Editing Price elasticity of demand (section)

===Constant elasticity and optimal pricing===

If one point elasticity is used to model demand changes over a finite range of prices, elasticity is implicitly assumed constant with respect to price over the finite price range. The equation defining price elasticity for one product can be rewritten (omitting secondary variables) as a linear equation.

:<math>LQ = K + E \times LP</math>
where

:<math>LQ = \ln(Q), LP = \ln(P), E</math> is the elasticity, and <math>K</math> is a constant.

Similarly, the equations for cross elasticity for <math>n</math> products can be written as a set of <math>n</math> simultaneous linear equations.

:<math>LQ_\ell = K_\ell + E_{\ell,k} \times LP^k</math>
where
:<math>\ell</math> and <math>k= 1, \dotsc, n,\,\, LQ_\ell = \ln(Q_\ell), LP^\ell = \ln(P^\ell)</math>, and <math>K_\ell</math> are constants; and appearance of a letter index as both an upper index and a lower index in the same term implies summation over that index.

This form of the equations shows that point elasticities assumed constant over a price range cannot determine what prices generate maximum values of <math>\ln(Q)</math>; similarly they cannot predict prices that generate maximum <math>Q</math> or maximum revenue.

Constant elasticities can predict optimal pricing only by computing point elasticities at several points, to determine the price at which point elasticity equals −1 (or, for multiple products, the set of prices at which the point elasticity matrix is the negative identity matrix).