“Skills, Occupation Inequality and Development” is a theoretical study. There is no general agreement about how income inequality will affect development in the long run. Classic growth models show that income inequality is beneficial to development due to agent’s heterogeneity and marginal propensity to save increasing with wealth. Neoclassical growth models present that income distribution plays no significant role on development assuming representative agents and decreasing marginal returns in investment. New classical growth theory demonstrates that income inequality impedes growth due to credit constraints and indivisibility of investment in human capital. This paper studies income inequality through the channel of complementary skills and occupations in aggregate production. In a new classical model economy with two complementary occupations, CES production technology, skills in utility, and uncertainty of completing high-skilled occupations, we find a continuum of equilibria denoted by a correspondence between aggregate capital stock and the low-skilled population share regardless of the distribution in initial endowments. Aggregate capital stock and aggregate income per capita are non-monotonically related to the low-skilled population share. Aggregate income per capita will be maximized at a certain distribution of occupations on the continuum of equilibria. Therefore, the correlation between development and inequality of occupation distribution can be both positive and negative which depends on the position of occupation division on the continuum of equilibria. The correlation between low skills and occupation inequality is monotonic within a country, but the correlation is opposite between developed and developing economies. The low skills will move up on the continuum of equilibria if the occupation inequality is smaller (larger) in developed (developing) economies. The study also shows that inequality of the occupation distribution plays different effects in developed economies from those in developing economies due to the assumption that skills affect the completion of occupations. Developing economies also present two patterns of equilibria, in which one has higher optimum inequality of occupations, another one has lower optimum inequality of occupations. The cause of two patterns of equilibria for developing economies comes from the assumption of Cobb-Douglas production function. Shifts of equilibrium lead to new levels of development due to a change of inequality in other characteristics of the economy. “Fair Division of Income Distribution, Development and Growth: Evidence from a Panel of Countries” is an empirical exercise. I use an unbalanced panel data to explore the correlation between aggregate income per capita and income inequality. A lot of studies document controversial results using the Gini index or other summary measurements of income inequality. I measure income inequality by the two dimensions of a point on the Lorenz Curve, where the Lorenz curve has unit slope. It is called fair division point, which involves the fair population share and the fair income share. The difference between the fair population share and the fair income share approximates the Gini index of an income distribution. My analysis shows that a country’s low income population relatively decreases (the fair population share drops slightly) as the economy grows; and at the same time, those low income households are relatively worse off (the fair income share falls even though the GDP per capita increases). Inversely, as an economy becomes rich, there are more low income households (the fair population share increases), but those low income households are better off (the fair income share goes up and GDP per capita increases as well). Overall, both the Gini index and the difference between the fair population share and the fair income share have been increasing during the last half century in the panel of countries. Therefore, income inequality increases as an economy is getting richer. The analysis presents strong evidence for optimum income inequality regarding both the aggregate productivity and the growth rate of GDP, where income inequality is measured by either the Gini index or the fair division shares. But no evidence has been found for the Kuznets’ hypothesis. Both high and low inequality of income distribution could harm an economy as we compare with its potential optimum inequality. Also developed economies show different optimum inequality from that in developing economies, and there is the growth-worst fair population share that results in the lowest growth in developed economies. Measurement of income inequality matters on its economic effects for the subsamples of the panel data.

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yes / An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced. It generalises the Birkhoff-von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions). An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to de ne locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented.

Yes / Coherent subspaces spanned by a finite number of coherent states are introduced, in a quantum system with Hilbert space that has odd prime dimension d. The set of all coherent subspaces is partitioned into equivalence classes, with d 2 subspaces in each class. The corresponding coherent projectors within an equivalence class, have the 'closure under displacements property' and also resolve the identity. Different equivalence classes provide different granularisation of the Hilbert space, and they form a partial order 'coarser' (and 'finer'). In the case of a two-dimensional coherent subspace spanned by two coherent states, the corresponding projector (of rank 2) is different than the sum of the two projectors to the subspaces related to each of the two coherent states. We quantify this with 'non-addditivity operators' which are a measure of quantum interference in phase space, and also of the non-commutativity of the projectors. Generalized Q and P functions of density matrices, which are based on coherent projectors in a given equivalence class, are introduced. Analogues of the Lorenz values and the Gini index (which are popular quantities in mathematical economics) are used here to quantify the inequality in the distribution of the Q function of a quantum state, within the granular structure of the Hilbert space. A comparison is made between Lorenz values and the Gini index for the cases of coarse and also fine granularisation of the Hilbert space. Lorenz values require an ordering of the d 2 values of the Q function of a density matrix, and this leads to the ranking permutation of a density matrix, and to comonotonic density matrices (which have the same ranking permutation). The Lorenz values are a superadditive function and the Gini index is a subadditive function (they are both additive quantities for comonotonic density matrices). Various examples demonstrate these ideas.

