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Chapter 15
Market Demand

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

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.

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.

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

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

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

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.

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

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).

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)

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.

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?

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*

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?

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?

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.

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.

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.

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.

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’?

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’?

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’?

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’?

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’?

Arc Own-Price Elasticity



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

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’?

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,

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,

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,

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,

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

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

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

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

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

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

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

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

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

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

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

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)

Point Own-Price Elasticity
E.g.
Then
so

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

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.

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.

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


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

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

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

Revenue and Own-Price Elasticity of Demand



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

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

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

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.

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.

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

Marginal Revenue and Own-Price Elasticity of Demand
and
so

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.

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

                                               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

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

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

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