Chapter 5 INCOME AND SUBSTITUTION EFFECTS MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning.
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Chapter 5
INCOME AND SUBSTITUTION EFFECTS
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
Demand Functions
• The optimal levels of
X
1 ,
X
2 ,…,
X
n can be expressed as functions of all prices and income • These can be expressed as n demand functions:
X
1 * =
d
1 (
P
1 ,
P
2 ,…,
P
n ,
I
)
X
2 * =
d
2 (
P
1 ,
P
2 ,…,
P
n ,
I
) • • •
X
n * =
d
n (
P
1 ,
P
2 ,…,
P
n ,
I
)
Homogeneity
• If we were to double all prices and income, the optimal quantities demanded will not change – Doubling prices and income leaves the budget constraint unchanged
X
i * =
d
i (
P
1 ,
P
2 ,…,
P
n ,
I
) =
d
i (
tP
1 ,
tP
2 ,…,
tP
n ,
tI
) • Individual demand functions are homogeneous of degree zero in all prices and income
Homogeneity
• With a Cobb-Douglas utility function utility =
U
(
X
,
Y
) =
X
0.3
Y
0.7
the demand functions are
X
* 0 .
3
I P X Y
* 0 .
7
I P X
• Note that a doubling of both prices and income would leave
X
* and
Y
* unaffected
Homogeneity
• With a CES utility function utility =
U
(
X
,
Y
) =
X
0.5
+
Y
0.5
the demand functions are
X
* 1 1
P X
/
P Y
I P X Y
* 1
P Y
1 /
P X
I P Y
• Note that a doubling of both prices and income would leave
X
* and
Y
* unaffected
Changes in Income
• An increase in income will cause the budget constraint out in a parallel manner • Since
P X
/
P Y
does not change, the
MRS
will stay constant as the worker moves to higher levels of satisfaction
Increase in Income
• If both
X
rises,
X
and and
Y Y
increase as income are normal goods Quantity of
Y
As income rises, the individual chooses to consume more
X
and
Y
A B C
U 1 U 2 U 3
Quantity of
X
Increase in Income
• If
X
decreases as income rises, inferior good
X
is an As income rises, the individual chooses to consume less
X
and more
Y
Quantity of
Y
C B
U 3
Note that the indifference curves do not have to be “oddly” shaped. The assumption of a diminishing
MRS
is obeyed.
U 2
A
U 1
Quantity of
X
Normal and Inferior Goods
• A good
X
i for which
X
i /
I
0 over some range of income is a normal good in that range • A good
X
i for which
X
i /
I
< 0 over some range of income is an inferior good in that range
Engel’s Law
• Using Belgian data from 1857, Engel found an empirical generalization about consumer behavior • The proportion of total expenditure devoted to food declines as income rises – food is a necessity whose consumption rises less rapidly than income
Substitution & Income Effects
• Even if the individual remained on the same indifference curve when the price changes, his optimal choice will change because the
MRS
must equal the new price ratio – the substitution effect • The price change alters the individual’s “real” income and therefore he must move to a new indifference curve – the income effect
Changes in a Good’s Price
• A change in the price of a good alters the slope of the budget constraint – it also changes the
MRS
at the consumer’s utility-maximizing choices • When the price changes, two effects come into play – substitution effect – income effect
Changes in a Good’s Price
Quantity of
Y
Suppose the consumer is maximizing utility at point
A
.
B
If the price of good
X
falls, the consumer will maximize utility at point
B
.
