2.1 Definition of MPS and MIS

As the percentage of the population which has an income below the mean, MPS can be defined in terms of the cumulative income distribution function (CDF), F(y).

Definition 1. Let y_i denote the ordered income for individual i, i = 1, ‥N, and μ the mean income. The mean population share, MPS, is the percentage value that the income CDF F(y) reaches at the mean μ: (1) with as the indicator function.

MPS takes on values in (0, 1]. Note that MPS = 0 can only be reached asymptotically: If N − 1 individuals each have an income of y, and only one individual has a positive income of a < y, then μ > a and MPS = 1/N. As N → ∞, MPS → 0. MPS = 1 is attained in the extreme case of income equality. With income y = μ for all individuals, F(μ) = MPS = 1.

For a symmetric distribution, in which the mean and median coincide, it holds that MPS = 0.5. A positively (negatively) skewed distribution has an MPS above (below) 0.5, so that [0.5, 1] is the prevalent range for most empirical applications. Higher values of MPS indicate that more individuals have an income below μ. For instance, if the mean is lifted up by rich earners in the tail, there will be comparatively many individuals below the mean.

But MPS is not an inequality measure. It captures the share of individuals earning less than the mean, which is not representative of inequality of the whole distribution. For example, it is evident that MPS does not satisfy the Pigou-Dalton transfer principle [22]: An income transfer from a richer to a poorer individual on either side of the mean will leave MPS unaffected. Therefore, a high MPS value does not necessarily go in line with a high Gini coefficient. This becomes evident when considering the extreme case where 99% of the population have an income of y and 1% of individuals have an income of 2y. The mean μ = 1.01y is higher than y and therefore MPS = 0.99. So, 99% of the population have an income below the mean but the Gini coefficient is very low at 0.0098.

Closely related to MPS is the mean income share, MIS, which is the share of total income that the individuals below the mean hold. It can be computed as (2) where [x] denotes the integer part of the number x, and is the mean of all incomes below the mean μ. MIS can also take on values in (0, 1], but it is smaller than 0.5 in positively skewed income distributions which are empirically relevant. MPS simply counts individuals below the mean, whereas MIS supplements this figure with the respective incomes of the individuals. A higher MIS ceteris paribus suggests that individuals below the mean are holding a larger share of total income. Together, MPS and MIS provide a full perspective of the relation between mean income and various parts of the distribution. It is important to note the focus of the two measures on different parts of the distribution when tracking changes over time: In a dynamic setting, MPS is particularly sensitive to changes in the middle of the income distribution; only individuals moving above or below the mean have a direct impact on MPS according to Definition 1. By contrast, MIS is equally sensitive to all individuals at the bottom and middle of the distribution, as long as their incomes y_i are below the mean, see (2).

This characterization suggests that MPS and MIS are informative statistics for cross-country comparisons on relative welfare. Looking only at mean income neglects that in some countries a vast majority of individuals might be poorer, so that this number does not capture their living standards. MPS and MIS help to put mean income as a welfare indicator into a relative perspective to the income distribution. Also, while inequality measures such as the Gini coefficient do not distinguish between top- and bottom- inequality, MPS and MIS focus specifically on individuals below the mean compared to those above. This is why two different distributions might have the same Gini coefficient but very different MPS and MIS.

Still, there are connections between the two measures and inequality indices: For a given MPS value, a higher MIS goes along with a lower Gini coefficient, because of lower bottom inequality. For a given MIS, a higher MPS will be associated with a higher Gini coefficient because it implies that the mean is driven up by rich earners. Furthermore, the difference between MPS and MIS can also be used as an inequality measure; it is equivalent to the Pietra Index (also called the Hoover index, Robin Hood index, or the Schutz index). The Pietra Index can be interpreted as the income share that would have to be redistributed to achieve an egalitarian distribution. It is defined as the maximum vertical deviation between the Lorenz curve and the egalitarian line, . Sarabia [23] shows that this holds at y = F(μ). By Definition 1, this is MPS. With MIS = L(MPS), it follows that P = MPS − MIS. These relationships are illustrated in Fig 1. For the Pietra index as a measure of intradistributional income inequality, see [24]. Also, it is possible to rewrite the Bonferroni index as in the special case of a triangular Lorenz curve. This shows that it is the interplay of MPS and MIS that drives various measures of income inequality. Different combinations of MPS and MIS might yield the same inequality in terms of the Gini coefficient, Pietra or Bonferroni index. However, tracing MPS and MIS allows us to look below the surface to see what is driving changes in inequality, as we will see in the following.

