Applied Mathematics for Business and Economics

Lecture Note Differentiation

It means that, at this price (p= 60 ), a one-percent increase in price will result in
a decrease in demand of approximately the same percent.

8.2 Levels of Elasicity of Demand
Ingeneral, the elasticity of demand ηis negative, since demand decreases as price

increases. If η > 1 , the percentage decrease in demand is greater than the percentage

increase in price that caused it. In this situation, economists say that demand is elastic

with respect to price. If η< 1 , the percentage decrease in demand is less than the

percentage increase in price that caused it. In this situation, economists say that
demand is inelastic with respect to price. If η = 1 the percentage changes in price and

demand are equal, the demand is said to be of unit elasticity.

Ifη > 1 , demand is said to be elastic with respect to price.

If η < 1 , demand is said to be inelastic with respect to price.

If η = 1 , demand is said to be unit elasticity with respect to price.

8.3 Elasticity and the Total Revenue
If R denotes the total revenue, p the price per unit, and q the number of units sold (i.e.
the demand), then we can obtain ൌܴݍ݌.

The level of the elasticity of demand with respect to price gives useful information
about the total revenue obtained from the sale of the product. In particular, if the

demand is inelastic (η < 1 ), the total revenue increases as the price increases

(although demand drops). The idea is that, in this case, the relatively small
percentage decrease in demand is offset by the larger percentage increase in price, and
hence the revenue, which is price times demand, increases. If the demand is elastic (

η > 1 ), the total revenue dereases as the price increases. In this case, the relatively

large percentage decrease in demand is not offset by the smaller percentage increase
in price. We summary the situation as follows

If demand is inelastic (η < 1 ), total revenue increases as price increases.

If demand is elastic (η > 1 ), total revenue decreases as price increases.

(The proof is omitted)
Example 2
Suppose the demand q and price pfor a certain commodity are related by the equation

qp=− (^3002) (for 0 ≤≤p 300 )
a. Determine where the demand is elastic, inelastic, and of unit elasticity with
respect to price.
b. Use the results of part a. to describe the behavior of the total revenue as a
function of price.
c. Find the total revenue function explicitly and use its first derivative to
determine its intervals of increse and decrease and the price at which revenue
is maximized.
Solution