Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
Economics 3030: Intermediate Microeconomic Theory
Topic 2:
Consumer Theory
The Slutsky Equation
Reference: Varian, Chapter 8
Outline:
1.
1.
2.
3.
4.
5.
6.
Introduction
Slutsky Equation
The Total Change in Demand
Example – Calculating Income and Substitution Effects
Rates of Change
Deriving the Slutsky Equation by Calculus
Introduction
We have seen that a change in price exerts both an income effect and a substitution effect and that
these may work with each other, as in the case of Normal goods, or against each other, as in the
case of Inferior and Giffen goods. We now examine these effects more formally via the Slutsky
Equation
2.
Slutsky Equation
Consider Figure 1 following:
x2
Original Choice
m p2
Final Choice
A
x2
C
x2′
B
I1
I0
0
x1
m p1
x1′
m′ p1′
I2
m p1′
Figure 1: Sltutsky Compensating Variation (Price Fall)
1
x1
Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
Thus in Figure 1 we have a fall in the price of good 1 from p1 to p1′ < p1 , a pivot outwards of the
budget line and a change in the consumer’s optimal bundle from A to C.
The fall in the price of good 1 has impacted on the consumer in two ways: First, there is the
substitution effect whereby the fall in the economic rate of substitution (ERS) between the two
goods means that the consumer does not have to sacrifice as many units of good 2 for additional
units of good 1. Second, there is an income effect whereby the fall in the price of good 1 changes
leads to an increase in the purchasing power (i.e. real income) of the consumer.
In terms of Figure 1, we measure the substitution effect from A-B and the income effect
from B-C. Under the Slutsky decomposton, the substitution effect is found by adjusting the
consumer’s income following the price change such that the consumer’s original consumption
bundle is affordable. Thus, if the price of a good falls we have to reduce money income, and vice
versa.
By how much do we need to adjust money income? Let m′ denote the amount of money
income that will make the original consumption bundle affordable at the new price of good 1 vis:
p1′x1 + p2 x2 = m′
(1)
Since x = ( x1 , x2 ) is affordable at both ( p1 , p2 , m ) and ( p1′, p2 , m′ ) , then we also have:
p1 x1 + p2 x2 = m
(2)
Subtracting (2) from (1) implies:
( m′ − m ) = x1 ( p1′ − p1 )
⇒
Δm = x1Δp1
(3)
where Δm = m′ − m and Δp1 = p1′ − p1 . Note that the change in income and the change in price
will always move in the same direction. For example, if the consumer is originally consuming 5
units of good 1, and the price of good 1 rises by $2 per unit, then money income must be increased
by 5*$2 = $10 for the consumer to be able to continue to consume his original bundle (i.e. the price
of good 2 has not changed). If the price of good 1 fell by $2 per unit, then money income must be
decreased by 5*$2 = $10 for the consumer to be able to continue to consumer his original bundle.
The Substitution Effect
The substitution effect from A to B measures how the consumer ‘substitutes’ one good for the other
when a price changes but purchasing power remains constant. It is calculated as:
Δx1s = x1 ( p1′, m′ ) − x1 ( p1 , m )
(4)
That is, the consumer’s demand for good 1 when he faces the new price of good 1, p1′ , and has the
level of money income, m′ , that would permit him to purchase his original bundle, x = ( x1 , x2 ) ,
less his demand for good 1 when he has his original level of money income, m , and faces the
original price of good 1, p1 . The substitution effect is sometimes called the change in the
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Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
consumer’s ‘compensated demand’ for a good. The idea is that the consumer is just being
‘compensated’ for a change in price.
