Elasticity

The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price: E_d = = where Q₀ and P₀ are the original quantity and price, respectively. In general, we don’t know which price or quantity is the original; hence the use of the "midpoint" formula: E_d = where and represent the average quantity and price. To simplify notation, we will substitute "Q" and "P" for their respective average values. In that case, after inverting and multiplying the denominator and rearranging, we can rewrite the formula as E_d = .

The first term in this product, , is the inverse of the slope of the demand curve as it is traditionally drawn. Along a straight-line demand curve, this value is constant. However the second term, , clearly declines as price falls and quantity increases (moving southeast along a demand curve.) Accordingly, the elasticity of demand declines along a straight-line demand curve.

In fact, suppose demand is given by the equation P = a -bQ so that a is the intercept on the price axis and b is the slope. The elasticity of demand is then E_d = . Consider a segment of the demand curve centered at the midpoint of the curve. At this point, P = a/2 and Q = a/2b. Inserting these values into the formula for the elasticity gives E_d = = 1. That is, demand is unit elastic at the midpoint of a straight-line demand curve. Since we know that elasticity continuously declines, this also tells us that demand is elastic above the midpoint and inelastic below.