A Model of Income Evaluation:

Income Comparison on Subjective Reference

Income Distribution

Atsushi Ishida (Kwansei Gakuin University)

∗

July 3, 2021

Abstract

People’s evaluation of the relative position of their income is not as

accurate as the relative income hypothesis assumes. It is observed from

empirical survey data that income evaluation is concentrated in the mid-

dle. We develop a model that assumes income comparison on a subjective

income reference distribution to explain the centralization phenomenon

of income evaluation. We conduct theoretical analysis and empirical pa-

rameter estimation using Bayesian statistical modeling. The theoretical

analysis shows that the centralization of income evaluation distribution oc-

curs when the subjective reference distribution is more dispersed than the

objective distribution. Empirical analysis using Japanese data from 2015

shows that the relationship between subjective and objective distributions

diﬀered depending on social categories with diﬀerent social experiences.

Women had a more ambiguous distribution than men. Among men, those

aged 45–54 had a subjective distribution closest to the objective distribu-

tion. Thus, the subjective reference income distributions that potentially

deﬁne people’s evaluation of their income and their diﬀerences based on

so cial category were only clariﬁed by constructing the model.

Keywords: income distribution; relative income hypothesis; Bayesian

statistical modeling

1 Introduction

This article focuses on individuals’ evaluations of their own income in relation

to others within the income distribution. Income evaluation can be both the

basis for satisfaction or dissatisfaction in terms of one’s economic situation. It

can also create a sense of fairness or unfairness regarding income distribution,

which could lead to social change or stabilization. It is assumed that evaluation

of income would be obtained by recognizing the relative position of one’s own

income in the distribution through comparison with others. Hence, this is also

related to the empirical validity of the “relative income hypothesis” in economics.

The relative income hypothesis was ﬁrst explicitly proposed by Duesenberry

(1949). Duesenberry proposed the idea that an individual’s consumption func-

tion depends on others’ income and the relative position among them. He

∗

aishida@kwansei.ac.jp

1

derived the theorem that “for any given relative income distribution, the per-

centage of income saved by a family will tend to be unique, invariant, and

increasing function of its percentile position in the income distribution. The

percentage saved will be independent of the absolute level of income. It fol-

lows that the aggregate saving ratio will b e independent of the absolute level

of income”(Duesenberry, 1949, 3). Duesenberry’s idea has recently been recon-

sidered, especially in the ﬁeld of subjective well-being studies. For example,

the well-known Easterlin Paradox was proposed (Easterlin, 1974, 1995, 2005);

it was empirically derived from longitudinal data in the United States, Japan,

and European nations. It states that the average happiness of citizens remains

constant over time, despite a sharp increase in national income per capita. The

relative income hypothesis is a prominent supposition, explaining this type of

paradox (Clark et al., 2008). Furthermore, there have been several empirical

survey data analytical studies using relative income variables as explanatory

variables (Clark and Oswald, 1996; Mcbride, 2001; Ferrer-i Carbonell, 2005;

Senik, 2008; Clark and Senik, 2010).

As seen in Dusenberry’s theorem, the relative income hypothesis generally

assumes that people correctly perceive their relative position in the income dis-

tribution. This study focuses on the empirical validity of this assumption. As

empirical evidence, we present survey data conducted in Japan in 2015 (see

Section 4 for details of the data). Figure 1 shows the distribution of individ-

ual annual income

1

, and Figure 2 shows the distribution of relative evaluation

of income, which ranges from 1 (lowest) to 10 (highest)

2

. Figure 1 shows the

income distribution is well ﬁtted by a lognormal distribution as theoretically

expected (Hamada, 2003, 2004). As for theoretical expectation of the distribu-

tion of income evaluation, if people perceive their income levels correctly, the

distribution is expected to be uniform as will be explained in the next section.

However, Figure 2 shows concentration slightly below the middle point and is

far from a uniform distribution.

[Figure 1 about here.]

[Figure 2 about here.]

