An anatomy of the distribution of urban income : a tale of two cities in Colombia

SWP650 An Anatomy of the Distribution of Urban Income A Tale of Two Cities in Colombia Rakesh Mohan WORLD BANK STAFF WORKING PAPERS Number 650   WORLD BANK STAFF WORKING PAPERS Number 650 An Anatomy of the Distribution of Urban Income A Tale of Two Cities in Colombia Rakesh Mohan The World Bank Washington, D.C., U.S.A.  Copyright ,' 1984 'he International 13ank for Reconstruction anid Development / THE WORLD BANK ISS H Street, N.W. Washington, D.C. 20433, U.S.A. All rights reserved Manufactured in the United States of America Fit st printing July 1984 rhis is a working document published informally by the World Bank. To present the results of research with the least possible delay, the typescript has n t been prepared in accordance with the procedures appropriate to formal p inted texts, and the World Bank accepts no responsibility for errors. The publication is supplied at a token charge to defray part of the cost of rnAanufacture and distribution. 'T'he views and interpretations in this document are those of the author(s) and should not be attributed to the World Bank, to its affiliated organizations, or to any individual acting on their behalf. Any maps used have been prepared solely for the convenience of the readers; the denominations used and the boundaries shown do not imply, on the part of the World Bank and its affiliates, any judgment on the legal status of any territory or any endorsement or acceptance of such boundaries. The full range of World Bank publications is described in the Catalog of World Bank Publications; the continuing research program of the Bank is outlined in World Bank Research Program: Abstracts of Current Studies. Both booklets are updated annually; the most recent edition of each is available without charge from the Publications Sales Unit of the Bank in Washington or from the European Office of the Bank, 66, avenue d'Iena, 75116 Paris, France. Rakesh Mohan is an economist in the Development Research Department of the World Bank. Library of Congress Cataloging in Publication Data Mohan, Rakesh, 1948- An anatomy of the distribution of urban incomc. (World Bank staff working papers no. 650) Bibliography: p. 1. Income distribution--Colombi-a--Bogota. 2. Income distribution--Colombia--Cali. I. Title. II. Series. HC198.B5M62 1984 339.2'2'098614 84-11989 ISBN 0-8213-0381-3  ABSTRACT This paper suggests an approach for analysis of distributional and other change in incomes in fast growing cities in the developing world. It notes that between 1973 and 1978 there appeared to be a tendency toward some improvement in the distribution of income in the two Colombian cities of Bogota and Cali. However, the level of inequality in the two cities remained very high, possible among the highest in the world. Among the distinguishing features of this study is the utilisation of primary data from different household surveys, of comparable quality spread over six years. Hence there is a high degree of confidence in the results obtained. The paper uses different samples, inequality indices, income concepts and ranking procedures in its analysis. It is found that, overall, there is a higher level of inequality in Bogota than in Cali. Particular attention is paid to the existence of spatial inequality among different parts of the two cities by using an index of spatial income segregation (ISIS). ISIS is found to be very high when the cities are disaggregated spatially into radical sectors rather than into concentric rings. The main disquieting feature of the findings is that ISIS appears to be increasing over time while inequality within the sectors is declining; this suggests that the cities are becoming more spatially segregated over time. The inequality in labor earnings is also calculated and decomposed. The paper finds that the industry of activity and the size of the firm of the worker contribute very little to overall inequality in earnings, in contrast to much of the labor segmentation literature. Instead, education, occupation and the location of residence (or background) of the worker contributed most to inequality in earnings. The results of the study suggest that increases in spatial inequality in household as well as labour earnings could have deleterious effects on the distribution of income in the future. Finally, in addition to sixteen tables and eight appendices, the paper carries a special computer program call EQUALISE which can compute income distribution indices directly from household data and decompose income inequality accordng to any characteristics.  