Copyright © Cengage Learning. All rights reserved. 12 Further Applications of the Derivative

Copyright © Cengage Learning. All rights reserved Elasticity

33 You manufacture an extremely popular brand of sneakers and want to know what will happen if you increase the selling price. Common sense tells you that demand will drop as you raise the price. But will the drop in demand be enough to cause your revenue to fall? Or will it be small enough that your revenue will rise because of the higher selling price?

44 Elasticity For example if you raise the price by 1%, you might suffer only a 0.5% loss in sales. In this case, the loss in sales will be more than offset by the increase in price and your revenue will rise. In such a case, we say that the demand is inelastic, because it is not very sensitive to the increase in price. On the other hand, if your 1% price increase results in a 2% drop in demand, then raising the price will cause a drop in revenues.

55 Elasticity We then say that the demand is elastic because it reacts strongly to a price change. We can use calculus to measure the response of demand to price changes if we have a demand equation for the item we are selling. We need to know the percentage drop in demand per percentage increase in price. This ratio is called the elasticity of demand, or price elasticity of demand, and is usually denoted by E.

66 Elasticity Let’s derive a formula for E in terms of the demand equation. Assume that we have a demand equation q = f (p) where q stands for the number of items we would sell (per week, per month, or what have you) if we set the price per item at p. Now suppose we increase the price p by a very small amount,  p.

77 Elasticity Then our percentage increase in price is (  p/p) × 100%. This increase in p will presumably result in a decrease in the demand q. Let’s denote this corresponding decrease in q by –  q (we use the minus sign because, by convention,  q stands for the increase in demand). Thus, the percentage decrease in demand is (–  q/q) × 100%.

88 Elasticity Now E is the ratio so Canceling the 100%s and reorganizing, we get

99 Elasticity 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 the formula We say that the demand is elastic if E > 1, is inelastic if E < 1, and has unit elasticity if E = 1.

10 Elasticity Quick Example Suppose that the demand equation is q = 20,000 – 2p where p is the price in dollars. Then If p = $2,000, then E = 1/4, and demand is inelastic at this price. If p = $8,000, then E = 4, and demand is elastic at this price.

11 Elasticity If p = $5,000, then E = 1, and the demand has unit elasticity at this price. In the above example, if p = $2,000 then the demand would drop by only for every 1% increase in price. To see the effect on revenue, we use the fact that, for small changes in price, Percentage change in revenue ≈ Percentage change in price + Percentage change in demand

12 Elasticity Thus, the revenue will increase by about 3/4%. Put another way: If the demand is inelastic, raising the price increases revenue.

13 Elasticity On the other hand, if the price is elastic (which ordinarily occurs at a high unit price), then increasing the price slightly will lower the revenue, so: If the demand is elastic, lowering the price increases revenue. The price that results in the largest revenue must therefore be at unit elasticity.

14 Example 1 – Price Elasticity of Demand: Dolls Suppose that the demand equation for Bobby Dolls is given by q = 216 – p 2, where p is the price per doll in dollars and q is the number of dolls sold per week. a. Compute the price elasticity of demand when p = $5 and p = $10, and interpret the results. b. Find the ranges of prices for which the demand is elastic and the range for which the demand is inelastic c. Find the price at which the weekly revenue is maximized. What is the maximum weekly revenue?

15 Example 1 – Solution a. The price elasticity of demand is Taking the derivative and substituting for q gives When p = $5,

16 Example 1 – Solution cont’d Thus, when the price is set at $5, the demand is dropping at a rate of 0.26% per 1% increase in the price. Because E < 1, the demand is inelastic at this price, so raising the price will increase revenue. When p = $10,

17 Example 1 – Solution cont’d Thus, when the price is set at $10, the demand is dropping at a rate of 1.72% per 1% increase in the price. Because E > 1, demand is elastic at this price, so raising the price will decrease revenue; lowering the price will increase revenue. b. and c. We answer part (c) first. Setting E = 1, we get

18 Example 1 – Solution cont’d Thus, we conclude that the maximum revenue occurs when We can now answer part (b): The demand is elastic when p > $8.49 (the price is too high), and the demand is inelastic when p < $8.49 (the price is too low). Finally, we calculate the maximum weekly revenue, which equals the revenue corresponding to the price of $8.49: R = qp

19 Example 1 – Solution cont’d = (216 – p 2 )p = (216 – 72) = 144 ≈ $1,222.