Demand Elasticity
Elasticity & Revenue
Market Demand cont.
Chapter 15
Demand Elasticity
Elasticity & Revenue
Outline
Deriving market demand from individual demands (done)
How responsive is qd to a change in price? (elasticity)
What is the relationship between revenue and demand
elasticity?
Demand Elasticity
Elasticity & Revenue
Clicker Vote
When deriving market demand from individual demand curves, we
add them up
A) Vertically
B) Diagonally
C) Horizontally
D) It depends
Demand Elasticity
Elasticity & Revenue
Price Elasticity of Demand
How sensitive is D(p) to price?
How much will quantity demanded change in response to a
given price change? (think in terms of percents)
Define price elasticity of demand, , as
=
∆q
q
∆p
p
=
p ∆q
,
q ∆p
or p/q times the slope of the demand curve.
At a particular point on the demand curve:
=
p ∂q
q ∂p
Demand Elasticity
Elasticity & Revenue
Price Elasticity of Demand
Example: Workout 15.4
Demand for kitty litter: ln D(p, m) = 1000 − p + ln m, where
p is price and m is income
Rewrite demand: D(p, m) = e 1000 e −p e m = me 1000 e −p
What is the price elasticity of demand for kitty litter when
1
p = 2 and m = 500?
Differentiate to find:
=
2
3
∂q
∂p
= −me 1000 e −p = −D(p, m) So
p
(−D(p, m)) = −p
D(p, m)
So = −2
p = 3 and m = 500?
= −3
p = 4 and m = 1500?
= −4
Demand Elasticity
Elasticity & Revenue
Price Elasticity of Demand
Demand curve slopes downward ( dq
dp < 0) so ≤ 0.
|| > 1 =⇒ demand is elastic
|| < 1 =⇒ demand is inelastic
|| = 1 =⇒ demand is unit elastic
Price
Elastic
(Є < -1)
Unit elastic
(Є = -1)
Inelastic
(-1 < Є < 0)
Bats
Demand Elasticity
Elasticity & Revenue
Price Elasticity of Demand
With linear demand: q = 20 − p (inverse: p(q) = 20 − q)
Above midpoint =⇒ demand is elastic
Below midpoint =⇒ demand is inelastic
At midpoint =⇒ demand is unit elastic
Price
20
Elastic
Unit elastic
10
Inelastic
10
20
Bats
Demand Elasticity
Elasticity & Revenue
Price Elasticity of Demand
Iso-elastic demand: q = ap −b
p dq
−b−1
q dp = a · (−b)p
= pq dq
dp =
|| = b
p
a
ap −b
· (−b)p −b−1 = −b
Price
Elasticity = -b
Bats
Demand Elasticity
Elasticity & Revenue
Other Elasticities
Suppose F = F (x, y )
How sensitive is F to a change in x?
Elasticity of F w.r.t. x (or x elasticity of F ) is given by
∂F
x
F (x, y ) ∂x
Example: income elasticity of demand
How sensitive is D to a change in income?
m
∂D
m = D(p,m)
∂m
m ≥ 0 =⇒ normal good
m < 0 =⇒ inferior good
Demand Elasticity
Elasticity & Revenue
Other Elasticities
Example: Workout 15.4 continued
Recall that ln D(p, m) = 1000 − p + ln m, so
D(p, m) = me 1000 e −p
What is the income elasticity of demand?
Differential w.r.t m:
∂D
D(p, m)
= e 1000 e −p =
∂m
m
m =
D(p,m)
m
m
D(p,m)
=1
Interpretation: m ↑ by $1 =⇒ D ↑ $1? NO! m ↑ by 1%
=⇒ D ↑ 1%
Demand Elasticity
Elasticity & Revenue
Elasticity & Revenue
What happens to revenue when you change p?
Revenue: R = pq
Change in revenue w.r.t. p:
∂R
∂q
=p·
+ q · 1 = q + q = q(1 + )
∂p
∂p
How does a price increase change revenue?
R ↑ if demand is inelastic
R ↓ if demand is elastic
R is unchanged if demand is unit elastic
Demand Elasticity
Elasticity & Revenue
Elasticity & Revenue
Q: What price maximizes revenue?
Price
Elastic
Unit elastic
Inelastic
Bats
∂R
= q(1 + ∗ ) = 0 ⇐⇒ ∗ = −1
∂p
A: The price at which demand is unit elastic
Example: D(p) = 40 − 2p. Unit elasticity occurs at
∗ =
p∗
∗ (−2) = −1 =⇒ p ∗ = 10
40 − 2p ∗
Demand Elasticity
Elasticity & Revenue
Marginal Revenue
What happens to revenue when quantity q changes?
Marginal Revenue:
∂R
∂p
=p·1+q
∂q
∂q
1
q ∂p
= p(1 + )
= p+p
p ∂q
MR =
Example: if = −1/2 then MR = −p < 0, so reducing the
quantity will increase revenue.
Demand Elasticity
Elasticity & Revenue
Marginal Revenue
Linear demand: p(q) = a − bq (inverse demand)
Price
Elastic
(Є < -1)
Unit elastic
(Є = -1)
Inelastic
(-1 < Є < 0)
Bats
Demand Elasticity
Elasticity & Revenue
Marginal Revenue
Linear demand: p(q) = a − bq (inverse demand)
Price
Unit
elasticity
a
a/2
MR
a/(2b)
Bats
MR = a − 2bq, so revenue maximizing (p, q) = (a/2, a/2b).