How Do You Maximize Revenue Using Elasticity?

There are several ways we can define elasticity E. Each indicates how the quantity demanded changes as the price is changed. In the examples below, we’ll utilize elasticity defined as

\displaystyle \text{E}\approx \frac{P}{Q}\,\frac{dQ}{dP}

With this definition, the revenue is maximized when E = -1.

Example 1 Suppose the demand function is defined by

\displaystyle Q=30-\frac{P}{5}

where Q units are demanded at a price of P dollars each.

a. Find an expression for the elasticity in terms of P.

b. Find the quantity at which the revenue is maximized.

m212_elasticity_1

Example 2 Suppose the demand function is defined by

\displaystyle Q=40-\frac{P}{4}

where Q units are demanded at a price of P dollars each.

a. Find an expression for the elasticity in terms of P.

b. Find the quantity at which the revenue is maximized.

m212_elasticity_2

Example 3 Suppose the demand function is defined by

\displaystyle Q=39300-7{{P}^{2}}

where Q units are demanded at a price of P dollars each.

a. Find an expression for the elasticity in terms of P.

b. Find the quantity at which the revenue is maximized.

m212_elasticity_3