Elasticity of Substitution - Economic Interpretation

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 denote expenditure on relative to that on . That is:

S_{21} \equiv \frac{p_2 c_2}{p_1 c_1}

As the relative price 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 = \frac{c_2}{c_1}\left(1 - E_{21} \right)

Thus, whether or not an increase in the relative price of leads to an increase or decrease in the relative expenditure on 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 is to increase expenditure on, since a given quantity of is more costly. On the other hand, assuming the goods in question are not Giffen goods, a rise in the relative price of leads to a fall in relative demand for, so that the quantity of purchased falls, which reduces expenditure on .

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 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 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 relative to is independent of the relative prices.

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