In this article the research on two-parameter Lorenz curve approximants is continued. Several characteristic properties of one such family of two-parameter functions used to approximate the Lorenz curve is studied. Several interpretations of the functional parameters are introduced. Distribution density functions that correspond to various values of parameters of the class of functions used to approximate the Lorenz curve are analysed. The implications and the value brought to the study of nonuniformity of resource distributions by this introduction is discussed. The ties between the values of the parameters and statistic estimators are tested. It is concluded that the values of the approximant parameters are connected to the coefficient of kurtosis and the value of the interquartile range. An interpretation of the approximant parameters with regards to economic inequality is suggested. The special case corresponding to uniform distribution of the shares of the resource allocated to separate agents is distinguished and discussed. It is concluded that it is of special value with regards to solving the problem of finding the optimal distribution of a resource within an economic system.

One possible representation of a empiric distribution of resources within an economic system is a Lorenz curve. Various single parameter and multiple parameter approximants of the Lorenz curve, the corresponding probability distribution functions and the probability density functions have been studied extensively [1][2][3][4][5][6][7][8]. As the Lorenz curve is currently the predominant apparatus used to characterise and simulate inequality in economic and social systems, both its analysis and applications are of great interest. The present paper constitutes an inquiry into one of its many proposed approximants and its properties. This is a follow-up research from previous elaboration on the class of two-parameter functions [3,4].

Let 𝐺 be a resource distributed among 𝑁 economic agents. Let 𝑖 = 1, . . . , 𝑛, . . . , 𝑁 be the indices of 𝐺 𝑖 -the amount of resource 𝐺 allocated to the,

A resource distribution 𝐺 𝑖 , 𝑖 = 1, . . . , 𝑁 may be represented by Lorenz curve [7,8]. For that purpose the original series is ranged in ascending order: 𝐺 1 𝐺 2 . . . 𝐺 𝑁 .

Next the accumulated sums

are calculated and plotted on a plane with axes 𝑥 = 𝑛/𝑁 and 𝑦 = 𝑆𝑛/𝑆 𝑁 . Previously [3,4] it has been demonstrated that the Lorenz curves may be approximated my the two-parameter curve belonging to the class of functions

It has been previously demonstrated [5,6] that parameters 𝛼 and 𝛽 may serve as indicators of resource distribution non-uniformity. At 𝛼 = 𝛽 = 1 the accumulated sums 𝑦(𝑥, 𝛼, 𝛽) = 𝑥 and the corresponding resource distribution is absolutely uniform, which means that each economic agent acquires possesses exactly 𝐺/𝑁 of the total accumulated wealth. In the opposite scenario, while 𝛼 → ∞, 𝛽 → ∞ the resource distribution approaches an extremely non-uniform distribution, where the agent of the highest rank has exclusive control over the entire stock of wealth and the agents of lower rank have nothing.

By manipulating the values of the parameters 𝛼 and 𝛽 various distribution density functions may be attained. It has previously been shown [7,8] that even the single parameter class of curves given by 𝑦(𝑥, 𝛼, 𝛽) = 1 − (1 − 𝑥 𝛼 ) 1/𝛼 , demonstrate a good fit for empiric wealth distributions. However, the implementation of the two-parameter class of functions 2 enables to improve the quality of the fit by allowing for asymmetric resource distributions, while the class of functions ?? can only simulate a symmetric (with regards to the main diagonal of the Lorenz square) distribution [9,10]. Besides, the use of the two-parameter approximant allows to more closely fit the intervals that correspond to groups of the highest and the lowest rank, while sustaining the ease of interpretation and calculation that were part of the incentive for the use of the single parameter approximant [11][12][13][14].

