# Chapter 8 Slutsky Equation

```Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics

In Chapter 6, we talk about how demand
changes when price and income change
individually.
In this chapter, we want to further analyze how
the change in price changes the demand.
 In particular, we decompose the change in
quantity demanded due to price change into
substitution effect and income effect.

2

What happens when a commodity’s price
decreases?
 Substitution effect: the commodity is
relatively cheaper, so consumers use more of
it, instead of other commodities, which are
now relatively more expensive.
 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.
3
x2
m
p2
Consumer’s budget is \$m.
Original choice
x1
4
x2
m
p2
Consumer’s budget is \$y.
Lower price for commodity 1
pivots the constraint outwards.
x1
5
x2
m
p2
m'
p2
Consumer’s budget is \$m.
Lower price for commodity 1
pivots the constraint outwards.
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
6

Slutsky asserted that if, at the new prices,
If less income is needed to buy the original
bundle then “real income” is increased
 If more income is needed to buy the original
bundle then “real income” is decreased

7


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.
8

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
9
x2
x2’
x1’
x1
10
x2
x2’
x1’
x1
11
x2
x2’
x1’
x1
12
x2
x2’
x2’’
x1’
x1’’
x1
13
x2
x2’
x2’’
x1’
x1’’
x1
14
x2
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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


Substitution effect is always negatively related to
the price change.
Note that the portion of the yellow 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,
and as a result, the substitution effect is positive for
a price decrease.
16
x2
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
17
x2
The income effect is
(x1’’,x2’’)  (x1’’’,x2’’’).
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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The change to demand due to
lower p1 is the sum of the
income and substitution effects,
(x1’,x2’)  (x1’’’,x2’’’).
x2
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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

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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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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

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.
22


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.
23
x2
x2’
x1’
x1
24
x2
x2’
x1’
x1
25
x2
x2’
x1’
x1
26
x2
x2’
x2’’
x1’
x1’’
x1
27
x2
The pure substitution effect is as for
a normal good. But, ….
x2’
x2’’
x1’
x1’’
x1
28
The pure substitution effect is as for a
normal good. But, the income effect is
in the opposite direction.
x2
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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x2
The overall changes to demand are
the sums of the substitution and
income effects.
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’
x1
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

In rare cases of extreme income-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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x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2’
x1’
x1
32
x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2’’’
x2’
x1’’’
x1’
x1
33
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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


Giffen good can only result when the income effect
of an inferior good is so strong that it dominates the
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.
35
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
36

The substitution effect is
x  x1 ( p'1 , m' )  x1 ( p1 , m)
s
1

The Income effect is
x1n  x1 ( p1 ' , m)  x1 ( p1 ' , m' )

Total Effect
x1  x1 ( p1 ' , m)  x1 ( p1, m)  x  x
s
1
n
1
37

In terms of derivative (or rate of change):
x1
x1s x1 m


p1
p1 m p1
x1
x1s x1


(  x1 )
p1
p1 m
x1
x1s x1


( x1 )
p1
p1 m


Which is known as the Slutsky Equation.
(This is just a rough presentation. The tools need
for formal derivations is not covered in this class.)
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39
40
41
Slutsky’s method of decomposition is not the
only reasonable way.
 Hicks proposed another way of holding “real
income” constant.
back the original bundle, Hicks method
compensates the consumer to buy back a
bundle that gives him the same utility as before.

42
43


Hicks Substitution Effect is also negative,
because of the convex preference. (It can also be
shown by revealed preference.)
The nominal income required to maintain the
utility constant is less than the one required to
buy back the same bundle. It implies a larger
income effect for a price decrease, but a smaller
income effect for a price increase.
44


If government wants to impose tax to support
public expenditure, or to ‘punish’ consumption of
a good, say, due to pollution, various means can be
used.
Here, given the revenue are the same in
equilibrium, how do the effects of income tax and
quantity tax on good 1 differ?
(Note: Income and tax rates are regarded as given
for consumers. The tax rate is adjusted so that at
equilibrium the tax revenue is the same.)
45
46
Income tax corresponds to an inward shift of
budget line.
 Quantity tax corresponds to an inward rotation
of the budget line.
 When the revenues are held the same for
comparison, the budget line for the income tax
must pass through the optimal point for
quantity tax.
(Note: The tax revenue is tx1*, where x1* is the
optimal quantity under quantity tax.)

47


With the same tax revenue, the utility level
attained is higher with income tax than with
the quantity tax.
But quantity tax has a stronger effect in
reducing the consumption of good 1 than
income tax.
48


Consider a similar case. Now a tax is imposed
to reduce consumption of certain good, but at
the same time, an equivalent amount is rebated
(given back) to the consumer.
Again, consumer has to take the rebate and tax
rate constant for his decision.
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50


Note that the new consumption bundle must
be on the original budget line, because in
equilibrium, the tax amount and rebate are the
same.
The consumer has become worse off after this
quantity tax and rebate program.
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





In this chapter, a decomposition of price effect on
quality demand 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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