Application to
the theory of consumption
Lectures in Behavioral economics
Fall 2011, Part 3
13.09.2011
G.B. Asheim, ECON4260, #3
1
Relevance of behavioral economics
Acc. to Vernon Smith (”Rational choice: The contrast between economics and psychology”, JPE 99, 1991, 877-89)
much beh. economic critique stems from the premises that
Two-sided auction: buyers announce bids, sellers announce asking prices.
1.
... rationality in the economy … derives from the rationContracts on acceptance. Some information but incomplete information
alitytoof
theExperiments
individual decision
makers
in theconverges
economy.
available
each.
show that such
a market
to price
represented
by theoretical
intersection intensive,
of supply and
demand
2and volume
2.
... individual
rationality
is a cognitively
intensive
calculating
curves--competitive
equilibrium. "This
is some
kind of magic!" What it shows
process of maximization
of the
self-interest.
is that people are quite good at discovering prices and quantities that clear a
3.
... even
economic
theory
be tested
by testing
directly the
market
if they have
onlycan
incomplete
market
information.
economic rationality of the individuals isolated from
interactive experience in social and economic environments.
Instead, V. Smith thinks that beh. economic critique is important
if it predicts different economic behavior at an aggregate level.

13.09.2011
G.B. Asheim, ECON4260, #3
2
Observed saving behavior

Present-biased pref. can explain these phenomena.

Illiquid assets combined with credit card debt
Illiquid assets are commitments.
Credit cards provide consumption smoothing,
b t facilitate
but
f ilit t tempting
t
ti shopping
h
i splurges.
l

Co-movement between income and consumption
Consequence of self-imposed liquidity constraints.
Becomes more binding at the time of retirement.

Laibson: A model that explains empirical evidence.
13.09.2011
G.B. Asheim, ECON4260, #3
3
1
Outline

Cons.-saving model w/illiquid asset
Laibson (1997).
Application of the multi-self approach, where
sophisticates use illiquid asset as commitment.
Yields explanation of observed saving behavior.

A theoretical result in the cons.-saving model
Laibson (1998).

Empirical test of the cons.-saving model
Angeletos et al. (2001).
Show that present-biased pref. fit data better.
13.09.2011
4
G.B. Asheim, ECON4260, #3
Laibson’s (1997) consumption-saving model
Consumer makes decisions in periods 1, , T
One liquid asset x
One illiquid asset z
Exogenous initial asset holdings x0 , z0  0
In period t :  earn labor inco me yt
 earn
e rn asset
sset inco
in me Rt ( xt 1  zt 1 )
 choose consumptio n ct and a new
asset allocation xt and zt such that
ct  xt  zt  yt  Rt ( xt 1  zt 1 )
ct  yt  Rt xt 1
zt 1 cannot be con - 
x
,
sumed at time t .
t zt  0
The illiquid asset
13.09.2011
5
G.B. Asheim, ECON4260, #3
Laibson’s (1997) consumption-saving model (2)
Period - t intertemp oral preference s :

U t  Et u (ct )    t 1   t u (c )
T

3-period consumption-saving example
Budget constraint : c1 
c2 c3
y
y

 y1  2  32  W
R R2
R R
Instantane ous utility : u (c )  ln c
Numerical illustrati on :
  0.8,   0.9, R  1.1, and W  $1000.
13.09.2011
G.B. Asheim, ECON4260, #3
6
2
Optimal period-0 behavior

max  ln(c1 )   ln(c2 )   2 ln(c3 )

c
c
y
y
subject to c1  2  32  y1  2  32  W
R R
R R
Solution :
c10 
W
RW
(R ) 2 W
0
0
,
c

,
c

2
3
1    2
1    2
1    2
Numerical illustrati on :
c10  $369.00, c20  $365.31, c30  $361.66
13.09.2011
G.B. Asheim, ECON4260, #3
7
Optimal period-1 behavior
max ln(c1 )   ln(c2 )   2 ln(c3 )
subject to c1 
c2 c3
y
y

 y1  2  32  W
R R2
R R
Solution :
c11 
W
RW
 (R ) 2 W
, c12 
, c31 
2
2
1    
1    
1     2
Numerical illustrati on (optimal period - 1)
0) :
c110  $$422
369..30
00,, cc1220  $$334
365..46
31, c3130  $$331
361..11
66
13.09.2011
G.B. Asheim, ECON4260, #3
8
Naive period-2 behavior
max ln(c2 )   ln(c3 )
subject to c2 
c3
 R (1   )W
 W2N  R (W  c11 ) 
R
1     2
Solution :
c2N 
W2N
RW2N
, c3N 
1  
1  
Numerical illustrati on (optimal
(naivete) period
:
- 0) :
c110  $$422
369..30
00,, cc2N20  $$365
369..31
46,, cc303N$$361
292.66
.61
13.09.2011
G.B. Asheim, ECON4260, #3
9
3
Sophisticated period-2 behavior
max ln(c2 )   ln(c3 )
subject to c2 
c3
 W2  R (W  c1 )
R
Solution :
c 2 (c1 ) 
W2
R (W  c1 )

1  
1  
c 3 (c1 ) 
RW2 R 2 (W  c1 )

