*To achieve full income equality, the factors that determine individual income, such as intelligence, inherited wealth, personalities and social skills, must be the same for everyone. It is an infeasible ideal. Chae Un Kim and Ji-Won Park propose a more feasible and realistic concept of income equality that could be incorporated into the Gini coefficient, the most widely used measure of inequality, ensuring the maximization of overall social welfare without hampering overall economic efficiency.*

Income inequality is well known to negatively affect any economic system, and it has been the subject of political disputes for the past 100 or more years. Many methods have been proposed to quantitatively measure income inequalities; of these, the Gini coefficient â€” which ranges from 0 (perfect equality) to 1 (perfect inequality) â€”is the most widely used measure. Perfect equality is achieved when all members of a given nation or society share the same level of income. It is often argued in political conflicts that perfect equality leads to economic inefficiency, and therefore to a less productive society. Perfect income equality can only be achieved when everyone is the same and equally able to contribute economically. To achieve full income equality, the factors that determine individual income, such as intelligence, inherited wealth, personalities and social skills, must be the same for everyone. However, these conditions cannot be fulfilled, even in the Smurfs’ village. Therefore, it is evident that the concept of perfect income equality is rather idealistic and practically infeasible in the real world.

If a more achievable and realistic concept of income equality can be incorporated into the Gini coefficient, it can provide a useful guideline for a realistic and practical distribution of income, ensuring the maximization of overall social welfare without hampering social welfare. overall economic efficiency.

Figure 1 shows the Lorenz curve for a typical nation. The Lorenz curve illustrates the distribution of national income (to the right of the perfect equality line). The perfect tie line cannot be a useful benchmark for the real world, so we need a more practical and achievable tie line. If it exists, this line provides a valuable benchmark for government policies (eg, income taxes) to redistribute income and reduce income inequality.

**Figure 1. The Lorenz curve of a typical nation**

Achievable income equality is defined as an optimal distribution of income that maximizes total social well-being without hampering the sustainable economic growth of a given nation or society. Moreover, within the framework of achievable income equality, income must be distributed equitably to individuals by correctly reflecting the realistic factors influencing their economic contributions. In our study, an optimal income distribution (i.e. achievable equality) could be modeled using the sigmoid welfare function and the Boltzmann income distribution. A sigmoid function is an S-shaped nonlinear function and is used in a wide range of research fields such as physics and economics. The Boltzmann income distribution is adapted from the physical sciences, where the underlying principle is based on maximizing entropy and provides the most probable, natural, and unbiased distribution of a physical system at thermal equilibrium. The notion of *most likely* in physical sciences has been translated into *fair* in our study.

**Figure 2. Function of individual well-being and total social well-being** *Notes: a) The nonlinear sigmoid function, reflecting more realistic individual well-being as income increases. With the critical values â€‹â€‹of low and high income (L and H), the two constants (Âµ and) in the sigmoid function can be determined. b) The total social welfare function is maximized at Î² = Î²*^{*} (the optimal distribution of income).

^{*}(the optimal distribution of income).

As shown in Figure 2-a, the individual sigmoid well-being function represents the realistic well-being, such as well-being, happiness, and satisfaction, that a person experiences as their income increases. If their income is close to zero, the welfare value should be at a minimum. The value of welfare will increase as income increases, but not rapidly below the critical low income value (i.e. the minimum cost of living). Because in this case the income is still insufficient to support a basic standard of living, the value of well-being increases slowly and gradually. But if the income exceeds the critical value of low income, people start to experience some economic freedom. Therefore, the value of well-being will suddenly increase rapidly. As income increases, the degrees of economic freedom increase and become saturated at a critical value of sufficient income. At the critical value of large incomes, the value of welfare would also be saturated; thereafter, the value of welfare would increase again only gradually.

As shown in Figure 2-b, the Boltzmann income distribution (Pi) is expressed as an exponential function of the income distribution factor (áº¼i), which is a measure of economic contributions and depends on factors such as intelligence , personality and social skills. Depending on the unique value (Î²) in the Boltzmann distribution, the income distribution can represent a wide range of income distributions, from perfect equality (Î² = 0) to perfect inequality (Î² = âˆž). When the Boltzmann income distribution is inserted into the sigmoid welfare function, the total social welfare function becomes a function of the value Î² and can be maximized to Î² *. The Boltzmann income distribution with Î² * represents the achievable equality (the optimal income distribution).

Based on the model, we conducted an empirical analysis of four countries (the United States, China, Finland and South Africa) and demonstrated how optimal income distributions could be assessed. Figure 3 shows the Lorenz curves for the real and optimal income distributions for each of these countries. In all countries, the Lorenz curves for the real income distribution are very scattered. In contrast, the Lorenz curve for the optimal income distribution (the line of equality of achievable incomes) lies between the diagonal (ideal perfect equality) and the lines of real incomes. In addition, the shape of the achievable tie line is very similar for the four countries. This observation is corroborated by the calculations of the Gini coefficient. The Gini coefficients for real income distributions are relatively widely distributed, while those for optimal income distributions are distributed tightly. Therefore, this result raises the possibility that a universal and workable line of equality can be found and applicable to all countries of the world.

**Figure 3. Lorenz curves for real and optimal income distributions for four countries**

*Note: The achievable tie lines (blue) are similar for all four countries. The hypothetical achievable equality line (blue) can serve as a practical guide for government policies and interventions (red arrow).*

In conclusion, our results show that achievable income equality could provide a practical guideline for government policies (eg, income taxes) and interventions for income and wealth redistribution. We strongly believe that our work provides a direct contribution for future theoretical and empirical studies on reducing income inequalities or suggesting government policies, and we plan to open a new window to real-world equality.

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*Notes:*