Transcript Consumer Theory

Introductory Microeconomics
(ES10001)
Topic 2: Consumer Theory
1. Introduction

We have seen how demand curves may be used to
represent consumer behaviour.

But we said very little about the nature of the
demand curve; why it slopes down for example.

Now we go ‘behind’ the demand curve

i.e. we investigate how buyers reconcile what they
want with what they can get
2
1. Introduction

N.B. We can use this theory in many ways - not
simply as household consumer buying goods.

For example:

Modelling decision of worker as regards his supply
of labour (i.e. demand for leisure)

Allocation of income across time (saving and
investment)
3
2. Theory of Consumer Choice

Four elements:
(i)
Consumer’s income
(ii)
Prices of goods
(iii)
Consumer’s tastes
(iv)
Rational Maximisation
4
3. The Budget Constraint

The first two elements define the budget constraint

The feasibility of the consumer’s desired
consumption bundle depends upon two factors:
(i) Income
(ii) Prices

Note: We assume, for the time being, that both are
exogenous (i.e. beyond consumer's control)
5
3. The Budget Constraint

Example (N.B two goods)

Two goods - films and meals

Student grant = £50 per week (p.w.)

Price of meal = £5

Price of film = £10
6
3. The Budget Constraint

Thus student can ‘consume’ maximum p.w. of 10
meals or 5 films by devoting all of his grant to the
consumption of only one of these goods.

Alternatively, he can consume some combination of
the two goods

For example, giving up one film a week (saving £10)
enables student to buy two additional meals (costing
£5 each)
.
7
3. The Budget Constraint
qm
£5*qm
qf
£10*qf
M
0
0
5
50
50
2
10
4
40
50
4
20
3
30
50
6
30
2
20
50
8
40
1
10
50
10
50
0
0
50
Table 1: Affordable Consumption Bundles
8
Figure 1: Budget Constraint
Films
5
A
4
B
1
0
2
8
10
Meals
9
3. The Budget Constraint

The budget constraint defines the maximum
affordable quantity of one good available to the
consumer given the quantity of the other good that
is being consumed.

N.B. Trade-off!

Trade-off is represented slope of budget
constraint.
10
3. The Budget Constraint

Intercepts
Determined by income divided by the appropriate
price of the good
Define maximum quantity of a particular good
available to an individual

Slope
Independent of income
Determined only by relative prices
11
3. The Budget Constraint

If consumer is devoting all income to films (qf =
£50/£10 = 5), then 1 meal can only be obtained by
sacrificing consumption of some films.

How many films must consumer give up?

pm = £5; thus to obtain that £5, the consumer must
give up 1/2 a film
12
3. The Budget Constraint

The slope of the budget constraint in this
example is thus:
pm
£5
1
==pf
£10
2
13
Figure 2: Slope of Budget Constraint
Films
qm = 1
5
Δqf = - 0.5
4.5
0
1
10
Meals
14
3. The Budget Constraint

More generally:

Two goods (x,y), prices (px, py) and money income
(m)


m = pxx + pyy
Slope of budget constraint: - px/py
15
3. The Budget Constraint

Proof:
m = px x + p y y
Þ
p y y = m - px x
Þ
m px
y=
x
py py
16
3. The Budget Constraint

Thus:
æ mö æ p ö
y=ç ÷ -ç x÷ x
è py ø è py ø
a
b
Þ
y = a - bx

æp ö
Such that Dy = -bDx = - ç x ÷ Dx
è py ø
17
3. The Budget Constraint

That is:
æ mö æ p ö
y0 = ç ÷ - ç x ÷ x0
è py ø è py ø
æ mö æ p ö
y1 = ç ÷ - ç x ÷ x1
è py ø è py ø
Þ
æp ö
Dy = - ç x ÷ Dx
è py ø
18
Figure 3: Budget Constraint
y
y = m/py - (px/py)x
m/py
Δx
A
Δy = -(px/py)Δx
B
0
m/px
x
19
3. The Budget Constraint

Intuition:

If additional unit of x costs px

Then its purchase requires a change in
consumption of y of –(px/py) (i.e. a sacrifice of y)
in order to maintain the budget constraint.
20
4. Preferences

Consider now the consumer preferences

Given what consumer can do, what would he like
to do?