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Fig 1. Illustration of MPS and MIS on the LC.

https://doi.org/10.1371/journal.pone.0242803.g001

2.2 Inclusiveness of income growth

An analysis of μ, MPS, and MIS at any point in time is insightful for measuring relative living standards and for analyzing how institutional differences across countries are influencing different parts of the income distribution. However, even more relevant for policymakers is tracing MPS and MIS over time, in particular in periods of economic growth. In the following theorem, we derive that MPS shows unambiguous changes in reaction to growth at various parts of the income distribution.

Theorem 1. Consider a population at time t = 1 with ordered incomes y_i,1, i = 1, …, N, and mean income μ₁. Define three growth scenarios from period 1 to 2 as follows: where 0 < p << 0.5 is the population share to be affected by top or bottom income growth, lb and ub are the lower and upper bounds for incomes to be affected by middle income growth. MPS is then characterized by the following reactions in different growth scenarios:

1. (a). Invariant to relative income changes by a factor c
2. (b). Invariant to absolute income changes by an absolute amount a
3. (c). MPS (weakly) increases for the pure top income growth: MPS₂ ≥ MPS₁
4. (d). MPS (weakly) increases for the pure bottom income growth: MPS₂ ≥ MPS₁
5. (e). MPS (weakly) decreases for the pure middle income growth: MPS₂ ≤ MPS₁

Proof. See S1 File.

Hence, if we have inclusive growth across the whole income distribution, both in relative and absolute terms (scenarios a and b), MPS stays constant. If increases in mean income are accompanied by changes in MPS, it indicates that some parts of the distribution are gaining more than others. The direction of the change in MPS expresses the relative welfare effects on middle income earners: If incomes at the top or bottom grow faster than at the middle of the income distribution, MPS increases (scenarios c and d); middle income households cannot keep up with the rising mean and some fall below it. Middle income growth, however, decreases MPS (scenario e), as more households will have an income that exceeds the mean. The middle income focus of MPS becomes also clear when decomposing its reaction into the sum of two effects: Income growth by the factor c of the affected individuals has a direct effect on MPS; the overall increase in μ has an indirect effect. In scenarios (c) and (d), MPS changes only due to the indirect effect. However, we have both a direct effect and an indirect effect—and typically a stronger reaction—if the middle of the distribution exhibits income growth as in (e).

Table 1 compares the reactions of MPS to those of MIS, the Gini coefficient, and the skewness for the above growth scenarios. Neither the Gini coefficient nor the skewness can capture the evolution of MPS. The Gini coefficient reacts differently to growth at the top and bottom, and is ambiguous towards middle income growth. The reactions of skewness often depend on the underlying distributions and the precise changes, while the changes in MPS hold in general. For example, if the initial distribution is lognormal and the top 10% (20%) of incomes double, the skewness increases (decreases). By contrast, MPS will always (weakly) rise because mean income has increased and some individuals above the old mean might fall below the new mean.

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Table 1. Comparison of measures in the growth scenarios from theorem 1.

https://doi.org/10.1371/journal.pone.0242803.t001

When we observe heterogeneous changes across the income distribution rather than growth affecting just one part, the co-movements between MPS and MIS become important. On its own, MIS does not exhibit unambiguous reactions in all growth scenarios (Table 1), unlike MPS. But together, the two measures can express which part of the distribution is benefiting in particular. While changes in MPS are driven by middle income earners, its combination with MIS can reflect what is happening to all individuals below the mean. The relative welfare changes of these individuals following mean income growth, as captured by the joint reactions of MPS and MIS, are shown in Table 2, a 3 × 3 welfare matrix. All effects should be compared to the baseline scenario of inclusive income growth (cell (2,2)), which is distribution-neutral and keeps MPS and MIS constant. If a constant MPS goes in line with an increasing (decreasing) MIS, individuals below the mean gain more (less) from economic growth than individuals above. Also, if more middle income individuals move above the mean (so that MPS decreases), but MIS stays constant (cell (3, 2)), those below the mean are better off because the same income share is now accumulated by fewer individuals. Obviously, the most strongly pro-poor and pro-middle income growth would be reflected in a decreasing MPS and simultaneously increasing MIS (cell (3,1)). Note that the welfare effects are in line with the effects on the Pietra ratio (MPS − MIS).

