Where It s Used. R. 1 Find f x. using the chain rule. d dx. Page 1 of 7
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1 Elasticity of Demand Learning Objectives: Find relative rates of change Find percentage rates of change Determine whether demand is elastic, inelastic, or has unit elasticity Find elasticity of demand Find values for which demand is elastic or inelastic Find revenue functions for a demand equation Solve applications involving elasticity of demand Retailers and manufacturers need to know how a small change in price will affect the demand for a product. In this section, we will learn about (price) elasticity of demand in which we will use calculus to predict how price change will affect demand, and hence, the revenue. If a small increase in price results in no change in demand, the price increase makes good business sense. However, if that small increase in prices results in a drop in demand, the increase is not wise. Where It s Used Business: Purchasing Stocks You received $2000 to invest as a HS graduation gift and are considering two stocks, MedStat and Comptec. Your broker expects MedStat s price per share to increase $15 per year for the next several years, and Comptec s price per share to increase by only $2 per year. Which is a better deal? Business: Revenue CoolShades, a manufacturer of souvenirs is currently selling a particular style of sunglasses for $15 per pair. If the price p and the demand x is related by the equation x f p p how will a price change affect revenue? 9, , d dx R. 1 Find f x ln using the chain rule. Page 1 of 7
2 R. 2 Reminder of concepts from introduction to microeconomics: Price elasticity of demand see The degree to which demand for a good or service varies with its price. Normally, sales increase with drop in prices and decrease with rise in prices. As a general rule, appliances, cars, confectionary and other non-essentials show elasticity of demand whereas most necessities (food, medicine, basic clothing) show inelasticity of demand (do not sell significantly more or less with changes in price). If elasticity of demand is less than 1, then an increase in price will result in an in revenue. List an item or two that may have an elasticity of demand less than 1 and justify why. If elasticity of demand is greater than 1, then an increase in price will result in a in revenue. List an item or two that may have an elasticity of demand greater than 1 and justify why. If the elasticity of demand is exactly 1, then how will a change in price affect the revenue? Relative and Percentage Rates of Change Ex. 1 You received $2000 to invest as a HS graduation gift and are considering two stocks, MedStat and Comptec. Your broker expects MedStat s price per share to increase $15 per year for the next several years, and Comptec s price per share to increase by only $2 per year. Which is a better deal? What really matters here is the. Recall R. 1 fx and we define this as f x, d which is ln f x. Note, this value is rate of change divided dx by the current value. Page 2 of 7
3 Ex. 2 For the situation described in Ex. 1, the MedStat s price per share is currently $200; and Comptec s price per is currently $8. Which is a better deal? Interestingly in this analysis, the amount of money invested does not matter in determining the best outcome. DEFINITIONS 1) The relative rate of change of a function f x is f x 2) The percentage rate of change is 100 %. f x f f x x. x f x 18 4 e. Ex. 3 Find the relative rate of change of 0.5 Page 3 of 7
4 f x 6x ln x; Ex. 4 Given the function: 2 A) Find the relative rate of change function. B) Find the relative rate of change at x 7. C) Find the percentage rate of change at x 7. Page 4 of 7
5 Ex. 5 Burger World can produce a hamburger for $1.25. If BW s daily sales are increasing at a rate of 20 hamburgers per day, how fast is its daily cost for hamburgers increasing? Elasticity of Demand Ex. 6 CoolShades, a manufacturer of souvenirs has determined that increasing the price of tourist beach towels by $1 will decrease the demand by 100 units. In-season, when they charge $10 per towel, they can sell 1,200. Off-season, when they charge $20 per towel, they sell 700. Assuming that the company wants to maximize revenue, when should they increase the price, in-season or off-season or never? Economists use elasticity of demand to answer the question When does an increase in price lead to an increase in revenue? DEFINITIONS 1) Price Elasticity of Demand Let the price p and demand x for a product be related by a pricedemand equation of the form x f p. Then the price elasticity of demand is E p relative rate of change of demand. relative rate of change of price THEOREM If price and demand are related by x f p, then the elasticity of demand is given by p p. pf E p This will be referred to as the point elasticity of demand. f Page 5 of 7
6 Ex. 7 CoolShades, a manufacturer of souvenirs currently sells a particular style of sunglasses. x f p p The price p and the demand x is related by the equation 9, A) Find the elasticity of demand. Elasticity of Demand Table: Value of E p Term Effect on Revenue 0E p 1 Inelastic A price increase will increase revenue. E p 1 Elastic A price increase will decrease revenue. E p 1 Unit A % change in price produces the same % change in demand. B) Find elasticity of demand when price is $15 per pair and interpret. Page 6 of 7
7 C) Find elasticity of demand when price is $21 per pair and interpret. D) Use the demand equation to find the revenue function. Sketch the revenue function, and indicate the regions of inelastic and elastic demand on the graph. Page 7 of 7
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