Amercan Journal of Operaons Research, 3, 3, 487-496 Publshed Onlne November 3 (hp://www.scrp.org/journal/ajor hp://dx.do.org/.436/ajor.3.3647 Forecasng he Convergence Sae of per Capal Income n Venam Nguyen Khac Mnh, Pham Van Khanh Naonal Economcs Unversy, Hano, Venam Mlary echncal Academy, Hano, Venam Emal: hacmnh@gmal.com, van_hanh78@yahoo.com Receved Augus 7, 3; revsed Sepember 7, 3; acceped Sepember 6, 3 Copyrgh 3 Nguyen Khac Mnh, Pham Van Khanh. hs s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal wor s properly ced. ABSRAC Convergence problem of an economc varable represens an underlyng forecas of neoclasscal economc growh model. hs paper ams o analyze he convergence of provncal per capa GDP sably n Venam over he perod of 99-7. hs can be done by wo approaches ncludng bas daa-based regresson mehod for esng convergence and Marov chan model for descrbng feaures of long-erm endency of per capa ncome n Venam growh n provnces. he regresson mehod resuls n he sgns of convergence. o apply Marov process, we dvde oal paern no 5 per capa ncome classes. Resul esmaed from he Marov chan model shows he poor convergence. Keywords: Convergence; Regresson Mehod; Marov Proces. Inroducon Suppose ha we observe an economc varable η η ( x, beng a sochasc process dependen on parameer (me, x X (space wh consdered area X. Observaons y : η ( x, a me (perod ( - are nown, and we consder followng convergen concepon of economc varable y : E{ η (, x } ( s unlmed over me as he convergence of funcon y for fne value y ( (called convergence sae a sub-regons (provnce x x X ( - N lm y y( + Growh coeffcen ŷ of he economc varable reveals varable rae (he varable level of a un, varable y n a un of me a me : y( +Δ y Δ y y y y ( Δ y For he model wh y assgned o ncome or producvy, he convergence n ncome and producvy s among he mos-recevng aenon economc ssues n recen years. here s an urgen need for research on he convergence due o s heory and praccal value. heorecally, analyzng he convergence can help o dsngush he dfferen growh heores based on s forecas on economc growh. Oherwse praccally, convergence researchng can conrbue o plannng and evaluang he provncal polcy measures a me when he economc dfferences beween sub-regons n a regon are profoundly ganed. herefore, convergence researches are conduced wdely beween many naons and regons. Many auhors focused on he convergence n ncome, however, sudyng he convergence n GDP accordng o regons also provdes many such mporan nformaon. Globally, mplemenng he Barro recurren n convergence researches was menoned n many counres bu only he nformaon of he nal and fnal sages of he researches was analyzed. Explong hs nformaon a all sages was no se forh ye. In Venam, as he auhors recognzed, he mplemenaon of classc Marov mehod n examnng he convergence n ncome per capa and producvy, growh rae hasn been suded ye. herefore, hs sudy s desgned o nroduce a new mehod o Venam s economc growh research a he regon levels. hs paper amed o evaluae he convergence level n provnces of Venam (sub-regons based on ncome per capa whch s examned hrough her daa on GDP,
488 N. K. MINH, P. VAN KHANH populaon and manpower n he perod 99-7. he dea of convergence n economcs (also somemes nown as he cach-up effec s he hypohess ha poorer economes per capa ncomes wll end o grow a faser raes han rcher economes. As a resul, all economes should evenually converge n erms of per capa ncome. Developng counres have he poenal o grow a a faser rae han developed counres because dmnshng reurns (n parcular, o capal are no as srong as n capal-rch counres. Furhermore, poorer counres can replcae he producon mehods, echnologes, and nsuons of developed counres. In economc growh leraure, he erm convergence can have wo meanngs. he frs nd (somemes called sgma-convergence refers o a reducon n he dsperson of levels of ncome across economes. Bea-convergence on he oher hand, occurs when poor economes grow