slide 3.6 - 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

An Economics Application: Elasticity of Demand

OBJECTIVES Find the elasticity of a demand

function. Find the maximum of a total-revenue

function. Characterize demand in terms of

elasticity.

3.6


DEFINITION:

The elasticity of demand E is given as a function of price x by

3.6 An Economics Application: Elasticity of Demand

E(x) x D (x)

D(x).


Example 1: Lake Shore Video has found that demand for rentals of its DVDs is given by

where q is the number of DVDs rented per day at x dollars per rental. Find each of the following:

a) The quantity demanded when the price is $2 per rental.

b) The elasticity as a function of x.


q D(x) 120 20x,


Example 1 (continued): c) The elasticity at x = 2 and at x = 4. Interpret the

meaning of these values of the elasticity.d) The value of x for which E(x) = 1. Interpret the

meaning of this price.e) The total-revenue function,f) The price x at which total revenue is a maximum.


R(x) x D(x).


Example 1 (continued): a)Thus, 80 DVDs per day will be rented at a price of $2per rental.

b) To find the elasticity, we must first find

Then we can substitute into the expression for elasticity.


D(2) 120 20(2) 80

D (x) 20D (x).


Example 1 (continued):


E(x) x D (x)

D(x)

E(x) x 20

120 20x

E(x) 20x

120 20x

E(x) x

6 x


Example 1 (continued): c) Find E(2).

At x = 2, the elasticity is 1/2 which is less than 1. Thus, the ratio of the percent change in quantity to the percent change in price is less than 1. A small percentage increase in price will cause an even smaller percentage decrease in the quantity.


E 2 x

6 x

2

6 2

1

2


Example 1 (continued): c) Find E(4).

At x = 4, the elasticity is 2 which is greater than 1. Thus, the ratio of the percent change in quantity to the percent change in price is greater than 1. A small percentage increase in price will cause a percentage decrease in the quantity that exceeds the percentage change in price.


E 4 x

6 x

4

6 42


Example 1 (continued): d) We set E(x) = 1 and solve for x (price, p).

Thus, when the price is $3 per rental, the ratio of the percent change in quantity to the percent change in price is $1.


x

6 x 1

x 6 x

2x 6

x 3


Example 1 (continued): e)

f) To find the price x that maximizes R(x), we find


R(x) x D(x)

R(x) x(120 20x)

R(x) 120x 20x2

R (x) 120 40x

R (x).


Example 1 (concluded): Note that R(x) exists for all values of x. Thus, we solve R(x) = 0.

Since there is only one critical value, we can use the second derivative to see if we have a maximum.

Since R(x) is negative, R(3) is a maximum. That is, total revenue is a maximum at $3 per rental.


120 40x 0

40x 120

x 3

R (x) 40 0


THEOREM 15

Total revenue is increasing at those x-values for which E(x) < 1.

Total revenue is decreasing at those x-values for which E(x) > 1.

Total revenue is maximized at the value(s) for which E(x) = 1.



Elasticity and RevenueFor a particular value of the price x.1. The demand is inelastic if E(x) > 1. An increase in

price will bring an increase in revenue. If demand is inelastic, then revenue is increasing.

2. The demand has unit elasticity if E(x) > 1. The demand has unit elasticity when revenue is at a maximum.

3. The demand is elastic if E(x) > 1. An increase in price will bring a decrease in revenue. If demand is elastic, then revenue is decreasing.


slide 3.6 - 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

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