Estimating the parameter of inequality aversion on the basis of a parametric distribution of incomes

Research background: In applied welfare economics, the constant relative inequality aversion function is routinely used as the model of a social decisionmaker’s or a society’s preferences over income distributions. This function is entirely det ermined by the parameter, ε, of inequality aversion. However, there is no authoritative answer to the qu stion of what the range of ε an analyst should select for empirical work. Purpose of the article: The aim of this paper is elaborating the method of deriving ε from a parametric distribution of disposable incomes. Methods: We assume that households’ disposable incomes obey the generalised beta distribution of the second kind GB2( a,b,p,q). We have proved that, under this assumption, the social welfare function exists if and only if ε belongs to (0, ap+1) interval. The midpoint εmid of this interval specifies the inequality aversion of the median soc ial-decisionmaker. Findings & Value added: The maximum likelihood estimator of εmid has been developed. Inequality aversion for Poland 1998–2015 has been esti mated. If inequality is calculated on the basis of disposable incomes, the standard inequality–deve lopment relationship might be complemented by inequality aversion. The “augmented” inequality– development relationship reveals new phenomena; for instance, the stage of economic develop ment might matter when assessing the impact of inequality aversion on income inequality. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(3), 391–417


Introduction
In this paper, we propose a new method of estimating the parameter, ε, of the constant relative inequality aversion function (CRIA) (Atkinson, 1970). In applied welfare economics, CRIA is the routinely-used mathematical tool of encompassing societal preferences over income distributions. The expected value of CRIA, i.e. the social welfare function (SWF), is the basic maximand of social policy. The parameter ε measures inequality aversion, i.e. the rate at which a society trades-off economic efficiency for income equality. 1 However, ε cannot be directly measured because it concerns unobservable social preferences.
In the literature, there is no consensus among economists concerning what empirical data can convincingly reflect a social attitude toward income inequality and how to elicit ε from such data. In Section 2, we present a review of some recent answers to these questions.
In this paper, we retrieve ε form the distribution of disposable income (DDI). The societal redistributive system transforms the distribution of market income (wages and capital interests) into DDI (market income minus tax, plus social transfers). Note that the current redistributive policy has no impact on the current distribution of market income; the policy shapes only current DDI. Thus social inequality aversion manifests itself in the form of the current DDI.
To be more specific, suppose m competitive redistributive policies which guaranty the same maximum SWF, but they differ concerning the level of inequality aversion, ε1,…,εm say. Thus the policies offer different solutions of the efficiency-equality trade-off. However, only one policy, say lth, wins such a competition, according to the legally binding rules of social choice, l=1,…,m. One may ask the question: What would εl be if the current DDP was the result of the wining redistributive policy?
To answer this question, we assume that DDP obeys the generalised beta distribution of the second kind (GB2) (MacDonald, 1984). Then, SWF will be the expected value of CRIA, with respect to GB2. We prove that SWF exists if and only if ε lies in a finite interval. We propose the midpoint of this interval as the estimate of social aversion to inequality. We develop the maximum likelihood estimator of ε.
To assess the usefulness of our method to retrieve unobservable inequality aversion, we estimate the parameter ε and related normative characteristics for Poland for the years 2000-2015. We use micro-data on DDP from the Polish Household Budget Surveys (PHBS). Then, we confront our empirical findings with relevant facts predicted by economic theories.
We organise the remainder of this paper as follows. At the beginning of Section 2, we introduce the basic welfare terms. Next, we review recent approaches to retrieving ε. In Section 3, we present the details of our method of estimating inequality aversion. Section 4 contains empirical results, namely the estimates of ε and related normative characteristics for Poland, for the years 2000-2015. In Section 5, we assess the usefulness of our method to retrieve inequality aversion. Here, we also verify some prominent economic hypotheses. Section 6 concludes.