A
U 1 U 2 Quantity of
X
Total increase in
X
Changes in a Good’s Price
Quantity of
Y
To isolate the substitution effect, we hold “real” income constant but allow the relative price of good
X
to change
A C B
The substitution effect is the movement from point
A
to point
C
U 1 U 2 The individual substitutes good
X
for good
Y
because it is now relatively cheaper Quantity of
X
Substitution effect
Changes in a Good’s Price
Quantity of
Y
The income effect occurs because the individual’s “real” income changes when the price of good
X
changes The income effect is the movement from point
C
to point
B B A C
U 1 U 2 If
X
is a normal good, the individual will buy more because “real” income increased Quantity of
X
Income effect
Changes in a Good’s Price
Quantity of
Y
An increase in the price of good
X
means that the budget constraint gets steeper
C
The substitution effect is the movement from point
A
to point
C A B
U 1 The income effect is the movement from point
C
to point
B
U 2 Quantity of
X
Substitution effect Income effect
Price Changes for Normal Goods
• If a good is normal, substitution and income effects reinforce one another – When price falls, both effects lead to a rise in Q D – When price rises, both effects lead to a drop in Q D
Price Changes for Inferior Goods
• If a good is inferior, substitution and income effects move in opposite directions • The combined effect is indeterminate – When price rises, the substitution effect leads to a drop in Q D , but the income effect leads to a rise in Q D – When price falls, the substitution effect leads to a rise in Q D , but the income effect leads to a fall in Q D
Giffen’s Paradox
• If the income effect of a price change is strong enough, there could be a positive relationship between price and Q D – An increase in price leads to a drop in real income – Since the good is inferior, a drop in income causes Q D to rise • Thus, a rise in price leads to a rise in Q D
Summary of Income & Substitution Effects
• Utility maximization implies that (for normal goods) a fall in price leads to an increase in Q D – The
substitution effect
causes more to be purchased as the individual moves along an indifference curve – The
income effect
causes more to be purchased because the resulting rise in purchasing power allows the individual to move to a higher indifference curve
Summary of Income & Substitution Effects
• Utility maximization implies that (for normal goods) a rise in price leads to a decline in Q D – The
substitution effect
causes less to be purchased as the individual moves along an indifference curve – The
income effect
causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve
Summary of Income & Substitution Effects
• Utility maximization implies that (for inferior goods) no definite prediction can be made for changes in price – The
substitution effect
in opposite directions and
income effect
move – If the income effect outweighs the substitution effect, we have a case of
Giffen’s paradox
The Individual’s Demand Curve
• An individual’s demand for
X
1 depends on preferences, all prices, and income:
X
1 * =
d
1 (
P
1 ,
P
2 ,…,
P
n ,
I
) • It may be convenient to graph the individual’s demand for
X
1 assuming that income and the prices of other goods are held constant
The Individual’s Demand Curve
Quantity of
Y
As the price of
X
falls...
P X
…quantity of
X
demanded rises.
U 3 U 1 U 2 X 1 I = P X1 + P
Y
X 2 X 3 I = P X2 + P
Y
Quantity of
X
I = P X3 + P
Y P X1 P X2 P X3
X 1 X 2 X 3
d X
Quantity of
X
The Individual’s Demand Curve
• An
individual demand curve
shows the relationship between the price of a good and the quantity of that good purchased by an individual assuming that all other determinants of demand are held constant
Shifts in the Demand Curve
• Three factors are held constant when a demand curve is derived – income – prices of other goods – the individual’s preferences • If any of these factors change, the demand curve will shift to a new position
Shifts in the Demand Curve
• A movement along a given demand curve is caused by a change in the price of the good – called a
change in quantity demanded
• A shift in the demand curve is caused by a change in income, prices of other goods, or preferences – called a
change in demand
Compensated Demand Curves
• The actual level of utility varies along the demand curve • As the price of
X
falls, the individual moves to higher indifference curves – It is assumed that nominal income is held constant as the demand curve is derived – This means that “real” income rises as the price of
X
falls
Compensated Demand Curves
• An alternative approach holds real income (or utility) constant while examining reactions to changes in
P X
– The effects of the price change are “compensated” so as to constrain the individual to remain on the same indifference curve – Reactions to price changes include only substitution effects
Compensated Demand Curves
• A compensated (Hicksian) demand curve shows the relationship between the price of a good and the quantity purchased assuming that other prices and utility are held constant • The compensated demand curve is a two dimensional representation of the compensated demand function
X
* =
h X
(
P X
,
P Y
,
U
)
Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of
Y P X slope
P X
1
P Y
…quantity demanded rises.