Yes / Lorenz values and the Gini index are popular quantities in Mathematical Economics, and are used here in the context of quantum systems with finite-dimensional Hilbert space. They quantify the uncertainty in the probability distribution related to an orthonormal basis. It is shown that Lorenz values are superadditive functions and the Gini indices are subadditive functions. The supremum over all density matrices of the sum of the two Gini indices with respect to position and momentum states is used to define an uncertainty coefficient which quantifies the uncertainty in the quantum system. It is shown that the uncertainty coefficient is positive, and an upper bound for it is given. Various examples demonstrate these ideas.

Background: Access to quality public transportation is critical for employment, especially for low-income and minority populations. This research contributes to previous work on equity analysis of the U.S. public transportation system by covering the 45 largest Metropolitan Statistical Areas (MSAs) and their counties. Objective: This study analyzes job accessibility of transit commuters in the 45 largest MSAs to assess the existing differences in accessibility between Census-defined socioeconomic status (SES) categories. Method: 2014 Census demographic data were matched to a previously published 2014 dataset of transit job accessibility at the Census Block Group level. Transit equality and justice analyses were performed based on population-weighted mean job accessibility and SES variables. Results: The findings suggest that within individual MSAs, the low-income populations and people of color have the highest transit job accessibility. However, in certain MSAs with high job accessibility, such as New York, Washington, D.C., Chicago, and Houston, there is a significantly disproportionate access to public transportation based on income. Variables such as income, and the use of personal vehicle, are found to have a statistically significant negative impact on job accessibility in almost all MSAs. The percentage of White workers has a significant impact on job accessibility in upper-mid-density MSAs and high-density MSAs. The percentage of the population with limited English speaking ability is not a significant determinant of job accessibility except in lower-mid-density MSAs. Disparities by income are greater than disparities by race. Racial disparities increase by MSA size and density controlling for income. The findings suggest that planning for public transportation should take into account risks, benefits, and other equally important aspects of public transportation such as frequency, connectivity, and quality of service. / Master of Urban and Regional Planning / In recent years, there has been a shift in focus from encouraging mobility to encouraging accessibility, along with the provision of more sustainable travel options (e.g., walking, cycling, public transport). Access to quality public transportation is critical for employment, especially for low-income and minority populations. This research contributes to previous work on equity analysis of the U.S. public transportation system by covering the 45 largest Metropolitan Statistical Areas (MSAs) and their counties. This study analyzes job accessibility of transit commuters to assess the existing differences in accessibility in terms of income, race, ability to speak English, etc. Transit equality and justice analyses were performed based on population-weighted mean job accessibility and SES variables. The findings suggest that within individual MSAs, the low-income populations and people of color have the highest transit job accessibility. However, in certain MSAs with high job accessibility, such as New York, Washington, D.C., Chicago, and Houston, there is a significantly disproportionate access to public transportation based on income. Variables such as income, and the use of personal vehicle, are found to have a statistically significant negative impact on job accessibility in almost all MSAs. The percentage of White workers has a significant impact on job accessibility in upper-mid-density MSAs and high-density MSAs. The percentage of the population with limited English speaking ability is not a significant determinant of job accessibility except in lower-mid-density MSAs. The findings suggest that planning for public transportation should take into account risks, benefits, and other equally important aspects of public transportation such as frequency, connectivity, and quality of service.

This diploma thesis deals with income inequality measurements and concentrates mostly on the subjective parameters that are used in inequality computations. The core of the thesis is an evaluation of data from a survey, in which a questionnaire was completed by 150 people, mostly students and recent graduates. The most surprising finding is that approximately one third of respondents support the absolute invariance principle; eventually, this means a denial of many types of measurements in welfare economics, including the Gini Index. In the questionnaire, the respondents were also supposed to estimate actual Czech income distribution. All groups of respondents, not excluding economists, substantially overestimated the lowest income class size.

Tarptautinių atsiskaitymų banko (BIS) Bazelio II susitarimo nuostatos dėl bankų minimalaus kapitalo apibrėžia reikalavimus kredito rizikos skaičiavimui. Kredito rizikos vertinimo metodai leidžia naudoti vidines įmonių reitingavimo sistemas. Vienas svarbiausių reitingavimo modelio uždavinių – į modelį parinkti tokius finansinius ar nefinansinius rodiklius, kurie geriausiai klasifikuotų įmones pagal jų finansinio pajėgumo lygį. Populiariausias statistinis atrankos rodiklis yra tikslumo koeficientas dar vadinamas Gini indeksu. Tradicinis Gini indeksas buvo apibrėžtas 1914 m. ir iki šiol yra naudojamas pajamų nelygybei populiacijoje apskaičiuoti. 1995 m. Mosleris ir Koševojus pristatė k-matį Gini indekso analogą kaip zonoido tūrį. Šio darbo tikslas – naudojantis zonoidų teorijos idėjomis sukonstruoti apibendrintą reitingavimo modelių Gini indeksą. Pirmoje darbo dalyje pateiktos tradicinės Lorenco kreivės bei Gini indekso sąvokos ir Gini indekso apibendrinimai. Antroje darbo dalyje pagal BIS naudojamas reitingavimo modelio galios sąvokas, apibrėžtas reitingavimo modelio Gini indeksas. Be to, apibrėžti Lorenco kreivės apatinės ir viršutinės aproksimacijų Gini indeksai bei sudaryti šių indeksų apibendrinimai – normos bei tūrio daugiamačiai Gini indeksai. Pabaigoje analizuojamas atskirų finansinių rodiklių Gini indeksų stabilumas bei bendras Gini indeksų – vienamčio, normos ir tūrio – stabilumas ir pateikiamos išvados. / Bank for International Settlements (BIS) Basel II resolutions on banks regulatory capital include requirements for credit risk calculation. Credit risk evaluation methods define the possibility of using the internal rating system. One of the main tasks to build the powerful scoring model is to select financial and non-financial factors that appropriately classify companies according to their financial situation. The most popular statistical measure used for discriminatory analysis is the accuracy ratio or Gini index. General Gini index presented in 1914 is still widely applied to measure income inequality in the population. The k-dimensional analogue of Gini index as volume of zonoid was defined only in 1995 by Mosler and Koshevoy. The main purpose of this paper is to build the generalized Gini index of scoring model following the theory of zonoids. In the first part of the paper the usual Lorenz curve, traditional Gini index and its summary measures are presented. The second part presents the definition of the scoring models Gini index according to scoring model power measures applied in BIS resolutions. Furthermore the Gini indexes of Lorenz curve bottom and top approximations are defined and two its summary measures – norm and volume Gini indexes are constructed. Finally the stability of separate financial ratios Gini indexes and the general stability of univariate, norm and volume Gini indexes are analysed and final conclusions are presented.

