Download Elasticity price elasticity of demand

September 12, 2008
Elasticity is a way to measure how much one variable changes in response
to another.
To measure how much demand changes in response to a change in price, use
the price elasticity of demand. Take two points (Q1 , P1 ) and (Q2 , P2 ) on a
demand curve.
The price elasticity of demand between those two points is defined as
−
(Q2 − Q1 )/[(Q1 + Q2 )/2]
.
(P2 − P1 )/[(P1 + P2 )/2]
Note that (Q1 + Q2 )/2 is the average of Q1 and Q2 , and (P1 + P2 )/2 is the
average of P1 and P2 .
The elasticity is thus equal to the negative of the slope
ratio of the averages of the points.
Q2 −Q1
P2 −P1
divided by the
Some examples.
Find the elasticity of the demand curve Q = 60 − 10P from the point P = 0
to the point P = 6.
First find the corresponding Q for each P . At P = 0, Q = 60 and at P = 6,
Q = 0. So Q1 = 60, Q2 = 0, P1 = 0 and P2 = 6. Then
d = −
(0 − 60)/[(0 + 60)/2]
(Q2 − Q1 )/[(Q1 + Q2 )/2]
=−
= 1.
(P2 − P1 )/[(P1 + P2 )/2]
(6 − 0)/[(6 + 0)/2]
Now find the elasticity of the same line, Q = 60 − 10P from the point P = 0
to the point P = 5.
At P = 5, Q = 10.
So we have
d = −
(10 − 60)/[(10 + 60)/2]
= −(−50/35)/2 = 5/7.
(5 − 0)/[(5 + 0)/2]
This shows that the elasticity of a line between two points can vary depending on which points are chosen.
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Another general point is illustrated by this example: For a linear, downwardsloping demand curve, as the average of the the P -values decreases, the price
elasticity of demand decreases.
This is clear from the definition of elasticity: Elasticity equals slope divided
by the ratio of the averages of the points. For a line, the slope is constant. As
(P1 + P2 )/2 decreases, (Q1 + Q2 )/2 increases. Thus as (P1 + P2 )/2 decreases,
the elasticity decreases.
Consider the demand curve Q = 1/P. Find the elasticity from P = 2 to
P = 6.
At P = 2, Q = 1/2. At P = 6, Q = 1/6. So
d = −
(1/3)/(1/3)
(1/6 − 1/2)/[(1/6 + 1/2)/2]
=
= 1/1 = 1.
(6 − 2)/[(6 + 2)/2]
4/4
Find the elasticity from P = 4 (Q = 1/4) to P = 8 (Q = 1/8).
−
(1/8 − 1/4)/[(1/8 + 1/4)/2]
(−1/8)/(3/16)
=−
= 1.
(8 − 4)/[(8 + 4)/2]
4/6
It seems that the elasticity of the function Q = 1/P is constant at 1 for all
points.
Extra Credit (3 extra points on next homework): Show that the elasticity between any two points of the function Q = 1/P is constant and equal
to 1. Show that Q = 1/P is the only function that has a constant elasticity of 1.
Extra Credit (3 more extra points on next homework): Find all functions
which have constant elasticity.
Consider the demand curve Q = a, for a any nonnegative number. This demand curve means that the same amount of a good is bought whatever the price.
The price elasticity of demand between the points P1 and P2 6= P1 is
−
(a − a)/[(a + a)/2]
= 0.
(P2 − P1 )/[(P1 + P2 )/2]
So the price elasticity of demand for a constant quantity demand curve is
zero.
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Consider the demand curve P = b for some constant b. This demand curve
means that buyers will buy any quantity of the good at P = b, zero of the good
at P > b and an infinite amount of the good at P < b.
The price elasticity of demand between any two points on this demand curve
is infinite - to get the elasticity one divides by zero.
Demand is said to be elastic between two points when the price elasticity
of demand between those points is greater than 1.
It is inelastic when the price elasticity of demand between two points is less
than one.
It is unit elastic when the price elasticity of demand between two points
equals 1.
Determinants of the price elasticity of demand for a good.
The price elasticity of demand measures how willing buyers are to move demand away from a good as price rises.
These factors affect price elasticity of demand:
1. Availability of close substitutes. The elasticity tends to be higher for
goods with close substitutes because people can switch to the substitute without much problem when price of the good rises.
