Consumption & Savings
Romer Chapter 7
Topics
1.
2.
3.
4.
What is savings?
Consumption, savings and income
Savings and the Interest Rate
Uncertainty and Savings
The Data
• Data on Expenditure Categories are typically
obtained from the National Income and Product
Accounts gathered by the statistical authorities.
• USA: Bureau of Economic Analysis, Dept. of
Commerce
– The national income and product accounts provide an
aggregated view of the final uses of the Nation's output
and the income derived from its production; two of its
most widely known measures are gross domestic
product (GDP) and gross domestic income (GDI). BEA
also prepares estimates of the Nation's stock of fixed
assets and consumer durable goods.
Data
• In HK, data is collected by the Census and
Statistics Department: NIPA Tables
• The U.N. maintains statistical databases for a
wide variety of countries UN Main Aggregates
Database
Consumption in HK
• Four consumption
categories
1. Food
2. Non-Durables: Clothes,
Toys
3. Durables: White Goods,
Electronics
4. Services: Health, Rental
Consumption Shares in HK
140
120
100
80
60
40
20
0
1970 1975
1980 1985
FOOD
DURABLES
Source: CEIC Database
1990
1995 2000
NONDURABLES
SERVICES
Categories of
Spending
BEA NIPA Table 2.3.5
Gross domestic product.......
2005
12455.8
Personal consumption expenditures.
8742.4
Durable goods...................
Motor vehicles and parts......
Furniture and household
equipment....................
Other.........................
1033.1
448.2
Nondurable goods................
Food..........................
Clothing and shoes............
Gasoline, fuel oil, and other
energy goods.................
Other.........................
2539.3
1201.4
341.8
Services........................
Housing.......................
Household operation...........
Electricity and gas.........
Other household operation...
Transportation................
Medical care..................
Recreation....................
Other.........................
5170.0
1304.1
483.0
199.8
283.2
320.4
1493.4
360.6
1208.4
377.2
207.7
302.1
694.0
HK Short-term: Year to year growth
0.4
0.3
0.2
0.1
Durables
NonDurables
GDP
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
0
-0.1
-0.2
-0.3
Theory of consumption best explains non-durables, services and food
consumption. HK NIPA Table 038
Savings
• Output which is not devoted toward current
consumption
Gross Savings = Income – Personal Consumption
Expenditure – Government Consumption Expenditure
2005
BEA NIPA
Tables
Personal consumption expenditures
Government Expenditure
and gross investment
Less Government Investment
Gross Consumption
Gross domestic product
Less Gross Consumption
Gross Savings
As a share of GDP
8742.4
2372.8
397.1
10718.1
12455.8
10718.1
1737.7
14.0%
Personal
Savings
vs.
Gross
Savings
What’s
Missing?
Personal income
Compensation of employees, received
Proprietors' income
Rental income
Personal income receipts on assets
Personal current transfer receipts
Less: Personal current taxes
Equals: Disposable personal income
Less: Personal outlays
Personal consumption expenditures
Durable goods
Nondurable goods
Services
Personal interest payments\1\
Personal current transfer payments
Equals: Personal saving
Personal saving as a percentage of
disposable personal income
2004
2005
9731.4 10239.2
6665.3 7030.3
911.1
970.7
127.0
72.8
1427.9 1519.4
1426.5 1526.6
1049.8 1203.1
8681.6 9036.1
8507.2 9070.9
8211.5 8742.4
986.3 1033.1
2345.2 2539.3
4880.1 5170.0
186.0
209.4