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Table 2. Relative welfare effects of MPS&MIS comovements.

https://doi.org/10.1371/journal.pone.0242803.t002

2.3 MPS and MIS of parametric lorenz functions

With micro-level income data, one can simply compute MPS and MIS with the definitions above. In many cross-country applications, however, only grouped-level percentile data are available. Then one can exploit the analytical tractability of MPS and MIS in terms of the Lorenz curve (LC), which links the cumulative income share L to the cumulative population share p.

Theorem 2. For a continuously differentiable LC, L(p), MPS as defined in Definition 1 is the value p at which the first derivative of the LC equals unity: (3) Proof. The LC and the income CDF are related by the following formula [see 25, 26]: (4) At the value p = MPS, we can use Definition 1 and take the inverse of the function (5) By combination of (4) and (5), it follows that (3) holds.

Hence, MPS is located at the point where a 45 degree line is tangential to the LC, see Fig 1. Uisng the definition of MIS as the income share of all individuals below the mean, MIS can be written as the Lorenz curve ordinate of MPS: (6)

In empirical studies with grouped data, parametric Lorenz functions, such as the lognormal or the Pareto LC, are fitted to the percentile points. Theorem 2 allows to derive the MPS and MIS values that parametric LCs imply. In Table 3 we list the closed forms for MPS and MIS for the most-widely used uni-parametric and multi-parametric functional forms. For example, for a lognormal distribution with parameter σ > 0, we have , where Φ(.) denotes the cumulative normal distribution. With varying σ values, MPS can take on values on the whole realm (0.5, 1). But not all parametric forms are equally able to capture changes in MPS and MIS and therefore express the inclusiveness of economic growth. The Weibull LC, although known for its flexibility in fitting LCs associated with both unimodal and zero-modal income densities [27], can only represent MPS ∈ [0.43, 0.64]. In the following section we are going to examine the importance of these limitations and explore which parametric forms can best trace the recent empirical evolution of MPS and MIS.

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Table 3. MPS and MIS for widely-used Lorenz curves.

https://doi.org/10.1371/journal.pone.0242803.t003

2.4 Data for the empirical investigation

In our empirical investigation, we apply the methods presented above in a cross-country setting. Looking at empirical MPS and MIS values over time, our goal is to investigate whether recent economic growth has been relatively inclusive of individuals ordered below mean income. We trace MPS and MIS over four decades (1978-2016) for different countries, analyze their comovements with changes in mean income, and study to what extent the parametric forms from Section 2.3 can capture these developments. Our data are harmonized cross-country micro-level income data sets from LIS [11]. To examine the evolution of the statistics, a long time span with frequent reporting periods is crucial. We therefore settle on those 16 high- and middle income countries which report income data for all waves between at least the 2nd (around 1985) and the 9th wave (around 2014). Note that the precise years vary across countries; the S2 File provides an overview. We work with both total and disposable household income. Both measures are calculated as equivalized income, which is obtained by dividing the household income by the square root of the number of household members. We follow the literature in weighting observations by the number of household members times household weights.

3.1 Micro data: MPS and MIS across countries and time

The 16 countries in our data set are listed in Table 4 together with their MPS and MIS in the first and last available observation year. MPS, MIS, and the Gini coefficient for all countries and years are provided in the S2 File. The countries are known to differ in their levels of income inequality and social mobility [32], labor market structures [33] as well as societal attitudes towards redistribution and taxation [34]. Consequently, we observe a lot of cross-country variation in terms of MPS and MIS. For the last available year (around 2014), MPS ranges from 0.53 (the Netherlands) to 0.70 (Mexico).

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Table 4. Summary statistics of the data set.

https://doi.org/10.1371/journal.pone.0242803.t004

While a detailed analysis of policy differences between these countries is beyond the scope of our analysis, we shed some light by conducting a crude grouping of countries: anglo-saxon countries, nordic countries, mediterranean countries as well as the diverse remaining group. Table 5 shows that, as expected, MPS is highest in anglo-saxon countries. With mean incomes driven up by top earners, there is a rather high share of individuals below the mean. By contrast, MPS is lowest in the nordic countries, with the mediterranean countries falling in between these two groups. The nordic countries also have the highest MIS out of the four groups: Despite their relatively low share of individuals below the mean, these households hold a larger share of total income than the more numerous below-mean income earners in other countries.

