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
QuantityofY
XP
M
vertical
intercept
horizontal intercept
YP
M
Slope
Y
X
P
P

The equation for
the budget line:
X
P
P
P
M
Y
Y
X
Y

Relative price ratio
Budget 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
QuantityofY
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
Quantityoflemonade(week),Y
M’ = 20; PX = 2; PY = 1
M = 10; PX = 2; PY = 1
X
P
P
P
M
Y
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
M1
XP
M2
XP
M3
YP
M2
YP
M3
YP
M1
• 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
Y
Engel 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
Quantityoflemonade(week),Y
X
P
P
P
M
Y
Y
X
Y

M = 10; PX = 2; PY = 1
XY 210 
M = 10; P’X = 1; PY = 1Decrease in the price of X (50%) XY -10
YP
M
XP
M
XP
M


Response to Changes in a Good’s Price
MPY ,
1
XP 2
XP
Y
X
Price-Consumption
Curve
E1
E2
E3
Decrease in the price of X:
Price of Y and income are held constant:
3
XP> >
YP
M
1
XP
M
2
XP
M
3
XP
M
1U
2U 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
XP
2
XP
3
XP

The Consumer’s Demand Function
Y
Y
X
X
P
MU
P
MU

X
U
MUX



Y
U
MUY



• 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)
1 YMUX
1 XMUY
YX P
X
P
Y 11 


1)1( 
Y
X
P
P
XY
• Let’s suppose that the utility function is: U = X Y + X + Y

The Consumer’s Demand Function
1)1( 
Y
X
P
P
XY
PX X + PY Y = M M
P
P
XPXP
Y
X
YX 





 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
PPM
X
2



The Own-Price Demand
X
YX
P
PPM
X
2


),,( MPPXX YX
),,( MPPXX YX
M = $100; PY = $10
The own-price demand curve
(ordinary demand function for X):
X = f (PX), ceteris paribus
X
X
P
P
X
2
10100 

X
X
P
P
X
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
X
PX


5.0
55

The Engel Curve
X
YX
P
PPM
X
2


),,( MPPXX YX
),,( MPPXX YX
PX = $5; PY = $10
The Engel curve for X
52
105



M
X
10
5

M
X
2
1
10

M
X
X
M
elasticityIncome
M
X



If Income Elasticity is positive, then X is a
normal good
(quantity demanded increases as income
increases, ceteris paribus)
 positive
M
X
10
1



positive


elasticityIncome X is a normal
good

The Cross-Price Demand Curve
X
YX
P
PPM
X
2


),,( MPPXX YX
),,( MPPXX YX
PX = $5; M = $100
52
5100


 YP
X
10
95 YP
X


10
5.9 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
P
elasticityprice-Cross Y
Y


X If CPE is positive, then X,Y are substitutes
If CPE is negative, then X,Y are complements
)(
10
1
positive
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
U
MUX



Y
U
MUY



• The condition for utility maximization is:
U = U (X, Y)
2
1
2
1
2
1 
 XYMUX
2
1
2
1
2
1 
 YXMUY
YX P
YX
P
XY 2
1
2
1
2
1
2
1
2
1
2
1 

PX X + PY Y = M M
P
P
XPXP
Y
X
YX 
XP
M
X
2
MXPX 2
Consumer’s demand
function for X
• The utility function is: 2
1
2
1
YXU 
2
1
2
1
2
1
2
1
2
1
2
1



XY
YX
P
P
X
Y
Y
X
P
P
X
Y

Y
X
P
P
XY 
PX = 4; M = 800; PY = 1 100
8
800
X
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
XP
2
XP
3
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
M
2
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
TE
SE
total effect (TE) =
substitution effect (SE) +
income effect (IE)
IE

The Substitution Effect: Two Definitions in
the Literature
Eugene Slutsky
1880-1948
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
M
2
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
M
2
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*
1
MM 
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
M
X
10
10 
)(14
310
120
10*
1 weekquartsX 


)(16
210
120
10*
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  MMM
Level of income necessary to keep purchasing
power constant
)(3.15
210
106
10*
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
M
2
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
D
SE 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 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
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
Normal
Increase Increase Increase
Inferior (not
Giffen)
Increase Decrease Increase
Giffen (also
inferior)
Increase Decrease Decrease

Microeconomics: Income and Substitution Effects

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