elasticity of substitution

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.Chirinko, Robert (2006). [http://www.sciencedirect.com/science/article/pii/S0164070407001619 Sigma: The Long and Short of It]. Journal of Macroeconomics. 2: 671-86.

The general definition of the elasticity of X with respect to Y is $E^X\_Y\; =\; \backslash frac\{\backslash \%\backslash \; \backslash mbox\{change\; in\; X\}\}\{\backslash \%\backslash \; \backslash mbox\{change\; in\; Y\}\}$, which reduces to $E^X\_Y\; =\; \backslash frac\{dX\}\{dY\}\; \backslash frac\{Y\}\{X\}$ 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 $MP\_K$ and $MP\_L$ 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 $U(c\_1,c\_2)$ and let $U\_\{c\_i\}=\; \backslash partial\; U(c\_1,c\_2)/\backslash partial\; \{c\_i\}$. Then the elasticity of substitution is:

:$E\_\{21\}\; =\backslash frac\{d\; \backslash ln\; (c\_2/c\_1)\; \}\{d\; \backslash ln\; (MRS\_\{12\})\}$

=\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. (These differentials are taken along the isoquant that passes through the base point. That is, the inputs $c\_1$ and $c\_2$ 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 $MRS\_\{12\}\; =\; p\_1/p\_2$, where $p\_1,\; p\_2$ 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 $E\_\{21\}\; =\; E\_\{12\}$:

:$E\_\{21\}\; =\backslash frac\{d\; \backslash ln\; (c\_2/c\_1)\; \}\{d\; \backslash ln\; (U\_\{c\_1\}/U\_\{c\_2\})\}$

=\frac{d \left(-\ln (c_2/c_1)\right) }{d \left(-\ln (U_{c_1}/U_{c_2})\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:Given that:

:$\backslash \; \backslash frac\{d\; (x\_2/x\_1)\}\{x\_2/x\_1\}\; =\; d\backslash log\; (x\_2/x\_1)\; =\; d\backslash log\; x\_2\; -\; d\backslash log\; x\_1\; =\; -\; (d\backslash log\; x\_1\; -\; d\backslash log\; x\_2)\; =\; -\; d\backslash log\; (x\_1/x\_2)\; =\; -\; \backslash frac\{d\; (x\_1/x\_2)\}\{x\_1/x\_2\}$

an equivalent way to define the elasticity of substitution is:

:$\backslash \; \backslash sigma\; =-\backslash frac\{d\; (c\_1/c\_2)\}\{d\; MRS\}\backslash frac\{MRS\}\{c\_1/c\_2\}=-\backslash frac\{d\backslash log\; (c\_1/c\_2)\}\{d\backslash log\; MRS\}$.

:$E\_\{21\}\; =\backslash frac\{d\; \backslash ln\; (c\_2/c\_1)\; \}\{d\; \backslash ln\; (MRS\_\{12\})\}$

=-\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:

:$\backslash sigma\_\{21\}\; =\backslash frac\{d\; \backslash ln\; (x\_2/x\_1)\; \}\{d\; \backslash ln\; MRTS\_\{12\}\}$

=\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.

Consider Cobb–Douglas production function $f(x\_1,x\_2)=x\_1^a\; x\_2^\{1-a\}$.

The marginal rate of technical substitution is

:$MRTS\_\{21\}\; =\; \backslash frac\{1-a\}\{a\}\; \backslash frac\{x\_1\}\{x\_2\}$

It is convenient to change the notations. Denote

:$\backslash frac\{1-a\}\{a\}\; \backslash frac\{x\_1\}\{x\_2\}=\backslash theta$

Rewriting this we have

:$\backslash frac\{x\_1\}\{x\_2\}\; =\; \backslash frac\{a\}\{1-a\}\backslash theta$

Then the elasticity of substitution is{{Cite web|url=https://eric-roca.github.io/courses/mathematical_appendix/elasticity_of_substitution/|title=Elasticity of substitution|date=21 May 2025}}

:$\backslash sigma\_\{21\}\; =\; \backslash frac\{d\; \backslash ln\; (\backslash frac\{x\_1\}\{x\_2\})\}\{d\; \backslash ln\; (MRTS\_\{21\})\}\; =\; \backslash frac\{d\; \backslash ln\; (\backslash frac\{x\_1\}\{x\_2\})\}\{d\; \backslash ln\; (\backslash theta)\}\; =\; \backslash frac\{d\; \backslash frac\{x\_1\}\{x\_2\}\}\{\backslash frac\{x\_1\}\{x\_2\}\}\; \backslash frac\{\backslash theta\}\{d\; \backslash theta\}\; =\; \backslash frac\{d\; \backslash frac\{x\_1\}\{x\_2\}\}\{d\; \backslash theta\}\; \backslash frac\{\backslash theta\}\{\backslash frac\{x\_1\}\{x\_2\}\}\; =\; \backslash frac\{a\}\{1-a\}\; \backslash frac\{1-a\}\{a\}\; \backslash frac\{x\_1\}\{x\_2\}\; \backslash frac\{x\_2\}\{x\_1\}\; =\; 1$

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 $S\_\{21\}$ denote expenditure on $c\_2$ relative to that on $c\_1$. That is:

:$S\_\{21\}\; \backslash equiv\; \backslash frac\{p\_2\; c\_2\}\{p\_1\; c\_1\}$

As the relative price $p\_2/p\_1$ changes, relative expenditure changes according to:

:$\backslash frac\{dS\_\{21\}\}\{d\backslash left(p\_2/p\_1\backslash right)\}\; =\; \backslash frac\{c\_2\}\{c\_1\}\; +\; \backslash frac\{p\_2\}\{p\_1\}\backslash cdot\backslash frac\{d\backslash left(c\_2/c\_1\backslash right)\}\{d\backslash left(p\_2/p\_1\backslash right)\}$

= \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.

• Constant elasticity of substitution
• Marginal rate of technical substitution

{{Reflist}}

• {{cite book |author-link=John Hicks |first=J. R. |last=Hicks |year=1932 |title=The Theory of Wages |publisher=Macmillan |title-link=The Theory of Wages }}
• {{cite book |author-link=Andreu Mas-Colell |last1=Mas-Colell |first1=Andreu |last2=Whinston |last3=Green |year=2007 |title=Microeconomic Theory |publisher=Oxford University Press |location=New York, NY |isbn=978-0195073409 }}
• {{cite book |author-link=Hal Varian |first=Hal |last=Varian |title=Microeconomic Analysis |edition=3rd |year=1992 |publisher=W.W. Norton & Company |isbn=978-0-393-95735-8 |url-access=registration |url=https://archive.org/details/microeconomicana00vari_0 }}
• {{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 |volume=89 |issue=1 |pages=183–192 |doi=10.1162/rest.89.1.183 |s2cid=57570638 }}

• [https://web.archive.org/web/20070224121907/http://cepa.newschool.edu/het/essays/product/elastic.htm The Elasticity of Substitution], Gonçalo L. Fonsekca, essay, The New School for Social Research.

Category:Consumer theory

Category:Elasticity (economics)

Category:1932 in economic history