Chapter 5 INCOME AND SUBSTITUTION EFFECTS Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC.

Chapter 5

INCOME AND SUBSTITUTION EFFECTS

Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

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:

X1* = d1(P1,P2,…,Pn,I)

X2* = d2(P1,P2,…,Pn,I)•••

Xn* = dn(P1,P2,…,Pn,I)

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

Xi* = di(P1,P2,…,Pn,I) = di(tP1,tP2,…,tPn,tI)

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

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

I30.*

XPY

I70.*

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

Changes in Income

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

• Since PX/PY does not change, the MRS will stay constant as the worker moves to higher levels of satisfaction

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

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

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

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


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


U1

U2

Quantity of X

Quantity of Y

A

B

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

C

Substitution effect

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


U1

U2

Quantity of X

Quantity of Y

A

B

The income effect occurs because theindividual’s “real” income changes whenthe price of good X changes

C

Income effect

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


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

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 QD

– When price rises, both effects lead to a drop in QD

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 QD, but the income effect leads to a rise in QD

– When price falls, the substitution effect leads to a rise in QD, but the income effect leads to a fall in QD

Giffen’s Paradox• If the income effect of a price change is

strong enough, there could be a positive relationship between price and QD

– An increase in price leads to a drop in real income

– Since the good is inferior, a drop in income causes QD to rise

• Thus, a rise in price leads to a rise in QD

Summary of Income & Substitution Effects

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

– 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


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

– 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


• 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

The Individual’s Demand Curve

• An individual’s demand for X1 depends on preferences, all prices, and income:

X1* = d1(P1,P2,…,Pn,I)

• It may be convenient to graph the individual’s demand for X1 assuming

that income and the prices of other goods are held constant


Quantity of Y

Quantity of X Quantity of X

PX

X2

PX2

U2

X2

I = PX2 + PY

X1

PX1

U1

X1

I = PX1 + PY

X3

PX3

X3

U3

I = PX3 + PY

As the price of X falls...

dX

…quantity of Xdemanded rises.


• 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 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


• 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

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* = hX(PX,PY,U)

hX

…quantity demandedrises.


Quantity of Y

Quantity of X Quantity of X

PX

U2

X2

PX2

X2

Y

X

P

Pslope 2

X1

PX1

Y

X

P

Pslope 1

X1 X3

PX3Y

X

P

Pslope 3

X3

Holding utility constant, as price falls...

Compensated & Uncompensated Demand

Quantity of X

PX

dX

hX

X2

PX2

At PX2, the curves intersect becausethe individual’s income is just sufficient to attain utility level U2


Quantity of X

PX

dX

hX

PX2

X1*X1

PX1

At prices above PX2, income compensation is positive because the individual needs some help to remain on U2


Quantity of X

PX

dX

hX

PX2

X3* X3

PX3

At prices below PX2, income compensation is negative to prevent an increase in utility from a lower price


• 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 = U(X,Y) = X0.5Y0.5

• The Marshallian demand functions are

X = I/2PX Y = I/2PY

• The indirect utility function is

50502 ..),,( utility

YXYX PP

PPVI

I


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

50

50

.

.

X

Y

P

VPX

50

50

.

.

Y

X

P

VPY


• Demand now depends on utility rather than income

• Increases in PX reduce the amount of X demanded

– only a substitution effect

50

50

.

.

X

Y

P

VPX

50

50

.

.

Y

X

P

VPY

A Mathematical Examination of a Change in Price

• Our goal is to examine how the demand for good X changes when PX changes

dX/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


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

minimum expenditure = E(PX,PY,U)

• Then, by definition

hX (PX,PY,U) = dX [PX,PY,E(PX,PY,U)]

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


• We can differentiate the compensated demand function and get

hX (PX,PY,U) = dX [PX,PY,E(PX,PY,U)]

X

X

X

X

X

X

P

E

E

d

P

d

P

h

X

X

X

X

X

X

P

E

E

d

P

h

P

d


• The first term is the slope of the compensated demand curve

• This is the mathematical representation of the substitution effect

X

X

X

X

X

X

P

E

E

d

P

h

P

d


• The second term measures the way in which changes in PX affect the demand for X through changes in necessary expenditure levels

• This is the mathematical representation of the income effect

X

X

X

X

X

X

P

E

E

d

P

h

P

d

The Slutsky Equation• The substitution effect can be written as

constant

effect onsubstituti

UXX

X

P

X

P

h

• The income effect can be written as

XX

X

P

E

I

X

P

E

E

d

effect income

The Slutsky Equation

• 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

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

d

P

d

UXX

X

X

X

constant

effect income effect onsubstituti


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

diminishing– the slope of the compensated demand curve

will always be negative

I

XX

P

X

P

d

UXX

X

constant


• 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

d

UXX

X

constant

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


• 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


B

A

I1

I2

I3

Quantity of X

Quantity of Y

Suppose that, when the budget constraint isgiven by I1, A is chosen

A must still be preferred to B when incomeis I3 (because both A and B are available)

