Equilibrium and Convergence in Income Distribution: The Case of 28 European Countries in the Recent, Turbulent Past (1995–2019)

The central hypothesis which underlies this article is the concept of equilibrium in personal income distribution. The famous economist Vilfredo Pareto (Pareto, 1895) is already known to have believed that the social distribution of personal incomes moves towards a stable equilibrium over time. He based his statement on the observation that personal income dispersion fluctuates neither internationally nor inter-temporally. Much later, Hans Jürgen Ramser (Ramser, 1987) identified stationarity in the secondary distribution of personal incomes (i.e. the income net after government intervention with taxes and transfers) and not in the primary distribution of incomes (out of the market process). Recent studies support this empirical finding (Genc, Miller & Rupasingha, 2011).

The existing skewness of (personal) income distribution may be interpreted as a display of social preferences, thereby implying that preserving a specific degree of income inequality is intentional (Blümle, 1992, p. 224) and not arbitrary. Although distributional justice continues to be a fundamental goal of economic policies, it does not focus strictly on achieving a perfect equitable income distribution. Both short- and long-term scenarios accept the unwarranted existence of a certain degree of inequitable income distribution in society (Blümle, 1992, p. 225). In reality, such equilibrium would seldom be achieved to a full extent, although the policymakers have good reasons to push towards the ‘steady state’ and thereby help reach convergence.

When it comes to the question of breaking down this idea to a possible empirical analysis, the world seems to be more complex: Will countries be moving always towards a similar equilibrium level of inequality? This question is all but trivial. Works by Esping-Andersen (1990, 1994, 1998), by Esping-Andersen and Myles (2009) and by Hall and Soskice (2001) suggest something different and highlight the fact that among developed countries there exist long-run differences in the institutional structure and in so-called ‘ institutional complementarities in the macroeconomy’ (Hall & Gingerich, 2004), which can also be conceived as different equilibria in income distribution. These different equilibria may be attributed not only to the differences in the institutional set-up, but also to the observed variety of historical experiences in the respective countries. Taking into account the role of business in national economies and the fact that there is more than one path to economic success (‘liberal market economies’ vs. ‘coordinated market economies’) explain the further differences. However, different historical experiences and/or unalike institutional structures become much less important, once countries share a longer period of a common economic, social and political history, as is the case for the member countries in the EU. This insight holds even if countries realise to a different and necessarily often changing extent what Esping-Andersen calls the ‘three worlds of welfare capitalism’ (liberal, conservative, social-democratic systems; see Tiemann, 2006 for a critical empirical evaluation of this non-undisputed concept). Furthermore, it is understood that the institutional complementarity and implicit coordination between a ‘supra-national’ European Central Bank (ECB) on the one hand, and national institutions, such as national unions, pursuing a ‘national’ wage policy on the other hand, is a common challenge for members of the Eurozone.

The article is organised as follows: after a brief review of relevant literature, we present our equilibrium model in personal income distribution. An exhaustive empirical part—considering a period of extreme economic turbulence—follows, whose findings support the concept of steady state and convergence in personal income distribution. The article concludes with a discussion of the theoretical and empirical results of the present study, the limitations of the analysis and the possibilities for future research endeavours.

In principle, there exist two strands of the literature which are relevant to the subject under consideration in the present study. While one of the two perspectives focuses on the relationship between an economic crisis and personal income distribution, the other questions the existence of convergence in personal income distribution. However, to date, no study is available in the literature which connects these two perspectives, as the present study does. In both of these research perspectives, it is essential to (1) address the ex-post Gini coefficient or (2) at least compare the Gini coefficient on market income over time with the former.

In a pre-crisis study (Van Kerm & Alperin, 2013), the authors reported that the arrangement of the countries of the world in descending order of annual income inequality for the period of 2003–2007 puts “Portugal and Baltic states (such as Estonia, as reported by the authors) at the top, and most Scandinavian countries (such as Finland and Denmark, as reported by the authors) at the bottom” (p. 937). This result was overly unspecific. The research papers published by Dolls, Fuest and Peichl (2011), De Beer (2012) and Kaitila (2013) concerned the issue of economic crisis and income distribution in Europe. Dolls et al. (2011) conducted two controlled experiments (simulations) of macro shocks to income and employment and observed that “both shocks lead to higher differences between the Gini coefficients based on equivalent disposable and market income” (ibid., p. 240). This effect applied to all the 19 European countries considered by the authors in their study. De Beer (2012), in our view, utilised too short a period (2008–2009) to conclude that “the economic crisis has not so far led to a general widening of income disparities and a rise in poverty” (ibid, p. 23).