faser han rch ones. Economss say ha here s condonal bea-convergence when economes experence bea-convergence bu are condonal on oher varables beng held consan. hey say ha condonal bea-convergence exss when he growh rae of an economy declnes as approaches s seady sae.. heorecal Bass.. Economs Vew of Consdered Approaches Generally, convergence resuls from neoclasscal growh heory where for ceran se of economes, her economc growh s nfnvely unsable and lessenng a he end, and her speed s lely o come o saonarness as producon funcon drves downward performances by capal sze. If hese groups of economcs have smlar economc srucures, hey would converge a a same sable condon, narrowng he ncome gap. A ha case, an absolue convergence occurs. However, n he case he economcs see dfferences n her srucures, hey wness dverse sable condons and unceran decrease n ncome-gap, whch s called condonal convergence. Dfferences n her sable condons wll be parally explaned n some addonal varables (see []. hs paper only focuses on absolue convergence. When analyzng a sandard convergence, researchers nvesgae (he presence of he convergence whch presens a declnng ncome-gap and he convergence dsplayng wheher he poor naons wness a faser economc growh han he developed ones. I s sad ha absolue convergence aes place when nal ncome and s laer developmen have a negave relaonshp. From hs heorecal economc pon, he classcal Barro recurren models are wdely mplemened (see [-3], s reconed ha: n he observed me (,, he growh pace of economc varable y s defned as: Barro model α β y + ln y (3 Wh y ( : y s he nal value of economc varable; α, β server as parameer whch are esmaed based on he corresponden regresson equaons: y ln α + β ln y + ε - N y (Barro regresson equaon Wh ε -observed error, N sub-regon number (provnces, ces, locales survved n X (naons, zones.; y η (, x and y η (, x s he observed value n sub-regon x ( - N of economc varable a he sarer and n he suded me (,. Afer esmang he parameers n he above recurren formula, he Neoclasscal economcs paradgm s employed o provde suffcen sgns for varable y o converge (. Neverheless, before hs sgn s used (when economc varable presens for ncome or producvy, s a mus o chec he form of Cobb-Duglass n he funcon for manufacurng or he concave condon of combned funcon for manufacure. hs can always be acheved, especally n he Barro recurren when only he nformaon of s nal and fnal seps are suded (,. Addonally, hese menoned suffcen sgns can help he researchers come o a ceran concluson abou he nably of convergence of economc varable y. herefore, he group of auhors wll nroduce a new mehod named expanded Barro recurren mehod n he nex. o surmoun hs defec. A second mehod: Marov mehod (see [4,5], whch s usually used n publshed documen, wll be also nroduced n he.3. hs mehod requres he nformaon abou ransonal move of he researched uns ncluded n cross-selecon and emporal chan. he auhors come o a concluson ha convergence aes shape f long-erm forecass for hese movemens go oward o when forecas range ncrease. In Secon 3, he menoned mehods wll be ulzed o analyze he convergence n per capal ncome n Venam s provnces durng 99-7, and hen o pcure he ypcaly n he long-erm endency n he capal per ncome growh of Venam s provnces. Calculaon oucomes show he conssence beween mehods used n he hess and he over-advanages of expanded Barro recurren (compared o classc Barro recurren... Expandng Barro Regresson Model In he dfferenal equaon lnguscs, o deal wh problem, we need o buld Cauchy problem respondng o dfferenal Equaon under form: y y y >, y y (5 (4
