Elasticity of Substitution - Example

Example

Consider Cobb-Douglas production function .

The marginal rate of technical substitution is

It is convenient to change the notations. Denote

Rewriting this we have

Then the elasticity of substitution is

\sigma_{21} = \frac{d \ln (\frac{x_2}{x_1}) }{d \ln MRTS_{12}} = \frac{d \ln (\frac{x_2}{x_1}) }{d \ln (\frac{a}{1-a} \frac{x_2}{x_1})} = \frac{d \ln (\frac{1-a}{a}\theta) }{d \ln (\theta)} = \frac{d \frac{1-a}{a}\theta}{d \theta} \frac{\theta}{\frac{1-a}{a}\theta}=1

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