Chapter 5 INCOME AND SUBSTITUTION EFFECTS MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning.

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Transcript Chapter 5 INCOME AND SUBSTITUTION EFFECTS MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning.

Chapter 5

INCOME AND SUBSTITUTION EFFECTS

MICROECONOMIC THEORY

BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION

WALTER NICHOLSON

Demand Functions

• The optimal levels of

1 ,

2 ,…,

n can be expressed as functions of all prices and income • These can be expressed as n demand functions:

1 * =

1 (

1 ,

2 ,…,

n ,

)

2 * =

2 (

1 ,

2 ,…,

n ,

) • • •

n * =

n (

1 ,

2 ,…,

n ,

)

Homogeneity

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

i * =

i (

1 ,

2 ,…,

n ,

) =

i (

1 ,

2 ,…,

n ,

) • Individual demand functions are homogeneous of degree zero in all prices and income

Homogeneity

• With a Cobb-Douglas utility function utility =

(

) =

0.3

0.7

the demand functions are

*  0 .

I P X Y

*  0 .

I P X

• Note that a doubling of both prices and income would leave

* and

* unaffected

Homogeneity

• With a CES utility function utility =

(

) =

0.5

the demand functions are

*  1 1 

P X

P Y



I P X Y

*  1 

P Y

1 /

P X



I P Y

• Note that a doubling of both prices and income would leave

* and

* unaffected

Changes in Income

• An increase in income will cause the budget constraint out in a parallel manner • Since

P X

P Y

does not change, the

MRS

will stay constant as the worker moves to higher levels of satisfaction

Increase in Income

• If both

rises,

and and

Y Y

increase as income are normal goods Quantity of

As income rises, the individual chooses to consume more

and

A B C

U 1 U 2 U 3

Quantity of

Increase in Income

• If

decreases as income rises, inferior good

is an As income rises, the individual chooses to consume less

and more

Quantity of

C B

U 3

Note that the indifference curves do not have to be “oddly” shaped. The assumption of a diminishing

MRS

is obeyed.

U 2

U 1

Quantity of

Normal and Inferior Goods

• A good

i for which 

i / 

 0 over some range of income is a normal good in that range • A good

i for which 

i / 

< 0 over some range of income is an inferior good in that range

Engel’s Law

• Using Belgian data from 1857, Engel found an empirical generalization about consumer behavior • The proportion of total expenditure devoted to food declines as income rises – food is a necessity whose consumption rises less rapidly than income

Substitution & 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

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

Changes in a Good’s Price

Quantity of

Suppose the consumer is maximizing utility at point

If the price of good

falls, the consumer will maximize utility at point

U 1 U 2 Quantity of

Total increase in

Changes in a Good’s Price

Quantity of

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

to change

A C B

The substitution effect is the movement from point

to point

U 1 U 2 The individual substitutes good

for good

because it is now relatively cheaper Quantity of

Substitution effect

Changes in a Good’s Price

Quantity of

The income effect occurs because the individual’s “real” income changes when the price of good

changes The income effect is the movement from point

to point

B B A C

U 1 U 2 If

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

Income effect

Changes in a Good’s Price

Quantity of

An increase in the price of good

means that the budget constraint gets steeper

The substitution effect is the movement from point

to point

C A B

U 1 The income effect is the movement from point

to point

U 2 Quantity of

Substitution effect Income effect

Price Changes for Normal Goods

• If a good is normal, substitution and income effects reinforce one another – When price falls, both effects lead to a rise in Q D – When price rises, both effects lead to a drop in Q D

Price Changes for Inferior 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 Q D , but the income effect leads to a rise in Q D – When price falls, the substitution effect leads to a rise in Q D , but the income effect leads to a fall in Q D

Giffen’s Paradox

• If the income effect of a price change is strong enough, there could be a positive relationship between price and Q D – An increase in price leads to a drop in real income – Since the good is inferior, a drop in income causes Q D to rise • Thus, a rise in price leads to a rise in Q D

Summary of Income & Substitution Effects

• Utility maximization implies that (for normal goods) a fall in price leads to an increase in Q D – 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

Summary of Income & Substitution Effects

• Utility maximization implies that (for normal goods) a rise in price leads to a decline in Q D – 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

Summary of Income & Substitution Effects

• Utility maximization implies that (for inferior goods) no definite prediction can be made for changes in price – The

substitution effect

in opposite directions and

income effect

move – If the income effect outweighs the substitution effect, we have a case of

Giffen’s paradox

The Individual’s Demand Curve

• An individual’s demand for

1 depends on preferences, all prices, and income:

1 * =

1 (

1 ,

2 ,…,

n ,

) • It may be convenient to graph the individual’s demand for

1 assuming that income and the prices of other goods are held constant

The Individual’s Demand Curve

Quantity of

As the price of

falls...

