1. Income and Substitution Effects
Chapter 5
Income and Substitution Effects

2. 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

3. xi* = xi(p1,p2,…,pn,I) = xi(tp1,tp2,…,tpn,tI)
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

4. 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

5. 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.

6. Effect of a Budget Increase on an Individual’s Demand CurveY L 1
2.8 e
Budget Line, L 1 I 1
26.7 X I PX
Y = - X PY PY E1
Price of X 12
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $419. D 1
26.7 X
, Budget I
Income goes up! Y 1
= $419 E * 1
26.7 X

7. Effect of a Budget Increase on an Individual’s Demand Curvey L 2 Y L 1
4.8 e 2
2.8 e
Budget Line, L 1 I 2 I 1
26.7
38.2 X I PX Y= - X PY PY E E
Price of X 12 1 2
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $419. D 2 D 1
26.7
38.2 X
, Budget I
Income goes up!
$628 Y
= $628 2 E * 2 Y
= $419 1 E * 1
26.7
38.2 X

8. Effect of a Budget Increase on an Individual’s Demand Curve2 Y L 1
Income-consumption curve e 3
7.1
4.8 e 2
2.8 e I3
Budget Line, L 1 I 2 I 1
26.7
38.2
49.1 X I PX
Y = - X PY PY E E E
Price of X 12 1 2 3
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $837. D3 D 2 D 1
26.7
38.2
49.1 X
, Budget I
Engel curve for X Y
= $837 * 3 E 3
Income goes up again! Y
= $628 2 E * 2 Y
= $419 1 E * 1
26.7
38.2
49.1 X

9. 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 C U3 B U2 A U1
Quantity of x

10. 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.

11. 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 C U3 B U2 A U1
Quantity of x

12. Normal and Inferior Goods
Example: Backward Bending ICC and Engel Curve – a good can be normal over some ranges and inferior over others

13. Income Consumption Curve: Normal and Inferior Goods

14. • • • Price Consumption Curves
Is the set of optimal baskets for every possible price of good x, holding all other prices and income constant.
Y (units)
PY = $4
I = $40 10 •
Price Consumption Curve • •
PX = 1
PX = 2
PX = 4
X (units)
XA=2
XB=10
XC=16 20

15. • • • Individual Demand Curve PX
Individual Demand Curve For Commodity X •
PX = 4 •
PX = 2 •
Quantity increasing
PX = 1 X
XA XB XC

16. 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.

17. 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

18. 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.

19. 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).

20. Changes in a Good’s Price: Price of X fallsU1 A
Suppose the consumer is maximizing utility at point A.
Quantity of y U2 B
If the price of good x falls, the consumer will maximize utility at point B.
Total increase in x
Quantity of x

21. 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
The individual substitutes x for y because it is now relatively cheaper
Substitution effect U1
Quantity of x

22. 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
The income effect is the movement
from point C to point B
If x is a normal good,
the individual will buy more because “real” income increased
Income effect A C U2 U1
Quantity of x
Total Effect or Price Effect(AB) = Substitution Effect (AC) Income Effect (CB)

23. 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
The substitution effect is the
movement from point A to point C
Substitution effect A
Income effect
The income effect is the
movement from point C
to point B B U1 U2
Quantity of x
Total Effect or Price Effect (AB) = Substitution Effect (AC) +
Income Effect (CB)

24. 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 

25. 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 

26. INCOME AND SUBSTITUTION EFFECTS: INFERIOR GOOD
Consumer is initially at point A on RS.
With a ↓ PFood, the consumer moves to point B.
a substitution effect, F1E (associated with a move from A to D),
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.

27. 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

28. 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

29. 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

30. The Individual’s Demand Curve
Quantity of y
As the price
of x falls... px x’
px’ U1 x1
I = px’ + py
x’’
px’’ U2 x2
I = px’’ + py
x’’’
px’’’ x3 U3
I = px’’’ + py x
…quantity of x
demanded rises.
Quantity of x
Quantity of x

31. 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

32. 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.

33. 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

34. 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

35. 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* = xc(px,py,U)

36. Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y px x’
px’ Xc
…quantity demanded
rises. U2
x’’
px’’
x’’’
px’’’
Quantity of x
Quantity of x

37. Compensated & Uncompensated Demand
x’’
px’’
At px’’, the curves intersect because the individual’s income is just sufficient to attain a certain utility level px x xc
Quantity of x

38. Compensated & Uncompensated Demandx* x’
px’
At prices above px’’, income compensation is positive because the individual needs more income to remain on a certain utility level px
px’’ x xc
Quantity of x

39. Compensated & Uncompensated Demandpx
x***
x’’’
px’’’
At prices below px’’, income compensation is negative to prevent an increase in utility from a lower price
px’’ X xc
Quantity of x

