Describing Income Inequality

Transcription

1 Describig Icome Iequality Module 051 Describig Icome Iequality

2 Describig Icome Iequality by Lorezo Giovai Bellù, Agricultural Policy Support Service, Policy Assistace Divisio, FAO, Rome, Italy ad Paolo Liberati, Uiversity of Urbio, "Carlo Bo", Istitute of Ecoomics, Urbio, Italy for the Food ad Agriculture Orgaizatio of the Uited Natios, FAO About EASYPol EASYPol is a a o-lie, iteractive multiligual repository of dowloadable resource materials for capacity developmet i policy makig for food, agriculture ad rural developmet. The EASYPol home page is available at: EASYPol has bee developed ad is maitaied by the Agricultural Policy Support Service, Policy Assistace Divisio, FAO. The desigatios employed ad the presetatio of the material i this iformatio product do ot imply the expressio of ay opiio whatsoever o the part of the Food ad Agriculture Orgaizatio of the Uited Natios cocerig the legal status of ay coutry, territory, city or area or of its authorities, or cocerig the delimitatio of its frotiers or boudaries. FAO December 006: All rights reserved. Reproductio ad dissemiatio of material cotaied o FAO's Web site for educatioal or other o-commercial purposes are authorized without ay prior writte permissio from the copyright holders provided the source is fully ackowledged. Reproductio of material for resale or other commercial purposes is prohibited without the writte permissio of the copyright holders. Applicatios for such permissio should be addressed to: copyright@fao.org.

3 Describig Icome Iequality Table of cotets 1. Summary...1. Itroductio Coceptual backgroud Geeral issues The etropy class of iequality idexes A step-by-step procedure to calculate etropy iequality idexes A step-by-step procedure for E(0), E(1) ad RE(1) A step-by-step procedure for E() ad RE() A example of how to calculate etropy iequality idexes A umerical example for E(0), E(1) ad RE(1) A umerical example for E() ad RE() The mai properties of etropy iequality idexes Sythesis Readers otes Time requiremets EASYPol liks Frequetly asked questios Refereces ad further readigs Module metadata... 15

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5 Describig Icome Iequality 1 1. SUMMARY This module illustrates the etropy class of iequality idexes. I particular, it shows how differet iequality idexes may be obtaied by usig a geeral defiitio (class) of idexes by assigig differet values to a fixed parameter. A step-by-step procedure ad umerical examples the show how to move from coceptual to operatioal groud.. INTRODUCTION This tool will deal with a specific class of iequality idexes, the etropy class idexes to which oe of the most popular iequality idexes belogs, i.e. the Theil Idex. Objectives The objective of the tool is to explai the use of complex iequality measures to compare icome s ad to discuss their relative merits as well as their relative disadvatages. This tool will deal with the etropy class of iequality idexes. It will show how iformatio o iequality may be coveyed by idexes coformig to certai desirable properties 1. Usig complex iequality measures allows us to rakig icome s accordig to icome iequality. This is particularly useful i a operatioal cotext to derive iformatio o the effects of alterative public programs o the of icome ad o poverty. Target audiece This module targets curret or future policy aalysts who wat to icrease their capacities i measurig impacts of developmet policies o iequality. O these grouds, ecoomists ad practitioers workig i public admiistratios, i NGOs, professioal orgaisatios or cosultig firms will fid this helpful referece material. Required backgroud Users should be familiar with basic otios of mathematics ad statistics. Liks to relevat EASYPol modules, further readigs ad refereces are icluded both i the foototes ad i sectio 8.1 of this module. 1 See EASYPol Module 054: Policy Impacts o Iequality: Iequality ad Axioms for Its Measuremet. EASYPol hyperliks are show i blue, as follows: a) traiig paths are show i uderlied bold fot; b) other EASYPol modules or complemetary EASYPol materials are i bold uderlied italics; c) liks to the glossary are i bold; ad

