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Who is (More) Rational?
Syngjoo ChoiUniversity College London
Shachar KarivUC Berkeley
Wieland MüllerUniversity of ViennaTilburg University
Dan SilvermanUniversity of Michigan
NBER
To be presented at the Norwegian School of Economics
Nov 7, 2011
Prologue
• Uncertainty is a fact of life so people’s attitudes towards risk enter everyrealm of economic decision-making.
• We must study individual behavior with respect to choice involving uncer-tainty.
• Models of decision making under uncertainty play a key role in every fieldof economics.
Experiments à la Allais
• Each theory predicts indifference curves with distinctive shapes in theMarschak-Machina probability triangle.
• By choosing alternatives that theories rank differently, each theory can betested against the others.
• The criterion typically used to evaluate a theory is the fraction of choicesit predicts correctly.
The Marschak-Machina probability triangle
1
HP
Increasing preference
LP 0
1
H, M, and L are three degenerate gambles with certain outcomes H>M>L
A test of Expected Utility Theory (EUT)
A
B
CD
1
HP
LP 0
1
EUT requires that indifference lines are parallel so one must choose either A and C, or B and D.
Results have generated the most impressive dialogue between observationand theorizing (Camerer, 1995):
...EU violations are much smaller (though still statistically significant)when subjects choose between gambles that all lie inside the triangle...
...due to nonlinear weighting of the probabilities near zero (as the rankdependent weighting theories and prospect theory predict)...
...the only theories that can explain the evidence of mixed fanning,violation of betweeness, and approximate EU maximization inside thetriangle...
Although this practice is understandable, it limits the usefulness of thedata:
[1] While these experiments reveal that violations exist, they give us littlesense of how important they are or how frequently they occur.
[2] The analysis builds on ad hoc assumptions about an error generatingprocess and lacks any substantive econometric methodology.
[3] Choice scenarios are not very important to the applications of theoriesof choice under uncertainty in economics.
Research questions
Consistency
— Is behavior under uncertainty consistent with the utility maximizationmodel?
Structure
— Is behavior consistent with a utility function with some special struc-tural properties?
Recoverability
— Can the underlying utility function be recovered from observed choices?
Extrapolation
— Given behavior in the laboratory, can we forecast behavior in otherenvironments?
A new experimental design
An experimental design that has a couple of fundamental innovations overprevious work:
— A selection of a bundle of contingent commodities from a budget set(a portfolio choice problem).
— A graphical experimental interface that allows for the collection of arich individual-level data set.
The experimental computer program dialog windows
Rationality
Let {(pi, xi)}50i=1 be some observed individual data (pi denotes the i-thobservation of the price vector and xi denotes the associated portfolio).
A utility function u(x) rationalizes the observed behavior if it achieves themaximum on the budget set at the chosen portfolio
u(xi) ≥ u(x) for all x s.t. pi · xi ≥ pi · x.
Afriat’s Theorem The following conditions are equivalent:
— The data satisfy GARP.
— There exists a non-satiated utility function that rationalizes the data.
— There exists a concave, monotonic, continuous, non-satiated utilityfunction that rationalizes the data.
Goodness-of-fit
• Verifying GARP is conceptually straightforward but it can be difficult inpractice.
• Since GARP offers an exact test, it is necessary to measure the extent ofGARP violations.
• Measures of GARP violations based on three indices: Afriat (1972), Varian(1991), and Houtman and Maks (1985).
Afriat’s critical cost efficiency index (CCEI) The amount by whicheach budget constraint must be relaxed in order to remove all violationsof GARP.
The CCEI is bounded between zero and one. The closer it is to one, thesmaller the perturbation required to remove all violations and thus thecloser the data are to satisfying GARP.
The construction of the CCEI for a simple violation of GARP
2x
1x
D
C
B A
x
y
The agent is ‘wasting' as much as A/B<C/D of his income by making inefficient choices.
Decision-making quality
• In economics, heterogeneity in choices is attributed to heterogeneity inpreferences, information, beliefs, or constraints.
• Several strands of empirical research consider heterogeneity in choicesdriven, instead, by differences in the quality of decision-making.
• The revealed preference criterion: if decisions are high quality then thereexists a utility function that the choices maximize.
Judgment about the quality of decision-making is generally made difficultby twin problems of identification and measurement:
— The identification problem Distinguishing differences in decision-making quality from unobserved differences in preferences, information,beliefs or constraints.