Four assumptions:
(i) Completeness
(ii) Consistency
(iii) Non-satiation
(iv) Diminishing Marginal Rate of Substitution
21
4. Preferences

Completeness
Consumers can rank alternative bundles according
to the satisfaction or utility they provide
Thus given two bundles a and b, then a ∼ b , a ∼ b
or a ∼ b
Preferences assumed only to be ordinal, not
cardinal; i.e. consumer simply has to be able to
say he prefers a to b, not to say by how much.
22
IV. Preferences

Consistency
Preferences are also assumed to be consistent
Thus if a
a
b and b c, then we would infer that
c
We assume consumer is logically consistent
23
4. Preferences

Non-satiation
Consumers assumed to always prefer more
‘goods’ to less.
We can accommodate economics ‘bads’ (e.g.
pollution) in this assumption by interpreting then
as ‘negative’ goods

We can illustrate the first three assumptions
graphically as follows
24
Figure 4a: Preferences
y
a∼ b∼ c
a
b
c
0
x
25
Figure 4b: Preferences
y
a∼ b∼ c
b ∼ d, b ∼ e
d
b ∼ g, b ∼ f
a
g
e
b
c
f
0
x
26
Figure 4c: Preferences
a∼ b∼ c
y
b ∼ d, b ∼ e
d
h
b ∼ g, b ∼ f
a
h∼ b∼ i
g
e
b
c
i
f
0
x
27
Figure 4d: Preferences
y
d
h
a
g
e
b
c
i
f
0
Indifference Curve
x
28
4. Preferences

Marginal Rate of Substitution (MRS)
The quantity of y (i.e. the ‘vertical’ good) the consumer
must sacrifice to increase the quantity of x (i.e. ‘the
horizontal’ good) by one unit without changing total utility.

We generally assume (smooth) diminishing MRS

To hold utility constant, diminishing quantities of
one good must be sacrificed to obtain successive
equal increases in the quantity of the other good.
29
4. Preferences

Diminishing MRS derives from underlying
assumption of diminishing marginal utility

Marginal utility of a good is defined as the change
in a consumer’s total utility from consuming the
good divided by the change in his consumption of
the good

Diminishing MRS assumes that the increase in
utility from consuming additional units of a good
30
is declining
4. Preferences

Non-satiation implies downward sloping
indifference curves; increases in one good require
sacrifices in the other good to hold total utility
constant.

However, we can go further; diminishing MRS
implies that indifference curves are convex to
origin, becoming flatter as we move to the right.

Indeed, the MRS of x for y is simply the slope of
31
the indifference curve
Figure 5: Indifference Curves
y
I0
0
32
x
Figure 5: Indifference Curves
y
A
B
I0
0
33
x
Figure 5: Indifference Curves
y
A
A´
B
B´
0
I0
34
x
Figure 5: Indifference Curves
y
Δx = 1
A
Δy
A´
Δx = 1
Δy
B
B´
0
I0
35
x
4. Preferences

Diminishing MRS implies consumers prefer
consumption bundles containing mixtures of
goods rather than extremes

i.e. Bundle C = (5, 5) preferred to both Bundle A =
(2, 8) and Bundle B = (8, 2)

Diminishing MRS (i.e. diminishing marginal
utility)
36
Figure 6: Indifference Curves
y
A
B
I0
0
37
x
Figure 6: Indifference Curves
y
A
8
B
2
0
I0
2
8
38
x
Figure 6: Indifference Curves
y
A
8
C
5
B
2
0
I1
I0
2
5
8
39
x
4. Preferences

Note:
(i) Any point on the indifference map must lay on
an indifference curve.
(ii) indifference curves cannot cross

Thus every point on the indifference map must lay
on one and only one indifference curve.
40
Figure 7: Indifference Curves
y
I2
I1
I0
0
41
x
Figure 8: Indifference Curves Cannot Cross
y
a
b
I1
c
I0
0
42
x
5. Utility Maximisation

Budget line shows the consumer’s affordable
bundles given the market environment.

