Measurement of Inequality:

Measurement of Inequality: Technical and Methodological Issues in the Case of Mexico. David Vázquez-Guzmán 1 University of Stirling. (Spring 2007) Abstract. Measuring inequality in a country, as relative deprivation, has been seen mostly in the grounds of normative vs. descriptive discussion. I argue that that might be only the starting point. In development literature, besides the traditional Gini coefficient, there are other types of measures that have been increasingly used, such as the Theil Index, which belongs to a wider family of Entropy measures. Different properties of these measures have to be considered in regards of the methodological challenge that poses to the researcher. After this, when the technical issues are explained, it can be clearer what among whom is calculated, for instance, household versus individual inequality might be considered, the same as the use between wage-income data and the one obtained from consumption-expenditure surveys. We can also consider deflation of prices that reflects changes in regional income distribution, which has been almost neglected in recent literature, while the inclusion of people with zero or even negative income is seldom mentioned. The data used comes from the Mexican Government (ENE, 1991-2003, INEGI) in order to test different assumptions measuring inequality, using also the spatial analysis with the Geographical Information Systems tools (GIS) to express some results. 1 Introduction. This paper is about practical perspectives in the calculation of inequality, considering different methodologies and types of measures. Having income as an available variable in the majority of surveys, we can focus on the measurement of [wage and money] income inequality as a proxy for real or social inequality. 2 I will touch some methodological problems doing this measurement, and I will calculate directly for survey data using different methods, in order to obtain a better understanding of inequality, having as an example the country of Mexico. 1 I am especially thankful to David Bell, Dipak Ghosh and Prasanta Pattanaik, and also appreciate the informal comments and remarks by Harminder Battu, Sheila Dow, Nick Hanley, Bob Hart, Azizur R. Khan, Robin Ruffel, Jaehee Son, and the attendees of the 2007 SES annual conference. All errors remain mine. I acknowledge the support of the Mexican Institute CONACYT. Technical and Methodological Issues. David Vazquez-Guzman. Page 1 of 43

For assessing the amount of social inequality due to economic changes, the measurement is usually done mostly by the calculation of the Gini coefficient, which is very common to do in developing countries. Share deciles are also common, and sometimes the division of some groups of population is considered, such as natural geographical regions, or divisions between urban and rural areas. It is obvious that volatility of economic factors is very different between developed and developing countries. In the case of Mexico, it happens that economic factors in recent decades are important in the explanation of the prevalent increase and decrease of inequality, such as the two major crises in 1982 and 1994-5 (For a short review see the Appendix 7.1, also Lustig, 2001 & Lustig and Székely, 1998). On the technical side, assuming that wage-money-income inequality works as a proxy for social inequality, there is a set of measures that can help us to explain levels of inequality in Mexico. I will calculate traditional Gini estimates, explaining common problems that normally the researcher has to face, such as considerations of different prices per region, or deciding between household and the individual as a different recipient unit for calculating inequality. I will also use some of the general entropy measures, such as the Theil index and the Logarithmic deviation, because of their convenient decomposable properties. It could be desirable to consider the measure of Atkinson (1970), mostly because of the failure of the Lorenz dominance criteria, 3 but I suggest that to be explored in future research. Modern computational programs, and faster available equipment, made convenient and easier to explore different combinations of methods that assess inequality the same as poverty. So the task of showing the voluminous information, to policy developers in a friendly way, is easier using suitable Geographical Information Systems tools (GIS). 1.1 Overview. 2 Some worries about this issue are found in early writings, such as those of Ahluwalia and Chenery, in Chenery et al. (1974). 3 It can be seen that some Lorenz curves intersect each other across time, for a discussion of this problem see part 3.7. Technical and Methodological Issues. David Vazquez-Guzman. Page 2 of 43

The organization of the paper is as follows: after this short introduction, the notations and measures used are briefly introduced in part 2. The bulk of the paper consists of methodological considerations in part 3, where several possibilities are calculated using available data, such as household vs. individual inequality, survey limitations, objective (descriptive) and normative measures, and similar things. In part 4, I show some results of individual inequality, using GIS tools. Finally, part 5 has the conclusions and directions of research. 2 Measures and other tools. One of the first technical decisions that the researcher has to face, in regards of computation of inequality, is the kind of data and the type of tool to measure it. I will roughly mention in this part the type of data sets, followed by the tools to measure inequality, so I will first introduce technical details of the data and software, later on some notations, and finally a review of the used measures, if the reader is familiar with the Gini coefficient and with the Entropy family of indices, this part can be skipped without problem. 2.1 Data and software. There are many data sets available for the researcher for the assessment of well being. It is different the decision if we talk about inequality rather than about poverty. For poverty assessment, in general there are not many decisions to take: if the researcher accounts with an expenditure survey, this is usually preferred because this is better to approach consumption. With expenditure data is possible to draw a poverty line according with a particular basket of basic goods. But for inequality assessment, there is another story. There are different traditions in regards of the use of different types of data. In developed economies, usually are chosen surveys with wage income data. In developing economies, sometimes are used expenditure surveys, in the case of Asian countries, but the tradition is different in others, such as the Latin-American ones. So, this varies in regards of the region of the world (the issue about income vs. consumption Technical and Methodological Issues. David Vazquez-Guzman. Page 3 of 43

survey data will be explored in more detail in part 3.3). For this study the data comes from an extended version of the National Survey of Employment (ENE Encuesta Nacional de Empleo), which is provided by the Mexican Institute of Information INEGI (Instituto Mexicano de Estadistica, Geografia e Informatica). It records data per household at individual level, and includes data not only on urban, but also rural communities from the years 1991-2003 (Details in Appendix 7.2.) Regarding computations and software, it is used STATA version 9.1 and routines based on the code developed by Stephen P. Jenkins (1999), and the graphic version of the Lorenz curves by Jenkins and Van Kerm (1999; 2001). For the graphic tools of Geographic Information Systems (GIS), it is used ARCGIS (ArcMap and ArcCatalog), Version 9.1. The geospatial map of Mexico was kindly provided by Glenn Graham Hyman (CIAT) from data developed in research of the Institute CIMMYT (Bellon et al. 2004). 2.2 Measures. Among the objective measures, 4 also called descriptive measures, there are several of this kind, such as the variance, the coefficient of variation, or the relative mean deviation, but those are not used very much by practical reasons or by methodological constraints. Instead, it is traditionally used the Gini coefficient, broadly mentioned in development literature, and recently explored by its various statistical properties, particularly for some sort of decomposability. 5 On the other hand, it has been increased the use of entropy measures, such as the Theil index or the Logarithmic deviation measure, which are both part of this family indices. Entropy measures are useful, mainly because of their decomposability by population sub-groups, but they have other limitations which are explained later on (see part 3.5). Besides this two kinds, in the paper are also used some traditional shares, such as deciles with their related ratios, but those do not need to much explanation, just some intuition. But before defining the formulae, are needed some notations that will be used in the rest of the paper. 4 For a long review of inequality measures see Cowell (1995, pp. 54, 66 &139, 2000), Sen (1973, Chapter 2), Anand (1983, Appendixes) or Dutta (2002). 5 For a survey of the Gini coefficient literature, see Xu (2004). Technical and Methodological Issues. David Vazquez-Guzman. Page 4 of 43

Notations I will introduce some notations here. Let I be any general inequality measure. Let n be the number of persons, where every person is indicated by i=1, n. Let y be the income vector, so y i is the income of person i. The number of groups within population are k=1, K., such that n = K n k k = 1. The average overall level of income is µ, such that n i= 1 y i = nµ, and the relative share of income of person i is xi, such that y i = nµ xi. Average and share group income are defined analogously for every group k. When x and y are given income distribution vectors, then I (y) and I(x) will be the correspondent degrees of inequality measures for those income distributions. The weight of a sample is wi n i w i f i= 1 N, such that N =, based on this so N = n ). w i = (when data is unweighted =1 i, w i I Objective measures. The best known objective type of measure is the Gini coefficient, developed more than a hundred years ago (Gini, 1910, 1912; Pareto, 1896, 1897; Lorenz 1905). It satisfies some intuitive criteria, such that if all the people have the same income, we have a perfect distribution. There are several ways to calculate this ratio, and different formulae give us a different intuition or welfare interpretation (Sen 1973, p. 31). A well known way to measure the Gini coefficient is the following: 1 (1) I = G = + [ y + 2y +... + ] G 1 ny n n n 2 1 2 µ where y y.... (Sen 1973, p. 31) 1 y n II Entropy measures 2 2 Technical and Methodological Issues. David Vazquez-Guzman. Page 5 of 43

