CHAPTER 5
INCOME AND SUBSTITUTION
EFFECTS
th
References: Snyder and Nicholson, 10 ed.
1
Demand Functions
• The optimal levels of x1,x2,…,xn can be
expressed as functions of all prices and income
• These can be expressed as n demand functions
of the form:
x1* = d1(p1,p2,…,pn,I)
x2* = d2(p1,p2,…,pn,I)
•
•
•
xn* = dn(p1,p2,…,pn,I)
2
Demand Functions
• If there are only two goods (x and y), we can
simplify the notation
x* = x(px,py,I)
y* = y(px,py,I)
• Prices and income are exogenous
– the individual has no control over these parameters
3
Homogeneity
• If we were to double all prices and income,
the optimal quantities demanded will not
change
– the budget constraint is unchanged
xi* = di(p1,p2,…,pn,I) = di(tp1,tp2,…,tpn,tI)
• Individual demand functions are
homogeneous of degree zero in all prices and
income
4
Homogeneity
• With a Cobb-Douglas utility function
utility = U(x,y) = x0.3y0.7
the demand functions are
0 .3 I
x* 
px
0.7I
y* 
py
• Note that a doubling of both prices and
income would leave x* and y* unaffected
5
Homogeneity
• With a CES utility function
utility = U(x,y) = x0.5 + y0.5
the demand functions are
1
I
x* 

1  p x / py p x
1
I
y* 

1  py / px py
• Note that a doubling of both prices and
income would leave x* and y*
unaffected
6
Changes in Income
• An increase in income will cause the
budget constraint out in a parallel
fashion
• Since px/py does not change, the MRS
will stay constant as the worker moves
to higher levels of satisfaction
7
Increase in Income
• If both x and y increase as income rises,
x and y are normal goods
Quantity of y
As income rises, the individual chooses
to consume more x and y
B
C
A
U3
U1
U2
Quantity of x
8
Increase in Income
• If x decreases as income rises, x is an
inferior good
As income rises, the individual chooses
to consume less x and more y
Quantity of y
Note that the indifference
curves do not have to be
“oddly” shaped. The
assumption of a diminishing
MRS is obeyed.
C
B
U3
U2
A
U1
Quantity of x
9
Normal and Inferior Goods
• A good xi for which xi/I  0 over some
range of income is a normal good in that
range
• A good xi for which xi/I < 0 over some
range of income is an inferior good in
that range
10
11
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
12
Changes in a Good’s Price
• 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
13
Changes in a Good’s Price
Suppose the consumer is maximizing
utility at point A.
Quantity of y
If the price of good x falls, the consumer
will maximize utility at point B.
B
A
U2
U1
Quantity of x
Total increase in x
14
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
The substitution effect is the movement
from point A to point C
A
C
U1
The individual substitutes
good x for good y
because it is now
relatively cheaper
Quantity of x
Substitution effect
15
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
B
A
The income effect is the movement
from point C to point B
C
U2
U1
If x is a normal good,
the individual will buy
more because “real”
income increased
Quantity of x
Income effect
16
Changes in a Good’s Price
Quantity of y
An increase in the price of good x means that
the budget constraint gets steeper
The substitution effect is the
movement from point A to point C
C
A
B
U1
The income effect is the
movement from point C
to point B
U2
Quantity of x
Substitution effect
Income effect
17
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
quantity demanded
– when price rises, both effects lead to a drop
in quantity demanded
18
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 quantity demanded, but the
income effect is opposite
– when price falls, the substitution effect leads
to a rise in quantity demanded, but the
income effect is opposite
19
Giffen’s Paradox
• If the income effect of a price change is
strong enough, there could be a positive
relationship between price and quantity
demanded
– an increase in price leads to a drop in real
income
– since the good is inferior, a drop in income
causes quantity demanded to rise
20
A Summary
• Utility maximization implies that (for normal
goods) a fall in price leads to an increase in
quantity demanded
– 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
21
A Summary
• Utility maximization implies that (for normal
goods) a rise in price leads to a decline in
quantity demanded
– 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
22
A Summary
• Utility maximization implies that (for inferior
goods) no definite prediction can be made
for changes in price
– the substitution effect and income effect move
in opposite directions
– if the income effect outweighs the substitution
effect, we have a case of Giffen’s paradox
23
The Individual’s Demand Curve
• An individual’s demand for x depends
on preferences, all prices, and income:
x* = x(px,py,I)
• It may be convenient to graph the
individual’s demand for x assuming that
income and the price of y (py) are held
constant
24
The Individual’s Demand Curve
Quantity of y
As the price
of x falls...
px
…quantity of x
demanded rises.
px’
px’’
px’’’
U1
x1
I = px’ + py
x2
x3
I = px’’ + py
U2
U3
Quantity of x
I = px’’’ + py
x
x’
x’’
x’’’
Quantity of x
25
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
26
Shifts in the Demand Curve
• Three factors are held constant when a
demand curve is derived
– income
– prices of other goods (py)
– the individual’s preferences
• If any of these factors change, the
demand curve will shift to a new position
27
Shifts in the Demand Curve
• A movement along a given demand
curve is caused by a change in the price
of the good
– a change in quantity demanded
• A shift in the demand curve is caused by
changes in income, prices of other
goods, or preferences
– a change in demand
28
Demand Functions and Curves
• We discovered earlier that
0 .3 I
x* 
px
0.7 I
y* 
py
• If the individual’s income is $100, these
functions become
30
x* 
px
70
y* 
py
29
Demand Functions and Curves
• Any change in income will shift these
demand curves
30
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
31
Compensated and Uncompensated Demand
Function
• A compensated demand function = the
consumer’s income is adjusted as the
price changes, so that the consumer’s
utility remains at the same level
• Uncompensated demand function =
there is a change in real income, and it
is not uncompensated
32
Compensated Demand Curves
• An alternative approach holds real income
(or utility) constant while examining
reactions to changes in px
– 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
33
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 twodimensional representation of the
compensated demand function
x* = xc(px,py,U)
34
Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y
px
p '
slope   x
py
slope  
…quantity demanded
rises.
px ' '
py
px’
px’’
slope  
px ' ' '
py
px’’’
xc
U2
x’
x’’
x’’’
Quantity of x
x’
x’’
x’’’
Quantity of x
35
Compensated &
Uncompensated Demand
px
At px’’, the curves intersect because
the individual’s income is just sufficient
to attain utility level U2
px’’
x
xc
x’’
Quantity of x
36
Compensated &
Uncompensated Demand
At prices above px2, income
compensation is positive because the
individual needs some help to remain
on U2
px
px’
px’’
x
xc
x’
x*
Quantity of x
37
Compensated &
Uncompensated Demand
px
At prices below px2, income
compensation is negative to prevent an
increase in utility from a lower price
px’’
px’’’
x
xc
x***
x’’’
Quantity of x
38
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
39
Compensated Demand
Functions
• Suppose that utility is given by
utility = U(x,y) = x0.5y0.5
• The Marshallian demand functions are
x = I/2px
y = I/2py
• The indirect utility function is
utility  V ( I, px , py ) 
I
2px0.5 py0.5
40
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
Vpy0.5
px0.5
Vpx0.5
y  0 .5
py
41
Compensated Demand
Functions
x
Vpy0.5
px0.5
Vpx0.5
y  0 .5
py
• Demand now depends on utility (V)
rather than income
• Increases in px reduce the amount of x
demanded
– only a substitution effect
42
A Mathematical Examination
of a Change in Price
• Our goal is to examine how purchases of
good x change when px changes
x/px
• 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
43
A Mathematical Examination
of a Change in Price
• Instead, we will use an indirect approach
• Remember the expenditure function
minimum expenditure = E(px,py,U)
• Then, by definition
xc (px,py,U) = x [px,py,E(px,py,U)]
– quantity demanded is equal for both demand
functions when income is exactly what is
needed to attain the required utility level 44
A Mathematical Examination
of a Change in Price
xc (px,py,U) = x[px,py,E(px,py,U)]
• We can differentiate the compensated
demand function and get
x c
x
x E



