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*,
p0 0?a
a/b
Point Own-Price Elasticity
pi
Xi*
pi = a - bXi*? X p
i
ii i
p
a p*,
p0 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 a2 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 a2 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 if1then
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 if1then
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
If1 then MR q( ),? 0
If1 0?then MR q( ),? 0
If1 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
If1then MR q( ),? 0
If1 0?then MR q( ),? 0
If1then 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