The Income Effect
The income effect from B to C measures the change in the demand for good 1 when we change
income from m to m′ , where m′ is the amount of income that enables the consumer to purchase
the original consumption bundle, and we hold prices constant at p1′, p2 . It is calculated as:
(
)
Δx1n = x1 ( p1′, m ) − x1 ( p1′, m′ )
(5)
Note that when the price of a good decreases, we need to decrease income in order to keep
purchasing power constant. If the good is a normal good, then the decrease in income will lead to
a decrease in demand, If the good is an inferior good, then the decrease in income will lead to an
increase in demand. Thus, the income effect can be positive or negative depending on whether the
good is a normal good or an inferior good
Sign of the Substitution Effect
The substitution effect is always negative since the change in demand due to the substitution effect
is always opposite to that of the change in price. The theory of revealed preference implies that the
change in demand for the good due to the substitution effect must be non-negative. That is, if
p1 > p1′ then we must have x1 p1′, m′ ≥ x1 p1 , m such that Δx1s ≥ 0 . Similarly, assuming convex
(
)
(
)
preferences and invoking indifference curve analysis suggests that if p1 > p1′ then we must have
(
)
(
)
x1 p1′, m′ > x1 p1 , m such that Δx1s > 0 . We generally adopt the latter approach and say that the
‘own price substitution effect is negative’ since the change in demand due to the substitution effect
is opposite to the change in price: if the price increases then the substitution effect leads to a fall in
demand, and vice versa.
3.
The Total Change in Demand
The total change in demand for good 1, Δx1 , is the change in demand due to the change in price
holding income constant:
Δx1 = x1 ( p1′, m ) − x1 ( p1 , m )
(6)
We have seen how this change can be broken up into two changes: the substitution effect and the
income effect. Thus:
Δx1 = Δx1s + Δx1n
⇒
(7)
x1 ( p1′, m ) − x1 ( p1 , m ) = ⎡⎣ x1 ( p1′, m′ ) − x1 ( p1 , m ) ⎤⎦ + ⎡⎣ x1 ( p1′, m ) − x1 ( p1′, m′ ) ⎤⎦
Note that the Slutsky equation is actually an identity – it is true for all values of p1 , p1′ , m and m′
. To be sure, the first and the fourth term on the right-hand side cancel such that the right-hand side
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Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
is identically equal to the left-hand side. Note that when the price of a normal good rises, we have:
Δx1 = Δx1s +Δx1n
(−)
(−)
(8)
(−)
That is, if good 1 is a normal good then Δx1n = ⎡⎣ x1 ( p1′, m ) − x1 ( p1′, m′ ) ⎤⎦ < 0 since a rise in price,
p′ > p , implies m′ > m . In this case, the substitution and income effects reinforce each other such
that demand falls. For an inferior good, however, we have:
Δx1 = Δx1s +Δx1n
(? )
(−)
(9)
(+)
That is, if good 1 is an inferior good then Δx1n = ⎡⎣ x1 ( p1′, m ) − x1 ( p1′, m′ ) ⎤⎦ > 0 since a rise in price,
p′ > p , implies m′ > m . If the second term on the right-hand side – the income effect – is
sufficiently large, then the total change in demand could be positive. This is the case of a Giffen
good – the increase in price has reduced the consumer’s purchasing power to such an extent that
the consumer increases his consumption of the inferior good.
It is apparent from equations (8) and (9) that Giffen goods must be inferior, but inferior
goods need not be Giffen – see Figures 2-4 following.
x2
A
B
I0
0
x1
C
I1
I2
x1′′
x1′
Substitution
Income
Figure 2: Normal Good
4
x1
Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
x2
C
A
I2
I0
B
I1
0
x1′
x1
x1
x1′′
Substitution
Income
Figure 3: Inferior Non-Giffen Good
x2
C
I2
A
I0
B
I1
0
x1′ x1
x1
x1′′
Substitution
Income
Figure 4: Inferior Good
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Economics 3030: Intermediate Microeconomic Theory
4.
Topic 2: Consumer Theory - The Slutsky Equation
Example - Calculating Income and Substitution Effects
Assume the consumer has the following demand function for milk:
x1 = 10 +
m
10 p1
(10)
The consumer’s income is originally $120 per week and the price of milk is $3 per quart. Thus, his
demand for milk will be 10 + $120/(10*$3) = 14 quarts per week.