Welfare frameworks
Suppose that a positive valued random variable X describes income distribution. 2 The standard SWF is the mean value of personal welfare u(x), where u(x) is the utility of income x. When X is of the discrete type, with the probability mass function P(X=xi)=1/n, n<∞, SWF will have the form (Lambert & Naughton, 2009). The authors interpret SWF (1) as "(…) a person's expected utility, measured from behind a 'veil of ignorance', which is specified in a thought experiment in such a way that the person may be identified with any one of the individuals populating the income distribution with the same probability." When X is of the continuous type, with the density function f(x) (X~f(x), for short), SWF will have the form (Lambert & Naughton, 2009). Note that SWF (2) exists if and only if the integral on the right side is absolute convergent and finite, namely, (Fisz, 1967, p. 64). CRIA, whose single parameter ε is the object of our interest, has the form (Atkinson, 1970), where log x is a natural logarithm of x. In the literature, the name 'inequality aversion' is commonly used for ε. Geometrically, ε reflects the curvature of CRIA. When ε<0, u(x|ε) is convex and represents an inequality-loving society. When ε=0, u(x|ε) is linear and characterises an inequality neutral society. Such a society does not care about inequality, preferring one distribution X1 over another X2 if and only if under X1 the mean income is higher than under X2 (Lambert, 2001, p. 99). If ε>0, u(x|ε) is strictly concave and represents an inequalityaverse society.
In general, μ ε is the solution to the equation: u(μ ε )=SWF. For utility function (4) and SWF (1), μ ε has the form For utility function (4) and SWF (2), μ ε is equal to provided that the integrals on the right-hand side of (7) are absolute convergent and finite. The trade-off between equality and economic efficiency (Okun, 1975) is apparent in the abbreviated social welfare functions (ASWF) (see Lambert, 2001, Chapter 5, for a full presentation). The Atkinson ASWF is equal to EDEI, namely, The following ASWF is the descriptive counterpart of (8), namely where G is the Gini index of income inequality. The .. / was proposed by Sheshinski (1972) and popularised by Sen (1973). Equations (8) and (9) show that efficiency, as measured by μ, can be traded-off for equity, as measured by 1-A(μ,ε), or 1-G. The disincentive effects of redistributive taxation can be more than offset by the gains to the poor (Lambert, 2001, p. 107). The trade-off explains why politicians do not reduce inequality to the extent higher than that observed.

Recent methods of inequality aversion estimation
In typical applications of A(ε,μ) for the comparisons of inequality in distinct income distributions, an analyst assumes a fixed value for ε and uses this value to all compared distributions. However, little theoretical or empirical ground exists to impose such an approach (Aristei & Perugini, 2016). Moreover, there is no consensus among economists regarding the range of ε an analyst should select. The literature offers various methods of establishing ε.
In experimental economics, two approaches for retrieving ε can be observed (see Clark & D'Ambrosio, 2015, for a broader presentation). In the first approach, ε is elicited from data yielded by the leaky bucket experiment (Okun, 1975). When a transfer of an income, e.g., $1, is made from a person with income x 1 (a rich person) to a person with income x 2 (a poor person), a certain fraction of it, say d, is lost because of administrative costs. The basis of eliciting ε is the extent of losses, or leakages, which are accepted by participants of an experiment.
Formally, the leaky bucket experiment consists in deriving the posttransfer SWF and equating it to the pre-transfer SWF. The rate, d, of leakage that preserves the initial SWF will be equal to # = 1 − 1 2 1 3 (10) (Atkinson, 1980). Note that d depends on the ratio x 2 /x 1 of incomes. The participants assess an acceptable leakage d of income for various levels of the ratio. Inequality aversion ε is the solution to Eq. (10).
The leaky bucket experiments have usually provided relatively low estimates of ε. Amiel et al. (1999) experimented with large groups of students from various universities and found that the median of ε was between 0.1 and 0.22. Pirttilä and Uusitalo (2007) found the median of ε below 0.5 when performing the leaky bucket experiment in a representative survey of Finnish people.
In the second approach to eliciting ε, participants of an experiment choose between distinct income distributions in hypothetical societies. In research with Swedish students, Carlsson et al. (2005) found the median of ε between 1 and 2. Notably, 7% of respondents reported ε<0. Pirttilä and Uusitalo (2007) found the median ε larger than 3.
