Inequalities in the distribution of income have created fat tail distributions, where 20% of the population controls 80% of the wealth. A tail index that is greater than 0 results in fat tailed distributions, which produces a more modest value for unlikely situations. Knowing that the high income distribution for every country is modeled by a Pareto, the research was aimed at proving that the world’s distribution was also modeled by a Pareto, which would be proven by the world’s Pareto index. The Pareto function for each nation was discerned by finding its Pareto index and coefficient, and threshold. With these known, the Pareto function was calculated and added together, yielding the probability associated with an income, x. Five income values were chosen and their corresponding probabilities were plotted on a log-log graph, revealing the world’s Pareto index. The acquired results were that the model for the world’s and each country’s high incomes was a Pareto since their tail indexes were greater than 0. It was concluded that the sum of a sample of Pareto distributions yields a Pareto distribution.
The Pareto Distribution is a power law function whose tail fall slower than that of a normal distribution. The tail index is responsible for the thickness of the tails, such that as its value increases, the tail becomes thicker, yielding a distribution where there are not a lot of inequalities in the dispersion of wealth. A tail index less than 0 is bounded, one equal to 0 falls at an exponential rate and one greater than 0 is fat or long tailed. There is less inequality in the spread of wealth as the Pareto index increases.
The Pareto Principle says that 20% of the population controls 80% of the wealth. The relationship between the spread of the wealth of investors and their return from their stocks was seen to be proportional such that the Pareto coefficient is equal to that of the Levy distribution’s exponent. The objective is to discern the distribution that would result from a sample of high incomes in the world.
Galton experimented with peas to show how the distribution of weight among offspring is influenced by inheritance. He then used these distributions to show that the sum of a sample of normal distributions yields a normal distribution. Galton separated the 490 peas he used into seven groups according to their weights and distributed 10 peas from each of those seven groups to each of his friends. His friends then grew these peas which created seeds whose weights were measured and models were combined together to see which type of model he would yield. From his experiments, Galton found out that the group of seeds that he supplied to each of his friends to grow, yielded seeds with weights that had a normal distribution. In addition, the variances for each group were synonymous, which correlates to the line that Galton made to be AB on the Galton board, where the drops are separated into different sections and would rest on a line set to be AB. If the line AB is not near the top of the Galton board, these drops would have a normal distribution, and if one section was opened and the drops fell to the bottom, a normal distribution would composed. If the other sections were opened one by one, the outcome would be a normal distribution as well.
The research would allow people to see the distribution of the population of high incomes in the world based on the sample that comprises it. The conjecture is that “If the Pareto distribution for the incomes of each country were added together, the new distribution would also be a Pareto if the wealth is dispersed in the same fashion and the Pareto index is greater than 0.”
Materials needed were the total income and population, as well as the gini indexes for each country studied, and the equations for the Pareto function, ,Pareto mean, , and gini coefficient, . Since the gini coefficient was known, its equation was used to solve for the Pareto index. The mean is the total income over the total population, the Pareto index was plugged into the equation and the threshold was solved for.
With the Pareto index and threshold known, the Pareto function was extracted for each country. To determine the relative probability of a certain income, the Pareto functions for each country were multiplied by the ratio of its population to the sum of the populations of the countries used.
With the product determined, five random high income values were chosen, 1,000,000, 1,500,000, 2,000,000, 2, 500,000 and 9,000,000, calculating their corresponding probabilities. These values were plotted on a log-log graph, and the slope of the line was taken, finding the Pareto index for the model of income in the world.
Calculation of the Income Probability for the United States: Table 1: Pareto and Threshold Parameter A list of the Pareto indexes for each country studied, Japan has the highest index at 2.5, yielding fewer inequalities in the spread of wealth. With the mean income for each country shown, the threshold parameter for each can be seen, with China having the smallest threshold at $3,424.20 and Switzerland having the greatest at $20,919.62.
Table 2: Calculation of the Pareto probability for a certainincome value, x. The Pareto functions for each countryallowed the probabilities for an income value to becalculated, predicting that the United States and Chinawould yield the greatest probabilities since their Paretoindexes are the smallest.
The Probabilities for World Income Values 1 0.1 0.01 0.001 0.0001 Probabilities for WorldProbability 0.00001 Income Values 0.000001 0.000000 1E-08 1E-09 1E-10 1E-11 1E-12 1E-13 1,000,000 10,000,000 Income
The Pareto model of world incomes involves the collaboration of individual Pareto functions. Canada has the greatest equality in the distribution of wealth since it’s gini index is the smallest and Pareto index is the greatest. With $1,000,000, $1,500,000, $2,000,000, $2,500, 000 and $9,000,000 and their corresponding probabilities, the slope of the log-log graph ended up with an index of 1.74.
Recall the hypothesis: If the Pareto distribution for the incomes of each country were added together, the new distribution would also be a Pareto if the wealth is dispersed in the same fashion and the Pareto index is greater than 0.” The proposed hypothesis was valid because once the Pareto probabilities were added together, the model for the world’s incomes was concluded to be a Pareto.
Galton showed that the distribution of the weight’s of peas is affected by inheritance such that when he split them into seven groups, they yielded seeds with weights that had a normal distribution and when those groups were added together, a normal distribution came about. Galton showed that the drops in each section of a quincunx resting on a line above the bottom is a normal distribution such that when he opened each section one by one, they generate a normal distribution.
Although the research was able to show the weighted sum of a sample of Paretos yielded a Pareto, there were limitations to it. The Pareto optimal values cannot be compared to each other with regards to the Pareto principle and the median income cannot be determined since the Pareto is not symmetrical.
Further research would be utilizing another methodology to tackle this problem. Rather than pinpointing the probability for each income value, the moments for each country would be calculated and the gamma function would be used to find the moment generating function. From there the world’s Pareto index could be found.
The study demonstrated that the sum of a sample of Pareto distributions does in fact yield a Pareto. The Pareto model for the incomes in the entire world is a fat-tailed distribution with a thick tail since its tail index is 1.74, which shows that there is not a lot of inequality in the spread of wealth. By finding the total income and population of all the countries used, the mean income in the world was calculated to be $12,449.53, with a threshold of $5,294.63.
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