slope
P X
2
P Y
P X1
slope
P X
3
P Y
P X2 P X3 h X
U 2 X 1 X 2 X 3
Quantity of
X
X 1 X 2 X 3
Quantity of
X
Compensated & Uncompensated Demand
P X
At
P X
2 , the curves intersect because the individual’s income is just sufficient to attain utility level
U
2
P
X2
X
2
d X h X
Quantity of
X
Compensated & Uncompensated Demand
P X
At prices above
P X
2 , income compensation is positive because the individual needs some help to remain on
U
2
P
X1
P
X2
d X h X
X
1
X
1 *
Quantity of
X
Compensated & Uncompensated Demand
P X
At prices below
P X
2 , income compensation is negative to prevent an increase in utility from a lower price
P
X2
P
X3
d X h X
X
3 *
X
3
Quantity of
X
Compensated & Uncompensated Demand
• For a normal good, the compensated demand curve is less responsive to price changes than is the uncompensated demand curve – the uncompensated demand curve reflects both income and substitution effects – the compensated demand curve reflects only substitution effects
Compensated Demand Functions
• Suppose that utility is given by utility =
U
(
X
,
Y
) =
X
0.5
Y
0.5
• The Marshallian demand functions are
X
=
I
/2
P X Y
=
I
/2
P Y
• The indirect utility function is utility
V
(
I
,
P X
,
P Y
)
I
2
P X
0 .
5
P Y
0 .
5
Compensated Demand Functions
• To obtain the compensated demand functions, we can solve the indirect utility function for
I
and then substitute into the Marshallian demand functions
X
VP Y
0 .
5
P X
0 .
5
Y
VP X
0 .
5
P Y
0 .
5
Compensated Demand Functions
X
VP Y
0 .
5
P X
0 .
5
Y
VP X
0 .
5
P Y
0 .
5 • Demand now depends on utility rather than income • Increases in
P X
demanded reduce the amount of
X
– only a substitution effect
A Mathematical Examination of a Change in Price
• Our goal is to examine how the demand for good
X
changes when
P X
changes
d X
/
P X
• Differentiation of the first-order conditions from utility maximization can be performed to solve for this derivative • However, this approach is cumbersome and provides little economic insight
A Mathematical Examination of a Change in Price
• Instead, we will use an indirect approach • Remember the expenditure function minimum expenditure =
E
(
P X
,
P Y
,
U
) • Then, by definition
h X
(
P X
,
P Y
,
U
) =
d X
[
P X
,
P Y
,
E
(
P X
,
P Y
,
U
)] – Note that the two demand functions are equal when income is exactly what is needed to attain the required utility level
A Mathematical Examination of a Change in Price
h X
(
P X
,
P Y
,
U
) =
d X
[
P X
,
P Y
,
E
(
P X
,
P Y
,
U
)] • We can differentiate the compensated demand function and get
h X
P X
d
P X X
d
P X X
d
E X
E
P X
h X
P X
d
E X
E
P X
A Mathematical Examination of a Change in Price
d
P X X
h X
P X
d
E X
E
P X
• The first term is the slope of the compensated demand curve • This is the mathematical representation of the substitution effect
A Mathematical Examination of a Change in Price
d
P X X
h X
P X
d
E X
E
P X
• The second term measures the way in which changes in
P X
for
X
affect the demand through changes in necessary expenditure levels • This is the mathematical representation of the income effect
The Slutsky Equation
• The substitution effect can be written as substituti on effect
h X
P X
X
P X U
constant • The income effect can be written as income effect
d
E X
E
P X
X
I
E
P X
The Slutsky Equation
• Note that
E
/
P X
=
X
– A $1 increase in
P X
expenditures by
X
raises necessary dollars – $1 extra must be paid for each unit of
X
purchased
The Slutsky Equation
• The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by
d
P X X
d
P X X
substituti on effect income effect
X
P X U
constant
X
X
I
The Slutsky Equation
d
P X X
X
P X U
constant
X
X
I
• The first term is the substitution effect – always negative as long as
MRS
diminishing is – the slope of the compensated demand curve will always be negative
The Slutsky Equation
d
P X X
X
P X U
constant
X
X
I
• The second term is the income effect – if
X
is a normal good, then
X
/
I
> 0 – if • the entire income effect is negative
X
is an inferior good, then
X
/
I
• the entire income effect is positive < 0
Revealed Preference & the Substitution Effect
• The theory of revealed preference was proposed by Paul Samuelson in the late 1940s • The theory defines a principle of rationality based on observed behavior and then uses it to approximate an individual’s utility function
