Fig 1.
The Lorenz curve of a typical country.
The hypothetical feasible equality line (blue) can serve as a practical guideline for government policies and interventions (red arrow).
Fig 2.
Individual welfare functions as a function of income.
(a) The linear function used in the Utilitarian social welfare. (b) The non-linear sigmoid function, reflecting more realistic individual welfare as income increases. With the critical low- and high- income values (L and H), the two constants (μ and α) in the sigmoid function can be determined.
Fig 3.
Total social welfare as a function of β value.
The welfare function is maximized at β = β*, and the corresponding Lorenz curve (blue) represents the feasible equality line (the optimal income distribution).
Table 1.
Share of household income in four countries.
Table 2.
Parameter values for the individual social welfare functions.
Fig 4.
Total social welfare as a function of β value in the four countries.
The social welfare plots are normalized to the number of individual households in each subgroup (0.2N, where N is the total number of individual households in a country). The total social welfare functions of the four countries are maximized at ,
,
, and
.
Fig 5.
Actual and optimal income distributions in four countries.
Compared to the actual income distribution, the optimal income distribution shows higher income in the lower quintiles and less income in the highest quintile.
Table 3.
Actual and optimal income distributions in four countries.
Fig 6.
Lorenz curves for the actual and the optimal income distributions in four countries.
The corresponding Gini coefficients can be found in Table 3. Note that the feasible equality lines (blue) are similar in all countries.
Fig 7.
Lorenz curves for the actual and the optimal income distributions in China from 1990 to 2016.
The corresponding Gini coefficients can be found in S3 Table. Note that the feasible equality lines (blue) are similar, while the actual income distributions are widely dispersed over time.
Fig 8.
Evolution of Gini coefficients in China from 1990 to 2016.
The Gini coefficient values at the data points can be found in S3 Table.