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Appendix 4.A
A Formal Model of
Consumption and Saving
Micro-foundation of Macro
Abel & Bernake: Macro Ch3
Varian: Micro Ch10
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Optimization over time
Max U(C, C f )
st a f  (1  r )(a  Y  C )
C f  a f Y f
Cf
Yf
C (
)  a Y  (
)
1 r
1 r

Current income Y, future income Yf : Endowment point: (a+Y, Yf)
initial wealth a, wealth at beginning of future period af ;
Choice variables: C = current consumption; Cf = future consumption
Slope of lifetime BC = -(1+r)
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Figure 4.A.1 The budget line
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Present-Value Budget Constraint (PVBC)



Present value of lifetime wealth:
PVLW = a+ Y + Yf/(1+r)
Present value of lifetime consumption:
PVLC = C + Cf/(1+r)
(4.A.2)
The budget constraint means PVLC = PVLW
C + Cf/(1+r) =a+ Y + Yf/(1+r)
(4.A.3)
Slope of PVBC≡ (△Cf/△C)= -(1+r)
Price of current consumption=(1+r)：△Cf = -(1+r) △C
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Consumer Preferences: Indifference Curves


A person is equally happy at any point on an indifference curve
3 important properties of indifference curves
 Slope downward from left to right:
Less consumption in one period requires more consumption in
the other period to keep utility unchanged
 Indifference curves that are farther up and to the right
represent higher levels of utility,
because more consumption is preferred to less
 Indifference curves are bowed toward the origin,
because people have a consumption-smoothing motive,
they prefer consuming equal amounts in each period rather than
consuming a lot one period and little the other period
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Figure 4.A.2 Indifference curves
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The Optimal Level of Consumption

Optimal consumption point:
the budget line is tangent to an indifference curve
(Fig. 4.A.3)
Tangency condition: ?
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Fig 4.A.3 The optimal consumption combination
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Saving (S), a lender or a borrower

S≡Y-C
If C=Y, S=0
If C<Y, S>0
If C>Y, S<0
 af=
0, C=a+Y: no-borrowing, no-lending
af>0, C< a+Y: lender, with interest income
af< 0, C< a+Y: borrower, with interest payment
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Income Effect (IE) or Wealth Effect

When income or wealth (PVLW) increases,
PVBC shifts outward, the opportunity set increases,
the demand for normal goods (C and Cf ) increases.
a↑, Y↑, Yf ↑: PVLW↑
 C↑ and Cf ↑
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Comparative Statics: △a, △Y, △Yf

An increase in wealth: a↑
 Increases PVLW, so LRBC shifts out parallel to old BC.
 with consumption smoothing, both current and future
consumption increase
 IE: a↑ PVLW↑, C↑, Cf↑ (Y is unchanged, S:↓)

An increase in current income: Y ↑ (Fig. 4.A.4)
Y↑ IE: C↑, Cf ↑ (S↑)

An increase in future income: Yf ↑
Yf↑ IE: C↑, Cf ↑ (S↓)
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Fig 4.A.4 An increase in income or wealth
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Permanent vs. temporary increase in income

Different types of changes in income
 Temporary increase in income:
Y rises and Yf is unchanged
?
 Permanent
increase in income:
Both Y and Yf rise
?
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The permanent income theory

This distinction made by Milton Friedman in the 1950s
and is known as the permanent income theory
 Permanent changes in income lead to much larger
changes in consumption
 Thus permanent income changes are mostly consumed,
while temporary income changes are mostly saved
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Life-Cycle Model



developed by Franco Modigliani and associates in the 1950s
 Patterns of income, consumption, and saving over an
individual’s lifetime (Fig. 4.A.5)
Real income steadily rises over time until near retirement;
at retirement, income drops sharply
Lifetime pattern of consumption is much smoother than the
income pattern.
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Figure 4.A.5
Life-cycle consumption,
income, and saving
hump-shaped of Y, S
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Ricardian equivalence: two-period model


Suppose the government reduces taxes by 100 in the
current period, r = 10%,
and taxes will be increased by 110 in the future period
Cf
Y f T f
C (
)  a  (Y  T )  (
)
1 r
1 r
T  0, T f  (1  r )T  0
Then the PVLW is unchanged, and no change in C.
Or ?


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Fiscal policy:
△T<0 (a lump-sum tax↓ )  △Sd
Assume closed economy: NFP=0,
Assume TR=INT=0 for simplicity
 S= Y + NFP– C – G
Spvt= Y + NFP – T + TR + INT – C
Sgovt =T – TR – INT – G
???
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Fiscal policy: △G>0  △Sd
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Excess sensitivity of consumption

Generally, life cycle or permanent income theory
have been supported by looking at real-world data
 But data shows some excess sensitivity of
consumption to changes in current income
 This could be due to short-sighted behavior
 Or it could be due to borrowing constraints
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Borrowing constraints



If a person wants to borrow and can’t, the borrowing
constraint is binding
A consumer with a binding borrowing constraint spends all
income and wealth on consumption.
 So an increase in income or wealth will be entirely spent
on consumption as well
 This causes consumption to be excessively sensitive to
current income changes
Perhaps 20% to 50% of the U.S. population faces binding
borrowing constraints.
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Comparative statics: change in r

r↑(Fig. 4.A.6)


one point on the old BC is also on the new BC:
the no-borrowing, no-lending point
Slope of new budget line is steeper
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Fig 4.A.6 The effect of an increase in the real interest
rate on the budget line
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Intertemporal substitution effect (ISE)
r↑makes future consumption cheaper relative to
current consumption
 use cheaper Cf to substitute more costly C,
C↓(S↑), Cf ↑

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Fig 4.A.7 The substitution effect
of an increase in the real interest rate
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Income Effect (IE)
For a person consume at no-borrowing, no-lending
point,
r↑ no IE
 If the person originally a lender,
r↑ positive IE  C↑, Cf ↑
 If the person originally a borrower,
r↑ negative IE  C↓, Cf ↓

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Δr: Total effect =ISE +IE

IE and ISE together
 If a person consumes at no-borrowing, no-lending
point (Fig. 4.A.7),
?
 For a lender, (Fig. 4.A.8)
?
 For a borrower,
?
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Fig 4.A.8 An increase in the real interest rate with both
an income effect and a substitution effect
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Δr  aggregate saving

The effect on aggregate saving of r↑ is ambiguous
theoretically.
 Empirical research suggests that saving ↑
(Saving function is positively-sloped)
 But the effect is small
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Temporary vs. Permanent increase in wage
Optimization over time (Ch3)
Max U(C, L, Cf, Lf)
 ISE between current C and future Cf
ISE between current L and future Lf
 If temporary w↑: strong ISE + weak IE
ISE > IE => L↓, h ↑
 If permanent w↑ : weak ISE + strong IE
ISE < IE => L ↑, h ↓
 Empirical evidence support the implication.
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