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
X | |
E | |
Y |
=
\% changeinX | |
\% changeinY |
X | |
E | |
Y |
=
dX | |
dY |
Y | |
X |
MPK
MPL
Let the utility over consumption be given by
U(c1,c2)
U | |
ci |
=\partialU(c1,c2)/\partial{ci}
E21=
dln(c2/c1) | = | |
dln(MRS12) |
dln(c2/c1) | = | |||||||||
|
| = | ||||||||||||||||||||||
|
| ||||
|
where
MRS
c1
c2
MRS12=p1/p2
p1,p2
Note also that
E21=E12
E21=
dln(c2/c1) | = | |||||||||
|
d\left(-ln(c2/c1)\right) | = | |||||||||
|
dln(c1/c2) | ||||||||||
|
=E12
An equivalent characterization of the elasticity of substitution is:[5]
E21=
dln(c2/c1) | =- | |
dln(MRS12) |
dln(c2/c1) | =- | |
dln(MRS21) |
dln(c2/c1) | =- | |||||||||
|
| =- | ||||||||||||||||||||||
|
| ||||
|
In discrete-time models, the elasticity of substitution of consumption in periods
t
t+1
Similarly, if the production function is
f(x1,x2)
\sigma21=
dln(x2/x1) | = | |
dlnMRTS12 |
dln(x2/x1) | = | |||||||
|
| =- | |||||||||||||||||
|
| ||||||||||||||||||
|
MRTS
The inverse of elasticity of substitution is elasticity of complementarity.
f(x1,x2)=x
a | |
1 |
1-a | |
x | |
2 |
The marginal rate of technical substitution is
MRTS21=
1-a | |
a |
x1 | |
x2 |
It is convenient to change the notations. Denote
1-a | |
a |
x1 | |
x2 |
=\theta
Rewriting this we have
x1 | |
x2 |
=
a | |
1-a |
\theta
Then the elasticity of substitution is[6]
\sigma21=
| ||||||
dln(MRTS21) |
=
| ||||||
dln(\theta) |
=
| ||||||
|
\theta | |
d\theta |
=
| ||||||
d\theta |
\theta | |||
|
=
a | |
1-a |
1-a | |
a |
x1 | |
x2 |
x2 | |
x1 |
=1
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
S21
c2
c1
S21\equiv
p2c2 | |
p1c1 |
As the relative price
p2/p1
dS21 | |
d\left(p2/p1\right) |
=
c2 | |
c1 |
+
p2 | ⋅ | |
p1 |
d\left(c2/c1\right) | |
d\left(p2/p1\right) |
=
c2 | |
c1 |
\left[1+
d\left(c2/c1\right) | ⋅ | |
d\left(p2/p1\right) |
p2/p1 | |
c2/c1 |
\right] =
c2 | |
c1 |
\left(1-E21\right)
Thus, whether or not an increase in the relative price of
c2
c2
Intuitively, the direct effect of a rise in the relative price of
c2
c2
c2
c2
c2
c2
c2
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
c2
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
c2
Note that when the elasticity of substitution is exactly one (as in the Cobb–Douglas case), expenditure on
c2
c1
d(x2/x1) | |
x2/x1 |
=dlog(x2/x1)=dlogx2-dlogx1=-(dlogx1-dlogx2)=-dlog(x1/x2)=-
d(x1/x2) | |
x1/x2 |
\sigma=-
d(c1/c2) | |
dMRS |
MRS | =- | |
c1/c2 |
dlog(c1/c2) | |
dlogMRS |