Drawing on the notion of entropy in information theory, Theil (1967) developed an interesting type of measure, 6 which was later on generalized by Shorrocks (1980). 7 The family of measures for positive incomes (satisfying mean independence and population replication) given a particular value of c is such that: c n 1 1 y i (2) I GE( c) ( y) = 1, c 0, 1 n c( c 1) i= 1 µ n 1 µ (3) L = I GE 0) ( y) = log, c = n y ( 0 i= 1 i n 1 yi yi (4) T = I GE 1) ( y) = log, c = n µ µ ( 1 i= 1 where T is better known as the Theil measure, while L is the mean logarithmic deviation, also known as a Theil s second measure. (Shorrocks, 1980, p. 622; Sen, 1973, pp. 34; Foster and Sen, 1997, pp. 140 & 156). 3 Survey considerations. There has been a recent improvement in availability of data in developing countries, but not all the data is collected with the same methodology. If we know a little bit more about the data collection, it will also be easier to know its methodological basis, so those sets can be fully exploited, and used more properly. In the recent literature about income inequality, most of the discussion lies in the type of measurement that shall be used (e.g. Gini coefficient, Atkinson measure, etc.), but usually other kind of practical problems are not addressed when we run into computations. It is common that it is omitted in the analyses to make reference to operational procedures that are behind the comparative indicators and it is taken for granted as if all estimations were generated with 6 For the intuitive explanation of the entropy approach, see Theil (1996, Appendix A). Technical and Methodological Issues. David Vazquez-Guzman. Page 6 of 43

the same statistical quality, and all inequality calculation methods are provided from a standard algorithm (Medina, 2001, p. 29). Usually happens in developing countries, that there is no formal definition for the measurement of inequality, which is not the case for poverty measurement that is explored in more detail (World Bank, 2004, p. 8). Having this scenario, it is difficult the comparison of inequality measurement across papers, or the understanding of what among whom is calculated. 3.1 Regional Prices. Now the first practical issue. Price adjustment is not very common in development literature, even though it has been highlighted by some authors. That adjustment is not assessed sometimes, because of the unavailability of regional prices, and also because the changes due to those prices can be very small. Letting I being an inequality measure, as we previously defined, it will be scale invariant (or income homogenous, or mean independent) if it remains the same for proportional changes of income. Formally, I ( y, n) is homogenous of degree zero in the income vector y if and only if I( ky; n) = I ( y, n) k > 0 (Shorrocks, 1980, p. 621, 1984, p. 1372; Foster and Sen 1997, p. 139). This condition might be violated if the changes in incomes are not the same for the whole population, as happens when those changes affects differently according with the region. Therefore, in the presence of price changing (e.g. inflation), we have a change in y, but the change is not the same for all y i y, which usually is the case in reality. In this sense, De Ferranti et al. (2004) commented that If prices faced by all households were the same, the distinction would be irrelevant. However, prices usually differ by location. (p. 52). The intuition is explained as follows: considering a heterogeneous country in regards of its population and their distribution, which it is normally the case, if we consider this country as a whole without regional differences, our results can be biased, because the poor and the rich are not homogenously distributed within the country. We must recognize that changes in consumer goods prices will affect the purchasing power 7 Shorrocks credited Cowell for having a similar contribution (1980, p. 615, n. 5). Cowell credited himself as the source of this generalization in 1977, not to Shorrocks (Cowell, 2000, p. 110). Technical and Methodological Issues. David Vazquez-Guzman. Page 7 of 43

of the poor and of the rich in different ways. (Cowell, 1995, p. 100). Mexico is not an exception of the heterogeneous countries, so in order to solve the issue for this case, it happens that the Bank of Mexico publishes periodically a regional index (2006). The Bank used their own regional distribution based on different factors, such as the proximity of a municipality with respect to the U.S. border, state geographical closeness and density population (2002, p. 6), as we can see in Figure 1.8 Mexico: Regional Distribution (Bank of Mexico). Detail Regional Distribution (B of M) NREGION_B_ 1 Frontera Norte 2 Noroeste 3 Noreste 4 Centro Norte 5 Centro Sur 6 Sur 7 Area met. Cd. de Mexico Lake; Ocean Land 4 Detail, Central part of Mexico. 0 37.5 75 150 225 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2007). 175 87.5 0 175 350 525 Kilometers Figure 1. Regions (Bank of Mexico) In Figure 2 it can be seen a set of selected years of the Generalized Lorenz curves for the whole country, without price deflation. It is convenient to note that without transforming income into 2002 prices, it is not possible to compare real income distribution, neither to compare how economic conditions, during the crisis of 1994-5, 8 This distribution is useful for consideration of prices, but it might not be very much for other purposes. It has the inconvenience of mixing municipalities within states that belong to different regions, as is clearly noted in bordered states colored in gray in the map. Technical and Methodological Issues. David Vazquez-Guzman. Page 8 of 43

really affected the people s purchasing power. It can also be seen that differences across years are very despair, so we can be tempted to believe that, using the Generalized Lorenz Dominance criteria (discussed in part 3.7), the Lorenz curves do not intersect each other and can be ranked accordingly; maybe thinking that later years were better than previous in regards of income distribution. GLorenz(groups of yr) 0 1000 2000 3000 4000 0.2.4.6.8 1 Cumulative population proportion income[1993] income[1998] income[2003] income[1996] income[2001] Figure 2 Distribution of Income without price index consideration After the income transformation, the picture changes quite a bit in Figure 3. For instance, the curves changes order, and it is clear that the Mexican economy enjoyed higher real incomes before the 94-5 crisis, where the income distribution of 1993 was better off than other years. We can also note that, just recently, the distribution has becoming similar to the years before the cited crisis, because of the approaching of recent distributional curves (2001 and 2003) to the distribution before 1996. It still seems to be that curves do not intersect each other, but that implication will be discussed later on in part 3.7. By now, using the graphical Lorenz curves is clear that it will be better to consider levels of inequality with regional prices, because that allow us to compare the status of income distribution through time. Technical and Methodological Issues. David Vazquez-Guzman. Page 9 of 43

GLorenz(groups of yr) 0 1000 2000 3000 4000 2 Income def, cum population, raw income 0.2.4.6.8 1 Cumulative population proportion incomedef[1993] incomedef[1998] incomedef[2003] incomedef[1996] incomedef[2001] Figure 3 Distribution of Income in regards of regional price index. Still considering prices, we can carry out some regional computations. We know that most inequality measures are invariant to a change in prices, because they focus in relative variations of income (Sen 1973, p. 69; Foster and Sen, 1997, p. 139; Annand, 1983, p. 339), but in this case, the assumption of mean-independence can be violated, because of the prices that affect income in a different way in each state or region. This fact can almost be neglected in state comparisons (except in states in the border with the U.S.; detail in Appendix 7.4, Table 7). But in the assessment of regions shown in Table 1, 9 in the columns with differences, some values are different than zero. The regions that show values different from zero, is because of the small, but perceived change between the column of the deflated vs. the non deflated inequality index, due to the presence of price changing. 9 Regional classification of states is briefly presented in Appendix 7.3 Technical and Methodological Issues. David Vazquez-Guzman. Page 10 of 43