Experimental economics has provided ambiguous estimates of ε (see Levitt & List, 2007, for a broader discussion). Beckman et al. (2004, p. 19) noted that the apparent shortcoming of the methods is "(…) what people say in response to hypothetical questions and what they actually do when income is at stake may be quite different." Moreover, the methods in question are impractical in a retrospective analysis of inequality aversion; current economic experiments cannot provide data on revealed preferences over the past income distributions unless independence of time is assumed.
In the literature, ε has also been retrieved from tax policies. Richter (1983), Vitaliano (1977), and Young (1987) have estimated ε based on the equal sacrifice model. This model assumes that income taxes are set such that the loss in individual utility is equated across all income levels, given a plausible utility function of income. Suppose t(x) denotes a tax schedule that expresses the tax liability of a person with income x. The tax schedule is an equal absolute sacrifice for the utility function u(x) if and only if, for all x and some constant c>0, the following identity holds: For utility function (1), Cowell and Gardiner (1999) demonstrated that (11) can be expressed as where t' is the first derivative of t. The ordinal least squares method can be applied to estimate ε, assuming a null intercept. Stern (1977) used (12) and found ε=1.97 for the UK fiscal year 1973/74. Cowell and Gardiner (1999) presented lower estimates of ε for the UK, namely, 1.43 and 1.41 for the respective fiscal years 1998/99 and 1999/2000 when using data on personal income tax. The estimates of ε are substantially lower (1.29 and 1.28, respectively) when based on aggregated data from fiscal files and National Insurance Contributions. Young (1990) fit model (11) to federal US income taxes in the years 1957, 1967, and 1977 and obtained values of ε equal to 1.61, 1.52, and 1.72, respectively. These estimates are much higher than those provided by the leaky bucket experiment. Young's estimates of ε for other countries are also higher than 1, for instance, 1.59 for Japan in 1987, 1.63 for West Germany in 1984, and 1.40 for Italy in 1987. Gouveia and Strauss (1994) estimated the equal sacrifice model for the United States for 1979 and 1989 and found ε values between 1.72 and 1.94. Aristei and Perugini (2016) estimated the model in question for former communist countries and found ε ranged from 0.93 to 1.68.
However, the estimation of ε based on the equal sacrifice criterion poses some problems. Young (1990) and Mitra and Ok (1996) have demonstrated that the criterion may be violated in practice. Also, the equal sacrifice model does not account for the possible reduction of income inequality by social transfer policies. Lambert et al. (2003) estimate countries' inequality aversion assuming the natural rate of subjective inequality (the NRSI hypothesis). The authors ask the following question: What would a country-specific ε be if subjective inequality were established at a given level A0?
The main shortcoming of this method is that the estimates of ε are conditional on NRSI. In other words, Eq. (13) does not specify a single value of ε, but the family {ε(A)}AϵI of ε, indexed by AϵI, where I is the set of unknown NRSI. Lambert et al. (2003) predict the following empirical consequence of NRSI: "We present evidence consistent with the existence of a natural rate of subjective inequality by verifying that countries with low (high) tolerance for inequality have low (high) inequality as measured by the Gini coefficient as well." We shall verify the NRSI hypothesis in Section 4.2. Kot (2017) proposed the method of recovering parameter ε from the psychophysical measurement of household welfare, originated by Kot (1997). In a survey, a respondent is supposed to imagine a situation in which his/her actual household income (y) increases (decreases) by $1, $2, etc., until he/she would notice a just perceptible change in welfare. Denoting by tl and tu, the respective lower-and upper-income thresholds, the parameter ε of the utility function (4) is the solution to the nonlinear equations where 0<p<1. Kot (2017) developed criteria for the predetermined selection of version (14), namely, ε<0, ε=0, 0<ε<1, ε=1 and ε>1, based on thresholds tl, tu, and y, for all p. Eq. (14) can be solved numerically. Assuming p=0.5 4 , Kot (2017) estimated ε using archival statistical data from the survey conducted among Polish households by The Public Opinion Research Centre in October 1999. The author found that Polish households are predominantly inequality averse. Inequality aversion larger than 1 dominates other levels of ε. Only 2 per cent of households reveal inequality aversion in the interval (0,1). Notably, 7.64% of households show ε<0; namely, they appeared to be inequality-lovers. This figure is surprisingly close to Carlsson's et al. (2005) 7% of inequality loving respondents. The existence of non-positive inequality aversion suggests the violation of the Principle of Transfers (see also Amiel et al., 2004).
