# 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).[1] 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.[2] 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.[3]

## 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.[4]

## Mathematical definition

Let the utility over consumption be given by ${\displaystyle U(c_{1},c_{2})}$ and let ${\displaystyle U_{c_{i}}=dU(c_{1},c_{2})/d{c_{i}}}$. Then the elasticity of substitution is:

${\displaystyle E_{21}={\frac {d\ln(c_{2}/c_{1})}{d\ln(MRS_{12})}}={\frac {d\ln(c_{2}/c_{1})}{d\ln(U_{c_{1}}/U_{c_{2}})}}={\frac {\frac {d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac {d(U_{c_{1}}/U_{c_{2}})}{U_{c_{1}}/U_{c_{2}}}}}={\frac {\frac {d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac {d(p_{1}/p_{2})}{p_{1}/p_{2}}}}}$

where ${\displaystyle MRS}$ is the marginal rate of substitution. The last equality presents ${\displaystyle MRS_{12}=p_{1}/p_{2}}$ 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 ${\displaystyle E_{21}=E_{12}}$:

${\displaystyle E_{21}={\frac {d\ln(c_{2}/c_{1})}{d\ln(U_{c_{1}}/U_{c_{2}})}}={\frac {d\left(-\ln(c_{2}/c_{1})\right)}{d\left(-\ln(U_{c_{2}}/U_{c_{1}})\right)}}={\frac {d\ln(c_{1}/c_{2})}{d\ln(U_{c_{2}}/U_{c_{1}})}}=E_{12}}$

An equivalent characterization of the elasticity of substitution is:[5]

${\displaystyle E_{21}={\frac {d\ln(c_{2}/c_{1})}{d\ln(MRS_{12})}}=-{\frac {d\ln(c_{2}/c_{1})}{d\ln(MRS_{21})}}=-{\frac {d\ln(c_{2}/c_{1})}{d\ln(U_{c_{2}}/U_{c_{1}})}}=-{\frac {\frac {d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac {d(U_{c_{2}}/U_{c_{1}})}{U_{c_{2}}/U_{c_{1}}}}}=-{\frac {\frac {d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac {d(p_{2}/p_{1})}{p_{2}/p_{1}}}}}$

In discrete-time models, the elasticity of substitution of consumption in periods ${\displaystyle t}$ and ${\displaystyle t+1}$ is known as elasticity of intertemporal substitution.

Similarly, if the production function is ${\displaystyle f(x_{1},x_{2})}$ then the elasticity of substitution is:

${\displaystyle \sigma _{21}={\frac {d\ln(x_{2}/x_{1})}{d\ln MRTS_{12}}}={\frac {d\ln(x_{2}/x_{1})}{d\ln({\frac {df}{dx_{1}}}/{\frac {df}{dx_{2}}})}}={\frac {\frac {d(x_{2}/x_{1})}{x_{2}/x_{1}}}{\frac {d({\frac {df}{dx_{1}}}/{\frac {df}{dx_{2}}})}{{\frac {df}{dx_{1}}}/{\frac {df}{dx_{2}}}}}}=-{\frac {\frac {d(x_{2}/x_{1})}{x_{2}/x_{1}}}{\frac {d({\frac {df}{dx_{2}}}/{\frac {df}{dx_{1}}})}{{\frac {df}{dx_{2}}}/{\frac {df}{dx_{1}}}}}}}$

where ${\displaystyle MRTS}$ is the marginal rate of technical substitution.

The inverse of elasticity of substitution is elasticity of complementarity.

## Example

Consider Cobb–Douglas production function ${\displaystyle f(x_{1},x_{2})=x_{1}^{a}x_{2}^{1-a}}$.

The marginal rate of technical substitution is

${\displaystyle MRTS_{12}={\frac {a}{1-a}}{\frac {x_{2}}{x_{1}}}}$

It is convenient to change the notations. Denote

${\displaystyle {\frac {a}{1-a}}{\frac {x_{2}}{x_{1}}}=\theta }$

Rewriting this we have

${\displaystyle {\frac {x_{2}}{x_{1}}}={\frac {1-a}{a}}\theta }$

Then the elasticity of substitution is

${\displaystyle \sigma _{21}={\frac {d\ln({\frac {x_{2}}{x_{1}}})}{d\ln MRTS_{12}}}={\frac {d\ln({\frac {x_{2}}{x_{1}}})}{d\ln({\frac {a}{1-a}}{\frac {x_{2}}{x_{1}}})}}={\frac {d\ln({\frac {1-a}{a}}\theta )}{d\ln(\theta )}}={\frac {d{\frac {1-a}{a}}\theta }{d\theta }}{\frac {\theta }{{\frac {1-a}{a}}\theta }}=1}$

## 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 ${\displaystyle S_{21}}$ denote expenditure on ${\displaystyle c_{2}}$ relative to that on ${\displaystyle c_{1}}$. That is:

${\displaystyle S_{21}\equiv {\frac {p_{2}c_{2}}{p_{1}c_{1}}}}$

As the relative price ${\displaystyle p_{2}/p_{1}}$ changes, relative expenditure changes according to:

${\displaystyle {\frac {dS_{21}}{d\left(p_{2}/p_{1}\right)}}={\frac {c_{2}}{c_{1}}}+{\frac {p_{2}}{p_{1}}}\cdot {\frac {d\left(c_{2}/c_{1}\right)}{d\left(p_{2}/p_{1}\right)}}={\frac {c_{2}}{c_{1}}}\left[1+{\frac {d\left(c_{2}/c_{1}\right)}{d\left(p_{2}/p_{1}\right)}}\cdot {\frac {p_{2}/p_{1}}{c_{2}/c_{1}}}\right]={\frac {c_{2}}{c_{1}}}\left(1-E_{21}\right)}$

Thus, whether or not an increase in the relative price of ${\displaystyle c_{2}}$ leads to an increase or decrease in the relative expenditure on ${\displaystyle c_{2}}$ 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 ${\displaystyle c_{2}}$ is to increase expenditure on ${\displaystyle c_{2}}$, since a given quantity of ${\displaystyle c_{2}}$ is more costly. On the other hand, assuming the goods in question are not Giffen goods, a rise in the relative price of ${\displaystyle c_{2}}$ leads to a fall in relative demand for ${\displaystyle c_{2}}$, so that the quantity of ${\displaystyle c_{2}}$ purchased falls, which reduces expenditure on ${\displaystyle c_{2}}$.

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 ${\displaystyle c_{2}}$ falls, but by proportionally less than the rise in its relative price, so that relative expenditure rises. 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 ${\displaystyle c_{2}}$ falls. 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 ${\displaystyle c_{2}}$ relative to ${\displaystyle c_{1}}$ is independent of the relative prices.

${\displaystyle \ {\frac {d(x_{2}/x_{1})}{x_{2}/x_{1}}}=d\log(x_{2}/x_{1})=d\log x_{2}-d\log x_{1}=-(d\log x_{1}-d\log x_{2})=-d\log(x_{1}/x_{2})=-{\frac {d(x_{1}/x_{2})}{x_{1}/x_{2}}}}$
${\displaystyle \ \sigma =-{\frac {d(c_{1}/c_{2})}{dMRS}}{\frac {MRS}{c_{1}/c_{2}}}=-{\frac {d\log(c_{1}/c_{2})}{d\log MRS}}}$.