Gini Theil Log Dev. Region Deflated Non-def. Difference Deflated Non-def. Difference Deflated Non-def. Difference 2001 2001 % 2001 2001 % 2001 2001 % 1 0.4818 0.4817 0.0134% 0.4931 0.4929 0.0265% 0.4451 0.4449 0.0246% 2 0.4625 0.4626-0.0064% 0.4215 0.4216-0.0167% 0.4338 0.4341-0.0265% 3 0.4338 0.4338 0.0000% 0.3521 0.3521 0.0000% 0.3651 0.3651 0.0000% 4 0.5426 0.5425 0.0138% 0.5412 0.5411 0.0096% 0.6044 0.6040 0.0440% 5 0.5111 0.5112-0.0064% 0.5044 0.5046-0.0186% 0.5165 0.5166-0.0183% 6 0.4214 0.4215-0.0166% 0.3542 0.3544-0.0210% 0.3403 0.3407-0.0395% 7 0.4254 0.4253 0.0094% 0.3620 0.3618 0.0220% 0.3140 0.3139 0.0076% 8 0.4524 0.4524-0.0012% 0.3875 0.3875-0.0018% 0.3772 0.3772-0.0022% Table 1 Effect of regional prices in income distribution. 2001. My results are consistent with the available literature, where small, but perceived changes are present in experiments in Argentina and Chile (De Ferranti et al., 2004, p. 53). Therefore, it might be seen that results became comparable, and a little bit more accurate, so it can be useful to use income with some price index, as it is here the 2002 prices. It will have the advantage of making it easier for comparisons across time with a uniform base year. Summing up, inequality seems to be clearer when prices, if available, are considered. 3.2 Household vs. Individual Inequality. The next issue is the recipient unit. There are several reasons why inequality has been calculated mostly on a household basis. We might think that, historically, it was easier to deal with household inequality, because of the fewer observations on households than on individuals. In the past, it used to be a big constraint the computational capacity, given the rudimentary computation equipment, but nowadays, the availability of computing resources made easy to calculate both individual and household inequality in many ways. Medina said There exists in empirical work a debate about which should be the appropriate variable to evaluate concentration of income. In fact, the principal controversy arises at the moment to choose household income or per capita income. (2001, p. 22, emphasis in the original). He argues later that both variables can be used, depending on the objective of research. Of course that choosing one scope versus the other is not the same; this decision affects in the results obtained, so there is different information in this regards. Some argue that taking the individual as the recipient unit Technical and Methodological Issues. David Vazquez-Guzman. Page 11 of 43

made the family weight unnecessary, because all individuals are considered the same in society. At the end of the discussion, it will be a decision of the policy maker if h/she attaches more importance to one of the recipient units, either the individual or the household. However, it shall be explained by the researcher the kind of recipient unit used, and how it might be interpreted. But again, it will not be his judgment the final decision about the validity of the results. To make a comparison between household and individuals, it is necessary to do both calculations, as is shown in Figure 4 (Gini), Figure 5 (Theil) and Figure 6 (Log. deviation). Where household identification is available, it can be noted that Gini coefficient for households (yellow) is higher than the Gini coefficient for individuals (blue). 10 There are ambiguous results with Theil index, and again higher household inequality considering the measure of logarithmic deviation. It can be generally inferred that, ceteris paribus, household inequality is higher than individual inequality (but on Theil index), and to make clear the understanding of why inequality is like that, or if that is so, it will be necessary to assess it with weights that come from equivalizing scales. 11 It can also be shown, that using bootstrap techniques, the statistical properties of the inequality measures remains more or less similar, as it can be seen by the dotted lines that surround every inequality measure (some details in Appendix 7.4, Table 8). 10 The pink line is the individual income without the price transformation, but the perceived change is minimal in the graph, as was explained in part 3.1, so the blue and the pink lines are very close. 11 It is important the topic of equivalizing, but because it needs a theoretical explanation in regards of this data, I am doing it in other research. The first results about these equivalizing methods, using the Engel and Rothbarth method, the same as the Amsterdam scale, are that generally, household inequality remains somehow higher, but depends very much on the assumptions about how the family is considered. Technical and Methodological Issues. David Vazquez-Guzman. Page 12 of 43

0.51 0.5 0.49 0.48 0.47 0.46 0.45 0.44 0.43 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 Inequality per annum (GINI), different recipient unit. Source: Author's calculations based on ENE, 95%, bootstrap (100 rep.) Gini Income (Ct) Gini Inc. Def. (Ct) Gini Houhld.* (Ct) Gini Income (Ct) Gini Income (Ct) Gini Inc Def (Ct) Figure 4. Income Inequality in Mexico (Gini) 0.55 0.5 0.45 0.4 0.35 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 Inequality per annum (THEIL), different recipient unit. Source: Author's calculations based on ENE, 95%, bootstrap (100 rep.) Theil Income (Ct) Theil Inc. Def. (Ct) Theil Houhld.* (Ct) Theil Inc Def (Ct) Theil Inc Def (Ct) Theil Income (Ct) Figure 5. Income Inequality in Mexico (Theil Index) 0.53 0.51 0.49 0.47 0.45 0.43 0.41 0.39 0.37 0.35 0.33 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 Inequality per annum, different recipient unit. WHOLE country. Source: Author's calculations based on microdata (ENE). logdev Income (Ct) logdev Inc. Def. (Ct) logdev Houhld.* (Ct) Figure 6. Income Inequality in Mexico (Log. Deviation) Technical and Methodological Issues. David Vazquez-Guzman. Page 13 of 43

3.3 Income vs. Expenditure Surveys. It is both conceptual and practical the decision between using income or consumption as the indicators of welfare. The first arising problem is the difficulty to obtain a reliable quantification of consumption (CTMP, 2002, p. 34), which is usually approximated by the recorded expenditure in a household survey. Both expenditure and labor incomes are usually available in many surveys, but there are some basic constraints when we want to use them to make indicators of welfare. On the consumption s side, there are some practical issues in regards of its measurement. One can be the assumptions in regards of the basket of goods prices, and how those prices are considered endogenous or exogenous to the system. Other assumption about consumption can be, if it is needed an individual approach, the kind of weight attached to every individual within the household ( equivalizing ). There can be seen also the different tradition between Asian and non-asian countries in order to collect data: the former have a stronger tradition in consumption data (Deaton and Zaidi, 2002, p. 13). In regards of income data, we also have different things to consider, like the different kinds of income, where is usually privileged money-wage income over other sources, such as capital or asset income. We might consider also the more detailed income data in regards of regions, and of course, the problem of underreported and misreported income. In both sides, consumption and income, it should be considered the different kinds of fluctuations between them, and finally, that the use of monetary representations in this type of measurement has been considered narrow by the approach of Sen s capabilities. All the previous issues have different and ambiguous impact in the measurement of real inequality and real poverty, but this should not be a definitive obstacle to make an assessment on the size of the several possible biases. In order to make the policy maker aware of this discussion, it will be good to compare the results with other sources (Table 2). Looking at the results around the year 2000, in first instance, we can see a comparison with other source that uses a similar income survey (ENEU-ENET), so we can check our estimates of the inequality indices. The paper of Lopez-Acevedo (2006), for the year 2000, both Gini (0.4400) and Theil index (0.3990) are lower than our estimates of household inequality (0.4966, Gini; Technical and Methodological Issues. David Vazquez-Guzman. Page 14 of 43