6 EASYPol Module 051 Aalytical Tools 3. CONCEPTUAL BACKGROUND 3.1 Geeral issues The use of complex iequality measures such as the etropy class ca be associated to the use of a descriptive approach to measure iequality. I particular, the use of these idexes does ot ivolve welfare judgmets 3. The etropy class of iequality idexes gives a differet descriptio of iequality with respect to simple statistical idexes. I particular usig complex iequality measures will ot give ay iformatio about the characteristics of the like locatio ad shape. I fact most of these idexes are traslatio ivariat ad they do ot say aythig about the positio of the icome. It is worth otig that much of the discussio will be carried out keepig i mid (either implicitly of explicitly) axioms as a eligible criterio to evaluate the performace of differet members of this class of idexes The etropy class of iequality idexes This class of iequality idexes is based o the cocept of «etropy». I thermodyamics, etropy is a measure of disorder. Whe applied to icome s, etropy (disorder) has the meaig of deviatios from perfect equality. The defiitio of a geeralised iequality idex is the followig: [1] 1 y i E( ) = 1 ( ) i y Expressio [1] defies a class because the idex E() assumes differet forms depedig o the value assiged to. A positive captures the sesitivity of the E idex to a specific part of the icome. With positive ad large, the idex E will be more sesitive to what happes i the upper tail of the icome. With positive ad small, the idex E will be more sesitive to what happes at the bottom tail of the icome. is a parameter that i priciple may rage from mius ifiity to ifiity, i.e. it ca take all possible real values. However, from a operatioal poit of view, is usually d) exteral liks are i italics. 3 See EASYPol Module 050: Policy Impacts o Iequality: Welfare Based Measures of Iequality ad EASYPol Modules 001 ad 003, respectively: Social Welfare Aalysis of Icome Distributios: Rakig icome s with Lorez Curves, ad Social Welfare Aalysis of Icome Distributios:Rakig Icome Distributios with Geeralised Lorez Curves. 4 The reader is therefore strogly advised to look at the EASYPol Module 054: Policy Impacts o Iequality: Iequality ad Axioms for its Measuremet before proceedig with this module

7 Describig Icome Iequality 3 chose to be o-egative, as for <0 this class of idexes is udefied if there are zero icomes. 5 Two particular cases of [1] are of particular iterest for iequality measuremet: a) =0; b) =1. 6 With =0, expressio [1] becomes: 1 yi [] E( 0) = l y With =1, expressio [1] becomes: 1 yi yi [3] E( 1) = l y y i i E(0) idex is called the mea logarithmic deviatio. E(1) is called the Theil Idex, by the ame of the author who first proposed it i Both idexes, however, share a udesirable feature, i.e., ot beig defied if there are zero icomes. Therefore, i a with all zero icomes except for the last, their maximum value caot be calculated directly. Rather, it ca oly be calculated by replacig zero icomes with arbitrary «very small icomes». However, if we replaced zero icomes with very small icomes, while E(1) approaches the maximum value of l() 7, the maximum value of E(0) would deped o how small these icomes are defied. I other words, E(0) is ot boud). 5 y Like i formula [1], i raised to a egative power would be a udefied umber. y 6 It is worth otig that expressio [1] is ot defied for =0 ad =1, as the deomiator ( ) = 0. i both cases. Expressio [] ad [3] below are therefore calculated by usig a rule by de l Hôpital, by which the limit of a udefied ratio betwee two fuctios of the same variable is equal to the limit of the ratio of their first derivative. By usig the rule by where d give that ( ( ) = ( 1 d gives expressios i the text. ), this meas that we must evaluate: 7 This ca be appreciated through the fact that i the most uequal : 1 y y 1 E( 1) = l [ l( ) ] l( ) y y. d y i y = i y i l, ad d y y y y i y l i i lim y y, which ( 1) 1 0

