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15
Market Demand

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From Individual to Market Demand Functions
uThink of an economy containing n consumers, denoted by i = 1, … ,n.
uConsumer i’s ordinary demand function for commodity j is

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From Individual to Market Demand Functions
uWhen all consumers are price-takers, the market demand function for commodity j is
uIf all consumers are identical then
where M = nm.

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From Individual to Market Demand Functions
uThe market demand curve is the “horizontal sum” of the individual consumers’ demand curves.
uE.g. suppose there are only two consumers; i = A,B.

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From Individual to Market Demand Functions
p1
p1
20
15
p1’
p1”
p1’
p1”

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From Individual to Market Demand Functions
p1
p1
p1
20
15
p1’
p1”
p1’
p1”
p1’

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From Individual to Market Demand Functions
p1
p1
p1
20
15
p1’
p1”
p1’
p1”
p1’
p1”

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From Individual to Market Demand Functions
p1
p1
p1
20
15
35
p1’
p1”
p1’
p1”
p1’
p1”
The “horizontal sum”
of the demand curves
of individuals A and B.

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Elasticities
uElasticity measures the “sensitivity” of one variable with respect to another.
uThe elasticity of variable X with respect to variable Y is

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Economic Applications of Elasticity
uEconomists use elasticities to measure the sensitivity of
–quantity demanded of commodity i with respect to the price of commodity i (own-price elasticity of
demand)
–demand for commodity i with respect to the price of commodity j (cross-price elasticity of
demand).

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Economic Applications of Elasticity
–demand for commodity i with respect to income (income elasticity of demand)
–quantity supplied of commodity i with respect to the price of commodity i (own-price elasticity of
supply)

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Economic Applications of Elasticity
–quantity supplied of commodity i with respect to the wage rate (elasticity of supply with respect
to the price of labor)
–and many, many others.

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Own-Price Elasticity of Demand
uQ: Why not use a demand curve’s slope to measure the sensitivity of quantity demanded to a change
in a commodity’s own price?

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Own-Price Elasticity of Demand
X1*
5
50
10
10
slope
= - 2
slope
= - 0.2
p1
p1
In which case is the quantity demanded
X1* more sensitive to changes to p1?
X1*

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Own-Price Elasticity of Demand
5
50
10
10
slope
= - 2
slope
= - 0.2
p1
p1
X1*
X1*
In which case is the quantity demanded
X1* more sensitive to changes to p1?

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Own-Price Elasticity of Demand
5
50
10
10
slope
= - 2
slope
= - 0.2
p1
p1
10-packs
Single Units
X1*
X1*
In which case is the quantity demanded
X1* more sensitive to changes to p1?

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Own-Price Elasticity of Demand
5
50
10
10
slope
= - 2
slope
= - 0.2
p1
p1
10-packs
Single Units
X1*
X1*
In which case is the quantity demanded
X1* more sensitive to changes to p1?
It is the same in both cases.

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Own-Price Elasticity of Demand
uQ: Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to
a change in a commodity’s own price?
uA: Because the value of sensitivity then depends upon the (arbitrary) units of measurement used
for quantity demanded.

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Own-Price Elasticity of Demand
is a ratio of percentages and so has no
units of measurement.
Hence own-price elasticity of demand is a sensitivity measure that is independent of units of
measurement.

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Arc and Point Elasticities
uAn “average” own-price elasticity of demand for commodity i over an interval of values for pi is
an arc-elasticity, usually computed by a mid-point formula.
uElasticity computed for a single value of pi is a point elasticity.

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Arc Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the “average” own-price
elasticity of demand for prices
in an interval centered on pi’?

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Arc Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the “average” own-price
elasticity of demand for prices
in an interval centered on pi’?

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Arc Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the “average” own-price
elasticity of demand for prices
in an interval centered on pi’?

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Arc Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the “average” own-price
elasticity of demand for prices
in an interval centered on pi’?

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Arc Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the “average” own-price
elasticity of demand for prices
in an interval centered on pi’?

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Arc Own-Price Elasticity

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Arc Own-Price Elasticity
So
is the arc own-price elasticity of demand.

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Point Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the own-price elasticity
of demand in a very small interval
of prices centered on pi’?