0.4812, Theil), because of the different assumptions made on the calculations. 12. In second place, if we compare the results with other sources that uses consumptionexpenditure data, the reading of household inequality (.5460, Gini;.6160, Theil) is higher than our estimates. In the other years, either around 1990 or 1995, the results are similar. The readings of inequality using expenditure surveys, taking in account weights to individuals within a family (shown with *** and the tag equiv ) are a little bit smaller than the others without weights, and those might be compared with income individual inequality, but that is out of the scope of this paper, because it is needed to explain the theoretical assumptions in this regards, the same as is needed to do other kind of computations, which will be available in next research. Inequality Indices for the Distribution of Earnings, comparison of different sources and years. Bottom 20% Middle 40% M. High 30%. Top 10% Gini Theil R 10/20 1991 (Ind. Own)^ 4.3% 22.1% 35.5% 38.1% 0.4438 0.4777 8.92 1991 (Hou. Own)^ na na na na na na na 1992 (Hou. L-A 2006)* 6.5% 23.4% 33.5% 36.6% 0.4340 0.3960 6.74 1992 (Hou. DeF 2004)** 3.1% 18.3% 33.9% 44.8% 0.5590 0.6670 14.45 1992 (Hou. DeF 2004)*** (equiv) 3.4% 19.3% 34.1% 43.2% 0.5390 0.6120 12.71 1997 (Ind. Own)^ 2.2% 19.7% 34.5% 43.6% 0.4917 0.4795 19.72 1997 (Hou. Own)^ 1.7% 16.2% 38.8% 43.3% 0.5044 0.4726 25.82 1996 (Hou. L-A 2006)* 5.7% 22.1% 33.6% 38.6% 0.4640 0.4740 6.74 1996 (Hou. DeF 2004)** 3.2% 18.9% 34.6% 43.3% 0.5440 0.6160 13.53 1996 (Hou. DeF 2004)*** (equiv) 3.6% 19.8% 34.7% 41.8% 0.5250 0.5710 11.61 1999 (Ind. Own)^ 3.7% 22.5% 36.0% 37.8% 0.4804 0.4916 10.14 1999 (Hou. Own)^ 2.9% 21.2% 39.8% 36.1% 0.4966 0.4812 12.27 2000 (Hou. L-A 2006)* 5.9% 23.3% 34.5% 36.2% 0.4400 0.3990 6.11 2000 (Hou. DeF 2004)** 3.1% 18.9% 34.9% 43.1% 0.5460 0.6160 13.90 2000 (Hou. DeF 2004)*** (equiv) 3.4% 19.7% 35.3% 41.5% 0.5270 0.5580 12.21 2002 (Ind. Own)^ 2.8% 21.3% 38.1% 37.8% 0.4638 0.4169 13.56 2002 (Hou. Own)^ 3.7% 21.3% 38.3% 36.7% 0.4834 0.4242 9.89 2002 (Hou. L-A 2006)* 6.7% 25.5% 35.5% 32.3% 0.3960 0.3020 4.81 * Lopez-Acevedo (2006), (ENE) ^ Own calculations based on ENE (income survey). ** De Ferranti et al. (2004) based on ENIGH (expenditure survey). *** De Ferranti et al., (equivalized income.) Table 2 Comparisons between surveys, inequality indices and deciles. This table also shows some other measures, such as deciles and shares, and it can also be clear that it might be little different the inequality levels, mostly in regards of the different methodological assumptions made. 12 She dropped not only observations with zero income but very high incomes because those were considered unreliable. Technical and Methodological Issues. David Vazquez-Guzman. Page 15 of 43

3.4 Population Sub-groups. When the decision comes about how inequality has to be shown, there are many possibilities considering population sub-groups, and for the researcher it might be a problem to make alone this decision. The sometimes endless possibilities can be, for instance, inequality among social groups (i.e. traditional caste systems, or groups divided by income level such as deciles), inequality that considers gender or sexual orientation groups, inequality among different geographical areas (e.g. urban and rural), or indices by state or some pre-classified regional or political distribution within the country. Some people can be strongly opinionated in favor of one of the divisions, but practically, it is not possible to show all combinations of the mentioned inequality. There can be several reasons to believe that one classification is better than the other, and it might be also believed in the convenience to show those results in a particular way. The fact is that people, affected by their beliefs or concerns, can argue in favor of traditional or others ways to express these results. For Mexico, the tradition might lie in the classification by regional groups, either by location or by urban and rural areas. I acknowledge that in other regions of the world, the interest can lie sometimes either in the gender classification (because of possible female discrimination), or sometimes in social groups classified by income level (i.e. deciles). The gap between urban and rural areas is recognized in Latin-American literature. In spite of this, in Mexico, for instance, when is controlled for observable characteristics, the size of gender inequality is narrower than in the rest of the Latin- American region (De Ferranti et al., 2004, p. 62) and sometimes, with surprising reverse results in favor of females (Idem, p. 75, n. 60). Other development literature, in regards of poverty measurement, tests for gender differences, finding no statistical significance in favor of males or females (Teruel et al., 2005, p. 22), and finally, it is also known the case where Mexican females have different understanding of what is development, such that they challenge the traditional theory that supposedly make them always better of when, for instance, they get a paid job outside of their house, or they enroll in contraceptive method programs promoted by the state (Nazar-Beutelspacher et al. 2005, p. 236). Technical and Methodological Issues. David Vazquez-Guzman. Page 16 of 43

For the case of Mexico, in the above mentioned literature, it usually matters the geographical differences, and also the gap between urban and rural areas. In the Figure 7 (Gini), we can see that inequality is consistently higher in rural areas, while happens the opposite to urban areas. Total inequality remains in the middle of both. We can also observe higher dispersion (using the bootstrap technique) in years where economic or social shocks happened, as was the 94-5 crisis, or the year 1999, that was previous to the presidential election of 2000. If we compare the graph of the Individual data (left), with the one with the household data (right), the numbers are similar, just the dispersions appears to be smaller in the household data. The Figure 8 shows the same, but using the Theil index. 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 INDIVIDUAL data. 95% confidence interval (bootstrap, 100 rep.) Source: Author's calculations based on microdata (ENE). Gini Inc. Def. (Ct) Gini Inc. Def. (U) Gini Inc. Def. (Ru) Gini Inc Def (Ct) Gini Inc Def (Ct) Gini Inc Def (U) 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 HOUSEHOLD data. 95% confidence interval (bootstrap, 100 rep.) Source: Author's calculations based on microdata (ENE). Gini Houhld.* (Ct) Gini Houhld.* (U) Gini Houhld.* (Ru) Gi i H hld * (Ct) Gi i H hld * (Ct) Gi i H hld * (U) Figure 7 Household and Individual Inequality (Gini). Total, Urban & Rural areas. 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 INDIVIDUAL data. 95% confidence interval (bootstrap, 100 rep.) Source: Author's calculations based on microdata (ENE). Theil Inc. Def. (Ct) Theil Inc. Def. (U) Theil Inc. Def. (Ru) Theil Inc Def (Ct) Theil Inc Def (Ct) Theil Inc Def (U) 0.3 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 HOUSEHOLD data. 95% confidence interval (bootstrap, 100 rep.) Source: Author's calculations based on microdata (ENE). Theil Houhld.* (Ct) Theil Houhld.* (U) Theil Houhld.* (Ru) Th il H hld * (Ct) Th il H hld * (Ct) Th il H hld * (U) Figure 8 Household and Individual Inequality (Theil). Total, Urban & Rural areas. Technical and Methodological Issues. David Vazquez-Guzman. Page 17 of 43

3.5 Income with zero values. There is in the literature some concern about the existence of negative or zero incomes for both methodological issues and computational problems. (Cowell, 1995, p. 155). I will just mention the case of zero income, the negative one is out of my scope. The fact that sometimes we have a zero in the data does not mean that that person represented in that observation shall not be considered, but the opposite. The problem arises if we think that those zero observations might be ignored, as usually happens in practical computations. When the functional form that considers the income distribution is assumed in some kind of logarithmic representation (e.g. T or L measure), this transformation becomes a problem. It is known that logarithms have other convenient properties, as it is to consider in a similar way high and very high values, but to explain with more detail what happen in the lower level of the distribution. This property is convenient when we care more about the poor than about the rich. 13 However, we have a problem for the non-definition of those observations with zero values with this kind of transformation: those observations might not come from underreported income, but they might be the most destitute people, and it would be desirable that somehow should be taken into account when we measure inequality (Anand 1983, p. 308). More than a technical problem, the most important issue might be from ethical nature. This small problem can be considered crucial sometimes, such that the kind of measures that do not have this weakness might be considered better in general sense: This [Gini] gives it a decided advantage [accommodate non positive incomes] over the Atkinson and Generalized Entropy family since these are only defined for positive income. (Dutta, 2002, p. 615, [comments added]). In the available literature there are attempts to identify those individuals and to impute them some simulated income according to their characteristics, taking instead the average estimates from the income of similar people. This can also be made not only with people with zero income, but also with people with missing observations. (Székely et al., 2000). 13 The fact that a logarithmic transformation staggers the income levels tends to soften the blow in reflecting inequality since it reduces the deviation, but on the other hand it has the property as noted before of highlighting differences at the lower end of the scale. (Sen, 1973, p. 29). Technical and Methodological Issues. David Vazquez-Guzman. Page 18 of 43

