Second Handout: The Measurement of Income Inequality: Basic Concepts

Transcription

1 Scod Hadout: Th Masurmt of Icom Iqualty: Basc Cocpts O th ormatv approach to qualty masurmt ad th cocpt of "qually dstrbutd quvalt lvl of com" Suppos that that thr ar oly two dvduals socty, Rachl ad Mart ad that Rachl has a smallr com tha Mart. W ca rprst ths stuato graphcally as follows: Mart's com D R A C H L A C Total com M A R T I N X X B' M F WW O O 45 º Rachl A' B X X Mart G Rachl's com I th graph OA+OBODOG total com socty (th "sz of th cak". Th DCG- l rprsts all possbl combatos of th coms of Rachl ad Mart, assumg total com socty s OGOD. Th pots alog th l DG

2 corrspod to varous ways of dvdg th cak btw Rachl ad Mart. Thus f socty s D, Mart gts all th cak ad f socty s G, Rachl gts t all. Th pot of qual sharg s locatd th mddl of DG, o th 45 dgrs l OM. I M th two popl gt half of th cak, that s, O X whch s th avrag com. Sc socty s rght ow at pot C, th socal dffrc curv gog through C rprsts th prst lvl of wlfar socty. Th socal dffrc curv has th sam shap as th usual (dvdual dffrc curv. At pot C Mart has OA ad Rachl OB At pot F Rachl has OA OA' ad Mart OB OB' Not that pots C ad F ar symmtrc wth rspct to pot M ad hc ar o th sam socal dffrc curv. W assum o prfrc for m or wom ad oly th dstrbuto (togthr wth th sz of th cak mattrs. Ths s why C ad F rprst th sam lvl of socal wlfar W may thrfor wrt that th socal wlfar W as W f (sz of cak, dgr of qualty of th dstrbuto of th cak Ad w assum that socal wlfar crass wth th sz of th cak ad dcrass wth th dgr of qualty of ts dstrbuto. Assum two cass whr both cass th caks hav th sam sz. I th frst cas ach of th two dvduals gts half th cak whl th scod cas o dvdual gts 75% ad th othr 25% of th cak. Th clarly socal wlfar s hghr th frst cas. Assum ow two cass whr th sharg s 50/50. Th frst cak s howvr bggr. Th clarly ths frst cak corrspods to a hghr lvl of socal wlfar. As for pot : Ths pot s locatd o th prst socal wlfar curv (th o gog through pot C, ad t s also o OM. If socty wr at pot, both popl would ar th sam com X (th cak s dvdd 50/50 ad socal wlfar wll b th sam as at pot C whr th dstrbuto s uqual. Not howvr that at pot th sz of th cak s smallr bcaus f w draw a l through whch s paralll to DG, t wll b closr to th org O. Th comparso of pots C ad llustrats thus th tradoff btw th sz of th cak ad th dgr of qualty of ts dstrbuto. Th lvl of com corrspodg to pot s calld "th qually dstrbutd quvalt lvl of com" (X. Ths prsso of "qually dstrbutd quvalt lvl of com" was troducd by Atkso (970 but th cocpt had b proposd arlr, udr aothr am, by Kolm (966. 2

3 Mor grally w ca comput th "qually dstrbutd quvalt lvl of com" by sayg that f such a com was shard qually, socty would hav th sam lvl of socal wlfar as th o stg ow wth a uqual sharg. I othr words w may wrt: W (, 2, W(,,... tms As bfor : rprsts that lvl of com that, f rcvd by vryo, would gv lad actly to th sam lvl of socal wlfar as th o prstly stg wth a uqual com dstrbuto. As mtod prvously wh socty movs o th dffrc curv from C to t wll hav a "smallr cak" but a mor qual sharg. I a crta way w ca say that O masurs th wlfar whl M rprsts th dcras th sz of th cak that was allowd by ths tradoff btw th sz of th cak ad th dgr of qualty of ts dstrbuto. v though socty lost wth rspct to th sz of th cak, t dd up wth a mor qual sharg ad hc could kp ts orgal lvl of wlfar. Atkso (970 suggstd thrfor th followg masur of qualty: I ((OM-O/OM ( OX OX / OX Ths dffrc may b prssd as: Iqualty -(quvalt lvl of com /avrag com Not frst that th furthr away s from M, th gratr th qualty. I othr words ad as pctd, ctrs parbus, th mor uqual th sharg, th hghr th masur of qualty, as show th followg graph: D X X X ' ' M O 45 X ' X X 3

