Transcript L7

L07
Slutsky Equation
Previous Class
Demand function x1* ( p1 , p2 , m)
How the demand is affected by
a) p1 change, holding p2 and m constant
b) m change, holding p2 and m constant
c) p2 change, holding p1 and m constant
Geometric and analytical answer!
Classification of goods
Price Change
Fix p2=1 and m=10. Change p1=2, p1’=1,
x2
What happens to
1) a relative price?
2) a purchasing power (real income)?
Can we separate the two effects?
x1
Today
2 questions
 How to measure Real Income (PP)
 Decomposition of the change in demand
– The effect resulting from the change of
a relative price (substitution effect)
– The effect resulting from the change of
real income (income effect)
Change of Real Income
p2=1 and m=10
p1=2,  p1’=1,
and
What m’ makes x* just affordable at p1 '  1, p 2  1
x2
m' 
10
x *  ( 2.5,5)
if m '  m then ...
5
10
x1
Real Income Change
 If,
at the new prices,
– less income is needed to buy the
original bundle then “real income”
is increased m '  m
– more income is needed to buy the
original bundle then “real income”
is decreased m '  m
Real Income Changes
Suppose p1, p2 changes to p1’, p2’
x2
Original budget constraint and choice
New budget constraint
Q: Real income
A) Increases
B) Decreases
C)
Stays
the
same
x1
Real Income Changes
Suppose p1, p2, changes to p1’, p2’
x2
Original budget constraint and choice
New budget constraint
x1
Substitution and Income Effect
 If
P1 changes, both relative price and real income
are affected
 Slutsky isolates the change in demand due only
to the change in relative prices
KEY IDEA:
“What is the change in demand when the
consumer’s income is adjusted (to m’) so that, at
the new prices, her real income is the same?”
Total Change
p1  1,
p1  2,
x2
p2  1,
T.CH
m  10
m  10
( 2 . 5 ,5 )
( 5 ,5 )
p2  1,
m' 
x1
Income Effect
Substitution
effect
p1  1,
p1  12,,
x2
p2  1,
m'  10
7.5
( 2 . 5 ,5 )
SE
IE
p2  1,
m'  10
7.5
( 5 ,5 )
x1
Substitution and Income Effect
*
x
 What happens to the demand
1
p1  2,
p1  1,
p2  1,
p2  1,
Total Change
m  10
 Instead
m  10
of going directly, 2 steps:
p1  2,
p2  1,
m  10
p1  1,
SE
p2  1,
m'  7.5
p1  1,
IE
p2  1,
m  10
Cobb-Douglass example
Data U  x x ,
2 2
1 2
x ( p1 , p2 , m) 
*
1
p 2  1, m  20, change p1  4  2
x ( p1 , p2 , m) 
*
2
Cobb-Douglass example
Data U  x x ,
2 2
1 2
p 2  1, m  20, change p1  4  2
Perfect Complements
U  min( x1 , x2 )
x ( p1 , p2 , m) 
*
1
p 2  1, m  20, change p1  4  2
x ( p1 , p2 , m) 
*
2
Perfect Complements
U  min( x1 , x2 )
p 2  1, m  20, change p1  4  2
Normal, Inferior and Giffent goods
 Normal
Goods
 Inferior goods
 Effects:
Reinforce or cancel out?
Normal Goods
x2
x1
Slutsky’s Effects for Income-Inferior
Goods
 Normal
good: demand increases in
income.
Q: substitution and income effects
A) Always have the same sign
B) Always have opposite signs
C) Depends of assumed parameters
Inferior Goods
x2
x1
Slutsky’s Effects for Inferior Goods
 Some
goods are inferior (i.e.
demand is reduced by higher
income).
Q: substitution and income effects
A) Always have the same sign
B) Always have opposite signs
C) Depends of assumed parameters
Giffen Goods
x2
x1
Slutsky’s Effects for Giffen
Goods
 Slutsky’s
decomposition of the effect
of a price change into a pure
substitution effect and an income
effect thus explains why the Law of
Downward-Sloping Demand is
violated for extremely incomeinferior goods.