# A model of income evaluation: income comparison on subjective reference income distribution

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 concen...

Topics: Income distribution (69%)

## Summary (3 min read)

Jump to: [1 Introduction] – [2.1 Model Assumption] – [2.2 Model Derivations] – [3.2 Model Derivation] – [4.1 Data] – [4.2 Variables] – [4.3 Common Subjective Distribution Model] – [4.4 Different Subjective Distributions Model] and [5 Conclusion]

### 1 Introduction

- This article focuses on individuals’ evaluations of their own income in relation to others within the income distribution.
- 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.
- This study focuses on the empirical validity of this assumption.
- Figure 1 shows the income distribution is well fitted by a lognormal distribution as theoretically expected (Hamada, 2003, 2004).

### 2.1 Model Assumption

- The authors 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.
- 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.
- From equation (2), x can be expressed as the inverse CDF of the subjective reference distribution, i.e. x = F−1s (p).

### 2.2 Model Derivations

- First, the authors assume, as a special case, that the subjective reference income distribution is equal to the objective income distribution, because there is no encounter bias.
- 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, the authors move on to more general situations where there is a difference between the objective income distribution and subjective reference income distribution, which is biased from the objective distribution.
- (8) From these derivations, the condition for the appearance of the centralization effect in terms of income evaluation can be summarized as follows.

### 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, the authors specify the growth rate of the objective and subjective reference income distribution3.
- They also show that the larger δ is, the smaller the maximum point of p∗ becomes.
- Conversely, behind the middle-concentrated distribution of income evaluation, the authors can assume the existence of a subjective income distribution that is more dispersed than the objective distribution.

### 4.1 Data

- The data used for the analysis are from the Stratification and Social Psychology Project Survey (SSP2015), which is a Japanese national sampling survey of class identity, social images, and other related attitudes toward social inequality and social stratification.
- The survey was conducted between January and June 2015.
- The sampling procedure was a three-stage stratified random sampling.
- The sampling list was the Japanese electoral roll and the basic resident registration.

### 4.2 Variables

- As mentioned in the introduction, the main variables in the model are individual annual income and the 10-scaled income evaluation .
- In addition, the authors introduce gender (male and female) and age cohort as covariates to examine the differences in the income evaluation within different social categories and the differences in the subjective income distributions across the categories assumed to be behind the evaluation.

### 4.3 Common Subjective Distribution Model

- In the following, the authors construct the model as a Bayesian statistical model and estimate the parameters.
- First, as a baseline model, the authors analyze a common subjective distribution model in which all members of a society potentially have the same subjective distribution, regardless of their social category.
- The authors conducted four chains of sampling for 6,000 iterations each, which included 1,000 initial iterations as warm-up samples.
- Because R̂ of each parameter is approximately 1.000, the authors can safely conclude that the MCMC sampling converged (Gelman et al., 2013, 284–6).
- Table 1 and Figure 5 show that the subjective income distribution has a slightly larger mean and a much larger variance than the objective one.

### 4.4 Different Subjective Distributions Model

- Next, the authors prepare a more complex and more realistic model with the additional assumption that each social category shares a different subjective income distribution, reflecting differences in social experience.
- Figure 8 shows estimated parameters of objective and subjective income distribution for each social category represented by posterior mean and interval between 0.05 and 0.95 quantiles.
- This may reflect the fact that women’s participation in the labor market is still lower than that of men, and many women participate in the labor market as a marginal labor force in Japan.
- Again, the authors can see that women have a subjective income distribution that is farther from the objective one than men, and among men, the 45-54 age cohort has a subjective distribution that is closest to reality.

### 5 Conclusion

- Thus far, the authors have developed a model that assumes income comparison on a subjective income reference distribution to explain the centralization phenomenon of income evaluation and have conducted theoretical analysis and empirical parameter estimation using Bayesian statistical modeling.
- The theoretical analysis shows that income evaluation centralization occurs when the subjective distribution is more dispersed than the objective distribution.
- Furthermore, in a specific model assuming a lognormal distribution, the authors could parametrically analyze the effect of the relationship between the subjective and objective distributions on the distribution of income evaluation.
- Furthermore, the authors found that the relationship between the subjective and objective distributions differed depending on gender and age cohorts with different social experiences.
- This can lead to biased income assessments, which has important implications for understanding people’s satisfaction in unequal societies.

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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

##### References

More filters

••

01 Jan 1974Abstract: Publisher Summary This chapter discusses the association of income and happiness. The basic data consist of statements by individuals on their subjective happiness, as reported in thirty surveys from 1946 through 1970, covering nineteen countries, including eleven in Asia, Africa, and Latin America. Within countries, there is a noticeable positive association between income and happiness—in every single survey, those in the highest status group were happier, on the average, than those in the lowest status group. However, whether any such positive association exists among countries at a given time is uncertain. Certainly, the happiness differences between rich and poor countries that one might expect on the basis of the within-country differences by economic status are not borne out by the international data. Similarly, in the one national time series studied, for the United States since 1946, higher income was not systematically accompanied by greater happiness. As for why national comparisons among countries and over time show an association between income and happiness that is so much weaker than, if not inconsistent with, that shown by within-country comparisons, a Duesenberry-type model, involving relative status considerations as an important determinant of happiness, is suggested.

3,968 citations

••

Abstract: Today, as in the past, within a country at a given time those with higher incomes are, on average, happier. However, raising the incomes of all does not increase the happiness of all. This is because the material norms on which judgments of well-being are based increase in the same proportion as the actual income of the society. These conclusions are suggested by data on reported happiness, material norms, and income collected in surveys in a number of countries over the past half century.

2,757 citations

••

Abstract: This paper attempts to test the hypothesis that utility depends on income relative to a ‘comparison’ or reference level. Using data on 5,000 British workers, it provides two findings. First, workers' reported satisfaction levels are shown to be inversely related to their comparison wage rates. Second, holding income constant, satisfaction levels are shown to be strongly declining in the level of education. More generally, the paper tries to help begin the task of constructing an economics of job satisfaction.

2,730 citations

•

Abstract: The well-known Easterlin paradox points out that average happiness has remained constant over time despite sharp rises in GNP per head. At the same time, a micro literature has typically found positive correlations between individual income and individual measures of subjective well-being. This paper suggests that these two findings are consistent with the presence of relative income terms in the utility function. Income may be evaluated relative to others (social comparison) or to oneself in the past (habituation). We review the evidence on relative income from the subjective well-being literature. We also discuss the relation (or not) between happiness and utility, and discuss some nonhappiness research (behavioral, experimental, neurological) related to income comparisons. We last consider how relative income in the utility function can affect economic models of behavior in the domains of consumption, investment, economic growth, savings, taxation, labor supply, wages, and migration.

2,239 citations

••

Abstract: The well-known Easterlin paradox points out that average happiness has remained constant over time despite sharp rises in GNP per head. At the same time, a micro literature has typically found positive correlations between individual income and individual measures of subjective well-being. This paper suggests that these two findings are consistent with the presence of relative income terms in the utility function. Income may be evaluated relative to others (social comparison) or to oneself in the past (habituation). We review the evidence on relative income from the subjective well-being literature. We also discuss the relation (or not) between happiness and utility, and discuss some nonhappiness research (behavioral, experimental, neurological) related to income comparisons. We last consider how relative income in the utility function can affect economic models of behavior in the domains of consumption, investment, economic growth, savings, taxation, labor supply, wages, and migration.

2,005 citations

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