Exponential structure of income inequality: evidence from 67 countries

Abstract

Economic competition between humans leads to income inequality, but, so far, there has been little understanding of underlying quantitative mechanisms governing such a collective behavior. We analyze datasets of household income from 67 countries, ranging from Europe to Latin America, North America and Asia. For all of the countries, we find a surprisingly uniform rule: income distribution for the great majority of populations (low and middle income classes) follows an exponential law. To explain this empirical observation, we propose a theoretical model within the standard framework of modern economics and show that free competition and Rawls’ fairness are the underlying mechanisms producing the exponential pattern. The free parameters of the exponential distribution in our model have an explicit economic interpretation and direct relevance to policy measures intended to alleviate income inequality.

Appendices

Appendix

N-person non-cooperative game

Arrow–Debreu’s General Equilibrium Model (ADGEM) is based on the well-known two criteria of neoclassical economics: utility maximization and profit maximization. If there are N consumers, each of whom operates a firm, the ADGEM describing their optimal behavior uses the following principles (Tao 2015, 2016):

• (a) Profit maximization: For each firm $i=1,\ldots ,N$, $y_i^*\in Y_i$ maximizes profits such that $p\cdot y_i \le p\cdot y_i^*$ for all $y_i \in Y_i$.

• (b) Utility maximization: For each consumer $i=1,\ldots ,N$, $x_i^*\in X_i$ is the solution of maximizing the preference $\mathop \succ \limits _{\sim i}$ under the budget set: $\big \{ x_i \in X_i :p\cdot x_i \le p\cdot \omega _i +\mathop \sum \nolimits _{j=1}^N \theta _{ij} p\cdot y_j^*\big \}$.

• (c) Market clearing: $\mathop \sum \nolimits _{i=1}^N x_i^*=\mathop \sum \nolimits _{i=1}^N \omega _i +\mathop \sum \nolimits _{i=1}^N y_i^*$.

Here $x_i$ and $X_i$ represent consumption vector and consumption set of the $i \mathrm{th}$ consumer, respectively; $y_i$ and $Y_i$ represent production vector and production set of the $i\mathrm{th}$ firm, respectively (Mas-Collel et al. 1995); $\theta _{ij}$ represents an ownership share of each firm $j=1,\ldots ,N$ paid to the $i\hbox {th}$ consumer. The allocation $\left( {x_1^*,\ldots ,x_N^*;y_1^*,\ldots ,y_N^*} \right)$ and a price vector $p=\left( {p_1 ,\ldots ,p_L } \right)$ constitute a Pareto optimal solution to ADGEM (a)–(c).

Rawls’ fairness of “2-person allocation”

For illustration, let us consider a simple “2-person society” in which the GDP is denoted by $2 and each person can earn a possible equilibrium income with $0, $1 or $2. For the “2-person society”, the Eq. (1) can be expressed in the form:

$$\begin{aligned} \left\{ {{\begin{array}{l} I_i =0,1,2\quad for \quad i=1,2 \\ \mathop \sum \nolimits _{i=1}^2 I_i =2 \\ \end{array} }} \right. . \end{aligned}$$

(B.1)

By Eq. (B.1), the “2-person society” will have three equilibrium income allocation (EIA): $A_1 =\left\{ {0,2} \right\}$, $A_2 =\left\{ {2,0} \right\}$ and $A_3 =\left\{ {1,1} \right\}$. They have been shown as below:

By Rawls’ principle of fair equality of opportunity, each EIA should occur with an equal probability (Tao 2015, 2016); therefore, each person’s expected income equals $1. The detailed calculation is as below:

$$\begin{aligned} \hbox {Probability}\left( A_{1} \right)= & {} 1/3,\hbox {Probability}\left( {A_2 } \right) =1/3,\hbox {Probability}\left( {A_3 } \right) =1/3 \nonumber \\ \hbox {Woman's Expected Income}= & {} 0\times \left( {1/3} \right) + 2\times \left( {1/3} \right) +1\times \left( {1/3} \right) =1 \end{aligned}$$

(B.2)

$$\begin{aligned} \hbox {Man's Expected Income}= & {} 2\times \left( {1/3} \right) +0\times \left( {1/3} \right) +1\times \left( {1/3} \right) =1 \end{aligned}$$

(B.3)

This means that each person owns the equal opportunity of earning money. If we denote the equal income distribution by a and the unequal income distribution by b, we do have $a=\left\{ A_{3} \right\}$ and $b=\left\{ {A_1 ,A_2 } \right\}$. By Rawls’ principle of fair equality of opportunity, a will occur with probability 1 / 3 and b will occur with probability 2 / 3. Following the rule of “survival of the likeliest”, b will be a result of natural selection.

Density function of income distribution

For “N-person allocation”, Tao has shown that, by applying Rawls’ fairness into Eq. (1) where N and Y are large enough, one will get the exponential income distribution which occurs with the highest probability (Tao 2015, 2016):

$$\begin{aligned} a_k= & {} g_k e^{-\frac{( {\varepsilon _k -\mu } )}{\theta }}, \nonumber \\&\varepsilon _1<\varepsilon _2<\cdots <\varepsilon _n. \end{aligned}$$

(C.1)

Here $\mu$ and $\theta$ denote marginal labor-capital return and marginal technology return, respectively (Tao 2016); readers can find the origin of these two parameters in “Appendix D”.

The formula (C.1) indicates that there are $a_k$ consumers each of which obtains $\varepsilon _k$ units of revenue, and k runs from 1 to n. Because income distribution (C.1) will occur with the highest probability, Tao call it the “spontaneous economic order” (Tao 2016).