The indifference map shows the consumer’s
desired bundles

To complete the model we assume rational
maximisation - i.e. the consumer chooses the
affordable bundle that maximises his utility.
43
5. Utility Maximisation

This is a non-trivial point. We are implicitly
assuming that the consumer only derives utility
from the consumption of x and y.

Moreover, rational maximisation implies
consumer processes huge amount of information
before choosing his most preferred bundle

In reality, perhaps we ‘satisfice’
44
5. Utility Maximisation

The optimal choice bundle will be that point at
which an indifference curve just touches the
budget line

That is, where an indifference curve is tangent to
the budget line

In words, where the consumer’s marginal rate of
substitution (MRS) and economic rate of
substitution (ERS) are in accord
45
5. Utility Maximisation

Marginal Rate of Substitution (MRS)
Amount of y consumer willing to sacrifice for one
extra unit of x
Slope of indifference curve

Economic Rate of Substitution (ERS)
Amount of y the consumer is obliged to sacrifice
for one extra unit of x
Slope of budget line
46
Figure 9: Equilibrium (MRS = ERS)
y
E1
y1
I2
I1
I0
0
x1
47
x
Figure 10: Disequlibrium (MRS ≠ ERS)
y
Δx
E0
ΔyERS
ΔyMRS
I0
0
48
x
5. Utility Maximisation

Since preferences are unique, individuals will not
choose identical bundles, even when confronted
by same market environment

But they will all move to point where MRS = ERS

Even with different preferences, since ERS is the
same for everyone (i.e. we all face same relative
prices), it must be the case that in equilibrium:

MRS1 = ERS = MRS2
49
6. Comparative Statics

We now consider how the consumer responds to
changes in his market environment

That is, to changes in:
(i) Endowment income;
(ii) Prices.

N.B, Comparative Statics / Dynamics
50
6. Comparative Statics

Changes in Income

An increase in endowment income causes a
parallel shift out of the budget constraint

A decrease in endowment income causes a parallel
shift in of the budget constraint
51
Figure 11: Increase in Income
m¢ > m
y
m¢ p y
m py
0
m px
m¢ px
x
52
Figure 12: Increase in Income
m¢ > m
y
A
m¢ p y
I1
x Normal
y Normal
B
m py
E1
E0
C
I0
0
m px
D
m¢ px
x
53
Figure 12: Increase in Income
m¢ > m
y
A
m¢ p y
x Inferior
y Normal
E1
B
I1
m py
E0
C
I0
0
m px
D
m¢ px
x
54
Figure 12: Increase in Income
m¢ > m
y
A
m¢ p y
B
m py
I1
E0
C
E1
x Normal
y Inferior
D
0
m px
m¢ px
x
55
Figure 12: Increase in Income
m¢ > m
y
A
m¢ p y
x Inferior
y Normal
B
x Normal
y Normal
m py
E0
C
x Normal
y Inferior
I0
0
m px
D
m¢ px
x
56
6. Comparative Statics

Changes in Prices

An increase in price causes a pivot inwards of the
budget constraint

An decrease price causes a pivot outwards of the
budget constraint.
57
Figure 13: Fall in Price
y
px¢ < px
m py
0
m px
m px¢
x
58
7. Income & Substitution Effects

Price changes affects the optimal choice bundle in
two distinct ways:

First, there is a change in relative prices as
represented by a change in the slope of the
budget constraint.

