Estimating elasticities from linear regressions

By Enrique Saldarriaga

This post aims to show how elasticities can be estimated directly from linear regressions.
Elasticity measures the association between two numeric variables expressed in percentages, whose interpretation is the percentage increase in one variable associated with a 1% change in another one. Elasticities have served economics for a long time, basically because they allow comparison between very different settings when changes in levels (e.g. dollars) are difficult to interpret. Take for instance a company that produces both cars and chocolate bars. If they want to know how changes in the price of both products would impact their demand, in order to determine which price to increase and which to maintain, the comparison would be a mess. A \$100 change would mean nothing for the demand of cars but would destroy demand for chocolates. Similarly, a decrease in 100K people in the chocolate bars’ demand is probably just a dent, but the same amount could represent a significant portion of the car’s market. Changes in percentages are a way to standardize change and its consequences. In health economics, elasticities are becoming more common because they are able to show fair comparisons between variables with heterogenous behavior depending on the context.

The most common elasticities are price and income. The income elasticity of any variable would be the percentage change in that variable associated with a 1% change in income; the elasticity price would be the percentage change in a variable associated with a 1% change in its own price. In economics, that variable usually would be the demand of a given good. In health economics, that variable could be any number of things.

For example, let’s say we want to estimate the income elasticity of BMI. The elasticity would be expressed as:

$\displaystyle \epsilon_I = \dfrac{\Delta \% BMI}{\Delta \% Income} = \dfrac{\Delta BMI}{\Delta Income}* \dfrac{Income}{BMI}$

where ∆ stands for variation and shows the variation of BMI or Income from one point to another in their joint distribution. In order to make this change infinitesimal, and therefore estimate the elasticity in the whole distribution, we can express those changes with differentials: ε_I = dQ/dP*Income/BMI

Now, to obtain elasticities directly from linear regressions we should use the logarithmic forms of the variables:

$\displaystyle ln(BMI) = \beta_0 + \beta_1 * ln(Income) = f(Income)$

The BMI is a function of income. The function can be expressed as:

$\displaystyle f(Income) = BMI = e^{ \beta_0 + \beta_1 * ln(Income)}$

To find the change in BMI associated with a change in income we should derive this function by Income. Given the form of the function we use the chain rule:

$\displaystyle f(x)=g(h(x)) ; f'(x)=g'(h(x))*h'(x)$

Where: $\displaystyle g(h) = e^h , h(Income) = \beta_0 + \beta_1 *ln(Income)$

$\displaystyle g'(h)= e^h ,h'(Income) = \beta_1 * \dfrac{1}{Income}$

Then:

$\displaystyle f'(Income) = e^{ \beta_0 + \beta_1 * ln(Income)} * \beta_1 * \dfrac{1}{Income}$

$\displaystyle \dfrac{\partial BMI}{\partial Income} = e^{ln(BMI)} * \beta_1 * \dfrac{1}{Income} = BMI * \beta_1 * \dfrac{1}{Income}$

$\displaystyle \beta_1 = \dfrac{ \partial BMI}{ \partial Income} * \dfrac{Income}{BMI}$

Thus, the β_1 coefficient is the estimation of the income elasticity. The same procedure could be applied to find the price elasticity. It’s interesting to notice that if only the independent variable were in its logarithmic form, the derivation would be easier, and the coefficient would be:

$\beta_1 = \dfrac{ \partial BMI}{ \partial Income} * \dfrac{Income}{1}$

Then, to estimate the elasticity it would be necessary to divide the coefficient by a measure of BMI, probably the mean to account for the whole distribution, as Sisira et al. did in their paper. However, in this case would be necessary to carefully select a good average measure. This method would be useful if we prefer to interpret other covariates’ coefficients using BMI rather than ln⁡(BMI).