Optimal Consumption over Many Periods

Facts About ConsumptionConsumption Under CertaintyPermanent Income Hypothesis

Uncertainty and Rational Expectations

Facts About Consumption

• Conflicting Consumption Data!MPC out of long-term changes in income higher than short-term (over business cycle).

• Total consumption is smoother than income over time.

• Consumption of non-durables is much less volatile than income while consumption of durables is more volatile than income.

FIGURE 3: Consumer Spending and Disposable Income

Copyright © 2006 South-Western/Thomson Learning. All rights reserved.

Real Disposable Income0

$3,036

$5,619

$6,081$3,432

19291939

19411947

1942 19431945

19551960

1964

1970

1974

19781979

1984

1980

19851986 1987

1988 198919911990

1992

19941996

1976

1995

19971998

1999

20002001

20022003

2004R

eal

Co

ns

um

er

Sp

en

din

g

Figure 3.9 Percentage Deviations from Trend in Real Consumption (black line) and Real GDP

(colored line) 1947–2006

Figure 8.6 Percentage Deviations from Trend in GDP and Consumption, 1947–

2006

A Multi-Period Model

• Consider the case where individuals live for T periods.

• Let a0 = s0(1+r) represent initial wealth.

• Consumers choose {ct} for t = 1,2,…,T which solves

subject to

(BC)

),,,(max 21 TcccU

tttt scrsy )1(1

T

tt

tT

tt

t

r

crs

r

y

11

101 )1(

)1()1(

• Combining (BC) for each period gives the multi-period lifetime (intertemporal) BC:

PDV of Lifetime PDV of Lifetime

Income + Initial Wealth Consumption

Method 1

• LaGrangian:

FOC:

for t = 1,…,T

Combining gives

MRSct,ct+1 = (1+r)

T

t

T

tt

tt

tT r

crs

r

ycccUL

1 110121 )1(

)1()1(

),,(

1)1(

tc rU

t

)1(1

rUUtt cc

Method 2

• LaGrangian:

FOC:

for t = 1,…,T

for t = 1, … T-1

Combining gives

MRSct,ct+1 = (1+r)

T

ttttttT scrsycccUL

1121 })1({),,(

tctU

1)1( tt r

)1(1

rUUtt cc

• An optimal sequence {ct*} for t=1,…,T solves

(T-1 equations)

and the lifetime BC:

)1(1

rUUtt cc

T

tt

tT

tt

t

r

crs

r

y

11

101 )1(

)1()1(

• Example: Time-separable utility function

where < 1 is the time discount factor:

and is the rate of time preference.

)()()()(),,,( 13

22121 T

TT cucucucucccU

T

tt

t cu1

1 )(

)1/(1

• Hence an optimal sequence {ct*} for t=1,…,T solves

(T-1 equations)

and

)1)((')(' 1 rcucu tt

T

tt

tT

tt

t

r

crs

r

y

11

101 )1(

)1()1(

Consumption with Certainty

• The optimal consumption decision, given r, is {ct} for t = 1,2,…,T solving

for t = 0,1,…,T-1, and

)1(),,(

),,(

1

1

1

rccU

ccU

Tc

Tc

t

t

T

tt

tT

tt

t

r

ca

r

ywe

110

11 )1()1(

• Simple Example: (i) Time separable utility with = 0 =1)(ii) Zero interest rate: r = 0.

The permanent income hypothesis – consumption decisions are based upon a constant proportion of lifetime wealth (“permanent income”) – M. Friedman.

01

1ay

Tc

T

tt

• Consumption is smoother than income.• Conflicts with traditional view of the importance of current

income. (current income by itself doesn’t matter!)• Statistical estimation of simple consumption functions may

not be useful:

• Saving will be very sensitive to income (in example,

)• Problem – if future income is known with certainty, then

there should be no consumption fluctuations. (see Graph)

)( TYbaC

cyss ttt 1

PIH with Uncertainty

• Consumers have information about current income but not sure about future income.

• Let Et be the expectation based upon all information up to and including period t.

• Consumer’s Problem: In each period choose {c} and {s} to maximize

subject to

T

tt

t cuE

)(

scrsy )1(1

• The lifetime BC is given by:

• For simplicity, assume(i) = 1 (no time discounting)(ii) r = 0, (zero interest rate)(iii) (quadratic utility)

• FOC at date t=1:

T

tt

tT

ttt

r

cEs

r

yEwe

)1()1( 0

2)2/1()( cccu

)(')(' 211 cuEcu

• Law of Itterated Expecations: The expected value given time-t information of an expected value given time t+1 information of a future variable is the expected value of that future variable conditional on time-t information.

• Example:

221 ttttt zEzEE

• Substituting utility function into FOC:

or more generally

for t = 1, …, T-1

• Notice this implies:

211 cEc

1 ttt cEc

TcEcEcEc 121111

• The lifetime BC is

Substituting in the FOC from previous slide:

This is the “uncertainty” version of PIH.

T

t

T

ttt syEcE

1 1011

T

tt syE

Tc

1011

1

• Calculating c2 and using c1 from above gives:

• This says that changes in consumption over time are due to revisions in the expectation of future income (i.e. new information)

T

t

T

ttt yEyE

Tcc

2 21212 1

1

Random Walk Hypothesis• Robert Hall - “Stochastic Implications of the Life-

Cycle/ Permanent Income Hypothesis: Theory and Evidence,” 1978, Journal of Political Economy

• Recall household FOC:

• If consumers have rational expectations, then this implies consumption follows a random walk:

where

and212 cc 021 E

121 ccE

T

ttt yEyE

T 212 )]()([

1

12

• More generally, consumption follows

where • The random walk hypothesis (R. Hall) says

(i) The best predictor of future consumption is current consumption.

(ii) If current consumption is based upon efficiently utilizing all information about future income, changes in consumption are unpredictable (c = )

(iii) MPC out of permanent shocks to income are larger than MPC out of temporary shocks.

11 ttt cc 01 ttE

• Implications(i) Anticipated changes in income should have no

effect on consumption.(ii) All changes or “shocks” to consumption are permanent.(iii) Consumption Smoothing MPC out of temporary income shocks will be smaller than out of permanent shocks to income.

• Empirical Evidence of Random Walk – anticipated (predictable) changes in income do increase consumption but by much less than 1-1 (50 cents to the $).

• Reasons Random Walk Theory of Consumption does not hold 100%:

(1) Borrowing Constraints: Inability of consumers to have easy access to credit markets.

(2) Market Interest Rate Fluctuations: If all consumers want to borrow market rates are driven up!

• Resolves conflict between various consumption data:

MPC for temporary changes in income < MPC for permanent income changes.

FIGURE 3: Consumer Spending and Disposable Income

Copyright © 2006 South-Western/Thomson Learning. All rights reserved.

Real Disposable Income0

$3,036

$5,619

$6,081$3,432

19291939

19411947

1942 19431945

19551960

1964

1970

1974

19781979

1984

1980

19851986 1987

1988 198919911990

1992

19941996

1976

1995

19971998

1999

20002001

20022003

2004R

eal

Co

ns

um

er

Sp

en

din

g

Figure 3.9 Percentage Deviations from Trend in Real Consumption (black line) and Real GDP

(colored line) 1947–2006

Real GDP & Consumption: 2005-2009

Figure 8.10 Scatter Plot of Percentage Deviations from Trend in Consumption of Nondurables and

Services Versus Stock Price Index

Figure 8.9 Stock Prices and Consumption of Nondurables and

Services, 1985–2006