The demand function is given as \( \log Q = \log A + 0.5 \log P \), where \( Q \) is quantity, \( P \) is the unit price of the good and \( A \) is a positive real number. The own price elasticity of demand is:
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- Perfectly elastic
- Perfectly inelastic
- Elastic
- Inelastic
The Correct Option is D
Solution and Explanation
Step 1: Differentiate the demand function.
The demand function is: \[ \log Q = \log A + 0.5 \log P \] Taking the derivative with respect to \( P \), we get: \[ \frac{dQ}{dP} = 0.5 \cdot \frac{1}{P} \] Now, calculate the price elasticity of demand using the formula: \[ \varepsilon = \frac{dQ}{dP} \cdot \frac{P}{Q} \] Substitute the values: \[ \varepsilon = 0.5 \cdot \frac{P}{Q} \] From the given demand function, we see that the elasticity is less than 1 (because of the 0.5 coefficient), meaning the demand is inelastic.
Step 2: Analyze the result.
Since the price elasticity of demand is less than 1, it indicates inelastic demand, meaning the quantity demanded is not very responsive to changes in price.
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