1  
1  
13.09.2011
G.B. Asheim, ECON4260, #3
10
Sophisticated period-1 behavior
max ln(c1 )   ln(c 2 (c1 ))   2 ln(c 3 (c1 ))
  R 2 (W  c1 ) 
 R (W  c1 ) 

max ln c1   ln 
   2 ln
1  
 1   


Solution :
c1S  1 W  2 ,
R (1 )W
R R (1 )W
c2S  c 2 (c1 )  11  
, c3S  c 3 (c1 )  1
   1    2
1    2
Numerical illustrati on (optimal
(sophistic
(naivete) :period
ation) :- 0) :
c110S  $$422
369
422..30
00
30,, cc2N202S  $$365
369..31
46,, cc303NS $$361
292.66
.61
13.09.2011
G.B. Asheim, ECON4260, #3
11
Sophisticates can implement optimal period-1
behavior by using the illiquid asset
If   0.8,   0.9, R  1.1, Y1  1000, and Y2  Y3  0, then
In period 1 :  consume $422.30
 save $304.05 in the liquid asset
(which yields $334.46 in period 2)
 save $273.64 in the illiquid asset
(which yields $331.11 in period 3)

Sophisticates strictly prefer the use of illiquid assets.

Naifs do not recognize the commitment value of
illiquid assets.
13.09.2011
G.B. Asheim, ECON4260, #3
12
4
The illiquid asset is not a perfect commitm. techn.
3-period consumption-saving example
  0.5,   1, R  1, and W  $1200.
You cannot prevent yourself
from consuming current income.

● If

Optimal period-1
Income
Consumption
Liquid asset
  0.8,   0.9, R  1.1, Y1  500, Y2  550,
and Y3  0, then the illiquid asset does not help at all.
An illiquid asset does not work as a commitment
device if you can borrow against its future payoff.

● Credit cards (and other liquidity enhancing intruments)
may undermine the commitment value of illiquid assets.
13.09.2011
13
G.B. Asheim, ECON4260, #3
A theoretical result in the consum.-saving model

Assume no illiquid asset
Period - t intertemp oral preference s :

U  Et u (ct )    t 1 
t
T
Optimality condition if   1 :
 t

(Laibson, 1998)
u (c )

Proof of a theoretical result in the cons.-saving model
u(ct )   R
 cash on hand : xt  yt  R ( xt 1  ct 1 )
dU t 1
 choose consumptio n : ct  xt
dx t 1
dU t  2
dU t  2 u ( ct 1 )


u(ct 1 )  R
dxt  2
dxt 2
R
 dc  dU t  2
dU t 1 dct 1

u (ct 1 )  R1  t 1 
dxt 1 dxt 1
 dxt 1  dxt  2
 dc
 dc  1 
  t 1  1  t 1  u(ct 1 )
 dxt 1  dxt 1   
  dc 
 dc  
u(ct )  R   t 1     1  t 1  u(ct 1 )
 dxt 1  
  dxt 1 
u (ct )  Et Ru (ct 1 )  Et R s s u (ct  s )

Generalize d opt. condition for sophistica tes with   1 :

 dc 
 dc  
u (ct )  Et R  t 1    1  t 1  u (ct 1 )
 dxt 1  
 dxt 1 
A source of ”under-saving”.
13.09.2011
G.B. Asheim, ECON4260, #3
14
Empirical test of the consumption-saving model


Compare data with simulations Angeletos et al. (2001)
under exponential disc. and hyperbolic disc. (sophisticates)
Households with liquid
assets > 1 month’s income:

Exponent. simulation: 73%
Hyperbolic simulation: 40%
Data:
43%

Households with positive
credit-card borrowing:
Exponent. simulation: 19%
Hyperbolic simulation: 51%
Data:
70%
13.09.2011
Mean credit-card borrowing
(all households):
Exponent. simulation: $900
Hyperbolic simulation: $3400
Data:
>$5000

Cons.-income comovement
(income coefficent):
Exponent. simulation: 0.032
Hyperbolic simulation: 0.166
Data:
 0.2
G.B. Asheim, ECON4260, #3
15
5
13.09.2011
G.B. Asheim, ECON4260, #3
16
13.09.2011
G.B. Asheim, ECON4260, #3
17
13.09.2011
G.B. Asheim, ECON4260, #3
18
6
13.09.2011
G.B. Asheim, ECON4260, #3
19
Proof of a theoretical result in the cons.-saving model
 cash on hand : xt  yt  R ( xt 1  ct 1 )
dU t 1
 choose consumptio n : ct  xt
dxt 1
dU t  2
dU t  2 u (ct 1 )


u (ct 1 )   R
dxt  2
dxt  2
 R
u (ct )   R
 dc  dU t  2
dU t 1 dct 1
u (ct 1 )  R1  t 1 

dxt 1 dxt 1
 dxt 1  dxt  2
 dc
 dc  1 
  t 1  1  t 1  u (ct 1 )
 dxt 1  dxt 1   
  dc 
 dc  
u (ct )  R  t 1    1  t 1  u (ct 1 )
 dxt 1  
  dxt 1 
13.09.2011
G.B. Asheim, ECON4260, #3
20
7