The student will learn about: two applications to economics: relative rates and elasticity of demand.

You 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.
Assets $30,000,000
Perhaps it is the Board of Coke Cola International.
Assets $89,430,000,000
2

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.
3

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 ).
4

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.
5

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

f '(x) f (x)
If the gross domestic product in trillions of dollars t years from now is predicted by G (t )

8.2 e t 
8.2 e t
1
2
Find the relative rate of change 25 years from now.
We first need G’ (t)
G ‘ (t) = 8.2 e t
1
2  chain

8.2 e t
1
2 
1
2 t

1
2
RRC

G '(t)
G (t)

8.2 e t
1
2 
1
8.2 e t
2
1
2 t

1
2
 1
2 t

1
2
When t

25, RRC

1
2
25

1
2

1

1
2 25

1 1
2 5
1
10

10 %

f '(x) f (x)
If the gross domestic product in trillions of dollars t years from now is predicted by G (t )

8.2 e t 
8.2 e t
1
2
Find the relative rate of change 25 years from now.
I find the following method easier.
RRC

G '(t)
G (t)
We can use our calculators to find these two numbers.
RRC

G '(t)

121.70
G (t) 1217

0.10

10%

Price demand equations have in the past been used to express price as a function of demand. That is, p
 f ( x )

2000
400
 x
However, this function can be solved for x and thus one has demand as a function of price.
x
 f ( p )

2000

400 p
Indeed the function can be written in the form x + 400p = 2000

Remember that it is generally true that as price increases demand decreases and as price decreases demand increases .
x Price increase
Demand decrease
Demand increase
Price decrease p revenue .

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.
We need some way to evaluate what is happening to revenue.
Elasticity of Demand is the tool we need.
11

The question is whether the rise in one is enough to compensate for the fall in the other.
For example, if a 1% price decrease brings a 2% quantity increase, revenue will rise. These are related rates.
R = p
 x
1 = 1

1
1.01 = 0.99

1.02
Revenue increases by 1%.
12

The question is whether the rise in one is enough to compensate for the fall in the other.
For example, if the 1% price decrease brings only a ½% quantity increase, revenue will fall. These are related rates.
R = p
 x
1 = 1

1
0.995 = 0.99

1.005
Revenue decreases by ½ %.
13

14

Roughly speaking, we may think of elasticity as the percentage change in demand divided by the percentage change in price:
But there is a better way!
15

Given x = D (p) a price demand equation, then the elasticity of demand is
 relative rate of relative rate of change change of demand of price
But there is a better way!
Elasticity of Demand -
If x = D (p) then elasticity of demand is
E ( p )
  p
 f ' ( p ) f ( p )

Intuitively, we may think of elasticity of demand as measuring how responsive demand is to price changes: elastic means responsive and inelastic means unresponsive .
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.
17

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

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 consumers need them even if prices rise, while for luxuries (restaurant meals) demand is elastic since consumers can cut back or find substitutes in response to price increases.
19

Use 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
E ( p )

E ( p )


 p
 f ' ( p ) f ( p ) p



2 p

1875
 p
2

2 p
2
1875
 p
2 continued

E ( p )

continued
 p



2 p

1875
 p
2

2 p
2
1875
 p
2
If price goes up the revenue will go up.
If price goes up the revenue will go down.

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.
R (p)
Inelastic
E(p) < 1
Elastic
E(p) > 1
On previous slide
p

ASSIGNMENT
§3.6 on my website.
4, 5, 10, 11, 12.