Dynamic Poverty Measures

Transcription

1 heorecal Economcs Leers do:436/el34 Publshed Onlne November (h://wwwscrporg/journal/el) Dynamc Povery Measures Absrac Eugene Kouass Perre Mendy Dara Seck Kern O Kymn 3 Resource Economcs Wes Vrgna Unversy Morganown USA Faculy o Economcs Chekh Ana Do Unversy Dakar Senegal 3 Dvson o Fnance Economcs Wes Vrgna Unversy Morganown USA E-mal: kernkymn@malwvuedu Receved June 9 ; revsed Augus 7 ; acceed Augus 5 In hs aer we roose mehods or deecng he number o ores based on dynamc omzaon echnques An llusraon s rovded he resuls are dscussed based on Governmen s objecves conrol varables Keywords: Omzaon echnques Povery Measures Inroducon How overy s measured s a cenral oc n economc olcy analyses However recenly has clearly aeared ha s no only he deermnaon o arcular overy levels a arcular nsans (based on several ndces avalable n he leraure) ha maer he mos he ahs o overy levels over me are also crcal crucal ndcaors n assessng he ecency o overy measures (Carle 6 []) hs aer adds o he leraure on hs oc by rovdng mehods or measurng overy n dynamc envronmen (Da Poescu 996 []) he roosed aroach answers he ollowng queson: How moran are dynamc omzaon echnques or overy measure analyss? he remer o he aer s organzed as ollows: In Secon we address he roblem o overy measures n a dynamc conex based on omal conrol An llusraon s rovded dscussed n Secon 3 Fnally some concludng remarks aear n Secon 4 Dynamc Povery Measures In he conex o dynamc omzaon me does maer he ahs o overy ndexes are moran he Problem n Dscree me Consder an nal nsan We assume ha a sac model has been used o deermne he number o ores n a gven oulaon based on a gven overy lne Le be he me ndex Y he vecor o revenues o he ores who have been dened a me Le u be he vecor o he conrol varables whch reresen he se o comms he objecve s o measure he erormance o he sysem o hs end we consder an objecve uncon o omze subjec o some consrans he above roblem can be ormalzed as ollows (Rusag 997 [3]) O J J : k Y a u () Y g Y u () he second equaly reresens he consrans on he sae vecor whch s he vecor o revenues Y a he objecve a each erod o me u he conrol varables he roblem now s o choose he bes conrol vecor u a each erod o me accordng o avalable resources such ha he above sysem s sased (Rusag 997 [3] Mar 997 [4]) In hs ye o roblem s he nal sage whch s he mos moran snce he objecve s o reduce or elmnae overy hs o course deends on nermedary objecves More seccally he dynamc omzaon roblem can be se u as a mnmzaon roblem o J (rouman 98 [5]) mn J J : k Y a u Y a Y a uu How o jusy he choce o he uncon ky ( a u )? In he above roblem s he nal revenue (3) Coyrgh ScRes EL

2 64 E KOUASSI E AL whch s he mos moran e Y In ac he objecve s o ge he ndvduals n he vecor o revenues Y ou o overy hereore nermedary objecves o be reached any se o decsons a me should be such ha Y Z Z s he overy lne From a mahemacal on o vew a he Eucldan norm o a e a mus be a leas o he same order as he overy lne o he oulaon under consderaon o hs end suces o rove ha a (4) (5) or a o o s he Lau noaon Equaon (4) has an mmedae soluon a ; In addon anoher consran s ha when r he level o rchness o he ndvdual r mus be n adequacy wh he overy lne no hgher e Y mus be hgher han he overy lne hereore he consran on he nal objecves mus be such ha (6) a (7) Regardng he sysem o conrol Y g Y u or smlcy we consder a lnear sysem hereore Y AY Bu (8) A s a square marx o order e A aj B s a marx wh rows j columns e B b j j hese marces can be dened usng economc heory Summarzng he dscree me roblem we have (rouman 98 [5]) mn J J : Y ay a uu s Y AY Bu a a (9) he Problem n Connuous me Based on he above develomens by analogy wh some mnor modcaons n connuous me he dynamc roblem can be ormalzed as ollows mn J : J Y a u d s dy AY Bu d a ; () u: a vecor o connuous se uncons mn J maxj Se G J Y u a 3 Soluon Y a u o conserve sace he soluon gven here s relaed only o he connuous case Deducon o he dscree me soluon s hen sraghorward Le V be he maxmal value o he objecve uncon I s easy o very ha V Y gven u: V Y gven u: AsY s Bsus Ys Y V Y max s Y s u s a s ds V Y V Y mn s Y s u s a s ds dy s () d Le us assume ha V s derenable n Y hen V Y mn Y a u V A Y B u V Y Y o u Coyrgh ScRes EL