Second, there is a change in purchasing power
(i.e. real income). The same level of money
income is now worth more to the consumer in
59
terms of its ability to purchase both goods.
Figure 13: Fall in Price
y
px¢ < px
m py
0
m px
m px¢
x
60
Figure 14: Effects of Fall in Price
y
Fall in price of good x reduces slope of budget
constraint (ERS) - i.e. fall in the relative price of
good x
m py
Fall in price of good x increases consumer’s real
income - i.e. expansion of the budget set
0
m px
m px¢
x
61
Figure 15: Effects of a Fall in Price
y
px¢ < px
m py
E1
E0
0
m px
m px¢
x
62
Figure 15: Effects of a Fall in Price
y
px¢ < px
m py
E0
E1
0
m px
m px¢
x
63
Figure 15: Effects of a Fall in Price
y
px¢ < px
m py
E1
E0
0
m px
m px¢
x
64
Figure 15: Effects of a Fall in Price
y
A
m py
px¢ < px
Good x
is Giffen
B
Good x is
Non-Giffen
E0
C
0
m px
m px¢
x
65
Figure 15: Effects of a Fall in Price
y
p1x < px0
m py
E1
E0
0
m px0
px
px0
x
E0
p1x
0
m p1x
E1
x0
x1
pd ( x)
x
66
Figure 15: Effects of a Fall in Price
y
p1x < px0
m py
E1
E0
0
m px0
px
px0
x
E0
p1x
0
m p1x
E1
x0
x1
pd ( x)
x
67
Figure 15: Effects of a Fall in Price
y
p1x < px0
E1
m py
E0
0
m px0
px
p
0
x
pd ( x)
E0
px0
1
x
m p1x
E1
x1
x0
x
68
7. Income & Substitution Effects

We decompose total effect of price change into:
(i) Income Effect
(ii) Substitution Effect

The income effect is the adjustment of demand to
the change in real income.

The substitution effect is the adjustment of demand
69
to the change in relative prices.
Figure 14: Income and Substitution Effects
y
(Fall in px)
A
E0
I0
0
A
x
70
Figure 14: Income and Substitution Effects
y
(Fall in px)
A
E0
E1
I1
I0
0
A
B
x
71
7. Income & Substitution Effects

We decompose the overall change in demand into
income and substitution effects by (hypothetically)
adjusting the consumer’s income to restore him to
the level of real income he enjoyed before the
price change

Given the fall in px and the subsequent increase in
real income, we therefore reduce the consumer’s
real income; mechanically, we drag the new
budget line back until it is just tangent to the
original indifference curve.
72
Figure 14: Income and Substitution Effects
y
(Fall in px)
A
E0
E1
I1
I0
0
A
B
x
73
Figure 14: Income and Substitution Effects
y
(Fall in px)
A
C
E0
E1
E2
I1
I0
0
A
C
B
x
74
Figure 14: Income and Substitution Effects
y
(Fall in px)
A
C
E0
E1
E2
I1
I0
0
A
C
B
x
75
Figure 14: Income and Substitution Effects
y
(Fall in px)
E0-E1: Total Effect (x0-x1)
A
E0-E2: Substitution Effect (x0-x2)
E2-E1: Income Effect (x2-x1)
E0
E1
E2
I1
I0
0
x0
x2 A
x1
B
x
76
8. Inferior and Giffen Goods

In a two good model, a price change always
induces a substitution effect in the opposite
direction of the change in price

i.e: a rise (fall) in px induces a substitution away
(towards) good x ceteris paribus

We usually say that ‘… the own price substitution
effect is always negative.’
77
8. Inferior and Giffen Goods

The income effect, however, can be positive (i.e.
normal good) or negative (i.e. inferior good)

A rise in the price of a normal good induces a
negative substitution effect and a negative income
effect, both of which act to reduce the demand for
good x

A rise in the price of an inferior good, however,
induces a negative substitution effect but a positive
income effect, thus the overall effect is ambiguous78
8. Inferior and Giffen Goods

If, when the price of an inferior good rises, the
positive income effect dominates the negative
substitution effect, we have the case of a Giffen
Good

That is, a good for which demand rises (falls)
when price rises (falls)

Giffen goods are very inferior good
79
Figure 15: Income and Substitution Effects
y
A
Good x: Normal / Non-Giffen
I0
C
E1
E0
I1
E2
0
A
C
B 80
x
Figure 15: Income and Substitution Effects
y
Good x: Inferior / Non-Giffen
I1
A
I0
E1
C
E0
E2
0
A
C
B 81
x
Figure 15: Income and Substitution Effects
y
Good x: Inferior / Giffen
I1
A
E1
C
E0
E2
I0
0
A
C
B 82
x
9. Measuring Real Income

