Transcript Chapter Fifteen

Market Demand
Molly W. Dahl
Georgetown University
Econ 101 – Spring 2009
1
From Individual to Market Demand
Functions

Consumer i’s demand for commodity j
x*ji (p1 , p2 , mi )

Market demand for commodity j
n *i
X j (p1 , p2 , m ,, m )   x j (p1 , p2 , mi ).
i 1
1
n
2
From Individual to Market Demand
Functions
The market demand curve is the
“horizontal sum” of the individual
consumers’ demand curves.
 Suppose there are only two consumers,
i = A,B.

3
From Individual to Market Demand
Functions
p1
p1
p1’
p1”
p1’
p1”
20 x*A
1
15
*B
x1
4
From Individual to Market Demand
Functions
p1
p1
p1’
p1”
p1
p1’
p1”
20 x*A
1
15
*B
x1
p1’
x*1A  xB
1
5
From Individual to Market Demand
Functions
p1
p1
p1’
p1”
p1
p1’
p1”
20 x*A
1
15
*B
x1
p1’
p1”
x*1A  xB
1
6
From Individual to Market Demand
Functions
p1
p1
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
7
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
8
Own-Price Elasticity
What is the own-price elasticity
pi
of demand in a very small interval
of prices centered on pi’?
*
pi ' dXi
 X* ,p 

pi’
i i
Xi ' dpi
is the elasticity at the
point ( Xi ', pi ' ).
Xi '
Xi*
9
Own-Price Elasticity
dX*i
 X* ,p  * 
i i
dpi
Xi
pi
Consider a linear demand curve.
If pi = a – bXi then Xi = (a-pi)/b and
*
dXi
1
 .
dpi
b
Therefore,
pi
1
pi

 X* ,p 
    
.
i i
( a  pi ) / b  b
a  pi
10
Own-Price Elasticity
pi
pi = a - bXi*
pi
 X* ,p  
i
i
a  pi
a
a/b
Xi*
11
Own-Price Elasticity
pi
a
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
p 0  0
0
a/b
Xi*
12
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*
13
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*
14
Own-Price Elasticity
pi
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
a   
own-price elastic
a/2
  1 own-price unit elastic
own-price inelastic
0
a/2b
a/b
Xi*
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Revenue, Price Changes, and Own-Price
Elasticity of Demand

Inelastic Demand:
 Raising
a commodity’s price causes a small
decrease in quantity demanded


Sellers’ revenues rise as price rises.
Elastic Demand:
 Raising
a commodity’s price causes a large
decrease in quantity demanded

Seller’s revenues fall as price rises.
16
Revenue, Price Changes, and Own-Price
Elasticity of Demand
*
Sellers’ revenue is R(p)  p  X (p).
17
Revenue, Price Changes, and Own-Price
Elasticity of Demand
*
Sellers’ revenue is R(p)  p  X (p).
*
dR
dX
So
 X* (p)  p
dp
dp
18
Revenue, Price Changes, and Own-Price
Elasticity of Demand
*
Sellers’ revenue is R(p)  p  X (p).
*
dR
dX
So
 X* (p)  p
dp
dp
*

p dX
*
 X (p )1 

*
 X (p ) dp 
19
Revenue, Price Changes, and Own-Price
Elasticity of Demand
*
Sellers’ revenue is R(p)  p  X (p).
*
dR
dX
So
 X* (p)  p
dp
dp
*

p dX
*
 X (p )1 

*
 X (p ) dp 
 X* (p)1   .
20
Revenue, Price Changes, and Own-Price
Elasticity of Demand
dR
 X* (p)1   
dp
dR
0
so if   1 then
dp
and a change to price does not alter
sellers’ revenue.
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Revenue, Price Changes, and Own-Price
Elasticity of Demand
dR
 X* (p)1   
dp
dR
0
but if  1    0 then
dp
and a price increase raises sellers’
revenue.
22
Revenue, Price Changes, and Own-Price
Elasticity of Demand
dR
 X* (p)1   
dp
dR
0
And if   1 then
dp
and a price increase reduces sellers’
revenue.
23
Revenue, Price Changes, 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.
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Marginal Revenue, Quantity Changes, 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
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Marginal Revenue, Quantity Changes, 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 
26
Marginal Revenue, Quantity Changes, 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   .


27
Marginal Revenue, Quantity Changes, 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
of the own-price elasticity of demand.
28
Marginal Revenue, Quantity Changes, 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.
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Marginal Revenue, Quantity Changes, 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.
30
Marginal Revenue, Quantity Changes, 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.
31
Marginal Revenue and Own-Price Elasticity
of Demand
p
a
p( q)  a  bq
a/2b
a/b
q
MR( q)  a  2bq
32