Arc elasticity of demand

The arc elasticity of demand is a way of accurately calculating elasticity and is also known as the midpoint method.

Elasticity measures percentage change in one variable (usually quantity demanded) in response to a percentage change in another variable (usually price):

<math>\frac{% \Delta Q}{% \Delta P} = \frac{\frac{Q_{new}-Q_{old}}{Q_{old}}}{\frac{P_{new}-P_{old}}{P_{old}}}</math>

This creates an ambiguity because the same change, in different directions, would yield a different percentage change. For example, suppose that at price 9, 105 widgets are demanded; at price 10, 100 widgets are demanded; and at price 11, 95 units are demanded. Then, if the price rises from 9 to 11,

<math>Elasticity = \frac{\frac{95-105}{95}}{\frac{11-9}{9}} = \frac{\frac{-10}{95}}{\frac{2}{9}} = \frac{-9}{19} \approx -.47</math>

while the price falling from 11 to 9 yields

<math>Elasticity = \frac{\frac{105-95}{105}}{\frac{9-11}{11}} = \frac{\frac{10}{105}}{\frac{-2}{11}} = \frac{11}{-21} \approx -.52</math>

Economists resolve this by averaging the endpoints to use the midpoint between the two endpoints; that is,

<math>Elasticity = \frac{\frac{105-95}{\frac{95+105}{2}}}{\frac{9-11}{\frac{9+11}{2}}} = \frac{\frac{10}{100}}{\frac{-2}{10}} = \frac{-1}{2} = -.5</math>

The elasticity measured from 9 to 11 is then the same as the elasticity measured from 11 to 9.