Chapter Fifteen
Market Demand
市场需求
Structure
?From Individual to Market Demand
Functions
?Elasticities
?Revenue and own-price elasticity of
demand
?Marginal revenue and price elasticity
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
?If all consumers are identical then
where M = nm.
X p p m m x p p mj n j i i
i
n
(,,,,) (,,),*1 2 1 1 2
1
? ?
?
?
X p p M n x p p mj j(,,) (,,)*1 2 1 2? ?
From Individual to Market Demand
Functions
?The market demand curve is the
“horizontal sum” of the individual
consumers’ demand curves.
?E.g,suppose there are only two
consumers; i = A,B.
From Individual to Market Demand
Functions
p1 p1
x A1* xB1*20 15
p1’
p1”
p1’
p1”
From Individual to Market Demand
Functions
p1 p1
x A1* xB1*
x xA B1 1* ?
p1 20 15
p1’
p1”
p1’
p1”
p1’
From Individual to Market Demand
Functions
p1 p1
x A1* xB1*
x xA B1 1* ?
p1 20 15
p1’
p1”
p1’
p1”
p1’
p1”
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 （ 弹性）
?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,
%
%
.? ?
?
Economic Applications of Elasticity
?Economists 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,需求的自价格弹性 )
?demand for commodity i with
respect to the price of commodity j
(cross-price elasticity of demand，
需求的交叉价格弹性 ),
Economic Applications of Elasticity
?demand for commodity i with
respect to income (income
elasticity of demand 需求的收入弹
性 )
?quantity supplied of commodity i
with respect to the price of
commodity i (own-price elasticity
of supply 供给的自价格弹性 )
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
?Q,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 10slope= - 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 10slope= - 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 10slope= - 2 slope= - 0.2
p1 p110-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 10slope= - 2 slope= - 0.2
p1 p110-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
?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,
Own-Price Elasticity of Demand
?
x p
x
p1 1
1
1
*,
*%
%
?
?
?
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.
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*
pi = a - bXi*
a
a/b
Point Own-Price Elasticity
pi
Xi*
pi = a - bXi* ? X p
i
ii i
p
a p*,? ? ?
a
a/b
Point Own-Price Elasticity
pi
Xi*
pi = a - bXi* ? X p
i
ii i
p
a p*,? ? ?
p ? ? ?0 0?a
a/b
Point Own-Price Elasticity
pi
Xi*
pi = a - bXi* ? X p
i
ii i
p
a p*,? ? ?
p ? ? ?0 0?
? ? 0
a
a/b
Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
? X p i
ii i
p
a p*,? ? ?
p a aa a? ? ? ? ? ? ?2 2 2 1? / /
? ? 0
Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
? X p i
ii i
p
a p*,? ? ?
p a aa a? ? ? ? ? ? ?2 2 2 1? / /
? ? ?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*,? ? ?
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*,? ? ?
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 （ 有弹性）
own-price inelastic （ 缺乏弹性）
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)
单位弹性
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
decrease in quantity demanded,then
sellers’ revenues rise.
?Hence own-price inelastic demand
causes sellers’ revenues to rise as price
rises.
?If raising a commodity’s price causes a
large decrease in quantity demanded,then
sellers’ revenues fall.
?Hence own-price elastic demand causes
sellers’ revenues to fall as price rises.
Revenue and Own-Price Elasticity of
Demand
R p p X p( ) ( ).*? ?Sellers’ revenue is
Revenue and Own-Price Elasticity of
Demand
R p p X p( ) ( ).*? ?Sellers’ revenue is
So
dR
dp
X p p dX
dp
? ?*
*
( )
Revenue and Own-Price Elasticity of
Demand
R p p X p( ) ( ).*? ?Sellers’ revenue is
So
?
?
?
?
?
?
?
?
??
dp
dX
)p(X
p1)p(X *
*
*
dR
dp
X p p dX
dp
? ?*
*
( )
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 ?
Revenue and Own-Price Elasticity of
Demand
? ?dRdp X p? ?* ( ) 1 ?
so if ? ? ?1then
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 ? ? ?1then
dR
dp ? 0
and a price increase reduces sellers’
revenue.
Revenue and Own-Price Elasticity of
Demand
In summary:
? ? ?1 0?? Own-price inelastic demand;
price rise causes rise in sellers’ revenue.
? Own-price unit elastic demand;
price rise causes no change in sellers’
revenue.
? ? ?1
? Own-price elastic demand;
price rise causes fall in sellers’ revenue.
? ? ?1
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
MR q p q( ) ( )? ???? ???1 1?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
??
?
??
?
???
11)q(p)q(MR
If ? ? ?1 then MR q( ),? 0
If ? ? ?1 0?then MR q( ),? 0
If ? ? ?1 then MR q( ),? 0
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 ? ? ?1then MR q( ),? 0
If ? ? ?1 0?then MR q( ),? 0
If ? ? ?1then 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)
Income Elasticity
?Normal good,?>0
?Inferior good,?<0
?Luxury good,?>1
?Necessary good,0<?<1
*
*
*,i
i
Xm
i
dXm
X d m
? ??
A Special Property
?Weighted average of income
elasticities of all goods is one with
the weights being expenditure
shares of the goods.
?In the case of 2 goods,
1 1 2 2 1ss????