Math 1314
Lesson 10
Elasticity of Demand
Suppose you owned a small business and needed to make some decisions about the pricing of
your products. It would be helpful to know what effect a small change in price would have on
the demand for your product. If a price change will have no real change on demand for the
product, it might make good sense to raise the price. However, if a price increase will cause a
big drop in demand, then it may not be a good idea to raise prices.
There is a measure of the responsiveness of demand for product or service to a change in its
price: elasticity of demand. This is defined as
percentage change in demand
.
percentage change in price
To develop this formula, we’ll start by solving our demand function for x, so that we have a
function x = f ( p ) . Then we have a demand function in terms of price. If we increase the price
by h dollars, then the price is p + h and the quantity demanded is f ( p + h) .
The percentage change in demand is 100 ⋅
100 ⋅
f ( p + h) − f ( p )
and the percentage change in price is
f ( p)
h
.
p
100 ⋅
If we compute the ratio given above, we have
f ( p + h) − f ( p )
f ( p)
.
h
100 ⋅
p
We can simplify this to
p  f ( p + h) − f ( p ) 
 .
f ( p) 
h
For small values of h,
f ( p + h) − f ( p )
≈ f ′( p ) , so we have
h
p ⋅ f ′( p )
.
f ( p)
This quantity is almost always negative, and it would be much easier to work with a positive
quantity. So the negative of this ratio is the elasticity of demand.
Elasticity of Demand
p ⋅ f ′( p )
, where p is price and f ( p) is the demand function and is differentiable at
E ( p) = −
f ( p)
x=p.
Lesson 10 – Elasticity of Demand
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Revenue responds to elasticity in the following manner:
If demand is elastic at p, then
• An increase in unit price will cause revenue to decrease or
• A decrease in unit price will cause revenue to increase
If demand is unitary at p, then
• An increase/decrease in unit price will cause the revenue to stay about the same.
If demand is inelastic at p, then
• An increase in the unit price will cause revenue to increase
• A decrease in unit price will cause revenue to decrease.
We have these generalizations about elasticity of demand:
Demand is said to be elastic if E ( p ) > 1 .
Demand is said to be unitary if E ( p ) = 1 .
Demand is said to be inelastic if E ( p ) < 1 .
So, if demand is elastic, then the change in revenue and the change in price will move in opposite
directions.
If demand is inelastic, then the change in revenue and the change in price will move in the same
direction.
Example 1: Find E ( p ) for the demand function x + 2 p − 15 = 0 and determine if demand is
elastic, inelastic or unitary when p = 4.
Lesson 10 – Elasticity of Demand
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Example 2: Suppose the demand function for a product is given by p = −0.02 x + 400 . This
function gives the unit price in dollars when x units are demanded.
a. Find the elasticity of demand.
b. Find E (100) and interpret the results.
c. Find E (300) and interpret the results.
d. If the unit price is $100, will raising the price result in an increase in revenues or a decrease in
revenues?
e. If the unit price is $300, will raising the price result in an increase in revenues or a decrease in
revenues?
Lesson 10 – Elasticity of Demand
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What else can E ( p ) tell you?
Example 3: If E ( p ) =
1
when p = 250 , what effect will a 1% increase in price have on
2
revenue?
Example 4: If E ( p ) =
3
when p = 250 , what effect will a 1% increase in price have on
2
revenue?
Popper 1: If demand is elastic at p, then a decrease in unit price will cause revenue to increase.
a. True
b. False
Lesson 10 – Elasticity of Demand
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Math 1314
Lesson 11: Exponential Functions as Mathematical Models
Exponential models can be written in two forms: f ( x ) = a ⋅ e x or g ( x ) = a ⋅ b x . In the first
model, the base of the exponent is the number e, which is approximately 2.71828…. In the
second model, the base of the exponent is a positive number other than 1. In both cases, the
variable is located in the exponent, and that’s why these are called exponential models.
Exponential functions can be either increasing or decreasing.
The function f ( x ) = a ⋅ ebx is an exponential growth function and it’s increasing.
The function f ( x ) = a ⋅ e −bx is an exponential decay function and it’s decreasing.
In both equations the variable a is called the initial amount and b is called the growth or decay
constant, depending on the type of function.
For a function of the form g ( x ) = a ⋅ b x , the function is an exponential growth function if b > 1
and is an exponential decay function if 0 < b < 1 .
We can also compute the rate at which an exponential function is increasing or decreasing.
We’ll do this by finding a numerical derivative.
We can use the regression feature to find an exponential equation for data that’s given.
Popper 2 is D
Example 1: Identify each function as a growth function or a decay function. Find the initial
value. Calculate f (30) and f '(30) .
a. f (t ) = 12.8e −0.285t
b. g ( x) = 38.6(1.0489) x
Example 2: Suppose the points (0, 10000) and (2, 10940) lie on the graph of a function. Find
the equation of the function in the form f ( x ) = a ⋅ ebx using GeoGebra, assuming that the
function is exponential.
Begin by entering the two given points in the spreadsheet, then make a list.
Command:
Answer:
Lesson 11 – Exponential Functions as Mathematical Models
1
Uninhibited Exponential Growth
Some common exponential applications model uninhibited exponential growth. This means
that there is no “upper limit” on the value of the function. It can simply keep growing and
growing. Problems of this type include population growth problems and growth of investment
assets.
Example 3: A biologist wants to study the growth of a certain strain of bacteria. She starts
with a culture containing 25,000 bacteria. After three hours, the number of bacteria has grown to
63,000. Assume the population grows exponentially and the growth is uninhibited.
a. Find the equation of the function in the form f ( x ) = a ⋅ ebx using GGB.
State the two points given in the problem.
Enter the two points in the spreadsheet, then make a list.
Command:
Answer:
b. How many bacteria will be present in the culture 6 hours after she started her study?
Popper 3: Given the following function, state the initial value for the function.
= 54.65 .
a. 0.3541
b.
54.65
Lesson 11 – Exponential Functions as Mathematical Models
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Example 4: The sales from company ABC for the years 1998 – 2003 are given below.
1998 1999 2000 2001 2002 2003
Year
profits in millions of dollars 51.4 53.2 55.8 56.1 58.1 59.0
Rescale the data so that x = 0 corresponds to 1998. Begin by making a list.
a. Find an exponential regression model for the data.
Command:
Answer:
b. Find the rate at which the company's sales were changing in 2007.
Command:
Answer:
Exponential Decay
Example 5: At the beginning of a study, there are 50 grams of a substance present. After 17
days, there are 38.7 grams remaining. Assume the substance decays exponentially.
a. State the two points given in the problem.
Enter the two points in the spreadsheet and make a list.
b. Find an exponential regression model.
Command:
Answer:
c. What will be the rate of decay on day 40 of the study?
Command:
Answer:
Exponential decay problems frequently involve the half-life of a substance. The half-life of a
substance is the time it takes to reduce the amount of the substance by one-half.
Lesson 11 – Exponential Functions as Mathematical Models
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