chapter 5 income and substitution effects 1 references: snyder and nicholson, 10 th ed

CHAPTER 5

INCOME AND SUBSTITUTION EFFECTS

1

References: Snyder and Nicholson, 10th ed.

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

Homogeneity

• If we were to double all prices and income, the optimal quantities demanded will not change– the budget constraint is unchanged

xi* = di(p1,p2,…,pn,I) = di(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

• Note that a doubling of both prices and income would leave x* and y* unaffected

xpx

I3.0*

ypy

I7.0*

6

Homogeneity• With a CES utility function

utility = U(x,y) = x0.5 + y0.5

the demand functions are

• Note that a doubling of both prices and income would leave x* and y* unaffected

xyx pppx

I

/1

1*

yxy pppy

I

/1

1*

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

Increase in Income• 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

Increase in Income• 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

Note that the indifferencecurves do not have to be “oddly” shaped. Theassumption of a diminishing MRS is obeyed.

B

U2

AU1

10

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

12

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

13

Changes in a Good’s Price• 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

14

Changes in a Good’s Price

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


U1

Quantity of x

Quantity of y

A

To isolate the substitution effect, we hold“real” income constant but allow the relative price of good x to change

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


U1

U2

Quantity of x

Quantity of y

A

The income effect occurs because theindividual’s “real” income 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

17


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

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

22

A Summary• Utility maximization implies that (for normal

goods) a rise in price leads to a decline in quantity demanded– the substitution effect causes less to be

purchased as the individual moves along an indifference curve

– the income effect causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve

23

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

24

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 (py) are held

constant

25

x

…quantity of xdemanded rises.

The Individual’s 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...

26

The Individual’s Demand Curve

• An individual 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

27

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

28

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

29

Demand Functions and Curves

• If the individual’s income is $100, these functions become

xpx

I3.0*

ypy

I7.0*

• We discovered earlier that

xpx

30*

ypy

70*

30

Demand Functions and Curves

• Any change in income will shift these demand curves

31

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 and Uncompensated Demand Function

• A compensated demand function = the consumer’s income is adjusted as the price changes, so that the consumer’s utility remains at the same level

• Uncompensated demand function = there is a change in real income, and it is not uncompensated

32

33

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 constrain the individual to remain on the same indifference curve

– reactions to price changes include only substitution effects

34

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)

35

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...

36

Compensated & Uncompensated Demand

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

37


Quantity of x

px

x

xc

px’’

x*x’

px’

At prices above px2, income compensation is positive because the individual needs some help to remain on U2

38


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

39


• 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

40

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

5.05.02),,( utility

yxyx pp

ppVI

I

41


• To obtain the compensated demand functions, we can solve the indirect utility function for I and then substitute into the Marshallian demand functions

5.0

5.0

x

y

p

Vpx 5.0

5.0

y

x

p

Vpy

42


• Demand now depends on utility (V) rather than income

• Increases in px reduce the amount of x demanded– only a substitution effect

5.0

5.0

x

y

p

Vpx 5.0

5.0

y

x

p

Vpy

43

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

• However, this approach is cumbersome and provides little economic insight

44


• Instead, 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

45


• 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

46


• 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

47


• 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

48

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

p

E

E

x

I

effect income

49


• Note that E/px = x

– a $1 increase in px raises necessary expenditures by x dollars

– $1 extra must be paid for each unit of x purchased

50

The Slutsky Equation• The utility-maximization hypothesis

shows that the substitution and income effects arising from a price change can be represented by

