Properties of Lorenz Curves for Transformed Income Distributions

Transcription

1 Theoreticl Economics etters htt://ddoiorg/4236/tel22259 Published Online December 22 (htt://wwwscirporg/journl/tel) Proerties of orenz Curves for Trnsformed Income Distributions John Fellmn 2 Swedish School of Economics Helsinki Finl 2 Folkhälsn Institute of Genetics Helsinki Finl Emil: fellmn@hnkenfi Received Setember 24 22; revised October 25 22; cceted November ABSTRACT Redistributions of income cn be considered s vrible trnsformtions of the initil income vrible The trnsformtion is usully ssumed to be ositive monotone-incresing continuous but discontinuous trnsformtions hve lso been discussed recently If the trnsformtion is t or trnsfer olicy the trnsformed vrible is either the ost-t or the ost-trnsfer income A centrl roblem hs been the orenz dominnce between the initil the trnsformed income This study considers nlyses of other roerties of the trnsformed orenz curves esecilly its limits We tke in ccount minly two cses () the trnsformed vrible orenz domintes the initil one (b) the initil orenz domintes the trnsformed one For lictions the first cse is more imortnt thn the second The limits obtined re not ccurte for secific trnsformtion but do hold generlly for ll distributions brod clss of trnsformtions so tht if one ursues generl conditions the ineulities obtined cnnot be imroved Keywords: Preto Distribution; T Policy; Trnsfer Policy Introduction Redistributions of income ccording to t or trnsfer olicies cn be considered s vrible trnsformtions of the initil income The trnsformtion is usully ssumed to be ositive monotone-incresing continuous The initil results re given in Theorem [-3] Consider nonnegtive rom vrible with the distribution function F men orenz curve et u be continuous monotone incresing function ssume tht Eu eists Then orenz curve for u eists u is monotone decresing ) if u 2) if is constnt u 3) if is monotone incresing The imortnce of cse () is tht it gives the ineulity effect of rogressive ttion The cse (2) corresonds to flt tes The lst cse (3) is of minor economic imortnce but it is included in order to comlete the theorem Recently Fellmn [45] hs lso discussed discontinuous trnsformtions If the trnsformtion is considered s t or trnsfer olicy the trnsformed vrible is either the ost-t or the ost-trnsfer income Under the ssumtion tht Theorem should hold for ll income distributions the conditions re both necessry sufficient [24] Hemming Keen [6] hve given n lterntive version of the conditions In this study we consider other generl roerties of the trnsformed orenz curves 2 Bckground Consider income defined on the intervl b where b with the distribution function F density function f men ercentile defined s F orenz curve The generl formule re b f d () f d (2) where b We consider the trnsformtion u where u is non-negtive continuous monotone-incresing Since the trnsformtion cn be considered s t u or trnsfer olicy u the Coyright 22 SciRes TE

2 488 J FEMAN trnsformed vrible is either the ost-t or the osttrnsfer income The men the orenz curve for vrible re b u f d (3) u f d (4) A fundmentl theorem concerning orenz dominnce is [24] Theorem 2 et be n rbitrry non-negtive rom vrible with the distribution F men orenz curve et u be nonnegtive monotone-incresing function let u E of let eist The orenz curve eists the following results hold: ) if only if tone-decresing 2) if only if 3) if only if u u u is mono- is constnt is mono- tone-incresing In the following we consider dditionl roerties of the orenz curve If u is constnt then ccording to Theorem (2) the trnsformed orenz curve is identicl with the initil one cse which will be ignored 3 Results 3 The Rtio Decresing u Is Monotoniclly According to Theorem F y orenz domintes F We introduce the vlues M m such tht Conseuently u lim M u lim m b u M m et F F Assume tht tht b conseuently u u u M m Note tht oints re chosen rbitrrily tht the eulity signs cnnot be ignored becuse we lso include the functions u which re not uniformly strict decresing in the clss of trnsformtions Hence we hve to include members for which eulities hold for lmost the whole rnge in ddition sub-intervls in which strict ineulities hold cn be chosen rbitrrily short locted rbitrrily within the rnge b If one ursues generl conditions the ineulities (8) (9) obtined below cnnot be imroved If we ssume tht u is monotoniclly decresing then u() must be continuous otherwise u should hve ositive jums [] From u u it follows tht u u the intervl yields d u f u f d The integrtion over d d (5) u f u f u u Anlogously it follows from u u u u we obtin tht Coyright 22 SciRes TE