The formula (C.1) can be rewritten in the form of continuous function. To see this, let us first observe:

$$\begin{aligned} \mathop \sum \limits _{k=1}^n a_k =N, \end{aligned}$$

(C.2)

which leads to:

$$\begin{aligned} \mathop \sum \limits _{k=1}^n \frac{a_k }{N}=1. \end{aligned}$$

(C.3)

Here $\frac{a_k}{N}$ denotes the proportion of populations each of whom earns $\varepsilon _k$ units of income. Now we write $\frac{a_k }{N}$ in the form of continuous function: $f\left( x \right)$. To this end, let us order:

$$\begin{aligned} f\left( x \right) =w\cdot e^{\frac{- ( {x-\mu })}{\theta }}, \end{aligned}$$

(C.4)

where x, which replaces $\varepsilon _k$, denotes a continuous income level, and by Rational Agent Hypothesis one has (Tao 2010) $x\ge \mu$.

Here w is an undetermined constant, which will be determined by the sum formula (C.3). Let us replace $\frac{a_k }{N}$ by (C.4), and transform sum operation of formula (C.3) into integral operation:

$$\begin{aligned} \mathop \int \nolimits _\mu ^{+\infty } w\cdot e^{\frac{-({x-\mu })}{\theta }}dx=1, \end{aligned}$$

(C.5)

which leads to $w=\frac{1}{\theta }$.

Finally, we obtain the density function of income distribution:

$$\begin{aligned} f\left( x \right) =\frac{1}{\theta }e^{\frac{- ({x-\mu })}{\theta }}. \end{aligned}$$

(C.6)

Technological progress and entropy

Because the firm consists of labor and capital, the Cobb–Douglas aggregate production function (or GDP) of neoclassical economics can be written in the form (Tao 2010, 2016):

$$\begin{aligned} Y=Y\left( {N\left( {L,K} \right) ,H} \right) , \end{aligned}$$

(D.1)

where L and K denote labor and capital, whereas $N\hbox { and }H$ denote the number of firms and technological progress.

The complete differential of (D.1) yields [see also Eq. (9) in Banerjee and Yakovenko (2010)]:

$$\begin{aligned} \hbox {d}Y\left( {N\left( {L,K} \right) ,H} \right) =\mu dN\left( {L,K} \right) +\theta dH, \end{aligned}$$

(D.2)

where $\mu =\partial Y/\partial N$ and $\theta =\partial Y/\partial H$ denote the marginal labor-capital return and the marginal technology return (Tao 2016), respectively.

Here Tao identifies the entropy $ln\varOmega$ with the technological progress H (Tao 2010, 2016):

$$\begin{aligned} H=ln{\varOmega }, \end{aligned}$$

(D.3)

where ${\varOmega }$ denotes the number of equilibrium income allocations that a given income distribution contains [Furthermore, ${\varOmega }$ also measures the choice freedom of social members (Tao 2016)]. For example, for the 2-person society described by “Appendix B”, we have ${\varOmega } \left( a \right) =1$ and ${\varOmega } \left( b \right) =2$. By maximizing (Tao 2010, 2015, 2016) ${\varOmega }$ one can obtain the exponential income distribution (2). Consequently, the technological progress H can be regarded as the entropy of socio-economical systems.

Furthermore, the complete differential of Eq. (D.1) can be rewritten in the form:

$$\begin{aligned} dY=\omega dL+rdK+\theta dH, \end{aligned}$$

(D.4)

where $\omega =\partial Y/\partial L$ and $r=\partial Y/\partial K$ denote marginal labor return and marginal capital return (Tao 2017), respectively. On the one hand, we might as well assume that capital markets exhibit perfect competition, so r also denotes the interest rate. On the other hand, by the principle of diminishing marginal return in neoclassical economics, $\omega$ denotes the minimum wage. Comparing Eqs. (D.2) and (D.4), we can obtain (Tao 2017):

$$\begin{aligned} \mu =\omega \cdot \sigma -r\cdot \sigma \cdot MRTS_{LK} , \end{aligned}$$

(D.5)

where $\sigma =dL/dN$ and $MRTS_{LK} =-dK/dL$. Here $\sigma$ denotes the marginal employment level and $MRTS_{LK}$ denotes the marginal rate of technical substitution of labor and capital (Tao 2017).

Main propositions

To obtain the consistent estimate of $\mu$, we do the estimate analysis in terms of two cases: full sample and truncation sample. In this paper, $lim_{n\rightarrow \infty } a_n =a$ means $lim_{n\rightarrow \infty } P\left( {a_n =a} \right) =1$, where $P\left( \xi \right)$ denotes the probability of $\xi$ occurring.

1.1 Full sample

Let us first drop the constraint $x\ge \mu$. For the full data (i.e., population), the Eq. (4) can be written in the form:

$$\begin{aligned} y_j= & {} \beta ^{*}x_j +\alpha ^{*}+\varepsilon _j , \end{aligned}$$

(E.1)

$$\begin{aligned} \mu ^{*}= & {} -\frac{\alpha ^{*}}{\beta ^{*}}, \end{aligned}$$

(E.2)

where $\beta ^{*}=-\frac{1}{\theta ^{*}}$, $\alpha ^{*}=\frac{\mu ^{*}}{\theta ^{*}}$, and $\varepsilon _j \sim N\left( {0,\sigma ^{2}} \right)$ for $j=1,2,\ldots ,\infty$. Here $\left\{ {x_j } \right\} _{j=1}^\infty$ and $\left\{ {y_j } \right\} _{j=1}^\infty$ denote the full data. $\beta ^{*}$ and $\alpha ^{*}$ are obtained by regressing $\left\{ {y_j } \right\} _{j=1}^\infty$ on $\left\{ {x_j } \right\} _{j=1}^\infty$.

