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). 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.

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 changes as their relative prices change.

Alternatively:

$E_{21} =\frac{d \ln (c_2/c_1) }{d \ln (MRS_{12})} =-\frac{d \ln (c_2/c_1) }{d \ln (MRS_{21})} =-\frac{d \ln (c_2/c_1) }{d \ln (U_{c_2}/U_{c_1})} =-\frac{\frac{d (c_2/c_1) }{c_2/c_1}}{\frac{d (U_{c_2}/U_{c_1})}{U_{c_2}/U_{c_1}}} =-\frac{\frac{d (c_2/c_1) }{c_2/c_1}}{\frac{d (p_2/p_1)}{p_2/p_1}}$

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.

See also

• Constant elasticity of substitution
• Marginal Rate of Technical Substitution

References

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

External links

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