109.7
119.2
174.3
-34.8
2.0
BEA NIPA Tables
Retained Earnings and Depreciation are not counted
in Personal Savings
-0.4
Gross Saving
Gross saving
Net saving
Net private saving
Personal saving
Undistributed corporate profits
Net government saving
Consumption of fixed capital
Private
Government
Gross domestic investment,
capital account transactions, and net lending,
Bureau of Economic Analysis
2005
1612
7.2
319.7
-34.8
354.5
-312.5
1604.8
1352.6
252.2
1683.1
19
50
19
52
19
54
19
56
19
58
19
60
19
62
19
64
19
66
19
68
19
70
19
72
19
74
19
76
19
78
19
80
19
82
19
84
19
86
19
88
19
90
19
92
19
94
19
96
19
98
20
00
20
02
20
04
20
06
20
08
20
10
USA
1,950
1,900
1,850
1,800
1,750
1,700
1,650
1,600
1,550
Hours per Worker
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.
Keynesian Consumption Function
• Consumption Function
C = A + mpc×[GDP – TAX]
– C = Household Consumption Expenditure
– A = Autonomous Consumption { Consumption not
dependent on current income}
– mpc = Marginal propensity to consume
• {Fraction of extra income will be spent on consumption}
• mpc will be smaller than consumption to GDP ratio if A is
positive.
Why do Chinese Save so Much?
Why do Americans Save so Little?
China
UN Main Aggregates Data Base
USA
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
45.00%
40.00%
35.00%
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
East Asian Savings Rates
•
•
•
As a region, East Asia has high savings rates. These high
savings rates have helped finance high rates of capital
accumulation and growth.
Why have East Asian savings rates been so high? Culture?
Luck?
Period
Saving
5 Macao SAR of China(Patacas)
2003
55.02%
Will it last?
GDP  C  G
s
GDP
UN Main Aggregates
Data Base
9
12
13
22
23
25
30
41
55
68
108
Singapore(Singapore Dollars)
2003
China(Yuan Renminbi)
2003
Malaysia(Ringgit)
2003
Thailand(Baht)
2003
Republic of Korea(Wons)
2003
Hong Kong SAR of China(Hong Kong 2003
Dollars)
Vietnam(Dong)
2003
Japan(Yen)
2003
Canada(Canadian Dollars)
2003
Germany(Euros)
2003
United States(Dollars)
2003
44.89%
42.48%
42.34%
33.27%
33.02%
31.92%
28.21%
25.49%
24.30%
21.43%
13.50%
Cultural Reasons
• mpc simply depends on cultural factors and not
economic factors.
• Hayashi, 1989 Japan's Saving Rate: New Data
and Reflections
• Japan: 1960-1990 Savings Rate averaged
about 30%
• Japan 1880-1935 Savings Rate average less
than 15%!
Japanese Gross Saving Rate 1994-2004
Source: CEIC Database
0.32
0.3
0.28
0.26
0.24
0.22
0.2
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Income and Savings
Present Discounted Value
• Life cycle consumption functions assume that
households consider not just the current flow of
income but the present value of lifetime income.
• Consider a stream of income received over
time {y0, y1, …, yT}. This is equivalent in value
to a certain amount of current income,
pvy < y0+ y1+ …+yT.
• Funds available today are worth more than
equivalent funds which are not available until
the future.
Present value
• Reason: Today can earn interest.
– Q: How much do you need today to have yt in t
periods.
y
• Answer:
t
(1  r )
t
• A future payment discounted by the interest rate
raised by the number of periods that must be
waited until the payment is made is called the
present value.
Present value of a stream of
payments
• Households earn a stream of income over
their lifetime. {y0, y1, …, yT}.
• Present value of an income stream is the sum
of the present values of each payment.
y1
y2
yT
pv y  y0 

