Chapter 15
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

*i
i
x j (p1 , p2 , m )
2
From Individual to Market Demand
Functions

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.
3
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.

4
From Individual to Market Demand
Functions
p
p
1
1
p1’
p1”
p1
p1’
p1”
20 x*A
1
15
*B
x1
The “horizontal sum”
of the demand curves
of individuals A and B.
p1’
p1”
35
x*1A  xB
1
5
Elasticities
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
6
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)
 demand for commodity i with respect to the
price of commodity j (cross-price elasticity of
demand).
7
Economic Applications of Elasticity
 demand
for commodity i with respect to income
(income elasticity of demand)
 quantity supplied of commodity i with respect to
the price of commodity i (own-price elasticity of
supply)
 and so on.
8
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?
9
Own-Price Elasticity of Demand
p1 10-packs
p1
10
10
slope
=-2
5
Single Units
slope
= - 0.2
X1
*
In which case is the quantity demanded
X1* more sensitive to changes to p1?
It is the same in both cases.
50
X1
*
10
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.

11
Own-Price Elasticity of Demand
*
%  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.
12
Own-Price Elasticity
dX*i
 X* ,p  * 
i i
dpi
Xi
pi
E.g. Suppose pi = a - bXi. Then Xi = (a-pi)/b and
*
dXi
1
  . Therefore,
dpi
b
pi
1
pi

 X* ,p 
    
.
i i
( a  pi ) / b  b
a  pi
13
Own-Price Elasticity
pi
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a
a/b
Xi*
14
Own-Price Elasticity
pi
a
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
p 0  0
0
a/b
Xi*
15
Own-Price Elasticity
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*
16
Own-Price Elasticity
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*
17
Own-Price Elasticity
pi = a - bXi*
pi
a
a/2
pi
 X* ,p  
i
i
a  pi
  
own-price elastic
  1 (own-price unit elastic)
own-price inelastic
0
a/2b
a/b
Xi*
18
Own-Price Elasticity
pi
 X* ,p  *
i i
X
i
X*i  kpia .
E.g.
So,
 X* ,p 
i
i
pi
a
kpi
*
dXi

dpi
*
i
dX
a 1
 ka pi
Then
dpi
 kapia 1  a
pia
a
pi
 a.
19
Own-Price Elasticity
pi
k
*
a
2
Xi  kpi  kpi 
pi2
  2
everywhere along
the demand curve.
Xi*
20
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.

21
Revenue and Own-Price Elasticity of
Demand
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.

22
Revenue and Own-Price Elasticity of
Demand
Sellers’ revenue is
So
*
R(p)  p  X (p).
*
dR
dX
 X* (p)  p
dp
dp
*

p dX
*
 X (p )1 

*
 X (p ) dp 
 X (p)1   .
*
23
Revenue and Own-Price Elasticity of
Demand
dR
*
 X (p)1   
dp
so if
  1
then
dR
0
dp
and a change to price does not alter sellers’
revenue.
24
Revenue and Own-Price Elasticity of
Demand
dR
*
 X (p)1   
dp
but if  1    0
dR
0
then
dp
and a price increase raises sellers’ revenue.
25
Revenue and Own-Price Elasticity of
Demand
dR
 X* (p)1   
dp
And if
  1
then
dR
0
dp
and a price increase reduces sellers’ revenue.
26
Revenue and Own-Price Elasticity of
Demand
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.
27
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.
dR( q)
MR( q) 
.
dq
28
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
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 
29
Marginal Revenue and Own-Price
Elasticity of Demand
q dp( q) 

MR( q)  p( q) 1 
.

 p( q) dq 
and
so
dq p


dp q
1

MR( q)  p( q) 1   .


30
Marginal Revenue and Own-Price
Elasticity of Demand
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 own-price elasticity of
demand.
31
Marginal Revenue and 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.
32
Marginal Revenue and Own-Price
Elasticity of Demand
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.
33
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  2bq.
34
Marginal Revenue and Own-Price
Elasticity of Demand
p
a
p( q)  a  bq
a/2b
a/b
q
MR( q)  a  2bq
35
Marginal
Revenue and Own-Price
p
Elasticity
of Demand
a
MR( q)  a  2bq
p( q)  a  bq
$
a/2b
a/b
q
a/b
q
R(q)
a/2b
36
Elasticities

Cross Price Elasticity of demand:
p j dxi
dxi / xi
i 

dp j / p j xi dp j
j

Income Elasticity of demand:
dxi / xi m dxi
 

dm / m xi dm
m
i
37
Elasticities

Ordinary Good:  ii  0
i

Giffen Good:
i  0

Normal Good:  im  0
m
Inferior Good:  i  0
m

 Luxury Good:
i 1
Necessary Good: 0   im  1
38