Elasticity of substitution

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Elasticity of substitution is the elasticity of the ratio of two inputs to a production (or utility) function with respect to the ratio of their marginal products (or utilities).

[edit] Mathematical definition

Let the utility over consumption be given by <math>U(c_1,c_2)</math>.

Then the intertemporal elasticity of substitution is

<math> E = \frac{d \ln (c_2/c_1) }{d \ln (MRS)}

         = \frac{d \ln (c_2/c_1) }{d \ln (U_{c_2}/U_{c_1})}
         = \frac{\frac{d (c_2/c_1) }{c_2/c_1}}{\frac{d (U_{c_2}/U_{c_1})}{U_{c_2}/U_{c_1}}}

</math> where <math>MRS</math> is the marginal rate of substitution.

Similarly, if the production function is <math>f(x_1,x_2)</math> then the elasticity of substitution is

<math> \sigma = \frac{d \ln (x_2/x_1) }{d \ln (TRS)}

         = \frac{d \ln (x_2/x_1) }{d \ln (\frac{df}{dx_2}/\frac{df}{dx_1})}
         = \frac{\frac{d (x_2/x_1) }{x_2/x_1}}{\frac{d (\frac{df}{dx_2}/\frac{df}{dx_1})}{\frac{df}{dx_2}/\frac{df}{dx_1}}}

</math> where <math>TRS</math> is the technical rate of substitution.