 ....
1
2
T
(1  r ) (1  r )
(1  r )
Consumption, Savings, and Future
Consumption
• The decision of the household to spend money on
goods is a simultaneous decision not to save this
money in the form of financial assets.
• A decision not to save money for the future is
simultaneously a decision not to have that wealth
available in the future to purchase consumption
goods.
• The consumption decision is based on a trade-off
between the welfare gained from consumption today
and welfare from consumption based in the future.
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.
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
B0  Y0  F0  C0
B1  Y1  (1  r ) B0  C1 
B2  Y2  (1  r ) B1  C2 
B3  Y3  (1  r ) B3  C3 
B1

B2

1  r 
1  r 
2
B3
1  r 
3

Y1
 B0 
Y2

1  r 
1  r 
2
Y3
1  r 
3

C1
1  r 
B1
C2

1  r  1  r 2
B2
1  r 
2

C3
1  r 
3
Combine the period-by-period savings
equations.
• Present value of consumption equals present
value of financial & human wealth
C3
C1
C2
CT
C0 


 ... 
 W  HW  F
2
3
T
1  r 1  r  1  r 
1  r 
Y3
Y1
Y2
YT
HW  Y0 


 ... 
2
3
T
1  r 1  r  1  r 
1

r
 
Algebra Trick
• If x ≠ 1, then 1  x  x 2  ....xT  1  1  xT 1 

1 x 
– If x = 1, 1+ x +…+ xT = T+1
• If x ≠ 1
x
x  x  x  ....x 
 1  xT 
1 x
2
– If x = 1, x+ x +…+ xT = T
T
Annuities & Annuity Value
• Just as any stream of future payments has a
present value, so does any current sum have an
annuity value.
• An annuity is an asset that makes a constant
payment every period, for a number of years, T.
Such an asset has a present value.
• The annuity value of any current amount is the
annuity payment generated by an annuity whose
present value is equal that current amount.
Present Value of an Annuity Payment:
Annuity Value of Present Wealth
•
•
•
The real present
value of an
annuity with
payment YP.
Off-the-shelf
formula for
geometric sum
Solve for present
value of an
annuity Y
Vt  Y 
T
P
YP
1  r 

YP
1  r 

2
YP
1  r 
3
 ... 
YP
1  r 

1
1
1
1
 Y 1 



...

2
3
T
 1  r  1  r  1  r 
1  r 
P
1
Vt 
T
1
1  r 
T 1
1
1
1 r
Y
P
T



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)
1
1
P
1  r W
Y 
1
1
T 1
1  r 
Permanent Income
• We define a households, permanent income
as the annuity value of its wealth, W.
1
r
1
W
1  r W  1  r
YP 
1
1
1
1
T 1
T 1
1  r 
1  r 
• Conceptually, if the household borrowed on all
of its future income and added that to its
financial wealth, it could buy an annuity
generating perfectly smooth income.
Permanent Income and Average Income
• If FW = 0, and r = 0, then YP = W/T
W HW Y0  Y1  Y2  ...  YT
P
Y 


T
T
T
• If r > 0, then Annuity Value is a weighted
average of lifetime income with larger weights
on current income than on income in the far
future.
Permanent Income and Current Income
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
HW  Y0 


 ... 
2
3
T
1 r
1  r 
1  r 
1  r 

1

 1
1 g
1  1 r
 
1 g
1 r
T 1

1
P
Y0  Y 
1
1
1 r
1 g
1 r
 1  

1 g T 1
1 r
1  

1 T 1
1 r
Y

0
Permanent Income and Current Income
If Y is mean reverting
Yt  Y  Yt BC , Yt BC  Yt BC1   tY0BC
W
r  0, C0 
T 1
HW  T  1  Y  Y0BC  (  )Y0BC  (  ) 2 Y0BC  (  )3 Y0BC  ...  (  )T Y0BC


1
HW  T  1  Y 
 1   T 1  Y0
1 
1
1
P
Y Y 

 1   T 1 Y0
T  1 1  


Intratemporal Utility Function
• A household will exist for t = 0,…,T periods then
expire.
• Household will enjoy a stream of consumption
spending {c0, c1, c2,….cT}
• Households preferences over this stream can be
defined by a utility function
U = U(c0, c1, c2,….cT)
• Often a utility function is represented as a weighted
sum of utility in each period (called felicity functions).
Example: Felicity
• Agents get the same utility from consumption
in each period.
• Households lifetime utility is a weighted sum
of the felicity that they receive in each period.
• The per-period utility of the household is called
the felicity function, u(ct).
• Felicity displays diminishing returns from
consumption u’(C) > 0, u’’(C) < 0
Felicity Function
u’(c)
u(c)
c
Example: Time separable utility
• Weights are higher in earlier period due to
households impatience. Households discount
future utility.
U = u(c0) + β u(c1) + β2 u(c2) + β3u(c3)+….
• Rate at which the household discounts future
utility is time discount rate.
Maximize Discounted Utility
• Maximize
max u (C0 )   u (C1 )   u (C2 )  ...   u (CT )
2
T
T
t

