To determine the price elasticity of demand using the midpoint formula, we need to follow a structured process. Here's a detailed, step-by-step solution:
1. Identify the Initial and Final Prices and Quantities:
We are given that the price decreases from [tex]$95 to $[/tex]90. Along with these price points, the corresponding quantities demanded are 30 and 40, respectively.
2. Calculate the Percentage Change in Price:
The midpoint formula for the percentage change in price is:
[tex]\[
\text{Percentage change in price} = \frac{\text{Final price} - \text{Initial price}}{(\text{Final price} + \text{Initial price}) / 2}
\][/tex]
Plugging in the given prices:
[tex]\[
\text{Percentage change in price} = \frac{90 - 95}{(90 + 95) / 2} = \frac{-5}{92.5}
\][/tex]
[tex]\[
\text{Percentage change in price} = -0.0541 \quad \text{or} \quad -5.41 \%
\][/tex]
3. Calculate the Percentage Change in Quantity Demanded:
The midpoint formula for the percentage change in quantity demanded is:
[tex]\[
\text{Percentage change in quantity demanded} = \frac{\text{Final quantity} - \text{Initial quantity}}{(\text{Final quantity} + \text{Initial quantity}) / 2}
\][/tex]
Plugging in the given quantities:
[tex]\[
\text{Percentage change in quantity demanded} = \frac{40 - 30}{(40 + 30) / 2} = \frac{10}{35}
\][/tex]
[tex]\[
\text{Percentage change in quantity demanded} = 0.2857 \quad \text{or} \quad 28.57 \%
\][/tex]
4. Calculate the Price Elasticity of Demand:
The price elasticity of demand (PED) is calculated by dividing the percentage change in quantity demanded by the percentage change in price:
[tex]\[
\text{PED} = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}
\][/tex]
Plugging in our calculated values:
[tex]\[
\text{PED} = \frac{0.2857}{-0.0541} = -5.29
\][/tex]
5. Interpreting the Result:
The negative sign indicates that the relationship between price and quantity demanded is inverse, which aligns with the law of demand. The absolute value of 5.29 indicates that the demand for this product is highly elastic.
Given the choices:
- 3.4
- 5.29
- 0.19
- 4.5
The correct answer is 5.29.
Thus, the price elasticity of demand for the product, when the price decreases from [tex]$95 to $[/tex]90, is -5.29 (or simply 5.29 in absolute terms).
