Elasticity of substitution is the elasticity

Elasticity (economics)

In economics, elasticity is the measurement of how changing one economic variable affects others. For example:* "If I lower the price of my product, how much more will I sell?"* "If I raise the price, how much less will I sell?"...

 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

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 of an isoquant

Isoquant

In economics, an isoquant 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.

Mathematical definition

Let the utility over consumption be given by . Then the elasticity of substitution is:

where is the marginal rate of substitution

Marginal rate of substitution

In economics, the marginal rate of substitution is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of utility.-Marginal rate of substitution as the slope of indifference curve:...

. The last equality presents 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:


In discrete-time models, the elasticity of substitution of consumption in periods and is known as elasticity of intertemporal substitution

Elasticity of intertemporal substitution

Elasticity of intertemporal substitution is a measure of responsiveness of the growth rate of consumption to the real interest rate...

.

Similarly, if the production function is then the elasticity of substitution is:
where is the marginal rate of technical substitution.

The inverse of elasticity of substitution is elasticity of complementarity

Elasticity of complementarity

Elasticity of complementarity is the percentage responsiveness of relative factor prices to a 1 percent change in relative inputs.-Mathematical definition:...

.

Example

Consider Cobb-Douglas production function .

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.

External links

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