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Wize University Calculus 1 Textbook > Applications of Differentiation for Business & Econ

Elasticity of Demand

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Elasticity of Demand

Elasticity of demand is the quantity that measures how a change in the price of a product will affect the quantity demanded. In particular, we can compute exactly how different values of elasticity affects the revenue.

The price elasticity of demand (which is shortened to demand elasticity) is defined to be the percentage change in quantity demanded, divided by the percentage change in price.

elasticity of demand=percentage change in quantity demandedpercentage change in price\text{elasticity of demand}=\frac{\text{percentage change in quantity demanded}}{\text{percentage change in price}}
The formula for computing the elasticity is

ε=−pq⋅dqdp\boxed{\varepsilon=-\frac{p}{q}\cdot\frac{dq}{dp}}
Where qqis the quantity and ppis the price.

Wize Tip
Notice that in the formula for the elasticity of demand, we differentiated the quantity as a function of the price. Typically, this means that the elasticity will also be a function of the price.

  • When ε>1\varepsilon > 1,we say that the quantity in demand is elastic→\rightarrowrevenue will increase when the price decreases.
  • When ε<1\varepsilon< 1,we say that the quantity in demand is inelastic →\rightarrow the revenue will increase when the price increases.
  • When ε=1\varepsilon = 1,we say that the quantity in demand is unit elastic →\rightarrowthe revenue will be maximized.

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Example: Elasticity of Demand

Assume the relationship between the price (pp) and the number sold (qq) is p2+q=120p^2+q=120

Find the Elasticity of Demand. For what value of pp, is the revenue maximized?

First solve for qqin terms of pp

p2+q=120⇒q=120−p2p^2+q=120 \Rightarrow q=120-p^2

⇒q′=−2p\Rightarrow q'=-2p

Compute Elasticity using ε=−pq⋅dqdp\varepsilon=-\frac{p}{q}\cdot\frac{dq}{dp}

ε=−p120−p2×−2p\displaystyle \varepsilon=-\frac{p}{120-p^2}\times-2p

=2p2120−p2\displaystyle =\frac{2p^2}{120-p^2}

Revenue is maximized when ε=1\varepsilon=1

1=2p2120−p2\displaystyle 1=\frac{2p^2}{120-p^2}

⇒120−p2=2p2\Rightarrow 120-p^2=2p^2

⇒120=3p2\Rightarrow 120=3p^2

⇒40=p2\Rightarrow 40=p^2

⇒p=40\Rightarrow \boxed{p=\sqrt{40}}

Note: We don't consider −40-\sqrt{40} as a solution since it does not make sense to charge a negative price.
Assume the relationship between the price (pp) and the number sold (qq) is p2+q2=98p^2+q^2=98

For what value of pp, is the revenue maximized?





Extra Practice
Elasticity of Demand
Assume the relationship between the price (pp) and the number sold (qq) is p2+q2=98p^2+q^2=98
For what value of pp, is the revenue maximized?
Elasticity of Demand
Assume that the price pp and the demand qq of a certain good are related by the equation
q=2e1−p+4p+1q = 2e^{1- p} + \frac{4}{p + 1}
a) Find the price elasticity ε(p)\varepsilon(p) as a function of pp .
Elasticity of Demand
The demand for Tit-Tat chocolate bars at price p is f(p)=20−1.5pf(p)=20-1.5p
a) Write a formula for the elasticity of demand at price p
b) compute the elasticity of demand at price p=1p=1 and p=8p=8 .
Elasticity of Demand
The price and quantity of certain commodity are related by the following expression:
2p2+3q2=3602p^2 + 3q^2 = 360
Elasticity of Demand
Assume the relationship between the price of pizza (pp) and the number of them sold (qq) is p2+q2=98p^2+q^2=98
For what value of pp, is the revenue maximized?