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.

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