These ﬁndings lead us to the puzzle of this article which is why the distribu-

tion of the relative evaluation of income level is centralized. The centralization

of the distribution of class identiﬁcation, which is a multidimensional evaluation

of status, is well known. Several mathematical models have been proposed to

explain this phenomenon (Fararo and Kosaka, 2003; Ishida, 2018). However, to

the best of our knowledge, there are no studies on the mechanism of centraliza-

tion in unidimensional income evaluation. To solve this puzzle, in the following

sections, we introduce a model that assumes that comparisons are made on a

subjective reference distribution of income that is shared among members of a

group rather than the objective income distribution per se.

We ﬁrst introduce a general model of income evaluation and derive some im-

plications of the model. We then introduce a model imposing the assumption of

1

The unit of income is the Japanese yen. We treated 344 cases with no individual income

and three cases with 100 million yen as missing.

2

The question in the questionnaire was as follows: Thinking about present-day Japanese

society, how would you rate your own level of income below on a scale of 1 (the highest level)

to 10 (the lowest level)?

2

lognormal distribution as the theoretical distribution of income. Next, we con-

duct an empirical data analysis of Japanese survey data by employing Bayesian

statistical modeling, based on the lognormal distribution model. Finally, we

present the conclusions.

2 General Model

In this section, we introduce the model in a general form without specifying the

types of income distribution and subjective reference income distribution. We

then derive some implications.

2.1 Model Assumption

Let Y ∈ {0, ··· , m} be a discrete random variable of response to the m + 1-

scaled income evaluation question, which ranges from 0 (lowest) to m (highest).

Let x ∈ (0, ∞) be the income level of an individual and p be the probability

that x exceeds the income level z, that is, p = Pr(x ≥ z).

We assume that an individual evaluates their relative position of income by

repeatedly comparing themselves with others whom they randomly encounter

on a subjective reference income distribution. It is further assumed that the

subjective reference distribution reﬂects the biased pattern of an individual’s

daily encounters and/or their expectations about the distribution that are not

based on their experience but on media information or rumors. Here, we make a

baseline assumption that the subjective reference income distribution is identical

and commonly shared among members of a social category, presupposing that

their social experiences and perceptions are almost similar. In addition, when

evaluating their income, the individual is assumed to respond according to the

number of times they have outperformed others in the last m comparisons.

The general model can be composed as

Y ∼ Binomial(m, p), (1)

p = F

s

(x), (2)

x ∼ f

o

(x), (3)

where F

s

is the cumulative distribution function (CDF in short) of the subjective

reference income distribution and f

o

is the probability density function (PDF

in short) of the objective income distribution.

We are ultimately interested in the distribution of Y . However, if m is a

constant, the distribution of Y depends only on the parameter p of the binomial

distribution, so we are essentially interested in the distribution of p. Then, the

PDF of p, denoted as φ(p), is

φ(p) = f

o

(x)

dx

dp

= f

o

(x)

1

{F

s

(x)}

′

=

f

o

(x)

f

s

(x)

.

From equation (2), x can be expressed as the inverse CDF of the subjective

reference distribution, i.e. x = F

−1

s

(p). Finally, we obtain the full form of the

3

PDF of p as

φ(p) =

f

o

(F

−1

s

(p))

f

s

(F

−1

s

(p))

. (4)

By deﬁnition, we can conﬁrm the following properties of φ(p), which indicate

that φ(p) is surely a PDF:

φ(p) = f

o

(x)

dx

dp

≥ 0,

Z

1

0

φ(p)dp =

Z

1

0

f

o

(x)

dx

dp

dp

=

Z

∞

−∞

f

o

(x)dx = 1.

2.2 Model Derivations

Henceforth, we assume that f

o

, f

s

is a unimodal and second-order diﬀerentiable

PDF. Furthermore, we assume F

s

, which is the CDF of f

s

, is a strictly increasing

function, and F

−1

s

, which is the inverse CDF of f

s

, is also a strictly increasing

and diﬀerentiable function.

First, we assume, as a special case, that the subjective reference income

distribution is equal to the objective income distribution, because there is no

encounter bias. That is, f

o

= f

s

. Then, we obtain φ(p) = 1, which means

p is uniformly distributed from 0 to 1. That is, p ∼ Uniform(0, 1). This is an

ideal situation of income evaluation, assumed by the relative income hypothesis,

where people perceive the objective income distribution correctly and evaluate

their income with respect to the exact relative position in the objective income

distribution.