 u (Ct )
t 0
s.t.
C1
C2
CT
C0 

 ... 
 HW  F
2
T
1  r 1  r 
1  r 
Lagrangian Penalty
• Assume that there is some utility cost λ of
overspending the budget constraint. Maximize
utility including this cost and set λ as small as
necessary so that people exactly hit their
budget constraint.
T
T
max   u(Ct )  {
t
t 0
t 0
Ct
1  r 
t
W}
First-Order Conditions
• Budget Constraint Holds
C3
C1
C2
CT
C0 


 ... 
 HW  F
2
3
T
1  r 1  r  1  r 
1  r 
• For each period, discounted marginal utility
equals discounted cost of spending one more
good over the limit.
u '(C0 )   ,  u '(C1 ) 
 u '(C2 ) 
2

(1  r )

(1  r )
,  u '(C3 ) 
3
2
,

(1  r )
,...,  u '(CT ) 
T
3

(1  r )T
Euler Equation
• The marginal utility of consumption in one
period is equal to the marginal benefit of
waiting one period which is the consuming the
good plus interest times the extra utility gained
from extra future consumption discounted by
impatience.
u '(Ct )    (1  r )  u '(Ct 1 )
Permanent Income
• Permanent Income Hypothesis:
β(1+r)=1 then c0=c1
• 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
1
income, W.
1
C
1
1 r
1
1 r T 1
 [ HW  F ]  Y P
Example
• The fraction is referred to as the propensity to
1 11 r
consume out of wealth.
1
T 1
1
1 r

• A household lives for = 40 periods and the real
interest rate is .02. In every period they would
consume a fraction of their wealth equal to
1 11.02
1 11.02 
41
 .0353
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.
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
• 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
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.
Applied Consumption Function
• Optimal consumption is a linear function of
human wealth and financial wealth. Both
growth part and cyclical part of human wealth
is proportional to current income.
C  aHW  bFW   a  d  Y  b  FW
*
t
• Dynamics of consumption expenditure self
correct to the optimal level