8 4 EASYPol Module 051 Aalytical Tools All other members of the class, i.e. for >1, have as a upper limit ( ) 8. As ca be easily see, this upper limit depeds o, i.e. it is differetiated amog members of the class, ad the rage of values each member ca take does ot rage betwee zero ad oe. Therefore, for the purpose of a operatioal approach, it is worth defiig a class of relative etropy iequality idexes RE, defied as the ratio betwee the value of the origial etropy idex ad the maximum value each member of that class assumes for ay give positive. This excludes the possibility of cosiderig E(0), as it has ot a upper limit. For all other members, i.e. positive ad 0, it is worth defiig the followig relative idexes: [4] E(1) RE(1) = = max E(1) 1 i yi yi l y y l As E(1) has already bee defied as the Theil Idex, RE(1) ca be called the relative Theil Idex. I the same way, we ca defie a relative geeralised etropy idex as follows: [5] 1 y i ( ) ( ) 1 E i y 1 yi RE( ) = = = ( ) 1 max E i y ( ) This procedure will esure that RE(1) ad RE() will rage betwee zero ad oe. 4. A STEP-BY-STEP PROCEDURE TO CALCULATE ENTROPY INEQUALITY INDEXES I order to illustrate the step-by-step procedure required to calculate various etropy iequality idexes, it is worth distiguishig two groups: a) E(0), E(1) ad RE(1); b) E() ad RE() with >1. 8 Sice for the most uequal : E( ) = 1 ( ) ( 1 ) y + y 1 = 1 ( ) [ ]

9 Describig Icome Iequality A step-by-step procedure for E(0), E(1) ad RE(1) Figure 1 illustrates the step-by-step procedure to calculate for the first group of idexes. Step 1 as usual as us to sort icome s before proceedig with the calculatio of iequality measures. Step asks us to defie the average mea icome i the icome uder aalysis. Step 3 asks us to defie, for each icome, its ratio with the average level of icome i the icome as calculated i Step. Step 4 asks us to take the logarithm of each ratio defied i Step 3 Step 5 simply takes the sum of all terms defied i Step 4. Figure 1: A step-by-step procedure to calculate E(0), E(1) ad RE(1) E(0), E(1) ad RE(1) STEP Operatioal cotet 1 If ot already sorted, sort the icome by icome level Defie the average icome level i the icome 3 Defie the ratio betwee each icome ad the average icome level 4 5 Defie the logarithm of the terms calculated i Step 3 Take the sum of the terms calculated i Step 4 6 Divide the sum i Step 5 by ad take its egative, this give E (0) 7 Multiply the results i Step 3 by those i Step 4 ad take the sum of these values 8 Divide the sum i Step 7 by, this give E (1) 9 Divide E (1) by l. This gives RE (1)

10 6 EASYPol Module 051 Aalytical Tools I Step 6, the first idex of this family ca be calculated by simply dividig the sum take i Step 3 by the umber of observatios ad by takig its egative value. This gives E(0). I Step 7, we ca proceed with the calculatio of aother idex of this class, by multiplyig the results obtaied i Step with those obtaied i Step 3 ad by takig the sum of these products. I Step 8, by dividig the sum obtaied i Step 5 by the umber of observatios, we get the idex E(1). I Step 9, by dividig E(1) by l() which is the maximum value of E(1) we ca take the relative Theil Idex RE(1). 4. A step-by-step procedure for E() ad RE() Figure shows the step-by-step procedure to calculate other members of the etropy class of iequality idexes, whe >1. The first three steps are idetical to those discussed i Figure 1 for the previous case. Step 4 asks us to choose the value of. Step 5 asks us to raise all terms calculated i Step 3 to power - as chose i Step 4 ad to subtract 1 from all terms. Step 6 simply asks us to take the sum of all terms calculated i Step 5. I Step 7, the first iequality idex ca be calculated, by dividig the sum of Step 6 by. This gives rise to E(). the term ( ) I order to proceed with the calculatio of RE(), istead, we must calculate the maximum value of E() i Step 8. By dividig E() by the maximum value of E(), we get RE() i Step 9.

11 Describig Icome Iequality 7 Figure : A step-by-step procedure to calculate E() ad RE(). E(a) ad RE(a) with a>1 STEP Operatioal cotet 1 3 If ot already sorted, sort the icome by icome level Defie the average icome level i the icome Defie the ratio betwee each icome ad the average icome level 4 Choose a 5 6 Raise the results i Step 3 to power a ad subtract 1 Take the sum of the values calculated i Step 5 7 Divide the sum by (a -a), this gives E(a) 8 Calculate the maximum value of E(a) 9 Calculate RE(a), by dividig E(a) by maxe(a) as calculated i Step 8 5. AN EXAMPLE OF HOW TO CALCULATE ENTROPY INEQUALITY INDEXES 5.1 A umerical example for E(0), E(1) ad RE(1) It is also worth distiguishig, at this stage, the example for E(0), E(1) ad RE(1) from the other members of the etropy class.

