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). In a competitive market, it measures the percentage change in the ratio of two inputs used in response to a percentage change in their prices. 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.
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History of the concept
John Hicks introduced this concept in 1932. Joan Robinson independently discovered it in 1933 using a mathematical formulation that was equivalent to Hicks's, though that was not realized at the time.
Mathematical definition
Let the utility over consumption be given by
where
Note also that
An equivalent characterization of the elasticity of substitution is:
In discrete-time models, the elasticity of substitution of consumption in periods
Similarly, if the production function is
where
The inverse of elasticity of substitution is elasticity of complementarity.
Example
Consider Cobb–Douglas production function
The marginal rate of technical substitution is
It is convenient to change the notations. Denote
Rewriting this we have
Then the elasticity of substitution is
Economic interpretation
Given an original allocation/combination and a specific substitution on allocation/combination for the original one, the larger the magnitude of the elasticity of substitution (the marginal rate of substitution elasticity of the relative allocation) means the more likely to substitute. There are always 2 sides to the market; here we are talking about the receiver, since the elasticity of preference is that of the receiver.
The elasticity of substitution also governs how the relative expenditure on goods or factor inputs changes as relative prices change. Let
As the relative price
Thus, whether or not an increase in the relative price of
Intuitively, the direct effect of a rise in the relative price of
Which of these effects dominates depends on the magnitude of the elasticity of substitution. When the elasticity of substitution is less than one, the first effect dominates: relative demand for
Conversely, when the elasticity of substitution is greater than one, the second effect dominates: the reduction in relative quantity exceeds the increase in relative price, so that relative expenditure on
Note that when the elasticity of substitution is exactly one (as in the Cobb–Douglas case), expenditure on