Now suppose that the price of milk falls to $2 per quart. Then his demand at the new price
will be 10 + $120/($10*2) = 16 quarts per week. The total change in demand is +2 quarts per week.
We decompose this total change as follows:
Substitution Effect
Recall from (4) above that the substitution effect is defined as:
Δx1s = x1 ( p1′, m′ ) − x1 ( p1 , m )
(11)
Thus, we need to calculate by how much we would have to change the consumer’s income to make
the original consumption of 14 quarts of milk a week just affordable when the price of milk is $2
a quart. We therefore apply the formula set out in equation (3) vis:
Δm = x1Δp1 = 14 ( $2 − $3) = −$14
(12)
Thus, we would have to reduce the consumer’s income by $14 to hold his purchasing power
constant following the fall in the price of milk. Thus:
m′ = m + Δm = $120 − $14 = $106
(13)
Note that if money income were $106, then the consumer could purchase his original bundle of
goods. His original bundle was 14 units of good 1, which originally cost him $42, implying that he
spent $120 - $42 = $78 on other goods. With an income of $106, he can still purchase 14 units of
good 1 at the lower price of $2 per unit, thereby spending $28 on good 1 and leaving $106 - $28 =
$78 to purchase his original quantity of other goods.
The consumer’s demand for milk when he faces the new price of $2 per quart and has $106
income is:
x1 ( p1′, m′ ) = x1 ( $2,$106 ) = 10 +
$106
= 15.3
10$2
(14)
Thus the substitution effect is:
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Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
Δx1s = x1 ( p1′, m′ ) − x1 ( p1 , m )
⇒
Δx1s = x1 ( $2,$106 ) − x1 ( $3,$120 )
(15)
⇒
Δx1s = 15.3 − 14 = 1.3
Holding his purchasing power constant, the fall in the price of milk has induced the consumer to
substitute towards milk by 1.3 quarts.
Income Effect
Recall from (5) above that the income effect is defined as:
Δx1n = x1 ( p1′, m ) − x1 ( p1′, m′ )
(16)
Thus, in terms of our example, we have:
Δx1n = x1 ( $2,$120 ) − x1 ( $2,$106 ) = 16 − 15.3 = 0.7
(17)
Thus, milk is a normal good for this consumer – the fall in the price of milk has induced a positive
income effect of 0.7 quarts per week.
5.
Rates of Change
We have seen that income and substitution effects can be described graphically as in Figures 2-4
or algebraically. via the Slutsky equation (or identity) (7):
Δx1 = Δx1s + Δx1n
(18)
The Slutsky equation is expressed in equation (18) in terms of absolute changes. It is more
common, however, to express it in terms of rates of change. For this, it will be helpful to define
Δx1m as the negative of the income effect vis:
Δx1m = x1 ( p1′, m′ ) − x1 ( p1′, m ) = −Δx1n
(19)
Thus, the Slutsky equation (18) becomes:
Δx1 = Δx1s − Δx1m
(20)
Dividing equation (20) through by Δp1 implies:
Δx1 Δx1s Δx1m
=
−
Δp1 Δp1 Δp1
(21)
The first right-hand side of equation (21) is the rate of change in demand when prices change and
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Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
income is adjusted so as to keep the original bundle affordable (i.e. the substitution effect) and the
second term is the rate of change in demand caused by the change in the consumer’s real income
as a result of the price change (i.e. the income effect). Given that we have an income change in the
numerator of this latter term, we often re-write the Slutsky equation with an income term as the
denominator of the second right-hand side (i.e. income effect) term. To do this, recall from equation
(3):
Δm = x1Δp1
⇒
Δm
Δp1 =
x1
(22)
Thus, we can re-write the Slutsky equation (21) as:
Δx1 Δx1s Δx1m
=
−
x1
Δp1 Δp1 Δm
(23)
Equation (23) is the common representation of the Slutsky equation in terms of rates of change.
