Below are attached questions,
Exercise 1 * * * 1.1 A population is divided into four groups, each one with four individuals. The individual incomes are: x1 = [1, 1, 2, 4]; x2 = [1, 1, 2, 4]; ra = [2, 2, 4, 8]; x4 = [4, 4, 8, 16] Calculate the two Theil measures of inequality, verifying that the between and the within group components are the same for both measures. Solution* *1.2 The individual incomes for three groups are given. In group 1, there are six individuals with incomes $11 = 012 = 0.5; 213 = 114 = 15 = 1; 016 = 8 In group 2, there are five individuals with incomes 121 = 122 = 123 = 024 = 1; 2:25 = 16 Group 3 has only three individuals and their incomes are Cal = 132 = 13 = 16 The three groups together constitute a total population of 14 individuals. Calculate the mean, median, mode, amplitude and variance of the income taking into account the 14 individuals. Calculate the Theil-T index related to the inequality in each group, the index related to global inequality and its within and between groups componentes. Do the same for the Theil-L index. Do the same for the Gini index.#1.3 Consider two populations divided into three stratum each. In population A, the 40% poorest have 10% of total income, the 40% of the middle have 40% and the 20% richest have 50%. In population B. the three stratum (40% poorest, 40% of the middle and 20% richest) have 20%, 20% and 60% of total income, respectively. We suppose there is no inequality within each stratum. Calculate the Gini index for each one of the two populations. Do the same for the Theil-T index and for the Theil-L indexes. Based on these results, verify in each one of the two populations the income distribution is more unequal. Comment the results taking into account the Lorenz curve for each popula- tion.1.5 IBGE (National Bureau of Statistics and Geography) has recently lauched the dual of the Theil index taking into account the working population with positive income using data from PNADs 2002 and 2003. Calculate the evolution (per- centual) of inequality using the dual of the Theil-T index taking into account the active age population (therefore PIA - 15 to 65 years old). 2002 2003 Variation Dual of the Theil-T 0.563 0.555 -1.42% % of PIA with zero income 0.507 0.519 2.43% Calculate the Theil-T index for PNAD 2003Exercise 2 2.1 Calculate the inequality indexes seen in the course (Theil-T, Theil-L, Gini and the duals) according to the following sample of incomes: 12 r = [1, 1, 2, 6, 30] If we add one person with zero income to the sample, how do the indexes change?2.2 Suppose that per capita income of household A, composed of only one individual, is 8. Suppose also that there is only another household in the economy, with incomes {1, 1, 2, 6, 30}. Calculate the level of inequality according to the following concepts: i) Household per capita income between households ii) Household per capita income between individuals iii) Calculate the inequality component of individual income between groups of households (i.e., A and B). Assume now that income of household A is 7. Recalculate it. iv) Suppose there is no socialization of incomes inside the households. How much of total income inequality is going to be underestimated taking into ac- count both scenarios?Exercise 3 3.1 - Empirical Estimates For the model In (Y() =a + 8 x X; + w;, we have the following estimate In (Y,) = 0.8972 + 0.1543 x X; (0.01768) (0.0497) R- = 0.4456 where Y = income from the main activity Xi = years of schooling The numbers in brackets are the standard errors of the estimates. Interpret the slope coefficient (give the formula), its significance and the R? of the regression.4.4 - Empiric Consider the labor decomposition of individual income taking into account dif- ferent sources: a. What is the rate of unemployment in the PIA (Active Age Population - 15 to 65 years old)? b. What is the fraction of the growth of the mean labor income in the PIA that is explained for the rise in occupation? c. If we assume a 0.5% per year growth of the PIA as a result of the recent demographic transition, what should be the growth of income from all sources? d. Compare the impacts in total income of the demographic bonus with the impacts of the rise in average years of schooling of the occupied (educational bonus)