Revealed Preference & the Substitution Effect
• Consider two bundles of goods:
A
and
B
• If the individual can afford to purchase either bundle but chooses
A
, we say that
A
had been revealed preferred to
B
• Under any other price-income arrangement,
B
can never be revealed preferred to
A
Revealed Preference & the Substitution Effect
Quantity of
Y
Suppose that, when the budget constraint is given by
I
1 ,
A
is chosen
A B A
must still be preferred to
B
is
I
3 (because both
A
and
B
when income are available)
I 3 I 1 I 2
If
B
is chosen, the budget constraint must be similar to that given by
I
2 where
A
is not available Quantity of
X
Negativity of the Substitution Effect
• Suppose that an individual is indifferent between two bundles:
C
and
D
• Let
P X
C
,
P Y
C
be the prices at which bundle
C
is chosen • Let
P X
D
,
P Y
D
be the prices at which bundle
D
is chosen
Negativity of the Substitution Effect
• Since the individual is indifferent between
C
and
D
– When
C
is chosen,
D
much as
C
must cost at least as
P X
C
X
C
– When
D
+
P Y
C
Y
is chosen,
C C
≤
P X
D
X
D
+
P Y
D
Y
D
must cost at least as much as
D
P X
D
X
D
+
P Y
D
Y
D
≤
P X
C
X
C
+
P Y
C
Y
C
Negativity of the Substitution Effect
• Rearranging, we get
P X
C
(X
C
-
X
D
) +
P Y
C
(Y
C
-
Y
D
) ≤ 0
P X
D
(X
D
-
X
C
) +
P Y
D
(Y
D
-
Y
C
) ≤ 0 • Adding these together, we get (
P X
C
–
P X
D
)(X
C
-
X
D
) + (
P Y
C
–
P Y
D
)(Y
C
-
Y
D
) ≤ 0
Negativity of the Substitution Effect
• Suppose that only the price of X changes (
P Y
C
=
P Y
D
) (
P X
C
–
P X
D
)(X
C
-
X
D
) ≤ 0 • This implies that price and quantity move in opposite direction when utility is held constant – the substitution effect is negative
Mathematical Generalization
• If, at prices
P i
0
bundle
X
i
0
instead of bundle
X
i
1
is chosen (and bundle
X
i
1
affordable), then is
i n
1
P i
0
X i
0
i n
1
P i
0
X i
1 • Bundle
0
has been “revealed preferred” to bundle
1
Mathematical Generalization
• Consequently, at prices that prevail when bundle
1
is chosen (
P i
1
), then
i n
1
P i
1
X i
0
i n
1
P i
1
X i
1 • Bundle
0
bundle
1
must be more expensive than
Strong Axiom of Revealed Preference
• If commodity bundle
0
is revealed preferred to bundle
1
, and if bundle
1
is revealed preferred to bundle
2
, and if bundle
2
is revealed preferred to bundle
3
,…,and if bundle
k-1
is revealed preferred to bundle
k
, then bundle
k
cannot be revealed preferred to bundle
0
Consumer Welfare
• The expenditure function shows the minimum expenditure necessary to achieve a desired utility level (given prices) • The function can be denoted as where expenditure =
E
(
P X
,
P Y
,
U
0 )
U
0 is the “target” level of utility
Consumer Welfare
• One way to evaluate the welfare cost of a price increase (from
P X
0
to
P X
1
) would be to compare the expenditures required to achieve
U
0 under these two situations expenditure at
P X
0
=
E
0 =
E
(
P X
0
,
P Y
,
U
0 ) expenditure at
P X
1
=
E
1 =
E
(
P X
1
,
P Y
,
U
0 )
Consumer Welfare
• The loss in welfare would be measured as the increase in expenditures required to achieve
U
0 welfare loss =
E
0 • Because
E
1 negative >
E
0 –
E
1 , this change would be – the price increase makes the person worse off
Consumer Welfare
• Remember that the derivative of the expenditure function with respect to
P X
the compensated demand function (
h X
) is
dE
(
P X
,
P Y
,
U
0 )
h X
(
P X
,
P Y
,
U
0 )
dP X
• The change in necessary expenditures brought about by a change in
P X
by the quantity of
X
demanded is given
Consumer Welfare
• To evaluate the change in expenditure caused by a price change (from
P X
0
to
P X
1
), we must integrate the compensated demand function
P
1
X
P X
0
dE
P
1
X
P X
0
h x
(
P X
,
P Y
,
U
0 )
dP X
– This integral is the area to the left of the compensated demand curve between
P X
0
and
P X
1
P X
P X
1
P X
0 Consumer Welfare
When the price rises from
P X
0
to
P X
1
, the consumer suffers a loss in welfare
welfare loss
h X
X 1 X 0
Quantity of
X
Consumer Welfare
• Because a price change generally involves both income and substitution effects, it is unclear which compensated demand curve should be used • Do we use the compensated demand curve for the original target utility (
U
0 ) or the new level of utility after the price change (
U
1 )?
