Chapter 5
Income and Substitution
Effects
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)
• 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
Homogeneity
• If all prices and income were doubled, the
optimal quantities demanded will not
change
– the budget constraint is unchanged
xi* = xi(p1,p2,…,pn,I) = xi(tp1,tp2,…,tpn,tI)
• Individual demand functions are
homogeneous of degree zero in all prices
and income
Changes in Income
• An increase in income will cause the
budget constraint to shift out in a
parallel fashion
• Since px/py does not change, the MRS
will stay constant as the individual
moves to higher levels of satisfaction
Effects of a Rise in Income
• Income-consumption curve shows how
consumption of both goods changes
when income changes, while prices are
held constant.
• Engel curve - the relationship between
the quantity demanded of a single good
and income, holding prices constant.
Y
Effect of a Budget Increase on an
Individual’s Demand Curve
L1
2.8
Budget Line, L
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $419.
Income goes up!
Price of X
PX
X
PY
12
26.7
I1
X
E1
D1
0
26.7
X
I , Budget
Y=
I PY
0
e1
Y1 = $419
0
E1*
26.7
X
y
L2
Y
Effect of a Budget Increase on
an Individual’s Demand Curve
L1
4.8
2.8
Budget Line, L
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $419.
Income goes up!
$628
Price of X
PX
X
PY
12
I1
26.7 38.2
E1
I2
X
E2
D2
D1
0
26.7 38.2
X
I , Budget
Y=
I PY
0
e2
e1
Y2 = $628
Y1 = $419
0
E2*
E1*
26.7 38.2
X
L3
L2
Y
Effect of a Budget Increase on
an Individual’s Demand Curve
L1
7.1
4.8
2.8
Budget Line, L
PX
X
PY
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $837.
Price of X
0
12
e2
e1
I1
26.7 38.2 49.1
E1
E2
I3
X
D3
D2
D1
0
26.7 38.2 49.1
E3*
Y2 = $628
Y1 = $419
0
X
Engel curve for X
Y3 = $837
Income goes up again!
I2
E3
I , Budget
Y=
I PY
Income-consumption curve
e3
E2*
E1*
26.7 38.2 49.1
X
Increase in Income
• If both x and y increase as income
rises, x and y are normal goods
As income rises, the
individual chooses
to consume more x and y
Quantity of y
B
C
A
U1
U3
U2
Quantity of x
Normal and Inferior Goods
If the income consumption curve shows that the
consumer purchases more of good x as her income rises,
good x is a normal good.
• Equivalently, if the slope of the Engel curve is positive,
the good is a normal good.
•
• If the income consumption curve shows that the
consumer purchases less of good x as her income rises,
good x is an inferior good.
• Equivalently, if the slope of the Engel curve is negative,
the good is an inferior good.
10
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
Quantity of y
more y
C
B
U3
U2
A
U1
Quantity of x
Normal and Inferior Goods
Example: Backward Bending
ICC and Engel Curve – a
good can be normal over
some ranges and inferior
over others
12
Income Consumption Curve:
Normal and Inferior Goods
Price Consumption Curves
Y (units)
PY = $4
I = $40
Is the set of optimal baskets for every
possible price of good x, holding all
other prices and income constant.
10
Price Consumption Curve
•
•
•
PX = 1
PX = 2
PX = 4
0
XA=2
XB=10
XC=16
20
X (units)
14
Individual Demand Curve
PX
Individual Demand Curve
For Commodity X
PX = 4
•
PX = 2
PX = 1
XA
•
XB
•
XC
Quantity increasing
X
15
Individual Demand Curve: Properties
• 1) The level of utility that can be attained
changes as we move along the curve.
• 2) At every point on the demand curve, the
consumer is maximizing utility by satisfying
the condition that
• MRS of X for Y = the ratio of the prices of
X and Y.
Changes in a Good’s Price
• A change in the price of a good alters the
slope of the budget constraint
–It changes the MRS at the consumer’s
utility-maximizing choices
• When the price changes, two effects come
into play
–substitution effect
–income effect
The Substitution Effect
• As the price of x falls, all else constant, good x
becomes cheaper relative to good y and vice versa.
•This change in relative prices alone causes the
consumer to adjust his/ her consumption basket.
• This effect is called the substitution effect.
• The substitution effect always is negative.
Income Effect
• As the price of x falls, all else constant, purchasing
power rises and vice versa.
•This is called the income effect of a change in
price.
•The income effect may be positive (normal good)
or negative (inferior good).
Changes in a Good’s Price:
Price of X falls
Suppose the consumer is
maximizing utility at point A.
Quantity of y
B
A
If the price of good x falls,
the consumer will
maximize utility at point B.
U2
U1
Quantity of x
Total increase in x
Changes in a Good’s Price:
Price of X falls
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
Substitution effect
The individual substitutes
x for y because it is now
relatively cheaper
Quantity of x
Changes in a Good’s Price :
