11

The student will learn about:

§3.6 Applications to Economics.

two applications to economics: relative rates and elasticity of demand.

2

Relative Versus Absolute RatesYou are sitting in a board meeting and the CFO – Chief Financial Officer – reports that profits increased $1,000,000 last year. What is your response?

Perhaps it is the Board of York Educational Federal Credit Union.

Perhaps it is the Board of Coke Cola International.

Assets $30,000,000

Assets $89,430,000,000

3

Relative Versus Absolute Rates

We need to take into account the enormous difference between these two business organizations.

Clearly we need an additional tool to assist us in making decisions.

4

Relative Rates of Change

If f (t) is the price of an item at time t, then the rate of change is f ‘ (t), and the relative rate of change is f ′ (t)/f(t), the derivative divided by the function.

We will sometimes call the derivative f ′ (x) the “absolute” rate of change to distinguish it from the relative rate of change f ′ (x)/f(x).

5

Relative Rates of Change

Relative rates are often more meaningful than absolute rates.

For example, it is easier to grasp the fact that the gross domestic product is growing at the relative rate of 2.2% a year than that it is growing at the absolute rate of $345,000,000,000 per year.

66

Relative Rate of Change

The relative rate of change of a function f (x) is

f '(x)

f (x)

77

ExampleIf the gross domestic product in trillions of dollars t years from now is predicted by

Find the relative rate of change 25 years from now.

We first need G’ (t)

G ‘ (t) =

12t tG (t ) 8.2e 8.2e

12t8.2e chain

12

1t 2

18.2e t

2

G '(t)RRC

G (t)

12

12

1t 2

t

18.2e t

2

8.2e

1

21

When t 25, RRC 252

f '(x)

f (x)

1

21

t2

1 1

2 25

1 1 110%

2 5 10

88

ExampleIf the gross domestic product in trillions of dollars t years from now is predicted by

Find the relative rate of change 25 years from now.

We can use our calculators to find these two numbers.

I find the following method easier.

12t tG (t ) 8.2e 8.2e

f '(x)

f (x)

G '(t)RRC

G (t)

G '(t) 121.70RRC 0.10 10%

G (t) 1217

99

Price-Demand ReviewPrice demand equations have in the past been used to express price as a function of demand. That is,

Indeed the function can be written in the form x + 400p = 2000

However, this function can be solved for x and thus one has demand as a function of price.

p4002000)p(fx

400

x2000)x(fp

1010

Price-Demand Review

We need something more powerful to help us make price demand decisions.

Remember that it is generally true that as price increases demand decreases and as price decreases demand increases.

p

x Price increase

Demand decrease

Price decrease

Demand increase

We need something more powerful to help us make price demand decisions. We are really interested in revenue.

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Elasticity of Demand

Remember revenue is price times quantity,

R = p . q,

and based on the relationship between price and demand, when one of these quantities rises, the other falls.

Elasticity of Demand is the tool we need.

We need some way to evaluate what is happening to revenue.

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Elasticity of Demand

For example, if a 1% price decrease brings a 2% quantity increase, revenue will rise. These are related rates.

Revenue increases by 1%.

R = p x

1.01 = 0.99 1.02

The question is whether the rise in one is enough to compensate for the fall in the other.

1 = 1 1

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Elasticity of Demand

For example, if the 1% price decrease brings only a ½% quantity increase, revenue will fall. These are related rates.

Revenue decreases by ½ %.

R = p x

0.995 = 0.99 1.005

The question is whether the rise in one is enough to compensate for the fall in the other.

1 = 1 1

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The concept of elasticity of demand was invented to analyze

such problems.

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Roughly speaking, we may think of elasticity as the percentage change in demand divided by the percentage change in price:

Elasticity of Demand

But there is a better way!

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Elasticity of DemandGiven x = D (p) a price demand equation, then the elasticity of demand is

priceofchangeofraterelative

demandofchangeofraterelative

)p(f

)p('fp)p(E

Elasticity of Demand -

If x = D (p) then elasticity of demand is

But there is a better way!

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Intuitively, we may think of elasticity of demand as measuring how responsive demand is to price changes: elastic means responsive and inelastic means unresponsive.

Elasticity of Demand

That is, for elastic demand, a price cut will bring a large increase in demand, so total revenue will rise.

On the other hand, for inelastic demand, a price cut will bring only a slight increase in demand, so total revenue will fall.

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Elasticity of Demand - Interpretation

E (p) Demand Interpretation

E (p) < 1 Inelastic p↑ then R↑ OR p↓ then R ↓

E (p) > 1 Elastic p↑ then R ↓ OR p↓ then R ↑

E (p) = 1 Unit A change in price produces the same change in demand.

E

19

Economists calculate elasticity of demand for many products, and some typical elasticities are shown in the table.

Notice that for necessities(clothing, food), demand is inelastic since consumersneed them even if prices rise,while for luxuries (restaurant meals)demand is elastic since consumerscan cut back or find substitutesin response to price increases.

Elasticity of Demand

2020

ExampleUse the price-demand equation to determine whether demand is elastic, inelastic, or has unit elasticity at the indicated values of p. x = f (p) = 1875 - p 2

continued

)p(f

)p('fp)p(E

2

2

2 p1875

p2

p1875

p2p)p(E

2121

Example continued

If p = 15, then E (15) =

2

2

2 p1875

p2

p1875

p2p)p(E

If p = 15, then E (15) = 0.27 < 1; If p = 15, then E (15) = 0.27 < 1; demand is inelastic

If p = 25, then E (25) =If p = 25, then E (25) = 1; If p = 25, then E (25) = 1; demand has unit elasticity

If p = 40, then E (40) =If p = 40, then E (40) = 11.64 > 1If p = 40, then E (40) = 11.64 > 1; demand is elastic

If price goes up the revenue will go up.

If price goes up the revenue will go down.

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Revenue and Elasticity of Demand

If demand is inelastic, then

a price increase will increase revenue.

a price decrease will decrease revenue.

If demand is elastic, then

a price increase will decrease revenue.

a price decrease will increase revenue.

On previous slide

p

R (p)

Inelastic Elastic

E(p) < 1 E(p) > 1

2323

ASSIGNMENT

§3.6 on my website.

4, 5, 10, 11, 12.