px px
E px
x x c
x E



px px
E px
45
A Mathematical Examination
of a Change in Price
x x
x E



px px
E px
c
• The first term is the slope of the
compensated demand curve
– the mathematical representation of the
substitution effect
46
A Mathematical Examination
of a Change in Price
x x
x E



px px
E px
c
• The second term measures the way in
which changes in px affect the demand
for x through changes in purchasing
power
– the mathematical representation of the
income effect
47
The Slutsky Equation
• The substitution effect can be written as
x c
x
substituti on effect 

px px
U constant
• The income effect can be written as
x E
x E
income effect  

 

E px
I px
48
The Slutsky Equation
• Note that E/px = x
– a $1 increase in px raises necessary
expenditures by x dollars
– $1 extra must be paid for each unit of x
purchased
49
The Slutsky Equation
• The utility-maximization hypothesis
shows that the substitution and income
effects arising from a price change can be
represented by
x
 substituti on effect  income effect
px
x
x

px px
U constant
x
x
I
50
The Slutsky Equation
x
x

px px
U constant
x
x
I
• The first term is the substitution effect
– always negative as long as MRS is
diminishing
– the slope of the compensated demand curve
must be negative
51
The Slutsky Equation
x
x

px px
U constant
x
x
I
• The second term is the income effect
– if x is a normal good, then x/I > 0
• the entire income effect is negative
– if x is an inferior good, then x/I < 0
• the entire income effect is positive
52
A Slutsky Decomposition
• We can demonstrate the decomposition
of a price effect using the Cobb-Douglas
example studied earlier
• The Marshallian demand function for
good x was
0.5 I
x ( p x , py , I ) 
px
53
A Slutsky Decomposition
• The Hicksian (compensated) demand
function for good x was
x c ( px , py ,V ) 
Vpy0.5
px0.5
• The overall effect of a price change on
the demand for x is
x
 0.5 I

px
px2
54
A Slutsky Decomposition
• This total effect is the sum of the two
effects that Slutsky identified
• The substitution effect is found by
differentiating the compensated demand
function
x
substituti on effect 

px
c
 0.5Vpy0.5
p
1 .5
x
55
A Slutsky Decomposition
• We can substitute in for the indirect utility
function (V)
substituti on effect 
0.5
x
1.5
x
 0.5(0.5 Ip
p
p
0.5
y
)p
0.5
y
 0.25I

px2
56
A Slutsky Decomposition
• Calculation of the income effect is easier
 0.5I  0.5
x
0.25I
income effect   x
 


