ISSN (Print) 0473-453X

Discussion Paper No. 1195

ISSN (Online) 2435-0982

REVISITING CES UTILITY FUNCTIONSFOR DISTRIBUTIONAL PREFERENCES:
DO PEOPLE FACE
THE EQUALITY–EFFICIENCY TRADE-OFF?
Keigo Inukai
Yuta Shimodaira
Kohei Shiozawa

October 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Revisiting CES utility functions for
distributional preferences: Do people face theequality–efficiency trade-off?
Keigo Inukai, Yuta Shimodaira, and Kohei Shiozawa∗
October 18, 2022
Abstract
The constant elasticity of substitution (CES) function is widelyused to model distributional preferences in modified dictator games.
However, it has been pointed out that its parameter interpretations
are inconsistent and problematic in applications. We constructed amodel to address this issue by combining two formulations of the
CES function. We demonstrated that the proposed model provides
consistent interpretations of parameters. Our results clarified thatthe conventional interpretations of the standard CES function parameters for describing distributional preferences are inappropriate.
Notably, “the equality–efficiency trade-off,” a conventional interpretation of the substitution parameter, is unrelated to observed individual behaviors.
Keywords: CES, Distributional preferences, Equality, Efficiency,
Parameter recovery
JEL codes: C91, D63, D64, D90∗

Inukai: Faculty of Economics, Meiji Gakuin University, 1-2-37 Shirokanedai, Minato, Tokyo, 108-8636, Japan (email: inukai@eco.meijigakuin.ac.jp). Shimodaira: Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka,
560-0043, Japan (email: u064674i@ecs.osaka-u.ac.jp). Shiozawa: Faculty of Economics,
Takasaki City University of Economics, 1300 Kaminamie-machi, Takasaki, Gunma, 3700801, Japan (email: shiozawa@tcue.ac.jp). The authors are grateful to Taisuke Imai,
Nobuyuki Hanaki, and participants of 2021 ESA Global Online Around-the-Clock Meetings and the 24th Experimental Social Science Conference of Japan for helpful discussions. Inukai acknowledges financial support provided through a Grant-in-Aid for Young
Scientists (17H04780). Inukai and Shiozawa acknowledge financial support provided
through a Grant-in-Aid for Transformative Research Areas (21H05070) from the Japan
Society for the Promotion of Science.

1

1

Introduction

Modeling individual preferences is a central issue in understanding individuals’ other-regarding behavior observed in experiments. One of the mostemployed modeling strategies is an outcome-based model that focuses on
an individual’s concerns about the distribution of payoffs among that individual and others (Fehr and Schmidt, 2006, Section 3.1; Cooper and Kagel,
2016, Section II). Among those strategies, the constant elasticity of substitution (CES) utility function of Arrow et al. (1961) has been shown to bea flexible and useful specification in modeling behaviors in modified dictator game experiments (Andreoni and Miller, 2002; Fisman, Kariv andMarkovits, 2007).1
The attractiveness of the model based on CES utility is due to the theoretical flexibility that encompasses typical forms of other-regarding preferences such as Rawlsian inequality aversion, altruistic, and selfish preferences, each of which corresponds to extreme cases of the CES utility
function. More importantly, the CES utility function encompasses thesetypical forms in a continuous manner using two parameters α—distribution
parameter—and ρ—substitution parameter. The parameter α controls therelative importance of the own and others’ payoffs. The parameter ρ controls the elasticity of substitution between these payoffs. Modeling thatuses the CES utility function enables us to study the heterogeneous nature of individual preferences for “equality” and “efficiency.” Indeed, there
have been many papers documenting the heterogeneity of preferences andtheir association with other variables (Andreoni and Miller, 2002; Fisman,
Kariv and Markovits, 2007; Fisman, Jakiela and Kariv, 2014, 2015; Fisman
et al., 2015, 2022; Li, Dow and Kariv, 2017; Li et al., 2022).2 The standard

1

Cox, Friedman and Gjerstad (2007) employed a modified version of the CES utility
function to model behaviors observed in several experiments. In this paper, we focus
on the direct use of the standard CES utility function, specifically in modified dictator
game experiments.
2
Concern for the efficiency of distribution is also an essential element of the otherregarding preferences that have been studied in the literature (Charness and Rabin,
2002; Engelmann and Strobel, 2004, 2006; Fehr, Naef and Schmidt, 2006; Fehr and2

formulation of the CES utility function is also used to study distributional
preferences in different choice environments or contexts (Andreoni, 2007;
Becker, Häger and Heufer, 2015; Brown, Meer and Williams, 2019; Erkut,
2022; Heufer, van Bruggen and Yang, 2020; Hong, Ding and Yao, 2015;
Jakiela, 2013; Korenok, Millner and Razzolini, 2013; Müller, 2019; Porterand Adams, 2016; Robson, 2021).
Despite its appeal, the standard CES utility function is also known tohave a theoretical difficulty. That is, in the limit of parameters corresponding to the Rawlsian maximin-type form (i.e., Leontief function form), thedistribution parameter becomes ineffective in identifying differences in theutility level or behavior, and thus loses its natural interpretation. In thecontext of other-regarding preferences, this is a serious problem, because
the distribution parameter is supposed to correspond to one of the central
concerns, equality of the distribution behaviors. Although Thöni (2015)
has articulated it as an essential problem in this context, this issue has
not received much attention. On the other hand, in the field of macroeconomics, this theoretical problem is recognized from the very beginning(Arrow et al., 1961, p.231). It has been studied as a problem of how tonormalize the CES production function (Klump, McAdam and Willman,
2012; Embrey, 2019). Several variants of the CES function have been proposed, which can be regarded as variations of the normalization constantof the input and output variables (Klump, McAdam and Willman, 2012,
Section 2; Embrey, 2019, Section 4).3
Similar to theoretical studies, there has been a scarcity of studies on
the parameter interpretation of the CES function in analyzing experimental
data on distributional preferences. For example, Becker, Häger and Heufer(2015) briefly described potential problems of the standard CES function
and attempted to analyze the data more appropriately by combining es-

Schmidt, 2006, Section 4.2; Cooper and Kagel, 2016, Section II.C).
3
In this paper, we do not discuss in general which normalization is more desirable
to use in the analysis of experimental data. Instead, we address the parameter interpretation by combining CES specifications, which have been shown to provide partial
solutions in prior studies.

