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Chapter 5
INCOME AND SUBSTITUTION
EFFECTS
1
Objectives
• How will changes in prices and income
influence influence consumer’s optimal
choices?
– We will look at partial derivatives
2
Demand Functions (review)
• We have already seen how to obtain consumer’s optimal
choice
• Consumer’s optimal choice was computed Max
consumer’s utility subject to the budget constraint
• After solving this problem, we obtained that optimal
choices depend on prices of all goods and income.
• We usually call the formula for the optimal choice: the
demand function
• For example, in the case of the Complements utility
function, we obtained that the demand function (optimal
choice) is:
I
I
x* 
px  0.25 py
y* 
4 p x  py
3
Demand Functions
• If we work with a generic utility function (we do
not know its mathematical formula), then we
express the demand function as:
x* = x(px,py,I)
y* = y(px,py,I)
•We will keep assuming that prices and income is
exogenous, that is:
–the individual has no control over these
parameters
4
Simple property of demand functions
• If we were to double all prices and
income, the optimal quantities demanded
will not change
– Notice that the budget constraint does not
change (the slope does not change, the
crossing with the axis do not change either)
xi* = di(px,py,I) = di(2px,2py,2I)
5
Changes in Income
• Since px/py does not change, the MRS
will stay constant
• An increase in income will cause the
budget constraint out in a parallel
fashion (MRS stays constant)
6
What is a Normal Good?
• A good xi for which xi/I  0 over some
range of income is a normal good in that
range
7
Normal goods
• 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
What is an inferior Good?
• A good xi for which xi/I < 0 over some
range of income is an inferior good in
that range
9
Inferior good
• 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
C
B
U3
U2
A
U1
Quantity of x
10
Changes in a Good’s Price
• A change in the price of a good alters the
slope of the budget constraint (px/py)
– Consequently, it changes the MRS at the
consumer’s utility-maximizing choices
• When a price changes, we can decompose
consumer’s reaction in two effects:
– substitution effect
– income effect
11
Substitution and 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
12
Sign of substitution effect (SE)
SE is always negative, that is, if price increases, the
substitution effect makes quantity to decrease and
conversely. See why:
1) Assume px decreases, so: px1< px0
2) MRS(x0,y0)= px0/ py0 & MRS(x1,y1)= px1/ py0
1 and 2 implies that:
MRS(x1,y1)<MRS(x0,y0)
As the MRS is decreasing in x, this means that x
has increased, that is: x1>x0
13
Changes in the optimal choice when a price
decreases
Suppose the consumer is maximizing
Quantity of y
utility at point A.
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
Substitution effect when a price decreases
Quantity of y
To isolate the substitution effect, we hold
utility constant but allow the
relative price of good x to change.
Purple is parallel to the new one
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
Income effect when the price decreases
The income effect occurs because the
individual’s “real” income changes
(hence utility changes) when
the price of good x changes
The income effect is the movement
from point C to point B
Quantity of y
B
A
C
U2
U1
If x is a normal good,
the individual will buy
more because “real”
income increased
Quantity of x
Income effect
How would the graph change if the good was inferior?
16
Subs and income effects when a price
increases
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
How would the graph change if the good was inferior?
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
• Obvious relation hold for a rise in price…
21
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
22
Compensated Demand Functions
• This is a new concept
• It is the solution to the following problem:
– MIN
PXX+ PYY
– SUBJECT TO U(X,Y)=U0
• Basically, the compensated demand functions are the
solution to the Expenditure Minimization problem that we
saw in the previous chapter
• After solving this problem, we obtained that optimal
choices depend on prices of all goods and utility. We
usually call the formula: the compensated demand
function
• x* = xc(px,py,U),
• y* = yc(px,py,U)
23
Compensated Demand Functions
• xc(px,py,U0), and yc(px,py,U0) tell us what
quantities of x and y minimize the expenditure
required to achieve utility level U0 at current
prices px,py
• Notice that the following relation must hold:
• pxxc(px,py,U0)+ pyyc(px,py,U0)=E(px,py,U0)
– So this is another way of computing the expenditure
function !!!!
24
Compensated Demand Functions
• There are two mathematical tricks to obtain the
compensated demand function without the need to solve
the problem:
– MIN
PXX+ PYY
– SUBJECT TO U(X,Y)=U0
• One trick(A) (called Shephard’s Lemma) is using the
derivative of the expenditure function
• Another trick(B) is to use the marshallian demand and
the expenditure function
25
Compensated Demand Functions
• Sheppard’s Lema to obtain the compensated
demand function
E ( p x , p y , u )
x
dp x
y
E ( p x , p y , u )
dp y
Intuition: a £1 increase in px raises necessary
expenditures by x pounds, because £1 must be paid
for each unit of x purchased.
Proof: footnote 5 in page 137
26
Trick (B) to obtain compensated demand
functions
x  x( p x , p y , I ), demand function
I  E ( p x , p y , U ), expenditur e function
Substituin g ...
x  x( p x , p y , E ( p x , p y , U ))  x( p x , p y , U )
that is, the compensate d demand function
because it depends on prices and utility
27
Trick (B) to obtain 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 expenditure function is
E  2 px0.5 p 0y.5 U  I
28
Another trick to obtain compensated demand
functions
• Substitute the expenditure function into
the Marshallian demand functions, and
find the compensated ones:
x
Up 0y.5
p x0.5
Upx0.5
y  0.5
py
29
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 changes the amount of x
demanded, keeping utility V constant. Hence
the compensated demand function only
includes the substitution effect but not the
income effect
30
Roy’s identity
• It is the relation between marshallian demand
function and indirect utility function
V
V
dp y
dpx
x( px , p y , I )  
; y ( px , p y , I )  
V
V
dI
dI
Proof of the Roy’s identity…
31
Proof of Roy’s identity
V ( px , p y , I )  u
V ( p x , p y , E ( p x , p y , u ))  u
Taking derivatives wrt p x :
V px '(.)  VI '(.) E ' px  0
Using previous trick:
E ' px
E