40. 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

41. 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)]

42. 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

43. The Response to a Change in Price
The first term is the slope of the compensated demand curve
the mathematical representation of the substitution effect

44. The Response to a Change in Price
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

45. The Slutsky Equation The substitution effect can be written as
The income effect can be written as

46. The Slutsky Equation
The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by

47. The Slutsky Equation The first term is the substitution effect
always negative
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

48. 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)
Income elasticity of demand (ex,I)
Cross-price elasticity of demand (ex,py)

49. 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

50. 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)
compensated cross-price elasticity of demand (exc,py)

51. 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

52. Consumer Welfare 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)

53. Consumer Welfare Suppose the consumer is maximizing
Quantity of y
Suppose the consumer is maximizing
utility at point A. U1 B
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 U2
Quantity of x

54. Consumer Welfare Quantity of y
The consumer could be compensated so that he can afford to remain on U2 C
CV is the amount that the individual would need to be compensated CV A B U2 U1
Quantity of x

55. Consumer Welfare When the price rises from px0 to px1,
the consumer suffers a loss in welfare
welfare loss
px1 x1
px0 x0
xc(px…U0)
Quantity of x

56. 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)?

57. Consumer Welfare
Is the consumer’s loss in welfare best described by area px1BApx0 [using xc(...,U1)] or by area px1CDpx0 [using xc(...,U0)]? px
Is U1 or U0 the appropriate utility target? C B
px1 A
px0 D
xc(...,U1)
xc(...,U0) x1 x0
Quantity of x

58. Consumer Welfare
We can use the Marshallian demand curve as a compromise px
The area px1CApx0 falls between the sizes of the welfare losses defined by xc(...,U1) and xc(...,U0) C
x(px,…) B
px1 A
px0 D
xc(...,U1)
xc(...,U0) x1 x0
Quantity of x

59. 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

60. 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.

61. Compensating Variation
p1=p1’
p2 is fixed. x2 u1 x1

62. Compensating Variation
p1=p1’ p1=p1” (Price rise)
p2 is fixed. x2 u1 u2 x1

63. Compensating Variation
p2 is fixed.
p1=p1’ p1=p1” (Price rise) x2 u1
CV = I2 - I1. u2 x1

64. 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.

65. Equivalent Variation
p1=p1’
p2 is fixed. x2 u1 x1

66. Equivalent Variation p1=p1’ p1=p1” (Price rise) p2 is fixed. x2 u1 u2

67. Equivalent Variation EV = I1 - I2. p2 is fixed.
p1=p1’ p1=p1” (Price rise) x2 u1
EV = I1 - I2. u2 x1

68. CV and EV for a fall in the price of X

69. The effect on (behaviour and) welfare of a change in a price.
Let us now look at these two variations with quasi-linear preferences....

70. Compensating vs. Equivalent Variation Quasi –linear Preferences

71. 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.

72. Market Demand P P P 10 P = 10 - Q P = 4 – 0.2Q 4 Q Q Q Segment 1

73. 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.

74. 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.

75. 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

76. 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

77. Positive Network Externality: Bandwagon Effect
Price
($ per
unit)
D20 20 40
D40 60
D60 80
D80
100
D100
When consumers believe more
people have purchased the
product, the demand curve shifts
further to the the right.
Quantity
(thousands per month)

78. Positive Network Externality: Bandwagon Effect
Price
($ per
unit)
D20 20 40
D40 60
D60 80
D80
100
D100
The market demand
curve is found by joining
the points on the individual
demand curves. It is relatively
more elastic.
Demand
Quantity
(thousands per month)

79. Positive Network Externality: Bandwagon Effect
Price
($ per
unit)
D20 20 40
D40 60
D60 80
D80
100
D100
But as more people buy
the good, it becomes
stylish to own it and
the quantity demanded
increases further.
Suppose the price falls
from $30 to $20. If there
were no bandwagon effect,
quantity demanded would
only increase to 48,000
$30
Demand 48
$20
Bandwagon
Effect
Pure Price
Effect
Quantity
(thousands per month)

80. 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

81. Network Externality: Snob Effect
Price
($ per
unit)
Demand
Originally demand is D2,
when consumers think 2,000
people have bought a good.
$30,000 4 6 8 D4 D6 D8
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
Quantity (thousands
per month) 2 14

82. Network Externality: Snob Effect
Quantity (thousands
per month)
Price
($ per
unit) 2
Demand D2
$30,000
$15,000 14 4 6 8 D4 D6 D8
Pure Price Effect
The demand is less elastic and as a snob good its value is greatly reduced if more people own it.
Net Effect
Snob Effect

83. 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

84. 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.

85. 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.

86. 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)

87. 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).

88. 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

89. 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