3

timation methods. However, Becker, Häger and Heufer (2015) does notclarify the problems with the conventional method as we do in the present
paper.
Gauriot, Heger and Slonim (2020) is another such example, discussing
the potentially inappropriate interpretation of the distribution parameterin conventional group-level analysis using the CES utility function or equivalent functional systems. Specifically, the authors proposed a novel utility
function to generalize the CES function. However, its theoretical properties are less clear than the well-studied CES functions.4 The authors
also mainly discussed group-level rather than individual-level analyses. Finally, in Breitmoser (2013), the focus is on various stochastic modelingapproaches using the standard CES utility function as opposed to the relationship between parameter values and observed behavior or preferences.
Hence, despite its broad applications, the problem in interpreting parameter values of the CES utility function has not been resolved adequately inthe context of distributional preferences.
This paper aims to address the interpretation of parameter values of
the CES utility function in individual-level analysis. We adopt a strategy using two formulations from the theoretical studies of CES functions,the standard formulation of Arrow et al. (1961) and a variant of Senhadji(1997), combining them in a way that compensates for each other’s disadvantages. We formulated a statistical model using a combined class ofutility functions.
We reanalyzed the well-known data of Fisman, Kariv and Markovits(2007). The reanalysis showed that the problem of interpreting the distribution parameter is critical in this context, and that the proposed model

4

The authors proposed to generalize the curvature parameter so that different values
can be specified for the individual’s and others’ payoffs. However, as easily seen fromsome numerical simulations, it does not seem very promising in terms of solving the
problem of interpretation of the distribution parameter. For example, in their formulation, it is possible to move the two curvature parameters, keeping α = 1/2 fixed, torepresent behaviors ranging from always spending equally to always allocating the mostto oneself (or others). We believe this makes whatever α is expected to represent more
ambiguous.

4

provides a more consistent interpretation than the standard CES model.
Furthermore, the consistent interpretation of the distribution parameter revealed that the conventional interpretation of the substitution parameter—“the equality–efficiency trade-off”—may be misleading. In other words,“the equality–efficiency trade-off” interpretation of the substitution parameter is a mathematical artifact that stems from a shortcoming of thestandard CES function in that it cannot characterize a specific range of behaviors. Hence, it is not primarily rooted in the behavior of the subjects.
We also conducted a parameter recovery analysis using simulations to
examine the behavior of the model (Wilson and Collins, 2019). The results revealed that while both two parameter estimations were reasonably
recovered over a broad range of possible parameter values, the estimateddistribution parameter was more reliable than the estimated substitution
parameter. The instability of the substitution parameter estimation depends mainly on the value of the distribution parameter rather than thevalue of the substitution parameter. Based on these facts, we present specific points that should be considered by researchers using this method.
The remainder of the paper is organized as follows. Section 2 provides details of the shortcoming of the standard CES utility function anddescribes our model, which characterizes a broader range of behaviors. Section 3 presents the results of a reanalysis of the well-known data of Fisman,
Kariv and Markovits (2007) and Section 4 presents the parameter recovery
simulation. Section 5 concludes.

2

TheoryIn this paper, we propose a modified CES utility model to analyze distributional preferences based on the experimental methodology of Fisman, Karivand Markovits (2007, henceforth FKM). In their methodology, each subject
participates in a modified dictator game5 and is asked to allocate mone-

5

The modified dictator game was developed by Andreoni and Miller (2002) to test
whether the redistribution decisions were consistent with the utility maximization model.

5tary payoffs between themselves and another anonymous subject. Each
allocation decision is made by choosing a pair of monetary payoffs (x1 , x2 )from the budget line p1 x1 + p2 x2 = 1, where x1 and x2 correspond to the
payoffs to the self (the subject) and the other (an anonymous subject),
respectively. The ratio p2 /p1 represents the price of giving. Each subject
makes 50 allocation decisions defined by 50 different prices of giving.
The experimental data are analyzed by using the utility maximization
model, which is a classical economic model of consumer behavior.6 In theutility maximization model, it is assumed that there is a utility function
U (x1 , x2 ) and that the payoff allocation (x1 , x2 ) is generated from the following optimization problem:
max U (x1 , x2 )x1 ,x2s.t. p1 x1 + p2 x2 ≦ 1(1)

x1 , x2 ≧ 0.
As seen in the above formulation, a crucial methodological decision forproperly modeling the redistribution behavior is the specification of theparametric class of the utility function, which is explained in the following
subsection.

2.1

Utility function specification

In this subsection, we first describe the two conventional formulations andtheir shortcomings. Then, using these two conventional specifications, we
propose a new class of parametric utility functions.
In FKM, the standard CES function originally derived by Arrow et al.
(1961) was used as a class of utility functions. The standard CES utility

6

In the literature, a conventional data analysis procedure consists of two consecutive
parts. The first part is the revealed preference analysis (Afriat, 1967, 1972; Chambers and Echenique, 2016), and the second is the econometric analysis described in thepresent paper. However, in principle, these two procedures can be used independently.
Therefore, for simplicity, this paper focuses only on the econometric analysis.

6function is defined as follows:1−ρ − ρ
U (x1 , x2 ) = (αx−ρ
1 + (1 − α)x2 )(2)

where α ∈ [0, 1] is the distribution parameter and ρ ∈ [−1, ∞) \ {0} is
the substitution parameter. The distribution parameter α controls the relative importance of the payoffs. The substitution parameter ρ controls
the curvature of its indifference curves and characterizes the elasticity of
substitution σ ∈ (0, ∞) as
1
.
(3)
σ=
ρ+1
The standard CES utility function has three typical forms in the limit of
the substitution parameter. Specifically, the utility function (2) tends toa symmetric Leontief utility function U (x1 , x2 ) = min{x1 , x2 } as σ → 0(ρ → ∞), a Cobb–Douglas utility function U (x1 , x2 ) = xα1 x21−α as σ → 1(ρ → 0), and a linear utility function U (x1 , x2 ) = αx1 +(1−α)x2 as σ → ∞(ρ → −1). Figure 1 visualizes three limits.
It is known that special care must be taken when using CES-type functions for modeling (Arrow et al., 1961, p.231). As seen in the first case,about the limit of the function, which is visualized in the left panel of
Figure 1, the distribution parameter becomes less effective at describing
individual behaviors as σ approaches 0. In other words, as σ approaches
0, the ability of α to represent the relative importance of the payoffs decreases. This is an essential problem when studying altruistic preferences(Thöni, 2015). Indeed, the ineffectiveness of the distribution parametermeans that the behavioral tendency to give some of one’s money to others may not be represented by the parameter, even if it is present in theobserved allocation decisions.
Fortunately, there is another formulation of the CES function that overcomes this well-known shortcoming. In Senhadji (1997), a reformulated
CES function was defined as follows:11+ρ −ρ − ρ
x2 )U (x1 , x2 ) = (α1+ρ x−ρ
1 + (1 − α)7(4)where α ∈ [0, 1], ρ ∈ [−1, ∞) \ {0}. Figure 2 visualizes three limits of thisformulation. The reformulated CES utility function (4) tends to a Leontief
utility function U (x1 , x2 ) = min{x1 /α, x2 /(1 − α)} as σ → 0 (ρ → ∞)where σ is defined as (3). Moreover, for σ → 1 (ρ → 0), the limit isU (x1 , x2 ) = α−α (1 − α)−(1−α) xα1 x21−α , which is a monotonic transformationof the standard CES function (2) case. Hence, in this formulation, we cancircumvent the well-known deficit of the standard CES function. However,on the contrary, this function approaches the symmetric linear utility function U (x1 , x2 ) = x1 + x2 as σ → ∞ (ρ → −1), as visualized in the rightpanel of Figure 2. Therefore, unfortunately, the distribution parameter
becomes ineffective in describing individual behaviors as σ approaches ∞,which would also be an essential problem when studying altruistic preferences for the same reason discussed above.
Therefore, in this study, the class of utility functions was formulated by
the following combined definition of the above CES functions:
11+ρ −ρ
1+ρ −ρ − ρ

