Transcript RTDCh15 - UCSB Economics

Chapter Fifteen
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
x*ji (p1 , p2 , mi )
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
 If
n
all consumers are identical then
X j (p1 , p2 , M)  n  x*j (p1 , p2 , m)
where M = nm.
From Individual to Market Demand
Functions
 The
market demand curve is the
“horizontal sum” of the demand
curves of the individual consumers.
 For example, suppose there are only
two consumers; i = A,B.
From Individual to Market Demand
Functions
p1
p1
p1’
p1”
p1’
p1”
20 x*A
1
15
*B
x1
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
Elasticities
 Elasticity
is a measure of the
“sensitivity” of one variable with
respect to another.
 The elasticity of a variable X with
respect to a variable Y is defined as
% x
 x,y 
.
% 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)
demand for commodity i with
respect to the price of commodity j
(cross-price elasticity of demand).
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
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 then the value of
sensitivity depends upon the chosen
units of measurement.
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 measure of the
sensitivity of quantity demanded to
changes in own-price which is
independent of the choice of units of
measurement.
Arc and Point Elasticities
 If
we measure the “average” ownprice elasticity of demand for
commodity i over an interval of
values for pi then we compute an arcelasticity, usually by using a midpoint formula.
 Elasticity computed for a single
value of pi is called a point elasticity.
Point Own-Price Elasticity
dX*i
 X* ,p  * 
i i
dpi
Xi
pi
An example of computing a point own-price
elasticity of demand.
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
Point Own-Price Elasticity
pi
pi = a - bXi*
pi
 X* ,p  
i
i
a  pi
a
a/b
Xi*
Point Own-Price Elasticity
pi
a
pi
 X* ,p  
i
i
a  pi
pi = a - bXi*
p 0  0
0
a/b
Xi*
Point 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*
Point 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*
Point 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*
Point Own-Price Elasticity
dX*i
 X* ,p  * 
i i
dpi
Xi
A second example of computing a point
own-price elasticity of demand.
*
*
a
dX
i  apa 1
Suppose Xi  kpi . Then
i
dpi
so
a
pi
pi
pi
a 1
 X* ,p 
 kapi
a
 a.
a
a
i
i
kpi
pi
Point Own-Price Elasticity
pi
k
*
a
2
Xi  kpi  kpi 
2
pi
  2
everywhere along
the demand curve.
Xi*
Revenue and Own-Price Elasticity of
Demand
 If
raising a commodity’s price causes
almost no decrease in quantity
demanded then sellers’ revenues will
rise.
 Hence own-price inelastic demand
will cause sellers’ revenues to rise as
price rises.
Revenue and Own-Price Elasticity of
Demand
 If
raising a commodity’s price causes
a very large decrease in quantity
demanded then sellers’ revenues will
fall.
 Hence own-price elastic demand will
cause sellers’ revenues to fall as
price rises.
Revenue and Own-Price Elasticity of
Demand
*
Sellers’ revenue is R(p)  p  X (p).
*
dR
dX
So
 X* (p)  p
dp
dp
Revenue 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 
Revenue 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   .
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.
Revenue 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.
Revenue and Own-Price Elasticity of
Demand
dR
 X* (p)1   
dp
And if
  1
dR
0
then
dp
and a price increase reduces sellers’
revenue.
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.
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
Marginal Revenue and Own-Price
Elasticity of Demand
Let p(q) denote the seller’s inverse
demand function; that is, 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 
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   .


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
of the own-price elasticity of demand.
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.
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.
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.
Marginal Revenue and Own-Price
Elasticity of Demand
p
a
p( q)  a  bq
a/2b
a/b
q
MR( q)  a  2bq
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