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Table 5. MPS and MIS by country group.

https://doi.org/10.1371/journal.pone.0242803.t005

It is intuitive to expect a positive cross-country correlation between MPS and income inequality. However, the scatter plot between MPS and the Gini coefficient in Fig 2a shows a more complex pattern. Despite a number of countries close to the diagonal line, we also find a group of countries with low income inequality, in particular, Norway, Poland, and Taiwan, which have comparatively high MPS values. They have few relatively poor households, while mean income is driven up by rich households, so that many middle income households are located below the mean. This is confirmed by their rather high MIS of 0.42-0.44. Hence, households below the mean hold a larger share of total income than in many others countries, both egalitarian and inegalitarian, see Fig 2b. For example, the U.S. and Denmark show similar levels of MPS of 0.60-0.61, but these households hold a total income share of 0.43 in Denmark and only 0.35 in the U.S. When countries differ in their MIS, a welfare ranking is only possible when taken together with MPS. The low MIS of Finland and the Netherlands is due to their low MPS, so that the corresponding income shares are accumulated by fewer individuals.

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Fig 2. MPS and other statistics across countries and time.

https://doi.org/10.1371/journal.pone.0242803.g002

How has MPS evolved over the last three decades as mean incomes grew substantially? Fig 2c illustrates that MPS was higher in the last (ca. 2014) than in the first observation period (ca. 1981) for 13 out of 16 countries. The evolution of MPS over time in Fig 2d shows notable increases in many countries, with the only sustained decrease occurring in the Netherlands. These results suggest that mean incomes have become less representative of the middle of the distribution in most countries, as more middle income earners than before are now ordered below the national mean. Momentous structural changes have occurred during these decades, which have all impacted the income distribution of industrialized economies, most prominently along with skill-biased technological change [35, 36] and globalization [37, 38]. Top incomes have risen substantially in many countries [39, 40]. The literature has highlighted that the middle class, by contrast, has often been squeezed by wage polarization [41, 42]. Our results on the rise in MPS should be seen in this context: They reflect that many middle income households have been unable to keep up with rising mean income and have fallen behind in relative terms.

While changes in MPS are driven by the middle of the distribution, the co-movement with MIS can give an indication of the relative welfare effects for all households below the mean, including the poorest. Table 4 shows that MIS slightly increased for more than half of the countries from the beginning to the end of the sample period, but the changes are mostly small. Despite rising MPS and hence more individuals below the mean, their corresponding income shares have not tended to increase by much. The U.S. is a case in point: In 2016, 67% of individuals were below mean income rather than 61% in 1979. But their total income share stayed constant at 35%. Hence, in the U.S. not only the middle of the distribution, as shown by MPS, but more generally all individuals below the mean were gaining less from economic growth than the top earners. In Norway, by contrast, both MPS and MIS have increased substantially: Middle income earners have lost out but the welfare effects on lower income earners are ambiguous, in line with the welfare matrix in Table 2.

Income growth at the top always tends to increase both MPS and income inequality, but the distributional movements of other growth scenarios may not, so that inequality and MPS do not always move in the same direction. Fig 2e looks at the correlations between MPS and the Gini coefficient as well as between MPS and skewness for each country over time. For example, in the U.S. and Australia, the development of MPS correlates strongly with both the Gini coefficient and the skewness of the distribution. But for many others, including Germany, Poland, and Canada, the evolution of MPS shows either no significantly positive or even a moderately negative correlation with the Gini coefficient and skewness, see Fig 2e. Obviously, growth in top incomes can only be part of the story. The developments of the last decades have affected different parts of the income distribution in different ways across countries: Countries with labor market conditions and redistributive policies in favor of the poor might keep inequality constant, but MPS still rises as middle income earners lose out in relative terms.