(α
x
+
(1
−
α)
x
)

1
2


U (x1 , x2 ) = α log(x1 ) + (1 − α) log(x2 )−ρ − ρ1(αx−ρ
1 + (1 − α)x2 )if ρ > 0 ( ⇐⇒ σ < 1)
if ρ = 0 ( ⇐⇒ σ = 1) (5)if ρ < 0 ( ⇐⇒ σ > 1)

where α ∈ [0, 1], ρ ∈ [−1, ∞), and σ = 1/(ρ + 1). The validity of thisformulation is clear. As mentioned above, each of the two conventional
formulations has its advantages and disadvantages in the exact opposite
limit of the parameter. Therefore, if we combine the two properly, we canutilize the advantages of the two formulations while avoiding the disadvantages. Moreover, the combined formulation (5) has a well-defined andwell-known demand function, including the case σ → 1. Indeed, in bothof the conventional formulations (2) and (4), the corresponding demand
functions are exactly the same as those derived from the Cobb–Douglas
utility function.7 A visualization of indifference curves corresponding to

7

In Thöni (2015), another reformulation was proposed as a remedy for the short-8Symmetric Leontief

Cobb-Douglas

Linear
100

80

80

80

60

60

60

x2

x2

100

x2

100

40

40

40

20

20

20

0

0

0
0

20

40

60

80

100

0

20

40

x1

60

80

0

100

20

40

60

80

100

x1

x1

Figure 1: The standard CES function of Arrow et al. (1961)
Notes: Indifference curves of standard CES utility functions with α = 0.75. Eachpanel depicts an approximation of a limiting case: σ = 0.1, σ = 1 + 10−3 , and σ = 10
are depicted in the left, middle, and right panels, respectively. Dashed lines depict
equations x1 = x2 and x1 /α = x2 /(1 − α). Note that the Leontief case (the left panel)approximates the symmetric Leontief function while we set the distribution parameter
asymmetric.

Cobb-Douglas

Leontief

Symmetric Linear
100

80

80

80

60

60

60

x2

x2

100

x2

100

40

40

40

20

20

20

0

0
0

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40

60

80

100

0
0

20

40

x1

60

x1

80

100

0

20

40

60

80

100

x1

Figure 2: The CES function of Senhadji (1997)
Notes: Indifference curves of the reformulated CES utility functions of Senhadji (1997)
with α = 0.75. Each panel depicts an approximation of a limiting case: σ = 0.1,
σ = 1 − 10−3 , and σ = 10 are depicted in the left, middle, and right panels, respectively.
Dashed lines depict equations x1 = x2 and x1 /α = x2 /(1 − α). Note that the linear
case (the right panel) approximates the symmetric linear function while we set the
distribution parameter asymmetric.

9

this extended formulation would be a figure made from the above two figures in the following order: left of Figure 2, middle of Figure 2, middle of
Figure 1, and right of Figure 1.
In the analysis, the share form of the demand function, which is derived
from the utility maximization model (1) with the utility function specified
by (5), is used as in FKM:

s = p 1 x1 =


α
)
( 1−α


 1−σp2α ( 1−α)+ pif σ ≦ 1

1

(6)

σ






α
( 1−α
)
 1−σσα+( 1−α ) pp21

otherwise

where α ∈ [0, 1] and σ ∈ (0, ∞). Note that the amount of wealth isnormalized to one.
One can easily see that the distribution parameter α is always effective
in describing individual behaviors regardless of the elasticity of substitution σ, including its limits as σ tends to 0, 1, or ∞. Moreover, in thisformulation, the behavioral interpretation for the distribution parameterα is clear for all three limiting cases. First, in the Leontief function case,i.e., σ → 0, the distribution parameter characterizes the ratio of payoffs as
x1 /x2 = α/(1 − α). Second, in the Cobb–Douglas function case (σ = 1), it
characterizes the ratio of expenditure shares, i.e., p1 x1 /p2 x2 = α/(1 − α).
Finally, in the linear function case, i.e., σ → ∞, it characterizes the degenerated allocation for self as p1 x1 = 1 if p1 /p2 ≦ α/(1 − α) and p1 x1 = 0
otherwise.8
This gives us a good theoretical reason to interpret the distribution
parameter as a measure of preference for equality. Indeed, if we focus

comings of (2) and (4). However, the proposed formulation has another property thatwhen σ → 1, the demand function approaches that of the Cobb–Douglas utility function
with the distribution parameter α2 / α2 + (1 − α)2 , where α ∈ [0, 1]. Note that this
distribution parameter is equal to α only if α = 0, 0.5, 1. Therefore, to maintain theclassical interpretation of the limiting case in the center, i.e., when σ → 1, we adopt the
combined formulation (5).
8
In the linear case, if p1 /p2 = α/(1 − α), then any choice satisfying the budget
constraint is optimal.

10

on α = 1/2, we see that, in every case identified above, the behavior
generally treats some quantity as equal depending on the price—in thecase of Leontief, it is the amount of payoffs; in the case of Cobb–Douglas,
the amount of spending; and in the linear case, the opportunity to occupy
payoffs. These seem to belong to a special case of “proportional equality,”
the traditional framework since Aristotle for discussing equality in relation
to distributive justice (Gosepath, 2021, Section 2.2). More precisely, if we
set α = 1/2, then
 σ
p2
x1
=
,
(7)
x2
p1
hence, it satisfies the form of “proportional equality.” From this point ofview, the elasticity of substitution σ controls how strongly the decision
maker considers the relative price information as a valid signal to make the
distribution equal. As we will see in Section 3, this consistent interpretationof the distribution parameter reveals problems with “the equality–efficiency
trade-off” interpretation of the substitution parameter ρ.
Finally, the limitations of this formulation should also be noted. As caneasily be seen from the definition of the utility function (5), in the selfish
case: α = 1, the (elasticity of) substitution parameter, ρ (or σ), becomes
irrelevant.9 In other words, the model does not satisfy identifiability in thesense that more than one parameter pair corresponds to the same behavior
corresponding to the selfish case. This theoretical shortcoming is not a
problem posed only by our extension, but is a drawback associated withthe use of CES-type functions in general. In the following, we also report
the practical implications of these shortcomings to clarify the limitations.