Identification is important because welfare conclusions and thus (con-strained) optimal policy will depend on the sources of any systematic dif-ferences in choices.
— The measurement problem Defining (and implementing) a portable,practical, autonomous, quantifiable, and economically interpretable mea-sure of decision-making quality.
The CentERpanel
• A representative sample of over 2,000 Dutch-speaking households (5,000individual members) in the Netherlands.
• A wide range of individual socio-demographic and economic informationfor the panel members.
• The subjects in the experiment were randomly recruited from the entireCentERpanel body.
Participants DropoutsNon-
participants
Female 45.43 37.89 50.00Age
16-34 18.53 3.16 26.1435-49 26.14 12.11 32.1350-64 35.62 38.42 27.5865+ 19.71 46.32 14.15
EducationLow 33.59 42.63 30.99Medium 29.70 22.63 31.61High 36.72 34.74 37.40
Household monthly income€0-2500 22.42 34.73 21.28€2500-3499 25.13 26.32 18.90€3500-4999 28.85 16.32 28.93€5000+ 23.60 22.63 30.89
OccupationPaid work 53.13 39.47 62.91House work 11.59 7.89 8.78Retired 20.90 42.63 13.95Others 14.38 10.00 14.36
Household compositionPartner 80.88 67.89 82.64# of kids 0.84 0.32 1.09
# of obs. 1182 190 968
Types of analysis
[1] A descriptive overview of the average quality of decisions (consistency witheconomic rationality).
[2] The relationship between decision-making quality and demographic andeconomic characteristics.
[3] The correspondence between wealth accumulation and behavior in the lab-oratory.
Mean CCEI scores
CCEI scores Combined
CCEI CCEI.887*** .735***(.022) (.037)
-.024*** -.011(.009) (.015)
Age-.016 -.007(.011) (.020)
-.052*** -.077***(.011) (.020)
-.051** -.081**(.020) (.032)
Education.009 .021
(.011) (.017).026** .060***(.011) (.018)
Income.026** .026(.012) (.019).020 .006
(.013) (.020).033** .017(.014) (.022)
Occupation.028 .030
(.018) (.026).047** .039(.021) (.030).037* .035(.019) (.030)
Household composition-.026** -.023(.011) (.018).001 .001
(.004) (.007)R 2 .068 .058# of obs. 1182 1182
# of children
65+
€2500-3499
€3500-4999
€5000+
Partner
High
Paid work
Medium
House work
Others
Constant
Female
35-49
50-64
Dominance
• Violations of monotonicity with respect to first-order stochastic dominance(FOSD) are errors, regardless of risk attitudes.
• A decision to allocate less tokens to the cheaper account violates domi-nance but need not involve a violation of GARP.
• We use expected payoff calculations (largest upward probabilistic shift) toassess how nearly choice behavior complies with dominance.
A scatterplot of CCEI and FOSD scores
Wealth differentials
=⇒ The heterogeneity in wealth is not well-explained either by standard observ-ables (income, education, family structure) or by standard unobservables(intertemporal substitution, risk tolerance).
=⇒ If consistency with utility maximization in the experiment were a goodproxy for (financial) decision-making quality then the degree to which con-sistency differ across subjects should help explain wealth differentials.