12 8 EASYPol Module 051 Aalytical Tools Table 1: A umerical example to calculate E(0), E(1) ad RE(1) STEP 1 STEP STEP 3 STEP 4 STEP 5 STEP 6 STEP 7 STEP 8 STEP 9 Sort the icome Calculate the average level of icome Defie the ratio betwee each icome ad the average icome level Defie the logarithm of Step 3 Take the sum of the terms calculated i Step 4 Divide the sum i Step 5 by ad take its egative, this give E (0) Multiply the results i Step 3 by those i Step 4 ad take the sum Divide the sum i Step 7 by, this give E (1) Divide the value i Step 8 by the maximum value l( ), this gives RE (1) Idividual A - A typical icome Mea icome , , l(5) = , , ,000 3, Total idividuals ( ) Total icome 15,000 Table 1 reports the case for E(0), E(1) ad RE(1), calculated o a stadard icome A. Steps illustrated with umerical outcomes, closely follow the steps discussed i Figure 1. Step calculates the average icome i the assumed icome, which is 3,000 icome uits. Step 3 ad Step 4, istead, provide for the trasformatios required by formula [] ad formula [3] i the text. The result of this process is a sigle umber (-0.706) i Step 5. The level of E(0) would the be calculated as (Step 6). Aother trasformatio (Step 7) is istead required to calculate E(1). The result of this trasformatio is agai a sigle umber (0.598). This is the basis to calculate E(1) i Step 8, which is equal to More importat is the calculatio of RE(1), i.e. the relative Theil Idex i Step 9. The outcome (0.074) meas that iequality i the simulated icome is about 7.4 per cet of the maximum iequality as measured by the Theil Idex. Just recall that the maximum level of E(1) is l(). I the specific case, the maximum level of the Theil Idex is l(5) = A umerical example for E() ad RE() Table reports a umerical example for the other members of the etropy class, assumig =.

13 Describig Icome Iequality 9 Table : A umerical example to calculate E() ad RE() STEP 1 Sort the icome STEP STEP 3 STEP 4 STEP 5 STEP 6 STEP 7 STEP 8 STEP 9 Calculate the average level of icome Defie the ratio betwee each icome ad the average icome level Choose a>1 Raise the results of Step 3 to power a ad subtract 1 Take the sum of the values calculated i Step 5 Divide the sum i Step 6 by (a -a). This gives E(a) Calculate the maximum value of E(a) Divide the value i Step 7 by the maximum value i Step 8, this gives RE (a) Idividual A - A typical icome Mea icome a , , ( ) 3, , ,000 3, Total idividuals ( ) 5 Total icome 15,000 Steps 1 to 3 are exactly the same as those i Table 1. Step 4 is characteristic of this process, as it asks us to choose. Oce has bee chose, the umbers calculated i Step 3 must be raised to. Subtract 1 from the results (Step 5). The sum of all these values is a sigle umber (1.111) calculated i Step 6. To get E(), we have to divide the sum of Step 6 for ( ). I this specific case, this term is equal to 10. Therefore, the value of E() is (Step 7). If we wat to calculate the relative etropy idexes, the maximum value of the idex must be calculated by dividig by ( ). The first term, i the specific case, is 0; the secod, as already oted, is 10. Their ratio is therefore equal to (Step 8). Step 9 reports the calculatio of the RE() idex for =, which is As before, this idex meas that the measured iequality i the simulated icome is about 5.6 per cet of the maximum iequality level. 6. THE MAIN PROPERTIES OF ENTROPY INEQUALITY INDEXES Some properties are commo to all members of the etropy iequality class. Therefore, they will be dealt with joitly. All members of both E ad RE class have zero as lower limit. For =0 ad =1, whe all icomes are equal, the ratio betwee each icome ad mea icome is 1. Therefore, l(1)=0 for all icomes, so that this sum is zero ad E(0)=E(1)=0. For all other members of the E class, raisig the ratio betwee each icome ad average icome to power gives a vector of oe. Subtractig 1 agai gives zero for all icomes. E() is therefore zero. This meas that the umerators of all RE idexes is also zero.