Now, we move on to more general situations where there is a diﬀerence

between the objective income distribution and subjective reference income dis-

tribution, which is biased from the objective distribution. That is, f

o

= f

s

.

Diﬀerentiating φ(p), which is expressed as equation (4), we obtain the deriva-

tive of φ(p) as

φ

′

(p) =

f

′

o

(x)f

s

(x) − f

o

(x)f

′

s

(x)

f

s

(x)

2

F

−1

s

(p)

′

. (5)

Because f

s

(x) > 0, {F

−1

s

(p)}

′

> 0 on their support from the assumption, the

sign condition of φ

′

(p) solely depends on the relation between the magnitudes

of f

′

o

(x)/f

o

(x) and f

′

s

(x)/f

s

(x) which are the growth rates of PDF of objective

income distribution and subjective income distribution, respectively, that is:

φ

′

(p) ⋛ 0 ⇐⇒

f

′

o

(x)

f

o

(x)

⋛

f

′

s

(x)

f

s

(x)

. (6)

Hence, if there is a point x

∗

that makes the growth rates of both f

o

and f

s

equal,

then the point p

∗

= F

s

(x

∗

) is a lo cal maximum, or minimum, point of φ(p).

For a simple example, if f

′

o

(x

∗

) = f

′

s

(x

∗

) = 0 which indicates both distributions

have the same mode, then the point p

∗

= F

s

(x

∗

) yields φ

′

(p

∗

) = 0.

4

We assume that there is a single point p

∗

such as φ

′

(p

∗

) = 0. The second

derivative at p oint p

∗

is

φ

′′

(p

∗

) =

f

′′

o

(x

∗

)f

s

(x

∗

) − f

o

(x

∗

)f

′′

s

(x

∗

)

f

s

(x

∗

)

2

F

−1

s

(p

∗

)

′

2

. (7)

The sign condition of φ

′′

(p

∗

) depends on the relation between the magnitudes

of f

′′

o

(x)/f

o

(x) and f

′′

s

(x)/f

s

(x), which might be called the growth rate in terms

of second derivative. That is,

φ

′′

(p

∗

) ⋛ 0 ⇐⇒

f

′′

o

(x)

f

o

(x)

⋛

f

′′

s

(x)

f

s

(x)

. (8)

From these derivations, the condition for the appearance of the centraliza-

tion eﬀect in terms of income evaluation can be summarized as follows. There

is a single maximum point that satisﬁes f

′

o

(x

∗

)/f

o

(x

∗

) = f

′

s

(x

∗

)/f

s

(x

∗

) and

f

′′

o

(x

∗

)/f

o

(x

∗

) < f

′′

s

(x

∗

)/f

s

(x

∗

), and p

∗

= F

s

(x

∗

) is around 0.5.

3 Lognormal Distribution Model

Next, we analyze a more restrictive model, in which both the objective and

subjective reference income distributions are assumed to be lognormally dis-

tributed.

3.1 Model Assumption

Let z ∈ (0, ∞) be an individual’s income, and x = log z be the logged income

ranging from −∞ to ∞. We assume that income z is lognormally distributed as

a common assumption in the ﬁeld of income distribution studies. Accordingly,

x is normally distributed. That is,

z ∼ Lognormal(µ, σ),

x ∼ Normal(µ, σ).

The PDF of the objective distribution of logged income x, which is a normal

distribution with parameters µ

o

, σ

o

, is denoted by f

o

(x|µ

o

, σ

o

), and that of the

subjective reference distribution, which is a normal distribution with parameters

µ

s

, σ

s

is denoted by f

s

(x|µ

s

, σ

s

). In concrete terms, the PDF can be expressed

as

f

k

(x|µ

k

, σ

k

) =

1

√

2πσ

k

exp

−

(x − µ

k

)

2

2σ

2

k

, (k = o, s). (9)

3.2 Model Derivation

Let us perform some derivation from the model.

First, to determine the condition of a local maximum point of the distribution

p, we specify the growth rate of the objective and subjective reference income

5