We find that the same conclusions from total income carry over to disposable income in most countries: The disposable income MPS is only marginally lower than the one for total income (Fig 2f); note that this scatter plot omits Mexico and Italy, where the data does not distinguish between total and disposable income. Policies that redistribute from the top to the bottom decrease the Gini coefficient, but they do not tend to have a large impact on households near the income mean and hardly affect MPS. We also calculate the difference of the average distributional metrics (MPS, MIS, Gini) in terms of gross and disposable income for each country, finding mean squared errors (MSE) of MPS, MIS and Gini between gross income and disposable income across countries to be, respectively, 0.014, 0.017, and 0.037. Hence, the difference between disposable and gross income is more than twice as large for the Gini coefficient as for MPS and MIS. Disposable income MIS is smaller than its gross income counterpart in 12 countries, while the relationship is reversed in Taiwan. Disposable Gini is always smaller than gross Gini—in line with the literature [43]—, and disposable MPS is smaller than gross MPS in all countries except Poland.

We dig deeper in S18 Table in the S2 File, where we provide growth rates of mean income g_μ, the growth rate of mean income for all individuals below the mean, g_subμ, as well as distributional metrics between the first and last available year. Note that there is a direct relation between the changes in these variables: With μ as national mean income and μ_sub as the average income for individuals included in MPS, (2) can be rewritten as (7) In terms of growth rates, this yields (8) (9) where g_subμ denotes growth of mean income for individuals below national mean income, and g_μ, g_MPS, g_MIS are the growth rates of μ, MPS, and MIS, respectively.

If the mean income of individuals at the bottom and middle of the distribution grows at a rate that is not slower than the growth rate of national mean income, the rising mean income will be unambiguously beneficial to individuals below the national mean income. It is worth mentioning that the equations above apply to any point on the Lorenz curve. Repeating the analysis with different thresholds than the mean, one can conduct a study that would be directly complementary to growth incidence curves.

In S18 Table in the S2 File we can see that the group-specific mean income of individuals below the national mean increased more than national mean income for both gross and disposable incomes in three countries (Denmark, Mexico, and Poland), while both MIS and MPS increased in Denmark and Poland, and decreased in Mexico. So, individuals below mean income were relatively better off in the three countries because their MIS increased more than MPS did. But in the other 13 countries MPS always increased by more than MIS, making individuals at the middle and the bottom of the distribution worse off, as discussed above. An interesting situation happened in Germany and the USA, where gross MIS growth was negative, but the growth of disposable MIS, MPS and Gini had become positive. One possible interpretation of this would be that government policies were effective in propping up the poorest earners, while those at the middle benefited less from redistributive policies and fell below mean income, leading to an increase in MPS.

S19 Table in the S2 File provides standard deviations of the 5-percentile income shares of each country over time, highlighting the dynamics at the top, bottom and middle of the distribution. We find that the largest variation always happened to the top 5% income share, and the second largest happened to either the bottom or the second-highest 5% income share. Meanwhile, there is also substantial variation around the middle of the distribution.

While the analysis so far considers summary statistics, let us now investigate the effect of mean income growth on MPS and other measures in a panel regression. For each country i and available year t, we compute the changes in distributional measure M from the previous available year. We regress it on the difference in log mean income between the same years. We measure log mean income both in nominal and real terms, using the LIS income means in national currency for the former and real GDP per capita data from the World Bank’s World Development Indicators for the latter. Table 4 shows the heterogeneity of annualized growth rates over countries, ranging from 0.75% in Italy to 4.10% in Poland in real terms. There is also a sizable variation over time. Controlling for country-fixed effects and time specifics either via a time trend or year-fixed effects, we estimate (10) (11) where the dependent variable M is, respectively, MPS, MIS, the Gini coefficient, skewness, or the Pietra ratio, MPS − MIS.

Note that we consider both the distributional effects and growth effects on ΔM by including ΔMPS on the right hand side of Eqs (10) and (11). However, this means that one should be cautious about a causal interpretation of the coefficients, as changes in MPS and MIS occur simultaneously.

The panel regression results in Table 6 below confirm that mean income growth was partially associated with increases in MPS (Panels A and B). In the specifications where the coefficient is statistically significant, it is always positive. MIS shows a tendency for positive reactions to growth (Panels C and D), but the magnitude is more subdued. When controlling for changes in MPS, the reactions of MIS to income growth lose significance in all specifications. This suggests that MPS moves more strongly with income growth than MIS.

