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University of Winnipeg

Statistics

STAT-2301

Hai Ta

Fall

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Principles of Microeconomics: Elasticity
ELASTICITY
Elasticity
Definition: Elasticity measures the responsiveness of a change in a variable to the change in
another variable
=> ε = %ΔQ/%ΔP
Economists measure Elasticity as the ratio of percentages changes because a ratio of
absolute change does not indicate the relative importance of a change (e.g., a $1 increase in the
price of a car would hardly affect car sales but a $1 increase in the price of gasoline would
significantly affect the sale of gasoline)
Price Elasticity of Demand:
Definition: Price Elasticity of Demand (η) is equal to the ratio of the percentage change in
quantity demanded to the percentage change in price (responsible for that percentage
change in quantity demanded)
ε = %ΔQ /%DP
e.g. If a 20% increase in the price of gasoline causes a 10% decrease in the quantity demanded
of gasoline, then ε = -10%/+20% = -0.5
Note: Price Elasticity of Demand is usually defined for first year students as the absolute value of
the ratio of the percentage changes of quantity demanded and price because the ratio is almost
invariably negative (unless Demand is upward sloping, a rare and possibly non-existent case) and
expression as an absolute value simplifies the relation between elasticity and changes in total
revenue. I will not use the absolute value expression of elasticity because the sign has
significance, particularly in relation to other types of elasticity.
- 1 - Principles of Microeconomics: Elasticity
Define:
1. Inelastic => %ΔQ /%DP< 1
i.e. the percentage response of Quantity is less than the percentage change in Price
2. Unit elastic => %ΔQ /%DP= 1
i.e. the percentage response of Quantity equals the percentage change in Price
3. Elastic => %ΔQ /%ΔP> 1
D
i.e. the percentage response of Quantity is greater than the percentage change in Price
Since Demand data in reality is in the form of Prices and Quantities, not percentage
changes, we need to express elasticity in prices and quantities.
Elasticity of Demand
= %ΔQ/%ΔP
= (100% * ΔQ/Q)/(100% * ΔP/P)
= (ΔQ/Q)/(ΔP/P)
= (ΔQ/ΔP) * P/Q)
= 1/slope * P/Q (since (ΔQ/ΔP = 1/(ΔP/ΔQ) = 1/slope)
We can express ΔQ and ΔP in terms of Po and P to 1et
ε = (Q – Qo)/Q)/(P – Po)/P
1 1
What are the P and the Q that we divide by?
Point Elasticity
Point Elasticity calculates elasticity at a point, i.e., relative to the original Price and Quantity
ε = (Q 1 Qo)/Qo)/(P –1Po)/Po
- 2 - Principles of Microeconomics: Elasticity
e.g. Suppose that the price of Ipods falls from $250 to $200 and the quantity sold of Ipods
increases from 500,000 to 600,000. What is the point elasticity of demand?
ε = (600,000 – 500,000)/500,000)/(200 – 250)/250 = - 1 or unit elasticity
→ Unit elasticity suggests that there is no change in Total Revenue with the change in price
but Total Revenue actually falls from $125 million to $120 million
Note also that point elasticity is not the same for the opposite direction, i.e., an increase in
price from $200 to $250 with a decrease in quantity from 600,000 to 500,000. In this case,
elasticity is –0.67, which suggests that the increase in price should increase Total Revenue as it
does. .
The reason for the failure of point elasticity to correctly predict the change in Total Revenue
and for the difference between the elasticities for equal rises and falls in price is that point
