Elasticity of substitution

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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]

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. 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.

Notes

↑ Sydsaeter, Knut and Hammond, Peter, Mathematics for Economic Analysis, Prentice Hall, 1995, pages 561-562.
↑ Technicaly speaking, curvature and elasticity are unrelated, but isoquants with different elasticities take on different shapes that might appear to differ in a general sense of curvature (see Olivier de La Grandville. Curvature and elasticity of substitution: Straightening it out. Journal of Economics (1996).
↑ Given that:
$\ {\frac {d(x_{2}/x_{1})}{x_{2}/x_{1}}}=d\log(x_{2}/x_{1})=d\log x_{2}-d\log x_{1}=-(d\log x_{1}-d\log x_{2})=-d\log(x_{1}/x_{2})=-{\frac {d(x_{1}/x_{2})}{x_{1}/x_{2}}}$
an equivalent way to define the elasticity of substitution is:
$\ \sigma =-{\frac {d(c_{1}/c_{2})}{dMRS}}{\frac {MRS}{c_{1}/c_{2}}}=-{\frac {d\log(c_{1}/c_{2})}{d\log MRS}}$ .

References

Hicks, J. R. (1932). The Theory of Wages. Macmillan. First defined there.
Mas-Colell, Andreu; Whinston; Green (2007). Microeconomic Theory. New York, NY: Oxford University Press. ISBN 0195073401.
Varian, Hal (1992). Microeconomic Analysis (3rd ed.). W.W. Norton & Company. ISBN 0-393-95735-7.
Klump, Rainer; McAdam, Peter; Willman, Alpo (2007). "Factor Substitution and Factor-Augmenting Technical Progress in the United States: A Normalized Supply-Side System Approach". Review of Economics and Statistics 89 (1): 183–192. doi:10.1162/rest.89.1.183.

External links

The Elasticity of Substitution, Gonçalo L. Fonseca, essay, The New School for Social Research.

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