2.2

Statistical model specification

In this study, to estimate the two parameters of the extended CES utility
function, a Bayesian statistical model was formulated. The model involves

9

Although the exact opposite case (α = 0) raises similar issues, here, we only discuss
α = 1, which typically appears in the context of altruistic preferences.

11

novelties to Bayesian modeling but are otherwise straightforward extensions of standard methods such as Andreoni and Miller (2002) and FKM.
Specifically, an error term distribution of the share form demand and priordistributions for the parameters were specified as described below.
2.2.1

Residual error term distribution

A truncated normal distribution was used for the residual error term.
Hence, it is assumed that an observation of the expenditure share is distributed on the unit interval in proportion to the normal distribution withlocation s and scale σs . This formulation leads to a natural interpretationthat the expenditure share in the allocation decision is likely to be aroundthe theoretical share s.10
2.2.2

Prior distributions for parameters α, σ and σs

For the distribution parameter α, the uniform distribution on the unit
interval is used as a non-informative prior distribution.
For the elasticity of substitution σ, the prior distribution was specified
on log(σ) rather than on σ. Figure 3 visualizes the reason for this decision: the symmetric treatment of the three limiting cases. As we see inthe previous subsection, in the original scale (i.e., σ), the Leontief utility
function is placed at zero and the linear utility function at infinity, treating
both asymmetrically. On the other hand, this reparameterization allows
us to treat them symmetrically by placing the Leontief utility function at
negative infinity and the linear utility function at positive infinity, with the

10

In the literature, a censored normal distribution, i.e., the two-limit Tobit model, hasbeen used for the distribution of the error term. However, the two-limit Tobit model canlead to unnatural interpretations when the scale σs is moderately large. For example,
when s = 0.8 and σs = 0.1, the expenditure share is more likely to be in the right corner(s = 1) instead of a point closer to the theoretical share s = 0.8. For this reason, we
specified the error term with the truncated normal distribution instead of the censored
distribution. Note that the former is also a standard specification in related literature(Andreoni, Gillen and Harbaugh, 2013; Ahn et al., 2014; Echenique, Imai and Saito,
2020, Online Appendix).

12

Cobb–Douglas utility function at the center. This symmetry is visible if
we see the parameters in the (σ, ρ + 1)-plane as in Figure 3.
The prior distribution for log(σ) was set to the Student’s t-distribution
with degrees of freedom ν = 4. The reason for this choice lies in Figure 4.
The Student’s t-distribution prior on log(σ) (left panel) can treat eachof the three limits of the CES utility function more uniformly than can
the uniform prior (right panel). In other words, the uniform distribution
is not non-informative here because the uniform distribution on log(σ)
unintentionally emphasizes the Leontief and linear functions, as seen in theright panel. The Cauchy distribution (the Student’s t-distribution with
degree of freedom ν = 1) and the normal distribution (the Student’s tdistribution with degree of freedom ν = ∞) can be candidates for the priordistribution. However, the degree of freedom ν = 4 is adopted here to makethe statistical model weakly informative and as parsimonious as possible(Gelman, Simpson and Betancourt, 2017).
Finally, for the scale parameter σs of the residual error term, the exponential distribution with rate parameter λ = 0.5 was adopted as a priordistribution. The rationale for this choice is as follows. First, note that
because demand is modeled in the share form (6), which can be expressed
within a unit interval, an error term with a scale of σs > 1.5 is sufficient
to mimic the uniformly random choice model proposed in Bronars (1987).
Furthermore, the 0.95-quantile and 0.99-quantile of the exponential distribution with the rate parameter λ = 0.5 are about 1.5 and 2.3, respectively.
Therefore, this prior distribution assigns a positive probability value evento models that can be considered as almost completely random choice behavior, and eventually, it does not seem to be very restrictive. This choice
allows us to keep the model as parsimonious as possible (Gelman, Simpson
and Betancourt, 2017).

2.3

Rescaling log(σ) to τ

Although the model was formulated by three parameters α, log(σ), andσs , a different scale was used to analyze the data. In particular, log(σ)
was treated by converting to the value of F4 : the cumulative distribution
13

5

Leontief

4log(σ) = -1log(σ) = -0.5ρ+1

3log(σ) = 0 (Cobb-Douglas)2log(σ) = 0.5log(σ) = 1

1

Linear
0

0

1

2

3

4

5

σ

Figure 3: The symmetric reparameterization log(σ)
Notes: Equation ρ + 1 = 1/σ and points (σ, ρ + 1) are depicted. The plotted points
correspond to symmetric values log(σ) ∈ {−1, −0.5, 0, 0.5, 1}.

Student's t-distribution

Uniform distribution

80

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60

60

x2

100

x2

100

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0

0
0

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0

x1

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100

x1

Figure 4: Prior distributions on log(σ)
Notes: Indifference curves of the extended CES functions with α = 0.5 and 30 different
values of log(σ) (solid lines) and the Cobb–Douglas utility function (dotted lines). Left:
30 different values of log(σ) are the r-quantile points of the Student’s t-distribution with
degree of freedom ν = 4 where r takes a value corresponding to one of the points that
divide interval [0.005, 0.995] into 29 equal length intervals. Right: 30 different values oflog(σ) are the k/29-quantile points (k = 0, 1, . . . , 29) of the uniform distribution on theclosed interval between the 0.005-quantile and the 0.995-quantile points of the Student’s
t-distribution with ν = 4. Note that if we use the uniform distribution on a broader
interval, it gives more emphasis to the Leontief and linear functions.

