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More Applications of Derivatives

Elasticity (Waner Section 5.5 Page 337~345)

Lecture 16

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Price Elasticity of Demand

pricein increase percentage

demandin decrease percentageE

%100

%100

ppqq

E

q

p

p

qE

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Price Elasticity of Demand

The price elasticity of demand E, is the percentage rate of decrease of demand per percentage increase in price. E is given by:

dq pE

dp q

Demand is: Elastic if E > 1

Unit Elasticity if E = 1

Inelastic if E < 1

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Price Elasticity of DemandIf the demand is elastic at p (E > 1), then an increase in unit price causes a decrease in revenue. A decrease in unit price causes an increase in revenue.

If the demand has unit elasticity at p (E = 1), then with an increase in unit price the revenue will stay about the same.

If the demand is inelastic at p (E < 1), then an increase in unit price causes an increase in revenue. A decrease in unit price causes a decrease in revenue.

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Price Elasticity of DemandEx. The monthly demand for T-shirts is given by

22000 5 q p where p denotes the wholesale unit price in dollars and q denotes the quantity demanded monthly.

Find the price elasticity of demand when p = $5 and p = $15, and interpret the results.

2

2 2

1010

2000 5 2000 5

dq p p pE p

dp q p p

E(15) = 18/7 which is elastic.E(5) = 2/15 which is inelastic.

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Another Elasticity

Income Elasticity of DemandThe income elasticity of demand E, is the percentage rate of increase in demand per percentage increase in income. E is given by:

q

I

dI

dqE

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Demand for OrangesWaner pg. 343, #1

The weekly sales of Honolulu Red Oranges is given by q = 1000 – 20p. Calculate the price elasticity of demand when the price is $30/orange. Interpret your answer.

20 and201000

dp

dqpq

q

p

dp

dqE

5.1)30(201000

3020]

20100020[

p

pE

Since demand is elastic (E > 1), an increase in price will decrease revenue.

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Demand for OrangesWaner pg. 343, #1

The weekly sales of Honolulu Red Oranges is given by q = 1000 – 20p. Calculate the price that gives a maximum weekly revenue and find this maximum revenue.

2201000)201000( pppppqR

25$

0401000

p

pdp

dR First Derivative Test:

p = $24; dR/dp = $40

p = $26; dR/dp = -$40

Revenue is at maximum.

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Demand for OrangesWaner pg. 343, #1

The weekly sales of Honolulu Red Oranges is given by q = 1000 – 20p. Calculate the price that gives a maximum weekly revenue and find this maximum revenue.

Remember -- if E = 1 then revenue is maximized!

25$

20100020

1]201000

20[

p

pp

p

pE

Max revenue

500,12$

)25(20)25(1000)25($

2010002

2

R

R

pppqR

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Income Elasticity of Demand: Live DramaWaner pg. 344, #15

The likelihood that a child will attend a live theatrical performance can be modeled by q = 0.01(0.0078x2 + 1.5x + 4.1) (15 x 100). Here, q is the fraction of children with annual household income x thousand dollars who will attend a live dramatic performance at a theater during the year. Compute the income elasticity of demand at an income level of $20,000 and interpret the result. (Round your answer to two significant digits.)

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Income Elasticity of Demand: Live DramaWaner pg. 344, #15

041001500000780 Rewrite 2 .x.x.q

041001500000780

01500001560

04100150000078001500001560

that Recall

2

2

2

.x.x.

x.x.

.x.x.

x.x.

q

x

dx

dqE

770 0410200150200000780

200150200001560

Thus,

2

2

20

....

..

q

x

dx

dqE x

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Income Elasticity of Demand: Live DramaWaner pg. 344, #15

E 0.77

Your interpretation:

At a family income level of $20,000, the fraction of children attending a live theatrical performance is increasing by 0.77% per 1% increase in household income.