Elasticity Price of Demand
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How much would your roommate pay to watch a live fight?
How does Showtime decide how much to charge for a live fight?
What about Hank and Son's Concrete?
How much should they charge per square foot?
Can ISU raise parking revenue by raising parking fees?
Or will the increase in price drive demand down so far that revenue falls?
All of these pricing issues revolve around the issue of how responsive the quantity demanded is to price.
Elasticity is a measure of how responsive one variable is to changes in another variable?
The Law of Demand
The law of demand states that when the price of a good rises, and everything else remains the same, the quantity of the good demanded will fall.
The real issue is how far it will fall.
The demand function is given by
Q
D
D h(P, ZD )
QD = quantity demanded P = price of the good ZD = other factors that affect demand
The inverse demand function is given by
P D h (Q , ZD )
P D g(Q , ZD )
To obtain the inverse demand function we just solve the demand function for P as a function of Q
D
D1
D
Examples QD = 20 - 2P 2P + QD = 20 2P = 20 - QD P = 10 - 1/2 QD Slope = - 1/2
Examples QD = 60 - 3P 3P + QD = 60 3P = 60 - QD P = 20 - 1/3 QD Slope = - 1/3
One measure of responsiveness is slope
For demand
Q
D
D h(P, ZD )
The slope of a demand curve is given by the change in Q divided by the change in P
Q slopel P
D
For inverse demand
P D g(Q , ZD )
The slope of an inverse demand curve is given by the change in P divided by the change in Q
D
P slopel D Q
Examples QD = 60 - 3P Slope = - 3 P = 20 - 1/3 QD Slope = - 1/3
Examples QD = 20 - 2P Slope = - 2 P = 10 - 1/2 QD Slope = - 1/2
We can also find slope from tabular data
Q 0 2 4 6 8 10 P 10 9 8 7 6 5
Q
P
Q D l l 2 a a 2 slopel P 1
Demand for Handballs
Q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
P 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
Q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
P 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
Demand for Handballs
11
Price
10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22
P
Quantity
Price
Q
Q 0 1 2 3 P 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
P 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
Demand for Handballs
11 10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22
Q = 2 - 4 = -2 P = 9 - 8 = 1
P 1 slopel l l D 2 Q
Quantity
Problems with slope as a measure of responsiveness
Slope depends on the units of measurement The same slope can be associated with very different percentage changes
Examples QD = 200 - 2P 2P + QD = 200 2P = 200 - QD P = 100 - 1/2 QD
P l l 1 slopel D 2 Q
Consider data on racquets
Let P change from 95 to 96 P = 96 - 95 = 1 Q = 8 - 10 = -2 Q
Q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
P 100 99.5 99 98.5 98 97.5 97 96.5 96 95.5 95 94.5 94 93.5 93
P
A $1.00 price change when P = $95.00 is tiny
Graphically for racquets
Demand for Racquets
Price 102 100 98 96
Slope = - 1/2
94 92 90 88 0 2 4 6 8 10 12 14 16 18
Large % change in Q
Quantity
Small % change in P
Graphically for hand balls
Demand for Handballs
11 Price 10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22
P=7-6=1 Q = 6 - 8 = -2 Slope = - 1/2
P
Quantity
Large % change in Q
Large % change in P
So slope is not such a good measure of responsiveness
Instead of slope we use percentage changes
The ratio of the percentage change in one variable to the percentage change in another variable is called elasticity
The Own Price Elasticity of Demand D is given by Q
%Q D l l %P
D
Q P P
D
There are a number of ways to compute percentage changes
Initial point method for computing
The Own Price Elasticity of Demand
Price Elasticity of Demand (Initial Point Method)
%Q D D l l %P (8 a 10) 8 (6 a 5) 6
P2 8
Q D QD P P
P
P 6 5.5 5 4.5 4
Q 8 9 10 11 12
Q
D l
( 8 a 10) 6 a 8 (6 a 5)
P
6 a a 12 a a 1.5 8 1
Final point method for computing
The Own Price Elasticity of Demand
Price Elasticity of Demand (Final Point Method)
%Q D D l l %P ( 8 a 10) 10 (6 a 5) 5
P2 10
Q D QD P P
P
P 6 5.5 5 4.5 4
Q 8 9 10 11 12
Q
D l
( 8 a 10) 5 a 10 (6 a 5)
P
5 a a 10 a a 1.0 10 1
The answer is very different depending on the choice of the base point So we usually use The midpoint method for computing
The Own Price Elasticity of Demand
Elasticity of Demand Using the Mid-Point
%Q l D l %P
D
Q D D Q P P
Q D D Q1 P Q0 or Q0 P Q1
For QD we use the midpoint of the Q's Q
D
1 D (Q1 P Q0 ) 2
Similarly for prices
P D P1 P P0 or P0 P P1
For P we use the midpoint of the P's
1 P D ( P1 P P0 ) 2
%Q l D l %P
(Q1 P Q0 ) 1 ( Q1 P Q0 ) 2 P (P1 P P0 ) 1 ( P1 P P0 ) 2
D
Q D D Q P P
(Q1 P Q0 ) P (Q1 P Q0 ) (P1 P P0 ) ( P1 P P0 )
Price Elasticity of Demand (Mid-Point Method)
%Q D D l l %P
(Q1 P Q0 ) P (Q1 P Q0 ) (P1 P P0 ) (P1 P P0 )
Q D QD P P
P
Q 8 9 10 11 12
P 6 5.5 5 4.5 4
(Q1 P Q0 ) (P1 P P0 ) (P1 P P0 ) (Q1 P Q0 )
(8 P 10) (6 P 5) l (6 P 5) (8 P 10) (P 2) (11) P 22 P 11 P P P (1) (18) 18 9
Classification of the elasticity of demand
Inelastic demand
When the numerical value of the elasticity of demand is between 0 and -1.0, we say that demand is inelastic.
