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(Derivation)
 
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<math> \frac{d ln(x_{2}/x_{1})}{d ln | TRS | } = \frac{d(x_{2}/x_{1})/(x_{2}/(x_{1})}{d TRS / TRS} = \sigma </math>
 
<math> \frac{d ln(x_{2}/x_{1})}{d ln | TRS | } = \frac{d(x_{2}/x_{1})/(x_{2}/(x_{1})}{d TRS / TRS} = \sigma </math>
   
Note: The notation <math>\sigma</math> 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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Note: The notation <math>\sigma \,</math> 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.
 
[[Category:Microeconomics]]
 
[[Category:Microeconomics]]

Latest revision as of 01:49, November 13, 2011

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.

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