Fig. 1-2 illustrate the correspondence between selected values of the parameters 𝛼 and 𝛽 and the corresponding frequency histograms. In this paper the discrete values case is considered, for the continuous case and the analytical results see [3,4]. Here we scrutinise the distribution of the total resource 𝐺 = 1000 among 𝑁 = 1000 economic agents with regards to variation of parameters 𝛼 and 𝛽. In fig. 1-2 the relative shares of the resource 𝐺 are plotted along the horizontal axis and the relative number of agents that possess that portion of the shared resource are plotted along the vertical axis. The histograms are standardised so that the 1000 values of 𝐺 𝑖 , 𝑖 = 1, . . . , 𝑁 are always grouped into 20 equal-sized intervals. It is evident that for 𝛼 = 1, 𝛽 = 1 all agents acquire an equal share of the resource 𝐺 [15]. While 𝛼 increases while the value of 𝛽 is fixed, the uniformity of the distribution is compromised as the portion of the total resource allocated to single agents are varied. Provided that every percentile of the resource distribution in the context of societal wealth represents a class, at 𝛼 → 2, 𝛽 = 1, probability that any given agent 𝐺 𝑖 is positioned within any given class is uniform. This does not mean that the shares 𝐺 𝑖 are equal. In fact, when 𝛼 = 2, 𝛽 = 1, 𝐺 𝑖 varies from 0 to 2 • 𝐺.

It may be noted that the distribution that corresponds to the Lorenz curve

is of special interest for two reasons:

the maximal share 𝐺 𝑖 that may be allocated to an agent or a group of agents is limited by 𝐺 𝑖 = 2 • 𝐺; the probability that an agent will be positioned within any percentile, decile, quartile is equal. The density function of the considered distribution with respect to 𝐺 is as follows

The Gini coefficient 𝐾 𝐺 -an extensively implemented measure of distribution non-uniformity [16][17][18][19][20][21] -for the curve given by 2 is equal to 1 − 2 𝛼+1 , or, for the case in consideration 𝐾 𝐺 = 1 3 . It must be noted that high (𝐾 𝐺 > 0.5) values of 𝐾 𝐺 are rarely observed with regard to societal economic inequality and at present time only occur in undeveloped and some developing countries.

It is further proposed that the values of parameters 𝛼 and 𝛽 of the class of functions 2 specify distinct statistical properties of empiric distributions of total accumulated wealth within a society among its members.

The parameter 𝛼 is a characteristic of peakedness of the probability density function. The resource density function will become less flat and more pointed as the value of parameter 𝛼 increases. For lower values of 𝛼 the values of 𝐺 𝑖 are concentrated closer to the mean value

while for higher values of 𝛼 the values of 𝐺 𝑖 are scattered further from 𝐺. This means that as the value of 𝛼 increases, less and less agents are allocated the 'fair' portion of the resource 𝐺. As to the statistical properties, the leptokurtic probability density functions are classified by a higher 𝛼 coefficient of kurtosis, calculated as 4 ] is the kurtosis of the stochastic value 𝐺, and 𝜎 is its standard deviation.

On the other hand, the parameter 𝛽 is characteristic of the disparity between the proportion of the resource belonging to the agents of the lowest and the highest rank. From an economic point of view the increase in the value of 𝛽 is accompanied by societal polarity which is characterised by the formation of two segregated groups of ultra-rich and ultra-poor. The value of the interquartile range (IQR) is determined primarily by the value of 𝛽. 𝐼𝑄𝑅 is the difference between the median value of the resource allocated to the 50% of the highest ranking agents and the median value of the resource allocated to 50% lowest ranking agents. With regards to measuring economic inequality the interquartile range is a measure of the gap between the median wealth of the 50% of the richest and the median wealth of the 50% of the poorest. The value of the interquartile range is a robust measure of variation in comparison with integral range, a characteristic of spread of a stochastic value. It is clear that the value of the coefficient of kurtosis of the considered class of distributions is chiefly determined by the value of the parameter 𝛼. It can be concluded that the parameter 𝛽, in comparison to the weight of the parameter 𝛼, scarcely affects the value of the coefficient of kurtosis of the distribution density and is largely insignificant with regards to measuring the flatness of the distribution. It is also shown that the maximal value of the interquartile range is attained at large values of the parameter 𝛽 while 𝛼 = 1. Therefore we propose that the parameter 𝛼 may be interpreted as a characteristic of the gap between the wealth of those of the highest rank and those of the lowest rank, while the parameter 𝛽 is a measure of variation.

The attained results are useful for solving various problems associated with finding optimal resource allocation among several objects or agents within an economic system. The proposed interpretation of the parameters 𝛼 and 𝛽 of the two-parameter Lorenz curve approximant makes it possible to subsequently define classes and clusters of empiric economic resource distributions and the corresponding economic and and societal distributions. The question whether there exists an empiric relationship between the resource distribution that corresponds to the special case of the studied Lorenz curve approximant and the performance of real economic systems is a subject of further investigation.