14

function of the Student’s t-distribution with ν = 4.11 Let τ be the rescaled
substitution parameter defined as follows: 1 log(σ) log(σ)2 + 6(8)
τ = F4 log(σ) = +32
2 log(σ)2 + 4 2
where log(σ) ∈ (−∞, ∞). Note that τ is defined by using a one-to-one
correspondence from ρ: the Leontief limit ρ → ∞ corresponds to τ → 0,
the Cobb–Douglas limit ρ → 0 corresponds to τ = 1/2, and the linear limitρ → −1 corresponds to τ → 1. Eventually, data analysis was conductedbased on the parameters α, τ , and σs .
This rescaling has two methodological advantages. First, because τ isin the unit interval, it is normalized to be on the same scale as α. Using
the same scale helps us compare the variability of α and τ in the analysis.
Second, this rescaling enables a well-balanced treatment of the three limitsof the CES utility function. Indeed, as can be seen in Figure 4, if parameterlog(σ) is weighted by the Student’s t-distribution with ν = 4, then thethree limits of the CES utility function are more balanced than the original
scale that corresponds to a uniform weight on log(σ). That is to say, thecumulative distribution function F4 can map the three limits of the CES
utility function on the unit interval in a balanced way.12

3

Reanalysis of Fisman, Kariv and Markovits(2007)We reanalyzed FKM’s replication data (Fisman, Kariv and Markovits,2019).13 For comparison, we also estimated the standard CES version of

11

See, for example, Shaw (2006).
Note that the prior distribution set to log(σ) can also be interpreted as a uniform
prior distribution on the parameter τ . In other words, τ is a scale to normalize and,at the same time, balance the three limiting cases of the CES utility function. Notethat our Bayesian method can be modified to a maximum likelihood method or a leastsquares method by using some optimization methods (Gelman et al., 2013, Section 4.5).
However, such modifications are not essential for the purposes of this paper.
13
We used only the data of the two-person budget set experiment in FKM.
1215

our model, that is, a model specified by the standard CES utility function(2) and the statistical assumptions described in Subsection 2.2.
Parameter estimation is conducted by simulating the posterior distribution of parameters by Markov chain Monte Carlo (MCMC) methods. The
analysis is implemented in R (R Core Team, 2016) using MCMC methods
in Stan (Carpenter et al., 2017; Stan Development Team, 2022).

3.1

Meaningfulness of the distribution parameter

First, Figure 5 shows that our extended CES model is more appropriate
than the standard CES model for estimating the distribution parameter
α. When compared with the observed choices depicted in the top panelof each column, the posterior distribution of α and its estimate for our
extended CES model (the vertical dashed line depicted in the panel labeled“Extended”) seem to be more relevant than the standard CES version(labeled “Standard”). More specifically, the posterior distribution of thestandard CES version of our model spreads if the observed choices areconcentrated on the diagonal line, and shrinks around α = 1 if the observed
choice is not on the diagonal line. This is caused directly by the vanishingof the distribution parameter in the standard CES utility function if ρ ≈ ∞(or τ ≈ 0). By contrast, the extended CES model helps us interpret the
distribution parameter α consistently over individuals.

3.2“Equality–efficiency trade-off ”

Here, we point out that interpreting the CES parameter ρ as “the equality–
efficiency trade-off” is inappropriate. In the literature, the parameter ρ isoften taken to represent “the equality–efficiency trade-off” that individual
subjects face when distributing payoffs. This interpretation stems fromthe fact that in the standard CES model, the case ρ → ∞ (or τ → 0) corresponds to the symmetric Leontief function. According to the standard
CES model, as ρ positively diverges, choices are concentrated around the
diagonal line x2 = x1 , regardless of relative prices (see Figure 1). Thus,
the equality of payoff allocations increases, while at the same time, the effi16

^ = 0.8, ^τ = 0.1 (ID 43)
α
80

60

60

60

40

x2 (other)

80

20

40

20

0

20

60

80

0
0

20

40

0.50

Extended
1.00^ = 0.76α^τ = 0.05

80

0

20

0.75

0.50

0.75

0.50

40

60

80

x1 (self)

Standard
1.00^ = 0.8α^τ = 0.1

τ

0.75

τ

0.75

Standard
1.00^ = 0.53α^τ = 0.05

τExtended60

x1 (self)

Extended
1.00^ = 0.99α^τ = 0.14

0.75

0.50

Standard
1.00^ = 0.92α^τ = 0.03

0.75

τ

40

τ

20

x1 (self)

1.00

40

0
0

0.50

^ = 1
α
^τ = 0.11

0.50
0.25

0.00

0.00

0.00

0.00

0.00

0.00

0.
5
0.
6
0.
7
0.
8
0.
9
1.
0

0.25

0.
5
0.
6
0.
7
0.
8
0.
9
1.
0

0.25

0.
5
0.
6
0.
7
0.
8
0.
9
1.
0

0.25

0.
5
0.
6
0.
7
0.
8
0.
9
1.
0

0.25

0.
5
0.
6
0.
7
0.
8
0.
9
1.
0

0.25

0.
5
0.
6
0.
7
0.
8
0.
9
1.
0

τ

^ = 0.92, ^τ = 0.03 (ID 76)
α80

x2 (other)x2 (other)^ = 0.53, ^τ = 0.05 (ID 16)
α

α

α

α

α

α

α

Figure 5: Meaningfulness of the extended CES model
Notes: Each column corresponds to an individual subject, whose ID is presented on the
top panels. The posterior mean estimates α̂ and τ̂ using our extended model are alsopresented on the top panels. Top: The observed choices of an individual subject (redcircle), the estimated choices based on our model (blue square), and the residuals definedby the differences between the two (black arrow). Bottom: The posterior samples of(α, τ ) based on our model (labeled “Extended”) and the standard CES version of themodel (labeled “Standard”), i.e., a model specified with the utility function (2) and thestatistical assumptions described in Subsection 2.2. The dashed lines indicate posteriormean estimates, which are also presented at the top-right of each panel.

17

ciency of payoff allocations decreases. However, Figure 5 shows that thesesubjects did not face such a “trade-off.” While the τ estimates do not differ much between these subjects in both the standard and extended CES
models, the observed behavioral tendencies for minimizing the differences(maximizing equality) vary considerably between subjects—obviously, ID16 allocates equality-oriented, while ID-76 behaves quite selfishly, if not
perfectly, although they both give up efficiency. It is still reasonable tointerpret a large ρ (or a small τ ) as low-efficiency orientation as conventionally assumed. However, it is not appropriate to interpret a large ρ asa high-equality orientation because the value of ρ solely does not discriminate between equality-oriented ID-16 and selfish ID-76. The mathematicalshortcoming of the standard CES function—for ρ → ∞ (or τ → 0), theeffect of α vanishes, and the representation of the various behavioral patterns degenerates to the equality orientation—has led to the mistaken understanding of ρ as the “equality–efficiency trade-off.” Note that although
our extension of the CES function resolves α’s degeneracy, it is still notplausible to interpret a large ρ as a high-equality orientation.
Let us now consider the interpretation of the two parameters of theextended CES model. In the extended CES model, τ represents intensityof maximizing efficiency by appropriately controlling the strength of the
behavioral reaction to price changes regardless of the value of α, excluding
the extreme case α = 1. As noted in section 2.1, focusing on α = 1/2, thethree extreme cases of Leontief, Cobb–Douglas, and linear are special casesof “proportional equality,” so the distribution parameter α seems appropriate as a measure of preference for equality. From this fact, we can also saythat τ is a parameter that organizes the various forms of equality-focused
behavior in terms of their different responses to price changes (or differences in the intensity of maximizing efficiency). Therefore, the extended
CES model maps the distributional behavior of subjects—including egalitarian, non-egalitarian, and these intermediate—using two separate axesof “equality” and “efficiency.”14

