Elasticity of Substitution - Mathematical Definition

Mathematical Definition

Let the utility over consumption be given by . Then the elasticity of substitution is:

E_{21} =\frac{d \ln (c_2/c_1) }{d \ln (MRS_{21})} =\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}}} =\frac{\frac{d (c_2/c_1) }{c_2/c_1}}{\frac{d (p_2/p_1)}{p_2/p_1}}

where is the marginal rate of substitution. The last equality presents which is a relationship from the first order condition for a consumer utility maximization problem. Intuitively we are looking at how a consumer's relative choices over consumption items changes as their relative prices change.

Note also that :

E_{21} =\frac{d \ln (c_2/c_1) }{d \ln (U_{c_1}/U_{c_2})} =\frac{d \left(-\ln (c_1/c_2)\right) }{d \left(-\ln (U_{c_2}/U_{c_1})\right)} =\frac{d \ln (c_1/c_2) }{d \ln (U_{c_2}/U_{c_1})} = E_{12}

An equivalent characterization of the elasticity of substitution is:

E_{21} =\frac{d \ln (c_2/c_1) }{d \ln (MRS_{12})} =-\frac{d \ln (c_2/c_1) }{d \ln (MRS_{21})} =-\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}}} =-\frac{\frac{d (c_2/c_1) }{c_2/c_1}}{\frac{d (p_2/p_1)}{p_2/p_1}}

In discrete-time models, the elasticity of substitution of consumption in periods and is known as elasticity of intertemporal substitution.

Similarly, if the production function is then the elasticity of substitution is:

\sigma_{21} =\frac{d \ln (x_2/x_1) }{d \ln MRTS_{12}} =\frac{d \ln (x_2/x_1) }{d \ln (\frac{df}{dx_1}/\frac{df}{dx_2})} =\frac{\frac{d (x_2/x_1) }{x_2/x_1}}{\frac{d (\frac{df}{dx_1}/\frac{df}{dx_2})}{\frac{df}{dx_1}/\frac{df}{dx_2}}} =-\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}}}

where is the marginal rate of technical substitution.

The inverse of elasticity of substitution is elasticity of complementarity.

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