Elasticity and Expenditure
Definitions
• Elasticity = responsiveness of quantity
demanded to price.
• Coefficient of elasticity =
Percent change in quantity
Percent change in price
• Percent change in Q = Change in Q / Q
= Q2 - Q1 divided by Q1
• Percent change in P = Change in P / P
= P2 - P1 divided by P1
Relationship to expenditure
• % Δ (PQ) = % Δ P + % Δ Q
• If % Δ P = 3 percent and % Δ Q = -10 % ,
then % Δ (PQ) = - 7 % or revenue declines
• Coef of elas = ε = % Δ Q divided by % Δ P
hence in the above example
ε = 10 / 3 = 3.33
The coefficient of elasticity is greater than 1,
so demand is ELASTIC
Problem:
% Δ Q = - 10 %, and % Δ P = 4 %
a. what is the coefficient of elasticity?
b. is demand elastic or inelastic?
c. what is the percent change in consumer
expenditure?
[See next slide for answers…but not before trying
to solve the problem]
Solution:
% Δ Q = - 10 %, and % Δ P = 4 %
a. what is the coefficient of elasticity?
ε = % Δ Q / % Δ P = 10 / 4 = 2.5
b. is demand elastic or inelastic?
-- since ε > 1.0, demand is elastic.
We should expect that consumer expenditure will decline
with an increase in price.
c. what is the percent change in consumer expenditure?
Since % Δ PQ = % Δ P + % Δ Q, we have
% Δ PQ = 4 % - 10 % = - 6 %
Problem:
% Δ Q = + 10 %, and % Δ Revenue = 3 %
a. what is the percent change in price?
b. what is the coefficient of elasticity?
c. is demand elastic or inelastic?
d. what is the percent change in consumer expenditure?
[See next slide for answers…but not before trying to solve
the problem]
Solution:
% Δ Q = + 10 %, and % Δ Revenue = + 3 %
a. what is the percent change in price?
Since % Δ PQ = % Δ P + % Δ Q
3 % = % Δ P + 10 %
Price must have fallen by 7 percent
b. what is the coefficient of elasticity?
Since ε = % Δ Q / % Δ P = 10 % / 7 % = 1.43
c. is demand elastic or inelastic? Elastic demand
– a reduction in price leads to an increase in revenue
-- the coefficient of elasticity is greater than 1.
d. what is the percent change in consumer expenditure?
The same as the percent change in revenue, + 3 %
Problem:
% Δ P = + 10 %, and % Δ Revenue = - 3 %
a. what is the percent change in quantity?
b. what is the coefficient of elasticity?
c. is demand elastic or inelastic?
d. what is the percent change in consumer expenditure?
[See next slide for answers…but not before trying to solve
the problem]
Solution:
% Δ Q = + 10 %, and % Δ Revenue = - 3 %
a. what is the percent change in price?
Since % Δ PQ = % Δ P + % Δ Q
- 3 % = % Δ P + 10 %
Price must have fallen by 13 percent
b. what is the coefficient of elasticity?
Since ε = % Δ Q / % Δ P = 10 % / 13 % =
c. is demand elastic or inelastic? Inelastic demand
– a reduction in price leads to a decrease in revenue
-- the coefficient of elasticity is less than 1.
d. what is the percent change in consumer expenditure?
The same as the percent change in revenue, - 3 %
Problem:
% Δ P = + 10 %, and ε = 2.5
a. what is the percent change in quantity?
b. is demand elastic or inelastic?
c. what is the percent change in consumer
expenditure?
[See next slide for answers…but not before trying
to solve the problem]
Solution:
% Δ P = + 10 %, and ε = 2.5
a. what is the percent change in quantity?
Since ε = 2.5 = % Δ Q / % Δ P = % Δ Q / 10
We have % Δ Q = - 25 % (when price goes up, Q down)
b. is demand elastic or inelastic?
Since ε = 2.5, demand is ELASTIC; we should expect
consumer expenditure to decrease when price
increases.
c. what is the percent change in consumer expenditure?
Since % Δ PQ = % Δ P + % Δ Q, we have
% Δ PQ = + 10 - 25 % = - 15%
Problem:
% Δ P = + 5 %, and ε = 1/4
a. what is the percent change in quantity?
b. is demand elastic or inelastic?
c. what is the percent change in consumer
expenditure?
[See next slide for answers…but not before trying
to solve the problem]
Solution:
% Δ P = + 5 %, and ε = ¼
a. what is the percent change in quantity?
Since ε = ¼ = % Δ Q / % Δ P = % Δ Q / 5
We have % Δ Q = - 1.25 % (when price goes up, Q down)
b. is demand elastic or inelastic?
Since ε = ¼ , demand is INELASTIC; we should expect
consumer expenditure to increase when price increases.
c. what is the percent change in consumer expenditure?
Since % Δ PQ = % Δ P + % Δ Q, we have
% Δ PQ = + 5 % - 1.25 % = + 3.75%
Calculating elasticity from a table
Price
Quantity % Δ P
1
180
2
160
3
140
4
120
5
100
%ΔQ
Elasticity
Calculating elasticity from a table
Price
Quantity % Δ P
1
180
2
3
4
5
160
140
120
100
2–1/1
= 100 %
3–2/2
= 50 %
4–3/3
= 33 %
5–4/4
= 25 %
6–5/5
= 20 %
%ΔQ
Elasticity
20 / 180
= 11.1 %
20 / 160
= 12.5 %
20 / 140
= 14.3 %
20 / 120
= 16.7%
20 / 100
= 20 %
0. 111
(inelastic)
0. 25
(inelastic)
0 . 43
(inelastic)
0. 67
(inelastic)
1. 00
(unit elastic)
Elasticity and slope
The table was derived from the demand equation:
Q = 200 - 20 P
Note that for each dollar price goes up, Q goes
down by 20 units.
In mathematical symbolism,
Δ Q / Δ P = - 20
This is close to the formula for the coefficient of
elasticity, but not quite the same. What is the
difference? [Pause for thought…]
Elasticity and slope
Slope = Δ Q / Δ P = - 20
But coefficient of elasticity =
Δ Q / Q divided by Δ P / P
The expression for the coefficient of
elastictity can be rearranged to get
ε = Δ Q / Δ P times P / Q
Which can be a handy computational rule for
elasticity AT a point.
Problem: given the demand equation
Q = 1500 – 25 P
What is the coefficient of elasticity at:
P = 10
?
First note that Q = 1500 – 250 = 1250,
So that the coefficient of elasticity is:
ε = Δ Q / Δ P times P / Q
ε = 25 (10) / 1250 = 250 / 1250
ε = 0. 20
Problem: given the demand equation
Q = 1500 – 25 P
What is the coefficient of elasticity at:
P = 50 ?
First note that Q = 1500 – 1250 = 250,
So that the coefficient of elasticity is:
ε = Δ Q / Δ P times P / Q
ε = 25 (50 / 250) = 1250 / 250
ε = 5.0