14 10 EASYPol Module 051 Aalytical Tools E(1) has l() as upper limit, while E(), for >1 9, has ( ) as upper limit. As this upper limit depeds o, each member of the E class has its ow upper limit. However, all relative etropy iequality idexes, RE(), have 1 as upper limit, as each of them is ormalized o the maximum value of E(). All members of E ad RE class are scale ivariat. This is due to the fact that whe all icomes are multiplied by a factor β, the ratio betwee each icome ad mea icome remais the same, as both are multiplied by β. All members of the E ad RE class are ot traslatio ivariat. By addig (subtractig) the same amout of moey to all icomes, E iequality idexes would decrease (icrease). Give that the deomiator of RE idexes is costat, they also decrease (icrease). All members of the E ad RE class satisfy the priciple of trasfers. If icome is redistributed from relatively richer idividuals to relatively poorer idividuals, E() decreases. The opposite holds true if icome is redistributed from relatively poorer to relatively richer idividuals. It is worth otig a particular characteristic of the way i which these idexes satisfy the priciple of trasfers. For E(0), the chage i the idex depeds o both the populatio size ad the level of idividual icomes ivolved i re. 10 I particular, the higher the gap betwee the icome of the receiver ad the icome of the door, the greater the reductio of E(0). 11 For E(1), the chage also depeds o the populatio size ad the level of idividual icomes ivolved i re. 1 Fially, for the other members of the etropy class E(), the chage depeds ot oly o the populatio size ad the level of 9 Sice for =0 ad =1the E idex is ot defied wheever there is a idividual icome equal to zero. 10 de(0) 1 1 This ca be see by cosiderig that: =, which is obtaied usig the geeral rule that dyi y i d f '( y) l f ( y) =. dy f ( y) 11 Assumig two idividuals ad the re of a give amout of icome dy, the differetial of the idex would be: de( 0) = dy uder the hypothesis that dy y y { 1 { receiver door = dy1 dy. Whe y 1 is very low ad y is very high, the differece i square brackets is larger, as the secod term i the square brackets is very small. This gives rise to a more egative de(0). 1 de(1) 1 1 y Agai, this ca be see by the derivative: i = 1+ l. dyi y y

15 Describig Icome Iequality 11 idividual icomes but also o level of. 13 The relative etropy idexes have a similar behaviour, as the deomiator of all these idexes is i fact a costat (the maximum value of the correspodig idex). Table 3 illustrates these properties with examples for E(0), E(1) ad RE(1) Table 3: The mai properties of E(0), E(1) ad RE(1) Idividual A - A typical icome B - Icome with equal icomes C - Icome with oly oe idividual havig icome Origial icome with all icomes icreased by 0% Origial icome with all icomes icreased by $,000 Origial icome with a re of $ 100 from the richest to the poorest Origial icome with a re of $ 100 from two idividuals aroud the mea of the icome 1 1,000 3, ,00 3,000 1,100 1,000,000 3,000 0,400 4,000,000, ,000 3, ,600 5,000 3,000, ,000 3, ,800 6,000 4,000 4, ,000 3,000 15,000 6,000 7,000 4,900 5,000 Total icome 15,000 15,000 15,000 18,000 5,000 15,000 15,000 E(0) E(1) RE(1) Obtaied by replacig zero icomes with arbitrary small icomes. The value depeds o how small the arbitrary icomes are. Limit value: l( ) UNCHANGED DECREASED DECREASED DECREASED E idexes decrease less if re occurs betwee idividuals with closer icomes As we ca easily see, the lower limit is zero for all idexes, while the maximum value is 1 oly for the relative etropy idex RE(1). Note that for E(0) ad E(1), the upper limit is calculated by replacig zero icomes with arbitrary small values. But, while E(0) is basically ubouded from above, the maximum level of E(1) teds to 1.609, i.e. l(5). Other properties are illustrated o the right had side part of Table 3. All idexes have the same value whe icomes are icreased by, say, 0 per cet; at the same time, all of them decrease whe a absolute amout of moey is added (,000 icome uits i the example). The last two colums, istead, illustrate how the idexes satisfies the priciple of trasfers. I geeral, all of them decrease after a re of moey from a relatively richer to a relatively poorer idividual. However, this reductio is lower whe the icome trasfer occurs betwee idividuals with closer icomes compare the last colum with the colum immediately before it. Table 4 illustrates the same properties with examples for E() ad RE(). 13 The derivative, i this case, is: expressio collapses to: de( ) = dy de() 1 1 y i =. dyi y y i 1 y ( ) i y y 1. For the case of =, the previous

