Microeconomics: Income and Substitution Effects

Consumer Behavior (II):

Income and Substitution Effects

Dr. Manuel Salas-Velasco

University of Granada, Spain

1

Consumer Behavior (II)

Introduction

Dr. Manuel Salas-Velasco 2

The Budget Constraint

Quantity of X

Qu

an

tity

of

Y

XP

M

vertical

intercept

horizontal intercept

YP

M

Slope

Y

X

P

P

The equation for

the budget line:

XP

P

P

MY

Y

X

Y

Relative price ratioBudget set

The budget set consists of all bundles that are affordable at the given prices and income


The Consumer’s Utility Maximizing Choice

Quantity of X

Qu

an

tity

of

Y

E

• The consumer’s utility is

maximized at the point (E)

where an indifference

curve is tangent to the

budget line

• The condition for utility

maximization

Y

Y

X

X

P

MU

P

MU

X*

Y*

(X*, Y*) is the utility-maximizing bundle

• The optimum quantities (X*, Y*) obtained by solving the Lagrangean problem tell us how much of each good an individual consumer will demand, assuming that he/she behaves rationally and optimizes his/her utility within his/her budget.



The Consumer’s Reaction to a Change in Income


Shifts in the Budget Line

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

0 1 2 3 4 5

Quantity of ice-cream (week), X

Qu

an

tity

of

lem

on

ad

e (

we

ek

), Y

M’ = 20; PX = 2; PY = 1

M = 10; PX = 2; PY = 1

XP

P

P

MY

Y

X

Y

XY 210

XY 2-20

Prices are held constant and

income increases (e.g. the

consumer’s income doubles)

YP

M

XP

M

XP

M

YP

M

M’ > M


Response to Income Changes

1U

2U

3U

Y

X

Income-Consumption Curve

E1

E2

E3

X, Y, normal goods

Prices are held constant

Income increases: M1 < M2 < M3

• Increases in money income cause a parallel outward shift of the budget line

• The utility-maximizing point moves from E1 to E2

to E3

YX PP ,

XP

M 1

XP

M 2

XP

M 3

YP

M 2

YP

M 3

YP

M 1

• By joining all the utility-maximizing points, an income-consumption line is traced out

*

1X *

2X

*

3Y

*

3X

*

1Y

*

2Y


How Consumption Changes as Income

Changes

M

YEngel Curve

for good Y, with good Y as normal

M1 M2 M3

*

1Y

*

2Y

*

3Y

MPPYY YX ,,


Engel Curve or Engel’s Law

A general reference to the function which shows the relationship between various quantities of a good a consumer is willing to purchase at varying income levels (ceteris paribus)

Ernst Engel(1821-1896)

A German statistician who studied the spending patterns of groups of people of different incomes

People spent a smaller and smaller proportion of their incomes on food as those incomes increased



The Consumer’s Reaction to a Change in Price


Shifts in the Budget Line

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

0 1 2 3 4 5

Quantity of ice-cream (week), X

Qu

an

tity

of

lem

on

ad

e (

we

ek

), Y

XP

P

P

MY

Y

X

Y

M = 10; PX = 2; PY = 1

XY 210

M = 10; P’X = 1; PY = 1 Decrease in the price of X (50%) XY -10

YP

M

XP

M

XP

M


Response to Changes in a Good’s Price

MPY ,

1

XP2

XP

Y

X

Price-Consumption

CurveE1

E2 E3

Decrease in the price of X:

Price of Y and income are held constant:

3

XP> >

YP

M

1

XP

M2

XP

M3

XP

M

1U2U 3U

*

1X *

2X *

3X

*

1Y*

2Y*

3Y


How Consumption Changes as Price Ratio

Changes

Quantity, X

Price of X

Demand Curve for X

*

1X *

2X*

3X

1

XP2

XP3

XP


The Consumer’s Demand Function

Y

Y

X

X

P

MU

P

MU

X

UMU X

Y

UMUY

• We are interested in finding the individual demand curve for the good X; an expression for quantity demanded as a function of all prices and income

• The condition for utility maximization is:

U = U (X, Y)