P X

…quantity of

demanded rises.

U 3 U 1 U 2 X 1 I = P X1 + P

X 2 X 3 I = P X2 + P

Quantity of

I = P X3 + P

Y P X1 P X2 P X3

X 1 X 2 X 3

d X

Quantity of

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

Shifts in the Demand Curve

• Three factors are held constant when a demand curve is derived – income – prices of other goods – the individual’s preferences • If any of these factors change, the demand curve will shift to a new position

Shifts in the Demand Curve

• A movement along a given demand curve is caused by a change in the price of the good – called a

change in quantity demanded

• A shift in the demand curve is caused by a change in income, prices of other goods, or preferences – called a

change in demand

Compensated Demand Curves

• The actual level of utility varies along the demand curve • As the price of

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

falls

Compensated Demand Curves

• An alternative approach holds real income (or utility) constant while examining reactions to changes in

P X

– 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

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

* =

h X

(

P X

P Y

)

Compensated Demand Curves

Holding utility constant, as price falls...

Quantity of

Y P X slope

 

P X

P Y

…quantity demanded rises.

slope

 

P X

P Y

P X1

slope

 

P X

P Y

P X2 P X3 h X

U 2 X 1 X 2 X 3

Quantity of

X 1 X 2 X 3

Quantity of

Compensated & Uncompensated Demand

P X

2 , the curves intersect because the individual’s income is just sufficient to attain utility level

d X h X

Quantity of

Compensated & Uncompensated Demand

P X

At prices above

P X

2 , income compensation is positive because the individual needs some help to remain on

d X h X

1 *

Quantity of

Compensated & Uncompensated Demand

P X

At prices below

P X

2 , income compensation is negative to prevent an increase in utility from a lower price

d X h X

3 *

Quantity of

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

Compensated Demand Functions

• Suppose that utility is given by utility =

(

) =

0.5

• The Marshallian demand functions are

P X Y

P Y

• The indirect utility function is utility 

(

P X

P Y

) 

P X

0 .

P Y

0 .

Compensated Demand Functions

• To obtain the compensated demand functions, we can solve the indirect utility function for

and then substitute into the Marshallian demand functions



VP Y

0 .

P X

0 .



VP X

0 .

P Y

0 .

Compensated Demand Functions



VP Y

0 .

P X

0 .



VP X

0 .

P Y

0 .

5 • Demand now depends on utility rather than income • Increases in

P X

demanded reduce the amount of

– only a substitution effect

A Mathematical Examination of a Change in Price

• Our goal is to examine how the demand for good

changes when

P X

changes 

d X

/ 

P X

• 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

A Mathematical Examination of a Change in Price

• Instead, we will use an indirect approach • Remember the expenditure function minimum expenditure =

(

P X

P Y

) • Then, by definition

h X

(

P X

P Y

) =

d X

[

P X

P Y

(

P X

P Y

)] – Note that the two demand functions are equal when income is exactly what is needed to attain the required utility level

A Mathematical Examination of a Change in Price

h X

(

P X

P Y

) =

d X

[

P X

P Y

(

P X

P Y

)] • We can differentiate the compensated demand function and get 

h X



P X



P X X

 



P X X

 



E X

 



P X

 

h X



P X

 



E X

 



P X

A Mathematical Examination of a Change in Price



P X X

 

h X



P X

 



E X

 



P X

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

A Mathematical Examination of a Change in Price



P X X

 

h X



P X

 



E X

 



P X

• The second term measures the way in which changes in

P X

for

affect the demand through changes in necessary expenditure levels • This is the mathematical representation of the income effect

The Slutsky Equation

• The substitution effect can be written as substituti on effect  

h X



P X

 



P X U

 constant • The income effect can be written as income effect   



E X

 