For the full sample⁵, the sample estimates of Eqs. (E.1) and (E.2) yield:

$$\begin{aligned} \hat{y} _i= & {} \hat{\beta } x_i +\hat{\alpha } , \end{aligned}$$

(E.3)

$$\begin{aligned} \hat{\mu }= & {} -\frac{\hat{\alpha } }{\hat{\beta } }, \end{aligned}$$

(E.4)

where $i=1,2,\ldots ,n$.

Due to the absence of the constraint $x\ge \mu$, equation (E.1) differs slightly from Eq. (3); therefore, we don’t ensure if $\mu ^{*}=\mu$. In section E2, we will discuss the estimate of $\mu$ when $x\ge \mu$ holds. In this section, we mainly investigate the consistency of the estimate (E.4).

Taking the least squares estimation on Eq. (E.3) we have:

$$\begin{aligned} \hat{\beta }= & {} \frac{\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) \left( {y_i -\bar{y} } \right) }{\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) ^{2}}, \end{aligned}$$

(E.5)

$$\begin{aligned} \hat{\alpha }= & {} \bar{y} -\hat{\beta } \bar{x}, \end{aligned}$$

(E.6)

where $\bar{x} =\frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i$ and $\bar{y} =\frac{1}{n}\mathop \sum \nolimits _{i=1}^n y_i$.

Since the exponential distribution (3) is only suitable for the low and middle parts of the income data, we should drop the high income data. Moreover, due to the economic meanings of $x_i$ in the Eq. (3), $\left\{ {x_i } \right\} _{i=1}^n$ should be a monotonic increasing sequence. Thus, we can make the following assumptions.

Assumptions

• (a). $\left| {x_i } \right| <\infty$ and $\left| {y_i } \right| <\infty$ for $i=1,2,\ldots ,n$.

• (b). $\left\{ {x_i } \right\} _{i=1}^n$ is a strictly monotonic increasing sequence with $x_i \ge 0$ for $i=1,2,\ldots ,n$.

• (c). $\varepsilon _j$ are i.i.d. $N\left( {0,\sigma ^{2}} \right)$.

Next we verify that $\hat{\beta }$ and $\hat{\alpha }$ are consistent estimates.

Theorem 1

Assume that $\varepsilon _j$ are i.i.d. $N\left( {0,\sigma ^{2}} \right)$. If there is $lim_{n\rightarrow \infty } \left( {{\varvec{X}}^{T}{\varvec{X}}} \right) ^{-1}={\varvec{0}}$, then one has:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\beta }= & {} \beta ^{*}, \end{aligned}$$

(E.7)

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\alpha }= & {} \alpha ^{*}, \end{aligned}$$

(E.8)

where ${{\varvec{X}}}=\left( {{\begin{array}{lll} {x_1}&{} \cdots &{} {x_n } \\ 1&{} \cdots &{} 1 \\ \end{array} }} \right) ^{T}$.

Proof

See Lai et al. (1979). $\square$

To verify Eqs. (E.7) and (E.8), we can only prove the following proposition.

Proposition 1

$lim_{n\rightarrow \infty } \left( {{{\varvec{X}}}^{T}{{\varvec{X}}}} \right) ^{-1}=\mathbf{0}$.

Proof

It’s easy to compute:

$$\begin{aligned} \left( {{{\varvec{X}}}^{T}{{\varvec{X}}}} \right) ^{-1}=\frac{1}{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i } \right) ^{2}}\left( {{\begin{array}{cc} n &{} {-\mathop \sum \nolimits _{i=1}^n x_i } \\ {-\mathop \sum \nolimits _{i=1}^n x_i }&{} {\mathop \sum \nolimits _{i=1}^n x_i^2 } \\ \end{array} }} \right) , \end{aligned}$$

so proving $lim_{n\rightarrow \infty } \left( {{{\varvec{X}}}^{T}{{\varvec{X}}}} \right) ^{-1}=\mathbf{0}$ is equivalent to verifying:

$$\begin{aligned} lim_{n\rightarrow \infty } \frac{1}{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i } \right) ^{2}}\left( {{ \begin{array}{cc} n&{} {-\mathop \sum \nolimits _{i=1}^n x_i } \\ {-\mathop \sum \nolimits _{i=1}^n x_i }&{} {\mathop \sum \nolimits _{i=1}^n x_i^2 } \\ \end{array} }} \right) =\left( {{\begin{array}{cc} 0&{} 0 \\ 0&{} 0 \\ \end{array} }} \right) . \end{aligned}$$

(E.9)

Obviously, proving Eq. (E.9) is equivalent to verifying the following three equations:

$$\begin{aligned}&lim_{n\rightarrow \infty } \frac{n}{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i} \right) ^{2}}=0, \end{aligned}$$

(E.10)

$$\begin{aligned}&\quad lim_{n\rightarrow \infty } \frac{\mathop \sum \nolimits _{i=1}^n x_i }{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i} \right) ^{2}}=0, \end{aligned}$$

(E.11)

$$\begin{aligned}&\quad lim_{n\rightarrow \infty } \frac{\mathop \sum \nolimits _{i=1}^n x_i^2 }{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i} \right) ^{2}}=0. \end{aligned}$$

(E.12)

One can compute:

$$\begin{aligned} n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i } \right) ^{2}=n^{2}\left[ {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\bar{x} } \right) ^{2}} \right] . \end{aligned}$$

(E.13)

Furthermore, we have the following result:

$$\begin{aligned}&\frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\bar{x} } \right) ^{2}=\frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i^2 -2\left( {\bar{x} } \right) ^{2} \nonumber \\&\quad + \left( {\bar{x} } \right) ^{2}=\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) ^{2}. \end{aligned}$$

(E.14)

By Assumption (b) we must have $\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) ^{2}\ne 0$; otherwise, $x_i =\bar{x}$ for $i=1,2,\ldots ,n$, contradicting the strict monotonicity. On the other hand, by the strict monotonicity, there should be at most one number $x_l$ leading to $x_l =\bar{x}$. Thus, if we order $\mathop {\min }\limits _{i\ne l} \left| {x_i -\bar{x} } \right| =A$, then we have $\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) ^{2}\ge 0+\left( {n-1} \right) \cdot A^{2}$.

