Elasticity of Substitution

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
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.