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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 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

Slide 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

Slide 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

Slide 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 $2 per 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

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 (continued): 3.6 An Economics Application: Elasticity of Demand

Slide 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

Slide 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

Slide 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 $ An Economics Application: Elasticity of Demand

Slide 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

Slide 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

Slide 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) = An Economics Application: Elasticity of Demand

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Elasticity and Revenue For 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 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.6 An Economics Application: Elasticity of Demand

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.6 An Economics Application: Elasticity of Demand