N. K. MINH, P. VAN KHANH 489 where y s ncome per capa (GDP a he begnnng of research (, and ŷ represens growh rae of GDP a. For economc hess ha economc growh rae defnely develops, where per capa growh rae negavely depends on prmary ncome and gradually becomes lower and hen convergen a saonary sae a he end (f hese economes have very smlar srucures, we consder followng models from Barro models n [,] for growh rae as: Expandng Barro Model λ λ y e A ln y > (6 When consderng model, we pu: λ β : e ; α : β A (7 And we ge soluon for (6 under form: α β ln y y y e (8 hen, s easy o see he form of convergence condon for he problem: < β < λ > (9 From hs condon, we nfer: A y lm y e : o denfy regresson model for deermne parameers A, λ n he model a sudyng me (,, (8 s aen a logarhm and we oban: y ln α β ln y y And followng algorhm. Algorhm.. (Convergence of Expandng Barro model: Sep : For every perod (year - a me sudy, we se up N lnear regresson equaons n accordance wh parameers α, β based on : y ln α β ln y + ε - N (a y where y s observable daa ( x, vnce x ( - Le, ( - of parameers, ( - η a sub-regon (pro- wh respecve error. α β are leas square esmaors α β acheved from above regresson model. Sep : Apply (9 o chec convergence condon < β < : a f he condon (9 s unsasfed wh every -, problem wll be concluded no o be convergen and hen sop compung. b wh all ( β for nsance we nex o Sep 3 f he condon s sasfed. Sep 3: We oban leas square esmaors for model parameers based on (7: ( β ln λ λ : ; A A α : β (b where λ represens convergence speed of he economy and from, we refer: y ( e A. In comparson models n he algorhm wh classcal Barro model, we nally consder he regresson model, and defned parameers α, β n funcon: X y ye ; X X( ; αβ, : ( α+ βln y where y y ( x y y( x wh y, ;, y as observable varables; s me conrol varable; x x,, xn represen space conrol varables (sub-regon. Aachng resulng observaons, we se: y y, x ; y y η (, x, X ( : ln y (3 X y ye ; (4 X : X( ; αβ, : ( α+ βln y where, α β are leas square esmaors of α, β from he sysem of N regresson Equaons (4. We hen have saemen as follow: Lemma.: If y ³ and denoe: N N ln y M : ; N N ln y ln y σ σ M σ σ Wh all p ( p (5 < < we have: S P X X < p; (6 p where S : σ σ ln y,( +. Proof: When we pu f ( x : (,ln y ( x ; θ ( α, β ; θ ( α, β We have (see, (4: (7
49 N. K. MINH, P. VAN KHANH θ; X f x X f x θ (8 From (4, we also refer classcal Barro regresson Equaon (4 wh lnear form as: θ ε ε X f x + ; E N. Hence, for leas square esmaor θ of θ, we have (see (5: E α α, E β β; D θ M (9 Besdes, for symmerc and posve deermnaon of marx D, we also have: D α σ, D β σ, σ σ > σ. hen, from (8 and (9 we have: { } ( θ D X f x D f x ( σ σ ln σ ln + y + y { } ( σ σ ln D X < + y S Besdes, from, (4, (9, we refer: E X E α + E βln y { } ( α + βln y X hen, use Chebyshev nequaly from (3: D{ X } p P X X < p S P X X < (. p. hs esablshed he resul. Now, we consder Expandng Barro Model aachng defned parameers λ >, A n,. In hs case we have: Y e ; : ( α β ln λ : e, : A y y Y + y γ β α γ where y y y( x y( x,, wh y, y as observable parameers; represens me conrol parameer and x x,, xn are space conrol parameers (a sub-regon x x. For each ( we pu: η y η, x, y, x, Y y ln θ : α, β, y, α α γ γ β N regresson Equaon (a for specfyng α α, Y α γ ln y ε ; ( N γ β γ ( have lnear form as: + + (3 Le ( θ : α, β, s leas square esmaor of parameers θ : ( α, β, n (3. No o loose generaly, we suppose: < γ < ; Le λ, A are respecve leas square esmaors of λ, A n correspondng regresson equaons: Pu: γ λ+ ε ; A ( α γ + ε Y ye ; : ( α + γ ln :, : y Y y γ λ α γ A (4 (5 And we oban proposon as: Lemma.. If y and λ (small enough, we have: KS p P Y Y < ; (6 p where K : (, ( + ( + < p < and we oban approxmae value of λ, A for (b a < β : γ ( Proof: For vecors of f ( x n (7, he N regresson equaons are dsplayed under form: θ + ε ( Y f x N On he lnear form of parameers (, square esmaors θ ( α, γ θ α γ, leas for parameers θ are lely o be acheved, where (see (5: E α ; α Eγ γ, Dθ M D α, σ Dγ σ (7 Smlarly, for prmary regresson equaons n (4, as hey are under parameer λ-based lnear form and ob- servable varable γ ncludes varance D{ γ } σ leas square esmaor λ of λ sasfes: ;, E λ λ Dλ σ (8 When repeang he above provemen for of he second regresson equaons n (4, we can see ha hans D α σ (see o observable varable wh varance { }