Wealth regressions
1.425** 1.348* 1.781** 1.728**
(0.565) (0.714) (0.746) (0.750)0.078 -0.091 -0.038
(0.381) (0.381) (0.384)-1.361 -1.366(0.838) (0.840)
0.103(0.072)
0.601*** 0.602*** 0.520*** 0.514***(0.127) (0.127) (0.121) (0.121)-0.228 -0.229 -0.299 -0.321(0.164) (0.164) (0.168) (0.169)-0.286 -0.284 -0.310 -0.282(0.316) (0.316) (0.319) (0.316)0.006 0.006 0.007 0.006
(0.005) (0.005) (0.005) (0.005)0.000 0.000 0.000 0.000
(0.000) (0.000) (0.000) (0.000)0.682*** 0.682*** 0.733*** 0.714***(0.183) (0.183) (0.191) (0.191)0.103 0.103 0.095 0.090
(0.092) (0.093) (0.095) (0.095)Education controls Y Y Y Y
5.932 5.888 7.797 7.371(5.862) (5.879) (5.880) (5.841)
R 2 0.1794 0.1778 0.1801 0.1819
# of obs. 517 517 494 494
# of children
Constant
Ln(hhld wealth)
Female
Age
Age2
Age3
Partner
CCEI
Combined CCEI
Risk attitude
Conscientiousness
Ln(hhld income '08)
Sources of correlation Checking Saving Stocks House
1.907** 1.792** -0.100* -0.179* -0.003 0.336***(0.751) (0.705) (0.057) (0.096) (0.050) (0.129)
0.686*** 0.25 -0.030** -0.062*** 0.011 0.098***(0.133) (0.184) (0.013) (0.021) (0.010) (0.023)
0.467*(0.262)0.312*(0.176)
-0.144 -0.037 0.019 0.017 0.005 -0.039(0.188) (0.184) (0.018) (0.029) (0.012) (0.039)
0.790*** 0.740*** -0.033 -0.058* -0.008 0.136***(0.222) (0.222) (0.021) (0.033) (0.014) (0.044)0.111 0.123 -0.004 -0.043*** 0 0.069***
(0.107) (0.101) (0.009) (0.013) (0.007) (0.019)4.282 -0.415 0.045 1.42 -0.287 -0.777
(5.947) (6.032) (0.750) (1.197) (0.388) (1.289)R 2 0.236 0.269 0.051 0.104 0.029 0.146# of obs. 377 377 512 502 514 479
Ln(hhld income '04)
Constant
Female
partner
# of children
Fraction of hhld wealthLn(hhld wealth)
CCEI
Ln(hhld income '08)
Ln(hhld income '06)
Risk aversion – the fraction of tokens allocated to the cheaper asset
Risk aversion
All sample CCEI ≥ 0.8 CCEI ≥ 0.9
-0.020*** -0.021*** -0.026***(0.007) (0.008) (0.010)
Age-0.010 -0.011 -0.019(0.011) (0.012) (0.014)
-0.045*** -0.048*** -0.054***(0.010) (0.011) (0.013)
-0.052*** -0.054*** -0.067***(0.014) (0.018) (0.021)
Education0.018** 0.018* 0.025**(0.008) (0.009) (0.012)0.015** 0.016 0.026**(0.008) (0.010) (0.012)
Income-0.002 -0.005 -0.007(0.008) (0.011) (0.013)-0.003 -0.008 -0.006(0.009) (0.011) (0.014)0.013 0.009 0.003
(0.011) (0.013) (0.016)Occupation
-0.006 -0.009 -0.014(0.011) (0.014) (0.017)0.011 0.007 0.006
(0.013) (0.017) (0.020)0.003 -0.006 -0.010
(0.012) (0.016) (0.020)Household composition
0.004 0.007 0.006(0.009) (0.010) (0.013)
-0.007** -0.005 -0.004(0.003) (0.004) (0.005)
0.632*** 0.644*** 0.645***(0.015) (0.019) (0.023)
R 2 0.048 0.045 0.054# of obs. 1,182 901 681
Others
Constant
Partner
# of children
Female
65+
High
€5000+
€2500-3499
€3500-4999
Paid work
House work
35-49
50-64
Medium
Loss/disappointment aversion
The theory of Gul (1991) implies that the utility function over portfoliostakes the form
min {αu (x1) + u (x2) , u (x1) + αu (x2)} ,
where α ≥ 1 measures loss/disappointment aversion and u(·) is the utilityof consumption in each state.
If α > 1 there is a kink at the point where x1 = x2 and if α = 1 we havethe standard EUT representation.
