Problem Set #4: Elasticity
Key Concepts
Elasticity of Demand
= % change in Quantity Demanded
% change in Price
Income Elasticity of Demand
= % change in Quantity Demanded
% change in Income
Cross Price Elasticity of Demand = % change in Quantity Demanded of X
% change in Price of Y
Price Elasticity of Supply = % change in Quantity Supplied
% change in Price
Formulas to calculate elasticity depending on information:
Elasticity of Demand
= %  Qd
%P
= (Q)/Q
(P)/P
= Q * P
P Q
Arc Elasticity: Define P and Q in the usual formula as the average of the range
P = (P1 + P2)/2 and Q = (Q1 + Q2)/2
Symbols:

= elasticity
d
= elasticity of demand
y
elasticity of income
xz = elasticity of X with respect to price of Z
s
= elasticity of supply
Problems
1.a) Suppose that when the price of gasoline increases from $0.48 to $0.52 per litre, gasoline
consumption falls from 10.1 million litres per year to 9.9 million litres in a given town.
Compute the arc elasticity of demand for gasoline.
b) If the income elasticity of demand for a commodity is +0.5, is the commodity an inferior
good? Exactly what does an income elasticity of 0.5 mean?
2.
You are an analyst employed by an automobile manufacturer that last year sold 50,000
compact cars at $6,000 each. Your market research indicates that:
(i) price elasticity of demand for your cars is -3.0
(ii) income elasticity of demand for your cars is +0.7
(iii) cross price elasticity for your cars with respect to the price of a comparable car
made by a competitor is +1.2
a) Suppose that you expect a ceteris paribus decrease in average incomes of 5% this year
compared to last year. How many cars will your company will sell this year?
b) Assume now that you do not think incomes will change, but that you expect your competitor
will decrease price by 5%. Assuming that your company does not change the price of its
cars, how many cars would expect your company will sell this year?
c) Now estimate sales this year if the only change you expect is an increase in the price of your
car to $6,600.
3.
The short-run market demand and supply curves for good X are as follows:
P = 100 - 2.0 Q
P = 40 + 3.0 Q
(P is in $ per unit)
(NB: Graphs are not necessary to answer the questions in this problem and are much less
efficient than algebra. A graph, though, may clarify your understanding)
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Problem Set #4: Elasticity
a) Find the equilibrium price and quantity.
b) What is the equilibrium price and quantity due to a tax of $5 per unit on producers?
c) What is the arc elasticity of demand between the initial and after tax equilibria?
Is total revenue at the after tax equilibrium higher or lower than at the before tax
equilibrium? Try to answer the question before your calculate total revenue.
d) How much of the tax do consumers pay? How much of the tax do producers pay?
e) How would your answers to (c) change if the demand curve was vertical? horizontal?
How would your answers to (c) change if the supply curve was vertical? horizontal?
Given these results, what general conclusion can one make relating the relative elasticities of
the demand and supply curves to the fraction of the tax paid by consumers in the short run?
4.
A student claims that a firm's total revenue always increases when there is an supply
induced increase in the price of its product. Briefly explain the truth or falsity of this
statement.
5.
A politician argues that the benefit of a per unit subsidy will accrue entirely to producers if
the demand for a commodity is perfectly elastic and the supply curve is upward sloping.
Prove the truth or falsity of this statement by drawing a diagram incorporating this
information. Be sure to indicate the original equilibrium price (Po), the equilibrium price
with the subsidy (Ps), and the net price for the producers after the subsidy (Ps+s). Explain
your answer briefly.
Multiple Choice
1.
If a 10% increase in price results in a a 5% reduction in quantity demanded, the price
elasticity of demand (ignoring the minus sign) is
a) 5
b) 2
c) 1
d) 0.5
e) 0.2
2.
The price elasticity of demand for Product W is calculated at 3 (technically, -3) for a small
range on the downward sloping demand curve. Within this range, if the price were to
increase by 10%, then the quantity demanded for Product W would:
a) rise by 10% b) fall by 10%
c) rise by 30% d) fall by 30% e) none of these
3.
If the price elasticity of demand for tape decks is -0.5, then a 20% increase in price will
a/ reduce quantity demanded by 40%
b/ reduce quantity demanded by 5%
c/ increase total expenditure
d/ reduce total expenditure
e/ none of the above
4.
If the price for Good B were to decrease from $10 to $8 and as a result the quantity
demanded were to increase from 10 million units to 12 million units, then the demand curve,
in the range analyzed, using a formula for arc elasticity would be:
a) elastic b) unitary elastic
c) inelastic d) perfectly elastic
5.
Suppose that the quantity demanded of good X decreases by 2% when income increases by
5%. The income elasticity of good X is
a/ +10%
b/ -.1
c/ +.1
d/ -.4
6.
If two goods are complements in consumption, their cross price elasticity of demand is
a/ zero
b/ negative
c/ positive
d/ either positive or negative depending on whether the goods are normal or inferior
e) inelastic
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Problem Set #4: Elasticity
7.
If the demand for a good is elastic, then
a/ as price rises, total revenue of producers falls
b/ as price falls, total revenue of producers falls
c/ as price rises, total revenue of producers rises
d/ as price rises, total revenue of producers is constant
e) as price rises, total expenditure of consumers rises
8.
If the cross elasticity of demand for a good X relative to good Y is -0.5 and the price of good
Y falls by 10%, the change in the quantity demanded of good X is:
a/ +5% b/ -5%
c/ -0.5%
d/ +0.5
e/ none of the above
9.
