Editing Price elasticity of demand (section)

==Optimal pricing==

Among the most common applications of price elasticity is to determine prices that maximize revenue or profit.

===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).

===Non-constant elasticity and optimal pricing===

If the definition of price elasticity is extended to yield a quadratic relationship between demand units (<math>Q</math>) and price, then it is possible to compute prices that maximize <math>\ln(Q)</math>, <math>Q</math>, and revenue. The fundamental equation for one product becomes

:<math>LQ = K + E_1 \times LP + E_2 \times LP^2 </math>

and the corresponding equation for several products becomes

:<math>LQ_\ell = K_\ell + E1_{\ell,k} \times LP^k + E2_{\ell,k} \times (LP^k)^2</math>

Excel models are available that compute constant elasticity, and use non-constant elasticity to estimate prices that optimize revenue or profit for one product<ref name="Mshtprice1">{{cite web|url=http://templates.modelsheetsoft.com/modelsheettemplates/product-price-elasticity-templates.aspx|title=Pricing Tests and Price Elasticity for one product|access-date=2013-03-03|archive-url=https://web.archive.org/web/20121113100923/http://templates.modelsheetsoft.com/modelsheettemplates/product-price-elasticity-templates.aspx|archive-date=2012-11-13|url-status=dead}}</ref> or several products.<ref name="Mshtprice2">{{cite web|url=http://templates.modelsheetsoft.com/modelsheettemplates/price-elasticity-templates.aspx|title=Pricing Tests and Price Elasticity for several products|access-date=2013-03-03|archive-url=https://web.archive.org/web/20121113101022/http://templates.modelsheetsoft.com/modelsheettemplates/price-elasticity-templates.aspx|archive-date=2012-11-13|url-status=dead}}</ref>

===Limitations of revenue-maximizing strategies===

In most situations, such as those with nonzero variable costs, revenue-maximizing prices are not profit-maximizing prices.<ref>{{Cite journal |last=Nash |first=John F. |date=1975 |title=A Note on Cost-Volume-Profit Analysis and Price Elasticity |url=https://www.jstor.org/stable/244724 |journal=The Accounting Review |volume=50 |issue=2 |pages=384–386 |jstor=244724 |issn=0001-4826}}</ref> For these situations, using a technique for [[Profit maximization]] is more appropriate.