Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
Alternative to Utility Maximization
We have examined how individuals maximize their welfare by maximizing their utility.
Diagrammatically, it is like moderating the individual’s indifference curve until it is just
tangent to the budget constraint. The individual’s choice thus selected gives us a demand
for the goods in terms of their price, and income. We call such a demand function a
Marhsallian Demand function. In general mathematical form, letting x be the quantity of
a good demanded, we write the Marshallian Demand as x M ≡ x M ( p, y ) .
Thinking about the process, we can reverse the intuition about how individuals maximize
their utility. Consider the following, what if we fix the utility value at the above level, but
instead vary the budget constraint? Would we attain the same choices? Well, we should,
the demand thus achieved is however in terms of prices and utility, and not income. We
call such a demand function, a Hicksian Demand, x H ≡ x H ( p, u ) . This method means
that the individual’s problem is instead framed as minimizing expenditure subject to a
particular level of utility. Let’s examine briefly how the problem is framed,
min p1 x1 + p 2 x 2
subject to u (x1 , x 2 ) = u
We refer to this problem as expenditure minimization. We will however not consider
this, safe to note that this problem generates a parallel demand function which we refer to
as Hicksian Demand, also commonly referred to as the Compensated Demand.
Decomposition of Changes in Choices induced by Price Change
Let’s us examine the decomposition of a change in consumer choice as a result of a price
change, something we have talked about earlier. Before we discuss this in general form,
let us use the result from our earlier discussion using Cobb-Douglas utility function.
We found earlier that assuming a Cobb Douglas utility function, for the problem of
α
β
max x A , xB ( x A ) (x B )
subject to y = p A x A + p B x B
the consumer’s choice may be characterized by the following demand functions
1
1
α
β
x AM =
y and x BM =
y . However, at the same time, note that
p A (β + α )
p B (β + α )
income is actually a function of prices. Let the choices at the current prices and income
level be x A and x B . Then the fixed level of income yielding the choices can be written
as y = p A x A + p B x B . We can rewrite both the demand functions as
p A x A + pB x B
p A x A + pB x B
β
and x BS =
.
(β + α )
(β + α )
pA
pB
These demand functions are now known as Slutsky Demand functions. The question
now is how does a price change alter choices? We call this type of examination
comparative statics. Let the price of good A change. Differentiating x AS with respect to
p A , we get the following,
x AS =
α
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
 pA x A − y 
α  x A  
α  y  


=
+
−
 ( p )2  α + β  p   α + β  ( p )2  
 A 
A


 A 
α  x A 
From the last equality, the first term,
, is the income effect, and the second,
α + β  p A 



 − α  y   , is called the substitution effect. Why? (Note that income effects if
 α + β  ( p )2  
 A 

always positive, and the substitution effect is always negative.) Well, let us revert back to
α
1
y . Differentiating the Marshallian
the Marshallian Demand for good A, x AM =
p A (β + α )
∂x AS
α
=
∂p A α + β
 1 
 . That is the

