Your Suggestions
Sample problems and
examples in lecture.
Download recitation
problems before
recitation.
Complete exercises in
recitations.
Reorganize web site.
Have power point slides
available earlier.
Overview class at the
beginning.
Your Suggestions
Board/slides.
Too fast/too slow.
Book does not have
enough examples.
Market Demand
From individual to
market demand.
Price elasticity of
demand.
Income elasticity of
demand.
An example: the
Laffer curve.
From Individual to Market
demand
Individual
i ‘s demand function for good 1:
x1i  x1i p1, p2, mi 
Aggregate demand (market demand)
function for good 1:
n
X 1 p1, p 2, m1, m2,..., mn    x1i  p1, p 2, mi 
i 1
Market Demand: Example
Consider 2 consumers of CDs:
i  1,2
Each consumer has the demand function:
xi  mi  p
Consumers have different incomes:
m1  $100
m2  $200
Market Demand: Example
Individual demand functions:
x1  $100  p
x 2  $200  p
Market demand:
X  $300  2 p
for
p  $100
X  $200  p
for
$100  p  $200
Inverse Market Demand:
Example
Market demand:
X  $300  2 p
X  $200  p
for
for
p  $100
$100  p  $200
Inverse demand:
p  $150  X / 2
for
p  $200  X
for
p  $100
$100  p  $200
Market Demand Curve
p
Individual demand
curves:
Market demand curve:
p
200
200
100
100
0
100
200
xi
0
100
300
X
Aggregation
Q:Is the sum of our demands (aggregate
demand) for a good always equal to the
demand of one individual whose income is
given by the sum of our incomes?
In other words is aggregate demand equal to
the demand of some representative
consumer who has income equal to the sum
of all individual incomes?
Aggregation
A: No, for two reasons:
1. Individuals have different preferences
2. Even if individuals had the same
preferences, some goods are necessary
goods, and others are luxury goods.
Aggregation: Example with a
Necessary Good
2 consumers with same
preferences
m
Equal income
distribution:
$144
X 1  10 10  20
$100
Unequal income
distribution:
$56
X 1  12  7.5  19.5
0
m  x1
7.5 10 12
2
x1
Elasticity
Looking for a measure
of how “responsive”
individual and
aggregate demands are
to changes in price and
income.
This measure is
important to determine
effects of taxes on
prices.
One Candidate
One candidate measure of how
“responsive” demand is to price changes is
the slope of the demand function (at a
given point):
X 1 p1, p 2, m1, m 2,..., mn 
p1
Problem with Slope of Demand
Function
Example: X G  100  p where
represents gallons of gasoline and
of one gallon.
X1
p is the price
Change units and measure gasoline in quarts (1/4
of gallon).
Let X Q represent quarts of gasoline. Demand is:
X Q  400  4 p
Elasticity
Instead of using slope, use price elasticity of
demand  :
 X 1 p1, p 2, m1, m2,..., mn   p1 
 
  
p1

 X 1 
Advantage:  independent of units
Example Cont’d
Demand for gasoline: X G  100  p
p
 p 

Elasticity:   1
100  p
 XG 
Demand for gasoline: X Q  400  4 p
Elasticity:
4p
p
 p 
  4   

400  4 p
100  p
 XQ 
Properties of Elasticity
Elasticity changes
with demand:
p
 
100
p
 
100  p
 1
50
 0
0
50
100
X
Properties of Elasticity
A demand function is elastic if:
A demand function is inelastic if:
 1
 1
A demand function is unit elastic if:
 1
Example: Cobb-Douglas
m
Demand function: x  c
p
x
m
Slope:
 c 2
p
p
x p 
mp
   c 2 
 1
p x 
p  cm
2
Elasticity:
Income Elasticity of Demand
Describes how responsive demand is to
changes in individual or aggregate income.
Defined similarly to price elasticity:
 x1 p1, p 2, m   m 
 
 
m

 x1 
Income Elasticity of Demand
Normal goods:
Inferior goods:
Luxury goods:
x1 p1, p 2, m  m
0
m
x1
x1 p1, p 2, m  m
0
m
x1
x1 p1, p 2, m  m
1
m
x1
The Laffer Curve
How do government
tax revenue change
when the tax rate
changes?
The Laffer Curve
Tax R.
If t  0 : zero
revenues.
If t  1 : zero
revenues.
There exists a tax rate *
t
that maximizes
revenues.
0
t
*
1 t
The Laffer Curve
Consider a population of identical workers
*
Each worker earns an hourly wage w
Each worker has to pay a tax t on his/her
wage
Thus a worker’s net hourly wage is:
w  1  t w
*
The Laffer Curve
A worker decides how many hours to work
according to the following labor supply
function:

Tax revenue:
T  twxh

* a
xh  w  (1  t ) w
a
The Laffer Curve
Tax revenue:
T  twxh
How do revenues change with the tax rate:
T
xh
 wxh  tw
t
t
The Laffer Curve
How do revenues change with the tax rate:
T
xh
 wxh  tw
t
t
Compute:
xh ((1  t ) w)
a 1

 a((1  t ) w) w
t
t
a
The Laffer Curve
Compare
with
so that
xh
a 1
 a((1  t ) w) w
t
xh
a 1
 a ((1  t ) w) 1  t 
w
xh
xh w

t
w 1  t 
The Laffer Curve
We know that:
Then:
xh
xh w

t
w 1  t 
T
xh
xh w
 wxh  tw
 wxh  tw
t
t
w 1  t 
The Laffer Curve
We want tax revenues to decrease with the
tax rate:
T
xh w
 wxh  tw
0
t
w 1  t 
This occurs when:
xh w
xh  t
w 1  t 
The Laffer Curve
This occurs when:
Rearrange:
xh w
xh  t
w 1  t 
xh w 1  t 

w xh
t
The Laffer Curve
Condition:
xh w 1  t 

w xh
t
Compute elasticity of labor supply:
xh w
w
a 1
 a((1  t ) w) 1  t 
a
w xh
((1  t ) w)
a
The Laffer Curve
Thus we have that tax revenues increase
when government reduces tax rate if:

1 t
a
t
Elasticity of labor supply estimated to be at
most 0.2
Tax rate on labor income is at most 0.5
The Laffer Curve
Elasticity of labor supply estimated to be at
most 0.2
Tax rate on labor income is at most 0.5
Plug into our condition and check that it is
not verified:

1 t
1  0.5
a

 0.2 
1
t
0.5