The second group of contributions has a considerable tradition and follows the seminal paper of Ravallion (2003): as a long-time member of World Bank research groups, his variety of natural interest concentrated on developing countries’ fate. However, his 2003 paper also delivered important methodological aspects for measuring (conditional and unconditional) convergence in personal income distribution. The critical hypothesis that Ravallion tested is whether the trend in inequality depends on its initial level (ibid., p. 352). He finds inequality convergence, “with a tendency for within-country inequality to fall (rise) in countries with initially high (low) inequality” (ibid., p. 355).

A specific example of follow-up investigations is the contribution of Alfani and Ryckbosch (2015). In their long-run historical perspective (1500–1800), they compared changes in inequality between central and northern Italy on the one hand and the southern and northern Low Countries on the other hand. Similarly, Martinez-Carrion and Maria-Dolores (2017) explored inequality and regional convergence in Spain and Italy for the long period of 1850–2000. Lessmann and Seidel (2015) investigated regional inequality (based on Gini indexes and other measuring instruments) and convergence based on much more recent satellite night-time light data for a vast sample of developed and developing countries. Sell (2015, p. 15–20) also analysed both developed and developing countries and obtained as a result that “globalisation and possibly other forces linked to the revolution in communication and information technologies have contributed to an almost worldwide convergence in the distribution of personal incomes. More precisely, one can say that developing (developed) countries’ income distribution has become more equal (unequal)” (ibid, p. 16).

A smaller section of this body of literature deals directly with the convergence issue applied to overall and/or parts of Europe (for example, the EU and/or the Eurozone) and the recent past. For example, Paas and Schlitte (2007) analysed a cross-section of 861 EU-25 regions from 1995 to 2003 to detect between- and within-country disparities in income distribution—measured by the Theil index. Melchior (2008) studied regional inequality and convergence in Europe (1995–2005), that is, for a period which preceded exactly the financial and world economic crisis of 2008/2009. She found that for the EU-27 as a whole, there was a modest increase in within-country regional inequality, but convergence across countries (ibid, p. 31).

In 2013, Kaitila, in turn, reported a result that was considered close to the findings of the present study, although a slightly different approach (involving the calculation of the sigma convergence) was followed in that study. According to Kaitila’s findings, “For the EU–15 (a little less for the EU–27, the authors), we found that the national Gini coefficients have converged considerably during these (1995–2011, the authors) years” (ibid., p. 14). A most recent contribution to the EU inequality subject stems from Savoia (2019): his sample covers the years 1989–2013 (i.e. he necessarily misses part of the Eurozone crisis episode) and countries from the so-called ‘NUTS 2’ regions. He finds a clear tendency that supports the results of Kaitila: “inequality is converging, but to a higher level” (ibid., p. 29).

A minor methodological remark: It is surprising to notice that regressions between GDP per capita and Gini coefficients of disposable income are seldom run for the enormous projects concerning European inequalities, and the focus of these projects remains restricted to the role of redistributive policies (Medgyesi & Toth, 2009, p. 135; Paulus, Figari & Sutherland, 2009, p. 154). An obvious step towards convergence remained undetected between 2000 and 2005, a period when the EU-27 was without Cyprus, Malta, Slovakia and Luxembourg. As stated by a previous study, “the level of inequality at the beginning of the period does not seem to influence the direction and the magnitude of the change in inequality” (Medgyesi & Toth, 2009, p. 140). This finding, however, appears flawed. The discrepancy is attributed to the addition of countries to the EU, which turned EU-15 into EU-25 in 2004 and into EU-27 later.

Therefore, the question of how economic crisis, equilibrium and convergence in personal income distribution could be addressed scientifically in a comprehensive approach remains demonstrated so far. We describe this task in the following sections.