N. K. MINH, P. VAN KHANH 49 (7, we have: ; σ EA A DA (9 wh  s leas square esmaor of A. λ < λ λ ferred from : Due o, can be re- ( γ λ γ λ hen, from, (5, (8, (9, we have { α } α ; { α } γ E D DA D{ α } σ σ { γ } λ γ D{ γ ln y} E ln y ln y ; ln yσ ( ln yσ (3 (3 From (3 and (3, we can employ nequaly Chebyshev o gan: n successon. A ha me: Pα α γ ( (, Pγ, σ δ δ > δ σ ln y δ δ δ > { } δ { } δ P Ω <, P Ω < Ω α α σ δ (, δ Ω γ γ σ ln y δ Base on hs algorhm and D Morgan rule, we acheve: { Ω Ω } { Ω Ω } P{ } P{ } δ ( δ P{ } P P Ω + Ω Ω Ω Pα α + γ γ ln y ( σ + σ ln y δ P α + γ y α + γ ln y ln ( σ σ ln y + δ ha means (revew, (5, (6: S PY Y < δ ( δ so when 6 Because ( + ( + seng above formula as p δ, we ge (6 < β : γ we have: When ( ( ( γ ln + γ ln β ln β γ γ β A ha pon, recurren Equaon (4 s approxmaely a: ln ( β ( α ( A β ( : α ( λ + ε ; + ε ; α
49 N. K. MINH, P. VAN KHANH herefore, we can evaluae he leas square esmaors λ, A of λ, A n he menoned recurren equaons by Formula (b. Amng o ndcang he preemnen advanage of expanded Barro, we provde followng algebrac clause: heorem.3. If he condons of Lemma. are me and < + + (3 We can examne he relave error y, y δ δ of he approxmae funcon y (under classc Barro and y (under expaned Barro by he formulas: y y y S p P δy : < e + : δ y p y y y ( KS p Pδy : < e + : δ y p, (33 (34 Wh he relably p, he esmaed value of relave error under classc model sees small dspary compared o s of expanded model: where S p ( K S p δ y δ y e e > (35 > K ( + ( + (36 Proof: Frsly, from he value of consan K n (6 and condon (3, (36 can easly found. Besdes, (4 nfer: X X X X X y y y e e X X ( e y + δ y + A hs pon, (6 can lead o: e. p P X X < S p s X X p P e + < e + s p Pδy < + e δ y ha means we successfully prove (33. Smlarly, from and (5, we also ge Y Y Y y y y e e Y Y Y Y ( e y + δ + herefore, we have y e KS p PY Y < P δ y < y p { δ } from (6 and (34 s demonsraed. Fnally, from expresson of δ y (n (33 and δ y (n (34, we can use K < n (36 o ge (35. he algorhms o evaluae errors (35 and (36 nroduced n above heorem am o compare classc Barro wh s expanded one n he perod [, ] when hey fully mee he condons n (3 (close evdenly. For example, when, hs condon s sasfed wh 5. However, when solvng a problem, we need o ae some followng facors no accoun. Noe. Because he LS esmaed α, γ of parameers α, γ n lnear regresson model (6 s seady esmaon: P lm α α, P lm γ γ N N In he same way, for he LS esmaed λ, A n recurren models (4 we have: P lm λ λ, P lm A A Now, (revew (4, (5: P,lm. y y N ha means, N (enough bg leads o..3. Marov Chan Models Consder problem wh pa ncome process, ( y beng a per caη s regarded as a/an (sochasc ncome per capa process n fac such ha E{ η } y(. Dvde he ncome per capa no n levels a - ( - a n wh a < a < < a n and pu ( n F F,, F as probably dsrbuon of η followng menoned earnngs level: We have ( { } η < ( - P a a F n ; : ; n n { η η [ } F PF P p + p : P + aj, aj a, a (37
N. K. MINH, P. VAN KHANH 493 If Marov chans are hen homogeneous respondng o ergodc ranson probably marx P, we have: n ( lm { η } lm F F : F,, F + lm y E + n lm ( a + a F + n + ( a a F : y + (38 hs s he reason for us o conver convergen problem research no valuae he homogeneous and ergodc Marov chans. Homogensaon suggess ha he probably of some provnce belongs o a wll fall no j a + as consan over me. A maxmum lelhood esmaon s gven by: p N N Here, N s he number of provnces ransferrng from o j a, N s oal provnces n a and represens processes. o es he nvarably of he ranson probably over me, we evaluae wn of followng hypoheses: Hypohess: : H p p Wh all and s assumpon as: where : H p p p N N s ranson probably esmae a. For hese hypoheses, lelhood rao s defned by: where,, j ( p N N p p λ ΠΠ p N ΠΠ are deermned n hypohess H and ΠΠ are deermned n hypohess H. And logλ s dsrbuon of χ ( n ( n s rue. herefore,, χ N p p p, j Have dsrbuon f ( n( n f H square s free order. Here, s he number of processes, n s sae class of Marov chans. 