When we decomposed the change in demand
resulting from a change in price into an income
and substitution effect, we did so by varying
money income

Specifically, when the price of good x fell, we
‘varied’ the consumer’s money income to hold his
real income constant, where real income was
defined as the consumer’s ability to enjoy a
particular level of utility
83
9. Measuring Real Income

Varying money income is this way is known as a
Hicks Compensating Variation in money income
(HCV)

HCV allows consumer to enjoy original level of
utility at the new relative price ratio

We ‘compensate’ the consumer for the change in
price

Sounds odd in respect of a price fall.
84
Figure 16.1: Hicks Compensating Variation
(Price Fall)
y
A
C
B
I1
I0
0
x
85
Figure 16.2: Hicks Compensating Variation
(Price Rise)
y
B
C
A
I1
I0
0
x
86
9. Measuring Real Income

An alternative definition of real income is the
ability to consumer not a particular level of utility,
but a particular bundle of goods

i.e. we vary the consumer’s money income
following a change in price to permit him to
consumer his original bundle of goods at the new
relative price ratio

The is know as the Slutsky Compensating
Variation (SCV) in money income.
87
Figure 16.3: Slutsky Compensating Variation
(Price Fall)
y
A
C
I0
B
I1
0
I2
x
88
Figure 16.4: Slutsky Compensating Variation
(Price Rise)
y
B
C
I2
A
I0
0
I1
x
89
9. Measuring Real Income

Both Hicks and Slutsky compensating variations
adjust the consumer’s new level of income (i.e. the
level following the price change) such that he is
able to enjoy either his original level of utility
(Hicks) or his original consumption bundle
(Slutsky)

An alternative approach is to adjust the
consumer’s original level of income in such a way
that he is able to enjoy the level of utility (Hicks)
or the consumption bundle (Slutsky) that he would
have been able to enjoy were he to face the change
in prices
90
9. Measuring Real Income

That is, we vary the consumer’s money income at
the original relative price ratio to enable him to
enjoy the level of real income (i.e. utility or
consumption bundle) that he would have been able
to enjoy from the price change

i.e. we provide the consumer with an Equivalent
Variation in money income

A variation in money income that will adjust the
consumer’s real income in a manner analogous to
the price change
91
Figure 16.5: Hicks Equivalent Variation
(Price Fall)
y
B
A
C
I1
I0
0
x
92
Figure 16.6: Hicks Equivalent Variation
(Price Rise)
y
C
A
B
I0
0
I1
x
93
Figure 16.7: Slutsky Equivalent Variation
(Price Fall)
y
B
A
I2
C
I0
0
I1
x
94
Figure 16.8: Slutsky Equivalent Variation
(Price Rise)
y
C
A
I0
B
I1
0
I2
x
95
9. Measuring Real Income

To summarise, we have eight cases

Hicks / Slutsky

Compensating Variation / Equivalent Variation

Price Rise / Price Fall
96
Figure 16.1: Hicks Compensating Variation
(Price Fall)
y
A
C
B
I1
I0
0
x
97
Figure 16.2: Hicks Compensating Variation
(Price Rise)
y
B
C
A
I1
I0
0
x
98
Figure 16.3: Slutsky Compensating Variation
(Price Fall)
y
A
C
I0
B
I1
0
I2
x
99
Figure 16.4: Slutsky Compensating Variation
(Price Rise)
y
B
C
I2
A
I0
0
I1
x
100
Figure 16.5: Hicks Equivalent Variation
(Price Fall)
y
B
A
C
I1
I0
0
x
101
Figure 16.6: Hicks Equivalent Variation
(Price Rise)
y
C
A
B
I0
0
I1
x
102
Figure 16.7: Slutsky Equivalent Variation
(Price Fall)
y
B
A
I2
C
I0
0
I1
x
103
Figure 16.8: Slutsky Equivalent Variation
(Price Rise)
y
C
A
I0
B
I1
0
I2
x
104
9. Measuring Real Income:
Aide Memoire

Hicks moves budget line to indifference curve

Slutsky moves budget line to consumption bundle

Compensating Variation moves NEW budget line to
ORIGINAL indifference curve / bundle

Equivalent Variation moves ORIGINAL budget
line to NEW indifference curve / bundle
105
10. Applications

Two key areas:
(i) Labour Supply;
(ii) Intertemporal Choice.
106
10.1 Labour Supply

Consider individual’s role as a supplier of factor
services

Individuals sell their labour to firms in return for a
wage.