It is surprising to see that irrespective of the definition of income, the economy in question or the time period under consideration, the distribution of incomes is skewed positively (i.e. skewed towards the right-hand or steep on the left-hand).

This fact has significant consequences for the parameters of the density function, for which the maximum value, which is referred to as the modus (y_mo) and is the most frequent event, would usually be located to the left of the median (y_me), and the latter, in turn, is located to the left of the arithmetic mean (y_ar). The characteristics of this kind of density function we depict in Figure 1.

Fig. 1

The consequences of this are extensive. According to Blümle (2005), most economic agents would receive an above-modus income. Based on this observation, the agents would have the impression of being well-paid, and therefore their attitude towards a redistribution (the existing distribution) of incomes should be quite critical (benevolent). The density function depicted in Figure 1 may be approximated, rather accurately, using a log-normal distribution of incomes, which is represented as follows: Y=exp(X) with X=N(μ,σ2) Y\, = \,\exp \left( X \right)\,{\rm{with}}\,X = N\left( {\mu ,{\sigma ^2}} \right)

The expected or similar average wage rate is then obtained using the following expression (Beichelt & Montgomery, 2003, pp. 46–8): E(y)−ya=exp(μ+12σ2) E\left( y \right) - {y_a} = \exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right)

Taking the full differential of the above-mentioned expression from left to right yields the following expression: dE(y)=dya=(dμ+σdσ)exp(μ+12σ2) dE\left( y \right) = d{y_a} = \left( {d\mu + \sigma d\sigma } \right)\exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right)

Furthermore, we consider the following expression: ymo=exp(μ−σ2) {y_{mo}} = \exp \left( {\mu - {\sigma ^2}} \right)

Taking the full differential of the above-mentioned expression yields the following expression: dymo=(dμ−2σdσ)exp(μ−σ2) d{y_{mo}} = \left( {d\mu - 2\sigma d\sigma } \right)\exp \left( {\mu - {\sigma ^2}} \right)

Finally, we obtain the following expression: yme=exp(μ)dyme=exp(μ) \matrix{{y_{me}} = \exp \left( \mu \right) \cr d{y_{me}} = \exp \left( \mu \right)}

In Figure 2, an increase in the standard deviation, σ, has the effect derived in Propositions 1–3: the modus is shifted to the left and the arithmetic mean is shifted to the right while the position of the median remains unchanged.

Fig. 2

It is now assumed that an increase in inequality or a higher concentration of incomes is perceived by an individual i as a loss of utility. The utility function of the individual i then reads as follows: Ui=Ui(yi−ymo;σ) {U_i} = {U_i}\left( {{y_i} - {y_{mo}};\sigma } \right) where ∂Ui∂ymo<0;∂Ui∂σ<0 {{\partial {U_i}} \over {\partial {y_{mo}}}} < 0;{{\partial {U_i}} \over {\partial \sigma }} < 0 .

Assuming the law of diminishing increases of damage, one obtains: ∂2Ui∂ymo2>0;∂2Ui∂σ2>0. {{{\partial ^2}{U_i}} \over {\partial y_{mo}^2}} > 0;{{{\partial ^2}{U_i}} \over {\partial {\sigma ^2}}} > 0.

Therefore, the corresponding iso-damage curves are concave. It is essential to apply a kind of budget constraint to determine an optimal solution. Such a budget constraint may be found in the properties of the log-normal distribution. Its properties reveal that an increasing dispersion of incomes does not alter the median of the distribution (Proposition 3), reducing and shifting the modus to the left (Proposition 2). These findings also indicate a likely increase in households’ share with an income above the (new) modus whenever the concentration of incomes, as measured by the standard deviation σ, increases.

The following expression was used to determine the mathematical solution for the Cobb–Douglas utility function which is presented in Figure 3: Ui=yi−ymo1/σ {U_i} = \sqrt {{y_i} - {y_{mo}}} \sqrt {1/\sigma }

Fig. 3

In the diagram depicted in Figure 3, one may identify and locate the equilibrium in the personal income distribution. The modus (y_mo) and the dispersion of incomes (σ) are allocated along the axes. The non-linear budget constraint, representing the log-normal distribution of incomes, has been labelled VV. This schedule is confronted with a troop of iso-damage curves (I_i), which are concave towards the origin of the coordinate system. The farther these curves are located from the origin, the higher is the individuals’ loss of utility.