3 J FEMAN 489 u (6) Conseuently u u When one obtins in (7) then u M M u (7) (8) The lower bound gives n evlution of how much the orenz curve hs incresed The uer bound is of minor interest is commented on lter When in (7) then u m one obtins m u In order to comre these ineulities with the ineulities in (8) we chnge the rgument from to the ineulities re m u (9) The lower bound gives n evlution of how much the orenz curve hs incresed The uer bound is of minor interest is discussed lter Ineulity (8) is licble to smll vlues ineulity (9) to lrge vlues of For smll vlues of we consider the difference u D () for lrge we consider the difference u D2 () In generl The rtio y u d d d d u is decresing conseuently d u d y d y d d d d d Now we differentite obtin d D u u D d d u d d u d Conseuently D is incresing from zero t D for (sy) Now we differentite obtin to mimum d D 2 D2 d u u d u d d u d Conseuently is decresing from D D2 2 to zero when The oint t which the shift from () to () is erformed is ch osen so tht D D Now 2 tht is Conseuently u u u u D D 2 ; Coyright 22 SciRes TE

4 49 J FEMAN Since the rtio u is decresing the difference u shifts its sign from lus to minus t oi nt Hemming Keen ([6]) gve the condition for orenz dominnce tht crosses the u level once from bove Our results bove hve shown tht the crossing oint is The condition obtined cn lso be otherwise elined If we write it s u we obtin the formul d d d d tht is the orenz curves between the orenz curves is miml for We define the difference function s hve rllel tngents the distnce D the lower bound of D for (2) D2 for u for u for (3) is Figure shows the orenz curves the lower bound the difference D between the lower bound Remrks The vrible orenz domintes the uer b ounds in (8) (9) tells us nothing bout the reductions in the ineulity The uer bound contins the mimum vlue M one hs to tke it for grnted tht it is lso inccurte when M is finite In ddition there my be situtions in which M The minimum vlue m cn be zero in this cse the uer bound is one the obvious ineulity is obtined u 32 The Rtio Incresing Is Monotoniclly The nlysis of this cse follows similr trces to the erlier study the results re nlogous to our erlier results but in this cse u my be discontinuous Only the ineulity signs hve chnged their directions We introduce the vlues M m such tht u u lim m lim M b conseuently u m M Note tht in this cse the oints re lso chosen rbitrrily tht the eulity signs cnnot be ignored becuse we lso include functions u which re not uniformly strictly incresing in the clss of trnsformtions Hence we hve to include members for which eulities hold for lmost the whole rnge in ddition the subintervls where strict ineulities hold cn be rbitrrily short cn be locted rbitrrily within the rnge If one ursues generl conditions the ineulities (7) (8) obtined below cnnot be imroved If u is discontinuous the discontinuities cn only ( ) ~ ( ) the difference D the lower bound when t Figure A sketch of the orenz curves the lower bound vrible orenz domintes the initil one D ~ between he trnsformed Coyright 22 SciRes TE

