Chapter Fifteen
Market Demand
From Individual to Market Demand
Functions
Think of an economy containing n
consumers,denoted by i = 1,…,n.
Consumer i’s ordinary demand
function for commodity j is
x p p mj i i* (,,)1 2
From Individual to Market Demand
Functions
When all consumers are price-takers,
the market demand function for
commodity j is
X p p m m x p p mj n j i i
i
n
(,,,,) (,,),*1 2 1 1 2
1

From Individual to Market Demand
Functions
p1 p1
x A1* xB1*
x xA B1 1*?
p1 20 15
35
p1’
p1”
p1’
p1”
p1’
p1”
The,horizontal sum”
of the demand curves
of individuals A and B.
Elasticities
Elasticity measures the,sensitivity”
of one variable with respect to
another.
The elasticity of variable X with
respect to variable Y is
x y x
y,
%
%
.
Own-Price Elasticity of Demand
Q,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?
A,Because the value of sensitivity
then depends upon the (arbitrary)
units of measurement used for
quantity demanded,
Arc and Point Elasticities
An,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.
Elasticity 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’?
Xi'"Xi"
X p i
ii i
X
p*,
*%
%

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’?
X p i
ii i
X
p*,
*%
%

% '? p hpi
i
100 2% ( " ' " )
( " ' " ) /
*? X X X
X Xi
i i
i i
100 2
Xi'"Xi"
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’?
Xi'"Xi"
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’?
Xi'
X p i
i
i
ii i
p
X
dX
dp*,
*'
'

is the elasticity at the
point ( ',' ).X pi i
Point Own-Price Elasticity
E.g,Suppose pi = a - bXi,
Then Xi = (a-pi)/b and
X p i
i
i
ii i
p
X
dX
dp*,*
*

.
b
1
dp
dX
i
*
i
Therefore,
X p i
i
i
ii i
p
a p b b
p
a p*,( ) /
.?

1
Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
X p i
ii i
p
a p*,
p a aa a
1
0
a/2
a/2b

Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
X p i
ii i
p
a p*,
1
0
a/2
a/2b

own-price elastic
own-price inelastic
(own-price unit elastic)
Point Own-Price Elasticity
X p i
i
i
ii i
p
X
dX
dp*,*
*

dX
dp
a pi
i
i
a
*
1
X p i
i
a i
a i
a
i
ai i
p
kp
ka p a p
p
a*,. 1
X kpi ia*,?E.g,Then
so
Point Own-Price Elasticity
pi
Xi*
X kp kp k
pi i
a
i
i
* 2
2
2everywhere along
the demand curve.
Revenue and Own-Price Elasticity of
Demand
If raising a commodity’s price causes
little (a lot of ) decrease in quantity
demanded,then sellers’ revenues
rise.
Hence own-price inelastic (elastic)
demand causes sellers’ revenues to
rise (drop) as price rises.
Revenue and Own-Price Elasticity of
Demand
R p p X p( ) ( ).*Sellers’ revenue is
So
X p* ( ),1?

dp
dX
)p(X
p1)p(X *
*
*
dR
dp
X p p dX
dp
*
*
( )
Revenue and Own-Price Elasticity of
Demand
dRdp X p* ( ) 1?
so if1 then
dR
dp? 0
and a change to price does not alter
sellers’ revenue.
Revenue and Own-Price Elasticity of
Demand
dRdp X p* ( ) 1?
but if1 0?then
dR
dp? 0
and a price increase raises sellers’
revenue.
Revenue and Own-Price Elasticity of
Demand
dRdp X p* ( ) 1?
And if1 then
dR
dp? 0
and a price increase reduces sellers’
revenue.
Marginal Revenue and Own-Price
Elasticity of Demand
A seller’s marginal revenue is the rate
at which revenue changes with the
number of units sold by the seller.
MR q dR qdq( ) ( ),?
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
MR q dR qdq dp qdq q p q( ) ( ) ( ) ( )
R q p q q( ) ( )
so

p q qp q dp qdq( ) ( ) ( ),1
Marginal Revenue and Own-Price
Elasticity of Demand
MR q p q qp q dp qdq( ) ( ) ( ) ( ),

1
dqdp pqand
so MR q p q( ) ( ),1 1?
Marginal Revenue and Own-Price
Elasticity of Demand

11)q(p)q(MR
If1 then MR q( ),? 0
If1 0?then MR q( ),? 0
If1 then MR q( ),? 0
Marginal Revenue and Own-Price
Elasticity of Demand
An example with linear inverse demand.
p q a bq( ),
Then R q p q q a bq q( ) ( ) ( )
and MR q a bq( ), 2
Marginal Revenue and Own-Price
Elasticity of Demand
p q a bq( )
MR q a bq( ) 2
a
a/b
p
qa/2b
Marginal Revenue and Own-Price
Elasticity of Demand
p q a bq( )
MR q a bq( ) 2
a
a/b
p
qa/2b
q
$
a/ba/2b
R(q)