For our purpose, I will test the solution that is mentioned by Anand (1983), assigning some small positive value to those individuals who are willing to work, but they have no income because they are unemployed (Anand 1983, p. 308). It will be shown that, among other things, standard ranking tools (e.g. deciles) suffer some changes after that consideration. Therefore, if we consider individuals who are willing to work, but have no job, and because of that, no income, then the Gini coefficient, as expected, changes around 0 to +2%. This can be shown in the column of differences, located at the middle and bottom part of Table 3. In other columns we can see the effect of the change on indices with logarithmic transformations, where the impact is different. If values are set up equal to 1 instead of the zero income, the Theil index increases from 1% up to 5%. We can argue that the higher increase of inequality expressed by the Theil index is not surprising, if we consider a higher proportion of unemployed people during the years following the 94-95 crisis, when they were more affected. The rest of the logarithmic measures, as Log. Deviation, shows a higher impact, and much more when income is equalized to 1 E-10, making it difficult to interpret those results. In the last case, Theil and Gini remain almost the same as previously shown. Technical and Methodological Issues. David Vazquez-Guzman. Page 19 of 43

Original, no individuals with zero income yr Log Dev Theil Gini 1991 0.32884 0.41812 0.43652 1993 0.32371 0.37999 0.43386 1995 0.49434 0.68378 0.50842 1996 0.44345 0.48935 0.48683 1997 0.45828 0.47955 0.4917 1998 0.45823 0.47712 0.48794 1999 0.44459 0.49157 0.48041 2000 0.45588 0.47981 0.4872 2001 0.45665 0.45887 0.4801 2002 0.43505 0.41692 0.46379 2003 0.41414 0.39138 0.45429 Calc. when indiv. with no income are changed to 1 Differences yr Log Dev Theil Gini Log Dev Theil Gini 1991 0.32884 0.41812 0.43652 0.00% 0.00% 0.00% 1993 0.32371 0.37999 0.43386 0.00% 0.00% 0.00% 1995 0.63829 0.70583 0.51917 29.12% 3.22% 2.11% 1996 0.57133 0.50924 0.49696 28.84% 4.06% 2.08% 1997 0.55198 0.49413 0.49907 20.45% 3.04% 1.50% 1998 0.52156 0.48688 0.49293 13.82% 2.05% 1.02% 1999 0.49072 0.49866 0.48409 10.38% 1.44% 0.77% 2000 0.5033 0.48697 0.49087 10.40% 1.49% 0.75% 2001 0.49983 0.46536 0.48347 9.46% 1.41% 0.70% 2002 0.47883 0.42346 0.46729 10.06% 1.57% 0.75% 2003 0.46844 0.39946 0.45869 13.11% 2.06% 0.97% Calc. when indiv. with no income are changed to 1 E-10. Differences yr Log Dev Theil Gini Log Dev Theil Gini 1991 0.32884 0.41812 0.43652 0.00% 0.00% 0.00% 1993 0.32371 0.37999 0.43386 0.00% 0.00% 0.00% 1995 1.1419 0.7059 0.51918 130.99% 3.23% 2.12% 1996 1.0264 0.50931 0.49697 131.46% 4.08% 2.08% 1997 0.88658 0.49418 0.49908 93.46% 3.05% 1.50% 1998 0.74605 0.48691 0.49294 62.81% 2.05% 1.02% 1999 0.65396 0.49868 0.48409 47.09% 1.45% 0.77% 2000 0.66816 0.487 0.49087 46.56% 1.50% 0.75% 2001 0.64921 0.46537 0.48347 42.17% 1.42% 0.70% 2002 0.62934 0.42348 0.46729 44.66% 1.57% 0.75% 2003 0.65423 0.39948 0.45869 57.97% 2.07% 0.97% Table 3. Effect of zero income in levels of inequality. Now we can have in mind the size of the effect of those destitute people in our measures, considering that the difference has a visible impact. It can be used only positive incomes (for comparability purposes among measures), hoping that the consideration of the unemployed people, and their inclusion in the calculation of inequality, can be explored in detail with other measures in later research. The ad-hoc assignation of a small positive value proved to be useful to shed some light, but the extent and the size of that value remains unclear. Technical and Methodological Issues. David Vazquez-Guzman. Page 20 of 43

3.6 Survey coverage. The principle of population replication (or independence of population), provides an extension of the basic theorem of Lorenz dominance, because it allows the comparability of inequality statements when the size of the population is not the same, which it is regularly the case in reality (Dutta, 2002, p. 610). This principle also permits that our inequality measures, either coming from different regions or from other time periods, can be comparable if we assume some sort of equivalence between the same population over different periods of time. Formally, the principle of population replication says about the inequality measures something like this: I( y; r n) = I( y, n) r > 0 r {Integers} (Shorrocks, 1980, p. 619). 14 The replicator r works as an arbitrary multiplier of the size of the population, and this convenient assumption makes it possible to compare inequality measures across regions, or to do the comparison between populations across time. But the above axiomatic generalization is one matter, and the other matter is the trust that policy makers can have when they consider the limitations of the survey coverage. Sometimes they assume that all survey coverage are the same across periods and regions, and ignore that this assumption is not totally correct. They might assume the correct use of the statistical methods to define samples with a proper distribution, ignoring for instance about the possible budget limitations of the survey institution, just to mention one of the problems. It shall be convenient to explain to them the assumptions made when we calculate inequality, so results can be taken in this regards to make our comparisons reliable, letting the judgment to policy makers about the generalization of our results as well. In the case of Mexico, it is stated in the government institution s procedures that all survey projects should consider, among many things, sample size and coverage according to the budget (INEGI 2005, p. 16). Usually, in almost all surveys we can find a detail of the generation of the sample, its size, and the methodology used to ensure that 14 Generally I Rr y) I( y, y,..., y) = I( y) y Y, r > 0 r { Integers} ( 1 where R r is a replicator matrix of dimension n ( y) rn( y) with the form R r = [ E, E,..., E] for some identity matrix E. (Shorrocks 1984, p. 1369). Technical and Methodological Issues. David Vazquez-Guzman. Page 21 of 43

the sample is representative of the population.15 It is not very common to find in those explanations about the detail of the budget constraints, or sometimes technical issues that limited either the size of the sample, or the data collection in certain areas. When our inequality measures take into account the principle of population replication, it is making a convenient, but a strong assumption. In the first two boxes of Figure 9, we can see that survey coverage was limited, that is explained by the empty spaces that correspond to the non-surveyed municipalities. Later on, we can see that little by little, most of the country was represented by the sample. All comparative results of inequality should be taken in this regards. Mexico: ENE Coverage. (1991) Mexico: ENE Coverage. (1995) Detail Detail MEXBDY MEXBDY Lake; Ocean Lake; Ocean Land <all other values> Land <all other values> 4 4 Detail, Central part of Mexico. 0 37.5 75 150 225 Detail, Central part of Mexico. 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2006). 190 95 0 190 380 570 0 37.5 75 150 225 Kilometers 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2006). 180 90 0 180 Detail Detail MEXBDY MEXBDY Lake; Ocean Lake; Ocean Land <all other values> Land <all other values> 4 4 Detail, Central part of Mexico. Detail, Central part of Mexico. 37.5 75 150 225 540 Kilometers Mexico: ENE Coverage. (2003) Mexico: ENE Coverage. (2000) 0 360 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2006). 180 90 0 180 360 540 Kilometers 0 37.5 75 150 225 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2006). 180 90 0 180 360 540 Kilometers Figure 9 Geographical coverage of ENE survey. In order to know how these possible new samples have been changing across time, and if that has been affecting the size or the dynamics of real inequality, I calculated the inequality coefficients for the same years, but considering just the 215 municipalities 15 Some technical details are found similar in INEGI (2000 and 2003). Technical and Methodological Issues. David Vazquez-Guzman. Page 22 of 43