I

xx

p

x

p

x

p

x

Uxx

x

constant

effect income effect onsubstituti

51


• 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

52


• 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

53

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

),,(

54


• 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

55


• 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

56


• 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

57


• Calculation of the income effect is easier

2

25.05.05.0 effect income

xxx ppp

xx

III

• Interestingly, the substitution and income effects are exactly the same size

58

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)

x

p

p

x

pp

xxe x

xxxpx x

/

/,

59

Marshallian Demand Elasticities

• Income elasticity of demand (ex,I)

x

xxxex

IIIII

/

/,

• Cross-price elasticity of demand (ex,py)

x

p

p

x

pp

xxe y

yyypx y

/

/,

60

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

61

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

]1[)(

,

xpx

xx

x

x exxp

xp

p

xp

62

Price Elasticity and Total Spending

• The sign of this derivative depends on whether ex,px is greater or less than -1

– if ex,px > -1, demand is inelastic and price and total spending move in the same direction

– if ex,px < -1, demand is elastic and price and total spending move in opposite directions

]1[)(

,

xpx

xx

x

x exxp

xp

p

xp

63

Compensated Price Elasticities

• It is also useful to define elasticities based on the compensated demand function

64


• 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 (ex

c,py)

65

Compensated Price Elasticities• The compensated own price elasticity of

demand (exc,px) is

cx

x

c

xx

ccc

px x

p

p

x

pp

xxe

x

/

/,

• The compensated cross-price elasticity of demand (ex

c,py) is

c

y

y

c

yy

ccc

px x

p

p

x

pp

xxe

y

/

/,

66

Compensated Price Elasticities• The relationship between Marshallian

and compensated price elasticities can be shown using the Slutsky equation

I

x

xx

p

p

x

x

pe

p

x

x

p x

x

c

cx

pxx

xx,

I,,, xxc

pxpx eseexx

• If sx = pxx/I, then

67


• 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

68

Demand Elasticities• The Cobb-Douglas utility function is

U(x,y) = xy (+=1)

• The demand functions for x and y are

xpx

I

ypy

I

69

Demand Elasticities• Calculating the elasticities, we get

1 2,

x

x

x

x

xpx

p

p

px

p

p

xe

x I

I

00 ,

x

p

x

p

p

xe yy

ypx y

1 ,

x

xx

p

px

xe

I

IIII

70

Demand Elasticities• We can also show

– homogeneity

0101,,, Ixpxpx eeeyx

– Engel aggregation

111,, II yyxx eses

– Cournot aggregation

xpyypxx sesesxx

0)1(,,

71

Demand Elasticities• We can also use the Slutsky equation to

derive the compensated price elasticity

1)1(1,,, Ixxpxc

px eseexx

• The compensated price elasticity depends on how important other goods (y) are in the utility function

72

Demand Elasticities• The CES utility function (with = 2,

= 5) isU(x,y) = x0.5 + y0.5

• The demand functions for x and y are

)1( 1

yxx pppx

I)1( 1

yxy pppy

I

73

Demand Elasticities• We will use the “share elasticity” to

derive the own price elasticity

xxx pxx

x

x

xps e

s

p

p

se ,, 1

• In this case,

11

1

yx

xx pp

xps

I

74

Demand Elasticities• Thus, the share elasticity is given by

1

1

1121

1

, 1)1()1(

yx

yx

yx

x

yx

y

x

x

x

xps pp

pp

pp

p

pp

p

s

p

p

se

xx

• Therefore, if we let px = py

5.1111

11,,

xxx pspx ee

75

Demand Elasticities• The CES utility function (with = 0.5,

= -1) isU(x,y) = -x -1 - y -1

• The share of good x is

5.05.01

1

xy

xx pp

xps

I

76

Demand Elasticities• Thus, the share elasticity is given by

5.05.0

5.05.0

15.05.025.05.0

5.15.0

,

1

5.0

)1()1(

5.0

xy

xy

xy

x

xy

xy

x

x

x

xps

pp

pp

pp

p

pp

pp

s

p

p

se

xx

• Again, if we let px = py

75.012

5.01,,

xxx pspx ee

chapter 5 income and substitution effects 1 references: snyder and nicholson, 10 th ed

Documents

good y

income effects

range of income

goods x

yas income

yathe income effect

incomeif x

y increase