14

It is reasonable to assume that this trade-off exists in distributional behavior if one

18

Finally, although the CES parameters are a map that organizes various
distributional behaviors, this is not a complete list of possible behavioral
patterns. Which of these various behaviors is socially more desirable isnot self-evident. Indeed, the desirability issue seems to have a substantial
relationship with one of the main issues in political philosophy (Gosepath,2021, Section 3). Thus, in general, one should be cautious about judging
small (or large) values of CES parameters as desirable or just a priori.

3.3

Comparison with the standard CES specification

Figure 6 shows estimation differences between our model and the standard
CES version across all subjects. As intended in the utility function specification, the estimation differences for the distribution parameter α areprevalent for the case τ̂ < 0.5. Moreover, in most of these cases, the estimated α of our model is considerably smaller than that of the standard
CES model. The reason for this result can be inferred from Figure 5. If a
subject behaves so that the flexible CES estimate is around α = 0.5 (as in
the left panel of Figure 5), then the posterior mean of the standard CES
model must be higher than 0.5 because such behavior is compatible with abroad range of α > 0.5 for the standard CES model. On the other hand,
if a subject behaves so that the flexible CES estimate has α > 0.5 (as in
the middle and right panels of Figure 5), then the posterior mean of thestandard CES model must be higher than that value because such behavior
is only compatible with some narrow range around α ≈ 1 for the standard
CES model. Thus, by using the standard CES specification, each of these

intentionally designs a set of choices such that subjects face an “equality–efficiency tradeoff,” Engelmann and Strobel (2004, 2006) are examples of such experiments. However,whether this trade-off represents a typical real-world constraint is context-dependentand debatable. For example, it has long been argued that equality does not necessarily
sacrifice (to a large extent) efficiency (Okun, 2015, Section 3 and 4; Herzog, 2021, Section 4.2). Andersen and Maibom (2020) have noted that “empirical knowledge on the
precise form and slope of the trade-off is scant.” They have also argued for the trade-off
between equality (expressed using the Gini coefficient) and efficiency (described as average income per capita) using cross-country data. However, even in this existent case,it has been shown that many countries would not face this trade-off.

19

subjects would be considered more selfish than their behavioral tendencies
would indicate.
For the case τ̂ > 0.5, the estimation differences on α are much lessprevalent. This also reflects our intention because the utility function in
our model is defined by the standard CES function when τ > 0.5. However,note that even for τ̂ > 0.5, we see differences in the estimates, which areconcentrated around α = 1. More precisely, for some cases such that τ̂ >0.5 and α̂ ≈ 1, the estimated τ are smaller in our model than in the standard
CES model. Figure 7 shows the reason behind this result. As can be seenin the right panel, in the standard CES model, the posterior distribution ofτ is not consistent with the range τ < 0.5 because most of the parameters(α, τ ) in that range correspond to a symmetric Leontief function (i.e., atendency to maximize equality) that is far from the observed behavior
presented in the left panel (i.e., a tendency to minimize equality). On
the other hand, in our model, the posterior distribution of τ is consistent
with the range τ < 0.5 because the parameter α represents intensity ofminimizing inequality regardless of the value τ . Therefore, the posterior
distribution of τ becomes broader, and its posterior mean becomes smallerin our model than in the standard CES model.

3.4

Inferential uncertainty

First, note that the wide posterior distribution of τ , shown in Figure 7,
is not a drawback of our model compared with the standard CES model.
Rather, our model adequately represents the uncertainty because the observed behavior is consistent with a wide range of τ . In the standard CES
model, on the other hand, the observed behavior is mainly consistent with
τ > 0.5, but this is primarily due to the ineffectiveness of α in the τ < 0.5
area, rather than an observation of some clear tendency to respond to pricechanges.
However, it is also true that this variability is a problem that needs tobe addressed as long as we use the CES utility function framework in thecontext of altruistic preferences. Indeed, as can be seen from definition (5),for any value of τ , α = 1 corresponds to a pure selfish preference that only
20

1.0

α

0.8

0.6

0.00

0.25

0.50

τ

0.75

1.00

Figure 6: Estimation differences between models
Notes: Posterior mean estimates of our model (blue square) and the standard CES
version of it (black circle), and differences between the two estimations for each subject(black arrow).

^ = 0.97, ^τ = 0.51 (ID 45)
αExtended

80Standard

1.0

1.0

0.9

0.9

0.8

0.8

40

20

0

α

α

x2 (other)

60

0.7

0.7

0.6

0.6

0.5
0

20

40

60

80

0.5
0.00

0.25

0.50

τ

x1 (self)

0.75

1.00

0.00

0.25

0.50

τ

0.75

1.00

Figure 7: Estimation differences when α̂ ≈ 1
Notes: A case such that α̂ ≈ 1 causes smaller τ̂ in our model than in the standard CES
model. Left: The observed choices of an individual subject (red circle), the estimatedchoices based on our model (blue square), and the residuals defined by the differences
between these two (black arrow). The posterior mean estimates of α and τ are also
indicated on the top. Middle and Right: The posterior samples of (α, τ ) based on
our model (labeled as “Extended”) and the standard CES version of the model (labeled
as “Standard”). The dashed lines indicate posterior mean estimates.

21

considers x1 (the payoff to self) and ignores x2 (the payoff to the other).
Hence, when α = 1, the value of τ can be anything, and thus, it is not
identifiable. Moreover, in the context of distributional preferences, some
selfish preference-type subject may correspond to α ≈ 1 in the CES utility
function framework. Therefore, in such cases, an estimate of τ tends to be
noisy because many values of τ are consistent with the observed behavior
that indicates α ≈ 1. Figure 8 visualizes this point. As shown in the figureon the left, there are a non-negligible number of subjects who show α̂ ≈ 1
in this experiment; and in the figure on the right, the uncertainty in theposterior distribution of τ is large for those subjects.
Finally, Figure 8 suggests two more patterns of inferential uncertaintyof the model. First, the uncertainty of α is smaller than that of τ . Thiscan be seen from the fact that the vertical bars of each marker tend to
be relatively shorter than the horizontal bars.15 Second, the uncertainty islarger when the estimated scale of residual error σˆs is large. This can beseen from the fact that the bars of each marker tend to be relatively longer
for the red triangles than for the blue circles.
In the next subsection, we present simulation results to confirm thesegeneral patterns of our model’s inferential uncertainty in a more systematicway, and to provide practical guidelines for using this methodology in thecontext of distributional preferences.