Ct   0  1  Ct 1  ...   j  Ct  j  f  Ct 1  Ct*1  g  Ct* j   t
Interest Rates and Savings
Two period problem
• Intuition from this problem can be derived
using simplest version of the theory in which T
= 1. U = U(c0, c1)
– Thus, there is only a current period (t = 0) and a
future period (t = 1).
• Preferences can be represented with 2
dimensional indifference curves.
Preferences
• People prefer some combinations of present and
future consumption.
– More is better. If two combo’s have equal future
consumption, choose the combo with more present
consumption.
– Smooth over time. Households have diminishing returns
to consumption in any period.
– Consumption is a normal good If income goes up,
ceteris parabis, consumption goes up in all periods.
• Preferences are represented by indifference curves
– Smooth sets of combo’s amongst which the
household is indifferent.
Indifference Curves
c1
I3
I2
I1
c0
Savings and the Budget Constraint
• Agents start out with a certain amount of financial
wealth fw, that they carried over from the previous
period.
• In each period, household will earn income from
producing goods yt. Households will also have a
fixed tax obligation taxt. Household after tax income
is yt .
• Balance is any funds that are left over after
consumption bt = yt +fwt - ct
• Financial wealth will just be balance plus some
goods interest rate, fwt = (1+r)bt-1
Intratemporal Budget Constraint
• The savings in period 1, the last period of life
will be 0. We can write this as a budget
constraint for each period. Assume fw0 = 0
– Time 0: c0 = y0 + fw0 – b0
– Time 1: c1 = y1 +(1+r)b0
Intertemporal Budget Constraint
• We can combine these budget constraint into
one intertemporal budget constraint.
– Divide time 1 budget constraint by (1+r)1 and add
up the budget constraints.
c1
y1
c0 
 y0 
 fw  hw  fw
1 r
1 r
fw  0
c1
y1
 y0 
 hw
0
1 r
1 r
Intuition of Budget Constraint
• The intertemporal budget constraint says that
the present value of consumption must be
equal to the present value of after-tax income
plus initial financial wealth.
• The present value of after-tax income could be
referred to as human wealth as it is the value
of the households ability to produce goods in
the future.
Budget Constraint
• There is a trade-off between consumption today and
consumption and the future which can be
represented geometrically.
• If the household has zero future consumption, it can
consume c0= hw0.
• If the household has zero consumption today it can
consume c1 = (1+r)(hw0).
• For each good given up in period 0, the household
can get an extra (1+r) in period 1.
• The s0 point is on the budget constraint. Define au0 =
y0 and au1 = y1
Budget Constraint
c1
(1+r)w
1+r
au1
au0
w
c0
Preferences
• Principles describe consumer preferences.
1. More is better: Higher indifference curves are
preferred.
2. Diminishing returns to consume in any period. The
slope of the indifference curves is decreasing.
 Consumption in every period is a normal good. Increases
in income increase consumption of a normal good.
 2U  2U
, 2 0
2
c0 c1
Optimum
• Optimal consumption choice is:
– on an indifference curve tangent to the budget
constraint (so the slope of the indifference curve is
equal to 1+r).
– Where marginal rate of substitution is equal to real
interest rate
MU 0
MRS 

MU1
U
c0
U
c1
 (1  r )
Optimum
• The optimal choice is also the solution to a
maximization problem.
c1
y1
max U (c0 , c1 ) s.t. c0 
 y0 
c0 , c1
1 r
1 r
y1
c1
max U ( y0 

, c1 )
c1
1 r 1 r
dU
1
U
U
0
U1  U 2 
 (1  r ) 
dc1
1 r
c0
c1
• Marginal utility of consumption at time t is
marginal felicity discounted by the discount
factor MUC,t = βt × u’(ct)
• Marginal rate of substitution of consumption
between two periods is the ratio of marginal
utility.
MRS 
MU t  j
MU t
 
j
u '(ct  j )
u '(ct )
Specific Utility
• Felicity function is the natural logarithm, u(ct) =
ln(ct)
c
MRS 

c
• Two period case.
1
c0
1
1
c1
0
c1
max ln(c0 )   ln(c1 ) s.t. c0 
w
1 r
c1
1
1
1
1 r
1
max ln( w 
)   ln(c1 )   