elasticity is only accurate for small changes relative to the original price and quantity because
elasticity usually changes as P and Q change. It is really a calculus concept dQ/dP *P/Q
Arc Elasticity
Arc Elasticity calculates elasticity relative to the average price and quantity of the change.
= ΔQ/ ΔP * (Average P)/(Average Q) or ΔQ/(Average Q)/ ΔP/(Average P)
= (Q 1 Qo)/(P –1Po) * (Po + P )/21(Qo + Q )/2 1
e.g. The arc elasticity for a decrease in price from $250 to $200 causing an increase in quantity
from $500,000 to $600,000 is the same as the arc elasticity of an increase in price from
$200 to $250 with the opposite change in quantity
= (600,000 – 500,000)/(200 – 250) * (250 + 200)/2/(500,000 + 600,000)/2
= -0.818
- 3 - Principles of Microeconomics: Elasticity
Elasticity and Total Revenue
Since movement along a Demand function implies an inverse relationship between Price
and Quantity, what is the effect on Total Revenue of a change in Price or a change in Quantity?
Should a firm lower price and thus increase quantity to increase Total Revenue or increase price
and lower quantity to increase Total Revenue? The Elasticity of Demand gives us the answer.
Suppose that a change in Price causes no change in Total Revenue (TR).
ΔTR/ ΔP = 0
=> Δ(P*Q)/ ΔP = 0
→ ΔP/ΔP * Q + P * ΔQ/ΔP = 0 (Product rule)
→ P*ΔQ/ΔP = -Q
→ P/Q * ΔQ/ΔP = -1
→ P/Q * ΔQ/ΔP= ε = 1
=> Demand is unit elastic if a change in Price causes no change in Total Revenue.
By similar calculations, we find that
Δ (P*Q)/dP > 0 => P/Q * ΔQ/ΔP > -1 so that P/Q * ΔQ/ΔP= ε < 1
=> Demand is inelastic if a change in Price causes an increase in Total Revenue
Δ (P*Q)/ ΔP < 0 => P/Q * ΔQ/ΔP < -1 so that P/Q * ΔQ/ΔP= ε > 1
=> Demand is elastic if a change in Price causes a decrease in Total Revenue
Inverting this result (which we can do since elasticity is a monotonic function) means that
for small changes in Price,
1. Inelastic => Total Revenue moves in the same direction as Price
[Total Revenue increases (decreases) if Price increases (decreases) or Quantity decreases
(increases)]
- 4 - Principles of Microeconomics: Elasticity
2. Unit Elastic => Total Revenue does not change if Price changes
→Total Revenue is at a maximum
3. Elastic => Total Revenue moves in the opposite direction as Price
[Total Revenue increases (decreases) if Price decreases (increases) or Quantity increases
(decreases)]
An illustration of the relevance of price elasticity’s relevance for total revenue is the
paradoxical result that farmers of wheat, an inelastic good, benefit more from harvest failure on
an international scale than harvest success.
e.g. Let us examine the behaviour of Elasticity for a Linear Demand function.
Suppose that the Demand function is P = 6 – 0.5Q. We can compile the following table for
elasticity at the price/quantity combination relative to the next price/quantity combination.
Price Quantity Total Point Elasticity Point Elasticity Arc Elasticity
Revenue %ΔQ/Δ%P ΔQ/ΔP*P/Q
6 0 0
5 2 10 100%/-20% = -5 2/-1*5/2 = -5 2/-1*4.5/3 = -3
4 4 16 50%/-25% = - 2 2/-1*4/4 = -2 2/-1*3.5/5 = -7/5
3 6 18 33%/-33% = -1 2/-1*3/6 = -1 2/-1*2.5/7 = -5/7
2 8 16 25%/-50% = -1/2 2/-1*2/8 = -1/2 2/-1*1.5/9 = -1/3
1 10 10 20%/-100%=-1/5 2/-1*1/10 = -1/5 2/-1*0.5/11=-1/11
0 12 0
- 5 - Principles of Microeconomics: Elasticity
Demand: P = 6 - 0.5Q
Price ($s)
7
6
5
Elastic
4
3 Unit Elastic
Inelastic
2
1
0
0 2 4 6 8 10 12
Quantity
Note: A linear demand function has a constant slope but elasticity (-1/

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