Intertemporal Approach
to the Current Account
Part 2
Volatile Investment
Investment is the most volatile part of demand.
A 1% change in technology or increase in
employment will increase target capital stock.
 In any year, investment is one-tenth as large
as the capital stock.

 A 1%
change in the capital stock requires a 10%
change in investment.
Investment Curve
r
Q: Why does the curve slope
down? The greater is
interest rate, the more
profitable capital must be to
invest in it.
Q: What shifts the curve?
Increases in technology and
labor increases profitability
of capital.
I (r )

I
Investment high but falling in Japan
and Korea, Investment high but
increasing in China.
Investment as a % ogf GDP
45
40
35
30
1990-1994
25
1995-1999
20
2000-2003
15
10
5
0
China
Japan
Korea, Rep.
United States
Savings high but falling in Japan and
Korea, savings high but increasing in
China.
Savings as a % of GDP
45
40
35
30
1990-1994
&
25
1995-1999
20
2000-2003
15
10
5
0
China
Japan
Korea, Rep.
United States
Two Consumption Theories
Keynesian: Consumption is dependent on
current income.
 Permanent Income Theory: Consumption
decision is a savings decision so households
take into account future income as well as
outstanding financial wealth.

 People
prefer smooth consumption and save and
borrow to do so.
Why do People Save?


Life Cycle Motives – Income is Not Smooth Across
Time. Households save, in part, to transfer income
from high income periods to low income periods.
Precautionary Motives – Households like to achieve
a buffer stock of wealth in the case of a possible bad
outcome. If households have a buffer stock of
saving, bad outcomes in terms of income don’t result
in really bad outcomes in terms of consumption.
Life Cycle Motives: Two Period Model
To examine life-cycle theory, we use simplest
possible model.
 One good consumed by a household that lives
two periods, C0 and C1.
 Household lives and earns income Y0 and Y21
in each period.
 Household pays taxes in each period T0 and
T1.
 Household can buy/sell bonds, B, at real
interest rate r.

Temporal Budget Constrants

First period,
B0 = Y0 – T0 – C0

Second period,
C1=Y1 –T1+(1+r)B0

1)
2)
Note B can be either > or < 0. If B > 0, household is
a saver. If B < 0, household is a borrower.
Intertemporal Budget Constraint
C
Y

T
1
1
1
 Combine two budget


B
constraints
1 r 1 r
 Present Discounted Value of
Lifetime Income equals
B  Y0  T0  C0
Present Discounted Value of
Lifetime Consumption.
C1
C0 
W
1 r
Y1  T1
W  Y0  T0 
1 r
Consider 3 scenarios

Baseline Y1 = Y2 =YY implies
2 r 

Permanent Consumption Hypothesis
C1=C2
1 r
1 r  2  r 
C
[W ] 
Y  Y


2r
2  r  1 r 
W Y 

Y

1 r  1 r 
Temporary Rise in Income

The propensity to consume is a fraction of the
temporary extra income. The remainder is saved
for the future.
Y
 2r 
Y1  Y  , Y2  Y  W  Y   

Y 
1 r  1 r 
1 r
1 r  2  r  1 r
1 r
C
W 
Y 
 Y 


2r
2  r  1 r  2  r
2r
1 r
1
S  Y    (Y 
) 

2r
2r
Permanent Rise in Income
The propensity to consume is larger when the
increase in consumption is permanent. There is
no need to save a permanent rise in income for
the future.
Y   2r

Y1  Y  , Y2  Y  ,  W  Y   

Y    

1 r
1 r
1 r  2  r

C
W 
Y  Y 

2r
2  r  1 r


S  Y    (Y   )  0

 1 r

Future Rise in Income

Consumption may rise when future income
increases which will also increase W. Savings will
fall as people borrow to enjoy future income

 2r 
today.
Y  Y , Y  Y  ,  W  Y  
Y 


 1 r  1 r
1 r
1  r  2  r 
 

C
W
Y 
Y 



2r
2  r  1  r  1  r 
2r
1
2


S  Y  (Y 
)
2r
2r
Permanent Income Hypothesis

A simplified (and extreme) version of this
theory hypothesizes that consumption is equal
in each period. *
C2
C
C 
W  C 

1 r
1 r
1 r
C
[W ]
2r
*
1
Income Stream & Consumption

Consider three hypothetical increases in income of Δ.
1.
2.
3.


A Temporary Increase – Y1 increase by Δ, but Y2 is
unchanged. This will increase W by Δ.
A Future Increase – Y2 increases by 100, but Y1 is unchanged.
W increases by Δ /1+r≈ Δ
A Permanent Increase – Y1 & Y2 increase by 100. W increases
by Δ(2+r/1+r) ≈2∙ Δ
Cases 1 & 2 increase W by nearly identical amounts. But
current consumption depends only on W. Thus, cases 1 & 2
will increase C1* , C2* by similar amounts.
Case 3 increases W by nearly double the amount.
Income Stream and Savings



In the first case, future income does not rise but
optimal future consumption, C2* does . Current
savings must rise.
In the second case, current income does not rise, but
optimal current consumption. Current savings must
fall.
What happens to savings with a permanent change
in income?
Application: Life Cycle of Saving



Permanent Income Hypothesis suggests that
households like to keep a constant profile of
consumption over time.
Age profile of income however is not constant.
Income is low in childhood, rises during maturity and
reaches a peak in mid-1950’s and drops during
retirement.
This generates a time profile for savings defined as
the difference between income and consumption.
Time Path of Savings
C,Y
S>0
C
S<0
S<0
Y
time
East Asian Demographics
During last 25 years, East Asian Nations had a
sharp decrease in their ‘dependency ratio’.
 Dependency ratio is the % of people in their
non-working years (children & seniors.
 Dependents are dis-savers and nondependents are savers.