Note its constituent elements:
Δx1 x1 ( p1′, m ) − x1 ( p1 , m )
=
Δp1
Δp1
(24)
Equation (24) is the rate of change in demand as prices change and income is held constant (i.e.
the overall change in demand):
Δx1s x1 ( p1′, m′ ) − x1 ( p1 , m )
=
Δp1
Δp1
(25)
Equation (25) is the rate of change in demand when price changes and income is adjusted such that
the consumer’s original consumption bundle is just affordable (i.e. the substitution effect):
x ( p′, m′ ) − x1 ( p1′, m )
Δx1m
x1 = 1 1
x1
Δm
Δm
(26)
Equation (26) is the rate of change in demand holding prices fixed at the new level and adjusting
income (i.e. the income effect). The income effect is itself composed of two components; how
demand changes as income changes multiplied by the original level of demand. When the price of
good 1 changes by Δp1 , the change in demand due to the income effect is equal to:
Δx1m =
x1 ( p1′, m′ ) − x1 ( p1′, m )
Δm
x1Δp1
(27)
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Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
But from (22) above, this last term is simply the change in income necessary to keep the original
bundle feasible. That is, x1Δp1 = Δm such that the change in demand due to the income effect
reduces to:
Δx1m =
x1 ( p1′, m′ ) − x1 ( p1′, m )
Δm
Δm
6.
Deriving the Slutsky Equation by Calculus
(28)
We can derive the Slutsky equation (23) more formally by using calculus. Consider the Slutsky
definition of the substitution effect in which the consumer’s income is adjusted such that he can
just afford to purchase his original consumption bundle x = x1 , x2 . If the new prices are
(
)
( )
p′ = ( p′, p′ ) and x = ( x , x ) .1 Let us define this relationship as the Slutsky demand function – i.e.
p′ = p1′, p2′ , then the consumer’s actual choice with this adjustment will depend upon
1
2
1
2
the demand function that maps out the Slutsky substitution effect - and write it as:
(
x1s = x1s p1′, p2′ , x1 , x2
)
(30)
(
)
Suppose the original price vector is p = p1 , p2 and the original level of income is m . The Slutsky
demand function tells us what the consumer would demand facing some different prices
p′ = p1′, p2′ and having money income m′ = p1′x1 + p2′ x2 that allows him to just afford his original
(
)
(
)
bundle. Thus, the Slutsky demand function at p1′, p2′ , x1 , x2 is the ordinary demand function at
(
)
p′ = p1′, p2′ and money income m′ = p1′x1 + p2′ x2 . That is:
(
)
(
x1s p1′, p2′ , x1 , x2 = x1 p1′, p2′ , p1′x1 , p2′ x2
⇒
(
)
(
x1s p1′, p2′ , x1 , x2 = x1 p1′, p2′ , m′
)
(31)
)
(
)
Equation (31) states that the Slutsky demand for good 1 at prices p′ = p1′, p2′ is the amount that
the consumer would have demanded if he had enough money to purchase his original bundle of
goods x = x1 , x2 . This is just the definition of the Slutsky equation.
(
)
Differentiating equation (31) with respect to price implies:
1
We generally assume that p2′ = p2 such that only the price of good 1 has changed.
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Economics 3030: Intermediate Microeconomic Theory
Topic 2: Consumer Theory - The Slutsky Equation
∂x1s
∂x
∂x
p1′, p2′ , x1 , x2 = 1 p1′, p2′ , m′ + 1 p1′, p2′ , m′ x1
∂p1
∂p1
∂m
(
)
(
)
(
)
⇒
(32)
∂x1
∂x
∂x
p1′, p2′ , m′ =
p1′, p2′ , x1 , x2 − 1 p1′, p2′ , m′ x1
∂p1
∂p1
∂m
(
)
s
1
(
)
(
)
Thus the total change in demand is comprised of an income and substitution effect as per equation
(23), to which equation (32) is identical except that the Δ ’s have been replaced by derivative signs.
10