Point P signals towards a situation where a preferably low iso-damage curve is tangential to VV. In a sense, P represents an equilibrium in the income distribution. Note that point P stands for what Chiang (1984, p. 231–232) labels a ‘goal equilibrium’. The “equilibrium state is defined as the optimum position for a given economic unit (a household, a business firm, or even an entire economy) and in which the said economic unit will be deliberately striving for the attainment of that equilibrium” (ibid, p. 232). A ‘nongoal equilibrium’, on the contrary, “dictates an equilibrium state ... in which ... opposing forces (demand and supply, for example, the authors) are just balanced against each other, thus obviating any further tendency to change”. (ibid, p. 231).

In comparison, the points Q and R represent the suboptimal solutions. Although Q and R fulfil the ‘budget constraint’ of the log-normal distribution, they are located on the less favourable iso-damage curve I₂. The equilibrium level of inequality for the individual and the society, economy and political system as a whole is now also being derived formally. As after learning the optimal solution’s intuition through the graphical analysis, the study has to proceed to a straightforward mathematical solution of what has been described ahead.

The individual i receives his own income, y_i, and the distributional parameters μ and y_mo are provided. Although the distributional parameter σ is also provided to him, with the knowledge that this parameter of income dispersion could be altered through a re-distributional policy by the government, he may deliberate on which σ would be the best for him. He would accordingly suggest his personal desired level of income dispersion, σ*_i, to the decision makers, thereby attempting to influence the ultimately determined level of income dispersion, σ*, in his favour. Of course, hardly anyone can calculate the exact value of his σ*_i, and in the usual policy-making process, the individuals cannot choose the final decision in a very exact manner. However, suppose that everyone knows by and large what is best for him, and the final decision can roughly reflect the collective decisions, then our theoretical result is still a good approximation, around which the real decisions would centre.

First, the personal desired level of income dispersion, σ*_i, is considered. The maximisation problem may be expressed as follows: maxUis.t.ymo=exp(μ−σ2) \max {U_i}s.t.{y_{mo}} = \exp \left( {\mu - {\sigma ^2}} \right)

Inserting the expression for U_i in the abovementioned equation, one obtains the following expression: maxyi−ymo1/σs.t.ymo=exp(μ−σ2) \max \sqrt {{y_i} - {y_{mo}}} \sqrt {1/\sigma } s.t.{y_{mo}} = \exp \left( {\mu - {\sigma ^2}} \right)

The Lagrangian for this maximisation problem is: Li=Ui−λ(ymo−exp(μ−σ2))==yi−ymo1/σ−λ(ymo−exp(μ−σ2)) \matrix{ {{L_i} = {U_i} - \lambda \left( {{y_{mo}} - \exp \left( {\mu - {\sigma ^2}} \right)} \right) = } \hfill \cr { = \sqrt {{y_i} - {y_{mo}}} \sqrt {1/\sigma } - \lambda \left( {{y_{mo}} - \exp \left( {\mu - {\sigma ^2}} \right)} \right)} \hfill \cr }

Taking the first-order condition (FOC) yields the following expression: (1) ∂Li∂σ=−yi−ymo2σ−32−2λσ exp(μ−σ2)=0 {{\partial {L_i}} \over {\partial \sigma }} = - {{\sqrt {{y_i} - {y_{mo}}} } \over 2}{\sigma ^{-{3 \over 2}}} - 2\lambda \sigma \,\exp \left( {\mu - {\sigma ^2}} \right) = 0 (2) ∂Li∂ymo=−1/σ2(yi−ymo)−12−λ=0⇒⇒ λ=−1/σ2(yi−ymo)−12 \eqalign{& {{\partial {L_i}} \over {\partial {y_{mo}}}} = - {{\sqrt {1/\sigma } } \over 2}{\left( {{y_i} - {y_{mo}}} \right)^{-{1 \over 2}}} - \lambda = 0 \Rightarrow \cr & \Rightarrow \,\lambda = - {{\sqrt {1/\sigma } } \over 2}{\left( {{y_i} - {y_{mo}}} \right)^{-{1 \over 2}}} \cr}

