## Market Demand Demand Elasticity Elasticity & Revenue Marginal Revenue. Market Demand Chapter 15

Market Demand Demand Elasticity Elasticity & Revenue Market Demand Chapter 15 Marginal Revenue Market Demand Demand Elasticity Elasticity & Re...
Author: Homer Robinson
Market Demand

Demand Elasticity

Elasticity & Revenue

Market Demand Chapter 15

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Outline

• Deriving market demand from individual demands • How responsive is qd to a change in price? (elasticity) • What is the relationship between revenue and demand

elasticity?

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Clicker Vote

When deriving market demand from individual demand curves, we add them up A) Vertically B) Diagonally C) Horizontally D) It depends

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

From Individual Demands to Market Demand Let the (inverse) demand of agent 1 and agent 2 be P(q1 ) = 20 − q1 q2 P(q2 ) = 5 − 2 Price

Price

20 D1(p)

5

20 Demand of Agent 1

Cats

D2(p)

10 Demand of Agent 2

Cats

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

From Individual Demands to Market Demand To find market (total) demand, we must fix the price and add up the quantities. Easier to do with demand, as opposed to inverse demand.

Price

Price

20 D1(p)

5

20 Demand of Agent 1

Cats

D2(p)

10 Demand of Agent 2

Cats

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

From Individual Demands to Market Demand To find market (total) demand, we must fix the price and add up the quantities. Easier to do with demand, as opposed to inverse demand. D1 (p) = max{20 − p, 0} D2 (p) = max{10 − 2p, 0} Price

Price

20 D1(p)

5

20 Demand of Agent 1

Cats

D2(p)

10 Demand of Agent 2

Cats

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

From Individual Demands to Market Demand The market demand is the horizonal sum (for a given p) of all individual demand: X D(p) = Di (p) i

= D1 (p) + D2 (p) Price 20 D1(p)

Market Demand

kink

5

D2(p)

10

20

Demand of Agent 2

Cats

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Clicker Vote

When deriving market demand from individual demand curves, we add them up A) Vertically B) Diagonally C) Horizontally D) It depends

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal 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?

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal 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? • Look at slope of demand curve

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal 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? • Look at slope of demand curve • Serious drawback to just using slope

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal 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? • Look at slope of demand curve • Serious drawback to just using slope • Heavy dependence on arbitrary units • Solution?

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal 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? • Look at slope of demand curve • Serious drawback to just using slope • Heavy dependence on arbitrary units • Solution? • Think in terms of percent change

Market Demand

Demand Elasticity

Elasticity & Revenue

Price Elasticity of Demand How sensitive is D(p) to price? • 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 dq q dp

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Price elasticity: Example Workout 15.4 • Demand for kitty litter: ln D(p, m) = 1000 − p + ln m, where

p is price and m is income • What is the price elasticity of demand for kitty litter when 1. p = 2 and m = 500?

2. p = 3 and m = 500? 3. p = 4 and m = 1500?

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Price elasticity: 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 ln m = me 1000 e −p • What is the price elasticity of demand for kitty litter when 1. p = 2 and m = 500?

2. p = 3 and m = 500? 3. p = 4 and m = 1500?

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Price elasticity: 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 ln m = me 1000 e −p • What is the price elasticity of demand for kitty litter when 1. p = 2 and m = 500? 1000 −p e = −D(p, m) So Differentiate to find: dq dp = −me =

p (−D(p, m)) = −p D(p, m)

2. p = 3 and m = 500? 3. p = 4 and m = 1500?

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Price elasticity: 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 ln m = me 1000 e −p • What is the price elasticity of demand for kitty litter when 1. p = 2 and m = 500? 1000 −p e = −D(p, m) So Differentiate to find: dq dp = −me =

p (−D(p, m)) = −p D(p, m)

So  = −2 2. p = 3 and m = 500?  = −3 3. p = 4 and m = 1500?  = −4

Market Demand

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 (|Є| < 1) Demand Bats

Marginal Revenue

Market Demand

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

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Price Elasticity of Demand Iso-elastic demand: q = ap −b −b−1 • dq dp = a · (−b)p p −b−1 = −b •  = pq dq dp = ap −b a · (−b)p • || = b

Price

Elasticity = -b

Bats

Marginal Revenue

Market Demand

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

x dF F (x, y ) dx • Example: income elasticity of demand • • • •

How sensitive is D to a change in income? dD m m = D(p,m) dm m ≥ 0 =⇒ normal good m < 0 =⇒ inferior good

Marginal Revenue

Market Demand

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?