East Asian Demographics



Due to plummeting birth
rates, East Asia had a
plummeting ratio of youths
as a share of population
This put a large share of
population in high savings
years.
Share of prime age adults
has hit its peak in most
Asian countries and will fall
over the next half century.
China
Hong Kong
Indonesia
Japan
South Korea
Malaysia
Singapore
Taiwan
Thailand
Change in Age Shares
%Below 15
% Prime Age 20-59
1950-1990
2005-2025
-13.56
0.41
-20.64 NA
-7.26
5.52
-16.72
-4.03
-18
-4.12
-7.7
7.5
-20.22
8.35
-18.82 NA
-14.74
0.25
Interest Rates: Incentives and Effects



A rise in interest rates increases the payoff to savings and
increases the incentive to save. Substitution Effect (Plus
Factor for All)
A rise in the interest rate reduces the amount of savings you
need to do to meet target level of future consumption. Income
Effect (Minus Factor for Net Savers).
A rise in the interest rate reduces the amount of borrowing
you can do and still meet some target lever of future
consumption. Income Effect (Plus Factor for Net Borrowers)
Aggregate Savings & Interest Rates




Interest rates have a positive impact on savings by
borrowers, i.e. borrowers reduce their borrowing.
Interest rates have an ambiguous effect on savings
by savers.
Since there is positive net savings, interest rates
have ambiguous effect on aggregate savings.
Empirically, impact of interest rates on savings are
hard to detect.
Saving Curve
S (r )
r
Q: Why does the curve slope
up? Empirical work
suggests substitution effect
is slightly more powerful
than income effect.
Q: What shifts the curve?
Changes in current income
relative to future income.

I (r )

I
Household born in period 0 and lives until period T. (T+1
period lives)
Household begins with real financial wealth F

Present value of consumption equals present
value of financial & human wealth
C3
C1
C2
CT
C0 


 ... 
W  F
2
3
T
1  r 1  r 
1  r 
1  r 
Y3
Y1
Y2
YT
W  Y0 


 ... 
2
3
T
1  r 1  r 
1

r
1

r




If Y grows at constant rate

Yt = (1+g)tY0
(1  g )Y0 (1  g ) 2 Y0 (1  g )3 Y0
(1  g )T Y0
W  Y0 


 ... 
2
3
T
1 r
1

r
1

r
1

r







 (1  x  x 2  ...xT )Y0 

1

 1
1  11gr
 
1 g
1 r
T 1
Y

1
 1  xT 1 Y0
1 x
0
Annuity Value



Just as any stream of future payments has a
present value, so does it have an annuity value.
An annuity is an asset that makes a constant
payment every period, for a number of years, T.
Such an annuity has a present value.
The annuity value of any amount is the size of the
payment of an annuity whose present value is equal
that amount.
Present Value of an Annuity Payment:
Annuity Value of Present Wealth



The real present value
of an annuity with
payment Y.
Off-the-shelf formula
for geometric sum
Solve for present value
of an annuity Y
PVt T  Y 
Y
1  r 

Y
1  r 
2

Y
1  r 
3
 ... 
Y
1  r 
T

1
1
1
1
 Y 1 


 ... 
2
3
T

1  r  1  r  1  r 
1  r 
1
1
1
1  r 

1
1  r 
1
PVt T 
2
 ... 
1
1  r 
T




1
1  r 
1
T 1
1
1 r
1
1  r 
T 1
1
1
1 r
Y
5)
Annuity Value of a Present Value



If you have some
current lump sum, PV,
payment and you want
to buy a annuity for T
periods.
Q: How big an annuity
payment Y can you
get.
A: Invert Equation 5)
YT 
1
1
1
1 r
1
1  r 
T 1
 PV
Permanent Income

The permanent income theory says that
households keep consumption smooth
consuming the annuity value of their financial
wealth, F, plus the present value of lifetime
income, W.
C
1
1
1
1 r
1
1 r T 1
 [W  F ]
Example


1 11 r
The fraction is referred to as the propensity to
consume out of wealth.
A household lives for = 40 periods and the
real interest rate is .02. In every period they
would consume a fraction of their wealth
 .0353
equal to


T 1
1  11 r 
1 11.02
1
41
1
1.02
Applications: Wealth Effect




Changes in asset prices will change the current
value of financial wealth.
The effect of an increase in financial wealth on
consumption is called the wealth effect.
According to the PIH, a one dollar increase in the
value of a stock portfolio should lead to an increase
in consumption equal to the propensity to consume
out of wealth.
Econometricians estimate that the wealth effect to
be less than $.05 consistent with our theory.