An illustration of the derived demand
2
1lnxx
2
1lnpp
0ln2
1 =xx
0ln2
1 =pp
Scatterplot of the CARA NLLS estimates
0.0
2.0
4.0
6.0
8.1
.12
.14
.16
.18
.2R
isk
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2Loss
Risk aversion (CARA)
All sample CCEI ≥ 0.8 CCEI ≥ 0.9
0.060*** 0.086*** 0.110***(0.021) (0.026) (0.032)
Age0.020 0.025 0.047
(0.030) (0.036) (0.043)-0.007 0.011 0.026(0.029) (0.035) (0.042)0.085* 0.148** 0.187***(0.045) (0.058) (0.071)
Education-0.026 -0.049 -0.069*(0.024) (0.031) (0.038)0.002 -0.018 -0.053
(0.024) (0.030) (0.038)Income
-0.023 0.004 0.023(0.027) (0.035) (0.044)-0.010 0.009 0.010(0.027) (0.035) (0.044)0.003 0.022 0.041
(0.031) (0.039) (0.048)Occupation
0.098*** 0.141*** 0.146**(0.038) (0.050) (0.065)0.046 0.056 0.017
(0.042) (0.057) (0.072)0.091** 0.133** 0.108(0.043) (0.056) (0.072)
Household composition-0.033 -0.034 -0.031(0.026) (0.032) (0.039)0.005 0.008 0.005
(0.010) (0.013) (0.015)0.125** 0.088 0.113(0.054) (0.068) (0.088)
R 2 0.022 0.03 0.037# of obs. 1,182 901 681
House work
Others
Partner
# of children
Constant
Medium
High
€2500-3499
€3500-4999
€5000+
Paid work
Female
35-49
50-64
65+
Loss aversion (CARA)
All sample CCEI ≥ 0.8 CCEI ≥ 0.9
-0.021 0.024 0.151(0.166) (0.203) (0.247)
Age0.468* 0.543* 0.706**(0.241) (0.280) (0.333)0.013 0.121 0.301
(0.236) (0.277) (0.331)0.834** 1.230*** 1.413**(0.363) (0.455) (0.560)
Education0.155 0.078 0.116
(0.192) (0.239) (0.296)0.475** 0.523** 0.428(0.192) (0.235) (0.291)
Income0.396* 0.620** 0.773**(0.216) (0.270) (0.337)-0.118 0.046 -0.098(0.217) (0.272) (0.339)0.097 0.216 0.192
(0.244) (0.299) (0.369)Occupation
0.394 0.752* 0.782(0.306) (0.394) (0.508)0.293 0.752* 0.524
(0.340) (0.447) (0.561)0.750** 1.253*** 1.283**(0.347) (0.440) (0.563)
Household composition-0.565*** -0.742*** -0.799***
(0.207) (0.249) (0.305)0.180** 0.204** 0.215*(0.083) (0.100) (0.120)
1.663*** 1.316** 1.432**(0.436) (0.538) (0.692)
R 2 0.036 0.047 0.051# of obs. 1,182 901 681
House work
Others
Partner
# of children
Constant
Medium
High
€2500-3499
€3500-4999
€5000+
Paid work
Female
35-49
50-64
65+
Individual-level data
• A review of the full data set reveals striking regularities within and markedheterogeneity across subjects.
• A quick look at the individual-level data indicates that whatever is goingon is not consistent with the standard interpretation of EUT.
• Examples that reveal some of the unexpected features of the data andillustrate the role of heuristics.
Diagonal
Boundary
Smooth
Diagonal and boundary
Diagonal, boundary and smooth
Diagonal, boundary and smooth
Center
Vertex
Centroid (budget shares)
Edge
Bisector
Center and bisector
Edge and bisector
Center, vertex, and edge
Vertex and edge
Center and bisector
Procedural rationality
• How subjects come to make decisions that are consistent with an underlyingpreference ordering?
• Boundedly rational individuals use heuristics in their attempt to maximizean underlying preference ordering.
— There is a distinction between true underlying preferences and revealedpreferences.
— Preferences have an EU representation, even though revealed prefer-ences appear to be non-EU.
2x
1x
21 xx =
A
True preferences
Reveled preferences
Discussion
Suppose there are states of nature and associated Arrow securities andthat the agent’s behavior is represented by the decision problem
max u (x)s.t. x ∈ B (p) ∩A
where B (p) is the budget set and A is the set of portfolios correspondingto the various archetypes the agent uses to simplify his choice problem.
The only restriction we have to impose is that A is a pointed cone (closedunder multiplication by positive scalars), which is satisfied ifA is composedof any selection of archetypes except the Centroid.
We can derive the following properties of the agent’s demand:
1. Let pk denotes the k-th observation of the price vector and
xk ∈ argmaxnu (x) : x ∈ B
³pk´∩A
odenotes the associated portfolio. Then the data
npk,xk
osatisfy
GARP.
2. There exists a utility function u∗ (x) such that for any price vector p,
x∗ ∈ argmax {u (x) : x ∈ B (p) ∩A}⇔
x∗ ∈ argmax {u∗ (x) : x ∈ B (p)} .
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