Marginal Revenue

Market Demand

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? • Differentiate w.r.t m:

dD D(p, m) = e 1000 e −p = dm m

Marginal Revenue

Market Demand

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? • Differentiate w.r.t m:

dD D(p, m) = e 1000 e −p = dm m D(p,m) m • m = D(p,m) =1 m

Marginal Revenue

Market Demand

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? • Differentiate w.r.t m:

dD D(p, m) = e 1000 e −p = dm m D(p,m) m • m = D(p,m) =1 m • Interpretation: m ↑ by \$1 =⇒ D ↑ \$1?

Marginal Revenue

Market Demand

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? • Differentiate w.r.t m:

dD D(p, m) = e 1000 e −p = dm m D(p,m) m • m = D(p,m) =1 m • Interpretation: m ↑ by \$1 =⇒ D ↑ \$1? NO! m ↑ by 1%

=⇒ D ↑ 1%

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Elasticity & Revenue What happens to revenue when you change p? • Revenue: R = pq

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Elasticity & Revenue What happens to revenue when you change p? • Revenue: R = pq • Change in revenue w.r.t. p:

dR dq =p· + q · 1 = q + q = q(1 + ) dp dp

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Elasticity & Revenue What happens to revenue when you change p? • Revenue: R = pq • Change in revenue w.r.t. p:

dR dq =p· + q · 1 = q + q = q(1 + ) dp dp • How does a price increase change revenue? • R ↑ if demand is • R ↓ if demand is • R is unchanged if demand is

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Elasticity & Revenue What happens to revenue when you change p? • Revenue: R = pq • Change in revenue w.r.t. p:

dR dq =p· + q · 1 = q + q = q(1 + ) dp dp • 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

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Elasticity & Revenue Q: What price maximizes revenue? Price Elastic Unit elastic

Inelastic

Bats

dR = q(1 + ∗ ) = 0 ⇐⇒ ∗ = −1 dp 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 ∗

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Marginal Revenue What happens to revenue when quantity q changes? • Marginal Revenue:

dR dp =p·1+q dq dq q dp 1 = p+p = p(1 + ) p dq 

MR =

• Example: if  = −1/2 then MR = −p < 0, so reducing the

quantity will increase revenue.

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue Linear demand: p(q) = a − bq (inverse demand)

Price Elastic (|Є| > 1)

Unit Elastic (|Є| = 1)

Inelastic (|Є| < 1) Demand Bats

Marginal Revenue

Market Demand

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).

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue: Clicker Vote 1 Q: For a given quantity, how does MR(q) relate to P(q)? A) B) C) D)

Above Equal Below Depends

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue: Clicker Vote 1 Q: For a given quantity, how does MR(q) relate to P(q)? A) B) C) D)

Above Equal Below Depends

Correct! (Why?)

Price

MR Kale

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue Q: Why is the MR curve always below D?

Price

MR Kale

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue

Marginal Revenue Q: Why is the MR curve always below D? A: Lower price to sell additional unit; earn extra p on additional unit, but lose revenue w/ lower price on all previous units. dR dp R = pq =⇒ MR = =p·1+q dq dq Price

MR Kale

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue: Clicker Vote 2 How elastic is demand at the quantity at which MR = 0? A) Elastic B) Unit Elastic C) Inelastic D) Not enough info

Price

MR Kale

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Clicker Vote How elastic is demand at the quantity at which MR = 0? A) Elastic B) Unit Elastic C) Inelastic D) Not enough info

Price

Є = -1

MR Kale

Marginal Revenue

Market Demand

Demand Elasticity

Elasticity & Revenue

Marginal Revenue Linear demand: p(q) = a − bq (inverse demand)

Price Unit elasticity

a

a/2

MR a/(2b)

Bats

a MR = a − 2bq, so revenue maximizing (p, q) = ( 2a , 2b ).

Marginal Revenue