Consequently, by Eq. (E.14) we can obtain:

$$\begin{aligned} \left| {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\bar{x} } \right) ^{2}} \right| =\left| {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) ^{2}} \right| \ge \frac{n-1}{n}\cdot A^{2}. \end{aligned}$$

(E.15)

Using Eqs. (E.13) and (E.15) one has

$$\begin{aligned}&\left| {\frac{1}{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i } \right) ^{2}}} \right| \nonumber \\&\quad =\left| {\frac{1}{n^{2}\left[ {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\bar{x} } \right) ^{2}} \right] }} \right| \nonumber \\&\quad \le \frac{1}{n^{2}\cdot \frac{n-1}{n}\cdot A^{2}}=\frac{1}{n\cdot \left( {n-1} \right) \cdot A^{2}}. \end{aligned}$$

(E.16)

On the other hand, by Assumption (a), we can order ${\textit{max}}_i \left| {x_i } \right| =B$; therefore, we have:

$$\begin{aligned} \left| {\mathop \sum \nolimits _{i=1}^n x_i } \right| =\mathop \sum \nolimits _{i=1}^n \left| {x_i } \right| \le n\cdot B, \end{aligned}$$

(E.17)

$$\begin{aligned} \left| {\mathop \sum \nolimits _{i=1}^n x_i^2 } \right| =\mathop \sum \nolimits _{i=1}^n x_i^2 \le n\cdot B^{2}. \end{aligned}$$

(E.18)

Using Eqs. (E.16)–(E.18), we can obtain:

$$\begin{aligned}&\left| {\frac{n}{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i } \right) ^{2}}} \right| \nonumber \\&\quad \le \frac{n}{n\cdot \left( {n-1} \right) \cdot A^{2}}=\frac{1}{\left( {n-1} \right) \cdot A^{2}}. \end{aligned}$$

(E.19)

$$\begin{aligned}&\quad \left| {\frac{\mathop \sum \nolimits _{i=1}^n x_i }{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i} \right) ^{2}}} \right| \le \frac{n\cdot B}{n\cdot \left( {n-1} \right) \cdot A^{2}}=\frac{B}{\left( {n-1} \right) \cdot A^{2}}. \end{aligned}$$

(E.20)

$$\begin{aligned}&\quad \left| {\frac{\mathop \sum \nolimits _{i=1}^n x_i^2 }{n\mathop \sum \nolimits _{i=1}^n x_i^2 -\left( {\mathop \sum \nolimits _{i=1}^n x_i } \right) ^{2}}} \right| \le \frac{n\cdot B^{2}}{n\cdot \left( {n-1} \right) \cdot A^{2}}=\frac{B^{2}}{\left( {n-1} \right) \cdot A^{2}}. \end{aligned}$$

(E.21)

Imposing $n\rightarrow \infty$ on Eqs. (E.19)–(E.21) one can obtain Eqs. (E.10)–(E.12). $\square$

By using the Theorem 1, it’s easy to compute:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\mu } =-\frac{\alpha ^{*}}{\beta ^{*}}=\mu ^{*}. \end{aligned}$$

(E.22)

Equation (E.22) indicates that if there is no the constraint $x\ge \mu$, then the estimate $\hat{\mu }$ is consistent. However, the existence of the constraint $x\ge \mu$ may lead to the inconsistency of estimate $\hat{\mu }$.

1.2 Truncation sample

Now let us recover the constraint $x\ge \mu$. Since the constraint $x\ge \mu$ holds, we attempt to construct a truncation estimate of $\mu$. To this end, we might as well assume that $\mu$ has existed. Thus, the truncation of the full data $x_j$ can be written as:

$$\begin{aligned} x_j \ge \mu , \end{aligned}$$

(E.23)

where $j=g^{*},g^{*}+1,\ldots ,\infty$.

Using the truncation data (E.23), Eq. (4) can be written as:

$$\begin{aligned} y_k= & {} \beta x_k +\alpha +\varepsilon _k , \end{aligned}$$

(E.24)

$$\begin{aligned} x_k\ge & {} \mu , \end{aligned}$$

(E.25)

where $\beta =-\frac{1}{\theta }$, $\alpha =\frac{\mu }{\theta }$, and $\varepsilon _k \sim N\left( {0,\sigma ^{2}} \right)$ for $k=g^{*},g^{*}+1,\ldots ,\infty$. Here $\beta$ and $\alpha$ are obtained by regressing $\left\{ {y_j } \right\} _{j=g^{*}}^\infty$ on $\left\{ {x_j } \right\} _{j=g^{*}}^\infty$.

Thus, the sample estimates of Eqs. (E.24) and (E.25) yield:

$$\begin{aligned} \hat{y} _i= & {} \hat{\beta } _g x_i +\hat{\alpha } _g , \end{aligned}$$

(E.26)

$$\begin{aligned} x_i\ge & {} \hat{\mu } _g , \end{aligned}$$

(E.27)

where $i=g,g+1,\ldots ,n$ and $g=g\left( n \right)$. Here $\left\{ {x_i } \right\} _{i=g}^n$ and $\left\{ {y_i } \right\} _{i=g}^n$ denote truncation sample. It’s worth emphasizing that $g^{*}$ and $g=g\left( n \right)$ are undetermined.