Price of X falls
The income effect occurs because “real”
income changes when the price of good
x changes
Quantity of y
B
A
The income effect is the movement
from point C to point B
C
U2
U1
Income effect
If x is a normal good,
the individual will buy
more because “real”
income increased
Quantity of x
Total Effect or Price Effect(AB) = Substitution Effect (AC) +
Income Effect (CB)
Changes in a Good’s Price:
Price of X rises
Quantity of y
An increase in the price of good x means
that the budget constraint gets steeper
C
A
The substitution effect is the
movement from point A to point C
B
U1
U2
The income effect is the
movement from point C
to point B
Quantity of x
Substitution effect
Income effect
Total Effect or Price Effect (AB) = Substitution Effect (AC) +
Income Effect (CB)
Price Changes – Normal Goods
• If a good is normal, substitution and income
effects reinforce one another
– when pX :
• substitution effect  quantity demanded of X
• income effect  quantity demanded of X 
– when pX :
• substitution effect  quantity demanded of X 
• income effect  quantity demanded of X 
Price Changes – Inferior Goods
• If a good is inferior, substitution and
income effects move in opposite
directions
– when pX :
• substitution effect  quantity demanded of X 
• income effect  quantity demanded of X 
– when pX :
• substitution effect  quantity demanded of X 
• income effect  quantity demanded of X 
INCOME AND SUBSTITUTION EFFECTS:
INFERIOR GOOD
Consumer is initially at point A
on RS.
With a ↓ PFood, the consumer
moves to point B.
i) a substitution effect, F1E
(associated with a move
from A to D),
ii) an income effect, EF2
(associated with a move
from D to B).
In this case, food is an inferior
good because the income
effect is negative.
A Special Case: The Giffen Good
● Giffen
good Good whose demand curve slopes upward
because the (negative) income effect is larger than the
substitution effect.
•Food: an inferior good
• Consumer is initially at A
• After the ↓P of food, moves
to B and consumes less
food.
• The income effect F2F1 >
the substitution effect EF2,
•Income effect dominates
over the substitution effect
• The ↓Pfood leads to a
lower quantity of food
demanded.
UPWARD-SLOPING DEMAND CURVE:
THE GIFFEN GOOD
Inferior good and Giffen Paradox
• 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
• Giffen Paradox: If the income effect of a price change
is strong enough, there could be a positive
relationship between price and quantity demanded
– an decrease in price leads to a increase in real income
– since the good is inferior, a increase in income causes
quantity demanded to fall
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 i.e (py) are held constant
The Individual’s Demand Curve
Quantity of y
As the price
of x falls...
px
…quantity of
x
demanded rises.
px’
px’’
px’’’
U1 U2
x1
I = px’ + py
x2
x3
I = px’’ + py
U3
Quantity of x
I = px’’’ + py
x
x’
x’’
x’’’
Quantity of x
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
• Difference between a change in quantity
demanded and change in demand
Compensated Demand Curve
• The demand curves shown so far have all
been uncompensated, or Marshallian,
demand curves.
• Consumer utility is allowed to vary with
the price of the good.
• Alternatively, we have a compensated, or
Hicksian, demand curve.
• It shows how quantity demanded changes
when price increases, holding utility
constant.
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 px
–the effects of the price change are
“compensated” so as to force 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 twodimensional representation of the
compensated demand function
x* = xc(px,py,U)
Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y
px
p '
slope   x
py
…quantity
demanded
rises.
slope  
px ' '
py
px’
px’’
slope  
px ' ' '
py
px’’’
Xc
U2
x’
x’’
x’’’
Quantity of x
x’
x’’
x’’’
Quantity of x
Compensated & Uncompensated Demand
At px’’, the curves intersect
because the individual’s
income is just sufficient to
attain a certain utility level
px
px’’
x
xc
x’’
Quantity of x
Compensated & Uncompensated Demand
At prices above px’’,
income compensation is
positive because the
individual needs more
income to remain on a
certain utility level
px
px’
px’’
x
xc
x’
x*
Quantity of x
Compensated & Uncompensated Demand
px
At prices below px’’, income
compensation is negative to
prevent an increase in utility
from a lower price
px’’
px’’’
X
xc
x***
x’’’
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
The Response to a Change in Price
• We will use an indirect approach using the
expenditure function
minimum expenditure = E(px,py,U)
• Then, by definition
xc (px,py,U) = x [px,py,E(px,py,U)]
The Response to a Change in Price
xc (px,py,U) = x[px,py,E(px,py,U)]
• We can differentiate the function and get
x c
x
x E