If the price elasticity of demand for opera was inelastic, then total receipts from opera
a/ increase with a rise in price of opera
b/ decrease with a rise in price of opera
c/ increase with a fall in price of opera
d/ increase with a fall in price of ballet
e/ increase with an increase in income
10. If Supply is infinitely inelastic and Demand is downward sloping, then at equilibrium
a/ supply and demand both determine price and quantity
b/ supply determines price and demand determines quantity
c/ supply determines quantity and demand determines price
d/ supply determines price and quantity
e/ demand determines price and quantity
11. If the cross price elasticity of X in relation to Y is positive, then X in relation to Y is
a) a substitute good in consumption
b) a complement good in production
c) a normal good
d) an inferior good with a downward sloping Demand
e) a Giffen good
12. If a 10% increase in price results in a 10% reduction in total revenue, the elasticity of
demand with respect to price is:
a) elastic
b) inelastic
c) unit elastic
d) unknown without more information
13. With respect to price elasticity of demand, which one of the following statements is correct:
a) if unitary price elasticity of demand were reported for an industry, then small
changes in quantity demanded would not change the industry's total revenue
b) if a blizzard devastated a Florida orange grove and as a result the Florida orange
industry received more total revenue for its crop, then industry demand for
oranges must be elastic
c) if the price elasticity of demand for Industry Z were computed at 3.0 (more
technically, -3.0), then a 12% increase in quantity demanded would lead to a 4%
increase in price
d) if the price elasticity of demand for Industry Y were computed at 0.3 (more
technically, - 0.3), then the demand curve is judged to be elastic.
14. If quantity supplied is greater than zero when Price is zero, then linear supply is
a) inelastic
b) unit elastic
c) elastic
d) first elastic and then inelastic
e) first inelastic and then elastic
15. The demand curve for cod would shift outward (increase) if
a) the price of salmon (a substitute good) were to decrease
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Problem Set #4: Elasticity
b) disposable incomes were to decrease and cod has a negative income elasticity
c) the price of rice (a complementary good) were to increase
d) cod workers received a pay reduction and cod is a normal good
Solutions
1.a) d =
%__Qd
%P
Arc D =
Q*P
where P = (P1 + P2)/2
P Q
Q = (Q1 + Q2)/2
= 10,100,000-9,900,000
*
(.48 + .52)/2
0.48-0.52
(10,100,000 + 9,900,000)/2
= -0.25
(or note that quantity falls by 20% [-200,000/10,000,000] when price increases by 8%
[0.04/0.50] giving elasticity of -.25.
b) Income Elasticity is negative in the case of an inferior good because demand and income
change in opposite directions. A income elasticity of +0.5 means that demand changes by half
the percentage that income changes in the same direction that income changes. X is therefore
a normal good with inelastic income elasticity.
Y = +0.7 (1% increase in income => 0.7% increase in Qd)
Pxz = +1.2 (1% increase in Pz (a substitute for X)
=> 1.2% increase in Qdx)
d = -3.0 (1% increase in price => 3% fall in Qd
a) 5% fall in income => 5 (0.7)% fall in Qd
= -3.5% (50,000) = -1750 cars
=> 50,000 - 1,750 = 48,250 cars will sell this year.
b) 5% fall in Psubstitute => -5 (1.2)% = -6% change in Qd
=> 50,000 - .06(50,000) = 47,000 cars sold this year
c) If the car's price rises from $6,000 to $6,600 (+10%), then expect quantity to fall by +10 (3.0)% = -30% => Estimated sales = 50,000 - .3(50,000) = 35,000.
2.
Given:
3.a) Equilibrium => Pd=Ps => 100 - 2.0Q = 40 + 3.0Q
=> Q = 12 and P = $76
b) Tax = $5/unit => Supply shifts to P = 45 + 3.0Q
=> Equilibrium at P = $78 and Q = 11
c) d = (12-11) * (76+78)/2
= -3.35
76-78) (12+11)/2
Total Revenue falls because Price rises when
Demand is elastic. [Revenue goes from 76*12 =
912 to 78*11 = 858]
d) Change in price from $76 to $78 effectively means
that consumers pays $2 of tax.
Producers now receive $78 but have to remit $5 to
the government, leaving them with $73. They
receive $76-$73 = $$3 less than before and thus
'pay' $3 of the tax
e) Vertical Demand
=> d = 0, i.e., perfectly inelastic Demand
=> Consumer pays the entire tax
Horizontal Demand => d = -infinity (perfectly elastic Demand)
=> Consumer pays no tax.
Vertical Supply => s = 0, i.e., perfectly inelastic Supply
=> Producer pays the entire tax
Horizontal Supply => s = infinity (perfectly elastic Supply)
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Problem Set #4: Elasticity
=> Producer pays no tax.
The more inelastic Demand is, therefore, the more consumers pay of the tax because Qd is
little affected by an increase in Price.
The more inelastic Supply is, the more producers pay of the tax because Qs is little affected
by an increase in Price.
The division of the tax between consumers and producers therefore depends on the relative
elasticities of Demand and Supply.
4.
The statement is false. An increase in price due to a shift in supply will only increase total
revenue if price elasticity of demand is inelastic over the range of the price increase. If
demand is elastic, an increase in price will decrease total revenue.
(Note that an increase in price due to an increase in Demand will always increase the firm's
revenue because the shift in Demand causes movement up the Supply Curve which increases
both price and quantity)
5.
The politician (surprisingly) is correct. The
subsidy reduces cost, shifting Supply down.
Equilibrium price does not change since
Demand is perfectly elastic. Consumers pay
the same price so they are unaffected by the
tax except for increased consumption.
Producers receive the original price plus the
full amount of the subsidy.
Multiple Choice
1.
2.
3.
4.
5.
d)
d)
c)
c)
d)
6.
7.
8.
9.
10.
b)
a)
a)
a)
c)
11.
12.
13.
14.
15.
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a)
a)
a)
a)
b)