P X

  



 



P X

The Slutsky Equation

• Note that 

/ 

P X

– A $1 increase in

P X

expenditures by

raises necessary dollars – $1 extra must be paid for each unit of

purchased

The Slutsky Equation

• The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by 



P X X



P X X

 substituti on effect  income effect  



P X U

 constant 



The Slutsky Equation



P X X

 



P X U

 constant 



• The first term is the substitution effect – always negative as long as

MRS

diminishing is – the slope of the compensated demand curve will always be negative

The Slutsky Equation



P X X

 



P X U

 constant 



• The second term is the income effect – if

is a normal good, then 

/ 

> 0 – if • the entire income effect is negative

is an inferior good, then 

/ 

• the entire income effect is positive < 0

Revealed Preference & the Substitution Effect

• The theory of revealed preference was proposed by Paul Samuelson in the late 1940s • The theory defines a principle of rationality based on observed behavior and then uses it to approximate an individual’s utility function

Revealed Preference & the Substitution Effect

• Consider two bundles of goods:

and

• If the individual can afford to purchase either bundle but chooses

, we say that

had been revealed preferred to

• Under any other price-income arrangement,

can never be revealed preferred to

Revealed Preference & the Substitution Effect

Quantity of

Suppose that, when the budget constraint is given by

1 ,

is chosen

A B A

must still be preferred to

3 (because both

and

when income are available)

I 3 I 1 I 2

is chosen, the budget constraint must be similar to that given by

2 where

is not available Quantity of

Negativity of the Substitution Effect

• Suppose that an individual is indifferent between two bundles:

and

• Let

P X

P Y

be the prices at which bundle

is chosen • Let

P X

P Y

be the prices at which bundle

is chosen

Negativity of the Substitution Effect

• Since the individual is indifferent between

and

– When

is chosen,

much as

must cost at least as

P X

– When

P Y

is chosen,

C C

≤

P X

P Y

must cost at least as much as

P X

P Y

≤

P X

P Y

Negativity of the Substitution Effect

• Rearranging, we get

P X

) +

P Y

) ≤ 0

P X

) +

P Y

) ≤ 0 • Adding these together, we get (

P X

–

P X

)(X

) + (

P Y

–

P Y

)(Y

) ≤ 0

Negativity of the Substitution Effect

• Suppose that only the price of X changes (

P Y

) (

P X

–

P X

)(X

) ≤ 0 • This implies that price and quantity move in opposite direction when utility is held constant – the substitution effect is negative

Mathematical Generalization

• If, at prices

P i

bundle

instead of bundle

is chosen (and bundle

affordable), then is

i n

  1

P i

X i

0 

i n

  1

P i

X i

1 • Bundle

has been “revealed preferred” to bundle

Mathematical Generalization

• Consequently, at prices that prevail when bundle

is chosen (

P i

), then

i n

  1

P i

X i

0 

i n

  1

P i

X i

1 • Bundle

bundle

must be more expensive than

Strong Axiom of Revealed Preference

• If commodity bundle

is revealed preferred to bundle

, and if bundle

is revealed preferred to bundle

, and if bundle

is revealed preferred to bundle

,…,and if bundle

k-1

is revealed preferred to bundle

, then bundle

cannot be revealed preferred to bundle

Consumer Welfare

• The expenditure function shows the minimum expenditure necessary to achieve a desired utility level (given prices) • The function can be denoted as where expenditure =

(

P X

P Y

0 )

0 is the “target” level of utility

Consumer Welfare

• One way to evaluate the welfare cost of a price increase (from

P X

) would be to compare the expenditures required to achieve

0 under these two situations expenditure at

P X

0 =

(

P X

P Y

0 ) expenditure at

P X

1 =

(

P X

P Y

0 )

Consumer Welfare

• The loss in welfare would be measured as the increase in expenditures required to achieve

0 welfare loss =

0 • Because

1 negative >

0 –

1 , this change would be – the price increase makes the person worse off

Consumer Welfare

• Remember that the derivative of the expenditure function with respect to

P X

the compensated demand function (

h X

) is

(

P X

P Y

0 ) 

h X

(

P X

P Y

0 )

dP X

• The change in necessary expenditures brought about by a change in

P X

by the quantity of

demanded is given

Consumer Welfare

• To evaluate the change in expenditure caused by a price change (from

P X

), we must integrate the compensated demand function



P X





P X

h x

(

P X

P Y

0 )

dP X

– This integral is the area to the left of the compensated demand curve between

P X

and

P X

P X

0 Consumer Welfare

When the price rises from

P X

, the consumer suffers a loss in welfare

welfare loss

h X

X 1 X 0

Quantity of

Consumer Welfare

• Because a price change generally involves both income and substitution effects, it is unclear which compensated demand curve should be used • Do we use the compensated demand curve for the original target utility (

0 ) or the new level of utility after the price change (

1 )?