 x
dpx
Substituting:
V px '(.)  VI '(.) x  0
and solving for x, we find the Roy's identity
32
Demand curves…
• We will start to talk about demand
curves. Notice that they are not the
same that demand functions !!!!
33
The Marshallian 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
34
The Marshallian 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
35
The Marshallian Demand Curve
• The Marshallian 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
• Notice that demand curve and demand
function is not the same thing!!!
36
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
37
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
38
Compensated Demand Curves
• An alternative approach holds utility
constant while examining reactions to
changes in px
– the effects of the price change are
“compensated” with income so as to constrain
the individual to remain on the same
indifference curve
– reactions to price changes include only
substitution effects (utility is kept constant)
39
Marshallian 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
40
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)
41
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
42
Compensated & Uncompensated
Demand for normal goods
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
43
Compensated & Uncompensated Demand for
normal goods
At prices above p’’x, income
compensation is positive because the
individual needs some help to remain
on U2
px
px’
px’’
x
xc
x’
x*
Quantity
of x
As we are looking at normal goods, income and
substitution
effects go
in the same direction, so they are reinforced. X includes both while Xc
only the substitution effect. That is what drives the relative position of44
both curves
Compensated & Uncompensated
Demand for normal goods
px
At prices below px2, income
compensation is negative to prevent an
increase in utility from a lower price
px’’
px’’’
x
xc
x*** income
x’’’
Quantity
of x
As we are looking at normal goods,
and
substitution
effects go
in the same direction, so they are reinforced. X includes both while Xc
only the substitution effect. That is what drives the relative position of45
both curves
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
46
Relations to keep in mind
• Sheppard’s Lema & Roy’s identity
• V(px,py,E(px,py,Uo)) = U0
• E(px,py,V(px,py,I0)) = I0
• xc(px,py,U0)=x(px,py,I0)
47
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
48
A Mathematical Examination
of a Change in Price
• However, for our purpose, 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
49
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
50
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
51
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
52
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 p x
I p x
E
Using trick A :
x
p x
x E
x
income effect  


x
I p x
I
53
The Slutsky Equation
• A price change can be represented by
x
 substituti on effect  income effect
px
x
x

px px
U constant
x
x
I
54
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
55
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
56
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
57
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
58
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
59
A Slutsky Decomposition
• We can substitute in for the indirect utility
function (V)
substituti on effect 
0.5
x
1.5
x
 0.5(0.5Ip
p
p
0.5
y
)p
0.5
y
 0.25 I

px2
60
A Slutsky Decomposition
• Calculation of the income effect is easier
 0.5I  0.5
x
0.25I
income effect   x
 


2
I
px
 px  px
• By adding up substitution and income
effect, we will obtain the overall effect
61