Individual makes a choice between income and
leisure given the dual constraints of time and the
wage
107
Figure 17: Budget Constraint
Y
Ymax
Y0
0
w
T
108
L
Figure 18: Preferences
Y
I2
I1
I0
0
109
L
Figure 21: Labour Market Equilibrium
Y
Ymax
Y1 = Y0 + w(T – L1)
Ymax = Y0 + wT
Y1
E0
I1
w
Y0
0
L1
A
T
110
L
Figure 22: Increase in Unearned Income
Y
E2
Y2
I2
Y1
E1
B
I1
A
Y0
0
L1
L2
T 111
L
Figure 23: Increase in Wage Rate
Y
E2
E1
I2
I1
0
L2
L1
T 112
L
Figure 23: Increase in Wage Rate
Y
E2
E3
E1
I2
I1
0
L3
L2
L1
T 113
L
10.1 Labour Supply

Note that the income and substitution effects work
against one another

Because leisure is a normal good, the income
effect from the increase in wage increases the
demand for leisure

But the wage rate is the opportunity cost, or price,
of leisure. Thus, an increase in the wage rate /
price of leisure induces a substitution away from
leisure
114
Figure 23: Increase in Wage Rate
Y
E1-E2: Total Effect (L1-L2)
E1-E3: Substitution Effect (L1-L3)
E3-E2: Income Effect (L3-L2)
E2
E3
E1
I2
I1
0
L3
L2
L1
T 115
L
Figure 24: Labour Supply Curve
w
Ls
E2
w2
w1
0
E1
(T-L1)
(T-L2)
T
(T-L)
116
10.1 Labour Supply

If the income effect dominates the substitution
effect, then we have a situation in which an
increase in the wage (i.e. the price of leisure) leads
to an increase in the demand for leisure

That is:

Leisure is Giffen …

… but Normal!
117
Figure 25: Increase in Wage Rate
Y
I2
E1-E2: Total Effect (L1-L2)
E1-E3: Substitution Effect (L1-L3)
E3-E2: Income Effect (L3-L2)
I1
E3
E2
E1
0
L3
L1
L2
T
118
L
Figure 26: Labour Supply Curve
w
Ls
w2
E2
E1
w1
0
(T-L2)
(T-L1)
T
(T-L)
119
10.1 Labour Supply

This is possible because there is also an
Endowment Effect in operation …

The Individual is entering the market with an
endowment of leisure which he is selling to the
firm

The presence of endowment effects complicates
the relationship between inferiority and Gifffeness
120
10.1 Labour Supply

Empirically, we tend to see labour supply
curves bending backwards at high wage
rates

i.e.
121
Figure 27: Labour Supply Curve
w
0
Ls
H = (T-L)
122
10.1 Labour Supply

Rather than at low wage rates

i.e.
123
Figure 28: Labour Supply Curve
w
Ls
0
H = (T-L)
124
10.1 Labour Supply

Moreover, backward bending labour supply
curves are usually observed for males but
nor females

i.e.
125
Figure 29: Labour Supply Curve
w
0
Lsm
Lsf
H = (T-L)
126
10.1 Labour Supply

Implications of backward bending labour supply
curve

Multiple equilibria

Unstable equilibria

What happens to w if it is perturbed slightly above
/ below its equilibirum level, w*? Do forces of
excess demand / excess supply force w back to w*
127
Figure 29: Labour Supply Curve
w
Ld
Ls
Unstable Equilibrium
E1
Stable Equilibrium
E
2
0
H = (T-L)
128
10.2 Intertemporal Choice