Since its introduction by Max Otto Lorenz, the Lorenz curve has been utilizedin several financial contexts. By using regression analysis to approximate theclaim cost of policyholders, a vector consisting of policyholder characteristics canbe obtained. The ordered Lorenz curve can subsequently be used to understandwhat commonalities are shared between profitable policyholders. This allows forbetter management of the insurance portfolio and thus better customer relationstowards both the policyholders and the insurer, which is important for an insuranceconsultancy agency. The aim of this thesis was to investigate which attributesapproximate the policyholder claim costs and consequently obtain insight into whatattributes are shared among profitable portfolio clients. The results presented inthis thesis show that a multi-linear regression model, transformed using the Box-Cox method is insufficient to approximate the claim costs in a convincing manner.The model obtained in the thesis was capable of identifying significant regressorsbut the overall result displayed uncertainties in regards to overall goodness of fit.This means that the variability explained by the regression model only represents4.95% of the variability in the claim cost data. Thus, the relativity measureintroduced in section 2.1.1 was deemed uninterruptible in a meaningful way.Consequently, the empirical distribution functions presented in section 1.1 wouldbe based on a faulty order statistic, and in turn the visualization of an orderedLorenz curve with such a relativity measure is unnecessary. / Sedan Lorenzkurvan introducerades av Max Otto Lorenz 1904 har den använtsinom flera finansiella sammanhang. Genom att använda regressionsanalys föratt approximera försäkringstagares skadekostnader kan en vektor som består avförsäkringstagarnas attribut erhållas. Den sorterade Lorenzkurvan kan i sin turanvändas för att förstå vilka gemensamma attribut som delas mellan lönsammaförsäkringstagare. Detta möjliggör bättre hantering av försäkringsportföljen ochdärmed bättre kundrelationer mot både försäkringstagarna och försäkringsbolaget,något som är viktigt för försäkringsförmedlare. Syftet med denna avhandling var attundersöka vilka egenskaper som approximerar försäkringstagarnas skadekostnaderoch därmed få insikt i vilka attribut som delas bland lönsamma portföljkunder.Resultaten som presenteras i denna avhandling visar att en multilinjär regressionsmodell,som transformeras med Box-Cox-metoden, är otillräcklig för att approximeraskadekostnader på ett övertygande sätt. Modellen som erhölls i avhandlingenkunde identifiera signifikanta regressorer, men det övergripande resultatet visadeosäkerheter när det gäller den generella anpassning. Detta innebär att variabilitetensom förklaras av regressionsmodellen bara representerar 4,95% av variabilitetenbland skadekostnadsdatan. Därmed ansågs relativitetsmåttet som introduceras iavsnitt 2.1.1 vara oanvändbart på ett meningsfullt sätt. Följaktligen ger de empiriskafördelningsfunktionerna som presenteras i avsnitt 1.1 ett felaktig sorteringsmåttsom i sin tur medför att visualiseringen av en sorterad Lorenzkurva baserad påovannämnda mått är onödigt.

The goal of this research was to determine which population factors were associated with iPad pricing differences across countries. Specifically, this paper measured the relationship between iPad prices in a given country and its U.S. dollar exchange rate, amount of income inequality, Gross Domestic Product per capita, luxury good sales growth, Individualism Index score, and population density. Panel data was collected for the iPad 2, the iPad Retina, and the iPad Mini tablets from 38 countries of varying geographic locations, economic paradigms, and political structures. The pooled data set yielded 114 observations in total. Regressing iPad price as a percent of national average income revealed a positive relationship between price and status consciousness as well as cultural individualism. There existed a negative relationship between iPad price and luxury sales growth. These results indicated that the iPad served as a status symbol with higher demand in countries that promoted individualism and exhibited higher degrees of income inequality.