If B is chosen, the budget constraint must be similar to that given by I2 where A is not available

Negativity of the Substitution Effect

• Suppose that an individual is indifferent between two bundles: C and D

• Let PXC,PY

C be the prices at which

bundle C is chosen

• Let PXD,PY

D be the prices at which

bundle D is chosen


• Since the individual is indifferent between C and D– When C is chosen, D must cost at least as

much as C

PXCXC + PY

CYC ≤ PXDXD + PY

DYD

– When D is chosen, C must cost at least as much as D

PXDXD + PY

DYD ≤ PXCXC + PY

CYC


• Rearranging, we get

PXC(XC - XD) + PY

C(YC -YD) ≤ 0

PXD(XD - XC) + PY

D(YD -YC) ≤ 0

• Adding these together, we get

(PXC – PX

D)(XC - XD) + (PYC – PY

D)(YC - YD) ≤ 0


• Suppose that only the price of X changes (PY

C = PYD)

(PXC – PX

D)(XC - XD) ≤ 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 Pi

0 bundle Xi0 is chosen instead of

bundle Xi1 (and bundle Xi

1 is affordable), then

n

i

n

iiiii XPXP

1 1

1000

• Bundle 0 has been “revealed preferred” to bundle 1

Mathematical Generalization

• Consequently, at prices that prevail when bundle 1 is chosen (Pi

1), then

n

i

n

iiiii XPXP

1 1

1101

• Bundle 0 must be more expensive than bundle 1

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

Consumer Welfare

• The expenditure function shows the minimum expenditure necessary to achieve a desired utility level (given prices)

• The function can be denoted as

expenditure = E(PX,PY,U0)

where U0 is the “target” level of utility

Consumer Welfare

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

0 to PX1) would be

to compare the expenditures required to achieve U0 under these two situations

expenditure at PX0 = E0 = E(PX

0,PY,U0)

expenditure at PX1 = E1 = E(PX

1,PY,U0)

Consumer Welfare

• The loss in welfare would be measured as the increase in expenditures required to achieve U0

welfare loss = E0 – E1

• Because E1 > E0, this change would be negative– the price increase makes the person worse

off

Consumer Welfare• Remember that the derivative of the

expenditure function with respect to PX is the compensated demand function (hX)

),,(),,(

00 UPPh

dP

UPPdEYXX

X

YX

• The change in necessary expenditures brought about by a change in PX is given by the quantity of X demanded

Consumer Welfare• To evaluate the change in expenditure

caused by a price change (from PX0 to

PX1), we must integrate the compensated

demand function

1

0

1

0

0

X

X

X

X

P

P

P

P

XYXx dPUPPhdE ),,(

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

0 and PX

1

welfare loss

Consumer Welfare

Quantity of X

PX

hX

PX1

X1

PX0

X0

When the price rises from PX0 to PX

1,the consumer suffers a loss in welfare

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 (U0) or the new level of utility after the price change (U1)?

Consumer Welfare

Quantity of X

PX

hX(U0)

PX1

X1

When the price rises from PX0 to PX

1, the actual market reaction will be to move from A to C

hX(U1)

dX

A

C

PX0

X0

The consumer’s utility falls from U0 to U1

Consumer Welfare

Quantity of X

PX

hX(U0)

PX1

X1

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

1BAPX0 [using hX(U0)]

or by area PX1CDPX

0 [using hX(U1)]?

hX(U1)

dX

A

BC

DPX

0

X0

Is U0 or U1 the appropriate utility target?

Consumer Welfare

Quantity of X

PX

hX(U0)

PX1

X1

We can use the Marshallian demand curve as a compromise.

hX(U1)

dX

A

BC

DPX

0

X0

The area PX1CAPX

0 falls between the sizes of the welfare losses defined by hX(U0) and hX(U1)

Loss of Consumer Welfare from a Rise in Price

• Suppose that the compensated demand function for X is given by

50

50

.

.

),,(X

YYXX P

VPVPPhX

the welfare loss from a price increase from PX = 0.25 to PX = 1 is given by

1

250

50501

25050

50

2

X

X

P

PXYX

XY PVPP

dPVP.

..

..

.


• If we assume that the initial utility level (V) is equal to 2,

loss = 4(1)0.5 – 4(0.25)0.5 = 2• If we assume that the utility level (V)

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


• Suppose that we use the Marshallian demand function instead

XYXX P

PPdX2

I ),,( I

the welfare loss from a price increase from PX = 0.25 to PX = 1 is given by

1

250

1

250 22

X

X

P

P

XX

X

PdP

P ..

lnI

I


• Because income (I) 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 Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC.

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