Conducting a specially designed survey is a practical shortcoming of Kot's method. Moreover, further investigations are necessary to specify the 'share' parameter p.

Estimating ε when disposable income obey the GB2 distribution
Let the positive valued continuous random variable X, with the density function f(x), describe DDP. Suppose m competitive redistributive policies which provide the same maximum SWF, but they differ concerning the level of inequality aversion ε 1 ,…,ε m . In other words, every policy offers the different solution of the efficiency-equality trade-off. Thus, there could be m competing DDPs, f(x|ε 1 ),…,f(x|ε m ), as the result of redistribution. Every ith policy promises the same social welfare SWF (2) equal to under constraint (3). However, only one policy, lth say, l=1,…,m, wins the competition, according to the legally binding rules of social choice. We may recognise ε l as the social norm of redistribution. Thus the current DDP, the 'winner', with the density function f(x|ε l ), embodies the redistributive norm ε l . We ask the following question: What would the level of ε l be if the current DDP was f(x|ε l )? Lerner (1944, p.9) was the first to propose the mean value solution to the problem of assigning a utility function to a person while assuming a state of ignorance.
To answer this question, we assume that DDP obeys the GB2 distribution with the density function where a, b, p, q are positive parameters, and B(p,q) is Eulers' Beta function (McDonald, 1984). We also assume that the mean of X exists, namely, the condition aq>1 holds (Kleiber & Kotz, 2003, p. 188). The GB2 distribution is now widely acknowledged to provide an excellent description of income distributions while including many other models as particular or limiting cases (Jenkins, 2007a). The GB2 with a=1 is the beta distribution of the second kind. When p=1, the GB2 takes the form of the Burr (1942) XII-type or the Singh-Maddala (1976) distribution. The GB2 with q=1 is the Burr (1942) III-type, or the Dagum (1977) distribution. When p=q =1, GB2 will become the log-logistic or the Fisk (1961) distribution. Also, the log-normal distribution (Aitchison & Brown, 1957) is a limiting case of the GB2 with a=1 and q→∞. (4) The integral on the right side specifies the partial/negative moment Ef[X 1-ε ] of order 1-ε of the GB2 distribution. Kleiber (1997) showed that the moment exists if and only if ε ϵ (max{0, 1-aq}, ap+1). As 1-aq<0, max{0, 1-aq}=0. Then, we obtain ε ϵ (0,ap+1). QED.

Proposition 1. Suppose u(x|ε) is given by
Proposition 1 states that a social decisionmaker would have inequality aversion within the interval (0,ap+1) if and only if he/she performed a conclusive appraisal of social welfare, namely, if and only if he/she operated with a finite SWF. Thus the proposition excludes unrealistic policies, which would promise infinite social welfare.
When a decisionmaker acts 'behind a veil of ignorance', ε will have the uniform distribution within the interval (0,ap+1). 5 Aristei and Perugini (2016) argue that the ε value revealed by redistributive policies should correspond to the preferences of the voter in the median position of the inequality aversion ladder. The median position corresponds to the midpoint of the uniform distribution of inequality aversion within (0,ap+1), i.e. _ `= a (bK + 1) We propose εmid (17) as the estimate of socially tolerable inequality aversion. 6 We can get the midpoint estimate of inequality aversion also for the particular cases of the GB2 distribution. For the Dagum distribution, the midpoint formula (17) is valid. For the Singh-Maddala distribution and the Fisk distribution, we get _ `= a (b + 1). For the beta distribution of the second kind, we get _ `= a (K + 1). We can derive the maximum likelihood (ML) estimator of εmid using the ML estimators of the parameters of the GB2 distribution (16).
