Question

The equation for a demand curve is \(\mathrm{P}=2 / \mathrm{Q}\). What is the elasticity of demand as price falls from 5 to \(4 ?\) What is the elasticity of demand as the price falls from 9to \(8 ?\) Would you expect these answers to be the same?

Short answer

When the price falls from 5 to 4, the elasticity of demand is -2.5. When the price falls from 9 to 8, the elasticity of demand is approximately 3.86. These values are not the same because the price elasticity of demand can vary along the demand curve, and in our given demand curve, it is not constant. As the price decreases, the elasticity of demand becomes more elastic (higher value).

Step-by-step solution

Write down the price elasticity of demand formula

Let's write down the formula for the price elasticity of demand. We have, \(E = \frac{\% \text{ change in quantity demanded}}{\% \text{ change in price}}\).

Calculate the initial and final quantities for both price changes

We are given the demand curve equation: \(P = \frac{2}{Q}\). For the first price change, the corresponding quantities when price is 5 and 4 are: Initial price, \(P_1 = 5\), so the initial quantity is \(Q_1 = \frac{2}{5}\) Final price, \(P_2 = 4\), so the final quantity is \(Q_2 = \frac{1}{2}\) For the second price change, the corresponding quantities when price is 9 and 8 are: Initial price, \(P_3 = 9\), so the initial quantity is \(Q_3 = \frac{2}{9}\) Final price, \(P_4 = 8\), so the final quantity is \(Q_4 = \frac{1}{4}\)

Calculate the percentage change in quantity and price for both scenarios

We use the formula, \% change = \(\frac{\text{new value - old value}}{\text{old value}} \times 100\) For the first price change (price falls from 5 to 4): \% change in price = \(\frac{P_2 - P_1}{P_1} \times 100 = \frac{4 - 5}{5} \times 100 = -20\%\) \% change in quantity = \(\frac{Q_2 - Q_1}{Q_1} \times 100 = \frac{\frac{1}{2} - \frac{2}{5}}{\frac{2}{5}} \times 100 = 50\%\) For the second price change (price falls from 9 to 8): \% change in price = \(\frac{P_4 - P_3}{P_3} \times 100 = \frac{8 - 9}{9} \times 100 = -11.11\%\) \% change in quantity = \(\frac{Q_4 - Q_3}{Q_3} \times 100 = \frac{\frac{1}{4} - \frac{2}{9}}{\frac{2}{9}} \times 100 \approx 42.86\%\)

Calculate the price elasticity of demand for both scenarios

Now, we can plug in the percentage changes in price and quantity into the price elasticity of demand formula. For the first price change (price falls from 5 to 4): \(E_{1} = \frac{\%\,\text{change in quantity demanded}}{\%\,\text{change in price}}\) \(E_{1} = \frac{50\%}{-20\%} = -2.5\) For the second price change (price falls from 9 to 8): \(E_{2} = \frac{\%\,\text{change in quantity demanded}}{\%\,\text{change in price}}\) \(E_{2} = \frac{42.86\%}{-11.11\%} \approx 3.86\)

Evaluate and compare

We found the elasticity of demand \(E_1\) when price falls from 5 to 4 is -2.5, and the elasticity of demand \(E_2\) when price falls from 9 to 8 is approximately 3.86. As we can see, these values are not the same. This is because the price elasticity of demand can vary along the demand curve, and in our given demand curve, it is not constant. As the price decreases, the elasticity of demand becomes more elastic (higher value).

Key concepts

Understanding the Demand Curve

The demand curve is a foundational concept in economics, representing the relationship between the price of a product and the quantity demanded by consumers. It is typically downward sloping, indicating that as the price of an item decreases, the quantity demanded increases, and vice versa.

A demand curve can take different shapes, indicating various types of relationships between price and demand. In our exercise, the demand curve is represented by the equation \(P = \frac{2}{Q}\), which is an example of a rectangular hyperbola. This implies that the product's revenue (Price × Quantity) is constant for any point on the curve. Understanding this relationship is crucial for businesses setting prices and for economists studying market behavior.

Calculating Percentage Change

Percentage change calculation is integral when analyzing how quantities respond to price changes. In the field of economics, this calculation helps determine the sensitivity of the quantity demanded relative to a change in price.

To calculate the percentage change, you subtract the old value from the new value, divide by the old value, and then multiply the result by 100. This formula gives a percent that indicates the relative change. In this exercise, percentage changes for both price and quantity demanded are computed, showcasing how to assess the variations and their significance for price elasticity evaluation.

Evaluating Elasticity of Demand

Elasticity of demand measures how much the quantity demanded of a good responds to a change in the price of that good. It is calculated by taking the percentage change in quantity demanded and dividing it by the percentage change in price. The result can be a positive or negative number, with negative elasticity often ignored in absolute terms to focus on its magnitude.

The demand is considered elastic when the absolute value of elasticity is greater than 1, meaning consumers are responsive to price changes. In contrast, it is inelastic when the absolute value is less than 1, indicating that consumers are not as responsive to price changes. The exercise showed two different elasticity calculations at different points on the demand curve, highlighting that elasticity can change depending on the price level, which is valuable for making pricing and supply decisions.