# 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] 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.[2] 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_{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 $MRS$ is the marginal rate of substitution. The last equality presents $MRS_{12} = p_1/p_2$ 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 $E_{21} = E_{12}$:

$E_{21} =\frac{d \ln (c_2/c_1) }{d \ln (U_{c_1}/U_{c_2})}  =\frac{d \left(-\ln (c_1/c_2)\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:[3]

$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 $t$ 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_{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 $MRTS$ is the marginal rate of technical 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 marginal rate of technical substitution is

$MRTS_{12} = \frac{a}{1-a} \frac{x_2}{x_1}$

It is convenient to change the notations. Denote

$\frac{a}{1-a} \frac{x_2}{x_1}=\theta$

Rewriting this we have

$\frac{x_2}{x_1} = \frac{1-a}{a}\theta$

Then the elasticity of substitution is

$\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 $S_{21}$ denote expenditure on $c_2$ relative to that on $c_1$. That is:

$S_{21} \equiv \frac{p_2 c_2}{p_1 c_1}$

As the relative price $p_2/p_1$ changes, relative expenditure changes according to:

$\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 $c_2$ leads to an increase or decrease in the relative expenditure on $c_2$ 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 $c_2$ is to increase expenditure on $c_2$, since a given quantity of $c_2$ is more costly. On the other hand, assuming the goods in question are not Giffen goods, a rise in the relative price of $c_2$ leads to a fall in relative demand for $c_2$, so that the quantity of $c_2$ purchased falls, which reduces expenditure on $c_2$.

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 $c_2$ 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 $c_2$ 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 $c_2$ relative to $c_1$ is independent of the relative prices.

$\ \frac{d (x_2/x_1)}{x_2/x_1} = d\log (x_2/x_1) = d\log x_2 - d\log x_1 = - (d\log x_1 - d\log x_2) = - d\log (x_1/x_2) = - \frac{d (x_1/x_2)}{x_1/x_2}$
$\ \sigma =-\frac{d (c_1/c_2)}{d MRS}\frac{MRS}{c_1/c_2}=-\frac{d\log (c_1/c_2)}{d\log MRS}$.