that were included in the first sample of 1991. Figure 10 shows the similar, but lower inequality of the new computations, based only on those municipalities. In that sense, it can be clear that the new regional coverage of the survey, had been allowing a better understanding of the dynamics of inequality, and might be inferred that there is a (small) positive relationship between the area covered, and the levels of inequality. In that sense, our axiom of independence of population does not hold perfectly for the case of Mexico, and the small sample (215 munic.) across time have also a higher variance, using the bootstrap technique. The results for the household inequality are similar, and are omitted for obvious reasons. 0.51 0.55 0.5 0.49 0.5 0.48 0.47 0.45 0.46 0.45 0.4 0.44 0.43 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 0.35 1991 1993 1995 1996 1997 1998 1999 2000 2001 2002 2003 INDIVIDUAL Ineq. different samples. Whole (yel.), small (Blue), GINI INDIVIDUAL Ineq. different samples. Whole (yel.), small (Blue), THEIL Source: Author's calculations based on microdata (ENE). Source: Author's calculations based on microdata (ENE). Gini Inc. Def. (Ct) Gini Inc Def (Ct) Gini Inc. Def. (Ct) Gini Inc Def (Ct) Theil Inc. Def. (Ct) Theil Inc. Def. (Ct) Theil Inc Def (Ct) Theil Inc Def (Ct) Figure 10. Comparison of different samples, Gini and Theil. 3.7 Lorenz dominance. It is common to see in papers that include measures of inequality, different size of those inequality measures, which sometimes might lie very close to each other. In that case, it is difficult to look only into the numbers and assess which inequality level was better than the other. We can use other tools, mostly graphical, that can help us to see how much inequality there is and how is it. We can see not only different distributional diagnostic plots, with different assumptions in regards of the form of the distribution function, but also we can check the aforementioned Lorenz curves and their related theorems. So I will talk briefly about the latter. Unfortunately, the extent of the theorems about Lorenz dominance and its generalization does not solve an issue, which is that the Technical and Methodological Issues. David Vazquez-Guzman. Page 23 of 43

income distributions represented by the Lorenz curves, which we are interested to compare, often intersect. If we are rigorous, we can neither say that we prefer some distribution over another, nor that we are indifferent, simply we can not rank them (being x and y suitable distribution vectors, happens that x y ) (Cowell, 2000, p. 107). In empirical analysis, it is true that when Lorenz curves intersect, it is possible to apply the Generalization of the Lorenz criteria, and sometimes the problem disappears (Cowell, 1995, p. 43). But if that is not the case, it will be necessary to apply methods that consider some form of the Social Welfare Function, as it does the Atkinson family of indices. In the case of Mexico, the distribution of traditional Lorenz curves appears to be as if the more equal distributions Lorenz-dominate the unequal, such as for almost all years the curves do not intersect each other, as is seen in Figure 11. In this figure it is explained that, for the selected years, only the distribution that clearly intersects with others is the one corresponding to 1993, but after the crisis of 94-5, most of the distributions have a particular shape that make it possible to make the comparison among them, 16 without the necessity, in the first instance, of using measures that attach a particular form of utility function U. 17 It can be argued that curves are very close each other, that is because of the fact that the Gini estimates are in the range between 0.443~0.504, and shall be bear in mind that for some of the readers this dominance can not be seen very clear. 16 The purpose of selecting only some years in this exercise was because of clarity. Showing all years in one graph, even tough the same holds, make difficult to asses Lorenz dominance. 17 Not all the authors recognize that U is a utility function, but somehow there is the agreement that that is a representation of the individual s utility (Sen, 1973, p. 39). Technical and Methodological Issues. David Vazquez-Guzman. Page 24 of 43

Lorenz(groups of yr) 0.2.4.6.8 1 0.2.4.6.8 1 Cumulative population proportion 5 income incomedef[1993], lorenz curves incomedef[1998] incomedef[2003] incomedef[1996] incomedef[2001] Figure 11 Lorenz curves whole country across time. Knowing that the original Lorenz dominance criterion is limited to populations with the same mean and the same size, which is not the case in the above graph, we can use the extended version of the theorem characterized by the Generalized Lorenz dominance, previously seen in Figure 3 in part 3.1. Sometimes the problem can disappear scaling the curves by the mean income (Dutta, 2002, p. 610). Recalling that Figure 3, the curves appear neater than before, but still some intersection of curves remains in the bottom part of the scale. In future research this will be considered by SWF measures, the same as the comparison of regions and states, because of the heavy intersection of Lorenz curves across regions. Therefore, interregional or intertemporal comparison just in regards of Gini coefficient is difficult, and it will be desirable to use some of the suitable measures (e.g. Atkinson measure) in this case. 4 Individual inequality: some results. All the burden of methodological considerations is in vain if there are no results. Now it is more possible to show friendly results with the availability of modern equipment and software. There is a recent growing trend in poverty mapping, because Technical and Methodological Issues. David Vazquez-Guzman. Page 25 of 43

of the The plotting of such information on maps poverty mapping is useful to display information on the spatial distribution of welfare and its determinants. It is also useful to display simultaneously different dimensions of poverty and/or its determinants. (World Bank, 2007). The so called spatial analysis allow us to process in short time a lot of information. Now that we consider one of the limitations of expenditure surveys, as is their level of aggregation, we need something else to do a proper spatial analysis. It will be desirable to account with data in the level of aggregation of census data, but usually census data do not focus in income sources or expenditure details, and the time period is very long between one survey and the other. Expenditure surveys have a lot of detail in consumption, but the number of observations is very limited because they have other focuses, such as the assessment of poverty at a country level. The graphics of Gini coefficient (Figure 12) and Theil Index (Figure 13) at municipality level are consistent with some stylized facts, as is the high inequality in country side regions, or some regions with higher inequality in states with big industrial corporations, or in tourist areas with high investment. The analysis and the interpretation of the inequality maps are left to the expert reader. For the case of Mexico, will be very difficult to show the almost 2500 municipalities in a table, but in an inequality map is possible to do that. Now is clear that inequality is different across regions (regardless of the missing municipalities with no data) because of the country heterogeneity, and their explanation will be a matter of future research. Technical and Methodological Issues. David Vazquez-Guzman. Page 26 of 43

Mexico: Levels of income inequality (2003). Detail 2003, municipium level G_2003 0.10937-0.30000 0.30001-0.40000 0.40001-0.50000 0.50001-0.60000 0.60001-1.00000 Regional Distribution NREGION 1 Center (Centro) 2 Semi-Center (Semi-Centro) 3 West (Occidente) 4 South (Sur) 5 Gulf-Peninsula (Golfo Peninsular) 6 North (Norte) 7 Northest (Noreste) 8 Northwest (Noroeste) Lake; Ocean Land Detail, Central part of Mexico. 0 37.5 75 150 225 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2007). 4 170 85 0 170 340 510 Kilometers Figure 12 Gini per municipality (2003). Individual inequality in Mexico. David Vazquez-Guzman. Page 27 of 43

Mexico: Levels of income inequality (2003). Detail 2003, Theil T_2003 0.02680-0.14361 0.14362-0.27675 0.27676-0.39638 0.39639-0.57776 0.57777-1.35942 Regional Distribution NREGION 1 Center (Centro) 2 Semi-Center (Semi-Centro) 3 West (Occidente) 4 South (Sur) 5 Gulf-Peninsula (Golfo Peninsular) 6 North (Norte) 7 Northest (Noreste) 8 Northwest (Noroeste) Lake; Ocean Land Detail, Central part of Mexico. 0 37.5 75 150 225 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2007). 4 170 85 0 170 340 510 Kilometers Figure 13 Theil index per municipality (2003). Individual inequality in Mexico. David Vazquez-Guzman. Page 28 of 43