1YMU X

1 XMUY

YX P

X

P

Y 11

1)1( Y

X

P

PXY

• Let’s suppose that the utility function is: U = X Y + X + Y


The Consumer’s Demand Function

1)1( Y

X

P

PXY

PX X + PY Y = M MP

PXPXP

Y

XYX

1)1(

X = X (PX, PY, M)Consumer’s demand function(generalized demand function)

MPPXXP YXX )1( MPPXPXP YXXX

YXX PPMXP 2

X

YX

P

PPMX

2


The Own-Price Demand

X

YX

P

PPMX

2

),,( MPPXX YX

),,( MPPXX YX

M = $100; PY = $10

Consumer’s demand function

The own-price demand curve(ordinary demand function for X):

X = f (PX), ceteris paribus

X

X

P

PX

2

10100

X

X

P

PX

2

110

Suppose we use the following parametric values:

• However, economists by convention always graph the demand function with price on the vertical axis and quantity demanded on the horizontal axis

The inverse demand function

PX

X

XPX

5.0

55


The Engel Curve

X

YX

P

PPMX

2

),,( MPPXX YX

),,( MPPXX YX

PX = $5; PY = $10


The Engel curve for X

52

105

MX

10

5

MX

2

1

10

MX

X

MelasticityIncome

M

X

If Income Elasticity is positive, then X is a normal good

(quantity demanded increases as income increases, ceteris paribus)


positiveM

X

10

1

positive

elasticityIncome X is a normal

good


The Cross-Price Demand Curve

X

YX

P

PPMX

2

),,( MPPXX YX

),,( MPPXX YX

PX = $5; M = $100



52

5100

YP

X10

95 YPX

105.9 YP

X

Cross-price demand curve

for X

• We hold the own price of good X and money income constant; we focus on the relationship between the quantity demanded of good X and the price of good Y

X

P

Pelasticity price-Cross Y

Y

X If CPE is positive, then X,Y are substitutes

If CPE is negative, then X,Y are complements

)(10

1positive

P

X

Y

positive

elasticityprice-Cross

X is a substitute for Y


Cobb-Douglas Utility Function

Y

Y

X

X

P

MU

P

MU

X

UMU X

Y

UMUY

• The condition for utility maximization is:

U = U (X, Y)

21

21

21

XYMU X

21

21

21

YXMUYYX P

YX

P

XY 21

21

21

21

21

21

PX X + PY Y = M MP

PXPXP

Y

XYX

XP

MX

2MXPX 2

Consumer’s demand function for X

• The utility function is: 21

21

YXU

21

21

21

21

21

21

XY

YX

P

P

X

Y

Y

X

P

P

X

Y Y

X

P

PXY

PX = 4; M = 800; PY = 1 1008

800X

X* = 100 units





The Income Effect and the Substitution Effect

of a Price Change

Quantity, X

Price of X

Own-Price Demand Curve for X(Inverse Ordinary Demand Function for X)

*

1X *

2X*

3X

1

XP2

XP3

XP

• When price of good X falls, the optimal consumption level (or quantity demanded) of good X increases

• What are the underlying reasons for a response in the quantity demanded of good X due to a change in its own price?

• Substitution effect: the impact that a change in the price of a good has on the quantity demanded of that good, which is due to the resulting change in relative prices (PX/PY)

• Income effect: the impact that a change in the price of a good has on the quantity demanded of that good due strictly to the resulting change in real income (or purchasing power)

Total effect



YP

M

1

XP

M2

XP

M

Y

X

Price of Y and monetary income are held

constant: MPY ,

Decrease in the price of X: 1

XP >2

XP

*

1X *

2X

*

1Y*

2Y

1U2U

E1 E2

YP

PX

1

YP

PX

2

TE

SE

total effect (TE) = substitution effect (SE) + income effect (IE)

IE


The Substitution Effect: Two Definitions in

the Literature

Eugene Slutsky1880-1948

Sir John R. Hicks 1904-89

The Slutsky substitution effect

The Hicks substitution effect

The effect on consumer choice of changing the price ratio, leaving his/her initial utility unchanged

The effect on consumer choice of changing the price ratio, leaving the consumer just able to afford

his/her initial bundle


The Slutsky Substitution Effect

YP

M

1

XP

M2

XP

M

Y

X

Price of Y and monetary income are held

constant: MPY ,

Decrease in the price of X: 1

XP >2

XP

*

1X *

2X

*

1Y*

2Y

1U

2U

E1 E2

YP

PX

1

YP

PX

2

YP

PX

2

E3

3U

*

3X

*

3Y

• We do this by shifting the line AB to a parallel line CD that just passes through E1 (keeping purchasing power constant)