EXTRACTO En el presente documento se propone un metodo para el andlisis de los cambios en la distribuci6n y otros aspectos de los ingresos en las ciudades del mundo en desarrollo en proceso de rapido crecimiento. Se observa en 41 que, entre 1973 y 1978, parecia haber una tendencia hacia una mejor distribuci6n de los ingresos en las ciudades colombianas de Bogota y Cali. Sin embargo, el nivel de desigualdad en las dos ciudades sigui6 siendo muy grande, y posiblemente se cuenta entre los mAs altos del mundo. Una de las caracterfsticas distintivas de este estudio es la utilizaci6n de datos primarios de diversas encuestas de unidades familiares de calidad comparable esparcidas en el curso de seis affos. De allf que haya un alto grado de confianza en los resultados obtenidos. El documento usa en su anAlisis diferentes muestras, indices de desigualdad, conceptos de ingresos y procedimientos de jerarquizaci6n. Llega a la conclusi6n de que, en general, hay un mayor nivel de desigualdad en Bogota que en Cali. Dedica atenci6n preferente a la existencia de una desigualdad espacial entre diversas partes de las dos ciudades al utilizar un fndice de desagregaci6n espacial de los ingresos. Halla que este fndice es muy alto cuando las ciudades se desglosan espacialmente en sectores radiales y no en anillos concentricos. La principal caracterfstica perturbadora de las conclusiones es que el fndice de desagregaci6n espacial de los ingresos parece aumentar con el tiempo en tanto que la desigualdad dentro de los sectores esta disminuyendo; esto indica que con el correr del tiempo las ciudades se estan volviendo mgs segregadas espacialmente. Tambidn se calcula y desglosa en el documento la desigualdad de los ingresos de los trabajadores. Se llega a la conclusi6n de que el tipo de industria de la actividad y el tamaflo de la empresa a que pertenece el trabajador contribuyen muy poco a la desigualdad general de los ingresos, contrariamente a lo que se sostiene en gran parte del material escrito sobre el desglose de la mano de obra. En cambio la educaci6n, la ocupaci6n y el lugar de residencia (o los antecedentes) del trabajador son elementos que han contribuido mas a la desigualdad de lo ingresos. Los resultados del estudio indican que los aumentos de la desigualdad espacial de los ingresos tanto de las unidades familiares como de los trabajadores podrfan tener efectos perjudiciales sobre la distribuci6n de los ingresos en el futuro. Finalmente, ademas de 16 cuadros y 8 apendices, el documento presenta un programa especial de computadora llamado EQUALISE con el que pueden calcularse indices de distribuci6n de los ingresos directamente a partir de los datos de las unidades familiares y descomponer la desigualdad de los ingresos de acuerdo con cualquier caracterfsticas.  L'auteur du pr6sent document propose ici une methode pour analy- ser I16volution de divers elements caracteristiques des revenus et, notam- ment, leur r6partition dans des villes du monde en d6veloppement en expan- sion rapide. Il note qu'entre 1973 et 1978, la r6partition du revenu dans deux villes colombiennes, Bogota et Cali, semble s'etre am6lioree. Les inegalites constatees dans ces deux villes sont toutefois rest6es tres importantes et pourraient meme etre parmi les plus fortes du monde. La presente etude se caracterise, entre autres, par l'utilisa- tion de donnees primaires tirees de diverses enquetes sur les menages de qualit6 comparable, effectuees sur une periode de six ans. Les resultats obtenus sont donc juges tres fiables. L'auteur utilise differents 6chan- tillons, indices d'in6galite, concepts de revenu et procedures de classe- ment dans le cadre de son analyse. Selon lui, les inegalites sont, dans l'ensemble, plus prononcees a Bogota qu'a Cali. Il accorde une attention particuliere aux inegalites qui existent entre differents quartiers des deux villes et, a cette fin, utilise un indice de segr6gation spatiale des revenus. Cet indice prend une valeur tres elev6e lorsque les villes sont decoup6es en secteurs radiaux plutot qu'en cercles concentriques. Le resultat le plus inquietant est que cet indice parait augmenter dans le temps alors meme que les inegalites au sein de chacun des secteurs diminuent; la segregation g6ographique semblerait donc s'intensifier avec les annees.  - 2 - L'auteur a 6galement calcule et decompose les in6galites enre- gistrees au niveau des remunerations de la main-d"oeuvre. II en conclut que le type d'activit6 industrielle et la taille des entreprises ne con- tribuent que peu a l'inegalite globale des salaires contrairement a ce qu'avancent la plupart des etudes consacr6es a la segmentation de la main-d'oeuvre. Ce sont le niveau d'education, le type d'emploi et le lieu de residence (ou l'environnement) de l'employe qui contribuent le plus a l'inegalite des revenus salariaux. L'6tude suggere que tout accroissement des inegalites g6ographiques enregistrees au niveau des revenus des menages ainsi que des salaires pourrait avoir un effet d6favorable sur la repartition des revenus dans les annees a venir. Enfin, outre seize tableaux et huit annexes, le lecteur trouvera joint a la presente etude un programme informatique intitule EQUALISE, qui permet de calculer directement les indices de repartition de revenu a par- tir des donnees sur les menages et de d6composer les in6galit6s des reve- nus en fonction de n'importe quel facteur caracteristique.  