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elasticity of substitution : ウィキペディア英語版
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 and thus, the substitutability between inputs (or goods), i.e. how easy it is to substitute one input (or good) for the other.〔Technically speaking, curvature and elasticity are unrelated, but isoquants with different elasticities take on different shapes that might appear to differ in a general sense of curvature (see )〕 In the modern period, John Hicks is considered to have formally introduced this concept in 1932, however he had, by his own admission, introduced the inverse of the elasticity of substitution, or the elasticity of complementarity. The credit then, also by Hicks' own admission, should go to Joan Robinson.
==Mathematical definition==
Let the utility over consumption be given by U(c_1,c_2). Then the elasticity of substitution is:
: E_ =\frac
=\frac)}
=\frac})}}}
=\frac}}

where MRS is the marginal rate of substitution. The last equality presents MRS_ = p_1/p_2 which is a relationship from the first order condition for a consumer utility maximization problem in Arrow-Debreu interior equilibrium. Intuitively we are looking at how a consumer's relative choices over consumption items change as their relative prices change.
Note also that E_ = E_:
: E_ =\frac)}
=\frac)\right)}
=\frac)}
= E_

An equivalent characterization of the elasticity of substitution is:〔Given that:
:\ \frac = d\log (x_2/x_1) = d\log x_2 - d\log x_1 = - (d\log x_1 - d\log x_2) = - d\log (x_1/x_2) = - \frac
an equivalent way to define the elasticity of substitution is:
:\ \sigma =-\frac\frac=-\frac.〕
: E_ =\frac
=-\frac
=-\frac)}
=-\frac})}}}
=-\frac}}

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/\frac)}
=\frac}/\frac)}/\frac}}
=-\frac}/\frac)}/\frac}}

where MRTS is the marginal rate of technical substitution.
The inverse of elasticity of substitution is elasticity of complementarity.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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