2
I
px
 p x  px
• Interestingly, the substitution and income
effects are exactly the same size
57
Marshallian Demand
Elasticities
• Most of the commonly used demand
elasticities are derived from the
Marshallian demand function x(px,py,I)
• Price elasticity of demand (ex,px)
ex ,p x
x / x
x px



px / px px x
58
Marshallian Demand
Elasticities
• Income elasticity of demand (ex,I)
e x ,I
x / x x I



I / I I x
• Cross-price elasticity of demand (ex,py)
ex , py
x / x
x py



py / py py x
59
Price Elasticity of Demand
• The own price elasticity of demand is
always negative
– the only exception is Giffen’s paradox
• The size of the elasticity is important
– if ex,px < -1, demand is elastic
– if ex,px > -1, demand is inelastic
– if ex,px = -1, demand is unit elastic
60
Price Elasticity and Total
Spending
• Total spending on x is equal to
total spending =pxx
• Using elasticity, we can determine how
total spending changes when the price of
x changes
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
61
Price Elasticity and Total
Spending
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
• The sign of this derivative depends on
whether ex,px is greater or less than -1
– if ex,px > -1, demand is inelastic and price and
total spending move in the same direction
– if ex,px < -1, demand is elastic and price and
total spending move in opposite directions
62
Compensated Price Elasticities
• It is also useful to define elasticities
based on the compensated demand
function
63
Compensated Price Elasticities
• If the compensated demand function is
xc = xc(px,py,U)
we can calculate
– compensated own price elasticity of
demand (exc,px)
– compensated cross-price elasticity of
demand (exc,py)
64
Compensated Price Elasticities
• The compensated own price elasticity of
demand (exc,px) is
e
c
x ,px
x c / x c x c px


 c
px / px px x
• The compensated cross-price elasticity
of demand (exc,py) is
x / x
x py


 c
py / py py x
c
e
c
x , py
c
c
65
Compensated Price Elasticities
• The relationship between Marshallian
and compensated price elasticities can
be shown using the Slutsky equation
px x
px x
px
x

 ex , p x  c 

x
x px
x px x
I
c
• If sx = pxx/I, then
ex,px  exc,px  sx ex,I
66
Compensated Price Elasticities
• The Slutsky equation shows that the
compensated and uncompensated price
elasticities will be similar if
– the share of income devoted to x is small
– the income elasticity of x is small
67
Demand Elasticities
• The Cobb-Douglas utility function is
U(x,y) = xy
(+=1)
• The demand functions for x and y are
I
x
px
I
y
py
68
Demand Elasticities
• Calculating the elasticities, we get
ex ,px
x px
I p x


 2 
 1
px x
p x  I 
 
 px 
ex ,py
e x ,I
py
x py


 0
0
py x
x
x I 
I

 

1
I x px  I 
 
 px 
69
Demand Elasticities
• We can also show
– homogeneity
ex,px  ex,py  ex,I  1 0  1  0
– Engel aggregation
s x ex,I  sy ey ,I    1    1      1
– Cournot aggregation
s x ex,px  sy ey ,px    ( 1)    0    s x
70
Demand Elasticities
• We can also use the Slutsky equation to
derive the compensated price elasticity
exc,px  ex,px  sx ex,I  1 (1)    1  
• The compensated price elasticity
depends on how important other goods
(y) are in the utility function
71
Demand Elasticities
• The CES utility function (with  = 2,
 = 5) is
U(x,y) = x0.5 + y0.5
• The demand functions for x and y are
I
x
1
px (1  px py )
I
y
1
py (1  px py )
72
Demand Elasticities
• We will use the “share elasticity” to
derive the own price elasticity
esx ,px
s x px


 1 ex,px
px s x
• In this case,
px x
1
sx 

I
1  px py1
73
Demand Elasticities
• Thus, the share elasticity is given by
esx ,px
 py1
 px py1
s x px
px





1 2
1 1
px s x (1  px py ) (1  px py )
1  px py1
• Therefore, if we let px = py
ex,px  es x ,px
1
1
 1  1.5
1 1
74
Demand Elasticities
• The CES utility function (with  = 0.5,
 = -1) is
U(x,y) = -x -1 - y -1
• The share of good x is
px x
1
sx 

I
1  py0.5 px0.5
75
Demand Elasticities
• Thus, the share elasticity is given by
es x ,px
0.5 py0.5 px1.5
s x px
px




0.5  0 . 5 2
0.5 0.5 1
px s x (1  py px ) (1  py px )

0.5 py0.5 px0.5
1  py0.5 px0.5
• Again, if we let px = py
ex,px  es x ,px
0.5
1
 1  0.75
2
76