4

Parameter recovery simulation

To see general patterns of our model’s inferential uncertainty, we conducteda parameter recovery simulation, which is a simple yet important practicefor any model-based behavior analysis (Wilson and Collins, 2019).
We conducted a simulation study by generating synthetic data sets
fixing parameters α, τ , and σs . In particular, α and τ are fixed so that

15

Note that the scales of the axes are different in the right panel. In particular, if thelength of the vertical and horizontal bars on a marker is the same, it means that theformer is actually smaller than the latter.

22

the overall simulation covers a valid range such that α ≧ 0.5 and τ ∈[0, 1].16 Moreover, σs was fixed at σs = 0.01, 0.05, 0.1, and 0.15 to cover a
typical range obtained by our estimation in the FKM data. For any fixedparameters α, τ , and σs , we generated 200 data sets, each consisting of 50
choices. Hence, in total, we generated 6 × 11 × 4 × 200 = 52, 800 data setsin the entire simulation. In the following, we focus on the cases σs = 0.01
and σs = 0.1. Results for other cases are presented in the Appendix.
Each choice is generated by first computing the share demand fromequation (6) and then generating a noisy share demand from the truncatednormal distribution described in 2.2.1 in subsection 2.2.17 Note that forcomparison purposes, we used an identical price system with 50 specificbudget constraints throughout the entire simulation.

4.1

Parameter identification (Case σs = 0.01)

First, by focusing on the case of σs = 0.01, we observe the impact of theparameter identification problem. Figure 9 shows simulation results forthis case.
The first thing to note is that for most of the true parameters, we
can recover their values by our inference procedure. In particular, we areable to recover the distribution parameter α at a level that can be usedin practice. This is good news because the distribution parameter is amore fundamental parameter in the context of distributional preferences.
Moreover, the rescaled substitution parameter τ can also be recovered inlarge part of the entire combinations of true parameter values. However,note that the estimation of τ is relatively more variable than that of α ineach combination.

16

More precisely, α was set at α = 0.5, 0.6, 0.7, 0.8, 0.9, and 0.99, and τ was set atτ = 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.99. Note that we ignore α < 0.5
because it is symmetric to the case α > 0.5.
17
Parameter σ is computed from fixed τ by using the quantile function of the Student’s
t-distribution with ν = 4 and the exponential function. See the definition of τ in (8) in
subsection 2.3.

23

1.0

0.8

0.8

α

α

1.0

0.6

0.6

0.00

0.25

0.50

0.75

0.00

1.00

0.25

0.50

0.75

1.00

τ

τ

Estimated σs ≤ 0.1 :

True

False

Figure 8: Scatterplot of the extended CES estimates
Notes: Left: Posterior mean estimate of (τ, α) and marginal histograms. Right:
Posterior mean estimate (blue circle or red triangle) with 0.025- and 0.975-quantile ofthe marginal posterior distributions of α and τ (bars on a marker).

1.0

0.9

α

0.8

0.7

0.6

0.5

0.0

0.1

0.2

0.3

0.4

0.5

τ

0.6

0.7

0.8

0.9

1.0

Figure 9: Estimation results of the simulation with σs = 0.01
Notes: For each true parameter (τ, α), the true parameter value (black closed marker),
200 corresponding simulation results (colored open markers), and mean of the 200 simulations (blue open marker) are depicted. Note that true parameter values and corresponding means calculated by 200 simulations are connected by black lines.

24

On the other hand, as expected from definition (5), the parameter identification problem for τ is severe when α = 0.99. This is a common feature
for any CES specification and hence, inevitable. Even when α = 0.9, identifying τ is difficult for cases such that τ ≧ 0.7. It should also be notedthat this latter difficulty stems from not only the specification of the CES,
but also the finite nature of the experimental method. Specifically, because
the budget system consists of a finite number of price systems, there is alimited amount of information to identify subtle differences in τ .

4.2

Inferential uncertainty (Case σs = 0.1)

Finally, we focus on a more realistic case, σs = 0.1, to examine the characteristics of the inferential uncertainty of our estimation procedure. Figure 10 shows the result. The first thing to note is that, as expected in
the previous section, the inferential uncertainty is larger when the scale ofresidual error σs is large. As in the previous case, α can be recovered more
precisely than τ . Indeed, if we note that the horizontal and vertical axesin this figure are drawn on the same scale, we see that the α estimate isless likely to overlap with its neighbor 0.1 away when compared with τ .
We also see that while the estimates for τ are not so precise, they do not
overlap significantly with parameters that are 0.2 apart, which suggeststhat they hold reasonable information. However, it should be noted thatthe bias and variability of the estimates become more severe for α = 0.9
and 0.99.
Hence, to observe this uncertainty more quantitatively, Figure 11 summarizes the same results in terms of inferential error. For both parameters,
the magnitude of estimation error largely depends on the value of α rather
than that of τ . In general, the estimation error is relatively smaller for α
than for τ , but even for α, distinguishing subtle differences seems difficult
when α ≈ 1. However, estimation for α is reasonably reliable to distinguishlarge differences. Moreover, estimation for τ is also reliable when we focus
on the cases where the value of α is not extreme such as α ≦ 0.8. Note thatin the FKM data, a non-negligible number of subjects were in that reliable
range, and the variation of τ in those subjects was not trivial (see Fig25

ure 8). Hence, even though these parameter estimates are not perfect, wemay be able to extract valid and meaningful information about behavioral
differences and the implications of these differences.

4.3

Discussion of the Results

Overall, these results suggest that special care should be taken when using this experimental procedure in practice, especially when analyzing thevalue of τ . Especially when α ≈ 1, it is inevitable that we consider theselimitations in the analysis if we aim to obtain consistent results among different subjects. In the context of distributional preferences, such difficult
cases are not merely logical possibilities because most experiments must
involve some proportion of selfish preference-type subjects. For example,
in such cases, τ can be dropped in the subsequent analysis if α is above
a certain threshold, e.g., α > 0.8 in the current settings. It is also desirable to conduct a sensitivity analysis of the threshold value. However,this threshold will depend on the purpose of the analysis, the set of theprice systems, and the setting of the statistical model, so that no genericthreshold is likely to exist. Therefore, it would be desirable to conductsimulations and set the threshold in advance for each individual study.