px px
E px
x
x c
x E



px px
E px
The Response to a Change in Price
x x c
x E



px px
E px
• The first term is the slope of the compensated
demand curve
– the mathematical representation of the substitution
effect
The Response to 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
The Slutsky Equation
• The substitution effect can be written as
x c x
substitution effect 

p x p x
U  constant
• The income effect can be written as
x E
x E
income effect  

 

E px
I px
The Slutsky Equation
• The utility-maximization hypothesis shows that
the substitution and income effects arising
from a price change can be represented by
x
 substitution effect  income effect
px
x
x

px px
U constant
x
x
I
The Slutsky Equation
x
x

px px
U constant
x
x
I
• The first term is the substitution effect
– always negative
x
x

px px
U constant
x
x
I
• The second term is the income effect
– if x is a normal good, entire income effect is negative
– if x is an inferior good, entire income effect is
positive
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 ,px
x / x
x px



px / px px x
• Income elasticity of demand (ex,I)
x / x x I
e x ,I 


I / I I x
• Cross-price elasticity of demand (ex,py)
ex , p y
x / x
x p y



p y / p y p y x
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
Compensated Price Elasticities
• It is also useful to define elasticities based on the
compensated demand function
• If the compensated demand function is
xc = xc(px,py,U)
we can calculate
– compensated own price elasticity of demand (exc,px)
c
c
c

x
/
x

x
px
c
ex ,px 

 c
px / px px x
– compensated cross-price elasticity of demand (exc,py)
exc,py
x c / x c x c py