3 E KOUASSI E AL 65 V wh Y s he Y graden oeraor Dvdng by leng we ge he Hamlon-Jacob-Bellman aral dervave equaon o he orm Y a u V mn V AY Bu Y gven u () We hen have he ollowng heorem: heorem Assumng here exss a derenable uncon V : whch sases () Assumng ha: : wh a connuous uncon n Lschz n Y sasyng Y a Y V A Y B Y Y mn Y a u V AY Bu Y gven u (3) hen s a conrol omal eedback or roblem (3) e V s he mnmum o J Proo: (See he Aendx) In he connuous case by analogy o he dscree case he general model roosed here s O : J ky a u d s dy g Y u d he secc model roosed s mn J: J Y a u d s dy A Y Bu d Y Y Y Y (4) (5) a s dened s a square negrable uncon on such ha a a (5) Y Z u wh u j a connuous se uncon j heorem Under he assumon o conrollably o he above sysem he roblem o mnmzaon adms an omal soluon Pahs omal conrols are obaned by resolvng he ollowng sysem dy A Y B B d d Y a A d Y Y (6) Y KY Z Y s he vecor o Ponryagn mullers Y he sace conanng condons on he Pon- ryagn mullers Y s dened n he Aendx Proo: (See he Aendx) 3 A Paramerc Illusraon As an llusraon we consder a general roblem aces by a Governmen n deermnng dynamc overy measures over me o ge a more neresng case we consder a aramerc roblem 3 he Problem Consder a aramerc dynamc omzaon roblem governmen auhores have some lexbly on nermedary objecves as well as he conrol varables he aramerc dynamc omzaon roblem can be ormalzed as ollows mn J Y ma m u d s dy AY Bu d a Y Y YY Coyrgh ScRes EL

4 66 E KOUASSI E AL a a unconal such ha a d A M B M or smlcy we assume ha he coecens o A B are no me deenden For he sake o lexbly he objecves as well as he conrol varables are arameerzed so as o accoun or ossble changes durng he mlemen- aon o Governmen economc olces; m m 3 Soluon Usng he Ponryagn rncle we ge he ollowng omaly sysem Mn J: J Y ma m u d s dy AY BB u d m d Y A ma d Y Y YY Y Z Y Y Se X We ge dx A BB m X d ma I A Y Y Y Z he above derenal sysem can be wren as dx d A BB m X ma I A Fm Cm d Or smly as X Cm X F m d C M F M m m wo cases mus be consdered deendng on he ac ha Cm s dagonalzable or no o conserve sace we consder only he rs case when egenvalues are real dsnc Noe ha he omaly sysem even hough lnear s oo general n s exresson n he sgn o he second member he resulng general resuls wll hen be dcul o nerre Le us consder a smle case Agan or smlcy we assume ha A B he omaly sysem s hereore dy Y m d m d Y ma d Y Y o x deas consder ha a he underlyng marces are m C m Fm m he egenvalues o are m C m wh m m s dagonalzable he marx o assocaed egenvecors P s dened as P m m P m m m 4m m m m 4m m m he derenal sysem s now equvalen o du d u P Fm du u d Solvng we ge m u ce m m s a consan o negraon c C m Coyrgh ScRes EL