PREFACE This paper is part of the City Study research program (RPO 671-47) which has been carried out in Bogota and Cali, Colombia by the World Bank in collaboration with the Corporacion Centro Regional de Poblacion of Bogota. The goal of the City Study is to increase the understanding of the workings of five major urban sectors - housing, transport, employment location, labor markets, and public finance - in order that the impact of policies and projects can be assessed more accurately. A rather large amount of data handling, documentation, and preparation has gone into the work leading to this paper. I would like to thank Alan Carroll for the original cleaning and documentation of the 1975 and 1977 Household Surveys; Nelson Valverde for making the 1973 Population Census Sample ready for use; Nancy Hartline for wrestling with the 1977 Household Survey, preparing its final documentation, and putting the income imputing method into effect; Sungyong Kang and M.Wilhelm Wagner for making the 1978 sample ready for use. The computations appearing in this paper have been prepared by Sungyong Kang and Robert Marshall. A word of special thanks goes to them. My intellectual debt to Sudhir Anand is obvious from the text of the paper. Original programming of parts of the program EQUALISE appearing as appendix III was done by Sudhir Anand and Sherman Robinson. I am grateful to them for making available their programs. I have also benefited from discussions with and the pioneering works on income distribution in Colombia of Miguel Urrutia and R. Albert Berry. A special word of appreciation goes to the staff of DANE (the Colombian Statistical Office) for having painstakingly conducted all the surveys in Colombia and for making them available in almost raw form so readily. The high quality of the 1978 City Study DANE Household Survey owes much to the diligence of Roberto Pinilla and Maria Cristina Jimenez of DANE and Gary Losee Alfredo Aliaga, Alvaro Pachon and Jairo Arias for CCRP-World Bank. Other papers in the City Study Labor Market and Income Distribution series are: 1. Rakesh Mohan, The People of Bogota: Who They Are, What They Earn, Where They Live. World Bank Staff Working Paper No. 390. Washington, D.C., 1980 . 2. Gary Fields, How Segmented is the Bogota Labor Market. World Bank Staff Working Paper No. 434. Washington, D.C., 1980. 3. Rakesh Mohan and Nancy Hartline, The Poor of Bogota: Who They Are, What They Do, Where They Live. World Bank Staff Working Paper No. 635. Washington, D.C., 1980. 4. Rakesh Mohan, M. Wilhelm Wagner and Jorge Garcia, Measuring Urban Malnutrition and Poverty: A Case Study of Bogota and Cali, Colombia. World Bank Staff Working Paper No. 447, Washington, D.C., 1981.  5. Rakesh Mohan, The Determinants of Labor Earnings in Developing Metropoli: Estimates from Bogota and Cali, Colombia. World Bank Staff Working Paper No. 498, Washington, D.C., 1981.  Table of Contents Page No. I. INTRODUCTION .......................................................... 1 1.1 Objectives . ..1.......................................... 1.2 Income Concepts used . .................................. 3 1.3 Measures of Income Inequality . . 6 1.4 Decomposition of Inequality .. 6 II. OVERALL TRENDS IN THE DISTRIBUTION OF INCOME ................... 23 2.1 Trends in Colombia as a Whole ......................... 23 2.2 The Distribution of Income in Bogota and Cali ......... 28 III. THE SPATIAL DISTRIBUITON OF INCOME ............................ 40 IV. THE DECOMPOSITION OF LABOUR EARNINGS . .......................... 54 4.1 The Spatial Distribution of Personal Earnings ......... 55 4.2 The Decomposition of Labor Earnings According to Various Personal Characteristi cs ...................... 59 4.3 Summary . .............................................. 74 V. SUMMARY AND CONCLUSIONS ......................................... 77 APPENDIX I Statistical Appendix ......................... 82 APPENDIX II The Data ..................................... 89 APPENDIX III EQUALISE: A Program for Income Distribution Studies ................... ........ .. .... 99  List of Tables Table No Title Page No. 1. The Distribution of Income in Colombia 1964-1974. . . . . . . . . . . . . . . . . . . 25 2. The Distribution of Income In Bogota and Cali: Income Shares 1973-1978. . . . . . . . 31 3a. Inequality Indices for Bogota Households 1973-1978. . . . . . . . . . . . . . . . . . . 33 3b. Inequality Indices for Bogota's Population Ranked by Household Income Per Capita 1973-1978. . . . . . . . . . . . . . . . . . . 34 4. Inequality Indices for Cali 1973-1978. . . . . 36 5. Inequality Indices for Bogota 1978 .. . . . . . 38 Results form Using Different Numbers of Observations in the Lorenz Curve 6a. The Spatial Distribution of Income in Bogota 1973-1978. . . . . . . . . . . . . . . 45 6b. The Spatial Distribution of Income in Cali 1973-1978. . . . . . . . . . . . . . . 46 7. The Spatial Distribution of Income Distinguished by Rings and Sectors: Bogota and Cali 1978. . . . . . . . . . . . . 49 8a. Spatial Inequality in Bogota: Individuals Ranked by HINCAP 1973-1978. . . . . . . . . . 