Inserting Eq. (2) in Eq. (1) yields the following expression: (yi−ymo)−12exp(μ−σ2)σ−yi−ymo2σ−32=0 {\left( {{y_i} - {y_{mo}}} \right)^{ - {1 \over 2}}}\exp \left( {\mu - {\sigma ^2}} \right)\sqrt \sigma - {{\sqrt {{y_i} - {y_{mo}}} } \over 2}{\sigma ^{ - {3 \over 2}}} = 0

Substituting y_mo with exp (μ − σ²) yields the following expression: (yi−exp(μ−σ2))−12exp(μ−σ2)σ−−yi−exp(μ−σ2)2σ−32=0 \eqalign{ & {\left( {{y_i} - \exp \left( {\mu - {\sigma ^2}} \right)} \right)^{ - {1 \over 2}}}\exp \left( {\mu - {\sigma ^2}} \right)\sqrt \sigma - \cr & - {{\sqrt {{y_i} - \exp \left( {\mu - {\sigma ^2}} \right)} } \over 2}{\sigma ^{ - {3 \over 2}}} = 0 \cr}

Multiplying both the sides with (yi−exp(μ−σ2))12σ32 {\left( {{y_i} - \exp \left( {\mu - {\sigma ^2}} \right)} \right)^{{1 \over 2}}}{\sigma ^{{3 \over 2}}} yields the following expression: exp(μ−σ2)σ2−yi−exp(μ−σ2)2=0⇒σ*i=f(yi,μ) \exp \left( {\mu - {\sigma ^2}} \right){\sigma ^2} - {{{y_i} - \exp \left( {\mu - {\sigma ^2}} \right)} \over 2} = 0 \Rightarrow { \sigma^*_i} = f\left( {{y_i},\mu } \right)

Therefore, the desired level of income dispersion by the individual i, represented by σ*_i, is determined uniquely by his income, y_i, and the distributional parameter, μ. The ultimately determined overall level of income dispersion, σ*, depends on all the individually desired levels, σ*_i, and on the exact policy decision-making process. In a process in which all the individual desires are assigned same weightage, the ultimately determined overall level of income dispersion, σ*, is the parameter which maximises social welfare in the form of aggregated utilities. Therefore, one obtains the following optimisation problem: max∑i=1nUi s.t. ymo=exp(μ−σ2) and y¯a=exp(μ+12σ2) \max \sum\limits_{i = 1}^n {{U_i}\,{\rm{s}}.{\rm{t}}.\,{{\rm{y}}_{mo}} = } \exp \left( {\mu - {\sigma ^2}} \right)\,{\rm{and}}\,{{\rm{\bar y}}_a} = \exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right)

In the aggregated utility function, n denotes the number of individuals in the society. In addition to the known constraint y_mo = exp(μ − σ²), one obtains another constraint y¯a=exp(μ+12σ2) {{\rm{\bar y}}_a} = \exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right) , in which y¯a {{\rm{\bar y}}_a} denotes the value of average income, which is constant because it was assumed that the redistribution does not change the total income, and therefore, the average income, which is expressed as follows: ∑i=1nUi=∑i=1nyi−ymo1/σ=1/σ∑i=1nyi−ymo \sum\limits_{i = 1}^n {{U_i} = } \sum\limits_{i = 1}^n {\sqrt {{y_i} - {y_{mo}}} \sqrt {1/\sigma } = \sqrt {1/\sigma } \sum\limits_{i = 1}^n {\sqrt {{y_i} - {y_{mo}}} } }

When the utility function is represented by a first-order Taylor series approximation around y_a, one obtains the following expression: ∑i=1nUi=1/σ∑i=1n(y¯a−ymo+yi−y¯a2y¯a−ymo)==n1/σy¯a−ymo \eqalign{ & \sum\limits_{i = 1}^n {{U_i}} = \sqrt {1/\sigma } \sum\limits_{i = 1}^n {\left( {\sqrt {{{\bar y}_a} - {y_{mo}}} + {{{y_i} - {{\bar y}_a}} \over {2\sqrt {{{\bar y}_a} - {y_{mo}}} }}} \right) = } \cr & = n\sqrt {1/\sigma } \sqrt {{{\bar y}_a} - {y_{mo}}} \cr}

