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Chapter 8
Slutsky Equation
Introduction
Previously, we analyzed the effect of changes in
prices and income on a consumer’s demand.
 In this chapter, we want to further analyze the
price change effect. Specifically, we will
decompose the effect into substitution effect
and income effect.

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Effects of a Price Change

What happens when a commodity’s price
decreases?
 Substitution
effect: change in demand due to
the change in the rate of exchange between the
two goods (this commodity becomes relatively
cheaper while the other good becomes relatively
more expensive).
 Income effect: change in demand due to the
increase in the consumer’s purchasing power.
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Effects of a Price Change
x2
y
p2
Consumer’s budget is $y.
Original choice
x1
4
Effects of a Price Change
x2
y
p2
Consumer’s budget is $y.
Lower price for commodity 1
pivots the constraint outwards.
x1
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Effects of a Price Change
x2
Consumer’s budget is $y.
y
p2
y'
p2
Lower price for commodity 1
pivots the constraint outwards.
Now only $y’ are needed to buy the
original bundle at the new prices,
as if the consumer’s income has
increased by $y - $y’.
x1
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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.
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Effects of a Price Change

Changes to quantities demanded due to the
change in relative prices, keeping income just
enough to buy the original bundle, are the
(pure) substitution effect of the price change.

Changes to quantities demanded due to the
change in ‘real income’ are the income effect of
the price change.
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Effects of a 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.
xi  x  x
s
i
n
i
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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 afford the original bundle?”
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Pure
Substitution
Effect
Only
x
2
x2’
x1’
x1
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Pure
Substitution
Effect
Only
x
2
x2’
x1’
x1
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Pure
Substitution
Effect
Only
x
2
x2’
x1’
x1
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Pure
Substitution
Effect
Only
x
2
x2’
x2’’
x1’
x1’’
x1
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Pure
Substitution
Effect
Only
x
2
x2’
x2’’
x1’
x1’’
x1
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Pure
Substitution
Effect
Only
x
2
Lower p1 makes good 1 relatively
cheaper and causes a substitution
from good 2 to good 1.
(x1’,x2’)  (x1’’,x2’’) is the
pure substitution effect.
x2’
x2’’
x1’
x1’’
x1
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Pure Substitution Effect



Substitution effect is always negatively related to the
price change.
Note that the portion of the grey compensated budget
line below x’1 is inside the budget set of the original
budget, thus these bundles should be less preferred
than the original bundle.
As a result, the consumer must choose a point at or
more than x’1 with the compensated budget. As a
result, the substitution effect is positive for a price
decrease.
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Income Effect
x2
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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Income Effect
x2
The income effect is
(x1’’,x2’’)  (x1’’’,x2’’’).
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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The Overall Change in Demand
x2
The change to demand due to
lower p1 is the sum of the
income and substitution effects,
(x1’,x2’)  (x1’’’,x2’’’).
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
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.
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Slutsky’s Effects for Normal Goods
x2
Good 1 is normal because
higher income increases
demand
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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Slutsky’s Effects for Normal Goods
x2
Good 1 is normal because
higher income increases
demand, so the income
and substitution
(x1’’’,x2’’’) effects reinforce
each other.
x2’
x2’’
x1’
x1’’
x1
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Slutsky’s Effects for Normal Goods

Since both the substitution and income effects
increase demand when own-price falls, a normal
good’s ordinary demand curve slopes down.

The Law of (Downward-Sloping) Demand
therefore always applies to normal goods.
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Slutsky’s Effects for Inferior Goods

Some goods are inferior (i.e. demand is reduced
when income is higher).

The substitution and income effects oppose
each other when an inferior good’s own price
changes.
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Slutsky’s Effects for Inferior Goods
x2
x2’
x1’
x1
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Slutsky’s Effects for Inferior Goods
x2
x2’
x1’
x1
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Slutsky’s Effects for Inferior Goods
x2
x2’
x1’
x1
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Slutsky’s Effects for Inferior Goods
x2
x2’
x2’’
x1’
x1’’
x1
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Slutsky’s Effects for Inferior Goods
x2
The pure substitution effect is as for
a normal good. But, ….
x2’
x2’’
x1’
x1’’
x1
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Slutsky’s Effects for Inferior Goods
The pure substitution effect is as for a
normal good. But, the income effect is
in the opposite direction.
(x1’’’,x2’’’)
x2
x2’
x2’’
x1’
x1’’
x1
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Slutsky’s Effects for Inferior Goods
x2
x2’
x2’’
The pure substitution effect is as for a normal
good. But, the income effect is
in the opposite direction. Good 1 is
(x1’’’,x2’’’) inferior because an
increase in income
causes demand to
fall.
x1’
x1’’
x1
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Slutsky’s Effects for Inferior Goods
x2
The overall changes to demand are
the sum of the substitution and
income effects.
(x ’’’,x ’’’)
1
x2’
2
x2’’
x1’
x1’’
x1
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Giffen Goods

In rare cases of extreme inferiority, the income
effect may be larger than the substitution effect,
causing quantity demanded to fall as own-price
rises.

Such goods are Giffen goods.
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Slutsky’s Effects for Giffen Goods
x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2’
x1’
x1
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Slutsky’s Effects for Giffen Goods
x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2’’’
x2’
x1’’’ x1’
x1
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Slutsky’s Effects for Giffen Goods
x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2’’’
x2’
x2’’
x1’’’ x1’
x1’’
Substitution effect
Income effect
x1
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Slutsky’s Effects for Giffen Goods



Giffen good can only result when the income effect
of an inferior good is so strong that it dominates the
pure substitution effect.
This may be possible for poor households where
the low-quality necessity has taken up a large
portion of expenditure.
This case is very rare, even if exists, so we have
confidence that the Law of Demand almost always
holds.
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Mathematical Treatment
If we denote m’ as the income required to
obtain the original bundle at the new prices, so
that
m’=p’1 x1 + p2 x2 and m=p1 x1 + p2 x2 .
 Thus the change in real income is
m’– m = (p’1 – p1 ) x1
 Or

m  p1 x1
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Mathematical Treatment

The substitution effect is
x1s  x1 ( p'1 , m' )  x1 ( p1 , m)

The Income effect is
x  x1 ( p1 ' , m)  x1 ( p1 ' , m' )
n
1

Total Effect (Slutsky identity)
x1  x1 ( p1 ' , m)  x1 ( p1 , m)  x  x
s
1
n
1
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Perfect Complements
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Perfect Substitutes
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Quasi-Linear Preference
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Summary






In this chapter, a decomposition of price effect on
quantity demanded is introduced.
Substitution effect: effect of change of price holding
‘real income’ constant.
Income effect: effect of change in real income.
For normal goods, both effects are negative w.r.t. a price
rise.
For inferior goods, sub. effect is negative, but income
effect is positive w.r.t. a price rise.
Giffen goods can only be inferior goods with very
strong income effect.
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