Elasticity of substitution is the ratio of percentage change in capital-labour ratio with the percentage change in Marginal Rate of Technical Substitution.
In a competitive market, it measures the percentage change in the two inputs used in response to a percentage change in their prices.
[Bergstrom, Ted (2015)]
''Lecture Notes on Elasticity of Substitution''
p. 5. Viewed June 17, 2016. It gives a measure of the curvature of an
isoquant
An isoquant (derived from quantity and the Greek word iso, meaning equal), in microeconomics, is a contour line drawn through the set of points at which the same quantity of output is produced while changing the quantities of two or more inputs. ...
, and thus, the substitutability between inputs (or goods), i.e. how easy it is to substitute one input (or good) for the other.
History of the concept
John Hicks
Sir John Richards Hicks (8 April 1904 – 20 May 1989) was a British economist. He is considered one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economic ...
introduced the concept in 1932.
Joan Robinson
Joan Violet Robinson (''née'' Maurice; 31 October 1903 – 5 August 1983) was a British economist well known for her wide-ranging contributions to economic theory. She was a central figure in what became known as post-Keynesian economics.
B ...
independently discovered it in 1933 using a mathematical formulation that was equivalent to Hicks's, though that was not implemented at the time.
[Chirinko, Robert (2006)]
''Sigma: The Long and Short of It''
'' Journal of Macroeconomics. '' 2: 671-86.
Definition
The general definition of the elasticity of X with respect to Y is
, which reduces to
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
and
or of the rental price (r) and the wage (w). Another application is to the ratio of consumption goods 1 and 2 with respect to the ratio of their marginal utilities or their prices. We will start with the consumption application.
Let the utility over consumption be given by
and let
. Then the elasticity of substitution is:
:
where
is the
marginal rate of substitution
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no exte ...
. (These differentials are taken along the isoquant that passes through the base point. That is, the inputs
and
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.) The last equality presents
, where
are the prices of goods 1 and 2. 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.
Note also that
:
:
An equivalent characterization of the elasticity of substitution is:
:
In discrete-time models, the elasticity of substitution of consumption in periods
and
is known as
elasticity of intertemporal substitution Elasticity of intertemporal substitution (or intertemporal elasticity of substitution, EIS, IES) is a measure of responsiveness of the growth rate of consumption to the real interest rate. If the real interest rate rises, current consumption may de ...
.
Similarly, if the production function is
then the elasticity of substitution is:
:
where
is the
marginal rate of technical substitution
In microeconomic theory, the marginal rate of technical substitution (MRTS)—or technical rate of substitution (TRS)—is the amount by which the quantity of one input has to be reduced (-\Delta x_2) when one extra unit of another input is used ( ...
.
The inverse of elasticity of substitution is
elasticity of complementarity.
Example
Consider
Cobb–Douglas production function
In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly phy ...
.
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
denote expenditure on
relative to that on
. That is:
:
As the relative price
changes, relative expenditure changes according to:
:
Thus, whether or not an increase in the relative price of
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
is to increase expenditure on
, since a given quantity of
is more costly. On the other hand, assuming the goods in question are not
Giffen good
In economics and consumer theory, a Giffen good is a product that people consume more of as the price rises and vice versa—violating the basic law of demand in microeconomics. For any other sort of good, as the price of the good rises, the sub ...
s, a rise in the relative price of
leads to a fall in relative demand for
, so that the quantity of
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
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
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
relative to
is independent of the relative prices.
See also
*
Constant elasticity of substitution
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McK ...
*
Marginal rate of technical substitution
In microeconomic theory, the marginal rate of technical substitution (MRTS)—or technical rate of substitution (TRS)—is the amount by which the quantity of one input has to be reduced (-\Delta x_2) when one extra unit of another input is used ( ...
Notes
References
* First defined there.
*
*
*{{cite journal , first1=Rainer , last1=Klump , first2=Peter , last2=McAdam , first3=Alpo , last3=Willman , year=2007 , title=Factor Substitution and Factor-Augmenting Technical Progress in the United States: A Normalized Supply-Side System Approach , journal=
Review of Economics and Statistics
''The'' ''Review of Economics and Statistics'' is a peer-reviewed 103-year-old general journal that focuses on applied economics, with specific relevance to the scope of quantitative economics. The ''Review'', edited at the Harvard University’s K ...
, volume=89 , issue=1 , pages=183–192 , doi=10.1162/rest.89.1.183 , s2cid=57570638
External links
The Elasticity of Substitution Gonçalo L. Fonsekca, essay,
The New School for Social Research
The New School for Social Research (NSSR) is a graduate-level educational institution that is one of the divisions of The New School in New York City, United States. The university was founded in 1919 as a home for progressive era thinkers. NSS ...
.
Consumer theory
Elasticity (economics)
1932 in economics