# Elasticities in estimated linear models

Ever wondered how your estimation of a linear function relates to the elasticities of the estimated model? I always seem to forget, especially if I have taken the logarithm on one or both sides of the equation. Here are the four cases you can have:

Linear:
The function has the following form (if you have more variables on the right hand side, this doesn’t change the story):

$Y=a + bX$

The elasticity is given by:

$\epsilon= \frac{dY}{dX}\frac{X}{Y}=b\frac{X}{Y}$

and the coefficient b is the change in Y from a unit increase in X.

Log-linear

$log(Y)=a + bX$

and the elasticity is given by:

$\epsilon= be^{a+bX}\frac{X}{Y} = bY\frac{X}{Y} =bX$

and the coefficient b is the percentage increase in Y from a unit increase in X.

Linear-log

$Y=a + b*log(X)$

and the elasticity is:

$\epsilon= \frac{b}{X}\frac{X}{Y} =\frac{b}{Y}$

and b is the change in Y caused by a 1% increase in X.

Log-log

$log(Y)=a + b *log(X)$

and the elasticity is:

$\epsilon= \frac{bY}{X}\frac{X}{Y} =b$

Depending on your regression equation the elasticity is therefore either the estimated coefficient (double log), the coefficient multiplied divided by the left-hand variable (linear-log), multiplied by the right-hand variable (log-linear) or the fraction of right-hand and left-hand variable (linear).
By the way: the formulas were written using WordPress and the Youngwhan’s Simple Latex Plug-In for writing equations in WordPress.

## 8 Replies to “Elasticities in estimated linear models”

1. Komson

Theoretically, elasticity is percentage change in y over percentage change in x. log-level form is semi elasticity. Other than log-log form, in order to find elasticity, you need to multiply the beta by the initial point.

e = xdy/ydx

log-log:
d ln(y) = beta d ln(x)
dy/y = beta * dx/x
beta = xdy/ydx …which is e

log-level:
d ln(y) = beta dx
dy/y = beta dx
beta = dy/ydx is missing x in the denominator and so in order to find elasticity you need to multiply it by some value of x. This is similar for level-log form and level-level form

2. Luke

Great simple summary

3. HELOISA BURNQUIST

I believe you need a b coefficient in your linear log function.

4. hongchoi

thanks! this really helped me!