Assume individual lives for two periods with a
lifetime income endowment of y = (y1, y2)

Consumption over time is c = (c1, c2)

Now, £x saved today (i.e. period 1) will yield
£(1+r)x tomorrow (i.e. period 2)

The future value of £x today is thus £(1+r)x
129
10.2 Intertemporal Choice

Conversely, the present value of £x received
tomorrow (i.e. period 2) is:
æ 1 ö
$ç
x
÷
è 1+ r ø

Intuitively, if we receive £x tomorrow, can borrow
£z today, where:
æ 1 ö
$z 1 + r = $x Þ $z = $ ç
x
÷
è 1+ r ø
(
)
130
10.2 Intertemporal Choice

Thus, given an income endowment of:
y 0 = (y10 , y20 )

Then the maximum period 1 income is:
æ 1 ö 0
yˆ = y + ç
÷ y2
è 1+ r ø
0
1

0
1
And the maximum period 2 income is:
yˆ 20 = y20 + (1 + r ) y10
131
Figure 30: Intertemporal Budget Constraint
y2
æ 1 ö 0
yˆ = y + ç
÷ y2
è 1+ r ø
0
1
ŷ20
0
1
yˆ 20 = y20 + (1 + r ) y10
y 0 = (y10 , y20 )
y20
0
y
0
1
ŷ
0
1
y1
132
10.2 Intertemporal Choice

Assume individual consumes in both periods

If the value of consumption in period 1 is c10, then
can save y10 - c10 in period 1 for period 2
consumption in excess of period 2 income, y20:
)(
(
c20 = y20 + 1 + r y10 - c10
Þ
(
)
)
(
)
c20 = éë y20 + 1 + r y10 ùû - 1 + r c10
Intercept
Slope
133
10.2 Intertemporal Choice
(
)
(
)
c20 = éë y20 + 1+ r y10 ùû - 1+ r c10
Þ
(
)
c20 = ŷ20 - 1+ r c10
Þ
(
)
(
)
Dc20 = - 1+ r Dc10
Þ
Dc20
Dc
0
1
= - 1+ r
134
Figure 30: Intertemporal Budget Constraint
c20 = éë y20 + (1 + r ) y10 ùû - (1 + r )c10
Þ
c2 , y2
ŷ20
c20 = yˆ 20 - (1 + r )c10
y 0 = (y10 , y20 )
y20
0
y
0
1
ŷ
0
1
c1 , y1
135
Figure 30: Intertemporal Budget Constraint
(
c2 , y2
Dc10
ŷ20
)
c20 = ŷ20 - 1+ r c10
Þ
(
)
Dc20 = - 1+ r Dc10
(
)
Dc20 = - 1+ r Dc10
slope = -(1+r)
0
ŷ
0
1
c1 , y1
136
Figure 30: Intertemporal Budget Constraint
c2 , y2
(y
0
1
Þ
ŷ20
(
)
y20 - c20 < 0
c20
y 0 = (y10 , y20 )
y20
0
)
- c10 > 0
c
0
1
y
0
1
ŷ
0
1
c1 , y1
137
Figure 30: Intertemporal Budget Constraint
c2 , y2
(y
0
1
Þ
ŷ20
(
)
- c10 < 0
)
y20 - c20 > 0
y 0 = (y10 , y20 )
y20
c20
0
y
0
1
c
0
1
ŷ
0
1
c1 , y1
138
10.2 Intertemporal Choice

