This study evaluates the conformity of Iran’s income distribution with the Econophysics Two-Class Theory of income distribution (EPTC) using Household Expenditure-Income Survey data from 2006 to 2016. Through probability density functions (PDFs) and complementary cumulative distribution functions (CCDFs) it is demonstrated that Iran's income structure follows A distinct dual structure: the lower class (97–99.7% of the population) adheres to an exponential Boltzmann-Gibbs distribution, indicating labor-driven income, while the upper tail (0.3–3%) follows a fluctuating Pareto power-law distribution, aligning with capital income. Methodologically, we employ log-linear CCDF scaling to verify the Boltzmann-Gibbs distribution for the majority population and log-log scaling to confirm the Pareto distribution for the elite segment. The thermal class maintains remarkable stability over time despite gradual increases in effective income temperature, whereas the power-law tail exhibits significant volatility. Key findings include the thermal equilibrium of the lower class—evidenced by normalized PDF convergence across years—and a sharp minimum wage-induced peak in its PDF. Notably, the PDF reveals a sharp narrow peak at lower income levels, attributable to government-mandated minimum wage policies.

Aim and Introduction

This paper examines the distribution of income in Iran from 2006 to 2016 and evaluates the validity of one of the latest economic theories concerning income distribution, namely, the Econophysics Two-Class Theory of Income Distribution (EPTC).

According to this model, income distribution generally comprises two classes. The lower class of this distribution, typically representing 97 to 99% of th society, follows the exponential (thermal) Boltzmann-Gibbs distribution, primarily driven by labor income. This distribution remains stable over time and undergoes minimal fundamental changes. Conversely, the income distribution of the upper class, constituting approximately 1 to 3% of society, follows the Pareto distribution, recognized as a superthermal distribution in econophysics. Notably, this distribution exhibits high variability over time, closely mirroring fluctuations in the stock market.

For this study, a review of the theoretical literature on the statistical distribution of income is conducted, tracing its evolution from Pareto's initial attempts to the formulation of the two-class distribution of income. In the methodology section, emphasis is placed on delineating the characteristics of two Probability Density Functions (PDFs) and Complementary Cumulative Distribution Functions (CCDFs) associated with exponential and Pareto distributions. The methodology elaborates on the approach to detecting income distribution patterns within the framework of the aforementioned theory. Subsequently, in the data and findings section, an examination of the income data spanning the specified time period in Iran is undertaken. The section meticulously explores the compatibility of these data with the EPCT, offering detailed discussions on the observed patterns and their alignment with the theoretical framework. Finally, the implications of the EPCT are elucidated, and the paper's conclusions are presented in the concluding remarks section.

Methodology

In complex systems concluding big data or complex models, alternative approaches beyond conventional statistical tests may be employed to estimate distributions. Visual inspection and descriptive analysis, facilitated by histograms and distribution charts, serve as effective tools for approximating distributions without relying on statistical tests. The selection of distributions is informed by theoretical considerations that align with the underlying characteristics of the system. These alternative methods offer practicality and informativeness, particularly in scenarios where traditional statistical assumptions may not hold or when dealing with extensive and unconventional data. The present article adopts this methodological approach to analyze income distribution in Iran.

The initial step involves drawing the histogram and probability density function (PDF). The shape of the histogram guides the identification of distribution. Given the potential complexity arising from large datasets, and the ambiguity that may arise from visual inspection of merely the PDF, a Complementary Cumulative Distribution Function (CCDF) plot serves as a valuable aid. Subsequently, following the first step and the selection of candidate theoretical distributions, the CCDFs are plotted to ascertain the optimal fit with the experimental data distribution. Consequently, the combined use of PDF and CCDF serves as indispensable tools for delineating annual income distribution patterns.

The resemblance between the graphs of the PDF for both exponential and Pareto distributions on a linear-linear scale poses challenges in distinguishing between these distributions. Similarly, the CCDF curve lacks clarity on a linear-linear scale due to this similarity. However, employing a logarithmic-linear scale to plot the survival function related to the data of the lower part of society proves beneficial, as it reveals a smooth line representative of the exponential Boltzmann-Gibbs law. Similarly, plotting the survival function for the upper part of the society on a logarithmic-logarithmic scale serves to elucidate the Pareto power law. Consequently, plotting the survival function for the entire dataset on a logarithmic-logarithmic scale, as per the hypothesis of the EPTC, should unveil two distinct segments: exponential and Pareto.

Findings

The data utilized in this study were derived from the raw tables pertaining to the household expenditure-income (budget) plan, annually published by the Statistical Center of Iran. Specifically focusing on data sourced from the urban population, which constituted approximately three-quarters of the total population during the study period. Data preparation commenced with the meticulous removal of zero and negative values, followed by deflation adjustments based on the consumer price index. Subsequently, data normalization was conducted utilizing the slope of the line of the CCDF for the lower part of the dataset, plotted on a logarithmic-linear scale for each year. This normalization process was initiated based on the initial estimate of the border income, set at the 99.7th percentile. Finally, an appropriate binning strategy was selected, with a uniform value of 0.4 (∆r≈0.4T) applied to all data subsequent to the initial 0.2 portion.

Plotting the PDF of the income pertaining to the lower class of the society across three scales—linear-linear, logarithmic-linear, and logarithmic-logarithmic—alongside the fitting line of the exponential distribution function for the year 2016 revealed a notable alignment, indicative of a robust fit with the theoretical exponential distribution.

Alternatively, the survival function chart was employed to analyze the income distribution among the upper class of society. Presenting this data graphically across three scales—linear-linear, logarithmic-linear, and logarithmic-logarithmic—for the entirety of 2016 underscored two key findings. Firstly, the tail-end distribution of income follows the Pareto distribution. Secondly, and of paramount significance, these graphical representations unequivocally affirmed the appropriateness of dividing the dataset into two distinct segments.

Plotting the PDF for the 11-year period revealed that the data pertaining to the lower part of the society, representing 99.7% of the total population, converged onto a singular curve following normalization across the entire duration under study. Subsequently, depicting the survival functions for the aforementioned 11-year time frame in a unified graph, utilizing both logarithmic-linear and logarithmic-logarithmic scales, served as a more definitive validation of the two-class theory of income distribution.

Discussion and Conclusion

The analysis of income data in Iran from 2006 to 2016 reveals a distinct two-class structure in the country's income distribution.

Firstly, the lower class, encompassing approximately 97 to 99.7% of the population, follows the exponential Boltzmann-Gibbs distribution, primarily driven by labor income. This statistical distribution reflects a cumulative process characterized by a constant rate of decrease, as indicated by the exponential distribution's parameter. The consistency observed in the exponential fit graphs of the survival function and data histogram across different years suggests the stability of income distribution within the lower class over time. This stability parallels thermal equilibrium in physics, suggesting that the majority of the population is in a stable equilibrium. Notably, the high-resolution histogram of the PDF reveals a sharp and narrow peak at low incomes, attributed to governmental policies such as the imposition of minimum wage regulations.

Conversely, the upper class, constituting approximately 0.3 to 3% of the population, follows a Pareto distribution, predominantly influenced by capital income. However, unlike the lower class, the distribution of income within this part does not align along a single line in the power law segment. This part undergoes discernible fluctuations from year to year, indicating instability within this economic sector. These fluctuations are attributed to the variability of capital income