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EECM 3714
Lecture 6: Unit 6

Elasticity
Renshaw, Ch
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Price elasticity of supply

• Example
2
...


Relationship between MR and price elasticity of demand

4
...


Logarithmic scales and elasticities
• Example

ABSOLUTE, PROPORTIONATE AND PERCENTAGE CHANGES
• In economics we are more concerned with proportionate or percentage changes in a
variable than with absolute changes in that variable
...

• Absolute change Δy is current level 𝑦1 minus initial level 𝑦0
• Proportionate change:

Δ𝑦
𝑦0

(pure number, such as

1
10

)
...

• Proportionate and percentage change are so similar that they are almost interchangeable
...

•

𝐸𝑝𝑠

=

𝑑𝑞 𝑠
𝑑𝑝

×

𝑝
𝑞𝑠

• If 𝐸𝑝𝑠 is positive and larger, the more sensitive quantity supplied is to changes in price
• 𝐸𝑝𝑠 > 1 - price elastic supply
• 0 < 𝐸𝑝𝑠 < 1 - price inelastic supply

• 𝐸𝑝𝑠 = 0 - perfectly price inelastic supply
• 𝐸𝑝𝑠 = ∞ - perfectly price elastic supply
• Do Examples 9
...
2

EXAMPLE
• Suppose that 𝑝 = 3 + 10𝑞 𝑠
...

• Solution
Given 𝑝 = 3 + 10𝑞 𝑠 and 𝑞 𝑠 = 10
• If 𝑞 𝑠 = 10 ⟹ 𝑝 = 103
𝐸𝑝𝑠 =

𝑑𝑞𝑠
𝑑𝑝

• To find
•

𝑑𝑞𝑠
𝑑𝑝

𝑝
𝑞𝑠
𝑑𝑞𝑠
𝑑𝑝

×
...
1𝑝 − 0
...
1

• Or: inverse function rule for differentiation (monotonic:
• So,

𝑝−3
10

𝑑𝑞𝑠
𝑑𝑝

• So 𝑑𝑝

= 𝑑𝑝

1

• ∴𝐸 =

= 10 > 0 ∀𝑞 𝑠 )

1

ൗ𝑑𝑞𝑠

=

1
10

= 0
...
1 ×

103
10

= 1
...
03 > 1, the supply of this product is price elastic at 𝑞 = 10; 𝑝 = 103

PRICE ELASTICITY OF DEMAND (PED)
• Shows how sensitive quantity demanded is to changes price
𝐸𝑝𝑑

=

𝑑𝑞 𝑑
𝑑𝑝

×

𝑝

...
The larger the absolute value, the more sensitive is
quantity demanded to changes in price
• 𝐸𝑝𝑑 > 1 - price elastic demand

• 0 < 𝐸𝑝𝑑 < 1 - price inelastic demand
• 𝐸𝑝𝑑 = 0 - perfectly price inelastic demand
• 𝐸𝑝𝑑 = ∞ - perfectly price elastic demand
• Do examples 9
...
4

EXAMPLE
•
•
•
•

Suppose that 𝑞𝑑 = −5𝑝2 − 10𝑝 + 400
...


•

𝐸𝑝𝑑

Solution
Given 𝑞𝑑 = −5𝑝2 − 10𝑝 + 400 and 𝑝 = 5
=

𝑑𝑞 𝑑
𝑑𝑝

×

𝑝
𝑞𝑑

• If 𝑝 = 5 ⟹ 𝑞𝑑 = 225

• Furthermore,

𝑑𝑞 𝑑
𝑑𝑝

= −10𝑝 − 10 = −10(𝑝 + 1)

• For 𝑝 = 5: −10 𝑝 + 1 = −10 5 + 1 = −60
∴ 𝐸 𝑝 = −60 ×

5
225

= −1
...


• Interpretation: Since 𝐸 𝑝 = 1
...

