Section 12.5 Elasticity
Price elasticity of demand refers to the
way demand changes as price changes.
►If you raise the price of popcorn at
movies, people will buy less popcorn in
the short term. Raising the price will
cause a drop in revenue. Demand is
elastic because it reacts strongly to
price change.
The Law of Supply and Demand says
that if you raise the price of a
commodity, the demand will drop. If
demand drops, it makes sense to
assume that revenue will drop, but
sometimes it is safe to raise the price of
a commodity when demand is strong.
If you raise your price 2%, but demand
drops ½%, revenue will still increase.
Here the demand is inelastic because it
does not react strongly to raises in price.
Almost anything trendy is an example of
this.
►WV raised the price of the new bug
several times as people were placing
orders for them when they first came
out.
Elasticity is the ratio of the percent
decrease in demand to the percent
increase in price.
percent decrease in demand
E
percent increase in price or
Change in demand

demand
E
Change in price
price
We could write this as
p

 100%
p
E
q
 100%
q
q p
E

p q
Since q is a function of p, when we let
the change in p approach zero, we get
dq p
E 
dp q .
You don’t need to memorize this
process, but you need to know how to
use the definition.
When E < 1, demand is inelastic and
decreases less than 1% for each 1%
increase in price.
When demand is inelastic, raising the
price increases revenue.
When E > 1, demand is elastic and
decreases more than 1% for each 1%
increase in price.
When demand is elastic, lowering price
increases revenue.
When E = 1, demand is has unit
elasticity and decreases at the same
rate as the price.
This is the price where revenue is
maximized.
Ex: (like #9) Suppose the demand for
Borgia Bar candy is given by
q  100e
p 3 p 2 / 2
Where q is the quantity demanded and
p is the retail price in dollars.
a) Determine the price elasticity of
demand when the price is 3 dollars and
interpret the number.
dq p
E 
dp q
3 
p

p 3 p 2 / 2
E  1  2 p 100e
p 3 p 2 / 2
2 

100e
E  3 p  1 p
When p = 3, E = 24
So, at a price of $3, quantity demanded
is decreasing 24% for each 1% increase
in price. The price should be lowered in
order to increase revenue.
b) What price will maximize revenue?
Revenue is a maximum when E = 1, so
1 = 3p2 – p
3p2 – p – 1 = 0 can’t be factored, so the
quadratic formula gives p = .7675918…
So, the price that maximizes revenue is
$0.77 per bar.
c) About how many bars will be sold
each month at this price?
p 3 p 2 / 2
q  100e
0.773.77 2 / 2
q  100e
About 89 bars will be sold at a price of
$0.77.
Example: Demand for some items
depends on a consumer’s income.
#16 page 822 is an example.
The likelihood that a child will attend a
live theatrical performance can be
modeled by
2
q  0.01 0.0006 x  0.38x  35
15 ≤ x ≤ 100
Here, q is the fraction of children with
annual household income x in
thousands of dollars who will attend a
live musical performance during a year.
Compute the income elasticity of
demand at an income level of $30,000
and interpret the result.
Since we expect that demand will
increase as income increases,


dq x
E 
dx q
E  0.010.0012 x  0.38

x
0.01 0.0006 x 2  0.38 x  35
Use the calculator

Y1  0.01 0.0006 x  0.38x  35
2
Y2 = nDeriv(Y1,X,X)*X/Y1


Set ∆table to 10
TABLE shows
q = 0.4694 and
E = 0.26587
Note increase in E as x increases.
So, about 47 out of every 100 children
with a household income of $30,000 will
attend a musical performance during a
year, and the number increases by
about 27% for every 1% increase in
income.