 c
py / py py x
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
Consumer Surplus
• Suppose we want to examine the change in an
individual’s welfare when price changes
Consumer Welfare
• If the price rises, the individual would have to
increase expenditure to remain at the initial level of
utility
expenditure at px0 = E0 = E(px0,py,U0)
expenditure at px1 = E1 = E(px1,py,U0)
• In order to compensate for the price rise, this person
would require a compensating variation (CV) of
CV = E(px1,py,U0) - E(px0,py,U0)
Consumer Welfare
Quantity of y
Suppose the consumer is maximizing
utility at point A.
If the price of good x rises,
the consumer will maximize
utility at point B.
The consumer’s utility
falls from U2 to U1
A
B
U2
U1
Quantity of x
Consumer Welfare
Quantity of y
The consumer could be compensated so
that he can afford to remain on U2
CV is the amount that the individual
would need to be compensated
CV
C
A
B
U2
U1
Quantity of x
Consumer Welfare
px
When the price rises from px0 to px1,
the consumer suffers a loss in welfare
welfare loss
px1
px0
xc(px…U0)
x1
x0
Quantity of x
Consumer Welfare
• A price change generally involves both
income and substitution effects
–should we use the compensated
demand curve for the original target
utility (U0) or the new level of utility
after the price change (U1)?
Consumer Welfare
px
px1
Is the consumer’s loss in welfare best described
by area px1BApx0 [using xc(...,U1)] or by area
px1CDpx0 [using xc(...,U0)]?
C
B
A
px0
Is U1 or U0 the
appropriate utility target?
D
xc(...,U1)
xc(...,U0)
x1
x0
Quantity of x
Consumer Welfare
px
px1
We can use the Marshallian demand curve as
a compromise
The area px1CApx0 falls between
the sizes of the welfare losses
defined by xc(...,U1) and xc(...,U0)
C
B
A
px0
D
x(px,…)
xc(...,U1)
xc(...,U0)
x1
x0
Quantity of x
Consumer Surplus
• We will define consumer surplus as the area
below the Marshallian demand curve and above
the prevailing market price
– shows what an individual would pay for the
right to make voluntary transactions at this
price
– changes in consumer surplus measure the
welfare effects of price changes
Compensating Variation versus
Equivalent Variation
• p1 rises.
• Q: What is the least extra income that,
at the new prices, just restores the
consumer’s original utility level?
• A: The Compensating Variation.
Compensating Variation
x2
p1=p1’
p2 is fixed.
I1  p x  p x
' '
1 1
x'2
'
2 2
u1
x'1
x1
Compensating Variation
p1=p1’
p2 is fixed.
p1=p1” (Price rise)
' '
'
I1  p1 x1  p2 x2
x2
 p"1x"1  p 2x"2
x"2
x'2
u1
u2
"
x1
x'1
x1
Compensating Variation
x2
x2'"'"
2
p2 is fixed.
p1=p1’
p1=p1” (Price rise) I1  p1' x1'  p2 x2'
 p"1x"1  p2x"2
x"2
x'2
I2  p x  p x
u1
u2
x"1 x'"
1
x'1
" '"
1 1
'"
2 2
CV = I2 - I1.
x1
Equivalent Variation
• p1 rises.
• Q: What is the least extra income that, at the
original prices, just restores the consumer’s
final utility level?
• A: The Equivalent Variation.
Equivalent Variation
x2
p1=p1’
p2 is fixed.
I1  p1' x1'  p2 x2'
x'2
u1
x'1
x1
Equivalent Variation
p1=p1’
p2 is fixed.
p1=p1” (Price rise) I  p ' x '  p x '
1
1 1
2 2
x2
 p"1x"1  p2x"2
x"2
x'2
u1
u2
x"1
x'1
x1
Equivalent Variation
p1=p1’
p1=p1” (Price rise)
x2
p2 is fixed.
I1  p x  p x
' '
1 1
'
2 2
 p"1x"1  p2x"2
x"2
x'2
I2  p x  p x
' '"
1 1
u1
'"
x2'"
2
u2
x"1
'
x'"
x
1
1
EV = I1 - I2.
x1
'"
2 2
CV and EV for a fall in the price of X
The effect on (behaviour and) welfare of a
change in a price.
• Let us now look at these two
variations with quasi-linear
preferences....
Compensating vs. Equivalent Variation
Quasi –linear Preferences
From Individual Demand to Market Demand
The market demand function is the horizontal sum
of the individual (or segment) demands.
In other words, market demand is obtained by
adding the quantities demanded by the individuals
(or segments) at each price and plotting this total
quantity for all possible prices.
Market Demand
P
10
P
P = 10 - Q
P = 4 – 0.2Q
4
Segment 1
P
Q
Segment 2
Q
Market demand
Q
Market Demand
Two points should be noted:
• 1.The market demand curve will shift to the right
as more consumers enter the market.
• 2. Factors that influence the demands of many
consumers will also affect market demand.
The aggregation of individual demands into market
becomes important in practice when market
demands are built up from the demands of different
demographic groups or from consumers located in
different areas.
Network Externalities
• Up to this point we have assumed that people’s
demands for a good are independent of one another.
• This assumption has led to obtain the market
demand curve simply by summing individual’s
demand.
• For some goods, one person’s demand also depends
on the demands of other people
• If one consumer's demand for a good changes with
the number of other consumers who purchased the