16 1 EASYPol Module 051 Aalytical Tools Table 4: The mai properties of E() ad RE() Idividual A - A typical icome B - Icome with equal icomes C - Icome with oly oe idividual havig icome Origial icome with all icomes icreased by 0% Origial icome with all icomes icreased by $,000 Origial icome with a re of $ 100 from the richest to the poorest Origial icome with a re of $ 100 from two idividuals aroud the mea of the icome 1 1,000 3, ,00 3,000 1,100 1,000,000 3,000 0,400 4,000,000, ,000 3, ,600 5,000 3,000, ,000 3, ,800 6,000 4,000 4, ,000 3,000 15,000 6,000 7,000 4,900 5,000 Total icome 15,000 15,000 15,000 18,000 5,000 15,000 15,000 E() RE() This meas that iequality i the icome is 5.6 per cet of the maximum iequality with a= UNCHANGED DECREASED DECREASED DECREASED E idexes decrease less if re occurs betwee idividuals with closer icomes Thigs are basically the same as i Table 3. Note agai that the idexes decrease less if trasfers of icome occur amog idividuals with closer icomes. Of some importace is also how the RE idexes vary whe varies. This is best observed by recallig the derivative of E() with respect to icome, de( ) = dy i 1 y ( ) i y y 1. From this expressio it is clear that the chage of the RE() which is equivalet to the ratio of the chage of E() o the upper limit of E() depeds o the particular situatio addressed. I geeral, this chage depeds o, the average icome level ad also the dispersio of icomes as measured by the ratio betwee each icome level ad average icome. 7. SYNTHESIS It is worth havig a comparative picture of how the iequality idexes belogig to this class perform with respect to the desirable properties discussed i the previous sectio. Just recall that these desirable properties are axioms used for iequality measuremet 14. Table 5 reports this compariso, by illustratig how differet idexes of the same class behave with respect to desirable properties (axioms). 14 See EASYPol Module 054: Policy Impacts o Iequality: Iequality ad Axioms for its Measuremet.

17 Describig Icome Iequality 13 Table 5: Geeralised etropy idexes ad desirable properties LOWER LIMIT UPPER LIMIT Priciple of trasfers Scale ivariace Traslatio ivariace Priciple of populatio Idex of relative iequality Appeal Etropy E(0) 0 The idex is ot defied whe there are zero icomes. It is ot bouded from above YES, more sesible if idividuals have distat icomes YES NO YES YES Low Etropy E(1) 0 The idex is ot defied whe there are zero icomes. Its limit is l YES, more sesible if idividuals have distat icomes YES NO YES YES Low Etropy E(a) 0 ( ) ( ) YES, more sesible if idividuals have distat icomes YES NO YES YES High Relative etropy RE(1) 0 1 YES, more sesible if idividuals have distat icomes YES NO NO NO Medium Relative etropy RE(a) 0 1 YES, more sesible if idividuals have distat icomes YES NO NO NO Medium Note first that the lower limit of all idexes is zero, while the upper limit is very differetiated. Oly relative etropy idexes, that are ormalised o the maximum value of the correspodig idexes, have 1 as a upper limit. The behaviour of these idexes is also peculiar with respect to the priciple of trasfers. All of them respect this priciple, but their reactio is differetiated depedig o where the icome trasfer occurs. I particular, all of them react more if the icome trasfer occurs amog two idividuals havig wider icome gaps (ot raks!) 15. All idexes are scale ivariat, ad oe of them is traslatio ivariat. The implicit cocept of iequality is therefore oe of relative iequality. Oly etropy idexes, ad ot relative etropy idexes, satisfy the priciple of populatio, i.e. the ivariace of the idex to replicatio of the origial populatio. For this reaso, oly etropy idexes, ad ot the relative etropy idexes, belog to the category of Relative Iequality Idexes (RII). All idexes therefore have shortcomigs. The appeal of E(0) ad E(1) is low, as they are ot defied i the presece of zero icomes. The appeal of RE(1) ad RE() is medium, as they do ot respect the priciple of populatio ad do ot belog to the class of RII. Therefore, the group of E() has the highest appeal, i this class, eve though its upper limit depeds o the size of the icome. 15 Compare EASYPol Module 040: Iequality Aalysis: The Gii Idex.