Taking the least squares estimation on Eq. (E.26) we have:

$$\begin{aligned} \hat{\beta } _g= & {} \frac{\mathop \sum \nolimits _{i=g}^n \left( {x_i -\bar{x} _g } \right) \left( {y_i -\bar{y} _g } \right) }{\mathop \sum \nolimits _{i=g}^n \left( {x_i -\bar{x} _g } \right) ^{2}}, \end{aligned}$$

(E.28)

$$\begin{aligned} \hat{\alpha } _g= & {} \bar{y} _g -\hat{\beta } _g \bar{x} _g , \end{aligned}$$

(E.29)

where $\bar{x} _g =\frac{1}{n-g+1}\mathop \sum \nolimits _{i=g}^n x_i$ and $\bar{y} _g =\frac{1}{n-g+1}\mathop \sum \nolimits _{i=g}^n y_i$.

The main purpose of this section is to derive the estimate $\hat{\mu } _g$. Assume $g^{*}<\infty$, thus we will have the following theorem and proposition:

Theorem 2

Assume that $\varepsilon _j$ are i.i.d. $N\left( {0,\sigma ^{2}} \right)$. If there is $lim_{n\rightarrow \infty } \left( {{{\varvec{X}}}_{g^{*}}^T {{\varvec{X}}}_{g^{*}}} \right) ^{-1}=\mathbf{0}$, then one has:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\beta } _{g^{*}}= & {} \beta , \end{aligned}$$

(E.30)

$$\begin{aligned} lim_{n\rightarrow \infty }\hat{\alpha } _{g^{*}}= & {} \alpha , \end{aligned}$$

(E.31)

where ${{\varvec{X}}}_{g^{*}} =\left( {{\begin{array}{lll} {x_{g^{*}} }&{} \cdots &{} {x_n } \\ 1&{} \cdots &{} 1 \\ \end{array}}} \right) ^{T}$.

Proof

Same as the Theorem 1. $\square$

Proposition 2

$lim_{n\rightarrow \infty } \left( {{{\varvec{X}}}_{g^{*}}^T {{\varvec{X}}}_{g^{*}} } \right) ^{-1}=\mathbf{0}$.

Proof

Same as the Proposition 1. $\square$

Consistent with the form of Eq. (E.4), $\hat{\mu } _g$ can be defined as:

$$\begin{aligned} \hat{\mu } _g =-\frac{\hat{\alpha } _g }{\hat{\beta } _g }. \end{aligned}$$

(E.32)

Now we start to derive the consistent condition of guaranteeing the validity of estimate (E.32).

Substituting Eqs. (E.29) into (E.32) one has:

$$\begin{aligned} \hat{\mu } _g =\bar{x} _g -\frac{\bar{y} _g }{\hat{\beta } _g }, \end{aligned}$$

(E.33)

which guarantees that the constraint of Eq. (E.26) has been imposed on the estimate (E.32).

On the other hand, Eq. (E.27) indicates:

$$\begin{aligned} \bar{x} _g >\hat{\mu } _g +\delta , \end{aligned}$$

(E.34)

where we have used the Assumption (b) and $\delta >0$.

Inserting Eqs. (E.33) into (E.34) yield:

$$\begin{aligned} \frac{\bar{y} _g }{\hat{\beta } _g }>\delta >0, \end{aligned}$$

(E.35)

which guarantees that the constraint of Eq. (E.27) has been imposed on the estimate (E.32).

Thus, we can obtain the core proposition of this Appendix as below:

Proposition 3

For a strictly monotonic increasing sequence $\left\{ {x_j } \right\} _{j=1}^n$, if there exists an integer $g=g\left( n \right)$ to guarantee:

• (A). $x_{i-1}<\mu <x_i$ or $x_i =\mu$, where $i=g<n$ and $lim_{n\rightarrow \infty } \frac{g}{n}=0$;

• (B). $\frac{\bar{y} _g }{\hat{\beta } _g }>\delta >0$ for any n;

  then one has:

  $$\begin{aligned} lim_{n\rightarrow \infty } \hat{\mu } _g =lim_{n\rightarrow \infty } \left( {\bar{x} _g -\frac{\bar{y}_g }{\hat{\beta } _g }} \right) =\mu , \end{aligned}$$

  (E.36)

  where g is uniquely determined by n and $g<\infty$. This means:

  $$\begin{aligned} lim_{n\rightarrow \infty } g=g^{*}. \end{aligned}$$

  (E.37)

To verify the Proposition 3, we need to prove the following four lemmas:

Lemma 1

If $\left\{ {\xi _i } \right\} _{i=1}^n$ is a monotonic sequence and if $\left| {\xi _i } \right| <\infty$ for any i, then one has:

$$\begin{aligned} lim_{n\rightarrow \infty } \xi _n =\xi , \end{aligned}$$

(E.38)

where $\left| \xi \right| <\infty$.

Proof

See the theorem 3.14 in Rudin (1976). $\square$

Lemma 2

For the sequence $\left\{ {\xi _i } \right\} _{i=1}^n$, if $lim_{n\rightarrow \infty } \xi _n =\xi$, then one has:

$$\begin{aligned} lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{i=1}^n \xi _i =\xi . \end{aligned}$$

(E.39)

Proof

Since $lim_{n\rightarrow \infty } \xi _n =\xi$, by the definition of limit, for every $\upepsilon >0$ there always exists a positive integer N so that when $k>N$, one has:

$$\begin{aligned} \left| {\xi _k -\xi } \right| <\frac{\epsilon }{2}. \end{aligned}$$

(E.40)

To verify Eq. (E.39), we only need to prove:

$$\begin{aligned} lim_{n\rightarrow \infty } \left( {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \xi _i -\xi } \right) =0; \end{aligned}$$

