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Market Demand
市场需求


Think of an economy containing n consumers,
denoted by i = 1, … ,n.
Consumer i’s ordinary demand function for
commodity j is
x*ji (p1 , p2 , mi )

When all consumers are price-takers, the market
demand function for commodity j is
n *i
X j (p1 , p2 , m ,, m )   x j (p1 , p2 , mi ).
i 1
1

n
If all consumers are identical then
*
X j (p1 , p2 , M)  n  x j (p1 , p2 , m)
where M = nm.


The market demand curve is the “horizontal sum” of the individual
consumers’ demand curves.
Denoted by demand function D=D(P) or inverse demand function
P=P(D)
p1
p1
p1’
p1”
p1’
p1”
p1
20 x*A
1
15
p1’
p1”
35
x*1A  xB
1
x*1B


Elasticity measures the “sensitivity” of one variable
with respect to another.
The elasticity of variable X with respect to variable
Y is
% x
 x,y 
.
% y

Economists use elasticities to measure the
sensitivity of
quantity demanded of commodity i with
respect to the price of commodity i (ownprice 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).

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 (ownprice elasticity of supply 供给的自价格弹
性)


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?
p1
10
slope
=-2
5
p1
10
X1*
slope
= - 0.2
50 X *
1
In which case is the quantity demanded
X1* more sensitive to changes to p1?
p1
10
slope
=-2
5
p1
10
X1*
slope
= - 0.2
50 X *
1
In which case is the quantity demanded
X1* more sensitive to changes to p1?
p1
10
10-packs
slope
=-2
5
p1
10
X1*
Single Units
slope
= - 0.2
50 X *
1
In which case is the quantity demanded
X1* more sensitive to changes to p1?
p1
10
10-packs
slope
=-2
5
p1
10
X1*
Single Units
slope
= - 0.2
50 X *
1
In which case is the quantity demanded
X1* more sensitive to changes to p1?
It is the same in both cases.


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.
*
%  x1
 x* ,p 
1 1
% p1
•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.
Measuring increases in percentage terms
keeps the elasticity unit-free
x x

y y
或
dx y


dy x
Price elasticity of demand
X
*
i
*
i , pi
dX
pi

 *
dpi X i
E.g. Suppose pi = a - bXi.
Then Xi = (a-pi)/b and
dxi pi
 ( p) 
dpi xi ( pi )
dx i
1

dpi
b
pi  1 
pi
pi
 1
    
   
xi  b  (a  pi ) / b  b  a  pi
pi
pi = a - bXi*
a
a/b
Xi*
pi
pi = a - bXi*
pi
 X* ,p  
i
i
a  pi
a
a/b
Xi*
pi
a
pi = a - bXi*
pi
 X* ,p  
i
i
a  pi
p 0  0
a/b
Xi*
pi
a
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
p 0  0
0
a/b
Xi*
pi
a
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a
a/2
p   
 1
2
aa/2
0
a/b
Xi*
pi
a
a/2
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a
a/2
p   
 1
2
aa/2
  1
0
a/2b
a/b
Xi*
pi
a
a/2
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a
pa  
 
aa
  1
0
a/2b
a/b
Xi*
pi = a - bXi*
pi
a   
a/2
pi
 X* ,p  
i
i
a  pi
a
pa  
 
aa
  1
0
a/2b
a/b
Xi*
pi
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a   
own-price elastic （有弹性）
a/2
  1
own-price inelastic （缺乏弹性
0
a/2b
a/b
Xi*
pi
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a   
own-price elastic
(own-price unit elastic)


1
a/2
单位弹性
own-price inelastic
0
a/2b
a/b
Xi*
dX*i
 X* ,p  * 
i i
dpi
Xi
pi
E.g. X*i  kpia . Then
so
dX*i
 apia 1
dpi
a
pi
pi
a 1
 X* ,p 
 kapi
a
 a.
a
a
i
i
kpi
pi
pi
  2
everywhere along
the demand curve.
Xi*

If raising a commodity’s price causes little decrease in
quantity demanded, then sellers’ revenues rise.


Hence own-price inelastic (缺乏弹性) 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 (富有弹性) demand
causes sellers’ revenues to fall as price rises.
*
R
(
p
)

p

X
(p).
Sellers’ revenue is
*
R
(
p
)

p

X
(p).
Sellers’ revenue is
*
dR
dX
So
 X* (p)  p
dp
dp
*
R
(
p
)

p

X
(p).
Sellers’ revenue is
*
dR
dX
So
 X* (p)  p
dp
dp
*

p dX
*
 X (p )1 

*
 X (p ) dp 
*
R
(
p
)

p

X
(p).
Sellers’ revenue is
*
dR
dX
So
 X* (p)  p
dp
dp
*

p dX
*
 X (p )1 

*
 X (p ) dp 
 X* (p)1   .
dR
 X* (p)1   
dp
dR
 X* (p)1   
dp
so if   1
then
dR
0
dp
and a change to price does not alter
sellers’ revenue.
dR
 X* (p)1   
dp
dR
0
but if  1    0 then
dp
and a price increase raises sellers’
revenue.
dR
 X* (p)1   
dp
And if
  1
dR
0
then
dp
and a price increase reduces sellers’
revenue.
In summary:
Own-price inelastic demand;  1    0
price rise causes rise in sellers’ revenue.
Own-price unit elastic demand;   1
price rise causes no change in sellers’
revenue.
Own-price elastic demand;   1
price rise causes fall in sellers’ revenue.

A seller’s marginal revenue is the rate at which
revenue changes with the number of units sold
by the seller.
dR( q)
MR( q) 
.
dq
p(q) denotes the seller’s inverse demand
function; i.e. the price at which the seller
can sell q units. Then
R( q)  p( q)  q
so
dR( q) dp( q)
MR( q) 

q  p( q)
dq
dq
q dp( q) 

 p( q) 1 
.

 p( q) dq 
q dp( q) 

MR( q)  p( q) 1 
.
 p( q) dq 
and
so
dq p


dp q
1

MR( q)  p( q) 1   .


1

MR( q)  p( q) 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.
1

MR(q)  p(q)1  


If   1
then MR( q)  0.
If  1    0 then MR( q)  0.
If   1
then MR( q)  0.
If   1 then MR( q)  0. Selling one
more unit does not change the seller’s
revenue.
If  1    0 then MR( q)  0. Selling one
more unit reduces the seller’s revenue.
If   1 then MR( q)  0. Selling one
more unit raises the seller’s revenue.
An example with linear inverse demand.
p( q)  a  bq.
Then R( q)  p( q)q  ( a  bq)q
and
MR( q)  a  2bq.
p
a
p( q)  a  bq
a/2b
a/b
q
MR( q)  a  2bq
p
a
MR( q)  a  2bq
p( q)  a  bq
$
a/2b
a/b
q
a/b
q
R(q)
a/2b

Recall that price elasticity of demand is
x
1

, p1
%x1

%p1
Hence income elasticity of demand is
m ,p 
1
%m
%p1
X




*
i ,m
*
dX
m
i


*
Xi
dm
Normal good: >0
Inferior good: <0
Luxury good: >1
Necessary good: 0<<1




From Individual to Market Demand Functions
Elasticities
Revenue and own-price elasticity of demand
Marginal revenue and price elasticity

消费者对商品 x 和在其它商品上的开支 y（价
格为1）的效用函数为
1 2
u ( x, y )  x  x  y
2
1）市场上有完全同样的消费者100人，写出x
的市场需求函数。
2）x该如何定价使销售收入最大？此时价格弹
性是多少？