0

c1
1 r
c1 1  r w  c1
c1
c0
1 r
Optimal Consumption: Lender
c1
(1+r)w
c1*
au1
c0*
au0
W
c0
Optimal Consumption: Borrower
c1
(1+r)W
au1
c1*
au0
c0 *
W
c0
Implications
• Current and future income affect optimal consumption
only through their affect on fw. An increase in fw results
in a parallel shift in budget constraint.
• Normal Good: An increase in w results in an increase in
both present and future consumption.
– A temporary increase in current income alone will lead to an
increase in w, an increase in current consumption and saving.
– The expectation of an increase in future income will lead to an
increase in w, an increase in current consumption and a
decrease in savings.
– A permanent increase in income will increase both current and
future consumption as well as current and future income and
therefore have negligible impact on consumption and savings.
Increase in Current Income
c1
(1+r)(w’)
{c0**,c1**}
(1+r)w
c1
Current Autarky
income increases.
Future autarky does
not.
Savings must rise.
**
c1*
c0*
c0
**
w
w’
c0
Increase in Future Income
c1
(1+r)(w’)
{c0**,c1**}
(1+r)w
c1
Future Autarky
income increases.
Current autarky does
not.
Savings must rise.
**
c1*
au1
c0*
c0
**
w
w’
c0
Consumption Smoothing
• Because consumption faces diminishing
returns in any period, consumers have an
incentive to allocate temporary increases in
income to all periods.
• Consumption will be smoother than income at
given interest rate. This matches the reality.
However, quantitatively, consumption not
smooth enough.
– Interest rate moves endogenously
– Borrowing constraints.
Rise in Interest Rate
c1
(1+r)w
au1
au0
w
c0
Rise in Interest Rate
• A change in the interest rate results in a pivot
in the budget constraint around the no-savings
point.
• Two basic affects of a change in the interest
rate.
– Substitution Effect: The real interest rate is the
relative price of consuming today (relative to future
consumption).
– Income Effect: The real interest rate affects the
budget opportunities available to agents.
Before interest rate rise, optimal c0
= au0
After interest rate rise, steeper
budget constraint crosses
indifference curves along for
higher utility to left of au0. Optimal
consumption drops, savings rise.
au0,au1
Before interest rate rise, optimal c0 >
au0
After interest rate rise, new steeper
budget constraint is to the left of
previous consumption level.
Household must cut back on current
consumption just to get to affordable
consumption combination. Savings
rise.
au0,au1
Before interest rate rise, optimal c0 < au0.
After interest rate rise, new steeper
budget constraint is to the right of
previous consumption level. New set of
affordable consumption combinations
which make the household better off,
some of which can involve less
consumption in period 1. Savings may
rise or fall.
Income Effect
• Change in interest rates changes the value of your
savings.
– For savers, (i.e. consumption to left of autarky point), a rise
in interest will increase the future value of those savings
increasing lifetime income
– For debtors, (i.e. consumption to right of autarky point) a
rise in the interest rate will increase future costs of paying
debts reducing lifetime income.
• If income goes up, you will have a tendency to
consume more in both periods. If income falls, you
will have a tendency to consume less in both periods
and savings will rise.
Substitution Effect
• A rise in the interest rate will make
consumption today more expensive relative to
consumption in the future.
• A rise in the real interest rate will lead to a
reduction in consumption today relative to
consumption in the future.
How strong is the substitution effect?
• Constant Elasticity Intertemporal Substitution
Utility Function
1
u (c ) 
c
1