5 Conclusions. I think is the responsibility of the researcher to make clear the assumptions when inequality is measured, so the policy maker or the people in society can have not only the results, but the considerations behind the story of inequality. There are two sides in almost all the coins when we measure inequality. The so called objective vs. normative approach, with the consideration of entropy measures proved to be valuable, but it can not be asserted that any of those is better than the other, they simply just have different properties. For instance, if the focus is the property of decomposition by population subgroup, entropy measures are better suited, while if the purpose is comparability of results with other research, Gini estimates might be better. About regional prices, if those are available, those shall be better used. The epistemological focus of the individual vis-à-vis the nuclear family, means the household, have many shadows. The expression of inequality with individuals as the recipient unit is the objective of almost all research, but household interaction can not be ignored. We know that surveys still have many methodological problems to get properly this information, and sometimes this can be confused with the expenditure vs. the income approach. We know that usually the expenditure survey has the household as a basic unit, while the income surveys usually do having individuals as a basis. Despite of this, the individualhousehold and the expenditure-income discussions should be treated separated, and the audience should be informed about these dichotomies, the same as the effects on the results presented. The consideration of income with zero values, is more than a technical problem for some measures, and becomes an ethical problem because of the de facto exclusion of the representation of the most destitute in our society. About other properties widely used, there are sometimes very strong assumptions when calculations are made. That is the case of the principle of population replication that allows the comparisons of populations across regions and through time. So, even convenient, that property should be considered in regards of the different times when we deal with population, because not all the times that property holds. The same happens with the theorem of Lorenz dominance and its generalization. In presenting the results, the convenient tools of spatial analysis (GIS) allow the researcher to shed some light over the distribution of income. A final and Individual inequality in Mexico. David Vazquez-Guzman. Page 29 of 43

conclusive remark is that it is better if the researcher is aware of the challenges posed by the different methodologies and techniques, so the reader might be informed about the implications and assumptions made, so his or her judgment can have a better basis when they make decisions. Individual inequality in Mexico. David Vazquez-Guzman. Page 30 of 43

6 References. [1] Anand, Subdhir. (1983). Inequality and Poverty in Malaysia, measurement and decomposition. Published for the World Bank. Oxford University Press. [2] Arrow, K. J. (1965). Aspects of the Theory of Risk-Bearing. Yrjö Jahnssonim Säätiö, Helsinki [3] Atkinson, A. B. (1970). On the Measurement of Inequality. Journal of Economic Theory. No. 2. pp. 244-263. [4] Bank of Mexico. (2002). El Indice Nacional de Precios al Consumidor: Caracteristicas y Actualización de su Base al Año 2002. Banco de Mexico. (http://www.banxico.org.mx/inpc/pdf/inpc2.pdf). [5] Bank of Mexico. (2006). Indices de precios. Online resource available in (http://www.banxico.org.mx/polmoneinflacion/estadisticas/indicesprecios/indicespre ciosconsumidor.html) [6] Bellon, M. R., Hodson, D. P., Martinez-Romero E., Montoya, Y., Becerril J., and White, J. W. (2004). Geospatial Dimensions of Poverty and Food Security - A Case Study of Mexico. CIMMYT project report. (International Maize and Wheat Improvement Center -Centro Integral de Mejoramiento de Maiz y Trigo). (http://www.cimmyt.org/gis/povertymexico). [7] Chenery, H., Ahluwalia, M. S., Bell, C. L. G., Duloy, J. H., Jolly, R. (1974). Redistribution with Growth. Published by the World Bank and the Institute of Development Studies (University of Sussex). Oxford University Press. [8] Cowell, F. (1995). Measuring Inequality. Second Edition. Prentice-Hall/ Harvester. London. [9] Cowell, F. (2000). Measurement of Inequality, in the Handbook of Income Distribution. Atkinson and Bourguignon eds., (2000). North Holland, Amsterdam. [10] CTMP -Comité Técnico para la Medición de la Pobreza. (2002). Medición de la Pobreza: Variantes Metodológicas y Estimación Preliminar. Serie Documentos de Investigación. México: Secretaria de Desarrollo Social (SEDESOL). [11] Deaton, A., Zaidi, S. (2002). Guidelines for Constructing Consumption Aggregates for Welfare Analysis. World Bank Livin Standars Measurement Study Working Paper 135. Washington, D. C. [12] De Ferranti, D., Perry, G. E., Ferreira, F. H. G., Walton, M.. (2004). Inequality in Latin America: Breaking with History?. The International Bank for Reconstruction and Development/The World Bank. Washington, D. C. Individual inequality in Mexico. David Vazquez-Guzman. Page 31 of 43

[13] Dutta, Bhaskar. (2002). Inequality, Poverty and Welfare. In Handbook of Social Choice and Welfare, Vol. 1. Arrow, K.J., Sen, A.K., and Suzumura, K., eds. (2002). Elsevier, Amsterdam. [14] Foster, J., Greer, T.; Thorbecke, E. (1984), A class of decomposable poverty measures, Econometrica, Vol.52. [15] Foster, J. E., Sen, A.K. (1997). On Economic Inequality after a quarter century, in A.K. Sen, ed.. On Economic Inequality. 2 nd edition. Clarendon Press. Oxford. pp. 107-219. [16] Gini, C. (1910). Indici di concentrazione e di dependenza. Atti della III Riunione della Societa Italiana per il Progresso delle Siencze, in Gini (1955), 3-120. [17] Gini, C. (1912). Variabilita e mutabilita. Studi economi-giuridici, Universita di Cagliari III, 2ª, in Gini (1955), 211-382. [18] INEGI (2000). Documento metodológico de la Encuesta Nacional de Empleo Urbano. México. (www.inegi.gob.mx). [19] INEGI. (2003). Diseño Muestral de la Encuesta Nacional de Empleo Urbano. México. (www.inegi.gob.mx). [20] INEGI (2005). La administración de proyectos estadísticos. Internal discussion document. Series: Lineamientos para la generación de estadística básica. (www.inegi.gob.mx). [21] Jenkins, Stephen P. (1999). Analysis of income distributions. Stata Technical Bulletin, No. 48, sg104. [22] Jenkins, Stephen P., Van Kerm, Philippe. (1999). Generalized Lorenz curves and related graphs. Stata Technical Bulletin, sg107. [23] Jenkins, Stephen P., Van Kerm, Philippe. (2001). Generalized Lorenz curves and related graphs: an update for Stata 7. The Stata Journal, Vol 1, No. 1. pp. 107-12. [24] López-Acevedo, G. (2006). Mexico: Two Decades of the Evolution of Education and Inequality. World Bank Policy Research Working Paper 3919. The World Bank. Washington, D. C. [25] Lorenz, M.O. (1905). Methods for measuring concentration of wealth. Journal of the American Statistical Association, Vol. 9. [26] Lustig, Nora. (2001). Life is not Easy: Mexico s Quest for Stability and Growth. The Journal of Economic perspectives, Vol. 15, No. 1 (Winter, 2001), 85-106 Individual inequality in Mexico. David Vazquez-Guzman. Page 32 of 43

