A soft drink company product manager wants to calculate the price elasticity of his product. From historical data, he constructed a data matrix where the mean of the prices was $2.93 and the mean of the sales measures was 279.9. Carrying out a regression on this data resulted in a regression coefficient of price equal to –28.7. Given these numbers, what would be the manager’s price elasticity estimate?

Question

BlackTom AI · Accepted Answer

Restating the scenario: A product manager has historical data with mean price P̄ = 2.93 and mean quantity Q̄ = 279.9. A regression using price as the independent variable yields a coefficient (dQ/dP) of -28.7. The task is to compute the price elasticity of demand from these numbers.

Option by option analysis:

Option A: +0.390
- This value is positive, implying an increase in price would increase quantity demanded, which contradicts the negative slope dQ/dP = -28.7 from the regression. Since elasticity is (dQ/dP) * (P̄/Q̄) and (dQ/dP) is negative, the elasticity should be negative if P̄ and Q̄ are positive. Therefore, this option is inconsistent with the given regression coefficient and mean values.

Option B: –124.192
- This is a large negative number. Elasticity is computed as product of the slope (-28.7) and the ratio P̄/Q̄. The ratio P̄/Q̄ is roughly 2.93/279.9 ≈ 0.01046, which when multiplied by -28.7 yields approximately -0.300, not -124.192. The magnitude here is far too large, indicating a misapplication of units or a miscalculation of the ratio. Thus this option is incorrect.

Option C: –28.603
- This option mirrors the regression coefficient in magnitude but ignores the scaling factor (P̄/Q̄) used to convert the slope to elasticity. Elasticity should be dimensionless and typically much smaller in magnitude because P̄/Q̄ is a small positive fraction. Since -28.603 is roughly the slope itself, it cannot be the elasticity derived from the given mean values. Therefore, this is incorrect.

Option D: –0.300
- This aligns with the formula for price elasticity of demand: E = (dQ/dP) * (P̄/Q̄). Plugging in dQ/dP = -28.7 and P̄/Q̄ = 2.93/279.9 ≈ 0.01046, we get E ≈ -28.7 * 0.01046 ≈ -0.300. The sign is negative (as price increases, quantity demanded falls), and the magnitude is plausible given the small P̄/Q̄ ratio. This option is consistent with the provided numbers and the standard elasticity calculation.

Option E: +27.417
- This is a large positive elasticity, which would imply that higher prices increase quantity demanded, contradicting the negative slope dQ/dP. Additionally, even if one attempted to combine the slope with a P̄/Q̄ ratio, the result would be a small negative number, not a large positive one. Therefore this option is clearly incorrect.

Summary: Among the options, only the calculation that uses the slope (-28.7) and the mean ratio (P̄/Q̄ ≈ 0.01046) yields a plausible elasticity near -0.300. The other options either have the wrong sign, implausible magnitude, or disregard the P̄/Q̄ scaling required to convert the regression slope into elasticity.

类似问题

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