac{\frac{20-40}{\frac{20+40}{2}}}{\frac{11-9}{\frac{11+9}{2}}}\\ =\frac{\frac{-20}{\frac{60}{2}}}{\frac{2}{\frac{20}{2}}}\\ =\frac{\frac{-20}{30}}{\frac{2}{10}}\\ =\frac{-0.666}{0.2}\\ =-3.33\end{array}$

Price elasticity is 3.33.

It can be concluded that the cross price elasticities are not same in all three cases (for three different prices). As the price decreases, the percentage change in quantity increases.

In all three cases, the cross price elasticities are negative which indicates that golf and movie are complements. If the price of the movie increases, the quantity of movie falls and the consumption of golf games also decline. This implies that, the consumption of a movie and golf games decline as the price of movies rises and vice versa.

Economics Concept Introduction

Concept introduction:

Cross price elasticity of demand: The cross price elasticity of demand is the change in the quantity demanded for a good due to the price change in another good. The sign of the coefficients of cross price elasticity of demand tells that whether the good is a complement or substitute. If the cross price elasticity of demand is positive, both goods are substitutes. Two goods are complement to each other if the cross price elasticity of demand is negative.

Income elasticity of demand: The income elasticity of demand is the change in the quantity demand for a good due to the change in income. The sign of the coefficients of income elasticity of demand tells whether the good is normal or inferior. If the income elasticity of demand is positive, the good is considered to be a normal good. The good is an inferior good if the income elasticity of demand is negative.

To determine

Cross price elasticity of demand and income elasticity of demand.

Explanation of Solution

Income elasticity of demand can be calculated by using the midpoint formula:

$\text{Income elasticity of demand}=\frac{\frac{\text{New quantity}-\text{Old quantity}}{\frac{\text{New quantity}+\text{Old quantity}}{2}}}{\frac{\text{New income}-\text{Old income}}{\frac{\text{New income}+\text{Old income}}{2}}}$ (2)

Substitute the respective values in Equation (2), to calculate the cross elasticity when the quantity changes from 10 units to 15 units and the income changes from $50,000 to $70,000.

$\begin{array}{c}{\text{Income elasticity of demand (E}}_1^{\text{I}})=\frac{\frac{15-10}{\frac{15+10}{2}}}{\frac{70,000-50,000}{\frac{70,000+50,000}{2}}}\\ =\frac{\frac{5}{\frac{25}{2}}}{\frac{20,000}{\frac{1,20,000}{2}}}\\ =\frac{\frac{5}{12.5}}{\frac{20,000}{60,000}}\\ =\frac{0.4}{0.333}\\ =1.2012\end{array}$

Substitute the respective values in Equation (2), to calculate the cross elasticity when the quantity changes from 15 units to 30 units and the income changes from $50,000 to $70,000.

$\begin{array}{c}{\text{Income elasticity of demand (E}}_2^{\text{I}})=\frac{\frac{30-15}{\frac{30+15}{2}}}{\frac{70,000-50,000}{\frac{70,000+50,000}{2}}}\\ =\frac{\frac{15}{\frac{45}{2}}}{\frac{20,000}{\frac{1,20,000}{2}}}\\ =\frac{\frac{15}{22.5}}{\frac{20,000}{60,000}}\\ =\frac{0.666}{0.333}\\ =2.0\end{array}$

Substitute the respective values in Equation (2), to calculate the cross elasticity when the quantity changes from 20 units to 50 units and the income changes from $50,000 to $70,000, required income elasticity of demand is given by;

$\begin{array}{c}{\text{Income elasticity of demand (E}}_3^{\text{I}})=\frac{\frac{50-20}{\frac{50+20}{2}}}{\frac{70,000-50,000}{\frac{70,000+50,000}{2}}}\\ =\frac{\frac{30}{\frac{70}{2}}}{\frac{20,000}{\frac{1,20,000}{2}}}\\ =\frac{\frac{30}{35}}{\frac{20,000}{60,000}}\\ =\frac{0.8571}{0.333}\\ =2.5714\end{array}$

In all three cases, the income elasticity of demand is not the same. As the price declines, the change in the quantity demanded becomes more sensitive to the change in income. The demand for the golf games is more sensitive to the income at lower prices.

Here, signs of the coefficients of income elasticity of demand are positive which indicates that, as the income increases, the demand for golf games increases and as income decreases, the demand for the golf game falls. That is the income and the demand for golf is positively related. Thus, the golf game is a considered to be a normal good.

If the demand for a good increases with the decrease in income, the sign of the coefficients of income elasticity of demand is negative and thus the good is an inferior good.

In short, if the sign of the coefficient of income elasticity of demand is positive, the good is a normal good and if it is negative, the good is an inferior good.

The golf game is a normal good and not an inferior good.

Economics Concept Introduction

Concept introduction:

Cross price elasticity of demand: The cross price elasticity of demand is the change in the quantity demanded for a good due to the price change in another good. The sign of the coefficients of cross price elasticity of demand tells that whether the good is a complement or substitute. If the cross price elasticity of demand is positive, both goods are substitutes. Two goods are complement to each other if the cross price elasticity of demand is negative.

Income elasticity of demand: The income elasticity of demand is the change in the quantity demand for a good due to the change in income. The sign of the coefficients of income elasticity of demand tells whether the good is normal or inferior. If the income elasticity of demand is positive, the good is considered to be a normal good. The good is an inferior good if the income elasticity of demand is negative.