4 Not howvr that th qualty d dpds also o th typ of socal wlfar fucto that s slctd. If w slct aothr socal wlfar fucto wth th sam allocato of com as C, w wll drv a d of qualty that taks aothr valu. From th graphcal pot of vw, what happs s that socal dffrc curvs may hav varous dgrs of covty. D H C Ê M F O 45 B Th prvous graph thus shows that wth th sam tal allocato of com but dffrt socal dffrc curvs w obta dffrt "quvalt lvls of com" (corrspodg rspctvly to, ' ad ~. I othr words th valu of th qualty d wll b dffrt wth th blu, gr ad rd socal dffrc curvs. Ths provs that th valu tak by th qualty d dpds ot oly o th orgal allocato of coms but also o o's socal phlosophy. Lt us tak two trm cass. I o trm cas w drw a socal dffrc curv whos shap s vry smlar to th o w draw producto thory to rprst a producto fucto wth fd proportos. Lt us call ths curv W ~ W kow that C ad ~, w hav th sam lvl of socal wlfar. All th pots alog th scto C ~ corrspod to th sam lvl of socal wlfar. Sc C ~ s vrtcal, t mas that although wh o movs from ~ to C, th com of Mart crass whl that of Rachl dos ot chag, socal wlfar dos ot vary bcaus th com of th poorst dvdual (Rachl dos ot chag. Th socal wlfar fucto such a cas may hc b wrtt as: W ~ M{com of Mart, com of Rachl} Mor grally wh thr ar dvduals th socal wlfar fucto wll b prssd as W M {, 2, } 4

5 Ths approach assums thus that what should b mportat to socty s what th stadard of lvg of th poorst prso s. Ths s fact th phlosophcal pot of vw tak by th socal phlosophr Joh Rawls hs famous book A Thory of Justc orgally publshd 97. Not that f w accpt Rawls's approach, w wll always obta th hghst possbl valu for th qualty d for a gv allocato of coms btw th dvduals. Aothr trm approach s to assum that oly th sz of th cak s mportat (ths s what s mplctly assumd wh coutrs ar compard o th bass of th pr capta com. Such a approach corrspods to th followg graph (th two straght ls ar assumd to b dtcally locatd: D M 45 O B G Thrfor all th pots alog th straght l DG corrspod to th sam lvl of wlfar ad hc th socal dffrc curv s fact DG. I such a cas, wll b dtcal to M: qualty s always zro sc qualty s ot mportat. Oly th sz of th cak s. Th Atkso (970 qualty d: Assum that thr ar dvduals ad that socal wlfar s qual to th sum of th utlts of th dffrt dvduals. W u( whr 5

6 u ( s th utlty of dvdual ad s th com of dvdual. Not that t s assumd frst that th utlty fucto s th sam for all dvduals, scod that ths utlty dpds oly o th com of th dvdual ad thrd that socal wlfar s addtv th utlts of th dvduals. I othr words th utlty of ach dvdual dos ot dpd o th coms of th othr dvduals, whch s qut a strog assumpto. Atkso assumd that th utlty fucto could b wrtt as: u ε ε ( a+ ( b /( wth a, b, ε f 0. If w assum a cotuous fucto w prss th utlty as u( a+ ( b /(ε ε Margal utlty s th wrtt as u ε ε ε ε '( u / ( b /( ( b f 0 Th scod drvatv s th prssd as u ''( b( ε ε p 0 Lt us ow comput th prsso [ u ''( / u'(. W th drv that ε ε [ u ''( / u'( [ ( bε / b ε Not that w ca also prss [ u ''( / u'( as u' ( / u' ( / Th lattr prsso s howvr smply th lastcty of th margal utlty wth rspct to com. So f com crass by %, margal utlty wll dcras by ε%. I fac ε corrspods to what s calld th coffct of rlatv rsk avrso or Arrow-Pratt coffct. Lt us ow drv Atkso's qualty d whch may b wrtt as: I ( X / X By assumpto w may wrt that 6