The Lagrangian for the maximisation problem now becomes the following: L=n1/σy¯a−ymo−λ1(ymo−exp(μ−σ2))−−λ2(y¯a−exp(μ+12σ2)) \eqalign{ & L = n\sqrt {1/\sigma } \sqrt {{{\bar y}_a} - {y_{mo}}} - {\lambda _1}\left( {{{\rm{y}}_{mo}} - \exp \left( {\mu - {\sigma ^2}} \right)} \right) - \cr & - {\lambda _2}\left( {{{{\rm{\bar y}}}_a} - \exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right)} \right) \cr}

Taking the FOC yields the following: (3) ∂L∂σ=−n2σ−32y¯a−ymo−2σλ1exp(μ−σ2)++σλ2exp(μ+12σ2)=0 \eqalign{ & {{\partial L} \over {\partial \sigma }} = - {n \over 2}{\sigma ^{ - {3 \over 2}}}\sqrt {{{\bar y}_a} - {y_{mo}}} - 2\sigma {\lambda _1}\exp \left( {\mu - {\sigma ^2}} \right) + \cr & + \sigma {\lambda _2}\exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right) = 0 \cr} (4) ∂L∂ymo=−n21/σy¯a−ymo−λ1=0 {{\partial L} \over {\partial {y_{mo}}}} = - {n \over 2}{{\sqrt {1/\sigma } } \over {\sqrt {{{\bar y}_a} - {y_{mo}}} }} - {\lambda _1} = 0 (5) ∂L∂μ=λ1exp(μ−σ2)+λ2exp(μ+12σ2)=0 {{\partial L} \over {\partial \mu }} = {\lambda _1}\exp \left( {\mu - {\sigma ^2}} \right) + {\lambda _2}\exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right) = 0

Substituting σλ2exp(μ+12σ2)=−σλ1exp(μ−σ2) \sigma {\lambda _2}\exp \left( {\mu + {1 \over 2}{\sigma ^2}} \right) = - \sigma {\lambda _1}\exp \left( {\mu - {\sigma ^2}} \right) from Eq. (5) in Eq. (3) yields the following: (6) ∂L∂σ=−n2σ−32y¯a−ymo−3σλ1exp(μ−σ2)=0 {{\partial L} \over {\partial \sigma }} = - {n \over 2}{\sigma ^{ - {3 \over 2}}}\sqrt {{{\bar y}_a} - {y_{mo}}} - 3\sigma {\lambda _1}\exp \left( {\mu - {\sigma ^2}} \right) = 0

Substituting λ1=−n21/σy¯a−ymo {\lambda _1} = - {n \over 2}{{\sqrt {1/\sigma } } \over {\sqrt {{{\bar y}_a} - {y_{mo}}} }} from Eq. (4) in Eq. (6) yields the following: −n2σ−32y¯a−ymo+3σn21/σy¯a−ymoexp(μ−σ2)==−n2σ−32y¯a−ymo+3n2σy¯a−ymoexp(μ−σ2)=0 \eqalign{ & - {n \over 2}{\sigma ^{ - {3 \over 2}}}\sqrt {{{\bar y}_a} - {y_{mo}}} + 3\sigma {n \over 2}{{\sqrt {1/\sigma } } \over {\sqrt {{{\bar y}_a} - {y_{mo}}} }}\exp \left( {\mu - {\sigma ^2}} \right) = \cr & = {n \over 2}{\sigma ^{ - {3 \over 2}}}\sqrt {{{\bar y}_a} - {y_{mo}}} + 3{n \over 2}{{\sqrt \sigma } \over {\sqrt {{{\bar y}_a} - {y_{mo}}} }}\exp \left( {\mu - {\sigma ^2}} \right) = 0 \cr}

Dividing both sides by n/2 and then multiplying both the sides with σ32y¯a−ymo {\sigma ^{{3 \over 2}}}\sqrt {{{\bar y}_a} - {y_{mo}}} yields −(y¯a−ymo)+3σ2exp(μ−σ2)=0 - \left( {{{\bar y}_a} - {y_{mo}}} \right) + 3{\sigma ^2}\exp \left( {\mu - {\sigma ^2}} \right) = 0 , which is equivalent to −(y¯a−ymo)+3σ2ymo=0 - \left( {{{\bar y}_a} - {y_{mo}}} \right) + 3{\sigma ^2}{y_{mo}} = 0 , because y_mo = exp(μ – σ²).

