# 13Elasticity

In some contexts, it is useful to interpret the results of a regression model in terms of elasticity or semi-elasticity. One strategy to achieve that is to estimate a log-log or a semilog model, where the left and/or right-hand side variables are logged. Another approach is to note that $$\frac{\partial ln(x)}{\partial x}=\frac{1}{x}$$, and to post-process the marginal effects to transform them into elasticities or semi-elasticities.

For example, say we estimate a linear model of this form:

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \varepsilon$

Let $$\hat{y}$$ be the adjusted prediction made by the model for some combination of covariates $$x_1$$ and $$x_2$$. The slope with respect to $$x_1$$ (or “marginal effect”) is:

$\frac{\partial \hat{y}}{\partial x_1}$

We can estimate the “eyex”, “eydx”, and “dyex” (semi-)elasticities with respect to $$x_1$$ as follows:

$\eta_1=\frac{\partial \hat{y}}{\partial x_1}\cdot \frac{x_1}{\hat{y}}\\ \eta_2=\frac{\partial \hat{y}}{\partial x_1}\cdot \frac{1}{\hat{y}} \\ \eta_3=\frac{\partial \hat{y}}{\partial x_1}\cdot x_1,$

with interpretations roughly as follows:

1. A percentage point increase in $$x_1$$ is associated to a $$\eta_1$$ percentage points increase in $$y$$.
2. A unit increase in $$x_1$$ is associated to a $$\eta_2$$ percentage points increase in $$y$$.
3. A percentage point increase in $$x_1$$ is associated to a $$\eta_3$$ units increase in $$y$$.

For further intuition, consider the ratio of change in $$y$$ to change in $$x$$: $$\frac{\Delta y}{\Delta x}$$. We can turn this ratio into a ratio between relative changes by dividing both the numerator and the denominator: $$\frac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}$$. This is of course linked to the expression for the $$\eta_1$$ elasticity above.

With the marginaleffects package, these quantities are easy to compute:

library(marginaleffects)
mod <- lm(mpg ~ hp + wt, data = mtcars)

avg_slopes(mod)
#>
#>  Term Estimate Std. Error     z Pr(>|z|)    S   2.5 %  97.5 %
#>    hp  -0.0318    0.00903 -3.52   <0.001 11.2 -0.0495 -0.0141
#>    wt  -3.8778    0.63276 -6.13   <0.001 30.1 -5.1180 -2.6376
#>
#> Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
#> Type:  response

avg_slopes(mod, slope = "eyex")
#>
#>  Term Contrast Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
#>    hp    eY/eX   -0.285     0.0855 -3.34   <0.001 10.2 -0.453 -0.118
#>    wt    eY/eX   -0.746     0.1418 -5.26   <0.001 22.7 -1.024 -0.468
#>
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
#> Type:  response

avg_slopes(mod, slope = "eydx")
#>
#>  Term Contrast Estimate Std. Error     z Pr(>|z|)    S    2.5 %    97.5 %
#>    hp    eY/dX -0.00173   0.000502 -3.46   <0.001 10.8 -0.00272 -0.000751
#>    wt    eY/dX -0.21165   0.037851 -5.59   <0.001 25.4 -0.28583 -0.137461
#>
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
#> Type:  response

avg_slopes(mod, slope = "dyex")
#>
#>  Term Contrast Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
#>    hp    dY/eX    -4.66       1.32 -3.52   <0.001 11.2  -7.26  -2.06
#>    wt    dY/eX   -12.48       2.04 -6.13   <0.001 30.1 -16.47  -8.49
#>
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
#> Type:  response