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).

1 Mathematical definition
2 See also
3 References
4 External links

[edit] Mathematical definition

Let the utility over consumption be given by $U (c 1, c 2)$ . Then the intertemporal elasticity of substitution is

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

where $M R S$ is the marginal rate of substitution.

Similarly, if the production function is $f (x 1, x 2)$ then the elasticity of substitution is

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