Question

When Judy's income increased from \(\$ 130\) to \(\$ 170\) a week, she increased her demand for concert tickets by 15 percent and decreased her demand for bus rides by 10 percent. Calculate Judy's income elasticity of demand for (a) concert tickets and (b) bus rides.

Short answer

The income elasticity of demand for concert tickets is 0.487 and for bus rides is -0.325.

Step-by-step solution

- Determining % Change in Income

First, calculate the percentage change in Judy's income. The change in income is from \( \$ 130 \) to \( \$ 170 \). Use the formula for percentage change: \[ \text{Percentage Change in Income} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \] \[ = \frac{170 - 130}{130} \times 100 \] \[ = \frac{40}{130} \times 100 \] \[ = 30.77\% \]

- Calculating Income Elasticity for Concert Tickets

Next, calculate the income elasticity of demand for concert tickets. Use the formula \[ E_I = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Income}} \] For concert tickets, the percentage change in quantity demanded is given as 15%. So, \[ E_I = \frac{15\%}{30.77\%} = \frac{15}{30.77} = 0.487 \]

- Calculating Income Elasticity for Bus Rides

Now, calculate the income elasticity of demand for bus rides. Use the same formula \[ E_I = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Income}} \] For bus rides, the percentage change in quantity demanded is -10% (a decrease). So, \[ E_I = \frac{-10\%}{30.77\%} = \frac{-10}{30.77} = -0.325 \]

Key concepts

Percentage Change

To understand income elasticity of demand, we first need to comprehend the concept of percentage change. Percentage change helps us measure how much a value has increased or decreased over time. For Judy's income, this involves calculating the change from \( \$130 \) to \( \$170 \). The formula for percentage change is: \[ \text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \]Using this formula, Judy's income change calculation is: \[ \frac{170 - 130}{130} \times 100 = 30.77\% \]This means Judy's income increased by 30.77%. Understanding percentage change is essential for determining the sensitivity of quantity demanded to income changes.

Quantity Demanded

Quantity demanded refers to the total amount of a good or service that consumers are willing to purchase at a given price. In economic terms, it highlights how much more or less of a product is bought when certain factors, like income, change. For example, Judy's increased income leads her to buy 15% more concert tickets but 10% fewer bus rides. Changes in quantity demanded due to income variations are crucial in understanding income elasticity.

• Concert tickets: +15% change in demand
• Bus rides: -10% change in demand

This change could be due to various factors such as preferences, the necessity of the product, or available substitutes.

Formula for Elasticity

Income elasticity of demand measures how sensitive the quantity demanded of a good is to changes in income. The formula used is: \[ E_I = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Income}} \]
This ratio tells us whether the good is a normal good (positive elasticity) or an inferior good (negative elasticity). In Judy's case:

• For concert tickets, the calculation is: \[ E_I = \frac{15\%}{30.77\%} = 0.487 \] showing a positive relationship, meaning concert tickets are a normal good.
• For bus rides, the calculation is: \[ E_I = \frac{-10\%}{30.77\%} = -0.325 \] showing a negative relationship, indicating bus rides are an inferior good.

Understanding these calculations helps us analyze consumer behavior and predict how demand for goods will fluctuate with income changes.