good, there are network externalities
• Network externality: When each individual’s
demand depends on the purchases of other
individuals.
Network Externalities
• Network externalities can be positive or
negative
• A positive network externality exists if the
quantity of a good demanded by a
consumer increases in response to an
increase in purchases by other consumers
• Negative network externalities are just the
opposite
Positive Network Externalities
• Bandwagon effect: A positive network
externality that refers to the increase in each
consumer’s demand for a good as more
consumers buy the good
• E.g: Toys for kids
Positive Network
Externality: Bandwagon Effect
Price
($ per
unit)
D20
D40 D60 D80 D100
When consumers believe more
people have purchased the
product, the demand curve shifts
further to the the right.
Quantity
20
40
60
80
100
(thousands per month)
Positive Network
Externality: Bandwagon Effect
Price
($ per
unit)
D20
D40 D60 D80 D100
The market demand
curve is found by joining
the points on the individual
demand curves. It is relatively
more elastic.
Demand
Quantity
20
40
60
80
100
(thousands per month)
Positive Network
Externality: Bandwagon Effect
Price
($ per
unit)
D20
D40 D60 D80 D100
$30
But asthe
more
people
Suppose
price
falls buy
it becomes
from the
$30good,
to $20.
If there
to own it effect,
and
were stylish
no bandwagon
the quantity
demanded
quantity
demanded
would
increases
further.
only increase
to 48,000
Demand
$20
Bandwagon
Effect
Pure Price
Effect
Quantity
20
40 48 60
80
79
100
(thousands per month)
Network Externalities
• Snob effect: A negative network externality
that refers to the decrease in each consumer’s
demand as more consumers buy the good
• The snob effect refers to the desire to own
exclusive or unique goods
• The quantity demanded of a “snob” good is
higher the fewer the people who own it
Network Externality: Snob Effect
Price
($ per
unit)
Originally demand is D2,
when consumers think 2,000
people have bought a good.
Demand
$30,000
However, if consumers think 4,000
people have bought the good,
demand shifts from D2 to D6 and its
snob value has been reduced.
$15,000
D2
Pure Price Effect
D4
D8
2
4
6
8
D6
Quantity
14
per month)
(thousands
Network Externality: Snob Effect
Price
($ per
unit)
The demand is less elastic and as a snob
good its value is greatly reduced if more
people own it.
Demand
$30,000
Net Effect
Snob Effect
$15,000
D2
Pure Price Effect
D4
D8
2
4
6
8
D6
Quantity
14
per month)
(thousands
Revealed Preference Hypothesis
• The theory of revealed preference was proposed
by Paul Samuelson in the year 1938.
• Considered major breakthrough in the theory of
demand .
• It establishes law of demand without using IC and
their respective assumptions.
• Assumptions:
– 1) Rationality
– 2) Consistency
– 3) Transitivity
– 4) Revealed Preference axiom
Revealed Preference Axiom
• 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
• The chosen basket A maximizes
(axiomatically) his utility and rest of the
baskets on the budget line are revealed
inferior.
Revealed Preference Hypothesis: Derivation of
Demand Curve
• Initially consumer at
B with budget line RS.
• A ↓ PF shifts budget
line from RS to RT.
• The new market
basket chosen must lie
on line segment BT' of
budget line R′T' (which
intersects RS to the
right of B), and the
quantity of food
consumed must be
greater than at B.
The Weak and the Strong Axioms of
the Revealed Preference Theory
• While listing the assumptions of the RPT, the
ordering principles (consistency and transitivity)
of consumer preferences are mentioned
• We combine the ordering principles with the RP
axioms to state and establish
• 1) Weak Axiom of Revealed Preference (WARP)
• 2) Strong Axiom of Revealed Preference (SARP)
Weak Axiom of Revealed Preference
(WARP)
• If bundle (x1, y1) is directly revealed preferred
to bundle (x2, y2), the two bundles being
different from each other, it cannot happen
that bundle (x2, y2) would be directly revealed
preferred to bundle (x1, Y1).
• That is: If (x1,y1) is revealed preferred when
(x2,y2) was affordable then (x1,y1) is preferred
always (or at all prices).
Weak Axiom of Revealed Preference (WARP)
If (x1, y1) is directly revealed preferred to (x2,y2),
and the two bundles are not the same, then it
cannot happen that (x2,y2) is directly revealed
preferred to (x1, y1).
This is the weak axiom of revealed preference
(WARP)
If this occurs then
WARP is violated
2 , y2 
1, y1 
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
• If (x1,y1) is revealed preferred to (x3,y3) either
directly or indirectly) then (x3,y3) cannot be directly
or indirectly revealed preferred to (x1,y1)
• SARP is a necessary and sufficient condition for
observed behaviour to be consistent with the
underlying model of consumer choice