5 E KOUASSI E AL 67 m u ce m m c beng a consan o negraon as well Snce U P X X PU Y X we oban Y u m m u m m Aer a b o algebra we ge m Y ce ce m c e ce m he omal conrol u s gven by u m We now dscuss several cases 33 Dscusson Case : m m c c Y c c c u c Povery can be grealy mroved on he condon ha nermedary objecves conrol varables be realsc comrehensve Case : m m xed e e Y c c ce ce ce ce u m Povery can be gradually mroved nermedary objecves are reachable conrol varables reasonably seleced Case 3: m m c Y ce ce c u ce Povery can be gradually mroved bu many conrol varables can creae enroy n he sysem Case 4: m xed m c c Y c c c u c Realsc nermedary objecves well chosen conrol varables may resul n osve macs n erms o overy allevaon Case 5: m xed m xed he behavors o Y u deend on he secc values assgned o m m Case 6: m xed m c Y ce ce c c u c Fxed objecves can be benecary or overy mrovemen bu many conrol varables can negavely aec he sysem Case 7: m m Y u oscllae o a case o uncerany Case 8: m m xed Y u Many objecves wh reasonable conrol varables may resul n overy mrovemen Case 9: m m Y u oscllae o anoher case o uncerany 4 Fnal Remarks How moran are dynamc omzaon echnques or overy analyss? In dynamc sengs he ahs o ncomes are essenal he aer rovdes mehods accordngly I remans o esablsh omaly sably crera or he characerzaon o he varous ahs n a uure research 5 Reerences [] P G Carle Inroducon à l analyse Numérque Marcelle e à l omsaon Dunod Pars 6 [] J M Da D Poescu Comme Omale Conceon Omsée des Sysèmes Ddero Ars e Scences Pars 996 [3] J S Rusag Omzaon echnques n Sascs AP Harcour Brace Comany Publshers San Dego 997 [4] R Mar Omsaon Ineremorelles: Alcaon aux Coyrgh ScRes EL

6 68 E KOUASSI E AL Modèles Macroéconomques Economca Pars 997 [5] M rouman Calculus o Varaons wh Elemenary Convexy Srnger-Verlag New York 98 Aendx: heorems Proos heorem Le Y u : be a vecor o uncons whch reresens conrol varables Y he soluon o Y AYBu YY Le Y be he soluon o Y AY Bu Y Y u Y he assumon on nsures he exsence o he soluon o he above sysem o show ha s an omal eedback conrol we need o rove ha d Y a u Y a u d Se Y A Y B Y D Y V Y We have V Y Y YV Y A Y B Y mn Y Y gven hus YV Y A Y B Y u Y Y V Y Y YV Y A Y B Y Hence V Y V Y Y Y d Y u Y a u On he oher h V Y V Y Y u Y a u dv Y d On he one h d dv V V Y V Y Y Y dy d d YV Y d d dv V dy Y Y YV Y We can mmedaely noce ha u d d dy Y u YV Y d dy A Y B u dy mn d Y u YV Y d Hence u Y Snce by assumon Y Y Y D dy mn Y u YV Y gven u d V YV Y A Y B u Y V Y gven u Y u d Coyrgh ScRes EL

7 E KOUASSI E AL 69 We hen conclude usng he Hamlon-Jacob-Bellman equaon he ac ha Y YY ha d Y u Y u d heorem We have mn J : J Y a u d s dy A Y B u d Y Y Y Y Y Z u wh u a connuous se uncon j he Hamlonan s H Y u Y a u A Y B u s he vecor o Ponryagn mullers he rs order condons are hus H Y u d Y d H Y u d Y d H Y u Y Y Y a A Y Y a j j j j j A Fnally we ge d H Y u Y d lowng derenal sysem d d Y a j Y G Y We now have he ol- j j j In our case G Id G Y ; x b G g G Y Y x x G b : A he omum we have H u ) I j Y G Y u j B u u j j u B j hen u B ) I hen B By assumon we whch s conrol- dy have d lable on B u A Y Consder dy A Y he uncon o ranson marces o d conrollably s equvalen o B he assumon o We clam ha he sysem s conrollable hen d oherwse rom Y a A d we would have deduced B hus B Hence whch s one condon snce canno boh be equal o zero a a me hereore We can now normalze he sysem by seng he ah omal conrol are obaned by solvng he ollowng sysem dy A Y B B d d Y a A d Y Y Y K Z Z Y Z Coyrgh ScRes EL