Elasticity of Demand
Jane-Marie Wright
Fall 2000
Abstract.
Notes for WC Section 5.5 (Optional Section). Min/Max
economic problems
Demand is elastic if it reacts strongly to a price change. Demand is inelastic if
it is not very sensitive to changes in price.
De…nition 1. The elasticity of demand E, is the percentage rate of decrease of demand per percentage increase in price.
dq p
dp q
We say that the demand is elastic if E > 1; is inelastic if E < 1 and has unit
elasticity if E = 1
Revenue is maximized when E = 1
E=
Remark 1. If the demand is inelastic, raising the price increases revenue. If the
demand is elastic, lowering the price increases the revenue.
Example 1. Page 405 Demand for Oranges Problem 2 The weekly sales of Honolulu
Red Widgets is given by q = 1000 10p. Calculate the elasticity of demand for a
price of $30 per Widget. Calculate the price that gives a maximum weekly revenue
and …nd this maximum revenue
dq
=
dp
10
dq p
p
= 10
dp q
1000 10p
30
E(30) = 10
= 37
1000 10 30
Maximum revenue occurs at unit elasticity
10p
1=
, Solution is: p = 50
1000 10p
R = pq = p (1000 10p) = 50 (1000 500) = 25 000
E = 37 : the demand is going down 3% per 7% increase in price at that price level.
Revenue is maximized when p=$50
Weekly revenue at that price is $25000
E=
1
Elasticity of Demand
2
Example 2. Page 405 Problem 4 The consumer demand curve for Professor Stefan
Schwartenegger dumbbells is given by q = (100 2p)2 where p is the price per dumbbell and q is the demand in weekly sales. Find the price Professor Schwartenegger
should charge for his dumbbells in order to maximize revenue
dq
dp
= 2 (100 2p) ( 2) = 400 + 8p
p
dq p
4p(100 2p)
4p
= ( 400 + 8p)
E=
=
=
dp q
(100 2p)2
(100 2p)2
100 2p
4p
Maximize Revenue 1 =
! 100 2p = 4p, Solution is: p = 50
3
100 2p
Revenue is maximized when p = 50=3 = $16:67 per dumbbell
Example 3. Page 406 Problem 18. Income Elasticity of Demand: Live concerts.
The likelihood that a child will attend a live musical performance can be modeled by
q = 0:01(0:0006x2 + 0:38x + 35) : 15
x
100: Here q is the fraction of children
with annual household income x who will attend a live musical performance during
the year. compute the income elasticity of demand at an income level of $30,000 and
dq x
interpret the result. See page 405 for Income Elasticity equation E =
dx q
dq x
x
= 0:01(0:0012x + 0:38)
2
dx q
0:01(0:0006x + 0:38x + 35)
(0:0012x + 0:38)x
0:01(0:0012x + 0:38)x
=
=
2
0:01(0:0006x + 0:38x + 35)
(0:0006x2 + 0:38x + 35)
0:001 2x2 + 0:38x
=
0:0006x2 + 0:38x + 35
0:001 2 (30)2 + 0:38 (30)
x = 30 : E(30) =
= 0:265 87
0:0006 (30)2 + 0:38 (30) + 35
E =
At a family income of $30,000 the fraction of children attending a live musical
performance is increasing by 0:27% per 1% increase in household income.
Example 4. Page 407 Problem 20. Income Elasticity of demand:Internet Usage.
The demand for Internet connectivity also goes up with household income. The
following graph shows some data on Internet usage, together with the logarithmic
model q = 0:2802 ln(x) 2:505 where q is the probability that a home with an annual
household income x will have an Internet connection.
Elasticity of Demand
3
1. Compute the income elasticity of demand to two decimal places for a household
income of $60,000 and interpret the result.
dq x
0:2802
x
=
dx q
x
0:2802 ln(x) 2:505
0:2802
=
0:2802 ln(x) 2:505
0:2802
= 0:484 95
E(60000) =
0:2802 ln(60000) 2:505
E =
The demand for Internet connectivity is increasing by 0.49% per 1% increase
in household income.
2. As household income increases, how is income elasticity of demand a¤ected?
The denominator will get larger, so E will get smaller.
3. The logarithmic model shown above is not appropriate for incomes about $100,000.
Suggest a model that might be more appropriate. A logistic model (limited
growth) might work
4. In the model you propose, how does E behave for very large incomes? E should
become close to zero.
Exercise 1. Derivation: Page 406 Problem 12 A general exponential demand
function has the form q = Ae bp (A; b, non zero constants)
Obtain a formula for the elasticity of demand at a unit price of p .
4
Elasticity of Demand
dq
=
dp
bAe
bp
dq p
p
bpAe bp
= bAe bp
=
= bp
dp q
Ae bp
Ae bp
Obtain a formula for the price that maximizes revenue
1
E = 1 = bp ! p =
b
Exercise 2. Derivation: Page 406 Problem 14 A general quadratic demand functionn has the form q = ap2 + bp + c (a; b; c constants a 6= 0)
E=
Obtain a formula for the elasticity of demand at a unit price of p.
dq
= 2ap + b
dp
dq p
p (2ap + b)
E=
= 2
dp q
ap + bp + c
Obtain a formula for the price or prices that maximizes revenue
p (2ap + b)
ap2 + bp + c
2ap2 bp
E = 1=
ap2 + bp + c =
ap2 + bp + c + 2ap2 + bp = 0
3ap2 + 2bp + c = 0
p =
2b
p
4b2
6a
12ac
=
b
p
b2
3a
3ac
Exercise 3. Derivation: Page 406 Problem 11 A general linear demand function
has the form q = mp + b (m; b, constants m 6= 0 )
Obtain a formula for the elasticity of demand at a unit price of p.
dq
=m
dp
dq p
mp
E=
=
dp q
mp + b
Obtain a formula for the price that maximizes revenue
mp
mp + b
mp + b =
mp
2mp =
b
b
p =
2m
E = 1=
Elasticity of Demand
5
Exercise 4. Derivation: Page 406 Problem 13 A general hyperbolic demand
k
function has the form q = r (r; k, non zero constants)
p
Obtain a formula for the elasticity of demand at a unit price of p.
dq
d
kr
=
(kp r ) = kp r 1 r = r+1
dp
dp
p
kr
dq p
kr ppr
p
= r+1
=r
E=
= r+1
k
dp q
p
p
k
pr
E is independent of p!