Proposition 2. Let the random variables A and P be the ML estimators of the parameters a and p of the GB2 distribution (16). Let covap be the covariance between A and P. Then, the ML estimator ̂= deZ a of εmid (17) will have the mean equal to f@ Ĉ = a (bK + g h RQ + 1) and the standard deviation equal to i@jĈ = a 7bk Q a + K a k R a + 2bK • g h RQ + @g h RQ C a ; /a , (19) where k R a and k Q a are the variances of A and P, respectively.
5 Note that such a decisionmaker is in the state of maximum entropy. The uniform distribution on the interval [a,b] is the maximum entropy distribution among all continuous distributions which are supported in the interval [a, b] (Cover & Thomas, 1991, p. 269). 6 Kot (2012, p. 81) derived formula (17) from the mathematical conditions of the existence of EDEI (7) in the GB2 distribution.
The distribution of the product A•P is crucial for making statistical inference concerning ε. Aroian et al. (1978) demonstrated that if either µ1/σ1 or µ2/σ2 or both approach infinity then Z=X1X2 will be asymptotically normal. 7 This observation justifies Proposition 3: Proposition 3. If either a/σa or p/σp, or both, tend to infinity, ̂ will have the asymptotic normal distribution with the mean (18) and the standard deviation (19).
Proposition 3 enables obtaining the asymptotic confidence interval for ε. For the sake of convenience, we shall refer to the proposed method of estimating inequality aversion as the parametric method (PM).

Estimates of GB2 distribution
We estimate the parameters of the GB2 distribution using statistical microdata data from the PHBS 2000-2015. The household monthly disposable incomes, in constant 2010 prices, are adjusted by household sizes, which provides incomes per capita. We omit null and negative incomes. We use household sizes as weights.
We estimate the parameters of the GB2 distribution by the ML method by using our programme written in Fortran 99 because the gb2fit Stata module (Jenkins, 2007b) failed to converge for some years. We calculate the matrix of variances-covariances using Brazauskas' (2002) exact formula for the Fisher information matrix. The results are presented in Table 1.
Assessing goodness of fit of the GB2 distribution poses a severe problem. We apply the Pearson chi-squared test using 20 equiprobable cells. However, Chernoff and Lehmann (1954) demonstrated that the test is not χ 2 and depends on the true values of the parameters when applying the ML method for ungrouped (raw) data. D'Agostino and Stephens (1986, p. 68) noticed: "All that can be said in general is that the correct critical points fall between those of χ 2 (k-h-1) and those of χ 2 (k-1).", where k is the number of cells and h is the number of estimated parameters.
In our case, the critical values of the test are χ 2 (20-1)=30.144 and χ 2 (20-4-1)=24.966, for the 5% significance level. Thus the chi-squared test prescribes rejecting the GB2 distribution as the theoretical model of Polish DDP. This result is typical in applications involving large sample sizes (McDonald, 1984). 8 According to our knowledge, other tests of goodness of fit, for example, the tests based on empirical distribution functions, have not been established for composite hypotheses concerning the GB2 distribution.
We can check the validity of the GB2 parameter estimates indirectly by comparing some empirical characteristics of DDP with their GB2 predictions. Table 2 shows the results of the comparison of the mean, the Gini index and , i.e. the Sheshinski-Sen ASWF (9). Examining Table 2 shows that GB2 distribution predicts the selected characteristics of DDP quite accurately. Regression functions, presented in Table 3, confirm this qualification.
Examining Table 3 shows that the GB2 distribution predicts the empirical characteristics of DDP almost perfectly because the values of the adjusted R 2 are very close to one. It is worth adding that the time series of the characteristics do not exhibit linear trends. Thus we may neglect the effect of a 'hidden third variable' ('Year') on R 2 .

The estimates of inequality aversion and related normative characteristics
Having the estimates of GB2 parameters presented in Table 1, we calculate the mid-point estimates ̂ of inequality aversion (18), and its standard errors D[ ] (19). As the ratios / [ ] are very large; we can calculate the bounds of 95% confidence intervals, according to Proposition 3. Table 4 and Fig. 1 show the results of the calculations.