[27] Lustig, Nora and Miguel Szekely. (1998). Economic Trends, Poverty and Inequality in Mexico. Washington, D.C.: Inter-American Development Bank. [28] Medina, Fernando. (2001). Consideraciones sobre el índice Gini para medir la concentración del ingreso. Estudios Estadísticos y Prospectivos, Serie 9. CEPAL- ECLAC, Naciones Unidas. Santiago de Chile. [29] Nazar-Beutelspacher, A., Zapata-Martelo, E., Vázquez-García, V. (2005). Does Contraception Benefit Women? Structure, Agency, and Well-Being in Rural Mexico. In Amartya Sen s work and ideas, Edited by Bina Agarwal, Jane Humphries and Ingrid Robeyns. Routledge. New York. [30] Pareto, V. 1896. Ecrits sur la courbe de la repartition de la richesse. In Oeuvres completes de Vilfredo Pareto, ed. Giovanni Busino, Geneva: Librarie Droz, 1965 [31] Pareto, V. 1897. Cours d economie politique. Ed. G.H. Bousquet and G. Busino, Geneva: Librarie Droz 1964. [32] Sen, A. K. (1973). On Economic Inequality. Second edition (1997). Clarendon Press. Oxford. [33] Sen, Amartya, K. (1999). Development as Freedom. Oxford University Press. Oxford. [34] Shorrocks, A. F. (1980). The Class of Additively Decomposable Inequality Measures. Econometrica. Vol. 48. No. 3. pp. 613-26. [35] Székely, M., Lustig, N., Cumpa, M., Mejía, J.A. (2000). Do We Know How Much Poverty There Is? Research Working Paper #437, Inter-American Development Bank. Washinghton, D.C. [36] Teruel, G., Rubalcava, L., Santana, A. (2005). Escalas de equivalencia para México. Series: Documentos de Investigación. Secretaria de Desarrollo Social. Mexico, D. F. [37] Theil, H. (1967). Economics and Information Theory. North-Holland, Amsterdam. [38] Theil, H. (1996). Studies in Global Econometrics. Kluwer Academic Publishers. Dordrecht, The Netherlands. [39] The World Bank. (2004). Poverty in Mexico: An Assessment of Conditions, Trends and Government Strategy. By Gladys López Acevedo and Michael Walton, of the Poverty Reduction and Economic Management Division. [40] The World Bank. (2007). Poverty Mapping, in The World Bank website. (http://web.worldbank.org/wbsite/external/topics/extpoverty/0,,men upk:336998~pagepk:149018~pipk:149093~thesitepk:336992,00.html). Individual inequality in Mexico. David Vazquez-Guzman. Page 33 of 43

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7 Appendix. 7.1 The Mexican Economy in the last decades. In order to be familiar with the results of calculations, sometimes it is better to understand economic and social conditions. Not only endogenous and exogenous economic conditions, but also political shocks are responsible for economic (negative and positive) growth, and therefore, for the increase and decrease in levels of income inequality. Being far away from the import-substitution model that used to rule from 1930 until the early 70 s, the most important issues in the Mexican economy in the last 30 years are two major crises (Lustig, 2001). The first one in 1982, as a consequence of a debt crisis, and the second one, in 1994, due to the inability and bad luck of the Mexican government to make a smooth transition between the administration of President Salinas de Gortari (1986-1994) and the administration of President Zedillo (1994-2000). The first crisis finished with a major devaluation of the Mexican Peso, while the second, known as the Tequila crisis, ended with a major $50-billion rescue from the IMF and the Clinton administration. The adverse conditions and the inability of the Mexican Government to alleviate these complex situations exacerbated the crises, such that in both the poorest people suffered most of the consequences. This fact is reflected in a disproportional increase in inequality across the country during the 90 s, while for different reasons this tendency appeared to change during the next decade. Some people argue that that has been due to the boosting effect of international trade (where the best example is the NAFTA agreement), while others blame the huge out-migration of nationals to the United States. 7.2 Data The data comes from the Mexican Institute of Information INEGI (Instituto Mexicano de Estadistica, Geografia e Informatica) which provides information to the public. We focus in the National Survey of Employment (ENE Encuesta Nacional de Individual inequality in Mexico. David Vazquez-Guzman. Page 35 of 43

Empleo), a recent survey that had its foundation in previous attempts traced to 1971. 18 The ENE, which started in 1991 as a biannual survey, changed its periodicity per annum later on, and recently added up quarterly data, as shown in Table 4. Methodology it is found in INEGI (2000, 2003). National Survey of Employment (ENE ) Surveys Population % of country Year Records* per year represented population 1991 143,957 1 - - 1992 - - - - 1993 139,902 1 - - 1994 - - - - 1995 111,949 1 59,083,793-1996 365,525 1 62,302,968-1997 119,404 1 63,425,327-1998 375,134 1 69,537,053 72.60% 1999 164,550 1 70,818,567 72.86% 2000 436,344 3 72,137,537 73.12% 2001 450,577 4 73,577,159 73.54% 2002 443,035 4 75,352,912 74.31% 2003 414,785 4 76,863,320 74.84% Total Records 3,165,162 * People of more than 12 years age. Source: INEGI. Table 4. Main data set. We have quarterly data between the year 2000 and 2003 in a more detailed fashion, having even the household characteristics correspondent to the year 2003, not only the personal survey. That is shown in Table 5. 18 In 1971 was agreed by the President and the Secretary of Commerce to start a new employment survey. It was between 1973 and 1984 that the Continuous Survey of Occupation (ECSO Encuesta Contínua sobre Ocupación) was developed with some changes. A second major change started in 1979, when the Direction of General Statistics (DGE) started and inquiry regarding the assertiveness of ECSO. The new product that includes some changes regarding international standards was ENEU (National Survey of Urban Employment), and covered the period between 1981 through 1998. The present survey (ENE) again was an effort to upgrade past surveys (ENEU), and between the years of 1991 and 1998, both instruments were surveyed. Individual inequality in Mexico. David Vazquez-Guzman. Page 36 of 43

National Survey of Employment (ENE ) Population % of country Year Quarter Records* represented population 2000 2 443,354 72,137,537 73.12% 3 454,232 72,528,671 73.25% 4 454,971 72,929,882 73.40% 2001 1 461,928 73,264,167 73.48% 2 460,271 73,577,159 73.54% 3 454,708 74,141,311 73.85% 4 455,390 74,496,476 73.96% 2002 1 456,645 75,026,801 74.24% 2 451,687 75,352,912 74.31% 3 442,114 75,836,489 74.55% 4 441,256 76,224,891 74.69% 2003 1 448,590 76,746,666 74.96% 2** 422,523 76,863,320 74.84% 3 366,715 77,076,798 74.81% 4 354,484 77,448,509 74.93% Total Records 6,568,868 * People of more than 12 years age. Source: INEGI. ** Available household characteristics Table 5. Quarterly data (2000-2003) The survey is divided in batteries of questions, and it obtains information about employment, self-employment and unemployment conditions, status of residence (i.e. migrant from other region), income, traditional questions about related working sector, and all general characteristics of people like gender and age. 7.3 Regional distribution. Mexico, in the same way as many other countries in the world, presents a natural heterogeneity within it. It is easier to show results by regions than for every state in the country, but there are many ways to divide the country in regions. Even the federal government does not have a unique way to consider regions. It happens that several departments within the government make their own distribution of states in regards of their own objectives. The distribution considers not only geographical closeness of states, but also different tastes according to related cultural values, historical inheritance and native composition. This distribution (Figure 14) considers major geographical characteristics, not only state proximity, which guarantees that states that appear close in the map are really economically closer to each other. This also considers two sets of mountains that naturally divided the country, where it is difficult to get from states in the center to states Individual inequality in Mexico. David Vazquez-Guzman. Page 37 of 43

in the Pacific or in the Gulf of Mexico. It also has the convenience of not mixing municipalities within states, so each region contains complete states (details in this Appendix, Table 6). So unless stated in the paper, the classification of regional distribution of Mexico will be the following: Mexico: Regional Distribution. Detail Regional Distribution NREGION 1 Center (Centro) 2 Semi-Center (Semi-Centro) 3 West (Occidente) 4 South (Sur) 5 Gulf-Peninsula (Golfo Peninsular) 6 North (Norte) 7 Northest (Noreste) 8 Northwest (Noroeste) Lake; Ocean Land Detail, Central part of Mexico. 0 37.5 75 150 225 300 Kilometers Inequality in Mexico. Own calculations based on ENE (INEGI). David Vazquez-Guzman. University of Stirling (2007). 4 175 87.5 0 175 350 525 Kilometers Figure 14 Regional Distribution. Individual inequality in Mexico. David Vazquez-Guzman. Page 38 of 43