§ 5.3
Applications of the Natural Logarithm
Function to Economics
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 47
Section Outline

Relative Rates of Change

Elasticity of Demand
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 47
Relative Rate of Change
Definition
Example
Relative Rate of Change: An example will be
The quantity on either
given immediately
side of the equation
hereafter.
d
f t 
ln f t  
dt
f t 
is often called the
relative rate of change of
f (t) per unit change of t
(a way of comparing
rates of change for two
different situations).
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 47
Relative Rate of Change
EXAMPLE
(Percentage Rate of Change) Suppose that the price of wheat per bushel at time
t (in months) is approximated by
f t   4  0.001t  0.01e t .
What is the percentage rate of change of f (t) at t = 0? t = 1? t = 2?
SOLUTION
Since
we see that
f t   0.001  0.01e t ,
f 0 0.001  0.01  0.009


 0.22%.
f 0
4  0.01
4.01
f 1
0.001  0.01e 1
 0.026


 0.65%.
1
f 1 4  0.0011  0.01e
4.0047
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 47
Relative Rate of Change
CONTINUED
f 2
0.001  0.01e 2
 0.00035


 0.0087%.
2
f 2 4  0.001 2  0.01e
4.0034
So at t = 0 months, the price of wheat per bushel contracts at a relative rate of
0.22% per month; 1 month later, the price of wheat per bushel is still
contracting, but more so, at a relative rate of 0.65%. One month after that
(t = 2), the price of wheat per bushel is contracting, but much less so, at a
relative rate of 0.0087%.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 47
Elasticity of Demand
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 47
Elasticity of Demand
EXAMPLE
(Elasticity of Demand) A subway charges 65 cents per person and has 10,000
riders each day. The demand function for the subway is q  2000 90  p .
(a) Is demand elastic or inelastic at p = 65?
(b) Should the price of a ride be raised or lowered in order to increase the
amount of money taken in by the subway?
SOLUTION
(a) We must first determine E(p).
q  f  p   2000 90  p
q  f  p  
 1000
90  p
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 47
Elasticity of Demand
CONTINUED
  1000 

 p
90  p 
 pf  p 
p

E p 


f  p
2000 90  p 180  2 p
Now we will determine for what value of p E(p) = 1.
p
1
180  2 p
p  180  2 p
3 p  180
p  60
Set E(p) = 1.
Multiply by 180 – 2p.
Add 2p to both sides.
Divide both sides by 3.
So, p = 60 is the point at which E(p) changes from elastic to inelastic, or visa
versa.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 47
Elasticity of Demand
CONTINUED
Through simple inspection, which we could have done in the first place, we can
determine whether the value of the function E(p) is greater than 1 (elastic) or
less than 1 (inelastic) at p = 65.
E 65 
65
 1. 3  1
180  2  65
So, demand is elastic at p = 65.
(b) Since demand is elastic when p = 65, this means that for revenue to
increase, price should decrease.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 47