(E.41)

that is, for every $\upepsilon >0$ there always exists a positive integer $N_0$ so that when $n>N_0$, one has:

$$\begin{aligned} \left| {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \xi _i -\xi } \right| <\epsilon . \end{aligned}$$

(E.42)

It’s easy to compute:

$$\begin{aligned}&\left| {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \xi _i -\xi } \right| \nonumber \\&\quad =\left| {\frac{1}{n}\left[ {\mathop \sum \nolimits _{i=1}^N \left( {\xi _i -\xi } \right) +\mathop \sum \nolimits _{j=N+1}^n \left( {\xi _j -\xi } \right) } \right] } \right| \nonumber \\&\quad \le \frac{1}{n}\left| {\mathop \sum \nolimits _{i=1}^N \left( {\xi _i -\xi } \right) } \right| +\frac{1}{n}\left| {\mathop \sum \nolimits _{j=N+1}^n \left( {\xi _j -\xi } \right) } \right| . \end{aligned}$$

(E.43)

Because $lim_{n\rightarrow \infty } \xi _n =\xi$, it’s easy to verify that $\left| {\xi _i } \right| <\infty$ and $\left| \xi \right| <\infty$. Thus, one has ${\textit{max}}_i \left| {\xi _i -\xi } \right| <\infty$. Consequently, thanks to $j>N$, Eq. (E.43) can be written in the form:

$$\begin{aligned}&\left| {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \xi _i -\xi } \right| \nonumber \\&\quad \le \frac{N}{n} {\textit{max}}_i \left| {\xi _i -\xi } \right| +\frac{n-N}{n}\frac{\epsilon }{2} \nonumber \\&\quad <\frac{N}{n} {\textit{max}}_i \left| {\xi _i -\xi } \right| +\frac{\epsilon }{2}. \end{aligned}$$

(E.44)

where we have used Eq. (E.40).

It’s easy to compute $lim_{n\rightarrow \infty } \frac{N}{n} {\textit{max}}_i \left| {\xi _i -\xi } \right| =0$. This means that for every $\upepsilon >0$ there always exists a positive integer $N_1$ so that when $k>N_1$, one has:

$$\begin{aligned} \frac{N}{k} {\textit{max}}_i \left| {\xi _i -\xi } \right| <\frac{\epsilon }{2}. \end{aligned}$$

(E.45)

Let us order $N_0 =max\left\{ {N,N_1 } \right\}$; thus, substituting Eqs. (E.45) into (E.44) we conclude that for every $\upepsilon >0$ when $n>N_0$, there always holds:

$$\begin{aligned} \left| {\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \xi _i -\xi } \right| <\epsilon . \end{aligned}$$

$\square$

Lemma 3

If $lim_{n\rightarrow \infty } \frac{g}{n}=0$, one has:

$$\begin{aligned}&lim_{n\rightarrow \infty } \bar{x} _g =lim_{n\rightarrow \infty } \bar{x} =x, \\&\quad lim_{n\rightarrow \infty } \bar{y} _g =lim_{n\rightarrow \infty } \bar{y} =y, \end{aligned}$$

where $x=lim_{n\rightarrow \infty } x_n$ and $y=\beta ^{*}x+\alpha ^{*}$.

Proof

We first verify $lim_{n\rightarrow \infty } \bar{x} _g =lim_{n\rightarrow \infty } \bar{x}$. It’s easy to check:

$$\begin{aligned} \bar{x}= & {} \frac{1}{n}\mathop \sum \nolimits _{i=1}^n x_i =\frac{1}{n}\mathop \sum \nolimits _{i=1}^{g-1} x_i \nonumber \\&+\,\frac{n-g+1}{n}\frac{1}{n-g+1}\mathop \sum \nolimits _{j=g}^n x_j \nonumber \\= & {} \frac{1}{n}\mathop \sum \nolimits _{i=1}^{g-1} x_i +\frac{n-g+1}{n}\bar{x} _g. \end{aligned}$$

(E.46)

Since $lim_{n\rightarrow \infty } \frac{g}{n}=0$, imposing $n\rightarrow \infty$ on Eq. (E.46) one obtains:

$$\begin{aligned} lim_{n\rightarrow \infty } \bar{x} _g =lim_{n\rightarrow \infty } \bar{x} , \end{aligned}$$

where we have used $\left| {x_i } \right| <\infty$.

Since Assumptions (a) and (b) hold, by using Lemma 1 one has: $lim_{n\rightarrow \infty } x_n =x$. This means that by using Lemma 2 one obtains $lim_{n\rightarrow \infty } \bar{x} =x$. Therefore, we verify $lim_{n\rightarrow \infty } \bar{x} _g =lim_{n\rightarrow \infty } \bar{x} =x$.

Now we start to verify $lim_{n\rightarrow \infty } \bar{y} _g =lim_{n\rightarrow \infty } \bar{y} =y$.

Based on the same technique from Eq. (E.46), by Assumption (a) we can verify $lim_{n\rightarrow \infty } \bar{y} _g =lim_{n\rightarrow \infty } \bar{y}$. By using Eq. (E.1), one has:

$$\begin{aligned} \bar{y} =\beta ^{*}\bar{x} +\alpha ^{*}+\bar{\varepsilon }, \end{aligned}$$

(E.47)

where $\bar{\varepsilon } =\frac{1}{n}\mathop \sum \nolimits _{i=1}^n \varepsilon _i$.

