Elasticity of substitution

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). It measures the curvature of an isoquant.

Mathematical definition

Let the utility over consumption be given by $U(c\_1,c\_2)$. Then the elasticity of substitution is:$E\; =\; frac\{d\; ln\; (c\_2/c\_1)\; \}\{d\; ln\; |MRS\; =\; frac\{d\; ln\; (c\_2/c\_1)\; \}\{d\; ln\; (U\_\{c\_1\}/U\_\{c\_2\})\}\; =\; frac\{frac\{d\; (c\_2/c\_1)\; \}\{c\_2/c\_1\{frac\{d\; (U\_\{c\_1\}/U\_\{c\_2\})\}\{U\_\{c\_1\}/U\_\{c\_2\}$where $MRS$ is the marginal rate of substitution.In discrete time models, the elasticity of substitution of consumption in periods $t$ and $t+1$ is known as elasticity of intertemporal substitution.

Similarly, if the production function is $f(x\_1,x\_2)$ then the elasticity of substitution is:$sigma\; =\; frac\{d\; ln\; (x\_2/x\_1)\; \}\{d\; ln\; |TRS\; =\; frac\{d\; ln\; (x\_2/x\_1)\; \}\{d\; ln\; (frac\{df\}\{dx\_1\}/frac\{df\}\{dx\_2\})\}\; =\; frac\{frac\{d\; (x\_2/x\_1)\; \}\{x\_2/x\_1\{frac\{d\; (frac\{df\}\{dx\_1\}/frac\{df\}\{dx\_2\})\}\{frac\{df\}\{dx\_1\}/frac\{df\}\{dx\_2\}$where $TRS$ is the technical rate of substitution.

The inverse of elasticity of substitution is elasticity of complementarity.

Example

Consider Cobb-Douglas production function $f(x\_1,x\_2)=x\_1^a\; x\_2^\{1-a\}$.

The technical rate of substitution is :$TRS\; =\; -\; frac\{a\}\{1-a\}\; frac\{x\_2\}\{x\_1\}$

Then the elasticity of substitution is:$sigma\; =\; frac\{d\; ln\; (frac\{x\_2\}\{x\_1\})\; \}\{d\; ln\; |TRS\; =\; frac\{d\; ln\; (frac\{x\_2\}\{x\_1\})\; \}\{d\; ln\; (frac\{a\}\{1-a\}\; frac\{x\_2\}\{x\_1\})\}\; =\; 1$

ee also

*Constant elasticity of substitution

References

*Hal Varian, "Microeconomic Analysis", 3rd edition, 1992, W.W. Norton & Company, ISBN 0-393-95735-7.

External links

* [http://cepa.newschool.edu/het/essays/product/elastic.htm The Elasticity of Substitution] , Gonçalo L. Fonseca, essay, The New School for Social Research.