Examples: Butter and margarine are close substitutes (If margarine is cheaper,
it may be easier to switch from butter to margarine that from margarine to butter), Toyota Corolla and Honda Civic.
Eggs don’t have a close substitute – their demand is less elastic than demand
for butter.
2. Necessities versus luxuries. Necessities tend to have inelastic demands;
luxuries tend to have elastic demands.
Examples: Food, medicine, visits to doctor and housing demand would tend
to be inelastic, as well as demand for addictive goods. Concert tickets, all kinds
of entertainment demands, demand for belonging to a gym club would tend to
be elastic.
3. Definition of the Market. Narrowly defined markets tend to have more
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elastic demand than broadly defined ones.
Food has inelastic demand as a minimum amount of food is needed to survive. Apples have a more elastic demand – can be substituted by pears or
another fruit. A particular type of apples would have an even more elastic
demand.
4. Time horizon. Demand for goods tends to be more elastic the longer the
time horizon is, because buyers have time to adjust their lifestyles/production
processes to exclude a good whose price has risen.
Total revenue and the price elasticity of demand
Total revenue R is the amount of money received by sellers and paid by
buyers for a good. It equals R = P × Q.
Total revenue can be shown graphically as the area of the rectangle with
width Q∗ and height P ∗ in the diagram:
P
S
P*
D
Q*
Q
When the demand is inelastic, raising the price causes revenue to rise. The
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decrease in quantity bought is proportionally smaller than the increase in price.
Consider the example Q = 12 − 6P . Between P = 0.5 and P = 1,
d = −
(6 − 9)/[(6 + 9)/2]
= (3/7.5)/(0.5/0.75) = 6/10 = 3/5 < 1.
(1 − 0.5)/[(1 + 0.5)/2]
So between those two points the demand curve is inelastic.
At P = 0.5, Q = 9 so R = P Q = 4.5. At P = 1, Q = 6, so R = P Q = 6.
The revenue rises when price is increased from 0.5 to 1.
Now consider a further price rise from P = 1 (Q = 6) to P = 1.5 (Q = 3).
The elasticity of demand between these two points is
d = −
(3 − 6)/[(3 + 6)/2]
= (3/4.5)/(0.5/1.25) = (2/3)/(2/5) = 5/3 > 1.
(1.5 − 1)/[(1.5 + 1)/2]
At P = 1, R = P Q = 6. At P = 1.5, R = P Q = 4.5. So revenue decreases
when price is raised from P = 1 to P = 1.5.
Other demand elasticities are defined similarly.
The income elasticity of demand between two points I1 and I2 is
Id =
(Q2 − Q1 )/[(Q1 + Q2 )/2]
.
(I2 − I1 )/[(I1 + I2 )/2]
If a person’s income elasticity of demand is greater than 1 for a good, that
good is said to be a luxury for that person. With fixed prices, it means they
spend a bigger fraction of their budget on that good when they are richer.
If a person’s income elasticity of demand is less than 1 for a good, that good
is a necessity for that person. With fixed prices, it means they spend a smaller
fraction of their budget on that good when they are richer.
The first statistical evidence of necessities/luxuries was a study by Engle
(1887) which showed that rich households spend on average a smaller proportion of their income on food than poor households.
If a poor person who becomes rich behaves like the average rich person, then
this study shows that food is a necessity.
Cross-price elasticity of demand
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The cross-price elasticity of demand of good 1 with respect to good 2 measures how much demand for good 1 changes when price of good 2 changes. It
is defined as
(Q12 − Q11 )/[(Q11 + Q12 )/2]
1,2
=
.
d
(P22 − P12 )/[(P12 + P22 )/2]
If the cross-price elasticity of demand of good 1 with respect to good 2 is positive, then good 1 is a substitute for good 2.
If the cross-price elasticity of demand of good 1 with respect to good 2 is
negative, then good 1 is a complement for good 2.
OPEC and the oil price.
OPEC is a cartel that controls 40% of the world’s oil resources.
Suppose however for this analysis that OPEC controls all the world’s oil
resources.
In reading the article, consider the following questions:
If OPEC controls all the world’s oil, what does its supply curve look like?
What does the world demand curve for oil look like? Is it elastic or inelastic?
How does this depend on the time frame chosen?
Why might some OPEC members have different interests in keeping prices
high than others?
What effects of high prices might some OPEC members want to avoid?
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