Consumption & Savings

Romer Chapter 7

Topics

1. What is savings?

2. Consumption, savings and income

3. Savings and the Interest Rate

4. 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 categories1. Food

2. Non-Durables: Clothes, Toys

3. Durables: White Goods, Electronics

4. Services: Health, Rental 0

20

40

60

80

100

120

140

1970 1975 1980 1985 1990 1995 2000

FOODDURABLES

NONDURABLESSERVICES

Consumption Shares in HK

Source: CEIC Database

Categories ofSpending

2005 Gross domestic product....... 12455.8 Personal consumption expenditures. 8742.4 Durable goods................... 1033.1 Motor vehicles and parts...... 448.2 Furniture and household equipment.................... 377.2 Other......................... 207.7 Nondurable goods................ 2539.3 Food.......................... 1201.4 Clothing and shoes............ 341.8 Gasoline, fuel oil, and other energy goods................. 302.1 Other......................... 694.0 Services........................ 5170.0 Housing....................... 1304.1 Household operation........... 483.0 Electricity and gas......... 199.8 Other household operation... 283.2 Transportation................ 320.4 Medical care.................. 1493.4 Recreation.................... 360.6 Other......................... 1208.4

BEA NIPA Table 2.3.5

HK Short-term: Year to year growth

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.419

67

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

Durables

NonDurables

GDP

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

Personal consumption expenditures 8742.4Government Expenditure and gross investment 2372.8Less Government Investment 397.1Gross Consumption 10718.1 Gross domestic product 12455.8 Less Gross Consumption 10718.1Gross Savings 1737.7As a share of GDP 14.0%

BEA NIPA Tables

Personal Savings

vs.Gross

Savings

2004 2005

Personal income 9731.4 10239.2 Compensation of employees, received 6665.3 7030.3 Proprietors' income 911.1 970.7 Rental income 127.0 72.8 Personal income receipts on assets 1427.9 1519.4 Personal current transfer receipts 1426.5 1526.6Less: Personal current taxes 1049.8 1203.1Equals: Disposable personal income 8681.6 9036.1Less: Personal outlays 8507.2 9070.9 Personal consumption expenditures 8211.5 8742.4 Durable goods 986.3 1033.1 Nondurable goods 2345.2 2539.3 Services 4880.1 5170.0 Personal interest payments\1\ 186.0 209.4 Personal current transfer payments 109.7 119.2Equals: Personal saving 174.3 -34.8 Personal saving as a percentage of disposable personal income 2.0 -0.4

What’s Missing?

Retained Earnings and Depreciation are not counted in Personal Savings

BEA NIPA Tables

Gross Saving 2005 Gross saving 1612Net saving 7.2 Net private saving 319.7 Personal saving -34.8 Undistributed corporate profits 354.5 Net government saving -312.5Consumption of fixed capital 1604.8 Private 1352.6 Government 252.2 Gross domestic investment, 1683.1 capital account transactions, and net lending,

Bureau of Economic Analysis

USA

1,550

1,600

1,650

1,700

1,750

1,800

1,850

1,900

1,950

1950

1952

1954

1956

1958

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

2010

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?

0.00%

5.00%

10.00%

15.00%

20.00%25.00%

30.00%

35.00%

40.00%

45.00%19

70

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

China USA

UN Main Aggregates Data Base

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?

• Will it last?

GDP C Gs

GDP

Period Saving5 Macao SAR of China(Patacas) 2003 55.02%9 Singapore(Singapore Dollars) 2003 44.89%

12 China(Yuan Renminbi) 2003 42.48%13 Malaysia(Ringgit) 2003 42.34%22 Thailand(Baht) 2003 33.27%23 Republic of Korea(Wons) 2003 33.02%25 Hong Kong SAR of China(Hong Kong Dollars)2003 31.92%30 Vietnam(Dong) 2003 28.21%41 Japan(Yen) 2003 25.49%55 Canada(Canadian Dollars) 2003 24.30%68 Germany(Euros) 2003 21.43%

108 United States(Dollars) 2003 13.50%

UN Main Aggregates Data Base

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 1994-2004Source: CEIC Database

0.2

0.22

0.24

0.26

0.28

0.3

0.32

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Income and Savings

Present Discounted Value• Life cycle consumption functions assume that h

ouseholds 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.

• Answer:

• 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.

(1 )t

t

y

r

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.

1 20 1 2

....(1 ) (1 ) (1 )

Ty T

y y ypv y

r r 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

0 0 0 0

1 1 11 1 0 1 0

2 2 1 22 2 1 2 2 2 2

3 3 323 3 3 3 3 3 2 3

(1 )1 1 1

(1 )11

(1 )1

1

1 1 1

1

B Y F C

B Y CB Y r B

B Y CBB Y r B C

r r

C Br r r

B Y B CB Y r B C

rr r r

r r

Combine the period-by-period savings equations.

• Present value of consumption equals present value of financial & human wealth

31 2

0 2 3 ...1 1 1 1

TT

CC C CC W HW F

r r r r

31 2

0 2 3 ...1 1 1 1

TT

YY Y YHW Y

r r r r

Algebra Trick

• If x ≠ 1, then

– If x = 1, 1+ x +…+ xT = T+1

• If x ≠ 1

– If x = 1, x+ x +…+ xT = T

2 111 .... 1

1T Tx x x x

x

2 .... 11

T Txx x x x x

x

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

2 3

2 3

...1 1 1 1

1 1 1 11 ...

