Transcript Mutli-Period Model of Consumption and PIH

Optimal Consumption over
Many Periods
Facts About Consumption
Consumption Under Certainty
Permanent 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
2004
2003
2002
2001
2000
1998
1997
$5,619
1995
1994
1992
1990 1991
1989
1988
1987
1986
1999
1996
1985
1984
1979
1980
1978
1976
1974
$3,036
1970
1964
1960
1955
1947
1945
1941
1942 1943
1939
1929
0
$3,432
Real Disposable Income
$6,081
Copyright © 2006 South-Western/Thomson Learning. All rights reserved.
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
maxU (c1 , c2 ,, cT )
subject to
yt  st 1 (1  r )  ct  st
(BC)
• Combining (BC) for each period gives the
multi-period lifetime (intertemporal) BC:
T
T
yt
ct
 s0 (1  r ) 

t 1
t 1
t 1 (1  r )
t 1 (1  r )
PDV of Lifetime
PDV of Lifetime
Income + Initial Wealth
Consumption
Method 1
• LaGrangian:
T
T
yt
ct 
L  U (c1 , c2 , cT )   
 s0 (1  r )  
t 1
t 1 
(
1

r
)
(
1

r
)
t 1
 t 1

FOC:
U ct 

(1  r )t 1
for t = 1,…,T
Combining gives
U ct  U ct1 (1  r )
 MRSct,ct+1 = (1+r)
Method 2
• LaGrangian:
T
L  U (c1 , c2 , cT )   t { yt  st 1 (1  r )  ct  st }
t 1
FOC:
U ct  t
for t = 1,…,T
t  (1  r )t 1
for t = 1, … T-1
Combining gives
U ct  U ct1 (1  r )
 MRSct,ct+1 = (1+r)
• An optimal sequence {ct*} for t=1,…,T solves
U ct  U ct1 (1  r )
(T-1 equations)
and the lifetime BC:
T
T
yt
ct
 s0 (1  r ) 

t 1
t 1
t 1 (1  r )
t 1 (1  r )
• Example: Time-separable utility function
U (c1, c2 ,, cT )  u(c1 )  bu(c2 )  b 2u(c3 )   b T 1u(cT )
T
  b t 1u (ct )
t 1
where b < 1 is the time discount factor:
b  1 /(1  r )
and r is the rate of time preference.
• Hence an optimal sequence {ct*} for t=1,…,T
solves
u' (ct )  bu' (ct 1 )(1  r )
(T-1 equations)
and
T
T
yt
ct
 s0 (1  r ) 

t 1
t 1
(
1

r
)
(
1

r
)
t 1
t 1
Consumption with Certainty
• The optimal consumption decision, given r, is
{ct} for t = 1,2,…,T solving
U ct (c1 , , cT )
U ct 1 (c1 ,  , cT )
 (1  r )
for t = 0,1,…,T-1, and
T
T
yt
ct
we  
 a0  
t 1
t 1
t 1 (1  r )
t 1 (1  r )
• Simple Example:
(i) Time separable utility with r = 0 (b=1).
(ii) Zero interest rate: r = 0.
1T

c   yt  a0 
T  t 1

The permanent income hypothesis –
consumption decisions are based upon a
constant proportion of lifetime wealth
(“permanent income”) – M. Friedman.
• 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:
C  a  b(Y  T )
• Saving will be very sensitive to income (in example,
st  st 1  yt  c )
• Problem – if future income is known with certainty, then
there should be no consumption fluctuations. (see
Graph)
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 t choose {ct}
and {st} to maximize
subject to
 T t t

Et  b u (ct )
 t t

yt  st 1 (1  r )  ct  st
• The lifetime BC is given by:
T
T
Et yt
Et ct
we  
 s0  
t t
t t
t t (1  r )
t t (1  r )
• For simplicity, assume
(i) b = 1 (no time discounting)
(ii) r = 0, (zero interest rate)
2
(iii) u(c)  c  (1 / 2)c (quadratic utility)
• FOC at date t=1:
u' (c1 )  E1u' (c2 )
• 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:
Et Et 1 zt 2   Et zt 2
• Substituting utility function into FOC:
c1  E1c2
or more generally
ct  Et ct 1
for t = 1, …, T-1
• Notice this implies:
c1  E1c1  E1c2    E1cT
• The lifetime BC is
T
T
E c  E y
t 1
1 t
t 1
1 t
 s0
Substituting in the FOC from previous slide:
1T

c1  E1 yt  s0 
T  t 1

This is the “uncertainty” version of PIH.
• Calculating c2 and using c1 from above gives:
T
1 T

c2  c1 
 E2 ( yt    E1 ( yt 
T  1  t 2
t 2

• This says that changes in consumption over time are
due to revisions in the expectation of future income
(i.e. new information)
Random Walk Hypothesis
• Robert Hall - “Stochastic Implications of the LifeCycle/ Permanent Income Hypothesis: Theory and
Evidence,” 1978, Journal of Political Economy
• Recall household FOC:
E1c2  c1
• If consumers have rational expectations, then this
implies consumption follows a random walk:
where
c2  c1   2
E1 2  0
and
1 T

2 
[ E2 ( yt )  E1 ( yt )]
T  1  t 2

•
More generally, consumption follows
ct 1  ct   t 1 where Et t 1  0
•
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 (Dc = )
(iii) MPC out of permanent shocks to income are larger
than MPC out of temporary shocks.
• 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
2004
2003
2002
2001
2000
1998
1997
$5,619
1995
1994
1992
1990 1991
1989
1988
1987
1986
1999
1996
1985
1984
1979
1980
1978
1976
1974
$3,036
1970
1964
1960
1955
1947
1945
1941
1942 1943
1939
1929
0
$3,432
Real Disposable Income
$6,081
Copyright © 2006 South-Western/Thomson Learning. All rights reserved.
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