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 if ? ? ?1 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 if ? ? ?1 0?then
dR
dp ? 0
and a price increase raises sellers’
revenue.
Revenue and Own-Price Elasticity of
Demand
? ?dRdp X p? ?* ( ) 1 ?
And if ? ? ?1 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
If ? ? ?1 then MR q( ),? 0
If ? ? ?1 0?then MR q( ),? 0
If ? ? ?1 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)