18 14 EASYPol Module 051 Aalytical Tools 8. READERS NOTES 8.1 Time requiremets Time requiremets to deliver this module is estimated at about three hours. 8. EASYPol liks Selected EASYPol modules may be used to stregthe readers backgrouds ad to further expad their kowledge o iequality ad iequality measuremet. This module belogs to a set of modules that discuss how to compare, o iequality grouds, alterative icome s geerated by differet policy optios. It is part of the modules composig a traiig path addressig Aalysis ad moitorig of socio-ecoomic impacts of policies. The followig EASYPol modules form a set of materials logically precedig the curret module, which ca be used to stregthe the user s backgroud: EASYPol Module 000: Chartig Icome Iequality: The Lorez Curve EASYPol Module 001: Social Welfare Aalysis of Icome Distributios: Rakig Icome Distributio with Lorez Curves EASYPol Module 040: Iequality Aalysis: The Gii Idex EASYPol Module 054: Policy impacts o iequality: Iequality ad Axioms for its Measuremet 8.3 Frequetly asked questios What is etropy i iequality aalysis? What is the advatage of usig the etropy class compared with other iequality idexes? 9. REFERENCES AND FURTHER READINGS Aad S.,1983. Iequality ad Poverty i Malaysia, Oxford Uiversity Press, Lodo, UK. Cowell F., Measurig Iequality, Phillip Alla, Oxford, UK. Dalto H., 190. The Measuremet of Iequality of Icomes, Ecoomic Joural, 30. Gii C., 191. Variabilità e Mutabilità, Bologa, Italy. Pigou A.C., 191. Wealth ad Welfare, MacMilla, Lodo, UK. Se A.K., O Ecoomic Iequality, Calaredo Press, Oxford, UK. Theil H., Ecoomics ad Iformatio Theory, North-Hollad, Amsterdam, The Netherlads. Yitzhaki S., O the Extesio of the Gii Idex, Iteratioal Ecoomic Review, 4,

19 Describig Icome Iequality 15 Module metadata 1. EASYPol Module 051. Title i origial laguage Eglish Describig Icome Iequality Frech Spaish Other laguage 3. Subtitle i origial laguage Eglish Frech Spaish Other laguage 4. Summary This module illustrates the etropy class of iequality idexes. I particular, it shows how differet iequality idexes may be obtaied by usig a geeral defiitio (class) of idexes by assigig differet values to a fixed parameter. A step-by-step procedure ad umerical examples the show how to move from coceptual to operatioal groud. 5. Date December Author(s) Lorezo Giovai Bellù, Agricultural Policy Support Service, Policy Assistace Divisio, FAO, Rome, Italy Paolo Liberati, Uiversity of Urbio "Carlo Bo", Istitute of Ecoomics, Urbio, Italy 7. Module type Thematic overview Coceptual ad techical materials Aalytical tools Applied materials Complemetary resources 8. Topic covered by the module 9. Subtopics covered by the module Agriculture i the macroecoomic cotext Agricultural ad sub-sectoral policies Agro-idustry ad food chai policies Eviromet ad sustaiability Istitutioal ad orgaizatioal developmet Ivestmet plaig ad policies Poverty ad food security Regioal itegratio ad iteratioal trade Rural Developmet 10. Traiig path Aalysis ad moitorig of socio-ecoomic impacts of policies 11. Keywords capacity buildig, agriculture, agricultural policies, agricultural developmet, developmet policies, policy aalysis, policy impact alaysis, poverty, poor, food security, aalytical tool, agricultural policies, icome iequality, icome, icome rakig, welfare measures, etropy class idexes, iequality idex, theil idex, welfare measures, social welfare fuctios, social welfare

20 16 EASYPol Module 051 Aalytical Tools