Dividing both the sides by y_mo, one obtains the following expression: −(y¯a/ymo−1)+3σ2=0 - \left( {{{\bar y}_a}/{y_{mo}} - 1} \right) + 3{\sigma ^2} = 0

One should notice that both the constraints imply ymo=y¯aexp(−32σ2) {y_{mo}} = {\bar y_a}\exp \left( { - {3 \over 2}{\sigma ^2}} \right) or y¯a/ymo=exp(32σ2) {\bar y_a}/{y_{mo}} = \exp \left( {{3 \over 2}{\sigma ^2}} \right) , and therefore, the above-mentioned equation becomes −(exp(32σ2)−1)+3σ2=0 - \left( {\exp \left( {{3 \over 2}{\sigma ^2}} \right) - 1} \right) + 3{\sigma ^2} = 0 , which determines σ* to be equal to approximately 0.9. (The exact value can be produced by Mathematica using the command Solve [-(Exp[3/2*s^2]-1)+3*s^2==0,s,Reals].)

Of course, external shocks will destroy any earlier equilibrium and lead to deviations from the former equilibrium level of inequality. The question as to which income groups will favour a subsequent redistribution policy depends on the external shock’s nature. If the shock tends to an increase (decrease) in the concentration of personal incomes, modern theory of inequity (equity) aversion (Bolton & Ockenfels, 2000; Sell, 2015) would suggest that the group of inequity (equity) -averse agents will push government to progressive (regressive) redistribution policies. It is important to note that our model, now as it stands, captures only the preference of inequity aversion.

Convergence in inequality is a big issue in empirically oriented research on personal income distribution. The majority of contributions in our empirical part test the NGM/convergence theory and CCT, as well as our approach. While the first suggests that market forces (international and intranational trade, factor movements, etc.) push the tendency towards convergence, the latter (and theoretical strands close to it; see the introduction) explains why primarily institutional factors—similar within clubs, but not similar outside—tend to put forward some sort of ‘conditional convergence’ in income inequality. Our approach is close to CCT but stresses much more the political decision process as a determinant for the steady state in income distribution.

Our empirical research tends to support our hypothesis, but the findings can be differentiated as follows:

Before controlling for the impact of the financial crisis (and its aftermath) and possible assimilation—a sort of ‘institutional assimilation process’ is presumably significant among members of the EU—of the Gini coefficients, an insignificantly-different-from-zero change in the Gini coefficient was obtained for the majority of countries. After controlling for the assimilation and optionally for the financial crisis effect, all countries show a significant deviation from our simple modelling, ignoring country-specific effects. This deviation, however, is quite similar among the countries under study. Only for a small number of countries can one observe statistically significant differences. Therefore, we have not only found convergence in inequality—despite the significant economic turbulence during the period of observation (1995–2019)—but we have also found that there are non-negligible country-specific effects that can contribute to improving the fit of the model.

That convergence in the Gini coefficients can also be explained by the FE approach or country dummies, which are similar to CCT. Our empirical findings reveal that combining the convergence prediction of NGM with FE/CCT is a viable strategy.

We ran additional regressions that detected the impact of time trend, previous Gini coefficients, country dummies and their interaction terms and of real per capita GDP on the actual inequality of personal incomes. Furthermore, we could confirm the Lessmann and Seidel result (2015), according to which regional inequality depends on the respective per capita GDP, y, and follows an N-shape. Finally, as the NGM also implies stationarity in time series of inequality measures such as the Gini coefficient, we implemented ADF tests to inquire the existence of stationarity. Our results weakly support this view.

The present study’s limitations are related to the availability of consistent and comparable data on the personal income distribution’s statistical moments, such as the modus of incomes. Consequently, it was not possible to test, directly, all the implications of the theoretical model. Future research on personal income distribution in Europe could intend to deepen the knowledge regarding the convergence-stimulating effects of the economic crises or the change in the institutional setting and could also interconnect the analysis of personal income distribution with the development of macroeconomic shares of total income (profits and wages).