51  List of Tables (Cont.) Table No. Title Page No. 8b. Spatial Inequality in Cali Individuals Ranked by HINCAP 1973-1978. . . . . . . . . . 52 9a. The Spatial Distribution of Labour Earnings in Bogota 1973-1978. . . . . . . . . . . . . 56 9b. The Spatial Distribution of Labour Earnings in Cali 1973-1978. . . . . . . . . . . . . 57 10. The Decomposition of Labour Earnings 1978. . . 61 11. The Distribution of Earnings by Age Groups 1978. . . . . . . . . . . . . . . . 62 12. The Distribution of Earnings by Education Groups 1978. . . . . . . . . . . . . . . . 64 13. The Distribution of Earnings by Employment Status 1978. . . . . . . . . . . . . . . . 66 14. The Distribution of Earnings by Occupational Categories 1978. . . . . . . . . . . . . . 68 15. The Distribution of Earnings by Industry of Activity 1978. . . . . . . . . . . . . 70 16. The Distribution of Earnings by Firm Size 1978. . . . . . . . . . . . . . . . 72  List of Maps Map No. Title Page No. 1. Bogota: Ring and Sector System. . . . . . . . . . . 42 2. Cali: Ring and Sector System. . . . . . . . . . . . 43  I. INTRODUCTION 1.1 Objectives This paper is part of the program of research on urban labour markets and income distribution in the cities of Bogota and Cali that has been conducted under the rubric of the City Study. Earlier papersll have developed a profile of the distribution of income as well as the labour market somewhat descriptively. One of the striking results was the appearance of significant differences between different areas of the city in terms of income and occupation. The existence of these spatial differences was also utilized as an additional variable analysing the determinants of income in Mohan (1981). This paper attempts to measure the distribution of income somewhat more systematically and provides a spatial decomposition of the inequality indices estimated. In so doing a measure of the spatial inequality of income within cities suggests itself. The availability of micro data sets for 4 different years from 1973 to 1978 permits the estimation of inequality indices for this whole period. It is unusual to have access to such micro data and to be able to make such estimations across data sets consistently. The comparisons between different years give a sense of the robustness (or otherwise) of the estimates derived. They also suggest caution in the over- interpretation of small changes in such indices between different surveys. Nonetheless the estimates for these years are compared with those available for earlier years. Mohan (1981) examined in detail the determinants of personal earnings. Those findings are supplemented here by the decomposition of the earnings distribution according to various characteristics of the 1/ See Mohan (1980) and Mohan and Hartline (1980)  -2- labour force. The results are largely similar but the decompositions do give further insight into the distribution of labour earnings. Most income distribution studies use grouped data to calculate the inequality indices. Sometimes, only published data are available which give data for the distribution of the population by specific income groups: there is then no choice but to use such grouped data. Even when micro data are available, computation is usually done after the data are grouped because of the expense involved in making direct calculations from the micro data. A similar procedure is used here but Appendix III lists EQUALISE, the computer program used, which takes micro data directly as input and permits any level of dissagregation desired for the computation of inequality indices. An extension of the same program is used to do the decomposition of inequality.i/ The objective of this paper is therefore to document the distribution of income in Bogota and Cali and to give some indications of the pattern of change that may have occurred in the recent past. A novel feature of this paper is the computation of spatial inequality within a city. This has been done particularly because spatial inequality was found to be quite marked in the cities under investigation. The computation of inequality indices has been done for both household income as well as household income per capita (henceforth referred to as HINCAP) and later for earned incomes. The concepts and 1/ Parts of the program were originally developed by Sherman Robinson and Sudhir Anand in earlier studies of income distribution for the World Bank.  -3- methodology used are laid out in the next parts of this introduction. The sections following present the results that have been obtained. 