INCOME AND SUBSTITUTION EFFECTS

1
Chapter 5

• INCOME AND SUBSTITUTION EFFECTS

2
Objectives

• How will changes in prices and income influence
  influence consumers optimal choices?
• We will look at partial derivatives

3
Demand Functions (review)

• We have already seen how to obtain consumers
  optimal choice
• Consumers optimal choice was computed Max
  consumers 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

4
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

5
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)

6
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)

7
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

8
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
Quantity of x
9
What is an inferior Good?

• A good xi for which ?xi/?I lt 0 over some range of
  income is an inferior good in that range

10
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
Quantity of x
11
Changes in a Goods 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
  consumers utility-maximizing choices
• When a price changes, we can decompose consumers
  reaction in two effects
• substitution effect
• income effect

12
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 individuals real
  income and therefore he must move to a new
  indifference curve
• the income effect

13
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 px1lt px0
• 2) MRS(x0,y0) px0/ py0 MRS(x1,y1) px1/ py0
• 1 and 2 implies that
• MRS(x1,y1)ltMRS(x0,y0)
• As the MRS is decreasing in x, this means that x
  has increased, that is x1gtx0

14
Changes in the optimal choice when a price
decreases
Quantity of y
Quantity of x
15
Substitution effect when a price decreases
Quantity of y
The individual substitutes good x for good y
because it is now relatively cheaper
A
U1
Quantity of x
16
Income effect when the price decreases
The income effect occurs because the individuals
real income changes (hence utility changes)
when the price of good x changes
Quantity of y
If x is a normal good, the individual will buy
more because real income increased
A
C
U2
U1
Quantity of x
How would the graph change if the good was
inferior?
17
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
A
B
U1
U2
Quantity of x
How would the graph change if the good was
inferior?
18
Price Changes forNormal 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

19
Price Changes forInferior 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

20
Giffens 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

21
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

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 Giffens paradox

23
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)

24
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 !!!!

25
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 Shephards Lemma) is using
  the derivative of the expenditure function
• Another trick(B) is to use the marshallian demand
  and the expenditure function

26
Compensated Demand Functions

• Sheppards Lema to obtain the compensated demand
  function

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
27
Trick (B) to obtain compensated demand functions
28
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

29
Another trick to obtain compensated demand
functions

• Substitute the expenditure function into the
  Marshallian demand functions, and find the
  compensated ones

30
Compensated Demand Functions

• 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

31
Roys identity

• It is the relation between marshallian demand
  function and indirect utility function

Proof of the Roys identity
32
Proof of Roys identity
33
Demand curves

• We will start to talk about demand curves. Notice
  that they are not the same that demand functions
  !!!!

34
The Marshallian Demand Curve

• An individuals demand for x depends on
  preferences, all prices, and income
• x x(px,py,I)
• It may be convenient to graph the individuals
  demand for x assuming that income and the price
  of y (py) are held constant

35
The Marshallian Demand Curve
Quantity of y
As the price of x falls...
px
Quantity of x
Quantity of x
36
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!!!

37
Shifts in the Demand Curve

• Three factors are held constant when a demand
  curve is derived
• income
• prices of other goods (py)
• the individuals preferences
• If any of these factors change, the demand curve
  will shift to a new position

38
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

39
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)

40
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

41
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)

42
Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y
px
Quantity of x
Quantity of x
43
Compensated Uncompensated Demand for normal
goods
px
x
xc
Quantity of x
44
Compensated Uncompensated Demand for normal
goods
px
px
x
xc
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 of both curves
45
Compensated Uncompensated Demand for normal
goods
px
px
x
xc
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 of both curves
Quantity of x
46
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

47
Relations to keep in mind

• Sheppards Lema Roys 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)

48
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

49
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

50
A Mathematical Examination of a Change in Price
xc (px,py,U) xpx,py,E(px,py,U)

• We can differentiate the compensated demand
  function and get

51
A Mathematical Examination of a Change in Price

• The first term is the slope of the compensated
  demand curve
• the mathematical representation of the
  substitution effect

52
A Mathematical Examination of 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

53
The Slutsky Equation

• The substitution effect can be written as

• The income effect can be written as

54
The Slutsky Equation

• A price change can be represented by

55
The Slutsky Equation

• 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

56
The Slutsky Equation

• The second term is the income effect
• if x is a normal good, then ?x/?I gt 0
• the entire income effect is negative
• if x is an inferior good, then ?x/?I lt 0
• the entire income effect is positive

57
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

58
A Slutsky Decomposition

• The Hicksian (compensated) demand function for
  good x was

• The overall effect of a price change on the
  demand for x is

59
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

60
A Slutsky Decomposition

• We can substitute in for the indirect utility
  function (V)

61
A Slutsky Decomposition

• Calculation of the income effect is easier

• By adding up substitution and income effect, we
  will obtain the overall effect