Arc elasticity

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Arc elasticity is the elasticity of one variable with respect to another between two given points.

The y arc elasticity of x is defined as:

$E_{x,y} = \frac{\% \mbox{ change in } x}{\% \mbox{ change in } y}$

where the percentage change is calculated relative to the midpoint

$\% \mbox{ change in } x = \frac{x_2 - x_1}{(x_2 + x_1)/2}$

$\% \mbox{ change in } y = \frac{y_2 - y_1}{(y_2 + y_1)/2}$

The midpoint arc elasticity formula was advocated by R. G. D. Allen due to the following properties: (1) symmetric with respect to the two prices and two quantities, (2) independent of the units of measurement, and (3) yield a value of unity if the total revenues at two points are equal.^[1]

Arc elasticity is used when there is not a general function for the relationship of two variables. Therefore, point elasticity may be seen as an estimator of elasticity; this is because point elasticity may be ascertained whenever a function is defined.

For comparison, the y point elasticity of x is given by:

$E_{x,y} = \frac{\partial \ln x}{\partial \ln y}$

[edit] Application in economics

The P arc elasticity of Q is calculated as

$(\% \mbox{ change in }Q)/(\%\mbox{ change in }P)$

The percentage is calculated differently from the normal manner of percent change. This percent change uses the average of the points, in lieu of the original point as the base. It uses the midpoint formula.

[edit] Example

If Demand changed from 8 units to 12 units, the midpoint percent change would be (12-8)/((12+8)/2))=40%. Normal percentage change would equal (12-8)/8= 50%. The midpoint formula has the benefit that a movement from A to B is the exact negative of a movement from B to A. In our example, the midpoint percentage would be -40%, whereas our normal percentage change would be -33.3%.

In the above example, assume the change from 8 to 12 units demanded was caused by a change in price from $3 to $1. The midpoint percentage change of price would be -100%. Therefore, the price elasticity of demand would be: (40%/-100%) or -40%. Often when speaking of price elasticities, it is common to write it as the negative or absolute value of the elasticity, such that price elasticity becomes a positive number.