7 W ( X, X 2,, X W ( X, X..., X whch mpls that N [ a+ ( b /(ε ε [ a+ ( b /(ε ε a + ε ε ( b /(ε a+ ( b /(ε ε ε ε (/(ε [(/ Not that th last mprsso mpls fact that ma. Whε 0,. s what s calld a gralzd Wh ε, 2 (/ Fally th Atkso qualty d wll b wrtt as I ATKINSON ( [(/ / ε /(ε Not that sc th coffct of varato s qual to th rato of th stadard dvato ovr th avrag, wth th stadard dvato wrtt as 2 ( / (, thr s a lk btw th coffct of varato ad Atkso's d wh ε. Proprts of Atkso's d:. If w multply all th coms by a costat λ>0, thr wll ot b ay chag Atkso's d. 2. Atkso's d vars from 0 to. It s qual to 0 wh all th coms ar th sam. Atkso's d tds towards wh o dvdual "ats all th cak". 3. If w trasfr a (postv amout of moy from a rchr to a poorr dvdual, th d dcrass. 4. If w add th sam amout of moy b to all dvduals, th Atkso d wll dcras. 7

8 O th sgfcac of ε th dfto of th Atkso d To udrstad that ε masurs fact our sstvty to qualty ( fac ε corrspods to th coffct of rlatv rsk avrso ad so our cas t s a coffct of rlatv qualty avrso, w tak a smpl ampl wth two coms, y 0 ad y It s asy to chck th followg rsults: ε.5: W th gt : y [(/2( (whch s vdtly smallr tha th arthmtc ma of 0 ad 50 whch s 30. ε(2/3 W th gt : y [(/2(0 (/3 +50 (/ (whch s smallr tha what w got wh ε.5, ad hc s closr to th smallr of th two coms. ε2 W th gt : y [(/2(0 (- +50 ( (whch s smallr tha what w got wh ε(2/3. ε5 W th gt : y [(/2(0 ( (-4 (-(/4.887 Obvously o ca guss that wh ε, y y M{y, y 2 } Th partcular cas wh ε : I such a cas (ε w hav to rplac th utlty fucto U(y a + (b/(-εy -ε wth th utlty fucto U ( y c+ d l y. To drv th qually dstrbutd quvalt lvl of com wrt: y such a cas, w c + d l y c+ b[ l y l y (/ y l y l y l y (/ l y 8

9 whch mpls fact that y s th gomtrc ma of th coms. I our llustrato w would gt y ( Not that, as pctd, th gomtrc ma s smallr tha th arthmtc ma. Morovr th y w gt ths cas (ε ls btw th valus of y w foud for ε (2/3 ad ε 2. Th Kolm Id (976: Srg Kolm took a dffrt approach to qualty masurmt. Lk Atkso, Kolm assumd that th socal wlfar s addtv th utlty of th dvduals, that s, W u( But h dfd th utlty fucto as: u( c ( d / wth c, d, f 0. Usg a cotuous fucto w wrt ths utlty fucto as u( c ( d / Th margal utlty s th drvd as u' ( ( d / ( d f 0. Thrfor th scod drvatv wll b prssd as u ''( d( p 0 Not that hr also thr s dcrasg margal utlty. Now lt us cosdr th followg prsso u ''( / u'( [ d /[ d Thrfor u' '( / u' ( It appars clarly that ths cas th lastcty of margal utlty s ot ay mor costat but s a fucto of com. To drv th "quvalt com" w wrt as bfor that W (,.., W (,.., 9

10 0 so that w gt / ( [ / ( [ d c d c d c d c / ( / ( (/ l[(/ (/ Th (absolut Kolm qualty d s th dfd as KOLM I ma com mus "quvalt com", that s, l[(/ (/ + KOLM I Th ma proprts of Kolm's d ar as follows: If w trasfr moy from a rch dvdual to a lss rch dvdual, qualty dcrass. 2 If w add a fd sum to all dvduals, thr wll b o chag th Kolm d. 3 If w multply all coms by a costat, qualty crass.