5 J FEMAN 49 be countble number of finite ositive jums Under such circumstnces u is still integrble We use the sme nottions s bove ssume tht F F tht conseuently tht Now u u u Consider u u intervl yields u d u f f d d u f u f d The integrtion over the u u Anlogously if we consider u u obtin (4) we u (5) u Hence u u When in (6) then one obtins m u m (6) u (7) Now the initil vrible orenz domintes the trnsformed the uer bound is the interesting cse When in (6) then u M one obtins u M After shift from to we obtin u M (8) Now the uer bound is of interest Formul (7) is licble for smll vlues formul (6) for lrge vlues of In the following we consider the difference between the uer bound the orenz curve tht is for smll vlues of u D (9) For lrge vlues of we consider the difference u D2 (2) In generl d y d d d The rtio u is incresing conseuently d y d y d d d d Now we differentite note tht is incresing obtin d u D u D d u d u d d u d Conseuently D is incresing from zero to mimum for Now we differentite D obtin 2 Coyright 22 SciRes TE

6 492 J FEMAN u d D2 d u d d u d u d Conseuently D2 is decresing from mimum to zero The oint denoted t which the shift from to D2 is erformed stisfies D D2 Now u u tht is u u This condition is identicl with the condition given bove in which u is decresing Agin the condition u cn be written u we obtin the formul d d d d tht is the orenz curves hve rllel tngents the distnce between the orenz curves is miml We define the difference function s D for (2) D2for D the uer bound of is u for u for In Figure 2 we sketch the orenz curves (22) the uer bound the difference D between the uer bo und Now the lower bounds re of minor interest becuse the initil vrible orenz domintes Note tht m is ossible in some situtions the lower bound in (7) cn be zero No te tht M cn be gret even M is ossible in some situtions the lower bound in (8) cn be even negtive Emle The Preto distribution Consider in- come with the Preto distribution F f where Now the orenz curve F From we obtin et the trnsformtion be u u tht the function is decresing We obtin the orenz curve D D ( ) ( ) ~ Figure 2 A sketch of the orenz curves the uer bound uer bound the difference D rible is orenz dominte d by the initil one D ~ so between the when the trnsformed v- Coyright 22 SciRes TE

7 J FEMAN 493 D D for D 2 for for ( ) for For the rtio u formed one In lictions the first cse is more imortnt thn the second becuse it yields olicies which reduce the ineulity The cse (2) in Theorem 2 is not included in this study becuse the initil the trnsformed orenz curves re identicl The limits obtined hold generlly for ll distributions brod clss of trnsformtions If one ursues generl conditions the ineulities obtined cnnot be imroved 5 Acknowledgements We re grteful to n nonymous referee for comments suggestions on revious version of the mnuscrit This study ws in rt suorted by grnt from the Mgnus Ehrnrooths Stiftelse Foundtion REFERENCES is decresing this cse being sketched in Figure if [] J Fellmn The Effect of Trnsformtions on orenz Curves Econometric Vol 44 No the rtio 824 doi:237/9345 u [2] U Jkobsson On the Mesurement of the Degree of Progression Journl of Public Economics Vol 5 No is incresing this cse being sketched in Figure doi:6/ (76) Conclusion Redistributions of income hve commonly been defined [3] N C Kkwni Alictions of orenz Curves in Economic Anlysis Econometric Vol 45 No doi:237/9684 s vrible trnsformtions of the initil income vrible [4] J Fellmn Discontinuous Trnsformtions orenz Curves The trnsformtions re minly considered s t or Trnsfer Policies Socil Choice Welfre Vol 33 trnsfer olicies yielding ost-t or ost-trnsfer incomes therefore the trnsformtions re usully ssumed to be ositive monotone-incresing continuous Recently discontinuous trnsformtions hve lso been discussed The fundmentl concern hs been the orenz ordering between the initil the trnsformed [5] [6] No doi:7/s J Fellmn Discontinuous Trnsfer Policies with Given orenz Curve Advnces Alictions in Sttistics Vol 2 No R Hemming M J Keen Single Crossing Conditions in Comrisons of T Progressivity Journl of in come In t his study we constructed limits for he trnsformed orenz curves We considered the otiml cses Public Economics Vol 2 No doi:6/ (83)932-4 tht the trnsformed vrible orenz domintes the initil one the initil vrible orenz domintes the trns- Coyright 22 SciRes TE