Note the effects of changes in income endowment
or interest rate

Change in income endowment shifts the intertemporal budget constraint parallel

Changes in interest rate pivot the budget constraint
around the initial income endowment
139
Figure 30: Intertemporal Budget Constraint
c2 , y 2
æ 1 ö 0
yˆ = y + ç
÷ y2
è 1+ r ø
0
1
ŷ12
0
1
yˆ 20 = y20 + (1 + r ) y10
ŷ20
y0
y1
y
y
y20
0
0
1
1
1
ŷ
0
1
ŷ
1
1
y1
140
Figure 30: Intertemporal Budget Constraint
c2 , y 2
æ 1 ö 0
0
0
0
ŷ = y + ç
y
Û
ŷ
=
y
+
1+
r
y
2
2
1
è 1+ r ÷ø 2
0
1
ŷ21
ŷ10
y1
y21
y20
(
0
1
)
æ 1 ö 1
1
1
0
ŷ = y + ç
y
Û
ŷ
=
y
+
1+
r
y
2
2
1
è 1+ r ÷ø 2
1
1
(
0
1
)
y0
0
y
0
1
ŷ
0
1
ŷ
1
1
y1
141
Figure 30: Intertemporal Budget Constraint
c2 , y2
ŷ22
æ 1 ö 0
0
0
0
ŷ = y + ç
y
Û
ŷ
=
y
+
1+
r
y
2
2
1
è 1+ r ÷ø 2
ŷ10
æ 1 ö 2
2
2
2
ŷ = y + ç
y
Û
ŷ
=
y
+
1+
r
y
2
2
1
è 1+ r ÷ø 2
y2
0
1
2
1
y22
y20
(
0
1
)
(
2
1
)
y0
0
y
0
1
y
2
1
ŷ
0
1
ŷ
2
1
142
c1 , y1
Figure 30: Intertemporal Budget Constraint
c2 , y2
Increase in Rate of Interest
ŷ20
ŷ
æ 1 ö i
yˆ = y + ç
÷ y2
è 1+ r ø
i
1
i
1
yˆ 2i = y2i + (1 + r ) y1i
1
2
(
y 0 = y10 , y20
y20
0
y
0
1
)
ŷ
0
1
ŷ
1
1
c1 , y1
143
10.2 Intertemporal Choice

Consider a (period 1) borrower

That is:

(c1 - y1) > 0

How does he react to changes in interest rate?
144
Figure 30: Intertemporal Budget Constraint
c2 , y2
Period 1 Borrowing
ŷ20
y20
I0
y0
c20
0
E0
y
0
1
c
0
1
ŷ
0
1
c1 , y1
145
Figure 30: Intertemporal Budget Constraint
c2 , y 2
Borrower (Fall in Interest Rate)
ŷ20
I1
y20
I0
y0
E1
c
0
2
0
E0
y
0
1
c
0
1
ŷ
0
1
c1 , y1
146
10.2 Intertemporal Choice

Thus, if interest rate falls:
(i) (Period 1) Borrower remains a (period 1)
borrower;
(ii) Is better-off;
(iii) Increases (period 1) borrowing if c1 a normal
good
147
Figure 30: Intertemporal Budget Constraint
c2 , y 2
Borrower (Fall in Interest Rate):
ŷ
(i) Substitution Effect E0-E2;
(ii) Income Effect E2-E1
0
2
I1
y20
I0
y0
E0
c
0
2
0
E1
E2
y
0
1
c
0
1
ŷ
0
1
c1 , y1
148
10.2 Intertemporal Choice

If interest rate rises:
(i) Borrower is definitely worse off;
149
Figure 30: Intertemporal Budget Constraint
c2 , y 2
Borrower: Increase in Interest Rate:
ŷ20
I1
y20
I0
y0
E1
c
0
2
0
y
0
1
E0
c
0
1
ŷ
0
1
c1 , y1
150
10.2 Intertemporal Choice

Conversely, for savers (c1 - y1) <0:

Rise in interest rates: (i) Remain savers; (ii) Better
off; (iii) Increase saving if c1 is an inferior good

Fall in interest rate: (i) Definitely worse off
151
Figure 30: Intertemporal Budget Constraint
c2 , y 2
Saver (Rise in Interest Rate)
ŷ20
E1
c20
y20
0
I1
E0
I0
y0
c
0
1
y
0
1
ŷ
0
1
c1 , y1
152
Figure 30: Intertemporal Budget Constraint
c2 , y 2
Saver (Rise in Interest Rate)
ŷ20
E2
c20
y20
0
E1
I1
E0
I0
y0
c
0
1
y
0
1
ŷ
0
1
c1 , y1
153