Applications of derivative
Price Elasticity of Demand and Marginal
Analysis
MAT 1300 B
Fall, 2011
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Price Elasticity of Demand
One way economists measure the responsiveness of consumers to a change in
the price of a product is with what is called price elasticity of demand.
For example, changing the price on vegetables usually strongly affects the
demand while changing the price of milk or water doesn’t affect that demand
that much.
Definition 1. If p(x) is a differentiable demand function, then the price
elasticity of demand is given by
η=
p(x)/x
dp/dx
where η the lowercase Greek letter eta. For a given price, the demand is said
to be elastic if |η| > 1 and the demand is said to be inelastic if |η| < 1.
The demand is unit elasticity if |η| = 1.
The Price Elasticity of Demand η measures the ratio of the percentage
change of demand of a product to the percentage change of price of the
product. If demand is elastic (|η| > 1), the percentage increase in demand
is greater than the percentage increase in price, and hence the demand is
sensitive to changes in price. Likewise if demand is inelastic(|η| < 1), the
percentage increase in demand is less than the percentage increase in price,
and hence the demand is insensitive to changes in price. When elasticity is
equal to 1, the percentage changes are roughly equal.
For example, rough values of elasticity for some common commodities are:
Tomatoes
η = 4.60
Automobiles η = 1.35
Housing
η = 1.00
Mail
η = 0.05
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Examples:
√
1. Let the demand function for a product be modeled by p(x) = 21− 32 x.
Find the price elasticity of demand when x = 36 and x = 400.
solution: Here
dp
dx
= − 4√3 x . So for x = 36:
p(36) = 21 −
3√
36
2
= 12
3
p0 (36) = − √
4 36
1
= −
8
Thus
η=
12/36
8
=−
−1/8
3
Here |η| > 1 so the demand is elastic.
For x = 400:
p(400) = 21 −
3√
400
2
= −9
3
p0 (400) = − √
4 400
3
= −
80
Thus
η=
−9/400
3
=
−3/80
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Here |η| < 1 so the demand is inelastic.
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2. Suppose the demand function for ice cream bars is given by p(x) =
8 − 2x. For what values of x do ice cream bars have unit elasticity?
solution: Mmm, ice cream bars. At any rate, the formula for η is
p(x)/x
dp/dx
(8 − 2x)/x
=
−2
4
= − +1
x
η =
We want the values of x for which |η| = 1.
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|η| = 1 ⇒ − + 1 = 1 ⇒ − + 1 = ±1
x
x
If − x4 + 1 = 1 then − x4 = 0 which is impossible. If − x4 + 1 = −1 then
− x4 = −2 ⇒ x = 2. Thus x = 2 gives us unit elasticity.
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