• To remove the income effect, imagine reducing the consumer’s money income until the initial bundle is just attainable

A

B

C

D

• Although is still affordable, it is not the optimal purchase at the budget line CD

*

1

*

1 , YX

• The optimal bundle of goods is:

SE IE

YP

M

2

XP

M TE

X is a normal goodDr. Manuel Salas-Velasco 24

The Slutsky Substitution Effect

YP

M

1

XP

M2

XP

M

Y

X*

1X *

2X

*

1Y*

2Y

1U

2U

E1 E2

YP

PX

1

YP

PX

2

YP

PX

2

2

XP

M

E3

3U

*

3X

*

3Y

YP

M

A

B

C

D

MPYPX YX *

1

1*

1E1:

MPYPX YX *

1

2*

1MM

MMM

Change (reduction) in money income necessary to make the initial bundle affordable at the new prices

M’= amount of money income that will just make the original consumption bundle affordable:

MMM

E3:

MPYPX YX *

3

2*

3

SE IE

TE

)( 12*

1 XX PPXM


Example

XP

MX

1010

)(14310

12010*

1 weekquartsX

)(16210

12010*

2 weekquartsX

• The individual demand function for milk is:

• Consumer’s income is $120 per week and PX is $3 per quart:

• Let’s suppose that the price of milk falls to $2 per quart:

• The total change (total effect): 2*

1

*

2 XX

MMM 14)32(14)( 12*

1 XX PPXM

106$14120 MMMLevel of income necessary to keep purchasing power constant

)(3.15210

10610*

3 weekquartsX

• The substitution effect is: 3.1143.15*

1

*

3 XX

• The income effect is: 0.7 (16 – 15.3)


The Hicks substitution effect

YP

M

1

XP

M2

XP

M

Y

X

MPY , 1

XP >2

XP

*

1X *

2X

*

1Y*

2Y

1U 2U

E1 E2

YP

PX

1

YP

PX

2

YP

PX

2

2

XP

M

E3

*

3X

*

3Y

YP

M

• To remove the income effect, imagine reducing the consumer’s money income until the initial indifference curve is just attainable

• We do this by shifting the line AB to a parallel line CD that just touches the indifference curve U1 (the utility level is held constant at its initial level)

A

B

C

DSE IE

TE

• The intermediate point E3

divides the quantity change into a substitution effect (SE) and an income effect (IE)


Income and Substitution Effects:

Inferior Good

1U

2U

E1

E2

E3

*

1X *

2X*

3X

Y

X

MPY ,

1

XP >2

XP

A

B

C

D

substitution effect

income effecttotal effect

• The consumer is initially at E1 on budget line AF

F

• With a decrease in the price of good X, the consumer moves to E2; the quantity of X demanded increases (total effect)

• The total effect can be broken down into:

o A substitution effect (associated with a move from E1 to E3)

o An income effect (associated with a move from E3 to E2)

X is an inferior good

• The substitution effect exceeds the income effect, so the decrease in the price of good X leads to an increase in the quantity demanded


Income and Substitution Effects:

The Giffen Good

1U

2U

E1

E2

E3

*

1X*

2X*

3X

Y

X

MPY ,1

XP >2

XP

A

B

C

D

substitution effect

income effect

total effect

• The consumer is initially at E1 on budget line AF

F

• With a decrease in the price of good X, the consumer moves to E2; the quantity of X demanded decrease (total effect)

• The total effect can be broken down into:

o A substitution effect (associated with a move from E1 to E3)

o An income effect (associated with a move from E3 to E2)

X is a Giffen good• The income effect exceeds the substitution effect, so the decrease in the price of good X leads to a decrease in the quantity demanded


Income and Substitution Effects of a reduction in price of good

X holding income and the price of good Y constant

Good X is:

Substitution effect

Income effect Total effect

NormalIncrease Increase Increase

Inferior (not Giffen)

Increase Decrease Increase

Giffen (also inferior)

Increase Decrease Decrease


Education

Microeconomics: Income and Substitution Effects