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Table 6. Panel regressions of changes in MPS and MIS on mean income growth.

https://doi.org/10.1371/journal.pone.0242803.t006

The S2 File considers the results of three robustness checks and additional analyses. In particular, S20 Table in the S2 File reruns the panel regressions of changes in MPS and MIS on mean income growth, both contemporaneously and with a lag. With the lag we address the simultaneity issue. However, it is statistically insignificant in most specifications, yielding no evidence for a delayed impact of growth on MPS and MIS. The main coefficients tend to change very little with the inclusion of the lags.

In S21 Table in the S2 File, we rerun these regressions using the country group dummies of anglo-saxon, nordic and mediterranean countries defined in Table 5 as control variables instead of using country fixed effects. The results are remarkably similar, both qualitatively and quantitatively (for example, the coefficient for the effect of changes in MIS on MPS in Panel A has a coefficient of 1.049 rather than 1.043). This shows that country ideosyncracy as expressed by the fixed effects can to a large extent be captured by the country groups, which in turn, represent differences in the regulatory, labor market and welfare regimes.

Finally, in S23 Table in the S2 File we replace MPS or MIS as the dependent variable and instead consider the growth effects on different measures, namely the Gini coefficient, the skewness and Pietra ratio. The effects of growth on the Gini coefficient have different signs when nominal and real data are used but are never statistically significant. This heterogeneity of growth effects on income inequality is in line with the literature [44–46]. Only the Pietra index reacts to changes in MPS by definition. However, the increase in MPS in reaction to mean income growth is rather robust across specifications and data sources (LIS survey data and national accounts), in particular taking the relatively small sample size into account. Moreover, our panel regression results are robust to using random effects instead of fixed effects. As a further robustness check, we experiment with nonlinear specifications by including higher-order terms, which turn out to be statistically insignificant.

We draw three conclusions from the empirical analysis: (i) The growing mean incomes of official statistics have become less representative of the middle of the income distribution, as shown by the increase in MPS. Middle income earners are failing to keep up with rising mean incomes, as fits the theory on wage polarization. (ii) Increases in MIS are primarily associated with increasing MPS, and rising mean incomes did not lead to significant increases in MIS after the effects of MPS are controlled for. By contrast, the panel regressions suggest that growth has moved hand in hand with increases in MPS. (iii) Changes in MPS and MIS typically go into the same direction, with MIS changing more slowly than MPS.

3.2 Grouped data: Parametric functions and MPS over time

Can the parametric functions used for grouped data also capture the empirical characteristics of MPS and MIS which we have now analyzed? Let us mimic a setting in which only grouped data is available and use 5-percentile shares from our LIS data set. We fit the eight LCs from Table 3 to these twenty data points for each country and year. An often-used measure of fit criterion for parametric LCs is the mean squared error (MSE) at the 20 percentile points [27, 47]. As an alternative performance criterion, we compute the difference between the MPS (or respectively MPS + MIS) implied by the functional forms and their empirical values. This is in the vein of Chotikapanich [28] who compares functional forms based on the difference between the implied and empirical Gini coefficients. This allows us to answer the question if the parametric forms that perform best at the percentile points also capture MPS and MIS most accurately. Table 7 shows that the answer is no. The Kakwani LC achieves the closest fit at the percentile points for 139 out of all 175 country-year distributions, but the Rohde and Wang-Smyth LCs often imply MPS values which come closer to the empirical ones. This also holds when we look at the implied MPS and MIS together.

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Table 7. Performance of functional forms.

https://doi.org/10.1371/journal.pone.0242803.t007

While this analysis was static, let us now examine how accurately the empirical evolution of a country’s MPS over time can be traced by the parametric functions. Here we find mixed results. In countries where MPS, Gini, and skewness are strongly correlated, such as the U.S., the empirical and implied MPS are very close (Fig 3a). By contrast, the rising MPS in low-inequality Norway cannot be captured well by the functional forms (Fig 3b). Obviously, the structural links between inequality and MPS imposed by the limited degrees of freedom can come at a disadvantage, when both move into opposite directions. Researchers working with grouped data should be aware of the choice of functional forms. Only if the implied MPS comes close to the empirical one can it be an appropriate measure for the inclusiveness of economic growth.

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Fig 3. Empirical and implied MPS by parametric forms.

https://doi.org/10.1371/journal.pone.0242803.g003