1

c0
1


1
,  0  u '(c)  c 
1
1


  1  r  c1
1



c0
   1  r  
c1
• When ψ = 1, the CEIS felicity is natural log for
all intents and purposes. Natural log felicity is
sometimes referred to as unit elasticity of
intertemporal substitution.
Point Elasticities
• Elasticity is the % change in one variable
caused by a % change in another variable.
• Elasticity of substitution is the % change in the
demand for one variable relative to another.
• Functions with constant elasticities are log
linear.
Income Effect: Borrowers
• For borrowers, households to the right of au, an
increase in the interest rate offers a lower budget
constraint, which allows less present consumption if
we keep future consumption constant.
• Substitution effect and income effect work the same
way. Present consumption drops relative to future
consumption but at any given future income,
affordable present consumption will drop.
– Present consumption of borrowers will drop if real interest
rate rises.
Income Effect Lenders
• For lenders, households to the left of au0, an
increase in the interest rate offers a higher budget
constraint and allows higher present consumption if
we keep future consumption constant.
• Income and substitution effects will work in opposite
ways. A rise in the interest rate reduces current
consumption relative to future consumption, but at
any given future consumption a higher level of
present consumption is affordable.
– Effect of an increase in the interest rate on consumption is
ambiguous.
Effect of Interest Rate on Savings
• Empirically, opinion on the effect of interest
rate on savings varies in a range from zero to
mildly positive.
The Effect of Interest-Rate Changes on
Household Saving and Consumption: A
Survey Douglas W. Elmendorf 1996-27
Uncertainty and Savings
Taiwan, National Health Insurance
• In 1995, Taiwan implemented a scheme
providing national health insurance to all
islanders.
• This program raised coverage rates from 57%
to 97%
• Aggregate gross savings declined in Taiwan.
• Careful study shows this to be concentrated
among low income households who were not
previously covered.
National Health Insurance and precautionary saving: evidence from Taiwan
Shin-Yi Chou , Jin-Tan Liu , James K. Hammitt
Taiwan Gross Saving Rate
(Taiwan National Income Accounts)
0.345
0.335
0.325
0.315
0.305
0.295
0.285
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Precautionary Savings
• Decision making is taken under certainty.
• Most saving is done under a cloud of
uncertainty about the future.
• Question: How does the uncertainty
environment affect the willingness to save?
• Return to Polonius.
– Assume that β = (1+r) = 1
– If fw0 = tax0 = tax1 = 0, & y0 = y1 then c0 = y0
• Return to the two period life problem. Abstract from
taxes and initial financial wealth.
• When consumption decision is made, the household
knows its current income, y0. However, second
period income is a random variable.
• Assume that there are two equally likely future
outcomes, good and bad. If the outcome is good, the
household will have income y0 + x. If the outcome is
bad, the household will have income y0 – x.
• Expected household income is
.5 *(y0 + X) + .5* (y0 - X) = y0.
Decision making under uncertainty
• Most popular decision making paradigm is
maximize expected utility subject to the
budget constraint.
– Pick three variables, c0, c1,GOOD cBAD
• Expected utility is
– U(c0) + .5 * u(c1,GOOD) + .5*u(c1,BAD)
• Budget constraints
– C1,GOOD = y1 + x + (y0-c0)
– C1,BAD = y1 -x + (y0-c0)
Maximization problem
• Max u(c0) + .5 ∙ u(y1 + x + (y0-c0)) +
+
.5 ∙ u(y1 - x + (y0-c0))
• 0 = u’(c0) - .5 * u’(c1,GOOD)- .5*u’(c1,BAD)
• u’(c0) = E[ u’(c1)]
Under uncertainty, set marginal
utility today equal to expected
marginal utility tomorrow.
Marginal utility function
• Further assume utility is a diminishing function
of consumption.
• This is true if utility is a constant intertemporal
elasticity of substitution function.
• Expected value of marginal utility is greater
than marginal utility of expected value.
Expected Marginal Utility is greater
than Marginal utility of expected value
B: E[u’(y0)]
A: u’(c0)
u’(c)
y0 - x
y0
y0 + x
c
Precautionary Savings
• If household sets marginal utility of
consumption today equal to marginal utility of
expected value tomorrow, this would be less
than expected value of marginal utility
tomorrow.
• Must act to increase marginal utility today or
reduce marginal utility in the future (i.e. shift
income toward the future)
• The household will shift income away from
periods of certainty toward periods of
uncertainty or save as insurance.
Precautionary savers & spenders
u’(c)
Precautionary
Savers
B: E[u’(y0)]
A: u’(c0)
Precautionary
Spenders
Certainty
Equivalent
c
Optimal Consumption: Borrowing
Constraints c0 = au0
c1
(1+r)(w)
au1
c1*
au0
c0 *
w
c0
Buffer Stock Savings
• Borrowing constraints and precautionary
savings interact.
• If short-term income falls sharply and borrowing
constraints hold, then consumption in bad
states may fall dramatically.
• Expected marginal utility of consumption may
be high due to this downside risk.
• Precautionary savings should fall as income
rises because high income people have a
smaller chance of hitting liquidity constraint.
Social Insurance, Financial Credit &
Savings
• Various government programs may reduce the
uncertainty of income.
• Social welfare or health insurance may reduce
the individual unpredictability of insurance and
reduce the need for precautionary savings.
• A more smoothly operating financial system
may also reduce the need for precautionary
savings.
MPC
• Under certainty with perfect financial markets,
the marginal propensity to consume out of
temporary income must be very small (as
shown by the wealth effect in stock markets.
• Propensity to consume increases if large
share of consumers face borrowing
constraints or precautionary motives are large.