5Conclusion

Prior work has developed an experimental procedure that allows for collecting rich individual-level data to elicit distributional preferences (Andreoni
and Miller, 2002; Fisman, Kariv and Markovits, 2007). While this procedure has been adopted in several application studies, it has also been
pointed out that the estimated distribution parameter is not meaningful
over a range of possible parameter values (Thöni, 2015). This study extended the basic estimation method and demonstrated its properties toprovide some guidelines for its use in the context of distributional preferences.
We extended the class of CES utility functions by using a combination

26

1.0

0.9

α

0.8

0.7

0.6

0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

τ

0.7

0.8

0.9

1.0

Figure 10: Estimation results of the simulation with σs = 0.1
Notes: For each true parameter (τ, α), the true parameter value (black closed marker),
200 corresponding simulation results (colored open markers), and mean of the 200 simulations (blue open marker) are depicted. Note that true parameter values and corresponding means calculated by 200 simulations are connected by black lines.

0.5

Estimation error of τ

Estimation error of α

0.25

0.00

−0.25

τ

0.01
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.99

0.0

−0.5

0.5

0.6

0.7

α

0.8

0.9

0.99

0.5

0.6

0.7

α

0.8

0.9

0.99

Figure 11: Estimation error of the simulation with σs = 0.1
Notes: Left: Estimation error of α. Right: Estimation error of τ . In each panel, eachbox corresponds to 200 estimation errors defined by the estimated values minus true
parameter value α or τ . Dashed horizontal lines indicate ±0.05. Note that the scale ofthe vertical axis of the left panel is doubled compared with the right panel.

27

of known CES specifications to make interpretation of the distribution parameter, α, align better with individual behavior. Based on the extended
CES utility specification, we developed a parameter estimation method
using the Bayesian statistical modeling approach that also gives a morebalanced scale for the substitution parameter, τ , than the original scale, ρ.
Our reanalysis of data from previous studies demonstrates that theextended estimation method gives a straightforward interpretation of the
distribution parameter rather than the conventional method. This is due toour extended CES specification that separately controls the distributional
share and the substitution reaction of individual behavior. Moreover, thismakes clear that the substitution parameter does not necessarily represent“the equality–efficiency trade-off” because this interpretation is primarily
based on the limitation of the standard CES specification, and not on the
restriction of individual behavior. Note that our conclusion is not that
individuals never face such trade-offs. Instead, our results suggest thatto examine whether people face the trade-off, we should address it moreelaborately in the research design. Note that the existence of “the equality–
efficiency trade-off” is context-dependent and debatable (see footnote 14).
In our parameter recovery simulation study, we also found that the distribution parameter is more reliable to use in further analyses than thesubstitution parameter. The estimates for the substitution parameter may
hold reasonable information if we use them with special care. More precisely, τ may have valid information when the distribution parameter α
is not close to 1. Hence, for example, one possible option is to drop τ if
α is above some threshold value, and conduct sensitivity analysis on thethreshold value. Because the threshold depends on the study design, it isdesirable to perform simulation analysis in advance. Note that, in the context of distributional preferences, such difficult cases should not be mere
logical possibilities because most experiments should involve some selfish
preference-type subjects. In the Fisman, Kariv and Markovits (2007) data,
there was a non-trivial variation of τ (see Figure 8). Hence, even thoughthese parameter estimates are imperfect, we may be able to extract valid
and meaningful information about behavioral differences and their impli28

cations. Finally, we reemphasize here that this type of simulation study isa simple and important practice for any model-based behavioral analysis(Wilson and Collins, 2019).
While this study extends the basic estimation method and provides amore appropriate interpretation with some guidelines for its use, further
cautions are worth noting. First, as noted in section 3, we should take carein interpreting the distribution parameter beyond the “equality” interpretation. The equality-focused behavior represented by setting α = 1/2 isa subset of the broad notion of “proportional equality,” which can contain egalitarian and non-egalitarian ideas of justice (Gosepath, 2021, p.
8). Therefore, in particular, we should be cautious in making a priori
judgments that small (or large) values of the distribution parameter aredesirable or just. Note that this experiment focuses on a local distributional decision between a few people in an abstract setting. Hence, weshould be aware that concepts discussed in political philosophy, which focuses on more global and institutional problems in more specific contexts,
may not directly apply to this experimental setting. Future work should
examine appropriate meanings and contexts of “equality” and “efficiency”
represented by the CES parameters rather than assuming these a priori.
Second, our model cannot be a remedy for potential “experimenter demand
effects” (Zizzo, 2010). Hence, we should pay attention to when and how
to use this experimental procedure, even when we use our extended data
analysis method. Finally, because our focus in this study is on CES utility functions and their parameters, it is obvious that some other variables
remained unchanged in the method and can be improved. In particular,
future work will consider other aspects of the model, such as the residual error term or the class of utility function in the first place, to make the model
more general and flexible, better representing subjects’ behavior. Future
work should also examine the set of price systems of the experiment tomake informative, cost-efficient, and purpose-oriented inferences.

29

Appendix
We present the simulation results for the cases σs = 0.05 (Figure A1)and σs = 0.15 (Figure A2). The general patterns are similar to the cases
σs = 0.01 and σs = 0.1. In particular, the estimation error is relatively
smaller for α than for τ . Limitations of the estimation are more severe
when noise is large, such as σs = 0.15. That is, even for α, distinguishingsubtle differences seems difficult when α ≈ 1. Therefore, as discussed in themain text, to extract valid and meaningful information about behavioral
differences and their implications, they should be used with special care.

30

1.0

0.9

α

0.8

0.7

0.6

0.5

0.0

0.1

0.2

0.3

0.4

0.5

τ

0.6

0.7

0.8

0.9

1.0

Figure A1: Estimation error of the simulation with σs = 0.05
Notes: For each true parameter (τ, α), the true parameter value (black closed marker),
200 corresponding simulation results (colored open markers), and mean of the 200 simulations (blue open marker) are depicted. Note that true parameter values and corresponding means calculated by 200 simulations are connected by black lines.

31

1.0

0.9

α

0.8

0.7

0.6

0.5

0.0

0.1

0.2

0.3

0.4

0.5

τ

0.6

0.7

0.8

0.9

1.0

Figure A2: Estimation error of the simulation with σs = 0.15
Notes: For each true parameter (τ, α), the true parameter value (black closed marker),
200 corresponding simulation results (colored open markers), and mean of the 200 simulations (blue open marker) are depicted. Note that true parameter values and corresponding means calculated by 200 simulations are connected by black lines.

32References
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