1.2 Income Concepts Used Appendix II gives a brief description of the sources of data used in this study. The incomes used are all monthly incomes - derived in the different surveys as detailed in the appendix. All the incomes measured are current incomes and therefore some caution needs to be exercised in interpreting them as measures of welfare. Household consumption expenditures are, in general, regarded as a better proxy of permanent income and hence of welfare levels of households and individuals. There is little choice, here, since there have been no recent surveys of consumption in Colombia. The computation of inequality by using current household income or household income per capita (HINCAP) also suffers from yet another problem. It fails to take into account the differential tax incidence on different incomes according to the prevailing tax structure in the country. Similarly, no adjustsment is made of the differential accrual to different income groups of the benefits due to public expenditure on public services such as health, education, sanitation, water, etc L/. Although, as is detailed in Appendix II, somewhat different income questions were asked in the four sources used - the 1973 Census, 1/ See Selowsky (1979) for explicit consideration of these in the computation of inequality in Colombia.  -4- the 1975 and 1977 DANEi' household surveys and in the 1978 DANE-World Bank City Study Household Survey, the basic income concepts used here have been the Monthly Household Income (HHY) and the Monthly Household Income per Capita (HINCAP). HINCAP is merely the HHY divided by the household size (HHSIZE). In principle, HHY includes the labour earnings of all members of the household, including earnings in kind, as well as non-labour income. Earnings in kind are not taken to include imputed earnings for household work. The coverage of labour earnings in all the surveys is much better than the coverage of non-labor income. The 1978 survey included detailed questions on non-labour income which were addressed specifically according to types of non-labour income. However, it is suspected that the coverage was still less than desirable. HHY does not include imputed income from housing for owner occupying households. Section IV of the paper analyses the distribution of labour earnings of individuals. Here, naturally, only individual labour earnings are used in the computation. Non-labour earnings are completely ignored. One important point worthy of note is that all households with zero incomes have been excluded in the computation of income inequality. Similarly, other households with miscoded location were also excluded. For the workers, the definition included those individuals who gave their primary activity as working and others who worked at least 15 hours a week. The households with zero incomes were essentially those whose members happened to be unemployed. As is mentioned in the Appendix II on 2/ DANE - Departamento Administrativo Nacional de Estadistica - The National Statistical Agency, Colombia.  -5- data, income was systematically imputed to those workers who were employed but who did not give information on their incomes. Studies on inequality have to choose the population unit as well as the income concept by which they should be ranked in order to derive income distributions. The most common distribution used has been that of households ranked by household income. This distribution has come under attack increasingly in recent works (e.g. Anand, 1982; Datta and Meerman, 1980) as not providing a good indication of differences in the level of living in the population. As argued by Anand, our general concern is with the welfare of individuals rather than households. Secondly, households vary by size as well as by composition in terms of age and sex. Hence, for true cross household comparisons, adjustments should be made to account for these variations. In some of the literature concerning the measurement of poverty based on nutrition levels, adjustments have been attempted by deriving "adult equivalent" measures of consumption. While some norms may be found for deriving adult equivalents based on age and sex for food consumption purposes, it is difficult to do so for consumption as a whole. For example, it may well be that, while a child may require less nutrition than an adult, expenditures on his/her education and health might well exceed those for adults. It is therefore difficult to estimate adult equivalents for total consumption purposes. The next best strategy is followed therefore and HINCAP (household income adjusted for household size) is used to rank households and individuals for welfare comparisons. This is certainly better than the use of HHY even though economies of scale in consumption are also ignored in the  -6- computation of HINCAP. It is assumed implicitly that distribution within the household is even - though some might argue that this is usually not the case. In any case, there is no information which could be used to make any other assumption. Although it is being argued that it is individual welfare that we are interested in, the basic income concept from which HINCAP is derived is HHY. There is really no other choice since the household is the basic consumption unit where consumption allocations are typically made. Furthermore, non-labour income usually accrues to the household rather than the individual. The distributions that are analyzed then in sections 11 and III are: i) The distribution