By Assumption (c) $\varepsilon _j$ are i.i.d. $N\left( {0,\sigma ^{2}} \right)$, so by using the law of large numbers, it’s easy to obtain:

$$\begin{aligned} lim_{n\rightarrow \infty } \bar{\varepsilon } =E\left( {\varepsilon _i \hbox {|}x_1 ,\ldots ,x_n } \right) =0. \end{aligned}$$

Therefore, substituting the above equation into Eq. (E.47) leads to:

$$\begin{aligned} lim_{n\rightarrow \infty } \bar{y} =\beta ^{*}lim_{n\rightarrow \infty } \bar{x} +\alpha ^{*}+lim_{n\rightarrow \infty } \bar{\varepsilon } =\beta ^{*}x+\alpha ^{*}. \end{aligned}$$

$\square$

Lemma 4

If $lim_{n\rightarrow \infty } \frac{g}{n}=0$, one has:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\beta }_g= & {} lim_{n\rightarrow \infty } \hat{\beta } =\beta =\beta ^{*}. \end{aligned}$$

(E.48)

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\alpha } _g= & {} lim_{n\rightarrow \infty } \hat{\alpha } =\alpha =\alpha ^{*}. \end{aligned}$$

(E.49)

Proof

Here we only verify Eq. (E.48). By the same technique, one can verify Eq. (E.49).

It’s easy to check:

$$\begin{aligned} \hat{\beta }= & {} \frac{\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) \left( {y_i -\bar{y} } \right) }{\mathop \sum \nolimits _{i=1}^n \left( {x_i -\bar{x} } \right) ^{2}} \nonumber \\= & {} \frac{\frac{1}{n}\mathop \sum \nolimits _{i=1}^{g-1} \left( {x_i -\bar{x} } \right) \left( {y_i -\bar{y} } \right) +\frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -\bar{x} } \right) \left( {y_j -\bar{y} } \right) }{\frac{1}{n}\mathop \sum \nolimits _{i=1}^{g-1} \left( {x_i -\bar{x} } \right) ^{2}+\frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -\bar{x} } \right) ^{2}}. \end{aligned}$$

(E.50)

Imposing $n\rightarrow \infty$ on Eq. (E.50) one obtains:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\beta }= & {} \frac{lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -\bar{x} } \right) \left( {y_j -\bar{y} } \right) }{lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -\bar{x} } \right) ^{2}}. \nonumber \\= & {} \frac{lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -lim_{n\rightarrow \infty } \bar{x} } \right) \left( {y_j -lim_{n\rightarrow \infty } \bar{y} } \right) }{lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -lim_{n\rightarrow \infty } \bar{x} } \right) ^{2}}, \end{aligned}$$

(E.51)

where we have used $\left| {x_i } \right| <\infty$, $\left| {y_i } \right| <\infty$ and $lim_{n\rightarrow \infty } \frac{g}{n}=0$.

Using Lemma 3, Eq. (E.51) can be rewritten as:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\beta }= & {} \frac{lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -lim_{n\rightarrow \infty } \bar{x} _g } \right) \left( {y_j -lim_{n\rightarrow \infty } \bar{y} _g } \right) }{lim_{n\rightarrow \infty } \frac{1}{n}\mathop \sum \nolimits _{j=g}^n \left( {x_j -lim_{n\rightarrow \infty } \bar{x} _g } \right) ^{2}}. \nonumber \\= & {} lim_{n\rightarrow \infty } \frac{\mathop \sum \nolimits _{j=g}^n \left( {x_j -\bar{x} _g } \right) \left( {y_j -\bar{y} _g } \right) }{\mathop \sum \nolimits _{j=g}^n \left( {x_j -\bar{x} _g } \right) ^{2}} \nonumber \\= & {} lim_{n\rightarrow \infty } \, \hat{\beta }_g. \end{aligned}$$

(E.52)

Because $g^{*}<\infty$, by the same technique for deriving Eq. (E.52), we can obtain: $lim_{n\rightarrow \infty } \hat{\beta } =lim_{n\rightarrow \infty } \hat{\beta } _{g^{*}}$.

On the other hand, by Theorem 1 one has $lim_{n\rightarrow \infty } \hat{\beta } =\beta ^{*}$ and by Theorem 2 one has $lim_{n\rightarrow \infty } \hat{\beta } _{g^{*}} =\beta$. Therefore, we conclude that Eq. (E.48) holds. $\square$

Now we start to verify the Proposition 3.

Proof of Proposition 3

Imposing $n\rightarrow \infty$ on Eq. (E.33) one obtains:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\mu } _g =lim_{n\rightarrow \infty } \bar{x} _g -\frac{lim_{n\rightarrow \infty } \bar{y} _g }{lim_{n\rightarrow \infty } \hat{\beta } _g }. \end{aligned}$$

(E.53)

Using Lemmas 1–4, Eq. (E.53) equals:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\mu } _g =x-\frac{y}{\beta }. \end{aligned}$$

(E.54)

We have known

$$\begin{aligned} \mu =-\frac{\alpha }{\beta }. \end{aligned}$$

(E.55)

Imposing $n\rightarrow \infty$ on Eq. (E.29) one obtains:

$$\begin{aligned} \alpha =y-\beta x. \end{aligned}$$

(E.56)

Substituting Eqs. (E.55) and (E.56) into Eq. (E.54) yields:

$$\begin{aligned} lim_{n\rightarrow \infty } \hat{\mu } _g =x-\frac{y}{\beta }=\mu . \end{aligned}$$

(E.57)

Because $\frac{\bar{y} _g }{\hat{\beta }_g }>\delta >0$ for any n, by Lemmas 3 and 4 we have:

$$\begin{aligned} lim_{n\rightarrow \infty } \frac{\bar{y} _g }{\hat{\beta }_g }=\frac{y}{\beta }\ge \delta >0. \end{aligned}$$

(E.58)

Thus, by Eq. (E.57) we must conclude:

$$\begin{aligned} \mu<x<\infty , \end{aligned}$$

(E.59)

where we have used $\left| {x_i } \right| <\infty$.