%Q D P %P P< 1 P P P P
%Q D < %P
Classification of the elasticity of demand
Elastic demand
When the numerical value of the elasticity of demand is less than -1.0, we say that demand is elastic.
%Q D P %P P> 1 P P P P
%Q D > %P
Classification of the elasticity of demand
Unitary elastic demand
When the numerical value of the elasticity of demand is equal to -1.0, we say that demand is unitary elastic.
%Q D P %P PD 1 P P P P
%Q D l %P
Classification of the elasticity of demand Perfectly elastic - D = - horizontal vertical
Perfectly inelastic - D = 0
Elasticity of demand with linear demand
Consider a linear inverse demand function P D A D BQ D The slope is (-B) for all values of P and Q For example,
P D 12 P 0.5Q D
The slope is -0.5 = - 1/2
Demand for Diskettes
13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 Price
P
Quantity
Q D D D2.0 P
P
P 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3
Q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Q Q
The slope is constant but the elasticity of demand will vary
%Q D l D l %P
Q D QD P P
Q (P1 a P0 ) P P (Q1 a Q0 )
P
l (8 a 10) (8 a 7 ) (8 a 7) (8 a 10) (P 2 ) (15) P 30 P5 a P P (1) (18) 18 3
P 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3
Q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Q
The slope is constant but the elasticity of demand will vary
%Q D D l l %P
Q D QD P P
Q (P1 a P0 ) P (Q1 a Q0 )
l (14 a 16) (5 a 4) (5 a 4 ) (14 a 16) (P 2 ) (9 ) P 18 P3 a P P (1) (30) 30 5
P
P 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3
Q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Q
The slope is constant but the elasticity of demand will vary
A linear demand curve becomes more inelastic as we lower price and increase quantity
P smaller
%Q D l Q (P1 a P0 ) D l %P P ( Q1 a Q0 )
Q larger
The elasticity gets closer to zero
The slope is constant but the elasticity of demand will vary
Q 0 2 4 6 8 10 12 14 16 18 20 22 24 P 12 11 10 9 8 7 6 5 4 3 2 1 0 Elasticity 0 -23.0000 -7.0000 -3.8000 -2.4286 -1.6667 -1.1818 -0.8462 -0.6000 -0.4118 -0.2632 -0.1429 -0.0435 Expenditure 22 40 54 64 70 72 70 64 54 40 22 0
The slope is constant but the elasticity of demand will vary
Q 0 2 4 6 8 10 12 14 16 18 20 22 24 P 12 11 10 9 8 7 6 5 4 3 2 1 0 Elasticity 0 -23.0000 -7.0000 -3.8000 -2.4286 -1.6667 -1.1818 -0.8462 -0.6000 -0.4118 -0.2632 -0.1429 -0.0435 Expenditure 22 40 54 64 70 72 70 64 54 40 22 0
Note
We do not say that demand is elastic or inelastic ..... We say that demand is elastic or inelastic at a given point
Example
Q %Q D QD D l l %P P P Q (P1 a P0 ) l P (Q1 a Q0 )
D
Constant with linear demand
The Own Price Elasticity of Demand and Total Expenditure on an Item
How do changes in an items price affect expenditure on the item?
If I lower the price of a product, will the increased sales make up for the lower price per unit?
Expenditure for the consumer is equal to revenue for the firm
Revenue = R = price x quantity = PQ Expenditure = E = price x quantity = PQ
Modeling changes in price and quantity
P = change in price Q = change in quantity The Law of Demand says that as P increases Q will decrease
P
Q
So P = initial price P = change in price P + P = final price Q = initial quantity Q = change in quantity Q + Q = final quantity
So Initial Revenue = PQ P + P = final price Q + Q = final quantity Final Revenue = (P + P) (Q + Q)
= P Q + P Q + P Q + P Q
Now find the change in revenue R = final revenue - initial revenue = P Q + P Q + P Q + P Q - P Q = P Q + P Q + P Q %R = R / R = R / P Q
R P Q P P Q P P Q P PQ PQ
We can rewrite this expression as follows
R P Q P Q P Q P P P PQ PQ PQ PQ P Q P Q %R P P P P Q PQ
%R P %P P %Q
Classification of the elasticity of demand
Inelastic demand D %Q P %P P< 1 %Q D P P P P + %R P %P P %Q
< %P
% Q and % P are of opposite sign so %R has the same sign as %P
Classification of the elasticity of demand
Inelastic demand D %Q P %P P< 1 %Q D P P P P + %R P %P P %Q
< %P
% Q and % P are of opposite sign so %R has the same sign as %P Lower price lower revenue Higher price higher revenue
Classification of the elasticity of demand
Elastic demand %Q D %Q D P %P P> 1 P P P P + %R P %P P %Q
> %P
% Q and % P are of opposite sign so %R has the opposite sign as %P Higher price lower revenue Lower price higher revenue
Classification of the elasticity of demand
Unitary elastic demand %Q D %Q D P %P PD 1 P P P P + %R P %P P %Q
l %P
% Q and % P are of opposite sign so their effects will cancel out and %R = 0.
Tabular data
Q Price falls 0 2 4 6 8 ...