Elasticity

Elasticity measures how responsive one variable is to a one-percent change in another, expressed as a ratio of percentage changes. It is comparable across variables with different units or magnitudes.

ByHoang TruongUpdated June 30, 2026

See it move

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The scatter-line chart plots the natural logarithm of price on the horizontal axis against the natural logarithm of quantity demanded on the vertical axis, with an OLS regression line fitted through the data. In a log-log specification the slope coefficient β₁ equals the price elasticity of demand directly; the caption shows β₁ = −1.8, meaning a 1% rise in price predicts a 1.8% fall in quantity demanded, confirming elastic demand without any further transformation of the coefficient.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicMultiple Regression & InterpretationAdvanced

The formula

LaTeX

\varepsilon = \frac{\Delta Y / Y}{\Delta X / X}

Variables

$Δ Y / Y$: Proportional change in the outcome variable
$Δ X / X$: Proportional change in the explanatory variable

General definition; |ε| > 1 elastic, |ε| < 1 inelastic

LaTeX

\varepsilon = \frac{dY}{dX} \cdot \frac{X}{Y}

Variables

$d Y / d X$: Slope of the curve at the evaluation point (= β₁ in level–level regression)
$X, Y$: Values of regressor and outcome at the point of evaluation

Used in level–level models; varies at every point along the curve

LaTeX

\ln Y = \beta_0 + \beta_1 \ln X + u \implies \varepsilon = \beta_1

Variables

$β_{1}$: OLS slope in the log–log model; equals the constant elasticity everywhere on the curve

Constant elasticity specification; β₁ reads directly as the percentage change in Y per 1% change in X

Check yourself

PracticeCORE

A researcher estimates the model: ln(Revenue) = 3.2 + 0.85·ln(Advertising) using data for 120 retail firms. Which statement about the coefficient 0.85 is correct?

Select an answer to check your understanding.