Examining Table 4 and Figure 1 shows two remarkable features of inequality aversion. First, the estimates of ε for Poland are greater than zero. Thus Polish society was inequality averse in the years 2000-2015. Moreover, all estimates of ε are greater than one. Thus our method of retrieving ε provides higher levels of inequality aversion than those offered by the leaky bucket experiments discussed in Section 2. Second, inequality aversion varies over time. According to our model of the competitive redistributive policies, every year, a society may promote a distinct policy, according to current challenges of an economic and social environment. Table 5 presents some normative characteristics of DDP. Besides the Atkinson ASWF, με, and the Atkinson inequality index, A(ε,μ), Table 5 shows two additional characteristics, namely the absolute and relative benchmark incomes x* and z*. Hoffman (2001) observed that a small increase in low incomes decreases inequality, whereas a small increase in high incomes enhances inequality. Therefore, there must exist a specific income level, x*, which separates these effects. The author referred to x* as the relative poverty line or the dividing line between the rich and the poor. Lambert and Lanza (2006) proved the existence of x* for a general class of inequality indices. The authors call x* the absolute benchmark level of income. The relative benchmark level of income equals to z*=x*/μ. When one measures inequality by the Atkinson index (5) (21) (Lambert & Lanza, 2006). It is worth adding that x* does not seem to be a right candidate for a poverty line as Hoffman's (2001) term 'the relative poverty line' suggests. Kot (2009) argues that a poverty line, z, should satisfy the inequality z ≤ EDEI. A policy operating with z > EDEI would promise eradication of inequality on the price of overall poverty. It is easy to see that x* >EDEI= με.

The appraisal of the parametric method PM
For assessing the usefulness of PM to retrieve ε, we estimate regression functions which link selected normative variables, based on ε, with corresponding descriptive counterparts. We fit the regression functions to the time series of the variables in question. As the time series do not exhibit linear trends, we may ignore a possible impact of the time variable (Year) on R 2 . Table 6 presents the estimates of the parameters of regression functions.
Model 1 in Table 6 shows the relationship between the normative Atkinson ASWF, με, (8) and the descriptive ASWF, < , (9). R 2 close to one means that με predicts < almost perfectly. In economic terms, the ranking of income distributions, according to με, is the same as the ranking according to < . If the rankings differed remarkably, our method of estimating ε would be questionable.
We can also appraise the usefulness of PM basing upon a particular consequence of Lambert and Lanza's (2006) theory of the benchmark incomes. Lambert and Lanza's (2006) Theorem 3 specifies a general relationship between x* and inequality aversion. We reformulate this theorem in terms of the Atkinson index (5). A(μ1,ε1) and A(μ2,ε2) be the Atkinson inequality indices (5), where ε1>ε2. Then, for all unequal income distributions X1 and X2, * < a * . However, a general conclusion that x* is a declining function of inequality aversion seems to be unambiguous only for ε>1. To show this, we differentiate the logarithm of (20) with respect to ε, namely As all our estimates of ε are greater than 1, x* and z* ought to be a declining function of ε. Models (2) and (3) in Table 6 demonstrate that our findings are consistent with this theoretical consequence. One can also see that the model (3) for z* fits the data better than the model (2) for x*.

Theorem 3. Let
We can also use our estimates of inequality aversion for the verification of some prominent economic hypotheses. Frisch (1959) hypothesised higher inequality aversion in poorer countries. Contrary to Frisch and Atkinson (1970) hypothesised higher inequality aversion in rich countries. Suppose that GDP per capita measures countries' economic development. Thus, Frish's hypothesis states that ε is a declining function of GDP per capita, whereas Atkinson's hypothesis states that ε is an increasing function of GDP per capita. Examining model 4 in table 6 shows that Frisch's hypothesis is true in the case of Poland for the years 2000-2015.
