Question

The accompanying table shows the price and yearly quantity sold of souvenir T-shirts in the town of Crystal Lake according to the average income of the tourists visiting. $$ \begin{array}{c|c|c} & \begin{array}{c} \text { Quantity of T-shirts } \\ \text { demanded when } \\ \text { average tourist } \end{array} & \begin{array}{c} \text { Quantity of T-shirts } \\ \text { demanded when } \end{array} \\ \text { Price of } & \text { average tourist } \\ \text { T-shirt } & \text { income is } \$ 20,000 & \text { income is } \$ 30,000 \\ \hline \$ 4 & 3,000 & 5,000 \\ 5 & 2,400 & 4,200 \\ 6 & 1,600 & 3,000 \\ 7 & 800 & 1,800 \end{array} $$ a. Using the midpoint method, calculate the price elasticity of demand when the price of a T-shirt rises from \(\$ 5\) to \(\$ 6\) and the average tourist income is \(\$ 20,000 .\) Also calculate it when the average tourist income is \(\$ 30,000\). b. Using the midpoint method, calculate the income elasticity of demand when the price of a T-shirt is \(\$ 4\) and the average tourist income increases from \(\$ 20,000\) to \(\$ 30,000 .\) Also calculate it when the price is \(\$ 7\)

Short answer

Answer: The price elasticity of demand when the price of a T-shirt rises from $5 to $6 is -2.2 when the average tourist income is $20,000, and -1.8333 when the average tourist income is $30,000. The income elasticity of demand when the price of a T-shirt is $4 and the income increases from $20,000 to $30,000 is 1.25, and when the price is $7, it is approximately 1.923.

Step-by-step solution

Calculate the percentage change in quantity demanded

From the table, the quantity demanded at \(5 is 2400 T-shirts, and at \)6 it is 1600 T-shirts. Using the midpoint method, we calculate the percentage change in quantity demanded as follows: \(\frac{1600-2400}{(\frac{1600+2400}{2})} = \frac{-800}{2000} = -0.4\)

Calculate the percentage change in price

The price increased from \(5 to \)6. Calculate the percentage change in price using the midpoint method: \(\frac{6-5}{(\frac{6+5}{2})} = \frac{1}{5.5} = 0.1818\)

Calculate the price elasticity of demand

Divide the percentage change in quantity demanded by the percentage change in price: \(Elasticity = \frac{-0.4}{0.1818} = -2.2\) So, the price elasticity of demand when the price of a T-shirt rises from \(5 to \)6 and the average tourist income is $20,000 is -2.2. Now let's calculate the price elasticity of demand when the price of a T-shirt rises from \(5 to \)6 and the average tourist income is $30,000:

Calculate the percentage change in quantity demanded

From the table, the quantity demanded at \(5 is 4200 T-shirts, and at \)6 it is 3000 T-shirts. Using the midpoint method, we calculate the percentage change in quantity demanded as follows: \(\frac{3000-4200}{(\frac{3000+4200}{2})} = \frac{-1200}{3600} = -0.3333\)

Calculate the percentage change in price

The price increased from \(5 to \)6. So the percentage change in price is the same as before, which is 0.1818.

Calculate the price elasticity of demand

Divide the percentage change in quantity demanded by the percentage change in price: \(Elasticity = \frac{-0.3333}{0.1818} = -1.8333\) So, the price elasticity of demand when the price of a T-shirt rises from \(5 to \)6 and the average tourist income is $30,000 is -1.8333. b. Calculate the income elasticity of demand when the price of a T-shirt is \(4 and the average tourist income increases from \)20,000 to $30,000:

Calculate the percentage change in quantity demanded

From the table, the quantity demanded at \(\$20,000\) income is 3000 T-shirts, and at \(\$30,000\) income it is 5000 T-shirts. Using the midpoint method, we calculate the percentage change in quantity demanded as follows: \(\frac{5000-3000}{(\frac{5000+3000}{2})} = \frac{2000}{4000} = 0.5\) Repeat this step for the price of $7: \(\frac{1800-800}{(\frac{1800+800}{2})} = \frac{1000}{1300} \approx 0.7692\)

Calculate the percentage change in income

The income increased from \(\$20,000\) to \(\$30,000\). Calculate the percentage change in income using the midpoint method: \(\frac{30000-20000}{(\frac{30000+20000}{2})} = \frac{10000}{25000} = 0.4\)

Calculate the income elasticity of demand

Divide the percentage change in quantity demanded by the percentage change in income for both prices: \(Elasticity_{\$4} = \frac{0.5}{0.4} = 1.25\) \(Elasticity_{\$7} = \frac{0.7692}{0.4} \approx 1.923\) So, the income elasticity of demand when the price of a T-shirt is \(\$4\) and the income increases from \(\$20,000\) to \(\$30,000\) is 1.25, and when the price is \(\$7\), it is approximately 1.923.

Key concepts

Price Elasticity of Demand

Price elasticity of demand measures how sensitive the quantity demanded is to a change in the price of a product. It is a crucial concept in economics as it helps businesses understand how a change in price might affect their revenue. A higher elasticity implies that consumers are more responsive to price changes.

• If the price elasticity is greater than 1 (in absolute value), the demand is considered elastic, meaning a small price change leads to a significant change in quantity demanded.
• If it's less than 1, the demand is inelastic, meaning quantity demanded is less responsive to a price change.
• If it equals 1, demand is unitary elastic, meaning the percentage change in quantity demanded is equal to the percentage change in price.

To find the price elasticity of demand, we use the formula:\[ \text{Elasticity} = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}} \]This concept helps determine pricing strategies. For instance, if demand is elastic, lowering prices may lead to higher total revenue because the increase in quantity sold compensates for the lower price per unit.

Income Elasticity of Demand

Income elasticity of demand measures how the quantity demanded of a good changes as consumer income levels change. It is particularly useful for businesses to predict how changing economic conditions could impact their sales.

• If income elasticity is greater than 1, the product is a luxury good, implying that people spend more on it as their income increases.
• If it's between 0 and 1, the product is a necessity, meaning changes in income have a moderate effect on the quantity demanded.
• If it is negative, the product is considered an inferior good, meaning demand decreases as consumer incomes rise.

Income elasticity of demand is calculated by:\[ \text{Income Elasticity} = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in income}} \]Understanding this concept allows businesses to adjust their product lines according to economic trends and consumer purchasing power. For example, during economic growth, products with high income elasticity might see an increase in demand.

Midpoint Method

The midpoint method provides a consistent way to calculate elasticity, avoiding potential distortions from using different bases for percentage changes. This method calculates the average percentage change, making it more reliable when dealing with larger changes.

• This method is especially useful when the direction of the change (increase or decrease) is substantial because it treats the movement symmetrically regardless of starting or ending points.

To use the midpoint method, we apply the following formula for percentage change:\[ \text{Percentage change} = \frac{\text{New value - Old value}}{\left(\frac{\text{New value + Old value}}{2}\right)} \]For example, if calculating price elasticity using this midpoint formula, both price and quantity changes are based on their respective midpoints, offering a balanced view of elasticity which prevents exaggerated results due to asymmetrical price or quantity shifts.
This method is widely favored in economics for its ability to maintain equilibrium, improving accuracy in different elasticity calculations, making it a powerful tool for both students and analysts.