How to calculate expenditure elasticity

The expenditure elasticity is a measurement of how expenditure on a good changes in response to a price change. The formula for the expenditure elasticity \epsilon is defined:

\quad \epsilon = \frac{\frac{P_1 X_1 - P_0 X_0}{P_0 X_0}}{\frac{P_1 - P_0}{P_0}}

where P_1 and X_1 is the price and quantity of good X in period 1 and P_0 and X_0 is the price and quantity in period 0.

The expenditure elasticity is different to the usual measures of elasticity in that it measures how expenditure changes in response to a price change instead of how the quantity of a good changes in response to a price change. In essence, the expenditure elasticity measures how much consumers increase or decrease their spending after a price increase.

If the expenditure elasticity is positive, the total expenditure on good X increases after a price increase.

If the expenditure elasticity is negative, the total expenditure on good Y decreases after a price increase.

Consider the following hypothetical sales data presented in the table below

Price Quantity Expenditure
$2 8 $16
$4 6 $24
$6 4 $24
$8 2 $16

We can calculate the expenditure elasticity when the price increases from $2 to $4 as follows:

 \quad \epsilon = \frac{\frac{$24-$16}{$16}}{\frac{4-2}{2}}

\quad \epsilon = 0.5

The expenditure elasticity is larger than zero, which is confirmed by the fact that expenditure clearly rose when the price increased from $2 to $4.

We can also calculate the expenditure elasticity when the price increases from $4 to $6, as follows:

 \quad \epsilon = \frac{\frac{$24-$24}{$24}}{\frac{6-4}{4}}

\quad \epsilon = 0

Since the expenditure did not change, the expenditure elasticity is 0.

Finally, if the price increases from $6 to $8 we can calculate the expenditure elasticity as

 \quad \epsilon = \frac{\frac{$16-$24}{$24}}{\frac{8-6}{6}}

\quad \epsilon = -1

In this case, the expenditure elasticity is negative since expenditure decreased when the price increased.

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