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Slutsky Equation
1
Slutsky’s Identity



Let xi (pi , m) be consumer’s demand for good i
when price of good i is pi and income is m
holding other prices constant
Similarly for xi (pi , m)
If the price of good i changes from pi to pi
Total change in demand denoted by
∆xi = xi (pi , m) - xi (pi , m)
2
Slutsky’s Identity


Now let m′be the new level of income such that
the consumer is just able to buy the original bundle
of goods
Total change in demand
∆xi = xi (pi , m) - xi (pi , m)
can be rewritten as
)]
∆xi = [xi (pi ,m) - xi (pi ,m) ] +[xi (p′
i , m) - xi (p′
i , m′
or denote
∆xi = ∆xis + ∆xin
where ∆xis = substitution effect and ∆xin = income effect
3
Slutsky’s Identity


Note that m - m is the amount of the change in
money income such that the consumer is just
able to buy the original bundle of goods (i.e.
purchasing power is constant)
Denote ∆m = m - m and ∆pi = pi - pi
∆m = ∆pi xi (pi , m)
This is the amount of money that should be given to the
consumer to hold purchasing power constant
4
Slutsky’s Identity

In terms of the rates of change, we can write
Slutsky’s Identity as
∆xi
∆pi
∆xis
∆pi
where ∆xin =
∆xim xi(pi, m)
∆m
∆xim
5
Effects of a Price Change

What happens when a commodity’s
price decreases?
 Substitution
effect: the commodity is
relatively cheaper, so consumers substitute
it for now relatively more expensive other
commodities.
 Income effect: the consumer’s budget of $m
can purchase more than before, as if the
consumer’s income rose, with consequent
income effects on quantities demanded.

Vice versa for a price increase
6
Effects of a Price Change
x2
m
p2
Consumer’s budget is $m.
Original choice
x1
7
Effects of a Price Change
x2
m
p2
Lower price for commodity 1
pivots the constraint outwards
New Constraint:
purchasing power is increased
at new relative prices
x1
8
Effects of a Price Change
x2
m
p2
m'
p2
Now only $m' are needed to buy the
original bundle at the new prices,
as if the consumer’s income has
increased by $m - $m'.
x1
Imagined Constraint: Income is adjusted
to keep purchasing power constant
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Effects of a Price Change
Changes to quantities demanded due to
this ‘extra’ income ($m - $m') are the
income effect of the price change.
 Slutsky discovered that changes to
demand from a price change are always
the sum of a pure substitution effect and
an income effect.

10
Real Income Changes

Slutsky asserted that if, at the new
prices,
 less
income is needed to buy the original
bundle then “real income” is increased
 more income is needed to buy the original
bundle then “real income” is decreased
11
Real Income Changes
x2
Original budget constraint and choice
New budget constraint
x1
12
Real Income Changes
x2
Less income is needed to
buy original bundle.
Hence, ……………………..
x1
13
Real Income Changes
x2
Original budget constraint and choice
New budget constraint
x1
14
Real Income Changes
x2
More income is needed to
buy original bundle.
Hence, ………………………
x1
15
Real Income Changes

Absence of Money illusion
If money income and prices increase
(or decrease) by the same proportion,
e.g. double
→ budget constraint and consumer’s
choice remain unchanged
16
Pure Substitution Effect

Slutsky isolated the change in demand
due only to the change in relative prices by
asking “What is the change in demand
when the consumer’s income is adjusted
so that, at the new prices, she can only
just buy the original bundle?”
17
Budget Constraints and Choices
x2
x2
Original budget constraint and choice
Original Indifference Curve
x1
x1
18
Budget Constraints and Choices
x2
New budget constraint
when relative price of x1 is lower
x2
x1
x1
19
Budget Constraints and Choices
x2
x2
Imagined budget constraint
x1
x1
20
Budget Constraints and Choices
x2
x2
Imagined Budget Constraint,
Indifference Curve, and Choice
x2
x1
x 1
x1
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Pure Substitution Effect Only
x2
Lower p1 makes good 1 relatively
cheaper and causes a substitution
from good 2 to good 1.
(
,
) ( ,
) is the
pure substitution effect
x2
x2
x1
x 1
x1
22
The Income Effect
x2
The income effect is
(
,
) (
,
x2
)
( x1 , x2 )
x2
x1
x 1
x1
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Total Effect
x2
The change in demand due to
lower p1 is the sum of the
income and substitution effects,
(
,
) (
,
)
( x1 , x2 )
x2
x2
x1
x 1
x1
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Slutsky’s Effects for Normal Goods
Most goods are normal (i.e. demand
increases with income).
 The substitution and income effects
reinforce each other when a normal good’s
own price changes.

