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)

3. 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

4. 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

5. Homogeneity
With a Cobb-Douglas utility function
utility = U(x,y) = x0.3y0.7
the demand functions are
A doubling of both prices and income would leave x* and y* unaffected

6. Homogeneity With a CES utility function utility = U(x,y) = x0.5 + y0.5
the demand functions are
A doubling of both prices and income would leave x* and y* unaffected

7. 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

8. Income Consumption Curve

9. Effects of a Rise in Income
Engel curve - the relationship between the quantity demanded of a single good and income, holding prices constant.
Income-consumption curve shows how consumption of both goods changes when income changes, while prices are held constant.

10. Engel Curves “X is a normal good” I ($) Engel Curve 92 68 40 X (units)

11. 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

12. 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

13. 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

14. 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

15. 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

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

17. • • • Price Consumption Curves
Is the set of optimal baskets for every possible price of good x, holding all other prices and income constant.
This is the individual’s demand curve for good x.
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

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

19. The Individual Demand Curve
● Price-consumption curve Curve tracing the utility-maximizing combinations of two goods as the price of one changes.
● Individual demand curve
Curve relating the quantity of a good that a single consumer will buy to its price.
The individual demand curve has two important properties:
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 the marginal rate of substitution (MRS) of X for Y equals the ratio of the prices of X and Y.

20. Demand Curve for “X”
Algebraically, we can solve for the individual’s demand using the following equations:
1. pxx + pyy = I
2. MUx/px = MUy/py – at a tangency.
(If this never holds, a corner point may be substituted where x = 0 or y = 0)

21. 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

22. Changes in a Good’s Price
Even if the individual remains on the same indifference curve, his optimal choice will change because the MRS must equal the new price ratio
the substitution effect
The individual’s “real” income has changed and he must move to a new indifference curve
the income effect

23. 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

24. 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

25. 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)

26. 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: Price Effect (AB) = Substitution Effect (AC) +
Income Effect (CB)

27. Price Changes – Normal Goods
If a good is normal, substitution and income effects reinforce one another
when p :
substitution effect  quantity demanded 
income effect  quantity demanded 
when p :
substitution effect  quantity demanded 
income effect  quantity demanded 

28. Price Changes – Inferior Goods
If a good is inferior, substitution and income effects move in opposite directions
when p :
substitution effect  quantity demanded 
income effect  quantity demanded 
when p :
substitution effect  quantity demanded 
income effect  quantity demanded 

29. INCOME AND SUBSTITUTION EFFECTS: INFERIOR GOOD
Consumer is initially at A on RS.
With a ↓ P of food, the consumer moves to 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.
However, because the substitution effect exceeds the income effect, the decrease in the price of food leads to an increase in the quantity of food demanded.

30. 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.
UPWARD-SLOPING DEMAND CURVE: THE GIFFEN GOOD
Food: an inferior good
Income effect dominates over the substitution effect,
Consumer is initially at A
After the ↓P of food falls, moves to B and consumes less food.
The income effect F2F1 > the substitution effect EF2,
The ↓P of food leads to a lower quantity of food demanded.

31. 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 decrease in price leads to a increase in real income
since the good is inferior, a increase in income causes quantity demanded to fall

32. 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

33. 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

34. 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

35. 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

36. Compensated Demand Curve
The demand curves shown thus far have all been uncompensated, or Marshallian, demand curves.
Consumer utility is allowed to vary with the price of the good.
Alternatively, a compensated, or Hicksian, demand curve shows how quantity demanded changes when price increases, holding utility constant.

37. 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

38. 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

39. 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)

40. 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

41. Compensated & Uncompensated Demandpx
x’’
px’’
At px’’, the curves intersect because
the individual’s income is just sufficient to attain utility level U2 x xc
Quantity of x

42. Compensated & Uncompensated Demandpx x* x’
px’
At prices above px’, income compensation is positive because the individual needs more income to remain on U2
px’’ x xc
Quantity of x

43. 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

44. 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

45. 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

46. 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

47. Compensated Demand Functions
Demand now depends on utility (V) rather than income
Increases in px reduce the amount of x demanded
only a substitution effect

48. 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)]

49. The Response to a Change in Price
xc (px,py,U) = x[px,py,E(px,py,U)]
We can differentiate the compensated demand function and get

50. 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

51. 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

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

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

54. 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, income effect is negative
if x is an inferior good, income effect is positive

55. 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)

56. Marshallian Demand Elasticities
Income elasticity of demand (ex,I)
Cross-price elasticity of demand (ex,py)

57. 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

58. 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

59. Price Elasticity and Total Spending
If ex,px > -1, demand is inelastic
price and total spending move in the same direction
If ex,px < -1, demand is elastic
price and total spending move in opposite directions

60. Compensated Price Elasticities
It is also useful to define elasticities based on the compensated demand function

61. 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)

62. Compensated Price Elasticities
The compensated own price elasticity of demand (exc,px) is
The compensated cross-price elasticity of demand (exc,py) is

63. 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

64. Relationship among demand elasticities
1) Homogeneity
2) Engel aggregation
3) Cournot aggregation

65. Homogeneity
Demand functions are homogeneous of degree zero in all prices and income
Any proportional change in all prices and income will leave the quantity of x demanded unchanged

66. Engel Aggregation
Engel’s law suggests that the income elasticity of demand for food items is less than one
this implies that the income elasticity of demand for all nonfood items must be greater than one.
Income elasticity of demand for various goods
Automobiles
Books
Restaurant Meals
Tobacco
Public Transportation −0.36

67. Cournot Aggregation
The size of the cross-price effect of a change in px on the quantity of y consumed is restricted because of the budget constraint.
Differentiate the budget constraint with respect to Px.

68. 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)

69. 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

70. 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

71. Consumer Welfare
The derivative of the expenditure function with respect to px is the compensated demand function
The amount of CV required can be found by integrating across a sequence of small increments to price from one price to another.

72. 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

73. 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)?

74. The Consumer Surplus Concept
The area below the compensated demand curve and above the market price is called consumer surplus
the extra benefit the person receives by being able to make market transactions at the prevailing market price

75. 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

76. 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

77. Consumer Surplus
We will define consumer surplus as the area below the Marshallian demand curve and above 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

78. Welfare Loss from a Price Increase
Suppose that the compensated demand function for x is given by
The welfare cost of a price increase from px = $1 to px = $4 is given by

79. Welfare Loss from a Price Increase
If we assume that V = 2 and py = 4,
CV = 222(4)0.5 – 222(1)0.5 = 8
If we assume that the utility level (V) falls to 1 after the price increase (and used this level to calculate welfare loss),
CV = 122(4)0.5 – 122(1)0.5 = 4

80. Welfare Loss from a Price Increase
Suppose that we use the Marshallian demand function instead
The welfare loss from a price increase from px = $1 to px = $4 is given by

81. Welfare Loss from a Price Increase
If income (I) is equal to 8,
Loss = 4 ln(4) - 4 ln(1) = 4 ln(4) = 4(1.39) = 5.55
this computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functions

82. Revealed Preference Hypothesis
The theory of revealed preference was proposed by Paul Samuelson in the year 1938.
Assumptions:
1) Rationality
2) Consistency
3) Transitivity
4) Revealed Preference axiom

83. 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
Under any other price-income arrangement, B can never be revealed preferred to A

84. 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.

85. Revealed Preference Theory
Preferences  predict consumer’s purchasing behavior
Purchasing behavior  infer consumer’s preferences

86. 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