Arc elasticity

In mathematics and economics, the arc elasticity is the elasticity of one variable with respect to another between two given points. It is the ratio of the percentage change of one of the variables between the two points to the percentage change of the other variable. It contrasts with the point elasticity, which is the limit of the arc elasticity as the distance between the two points approaches zero and which hence is defined at a single point rather than for a pair of points.

The y arc elasticity of x is defined as:

where the percentage change in going from point 1 to point 2 is usually calculated relative to the midpoint:

The use of the midpoint arc elasticity formula (with the midpoint used for the base of the change, rather than the initial point (x₁, y₁) which is used in almost all other contexts for calculating percentages) was advocated by R. G. D. Allen for use when x refers to the quantity of a good demanded or supplied and y refers to its price, due to the following properties: (1) it is symmetric with respect to the two prices and quantities, (2) it is independent of the units of measurement, and (3) it yields a value of unity if the total revenues (price times quantity) at the two points are equal.

The arc elasticity is used when there is not a general function for the relationship of two variables, but two points on the relationship are known. In contrast, calculation of the point elasticity requires detailed knowledge of the functional relationship and can be calculated wherever the function is defined.

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

The arc elasticity of quantity demanded (or quantity supplied) Q with respect to price P, also known as the arc price elasticity of demand (or supply), is calculated as

Suppose that two points on a demand curve, ${\displaystyle (Q_{1},P_{1})}$ and ${\displaystyle (Q_{2},P_{2})}$, are known. (Nothing else might be known about the demand curve.) Then the arc elasticity is obtained using the formula