Arc elasticity of demand

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The arc elasticity of demand is a way of accurately calculating elasticity and is also known as the midpoint method (for example in Greg Mankiw's introductory texbook).

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

\frac{% \Delta Q}{% \Delta P} = \frac{\frac{Q_{new}-Q_{old}}{Q_{old}}}{\frac{P_{new}-P_{old}}{P_{old}}}

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,

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

while the price falling from 11 to 9 yields

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

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

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

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

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