of households ranked by household income. ii) The distribution of households ranked by household income per capita. iii) The distribution of individuals ranked by household income capita. Section IV gives the analysis of the distribution of workers ranked by labour income, which includes income in kind. 1.3. Measures of Income Inequality Various inequality measures have been utilized in this study and then two used for decomposition purposes. The measures displayed in the tables are: i) Gini coefficient ii) Theil inequality index  -7- iii) The standard deviation of logarithims iv) Atkinson index v) Coefficient of variation The Gini coefficient, the standard deviation of logs and the coefficient of variation need no explanation since they are the most commonly used and well known measures. Although TheiLL/ and Atkinson2/ are now well known, their use in the computation of inequality is still not common. Anand (1982) gives a comprehensive theoretical introduction to all these measures so only a brief explanation of Theil and Atkinson is offered here..! Two important properties of inequality measures that are regarded as desirable are: i) They be mean-independent (i.e. their value remain unchanged if everyone's income is increased by the same proportion). ti) They satisfy the "Pigon-Dalton" condition which requires that any transfer from a richer person to a poorer person (which does not change their relative ranks) be reflected in a reduction of the inequality index. Simply put, a transfer from a richer to a poorer person without changing their ranks should imply higher equality as measured by the index. The Gini coefficient satisfies these two conditions, as does the coefficient of variation (the square root of variance divided by the mean). 1/ First proposed in Theil (1967) and applied in Theil (1972). 2/ Atkinson (1970). 3/ The following is taken from Anand (1982) (Annex A) and Theil (1972).  -8- The variance of incomes, which would be an obvious measure of the inequality in a distribution, does satisfy the Pigon-Dalton condition but is obviously not mean independent. The log variance of income, (or the variance of the logarithm of incomes) does, however, satisfy both these conditions: 1 n2 Var (y) - n i (y. - hence Var (Xy) = X2 Var (y) but Var (log X y) = Var (log y) where yi is income of unit i, p is the mean and X is the a proportionate change in all incomes. The log variance of incomes is therefore a convenient measure to use-LI The main problem in its use is that the measure is not defined when there are zero incomes in the sample. This is the main reason for the exclusion of all zero incomes in the sample. It is also an attractive measure to use because of its decomposition properties - as will be detailed below. 1.31 An Introduction to the Theil Index The Theil index is based on the concept of expected information in a distribution. Essentially, when the probability of an event 1 The Pigon-Dalton condition is satisfied for log variance for all incomes except those over P e where p is the geometric mean income and e is the base of natural logarithms.  -9- occurring is low, the information contained in a message stating that it has occurred is high. Intuitively, then, information can be regarded as the converse of probability. As a precise measure of information contained in a message that states that an event with prior probability p has occurred, the function suggested is: h (p) = log - (1) p which ranges from (when p is small, the event is unlikely and hence when it does occur, information in the message stating that it has occurred is high) to zero (when p=l, the event is certain). For a distribution of events, each with probability pi (i = 1, ...n) and i-li Pi1 The expected information of the message on the occurrence of one of these events is H1= in pi log - (2) i.e. the weighted average of the information contained in each message. The minimum of this distribution is when p. = 1 and all Pi = 0 (i 3 j) i.e. H = log 1 =0 and the maximum is when  -10- p. = P. = n for all i j p 1- ~n Then H= - log i=1 n n = log n. H is also known as entropy. In its application to the measurement of income distribution, it can be interpreted as a measure of uncertainty. Higher the uncertainty in a distribution, greater the expected information in the message stating that an event has occurred from the distribution of possible events, hence the entropy of a distributton may be said to measure the "amount of uncertainty" in it. It is similar to variance which is the more common measure used to measure uncertainty. Now consider an income distribution, z1, z2. .zn i.e. n individuals with income zi. Let Z = n. z i=1 i z. Then nZ 11 =1. =1- Z The entropy of the income distribution can then be defined zi Let qi Z hence Eq= 1.  and all q. > 0 n 11 Then H(q) l q. logq q. as for any distribution of probabilities Hr (q) = Z -Z log - (all summations are from 1 to n) i/ z. but Z = n z where z is the mean of the distribution Then H(q) = Z log nz z./nz n =- 1 ,. log n;\ n z AI zi/ Iz z. n E