Elasticity of substitution is the ratio of percentage change in capital-labour ratio with the percentage change in Marginal Rate of Technical Substitution.
[1] In a competitive market, it measures the percentage change in the two inputs used in response to a percentage change in their prices.
[2] It gives a measure of 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.
[3] John Hicks introduced the concept in 1932.
Joan Robinson independently discovered it in 1933 using a mathematical formulation that was equivalent to Hicks's, though that was not implemented at the time.
[4] The general definition of the elasticity of X with respect to Y is
for infinitesimal changes and differentiable variables.
The elasticity of substitution is the change in the ratio of the use of two goods with respect to the ratio of their marginal values or prices.
The most common application is to the ratio of capital (K) and labor (L) used with respect to the ratio of their marginal products
Another application is to the ratio of consumption goods 1 and 2 with respect to the ratio of their marginal utilities or their prices.
(These differentials are taken along the isoquant that passes through the base point.
are not varied independently, but instead one input is varied freely while the other input is constrained to lie on the isoquant that passes through the base point.
Because of this constraint, the MRS and the ratio of inputs are one-to-one functions of each other under suitable convexity assumptions.)
This is a relationship from the first order condition for a consumer utility maximization problem in Arrow–Debreu interior equilibrium, where the marginal utilities of two goods are proportional to prices.
Intuitively we are looking at how a consumer's choices over consumption items change as their relative prices change.
: An equivalent characterization of the elasticity of substitution is:[5] In discrete-time models, the elasticity of substitution of consumption in periods
is the marginal rate of technical substitution.
The marginal rate of technical substitution is It is convenient to change the notations.
The elasticity of substitution also governs how the relative expenditure on goods or factor inputs changes as relative prices change.
leads to an increase or decrease in the relative expenditure on
depends on whether the elasticity of substitution is less than or greater than one.
Intuitively, the direct effect of a rise in the relative price of
leads to a fall in relative demand for
purchased falls, which reduces expenditure on
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
In this case, the goods are gross complements.
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
In this case, the goods are gross substitutes.
Note that when the elasticity of substitution is exactly one (as in the Cobb–Douglas case), expenditure on