25
Slutsky’s Effects for Normal Goods
x2
Good 1 is normal because .……
…………………………………….
x2
( x1 , x2 )
x2
x1
x 1
x1
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Slutsky’s Effects for Normal Goods
x2
so the income and
substitution effects
………… each other
x2
( x1 , x2 )
x2
Total
Effect
x1
x 1
x1
27
Slutsky’s Effects for Normal Goods
When pi decreases, ∆pi is negative (─)
∆pi → ∆xi = ∆xis + ∆xin
(─)
( )
( )
( )
both substitution and income effects
increase demand when own-price falls.
 Alternatively,
∆xi
∆xis
∆xim xi(pi, m)
=
─
∆pi
∆pi
∆m
( )
( )
( )x( )

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Slutsky’s Effects for Normal Goods
When pi decreases, ∆pi is positive (+)
∆pi → ∆xi = ∆xis + ∆xin
(+)
( )
( )
( )
both substitution and income effects
decrease demand when own-price rises.
 Alternatively,
∆xi
∆xis
∆xim xi(pi, m)
=
─
∆pi
∆pi
∆m
( )
( )
( )x( )

29
Slutsky’s Effects for Normal Goods
In both cases, a change is own price
results in an opposite change in demand
∆xi
is always…………
∆pi
→ a normal good’s ordinary demand curve
slopes down.
 The Law of Downward-Sloping Demand
therefore always applies to normal goods.

30
Slutsky’s Effects for Income-Inferior Goods
Some goods are income-inferior (i.e.
demand is reduced by higher income).
 The pure substitution effect is as for a
normal good. But, the income effect is in
the opposite direction.
 Therefore, the substitution and income
effects oppose each other when an
income-inferior good’s own price changes.

31
Slutsky’s Effects for Income-Inferior Goods
x2
x2
x1
x1
32
Slutsky’s Effects for Income-Inferior Goods
x2
( x1 , x2 )
x2
x2
x1
x1
Good 1 is incomeinferior because
………………………
………………………
………………………
x1
33
Slutsky’s Effects for Income-Inferior Goods
x2
( x1 , x2 )
x2
Substitution and
Income effects
……….. each other
x2
Total
Effect
x1
x1
x1
34
Slutsky’s Effects for Income-Inferior Goods
When pi decreases, ∆pi is negative (─)
∆pi → ∆xi = ∆xis + ∆xin
(─)
( )
( )
( )
substitution effect increases demand while
income effect reduces demand
 Alternatively,
∆xi
∆xis
∆xim xi(pi, m)
=
─
∆pi
∆pi
∆m
( )
( )
( )x( )

35
Slutsky’s Effects for Income-Inferior Goods
When pi decreases, ∆pi is positive (+)
∆pi → ∆xi = ∆xis + ∆xin
(+)
( )
( )
( )
both substitution and income effects
decrease demand when own-price rises.
 Alternatively,
∆xi
∆xis
∆xim xi(pi, m)
=
─
∆pi
∆pi
∆m
( )
( )
( )x( )

36
Slutsky’s Effects for Income-Inferior Goods

In general, substitution effect is greater than
income effect.

Hence, ∆xi is usually positive when pi decreases.
and ∆xi is usually negative when pi increases.
∆xi
 That is
is …………………..
∆pi
and Demand Curve slopes downward
37
Giffen Goods
In rare cases of extreme income-inferiority,
the income effect may be larger in size
than the substitution effect, causing
quantity demanded to fall as own-price
rises.
 Such goods are called Giffen goods.

38
Slutsky’s Effects for Giffen Goods
x2
Income effect …………
Substitution effect.
x2
x2
x2
x1 x 1
x 1
x1
Substitution effect
Income effect
39
Slutsky’s Effects for Giffen Goods
x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2
x2
Total
x1 x 1
x1
Effect
40
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 Giffen goods.
41
Hick’s Income and Substitution Effects

Previously, we learn
Slutsky’s Substitution Effect: the change in
demand when purchasing power is kept constant.
Hick proposed another type of
Substitution Effect where consumer is
given just enough money to be on the
same indifference curve.
 Hick’s Substitution Effect: the change in
demand when utility is kept constant.

42
Hick’s Income and Substitution Effects

Total change in demand when price changes
∆xi = xi (pi , m) - xi (pi , m)
can be rewritten as
∆xi = [x i (p′
i , e( p′
i , u)) - x i (pi , m) ]
+ [x i (p′
i , m) - x i (p′
i , e( p′
i , u)) ]
Where e( p′
i , u) is minimum income needed to
achieve the original utility u at price p′
i
[x i (p′
i , e( p′
i , u)) - x i (pi , m) ] = substitution effect
[x i (p′
i , m) - x i (p′
i , e( p′
i , u)) ] = income effect
43
Hick’s Income and Substitution Effects
x2
New budget constraint when p1 falls
x2
Original choice
New choice
Original budget constraint
x1
x1
44
Hick’s Income and Substitution Effects
x2
Substitution Effect is optimal
choice found on the original
indifference curve using the
new relative prices
x2
x2
x2
x1
x 1
x1
x1
Income Effect
45
Hick’s Income and Substitution Effects
x2
As before, Substitution
and Income effects
……….. each other
x2
x2
x2
x1
x 1
x1
x1
46
Demand Curves

Marshallian (Ordinary) Demand
shows the quantity actually demanded when
own price changes holding ……….. constant

Slutsky Demand
shows Slutsky substitution effect when own price
changes holding …………………… constant

Hicksian (Compensated) Demand
shows Hick substitution effect when own price
changes holding ……….. constant
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Comparison: Hick and Slutsky Substitution
Effects when own price falls
x2
……….. budget constraint
x2
x2
x1
……… budgetx constraint
1
S
xH
1 x 1 …… Substitution
………. Substitution
x1
48
Demand Curves for Normal Good when
Own Price Falls
p1
p1
…………… Demand
………. Demand
……….. Demand
p′
1
x1
H
x1
x1S
x1
x1
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