1

Chapter 5

INCOME AND SUBSTITUTION EFFECTS

2

Objectives• How will changes in prices and income

influence influence consumer’s optimal choices?– We will look at partial derivatives

3

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:

yx ppx

25.0*

I

yx ppy

4*

I

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 x

Quantity of y

C

U3

B

U2

A

U1

As income rises, the individual choosesto consume more x and y

9

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

10

Inferior good• If x decreases as income rises, x is an

inferior good

Quantity of x

Quantity of y

C

U3

As income rises, the individual choosesto consume less x and more y

B

U2

AU1

11

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

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 individual’s 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: px1< px

0

2) MRS(x0,y0)= px0/ py

0 & MRS(x1,y1)= px1/ py

0

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

14

Changes in the optimal choice when a price decreases

Quantity of x

Quantity of y

U1

A

Suppose the consumer is maximizing utility at point A.

U2

B

If the price of good x falls, the consumer will maximize utility at point B.

Total increase in x

15

Substitution effect when a price decreases

U1

Quantity of x

Quantity of y

A

To isolate the substitution effect, we holdutility constant but allow the relative price of good x to change. Purple is parallel to the new one

Substitution effect

C

The substitution effect is the movementfrom point A to point C

The individual substitutes good x for good y because it is now relatively cheaper

16

Income effect when the price decreases

U1

U2

Quantity of x

Quantity of y

A

The income effect occurs because theindividual’s “real” income changes (hence utility changes) whenthe price of good x changes

C

Income effect

B

The income effect is the movementfrom point C to point B

If x is a normal good,the individual will buy more because “real”income increased

How would the graph change if the good was inferior?

17

Subs and income effects when a price increases

U2

U1

Quantity of x

Quantity of y

B

A

An increase in the price of good x means thatthe budget constraint gets steeper

CThe substitution effect is the movement from point A to point C

Substitution effect

Income effect

The income effect is the movement from point C to point B

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

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

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 Giffen’s 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 Shephard’s 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

• Sheppard’s Lema to obtain the compensated demand function

y

yx

x

yx

dp

uppEy

dp

uppEx

),,(

),,(

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

utility and priceson dependsit because

function demand dcompensate theis,that

),,()),,(,,(

... gSubstituin

function eexpenditur ),,,(

function demand ,),,(

UppxUppEppxx

UppEI

Ippxx

yxyxyx

yx

yx

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

IU2 5.05.0 yx ppE

29

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

5.0

5.0

x

y

p

Upx 5.0

5.0

y

x

p

Upy

Another trick to obtain compensated demand functions

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

5.0

5.0

x

y

p

Vpx 5.0

5.0

y

x

p

Vpy

31

Roy’s identity

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

( , , ) ; ( , , ) yxx y x y

VVdpdp

x p p I y p p IV VdI dI

Proof of the Roy’s identity…

32

Proof of Roy’s identity

x

( , , )

( , , ( , , ))

Taking derivatives wrt p :

'(.) '(.) ' 0

Using previous trick:

'

Substituting:

'(.) '(.) 0

and solving for x, we find the Roy's identity

x x

x

x

x y

x y x y

p I p

px

p I

V p p I u

V p p E p p u u

V V E

EE x

dp

V V x

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 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

35

x

…quantity of xdemanded rises.

The Marshallian Demand Curve

Quantity of y

Quantity of x Quantity of x

px

x’’

px’’

U2

x2

I = px’’ + py

x’

px’

U1

x1

I = px’ + py

x’’’

px’’’

x3

U3

I = px’’’ + py

As the price of x falls...

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 individual’s 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

xc

…quantity demandedrises.

Compensated Demand Curves

Quantity of y

Quantity of x Quantity of x

px

U2

x’’

px’’

x’’

y

x

p

pslope

''

x’

px’

y

x

p

pslope

'

x’ x’’’

px’’’y

x

p

pslope

'''

x’’’

Holding utility constant, as price falls...

43

Compensated & Uncompensated Demand for normal goods

Quantity of x

px

x

xc

x’’

px’’

At px’’, the curves intersect becausethe individual’s income is just sufficient to attain utility level U2

44

Compensated & Uncompensated Demand for normal goods

Quantity of x

px

x

xc

px’’

x*x’

px’

At prices above p’’x, income compensation is positive because the individual needs some help to remain on U2

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

Quantity of x

px

x

xc

px’’

x*** x’’’

px’’’

At prices below px2, income compensation is negative to prevent an increase in utility from a lower price

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

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

• 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)

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

• We can differentiate the compensated demand function and get

xc (px,py,U) = x[px,py,E(px,py,U)]

xxx

c

p

E

E

x

p

x

p

x

xx

c

x p

E

E

x

p

x

p

x

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

xx

c

x p

E

E

x

p

x

p

x

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

xx

c

x p

E

E

x

p

x

p

x

53

The Slutsky Equation

• The substitution effect can be written as

constant

effect onsubstituti

Uxx

c

p

x

p

x

• The income effect can be written as

xx

p

Ex

xp

E

p

Ex

p

E

E

x

x

x

xx

II

I

effect income

:A trick Using

effect income

54

The Slutsky Equation• A price change can be represented by

I

xx

p

x

p

x

p

x

Uxx

x

constant

effect income effect onsubstituti

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

I

xx

p

x

p

x

Uxx constant

56

The Slutsky Equation

• 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

I

xx

p

x

p

x

Uxx constant

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

xyx p

ppxI

I5.0

),,(

58

A Slutsky Decomposition

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

5.0

5.0

),,(x

yyx

c

p

VpVppx

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

2

5.0

xx pp

x I

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

5.1

5.05.0 effect onsubstituti

x

y

x

c

p

Vp

p

x

60

A Slutsky Decomposition

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

25.1

5.05.05.025.0)5.0(5.0

effect onsubstitutixx

yyx

pp

ppp II

61

A Slutsky Decomposition

• Calculation of the income effect is easier

2

25.05.05.0 effect income

xxx ppp

xx

III

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