• Given an inverse demand function 𝑝 = f(𝑞) we can write TR as 𝑇𝑅 = 𝑝𝑞 = f(𝑞)𝑞
• Then, using the product rule of differentiation:
d𝑝

• where f ′ (q) = d𝑞 and

d𝑇𝑅
d𝑞

d𝑇𝑅
d𝑞

= f(𝑞) + 𝑞f ′ (q)

≡ 𝑀𝑅 and substituting these into

d𝑝

d𝑇𝑅
d𝑞

d𝑞

= f(𝑞) d𝑞 + 𝑞f ′ (q)

𝑞 d𝑝

• We have: 𝑀𝑅 = 𝑝 + 𝑞 d𝑞 and by factoring out p: 𝑀𝑅 = 𝑝 1 + 𝑝 d𝑞
𝑝 d𝑞

• Because 𝐸 𝐷 = 𝑝 d𝑝 and the inverse (or reciprocal) of 𝐸 𝐷 is
1

• Thus 𝑀𝑅 = 𝑝 1 + 𝐸𝐷

1
𝐸𝐷

𝑞 d𝑝

= 𝑝 d𝑞

and this expression shows the relationship between marginal

revenue, price, and the elasticity of demand
...
So, a small price
reduction, which leads to an increase in quantity demanded, will lead to an increase in
total revenue
...
So, a small price
reduction, which leads to an increase in quantity demanded, will lead to a decrease in total
revenue
...
A price change whether positive or negative, will lead to a decrease in total
revenue
...
youtube
...
lj
• Therefore,

d𝑝
d𝑞

= 0, implies that 𝑀𝑅 = 𝑝lj + 𝑞 0 = 𝑝lj and the elasticity of demand for

the firm's product tends to minus infinity
...


OTHER ELASTICITIES
• Given T𝐶 = 𝑓 𝑞
...
Following a small increase in output, the proportionate change in total cost is greater
than the proportionate change in output
...
Following a small increase in output, the proportionate change in total cost is less than
the proportionate change in output
...
Following a small increase in output, the proportionate change in total cost is equal to
the proportionate change in output
...


OTHER ELASTICITIES
• Given 𝐶 = 𝑓 𝑌
...
4, p
...

• Now log both sides: ln 𝑞 = ln 𝐴 − ln(𝑝−𝛼 ) = ln 𝐴 − 𝛼ln 𝑝
• Now differentiate:
𝑝

𝑑 ln 𝑞
𝑑 ln 𝑝

d𝑞

• So, 𝐸 𝑝 = 𝑞 × d𝑝 =

= −𝛼 (Note that this is the same as the 𝐸 𝑝 we found above)
...
5
• Find and interpret the price elasticity of demand if 𝑝 = 4 using logarithms

• Solution
• Given 𝑞 = 16𝑝−0
...
5 ln 𝑝

• For this function, 𝐸 𝑝 =

𝑑 ln 𝑞
𝑑 ln 𝑝

= −0
...
5 < 1, therefore the demand for this product is price inelastic at all
prices, quantities
...
Find the price elasticity of demand (𝑒 𝑝 ) if 𝑝 = 0
...

𝑝

• 𝐸 =

𝑑𝑞𝑑
𝑑𝑝

𝑝

×𝑞

•

−2
× 𝑒 −1 = 1
...
5 = 0
...
So, use product rule of

•

𝑑𝑞𝑑
𝑑𝑝

𝑒 −2𝑝

•

𝑞𝑑

• ∴

=𝑢⋅

𝐸𝑝

𝑑𝑣
𝑑𝑝

+𝑣⋅

𝑑𝑢
𝑑𝑝

𝑝=0
...
4715

=

𝑝−2

−2𝑒 −2𝑝

=

𝑑𝑞𝑑
𝑑𝑝

× = −8
...
5
1
...
5

= −3

• Another simply and quick method is to substitute 𝑝 = 0
...
5

𝑝+1
)
𝑝

= −2 (

𝑝

𝑝

× 𝑞 = −2𝑝−2 𝑒 −2𝑝 (1 + 𝑝−1) × 𝑝−2 𝑒 −2𝑝

× 𝑝 = −2 (0
...


= −8
...
e
...
last example)
• Do Progress Exercises 9
...
4
• Do Progress Exercise 13
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• Next lecture: Unit 7 (Financial Mathematics)

• Please note, we will be having a lecture of Friday 25 March 2022

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