Applied Mathematics for Business and Economics

Lecture Note Differentiation

A convinience measure of sentitivity of demand to changes in price is the percentage
change in demand that is generated by a 1 percent increase in price. If p denotes the price,
q the corresponding number of units demanded, and ݌∆ a (small) change in price, the
approximation formula for percentage change gives

( )

Percentage change in 100

dq dp p

q

q

Δ



ൌ0.01݌

In particula, if the change in p is a 1-percent increase, then ݌∆ and

( )(0, 01 )

Percentage change in 100

dq dp p pdq

q

qq

 =

dp

The expression on the right-hand side of this approximation is known in economics as the
elasticity demand. In summary,
If q denotes the demand for a commodity and p its price, the elasticity of demand,

is given by

pdq

qdp

η=. It is the percentage change in demand due to a 1 percentage

increase in price.

Example 1
Suppose the demand q and price p for a certain commodity are related by the linear
equationqp=−240 2 (for 01 ≤≤p 20 ).

a. Express the elasticity of demand as a function of p.

b. Calculate the elasticity of demand when the price is ݌ൌ100. Inteprete the

answer.

c. Calculate the elasticity of demand when the price is ݌ൌ50. Inteprete the

answer.

d. At what price is the elasticity of demand equal to െ1?

Solution
a. The elasticity of demand is

()

2

2

240 2 120

pdq p p p

qdp q p p

η==−=− =−

− −

b. When ݌ൌ100, the elasticity of demand is

100

5

120 120 100

p

p

η=− =− =−

−−

That is, when the price is ݌ , a 1 percent increase in price will produce a
decrease in demand of approximately 5 percent.

ൌ100

c. When ݌ൌ50, the elasticity of demand is

50

0.71

120 120 50

p

p

η=− =− =−

−−

That is, when the price is ൌ݌ , a 1 percent increase in price will produce a
decrease in demand of approximately 0.71 percent.

50

d. The elasticity of demand will be equal to െ1when

1

120

p

p

−=−

−

solving for p to get p= 60.