## FANDOM

15 Pages

The elasticity of substitution $\sigma \,$ is given by the percentage change in the factor ratio over the percentage change in the TRS, with output held fixed.

## Derivation Edit

By definition,

$\sigma = \frac{d(x_{2}/x_{1})/(x_{2}/(x_{1})}{d TRS / TRS} = \frac{d ln(x_{2}/x_{1})}{d ln | TRS | } \,$


An intuitive (albeit not rigorous) way to think of the second notation is to realize:

$d ln (x_{2}/x_{1}) = \frac{1}{x_{2}/x_{1}}\cdot d (x_{2}/x_{1})$
$d ln (|TRS|) = \frac{1}{TRS} \cdot d TRS$


Combining both equations, we get

$\frac{d ln(x_{2}/x_{1})}{d ln | TRS | } = \frac{d(x_{2}/x_{1})/(x_{2}/(x_{1})}{d TRS / TRS} = \sigma$


Note: The notation $\sigma \,$ is unfortunate as it collides with the standard deviation but widely used (e.g. Varian (1994)). Be careful not to confuse both fundamentally different concepts.

Community content is available under CC-BY-SA unless otherwise noted.