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):

The elasticity is given by:

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

**Log-linear**

and the elasticity is given by:

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

**Linear-log**

and the elasticity is:

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

**Log-log**

and the elasticity is:

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.