5.3 Applications of the Natural Logarithm
Function to Economics
In section 5.3, we will deal with two applications of
natural logarithms in economics
• Relative rates of change
• Elasticity of demand
Relative Rates of Change
Recall that the logarithmic derivative of a function
f(t) is defined by the equation
d
f ' (t )
ln f (t ) 
dt
f (t )
The quantity on either side of this equation is often
called the relative rate of change of f(t) per unit
change of t, since it compares the rate of change of
f(t) with itself.
The percentage rate of change is the relative rate
of change of f(t) expressed as a percentage.
Suppose that a certain school of economists
modeled the Gross Domestic Product of the United
States at a time t (measured in years from January 1,
1990) by the formula
f (t )  3.4  .04t  .13e
t
where the Gross Domestic Product is measured in
trillions of dollars.
What was the predicted rate of growth (or decline)
of the economy at t = 0 and t = 1?
Solution
f ' (t )  .04t  .13e
Then
t
,
f ' (0) .04  .13
.09


 2.6%
f (0) 3.4  .13
3.53
So, on January 1, 1990, the economy is predicted to
contract at a relative rate of 2.6% per year.
1
f ' (1)
.04  .13e
.00782


 .2%
1
f (1) 3.4  .04  .13e
3.4878
On January 1, 1991, the economy is predicted to
still be contracting but only at a relative rate of .2%
per year.
Suppose the value in dollars of a certain business
investment at time t may be approximated
empirically by the function
f (t )  750,000e
.6 t
Use a logarithmic derivative to describe how fast
the value of the investment is increasing when
t = 5 years.
f ' (t ) d
d
.6
 ln f (t )  ln 750,000e
f (t ) dt
dt
d
.6 t
 (ln 750,000  ln e )
dt
d
d
 ln 750,000  .6 t
dt
dt
1
d
 0  .6t 2
dt
1
 1   2 .3
 (.6) t 
t
2
t
When t = 5,
f ' (5) .3

 .134  13.4%
f (5)
5
When t = 5 years, the value of the investment is
increasing at the relative rate of 13.4% per year.
Elasticity of Demand
Recall that in section 2.7 we discussed demand
equations, which relate price and quantity of a
produced item for monopolists and an entire
industry.
We want to rewrite the demand equation so show
quantity as a function of price.
q = f(p)
Usually, raising price decreases demand. So, q =
f(p) has a negative slope and is decreasing
everywhere.
The derivative f’(p) compares the change in
quantity demanded to the price.
The concept of elasticity is designed to compare
the relative rate of change of the quantity
demanded with the relative rate of change in price.
Consider a particular demand function q = f(p) and a
particular price p.
The ratio of the relative rates of change of the
quantity demanded and the price is given by
d
ln f ( p)
[relative rate of change of quantity] dp

d
[relative rate of change of price ]
ln p
dp
f ' ( p)
pf ' ( p)
f ( p)


1
f ( p)
p
Since f’(p) is always negative for a typical demand
function, the quantity
pf ' ( p)
f ( p)
will be negative for all values of p.
Economists like to work with positive numbers, so
the elasticity of demand is the above quantity
multiplied by –1.
The elasticity of demand E(p) at price p for the
demand function q = f(p) is defined to be
pf ' ( p)
E ( p)  
f ( p)
Suppose that the demand function for a certain
metal is q = 100 – 2p, where p is the price per
pound and q is the quantity demanded In millions
of pounds.
1. What quantity can be sold at $30 per pound?
2. Determine the function E(p).
3. Determine and interpret the elasticity of demand
at p = 30.
4. Determine and interpret the elasticity of demand
at p = 20.
1. What quantity can be sold at $30 per pound?
q = f(p), f(p) = 100-2p
When p = 30, q = f(30) = 100 – (2)(30) = 40
2. Determine the function E(p).
pf ' ( p)
p(2)
2p
E ( p)  


f ( p)
100  2 p 100  2 p
3. Determine and interpret the elasticity of
demand at p = 30.
2(30)
60 3
E (30) 


100  2(30) 40 2
When price is set at $30 per pound, a small increase
in price will result in a relative rate of decrease in
quantity demanded of about 3/2 times the relative
rate of increase in price.
Example: If the price of $30 is increased by 1%,
then quantity demanded will decrease by (3/2)(1%)
= 1.5%
4. Determine and interpret the elasticity of demand
at p = 20.
2(20)
40 2
E (20) 


100  2(20) 60 3
When the price is set at $20 per pound, a small
increase in price will result in a relative rate of
decrease in quantity demanded by only 2/3 of the
relative rate of increase of price.
Example: If the price is increased from $20 by 1%,
the quantity demanded will decrease by (2/3)(1%).
Economists say that demand is elastic at price p0
if E(p0) > 1.
Economists say that demand is inelastic at price
p0 if E(p0) < 1.
To better understand the concept of elasticity, let us
consider revenue.
Recall the relationship between revenue and price
[revenue] = [quantity] x [price per unit] or
R( p)  f ( p)  p
If we differentiate R(p) using the product rule, we
find
f(p) will always be positive.
When demand is elastic at some price p0, E(p0) > 1
and 1 – E(p0) is negative. So, R’(p0) is negative and
R(p0) is decreasing.
That is, an increase in price will result in a
decrease in revenue, and a decrease in price will
result in an increase in revenue.
When demand is inelastic at some price p0, E(p0) <
1 and 1 – E(p0) is positive. So, R’(p0) is positive
and R(p0) is increasing.
That is, an increase in price will result in an
increase in revenue, and a decrease in price will
result in a decrease in revenue.
Summary
The change in revenue is in the opposite direction of
the change in price when demand is elastic and in the
same direction when demand is inelastic.