P X

P X

0 Consumer Welfare

When the price rises from

P X

, the actual market reaction will be to move from

The consumer’s utility falls from

0 to

C A X 1 X 0

d X h X

(

0 )

h X

(

1 ) Quantity of

P X

P X

0 Consumer Welfare

Is the consumer’s loss in welfare best described by area

P X

BAP X

or by area

P X

CDP X

[using

(

1 )]?

h X

(

0 )]

C B A

0 or

1 the appropriate utility target?

D X 1 X 0

d X h X

(

0 )

h X

(

1 ) Quantity of

P X

P X

0 Consumer Welfare

We can use the Marshallian demand curve as a compromise.

C B

The area

P X

CAP X

falls between the sizes of the welfare losses defined by

h X

(

0 ) and

h X

(

1 )

A D

d X h X

(

0 )

h X

(

1 )

X 1 X 0

Quantity of

Loss of Consumer Welfare from a Rise in Price

• Suppose that the compensated demand function for

is given by



h X

(

P X

P Y

) 

VP Y

0 .

P X

0 .

5 the welfare loss from a price increase from

P X

= 0.25 to

P X

= 1 is given by 1  0 .

VP Y

0 .

dP X P X

0 .

5  2

VP Y

0 .

P X

0 .

P X

 1

P X

 0 .

Loss of Consumer Welfare from a Rise in Price

• If we assume that the initial utility level (

) is equal to 2, loss = 4(1) 0.5

– 4(0.25) 0.5

= 2 • If we assume that the utility level (

) falls to 1 after the price increase (and used this level to calculate welfare loss), loss = 2(1) 0.5

– 2(0.25) 0.5

= 1

Loss of Consumer Welfare from a Rise in Price

• Suppose that we use the Marshallian demand function instead



d X

(

P X

P Y

)  I 2

P X

the welfare loss from a price increase from

P X

= 0.25 to

P X

= 1 is given by  1 0 .

P X dP X



P X

 1

P X

 0 .

Loss of Consumer Welfare from a Rise in Price

• Because income (

) is equal to 2, loss = 0 – (-1.39) = 1.39

• This computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functions

Important Points to Note:

• Proportional changes in all prices and income do not shift the individual’s budget constraint and therefore do not alter the quantities of goods chosen – demand functions are homogeneous of degree zero in all prices and income

Important Points to Note:

• When purchasing power changes (income changes but prices remain the same), budget constraints shift – for normal goods, an increase in income means that more is purchased – for inferior goods, an increase in income means that less is purchased

Important Points to Note:

• A fall in the price of a good causes substitution and income effects – For a normal good, both effects cause more of the good to be purchased – For inferior goods, substitution and income effects work in opposite directions • A rise in the price of a good also causes income and substitution effects – For normal goods, less will be demanded – For inferior goods, the net result is ambiguous

Important Points to Note:

• The Marshallian demand curve summarizes the total quantity of a good demanded at each price – changes in price prompt movemens along the curve – changes in income, prices of other goods, or preferences may cause the demand curve to shift

Important Points to Note:

• Compensated demand curves illustrate movements along a given indifference curve for alternative prices – these are constructed by holding utility constant – they exhibit only the substitution effects from a price change – their slope is unambiguously negative (or zero)

Important Points to Note:

• Income and substitution effects can be analyzed using the Slutsky equation • Income and substitution effects can also be examined using revealed preference • The welfare changes that accompany price changes can sometimes be measured by the changing area under the demand curve

Chapter 5 INCOME AND SUBSTITUTION EFFECTS MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning.

Transcript Chapter 5 INCOME AND SUBSTITUTION EFFECTS MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning.

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