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Point Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the own-price elasticity
of demand in a very small interval
of prices centered on pi’?
As h ® 0,

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Point Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the own-price elasticity
of demand in a very small interval
of prices centered on pi’?
As h ® 0,

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Point Own-Price Elasticity
pi
Xi*
pi’
pi’+h
pi’-h
What is the own-price elasticity
of demand in a very small interval
of prices centered on pi’?
As h ® 0,

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Point Own-Price Elasticity
pi
Xi*
pi’
What is the own-price elasticity
of demand in a very small interval
of prices centered on pi’?
As h ® 0,

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Point Own-Price Elasticity
pi
Xi*
pi’
What is the own-price elasticity
of demand in a very small interval
of prices centered on pi’?
is the elasticity at the
  point

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Point Own-Price Elasticity
E.g. Suppose pi = a - bXi.
Then Xi = (a-pi)/b and
Therefore,

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Point Own-Price Elasticity
pi
Xi*
pi = a - bXi*
a
a/b

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Point Own-Price Elasticity
pi
Xi*
pi = a - bXi*
a
a/b

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Point Own-Price Elasticity
pi
Xi*
pi = a - bXi*
a
a/b

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Point Own-Price Elasticity
pi
Xi*
pi = a - bXi*
a
a/b

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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b

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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
a/2
a/2b

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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
a/2
a/2b

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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
a/2
a/2b

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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
a/2
a/2b
own-price elastic
own-price inelastic

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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
a/2
a/2b
own-price elastic
own-price inelastic
(own-price unit elastic)

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Point Own-Price Elasticity
E.g.
Then
so

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Point Own-Price Elasticity
pi
Xi*
everywhere along
the demand curve.

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Revenue and Own-Price Elasticity of Demand
uIf raising a commodity’s price causes little decrease in quantity demanded, then sellers’ revenues
rise.
uHence own-price inelastic demand  causes sellers’ revenues to rise as price rises.

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Revenue and Own-Price Elasticity of Demand
uIf raising a commodity’s price causes a large decrease in quantity demanded, then sellers’
revenues fall.
uHence own-price elastic demand causes sellers’ revenues to fall as price rises.

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Revenue and Own-Price Elasticity of Demand
Sellers’ revenue is

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Revenue and Own-Price Elasticity of Demand
Sellers’ revenue is
So

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Revenue and Own-Price Elasticity of Demand
Sellers’ revenue is
So

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Revenue and Own-Price Elasticity of Demand
Sellers’ revenue is
So

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Revenue and Own-Price Elasticity of Demand

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Revenue and Own-Price Elasticity of Demand
so if
then
and a change to price does not alter
sellers’ revenue.

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Revenue and Own-Price Elasticity of Demand
but if
then
and a price increase raises sellers’
revenue.

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Revenue and Own-Price Elasticity of Demand
And if
then
and a price increase reduces sellers’
revenue.

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Revenue and Own-Price Elasticity of Demand
In summary:
Own-price inelastic demand;
price rise causes rise in sellers’ revenue.
Own-price unit elastic demand;
price rise causes no change in sellers’
revenue.
Own-price elastic demand;
price rise causes fall in sellers’ revenue.

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Marginal Revenue and Own-Price Elasticity of Demand
uA seller’s marginal revenue is the rate at which revenue changes with the number of units sold by
the seller.

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Marginal Revenue and Own-Price Elasticity of Demand
p(q) denotes the seller’s inverse demand function; i.e. the price at which the seller can sell q
units.  Then
so

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Marginal Revenue and Own-Price Elasticity of Demand
and
so

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Marginal Revenue and Own-Price Elasticity of Demand
says that the rate
at which a seller’s revenue changes
with the number of units it sells
depends on the sensitivity of quantity
demanded to price; i.e., upon the
of the own-price elasticity of demand.

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Marginal Revenue and Own-Price Elasticity of Demand
If
then
If
then
If
then

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                                               Selling one
more unit raises the seller’s revenue.
                                                 Selling one
more unit reduces the seller’s revenue.
                                               Selling one
more unit does not change the seller’s
revenue.
Marginal Revenue and Own-Price Elasticity of Demand
If
then
If
then
If
then

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Marginal Revenue and Own-Price Elasticity of Demand
An example with linear inverse demand.
Then
and

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Marginal Revenue and Own-Price Elasticity of Demand
a
a/b
p
q
a/2b

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Marginal Revenue and Own-Price Elasticity of Demand
a
a/b
p
q
a/2b
q
$
a/b
a/2b
R(q)