1. § 5.3
Applications of the Natural Logarithm Function to Economics

2. Section Outline
Relative Rates of Change
Elasticity of Demand

3. Relative Rate of Change
Definition
Example
Relative Rate of Change: The quantity on either side of the equation
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).
An example will be given immediately hereafter.

4. 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
What is the percentage rate of change of f (t) at t = 0? t = 1? t = 2?
SOLUTION
Since
we see that

5. Relative Rate of Change
CONTINUED
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 %.

6. Elasticity of Demand

7. 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
(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).

8. Elasticity of Demand
CONTINUED
Now we will determine for what value of p E(p) = 1.
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.

9. 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.
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.