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.