A relationship between the Gini index G and the Atkinson index A(μ,ε) (5) is crucial for Lambert's et al. (2003) NRSI hypothesis presented in Section 2.2. We can verify this hypothesis empirically. Model 5 in Table 6 shows that there is a statistically significant positive linear relationship between G and A(μ,ε). 9 Lambert et al. (2003) also hypothesised that the Gini index is a declining function of ε. Hereafter, we shall refer to this hypothesis as LMS, according to authors' names, to wit Lambert Millimet and Slottje. However, LMS has a competitor in the form of the well-known inequalitydevelopment relationship (IDR). IDR was originally proposed by Kuznets (1955), who presented the famous inverted-U hypothesis: during the development, inequality first increases and then declines.
Unfortunately, the standard IDR cannot be applied directly for testing LMS because Kuznets and his followers have applied inequality in the distribution of market income, thus ruling out all redistributive issues. As we argued in Section 1, a society's attitude towards inequality shapes DDI, not the distribution of market income.
We can verify LMS using the Gini index for DDP and complementing the standard IDR by ε. In this manner, we obtain the augmented inequalitydevelopment relationship (AIDR). More specifically, AIDR links the Gini index with GDP per capita and with inequality aversion ε.
We shall analyse AIDR non-parametrically using graphical visualisations. The impact of a single dimension on inequality can be determined when respecting the ceteris paribus rule. Thus, for a given degree of inequality aversion, we shall obtain the standard IDR curve. For a given level of economic development, we shall get G as a function of ε. We shall use this function for testing LMS. Figure 2 displays AIDR in the three-dimensional space, whereas Fig. 3 displays the contours of AIDR. We fit the surface of the Gini index by splines in order to avoid troubles with a parametric specification of AIDR. We use the graphic module of Statistica, 3.3, TIBICO Software Inc. Figure 3 shows that inequality seems to be a declining function of inequality aversion for GDP/capita above 1300 [PLN], ceteris paribus. For lower levels of GDP, inequality seems to trace out the U-shaped curve along with increasing inequality aversion, ceteris paribus. Thus the LMS hypothesis appears to be true only for a high stage of economic development. We also observe in Figs. 2 and 3 that inequality traces out the classical inverted U-shaped curve along with the development, ceteris paribus.

Conclusions
In this paper, we propose a parametric method, PM, of estimating a society's inequality aversion, ε, assuming that disposable income distribution, DDP, obeys the GB2(x;a,b,p,q) distribution. We argue that DDP embodies societal aversion to inequality. We prove that the social welfare function, SWF, takes on a finite value if and only if ε lies in the interval (0,ap+1). The values of ε outside this interval would characterise unrealistic policies offering infinite social welfare. We propose the midpoint, εmid, of this interval as the estimate of societal aversion to inequality. We develop the maximum likelihood estimator of εmid, which enables calculating the standard errors and the confidence intervals of inequality aversion. PM has some advantages over the methods developed until now. PM provides objective estimates of ε, in contrast to subjective estimates of ε offered by the leaky bucket experiments, or by Kot's (2017) method. PM also has an advantage over the methods based on the equal sacrifice model. The methods elicit ε from tax data which are scarce and imperfect, whereas PM requires data on DDP which are available for many countries and years. One can also calculate εmid using the parameters of the GB2 distribution, already estimated in many empirical studies. For the review of such studies, see Kleiber andKotz, (2003, pp. 195-196, 209-2010, 221-222), among many others. Obviously, the GB2 distribution, or its particular cases, should be fitted to disposable income data. 10 As inequality aversion is bounded from the above, passing with ε to infinity seems to be debatable. Some authors claim that CRIA could reflect the Rawlsian maximin when ε→∞ (see, e.g. Atkinson, 1970;Lambert, 2001, pp. 99-101). From Proposition 1 it follows that such a claim is unrealistic for the major theoretical models of income distributions since it assumes implicitly infinite social welfare.
The statistical analysis of inequality aversion for Poland provides empirical results which are consistent with some theoretical predictions. Such consistency confirms the usefulness of PM to retrieve a society's aversion to inequality. The augmenter inequality-development relationship shows that the stage of economic development might matter when assessing the impact of inequality aversion on income inequality. However, further empirical studies are necessary for confirming this supposition.   Note: -the Sheshinski-Sen abbreviated welfare function (9).