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:

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


[math]Y=a + bX[/math]


The elasticity is given by:


[math]\epsilon= \frac{dY}{dX}\frac{X}{Y}=b\frac{X}{Y} [/math]


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




[math]log(Y)=a + bX [/math]


and the elasticity is given by:


[math]\epsilon= be^{a+bX}\frac{X}{Y} = bY\frac{X}{Y} =bX[/math]


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




[math]Y=a + b*log(X)[/math]


and the elasticity is:


[math] \epsilon= \frac{b}{X}\frac{X}{Y} =\frac{b}{Y} [/math]


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



[math]log(Y)=a + b *log(X)[/math]


and the elasticity is:


[math]\epsilon= \frac{bY}{X}\frac{X}{Y} =b[/math]


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 thoughts on “Elasticities in estimated linear models

  1. 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

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

    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. The pictures are not displaying? This looks interesting but would love to see the images that go along with the breakdown…

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