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
Slide 3.6 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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).
Slide 3.6 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
q D(x) 120 20x,
Slide 3.6 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
R(x) x D(x).
Slide 3.6 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
D(2) 120 20(2) 80
D (x) 20D (x).
Slide 3.6 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 1 (continued):
3.6 An Economics Application: Elasticity of Demand
E(x) x D (x)
D(x)
E(x) x 20
120 20x
E(x) 20x
120 20x
E(x) x
6 x
Slide 3.6 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
E 2 x
6 x
2
6 2
1
2
Slide 3.6 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
E 4 x
6 x
4
6 42
Slide 3.6 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
x
6 x 1
x 6 x
2x 6
x 3
Slide 3.6 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 1 (continued): e)
f) To find the price x that maximizes R(x), we find
3.6 An Economics Application: Elasticity of Demand
R(x) x D(x)
R(x) x(120 20x)
R(x) 120x 20x2
R (x) 120 40x
R (x).
Slide 3.6 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
120 40x 0
40x 120
x 3
R (x) 40 0
Slide 3.6 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
Slide 3.6 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
3.6 An Economics Application: Elasticity of Demand
Slide 3.6 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
3.6 An Economics Application: Elasticity of Demand
Slide 3.6 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
3.6 An Economics Application: Elasticity of Demand
Top Related