Since $x_{g-1}<\mu <x_g$ or $x_g =\mu$, by Assumption (b) we have:

$$\begin{aligned} 0\le \mu<x<\infty . \end{aligned}$$

Now we further verify that, for a given n, there is no another $g^{\prime } \ne g$ to guarantee that $x_{{g^{\prime } - 1}}< \mu < x_{{g^{\prime }}}$ or $x_{{g^{\prime }}} = \mu$. We discuss this point in terms of two cases. First, if $x_{{g^{\prime } - 1}}< \mu < x_{{g^{\prime }}}$ holds, we have to conclude $x_{{g - 1}}< \mu < x_{{g^{\prime }}}$ and $x_{{g^{\prime } - 1}}< \mu < x_{g}$. For this case, we might as well assume $g^{\prime } > g$, which by Assumption (b) leads to $x_{{g^{\prime }}} > x_{g}$. This means $x_{g} \le x_{{g - 1}}$, contradicting Assumption (b). Likewise, we can refute $g^{\prime } < g$. Second, if $x_{{g^{\prime }}} = \mu$ and $g^{\prime } \ne g$, then by Assumption (b) the contradiction occurs. In summary, we must conclude $g^{\prime } = g$.

Finally, we verify $g<\infty$. If $g=\infty$, by $x_{g-1}<\mu <x_g$ or $x_g =\mu$, we have to conclude $lim_{l\rightarrow \infty } x_l =\mu =x$, which contradicts $\mu<x<\infty$.

Based on the results above, we should have $lim_{n\rightarrow \infty } g=g^{*}$. To see this, we might as well assume that $lim_{n\rightarrow \infty } g>g^{*}$. Then, by $x_{g-1}<\mu =x_{g^{*}} <x_g$, one has $lim_{n\rightarrow \infty } g-1=g^{*}$, which contradicts $x_{lim_{n\rightarrow \infty } g-1} <x_{g^{*}}$, where we have used $g<\infty$. $\square$

Description of data sources

Source	Countries	Link
Socio-Economic Database of Latin America and the Caribbean	ARG, BLZ, BOL, BRA, CHL, COL, CRI, DOM, ECU, SLV, HTI, HND, MEX, PRY, PER, URY, VEN	http://sedlac.econo.unlp.edu.ar/eng/statistics.php
Australian Bureau of Statistics	AUS	http://www.ausstats.abs.gov.au/ausstats/subscriber.nsf/0/B0530ECF7A48B909CA257BC80016E4D3/$File/65230_2011-12.pdf
Eurostat	AUT, BEL, BGR, HRV, CYP, CZE, DNK, EST, FIN, FRA, DEU, GRC, HUN, ISL, IRL, ITA, LVA, LTU, LUX, MKD, MLT, NLD, NOR, POL, PRT, ROU, SRB, SVK, SVN, ESP, SWE, CHE, TUR, GBR	http://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=ilc_di01&lang=en
Statistics Canada	CAN	http://www.statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/famil105a-eng.htm
Hong Kong	HKG	http://www.census2011.gov.hk/pdf/household-income.pdf
Nepal Rastra Bank	NPL	http://www.nrb.org.np/red/publications/study_reports/Study_ReportsHousehold%20Budget%20Survey%202008%20(Report).pdf
Russian Federal State Statistics Service	RUS	http://www.arcticstat.org/Table.aspx/Region/Russian_Federation/Indicator/Personal!Household_Income/2008-08-21-05/10874
Singapore Department of Statistics	SGP	http://www.singstat.gov.sg/docs/default-source/default-document-library/publications/publications_and_papers/household_income_and_expenditure/pp-s22.pdf
Korean Statistical Information Service	KOR	http://kosis.kr/statHtml/statHtml.do?orgId=101&tblId=DT_1L6E001&conn_path=I2&language=en
National Statistical Office of Thailand	THA	http://web.nso.go.th/en/survey/house_seco/data/Whole%20Kingdom_13_FullReport.pdf

Source	Countries	Link
United Kingdom National Statistics	GBR	https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/503472/SPI_National_Statistics_T3_1_to_T3_11.pdf
United States Census Bureau	USA	http://www2.census.gov/programs-surveys/demo/tables/p60/252/table3.pdf
Eurostat	BGR, CZE, DNK, HUN, ISL, LTU, LVA, NOR, POL, SWE	http://ec.europa.eu/eurostat/web/exchange-rates/data/database
OECD Statistics	AUT, BEL, BGR, CZE, CHE, DEU, DNK, ESP, EST, FIN, FRA, GRC, HUN, IRL, ISL, ITA, LTU, LUX, LVA, NLD, NOR, POL, PRT, SVK, SVN, SWE	http://stats.oecd.org/Index.aspx?DataSetCode=FIXINCLSA#
Federated States of Micronesia	FSM	http://prism.spc.int/images/documents/HEIS/2005_FSM_HIES_Report-Final.pdf
Department of Census and Statistics Ministry of Finance and Planning Sri Lanka	LKA	http://www.statistics.gov.lk/HIES/HIES2012PrelimineryReport.pdf
Bangladesh Bureau of Statistics—Ministry of Planning	BGD	http://catalog.ihsn.org/index.php/catalog/2257
Liberia Institute for Statistics and Geo-Information Services—Government of Liberia	LBR	http://microdata.worldbank.org/index.php/catalog/2563
Central Agency for Public Mobilization and Statistics (CAPMAS)—Arab Republic of Egypt	EGY	http://www.ilo.org/surveydata/index.php/catalog/1261
Namibia Statistics Agency	NAM	http://www.ilo.org/surveydata/index.php/catalog/320
China Institute for Income Distribution	CHN	http://www.ciidbnu.org/