cMarcia Drost August 15, 20071Math142 Lecture Notes4.7 Elasticity of DemandDefinition: Relative and Percentage Rates of Change• The relative rate of change of a function f (x)isf0(x)f(x).•The percentage rate of change is 100 ×f0(x)f(x).• Note: Sinceddxln f (x)=f0(x)f(x), then the relative rate of change of f(x) isthe derivative of the logarithm of f(x).Example 1: A broker is trying to sell you two stocks: Sunset Development and AggieGrowth Fund. The broker predicts the Sunset Development stocks will increase $2 per yearover the next few years, and Aggie Growth Fund will increase $1 per year over the sametime period. Which is the better buy?Do you need additional information?The cost of each stock is the missing information. Suppose the Sunset Development costs$50 per share and Aggie Growth Fund costs $20 per share.Stock Relative rate of change Percentage rate of changeSunset Development250= .044%Aggie Growth Fund120= .055%Example 2: Given the model for the population data is f (t)=2.5t+ 250 where t is thenumber of years since 1990. Find the relative rate of change of f(t) on the interval [5, 12].Year Real GDP in billions of dollars Population in millions1995 $7,540 262.7651996 $7,810 265.1901997 $8,150 267.7441998 $8,500 270.2991999 $8,850 272.8202000 $9,190 275.3062001 $9,210 277.8032002 $9,440 280.306cMarcia Drost August 15, 20072Elasticity of demand is a mathematical tool that can be used to measure the impact thata change in price has on the demand for a product. The term elasticity generally refers tohow sensitive the demand is to a change in price.Definition of Terms• Elastic: a small change in price produces a significant change in demand.E(p) > 1• Inelastic: a change in price does not affect demand.E(p) < 1• Unit: a change in price produces the same change in demand.E(p)=1Elasticity of DemandIf price and demand are related by x = f (p), then the elasticityof demand is given by E(p)=−p·f0(p)f(p)Suppose the price $p, and quantity x are related by the equation x+250p = 5275. In workingwith problems involving revenue, cost or profit, we usually use the demand equation, in otherwords, solve for x.x + 250p = 2000x = −250p + 2000f(p)=−250p + 2000Elasticity of demand if the negative of the ratio between relative rate of change of demandand the relative rate of change of price..r e l a t i v e r a t e o f c h a n g e o f d e m a n d r e l a t i v e r a t e o f c h a n g e o f p r i c e .E ( p ) = −− ____________________________________________E(p)=−p·f0(p)f(p)cMarcia Drost August 15, 20073Elasticity of DemandGiven the demand function x = d(p), the elasticity of demand, denoted E(p), of aproduct at a price p is given byE(p)=−p·f0(p)f(p)E(p)value demand is: change in price will cause: R0(p) is:Between 0 and 1 Inelastic relatively small changes in dema nd positive −→ raise price.Greater than 1 Elastic relatively large changes in demand negative −→ lower price.Equal to 1 Unitary relatively equal change in demand zero −→ keep price sameExample 3: A school’s junior business club is holding its annual raffle. Data collectedfrom raffles in the past indicate that the demand function for the tickets follows the modelp =√36 − x where p represents the price of a ticket and x represents the number of ticketseach member sells each day.(a) Solve to get the demand function in the form x = f (p)(b) Find the elasticity of demand E(p).(c) Evaluate E(3), interpret the results, and determine any adjustments the business clubshould make to the price of the tickets.(d) Evaluate E(4), interpret the results, and determine any adjustments the business clubshould make to the price of the tickets.cMarcia Drost August 15, 20074pR ( p )..Example 4: Find E(p) for the price-demand equation x = f (p)=10,000 − 500p andinterpret each of the following.• E(4)• E(16)•