P X
P X
1
P X
0 Consumer Welfare
When the price rises from
P X
0
to
P X
1
, the actual market reaction will be to move from
A
to
C
The consumer’s utility falls from
U
0 to
U
1
C A X 1 X 0
d X h X
(
U
0 )
h X
(
U
1 ) Quantity of
X
P X
P X
1
P X
0 Consumer Welfare
Is the consumer’s loss in welfare best described by area
P X
1
BAP X
0
or by area
P X
1
CDP X
0
[using
h
[using
X
(
U
1 )]?
h X
(
U
0 )]
C B A
Is
U
0 or
U
1 the appropriate utility target?
D X 1 X 0
d X h X
(
U
0 )
h X
(
U
1 ) Quantity of
X
P X
P X
1
P X
0 Consumer Welfare
We can use the Marshallian demand curve as a compromise.
C B
The area
P X
1
CAP X
0
falls between the sizes of the welfare losses defined by
h X
(
U
0 ) and
h X
(
U
1 )
A D
d X h X
(
U
0 )
h X
(
U
1 )
X 1 X 0
Quantity of
X
Loss of Consumer Welfare from a Rise in Price
• Suppose that the compensated demand function for
X
is given by
X
h X
(
P X
,
P Y
,
V
)
VP Y
0 .
5
P X
0 .
5 the welfare loss from a price increase from
P X
= 0.25 to
P X
= 1 is given by 1 0 .
25
VP Y
0 .
5
dP X P X
0 .
5 2
VP Y
0 .
5
P X
0 .
5
P X
1
P X
0 .
25
Loss of Consumer Welfare from a Rise in Price
• If we assume that the initial utility level (
V
) is equal to 2, loss = 4(1) 0.5
– 4(0.25) 0.5
= 2 • If we assume that the utility level (
V
) falls to 1 after the price increase (and used this level to calculate welfare loss), loss = 2(1) 0.5
– 2(0.25) 0.5
= 1
Loss of Consumer Welfare from a Rise in Price
• Suppose that we use the Marshallian demand function instead
X
d X
(
P X
,
P Y
,
I
) I 2
P X
the welfare loss from a price increase from
P X
= 0.25 to
P X
= 1 is given by 1 0 .
25
I
2
P X dP X
I
ln
P X
2
P X
1
P X
0 .
25
Loss of Consumer Welfare from a Rise in Price
• Because income (
I
) is equal to 2, loss = 0 – (-1.39) = 1.39
• This computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functions
Important Points to Note:
• Proportional changes in all prices and income do not shift the individual’s budget constraint and therefore do not alter the quantities of goods chosen – demand functions are homogeneous of degree zero in all prices and income
Important Points to Note:
• When purchasing power changes (income changes but prices remain the same), budget constraints shift – for normal goods, an increase in income means that more is purchased – for inferior goods, an increase in income means that less is purchased
Important Points to Note:
• A fall in the price of a good causes substitution and income effects – For a normal good, both effects cause more of the good to be purchased – For inferior goods, substitution and income effects work in opposite directions • A rise in the price of a good also causes income and substitution effects – For normal goods, less will be demanded – For inferior goods, the net result is ambiguous
Important Points to Note:
• The Marshallian demand curve summarizes the total quantity of a good demanded at each price – changes in price prompt movemens along the curve – changes in income, prices of other goods, or preferences may cause the demand curve to shift
Important Points to Note:
• Compensated demand curves illustrate movements along a given indifference curve for alternative prices – these are constructed by holding utility constant – they exhibit only the substitution effects from a price change – their slope is unambiguously negative (or zero)
Important Points to Note:
• Income and substitution effects can be analyzed using the Slutsky equation • Income and substitution effects can also be examined using revealed preference • The welfare changes that accompany price changes can sometimes be measured by the changing area under the demand curve