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] In a competitive market, it measures the percentage change in the ratio of two inputs used in response to a percentage change in their prices.^[2] 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.^[3]

History of the concept

John Hicks introduced this concept in 1932. Joan Robinson independently discovered it in 1933 using a mathematical formulation that was equivalent to Hicks's, though that was not realized at the time.^[4]

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:^[5]

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; Hammond, Peter (1995). Mathematics for Economic Analysis. Prentice Hall. pp. 561–562.
↑ Bergstrom, Ted (2015). Lecture Notes on Elasticity of Substitution, p. 5. Viewed June 17, 2016.
↑ Technically 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 de La Grandville, Olivier (1997). "Curvature and elasticity of substitution: Straightening it out". Journal of Economics. 66 (1): 23–34. doi:10.1007/BF01231465. )
↑ Chirinko, Robert (2006). Sigma: The Long and Short of It. Journal of Macroeconomics. 2: 671-86.
↑ 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:
$\ \sigma =-\frac{d (c_1/c_2)}{d MRS}\frac{MRS}{c_1/c_2}=-\frac{d\log (c_1/c_2)}{d\log MRS}$ .

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. Fonsekca, essay, The New School for Social Research.

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