α + β  p A 
∂x AM
∂x AM
first term is just
x A . Similarly, the second element is just
, that is it is the
∂p A y = y
∂y
Demand with respect to the fixed income of good A, y , we get,
α
derivative of the marshallian demand function, evaluated at the original income level, y .
Examining the scenario of a price increase in good A, and representing the change
diagrammatically, the substitution effect is the movement from point A to point B. It
involves the substitution between the consumption of good A for good B on account of
the price increase. Note that in Slutsky’s decomposition, we hold the income constant
since in moving from point A to point B along the new budget constraint, we have
ensured that the initial consumption point is still affordable. This is achieved by varying
the prices of the two goods, hence the change in slope of the budget constraint.
x
A
The first shift in budget constraint
involves pivoting on the original
consumption choice, point A, ensuring
that we are examining substitution at
the same income level.
A
B
x
The second shift is from the
second budget line, but in a
parallel fashion since we are
considering a pure income
A
U3
C
U2
U1
x
B
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
The movement from point B to point C involves the movement between two parallel
budget constraints, and hence involves the change (more precisely a decrease in income,
since the increase in the price of good A, without a commiserating change in the price of
good B implies a real change in income). Note that the two effects moves in opposing
directions as suggested by the decomposition of our marshallian demand function.
***There is an important point to be made here; the sequence of the decomposition
must follow with substitution effect, and then the income effect. This is because the
substitution effect considers the trade off between the consumption of goods holding
the initial consumption choice constant.
We can describe this decomposition in a more general manner. First, as above, let
xiM (P, y ) = xiS P, P X = y , where P and X are vectors of prices and goods respectively,
and where i indexes the good under consideration. As before the superscript M indexes
the Marshallian demand, while S indexes the Slutsky counterpart. Then the effects of a
price change in good i is
∂xiM P, y ∂xiM P, y
∂xiS P, P X
+
=
xi
∂y
∂pi
∂pi
(
)
(
)
( )
( )
(
( )
)
( )
∂xiM P, y ∂xiS P, P X
∂x M P, y
xi
=
− i
∂pi
∂pi
∂y
Which is the more general version of the decomposition we have before. This also tells us
why we typically refer to the substitution effect as the Slutsky substitution effect, where
income is held at the original choice level. From the above we can make several points
about the sign/direction of impact:
1. The substitution effect of always a negative one. That is as price increases (falls)
the change in quantity demanded falls (rises).
2. The income effect can be either positive or negative, that is an increase (fall) in
income can lead to either an increase or fall in demand of the good.
a. When the income effect of a good is positive, we call such a good a
normal good. That is the greater (lower) one’s income is the greater
(lower) one’s demand for the good. Further, for a normal good, the
direction of both income and substitution effect reinforces each other.
Example: When prices falls for a good, substitution effect means more of
the good is demanded, and income effect says the effective rise in income
raises the good’s consumption even further.
b. When the income effect is negative, we say a good is an inferior good. For
such a good, the greater the income, the lower the demand for the good,
and vice versa. In such a case, the net impact of a price change is
ambiguous since the ultimate change is dependent on the relative size of
both effects. We will examine this further shortly.
⇒
Decomposition of the Hicksian Demand
As noted before, the Hicksian Demand express the individual’s choice in terms of the
price of the good and the utility level attained, xiH ≡ xiH p1 , p 2 , u , in a two good
(
)
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
problem. The question then is if there is an equivalent to Slutsky substitution effect for
the Hicksian demand. Well, there is! The prove of which is outside the scope of this
course, but it relies on the following,
∂xiH P, u ∂xiS P, P X = y
=
∂pi
∂pi
Which says that the Slutsky substitution and the Hicksian substitution effect are the
same. The entire decomposition can be written as
∂xiM P, y ∂xiH P, P X ∂xiM (P, y )
=
−
xi
∂pi
∂pi
∂y
Note the slight difference also in the income effect in this decomposition, since we are
not considering the income effect from the original level of income. This is because the
substitution effect holds the utility and not income constant. The diagrammatic
representation is similar to the above, with the exception being that the substitution effect
holds the initial utility constant on which the choice is on.
( )
( )
x
(
(
A
)
)
The first shift in budget constraint
involves rotating around the original
indifference curve, U2, ensuring that
we are examining substitution at the
same level of utility.
The second shift is from the
second budget line, but in a
parallel fashion since we are
considering a pure income
A
B
x
A
C
U2
U1
x
B
Note that the sole difference in both the mathematical representation, and the
diagrammatic representation is the substitution effect. Again, note the sequence is first
substitution, then income effect.
Types of goods, and their characterization in terms of Income and Substitution
effect.
There are essentially 3 types of goods,
1. Normal Goods,
2. Inferior Goods,
3. Giffen Goods
We will use the ideas we learn above to characterize them. A diagram for a normal good
is as follows: Suppose the price of good B rises.
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
x
A
B
A
U3
C
U2
U1
Substitution Effect
x
B
Income Effect
As can be seen above, and as noted before, both effects reinforce each other for the
demand of good B, leading eventually from point A to point C, a fall in quantity
consumed of good B.
However, if good B is an inferior good, the income effect would be negative, that is
under the same scenario of an increase in price of good B, substitution would reduce
demand for good B, but income effect would increase the demand for good B since the
increase in price of good B reduces the real income.
x
A
B
A
U3
C
U2
U1
Substitution Effect
x
B
Income Effect
In the diagram above, the price increase still lead to a decrease in the quantity of good B
consumed, which means that the net effect is still a negative one. Put another way,
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
although the income effect raised the consumption of the inferior good B, the substitution
effect is still stronger such that in the final scene, demand still fell.
It is however possible to that the income effect is stronger than the substitution effect,
x
A
B
A
U3
U2
C
U1
Substitution Effect
x
B
Income Effect
And when that happens, we refer to such a good a Giffen Good. Note that Giffen good
must be a Inferior Good, but an inferior good is not a Giffen Good. The decomposition
can be similarly performed for a Hicksian Substitution effect.
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
Substitution and Income Effects under other forms of Indifference Curves we have
discussed thus far
We now examine the impact when we have a Leontieff Utility function: Because this
utility function assumes that goods are perfect complements, there is only one
combination for every income level, which then means that there is no substitution effect
as our diagram notes.
x
A
A
B
Income Effect
U3
U2
x
B
Similarly, for a perfectly linear utility function, there would not be any income effect, but
have instead a substitution effect only. You should verify this yourself.
Another interesting variation is that of a quasilinear utility function, and when the price
of the numeriare good changes. The diagram below depicts a decrease in price of a
numeriare good, and it effects. Just like the linear case, the effect is one that is purely
generated by the substitution effect.
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Intermediate Microeconomic Theory: ECON 251:21
Substitution and Income Effect
x
A
U3
U2
U1
Substitution Effect
x
B
8