# Management and Organizational Studies 1022F/G Lecture Notes - Product Rule, Monotonic Function, Arc Elasticity

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Published on 1 Feb 2013

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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 = %∆QD/%∆P

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.

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Define:

1. Inelastic => %∆QD/%∆P< 1

i.e. the percentage response of Quantity is less than the percentage change in Price

2. Unit elastic => %∆QD/%∆P= 1

i.e. the percentage response of Quantity equals the percentage change in Price

3. Elastic => %∆QD/%∆P> 1

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

= %∆Q/%∆P

= (100% * ∆Q/Q)/(100% * ∆P/P)

= (∆Q/Q)/(∆P/P)

= (∆Q/∆P) * P/Q)

= 1/slope * P/Q (since (∆Q/∆P = 1/(∆P/∆Q) = 1/slope)

We can express ∆Q and ∆P in terms of Po and P1 to get

ε = (Q1 – Qo)/Q)/(P1 – Po)/P

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 = (Q1 – Qo)/Qo)/(P1 – Po)/Po

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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.

= ∆Q/ ∆P * (Average P)/(Average Q) or ∆Q/(Average Q)/ ∆P/(Average P)

= (Q1 – Qo)/(P1 – Po) * (Po + P1)/2/(Qo + Q1)/2

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

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