Distinguishing Non-stationarity from Inconsistency in
Intertemporal Choice
Syngjoo Choi, UCL
Wieland Müller, Vienna
Shachar Kariv, UC Berkeley
Dan Silverman, Arizona State
RAND
November 2014
1 / 29
Background
Models of time-inconsistency have attracted attention
In particular, non-constant time discounting captures problems of
self-control to great e¤ect.
Hyperbolic or quasi-hyperbolic preference also …nd empirical support
in experiments
I
Frederick et al. (2002) review
Recent experimental studies argue this evidence (re: money rewards)
isn’t robust.
I
Andersen et al. (2011), Augenblick et al. (2013), Andreoni Sprenger
(2012), Gine et al. (2013), Halevy (2014)
Lean toward other forms of non-stationarity
2 / 29
This Paper
Presents a novel design for use on time preferences
Collects rich, individual-level data
Richness allows nonparametric analysis of consistency with utility
maximization and stationarity of time preferences.
Examine both traditional lab environment and a …eld enviroment
3 / 29
Preview of Results
Findings support a di¤erent view of what appears like
non-stationarities
Lots of individual heterogeneity, both in demand behavior and levels
of consistency
Inconsistency, per se, is quantitatively more important than any form
of non-stationarity.
4 / 29
Experimental Design: Environments
Experiments are conducted at:
I
I
Xlab at UC Berkeley
CentERpanel in the Netherlands
CentERpanel, online household panel survey with more than 2,000
hhlds (5,000 individuals).
1,636 participated
I
211 in Xlab, 1,425 from CentERpanel
5 / 29
Experimental Design: Basics
Now fairly standard method in choice under uncertainty
New for intertemporal choice problems
Problem is to select a bundle of time-dated income from a budget set.
Graphical interface allows a lot of rich data to be collected quickly.
6 / 29
7 / 29
Advantages
Choice of a bundle from a budget constraint provides more
information about preferences than discrete choice.
Large menu of decision problems gives economically complete and
statistically powerful picture.
Allows interpretation of behavior at individual level.
8 / 29
Experimental Design: Details
Subject chooses from budget line bundles consisting of some income
sooner and some income later
Choice problems (E) involve tradeo¤s between income today and 60
days later
Choice problems (L) involve tradeo¤s between income in 60 days and
60 days after that.
Static experiment with exact same budget lines – randomly ordered.
Subjects paid by direct deposit (NL) or virtual debit card (Berk.)
9 / 29
Theory: Consistency
Logcially prior, but understudied question about intertemporal
choices: Are they consistent with utility maximization?
GARP requires that if (x, t) is indirectly revealed preferred to (x0 , t0 )
then (x0 , t0 ) is not strictly and directly revealed preferred to (x, t) .
If (x, t) is revealed preferred to (x0 , t0 ), then (x, t) must cost at least
as much as (x0 , t0 ) at the prices that prevail when (x0 , t0 ) is chosen.
GARP implies there exists a complete, transitive preference ordering.
10 / 29
Theory: Consistency
Afriat’s (1967) Theorem: The following are equivalent
The choice data satisfy GARP
There exists a non-satiated utility function that rationalizes the data
There exists a concave, monotonic, continuous, and non-satiated
utility function that rationalizes the data.
11 / 29
Theory: Consistency
Afriat’s (1972) Critical Cost E¢ ciency Index (CCEI)
CCEI measures the smallest fraction by which all budget constraints
must be shifted to remove all violations of GARP
CCEI 2 [0, 1] , values closer to one are closer to perfect consistency
with GARP, and thus utility maximization.
Because subjects make choices from wide range of budget sets, data
provide a stringent test.
12 / 29
Constructing the CCEI
x2
D
C
B
A
y
x
x1
13 / 29
14 / 29
15 / 29
16 / 29
Correlates of Near-term CCEI
17 / 29
Correlates of Far-term CCEI
18 / 29
Stationarity, Time Invariance, and Time Consistency
Early experiments show time discount rates decline as tradeo¤s are
pushed into the temporal distance
Subjects often choose the larger and later of two rewards when both
are distant in time, but prefer smaller sooner as both draw near to the
present.
Interpreted as non-constant time discounting and problems of
self-control
19 / 29
Stationarity
De…nition
t
is stationary if for every t, t0
(x, t + ∆1 )
t
0 and ∆1 , ∆2
x0 , t + ∆2 () x, t0 + ∆1
0
t
x0 , t0 + ∆ 2
Ranking does not depend on the distance from t. This is what is tested in
a standard experiment.
20 / 29
Time-Invariance
De…nition
f
T
t g t=1
is time-invariant if for every t, t0
(x, t + ∆1 )
t
0 and ∆1 , ∆2
x0 , t + ∆2 () x, t0 + ∆1
t0
0
x0 , t0 + ∆ 2
Ranking does not depend on calendar time. Payments are evaluated
according to stopwatch time.
21 / 29
Time-Consistency
De…nition
f%t gTt=1 is time-consistent if for every t, t0
(x, t + ∆1 )
t
0 and ∆1 , ∆2
(x0 , t + ∆2 ) () (x, t + ∆1 )
t0
0
(x0 , t + ∆ 2 ).
Ranking does not change as the evaluation perspective changes from
t to t0 . Time consistency precludes dynamic preference reversals.
Properties are pair-wise independent, but any two properties imply
the third (Halevy, 2014)
22 / 29
Lots of Static Preference Reversals
23 / 29
GARP-based Test of Stationarity
A non-parametric approach for testing whether there is a single
preference ordering that can rationalize all intertemporal choices (for
a given subject)
Combine dataset E with dataset L.
Compute the CCEI for this combined dataset.
Compare that number to the minimum CCEI in each of the separate
treatments.
The CCEI for the combined dataset can be no bigger than the
minimum of the CCEIs for the separate datasets E and L.
24 / 29
25 / 29
26 / 29
Statistical (Permuation) Tests of Stationarity
.
Think of the treatment as the time frame
Null hypothesis is that there is a single preference ordering that can
rationalize choices in both near and far time frames.
Distribution for a test statistic under the null hypothesis is obtained
by permuting the two choice sets
I
Randomly reassign a choice from dataset E or L to each budget line.
We compare the actual CCEI, Varian (1991) and MPI scores with
their permutation distribution.
27 / 29
Results
Little evidence of non-stationarity
We reject a null of stationarity for only a small fraction of the
subjects.
Signi…cance level
CCEI
Varian
MPI
1
6.60
6.60
7.55
5
8.49
10.38
10.38
10
14.15
13.21
16.04
28 / 29
Summary
Experimental interface allows rapid collection of rich choice data
Permits individual-level analysis of intertemporal choice with new
depth.
Non-parametric analysis indicates that inconsistency is quantitatively
more important than non-stationarity.
Point to modeling inconsistency more than “exotic” preferences.
29 / 29