1 1 1 1

P P P PT P

t T

PT

Y Y Y YV Y

r r r r

Yr r r r

1

11

11

11

T

T Pt

rV Y

r

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

1111

11

P

T

rY W

r

Permanent Income• We define a households, permanent income

as the annuity value of its wealth, W.

• 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.

1 1

111 11 1

1 11 1

P

T T

rW

r rY W

r r

Permanent Income and Average Income

• If FW = 0, and r = 0, then YP = W/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.

0 1 2 ...P TY Y Y YW HWY

T T T

Permanent Income and Current IncomeIf Y grows at constant rate

• Yt = (1+g)tY0

2 30 0 0 0

0 2 3

111 111 1

0 011 1 111 1 1

(1 ) (1 ) (1 ) (1 )...

1 1 1 1

1111

1 1 1

T

T

TgrTg P r

rg g Tr r r

g Y g Y g Y g YHW Y

r r r r

Y Y Y

Permanent Income and Current IncomeIf Y is mean reverting

1 0

0

2 30 0 0 0 0

10

10

,

0,1

1 ( ) ( ) ( ) ... ( )

11 1

1

1 11

1 1

BC BC BC t BCt t t t

BC BC BC BC T BC

T

P T

Y Y Y Y Y Y

Wr C

T

HW T Y Y Y Y Y Y

HW T Y Y

Y Y YT

Intratemporal Utility Function

• A household will exist for t = 0,…,T periods then expire.• Household will enjoy a stream of consumption spendin

g {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

20 1 2

0

1 20 2

max ( ) ( ) ( ) ... ( )

( )

. . ...1 1 1

TT

Tt

tt

TT

u C u C u C u C

u C

C C Cs t C HW F

r r 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.

0 0

max ( ) { }1

T Tt t

t tt t

Cu C W

r

First-Order Conditions• Budget Constraint Holds

• For each period, discounted marginal utility equals discounted cost of spending one more good over the limit.

31 2

0 2 3 ...1 1 1 1

TT

CC C CC HW F

r r r r

0 1

2 32 32 3

'( ) , '( ) ,(1 )

'( ) , '( ) ,..., '( )(1 ) (1 ) (1 )

TT T

u C u Cr

u C u C u Cr r r

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.

1'( ) (1 ) '( )t tu C r u C

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 income, W.

1

1111

11

[ ]T

Pr

r

C HW F Y

Example

• 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 equal to

11.02

4111.02

1

1.0353

11

111

1

1

rT

r

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

time

C,Y

S>0

S<0C

Y

S<0

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.

Change in Age Shares%Below 15 % Prime Age 20-591950-1990 2005-2025

China -13.56 0.41Hong Kong -20.64 NAIndonesia -7.26 5.52Japan -16.72 -4.03South Korea -18 -4.12Malaysia -7.7 7.5Singapore -20.22 8.35Taiwan -18.82 NAThailand -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 non-dependents 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.

• Dynamics of consumption expenditure self correct to the optimal level

*tC aHW bFW a d Y b FW

* *0 1 1 1 1...t t j t j t t t j tC C C f C C g C

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

c0

c1

I1

I2

I3

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.

1 10 0

1 10 0

1 10

1 1

c yc y fw hw fw

r rfw

c yy hw

r 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

c0

c1

w

(1+r)w

1+r

au0

au1

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.

2 2

2 20 1

, 0U U

c c

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

0 0

1

1

(1 )

UMU cMRS rUMU

c

Optimum

• The optimal choice is also the solution to a maximization problem.

0 1

1

1 10 1 0 0

,

1 10 1

1 21 0 1

max ( , ) . .1 1

max ( , )1 11

0 (1 )1

c c

c

c yU c c s t c y

r ry c

U y cr r

dU U UU U r

dc r c c

• 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. '( )

'( )t j t jj

t t

MU u cMRS

MU u c

Specific Utility

• Felicity function is the natural logarithm, u(ct) = ln(ct)

• Two period case.

1

10 1 0

11

11 1 0

max ln( ) ln( ) . .11 1 1 1 1

max ln( ) ln( ) 01 1

1

c

cc c s t c w

rc r

w ccr c r c cw

r

0

1

11

10

c

c

cMRS

c

Optimal Consumption: Lender

c0

c1

W

(1+r)w

au0

au1

c0*

c1*

Optimal Consumption: Borrower

c0

c1

W

(1+r)W

au0

au1

c0*

c1*

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 in

crease in w, an increase in current consumption and saving.– The expectation of an increase in future income will lead to an i

ncrease 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

c0

c1

w

(1+r)w

c0*

c1**

w’

(1+r)(w’)

{c0**,c1

**}

c1*

c0**

Current Autarky income increases. Future autarky does not.

Savings must rise.

Increase in Future Income

c0

c1

w

(1+r)w

au1

c0*

c1**

w’

(1+r)(w’)

{c0**,c1

**}

c1*

c0**

Future Autarky income increases. Current autarky does not.

Savings must rise.

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

c0

c1

w

(1+r)w

au0

au1

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

• 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.

11 1

1 1

00 1

1

1( ) , 0 '( )

11

1 1

cu c u c c

cc r c r

c

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)

0.285

0.295

0.305

0.315

0.325

0.335

0.345

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

u’(c)

y0 y0 + xy0 - x

A: u’(c0)

B: E[u’(y0)]

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)

A: u’(c0)

B: E[u’(y0)]

c

Precautionary

Spenders

Precautionary

Savers

Certainty

Equivalent

Optimal Consumption: Borrowing Constraints c0 = au0

c0

c1

w

(1+r)(w)

au0

au1

c0*

c1*

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.