3. Expermenal Esmang Resuls Barro Model: Se ndependen varables are growng logarhm of per capa ncome n he provnce a lae y (, and dependen ones represen ncome per capa a early sages. Afer removng napproprae varables, fnal esmaon for he perod of 99-7 s as: y Ln.53.84Ln y Se R (.38 (.5.43; DW.747 ( y Esmaed resuls shows ha negave coeffcen β have no % lower sgnfcance level %. We also spl no small perods o esmae above equaon bu no ones see coeffcen β sascally beer han ha of he whole perod 99-7. Expandng Barro Model: From above heorecal models and per capa ncome daa of 59 provnces over Venam, we used analyss of regresson o analyze he convergence of economc varable GDP (ncome per capa. For each perod (year -7 ( 99-7, we buld 7 regresson equaons by vrue of cross daa: ln y α β ln y + ε y ( -59, -7 For hese equaons, we oban he esmaors for economc convergen speed: and value ( β 7 ln 7 λ.558 7 α Â.6 7 β Average earnngs a convergen sae s A e 63748( vnd y. We can use followng model o forecas average ncome n comng years:.558 A β β ln y ( e ( Aln y y y e y e (39 We oban forecased resuls as below able. Here, we denoe GDP s esmaor for GDP. Marov Chan Model: Consder parameer F whch represens provncal per capa GDP dsrbuon. o dsconnec F under form, we use an expermenal procedure whch
494 N. K. MINH, P. VAN KHANH able. Average earnngs forecass (un: vnd. 9 3 4 GDP 65.9 633.7 655.7 6775.9 76.4 74. 5 6 7 8 9 GDP 7483.3 779.7 798.6 838.9 85.6 8769.9 calculaes F n he prmary perod of 99-99 or 99-993 frs, and hen sors hem n progressve order. Wha more, we wll dvde F no several nervals so ha every nerval ncludes mnmum varances. Arranged urnng pons n F correspond o hresholds n he nervals, so we have C [ ;.5 ], C [.5;. ], C3 [.;.5 ], C.5;3., C 3.; [ ] [ ] 4 6 for gross rae. Expermenal resuls from Marov chans provde deep undersandng abou changng feaures of dynamc changng dsrbuon of provncal per capa GDP. mean of P n he perods s consdered as an esmaor of ranson probably marx P. Dsrbuon F consss of resdual values beween provncal average GDP and ha of Venam. F s calculaed and raned n order of each process. If a perod comprses 3 years, we wll have 6*59 observaons from 99 o 7. he economy wll dvde no 5 saes: Sae : Average GDP vnd.5 mllon. Sae : vnd.5 mllon < Average GDP vnd mllon. Sae 3: vnd mllon < Average GDP vnd.5 mllon. Sae 4: vnd.5 mllon < Average GDP vnd 3 mllon. Sae 5: Average GDP > 3 mllon. We have ranson marx sysem as:.449.575.477.463.464.545.57.773.4837.88.3333.6667 As n hs marx, Sae provnces wh lower VND.5 mllon per capa GDP wll conss of 4.49% percenage of hose who says a Sae and 57.5% percenage lef comes o Sae wh per capa GDP floang around VND.5 -. mllon. he remanng does he same. We can see from above marx ha ranson probables from Sae o Sae are greaer han Sae ranson probables comng o Sae 3 whch reman bgger han Sae 3 probables urnng o Sae 4 afer compleng a perod. hs s also convergen sgn of he economes. Afer 5 processes, we have ranson probably marx:.38.46.637.99.4879.49.63.84.7767.84.3.369.947.4.9959. hs s he upper rangular marx wh decreasng per capa GDP a Saes and whch ndcaes convergence mar of he economy. As n hs marx, provnces wh hgh per capa GDP are lely o fall no % saes afer saes afer 5 developmen perods. Smlarly, 48.79% of provnce wh of provnce wh lower VND.5 mllon per capa GDP ups o hgher VND 3 mllon. We connue o consder convergence a me by far. We oban followng ranson probably marx afer perods:..93.33.63.959.3.35.78.9865.5.7.6.997... herefore, 3 years behnd remans sae 5 called absorbed sae. hs means ha some process falls no sae 5 won be lely o come o ohers. Probably dsrbuon durng he perod of 5-7 as: F 5-7.339.864.3898.69.379 We have probably dsrbuon forecas for perod 8- as: F 8-.44.99.876.9.465 he daa ndcaes.5% percenage of provnce wh lower VND.5 mllon per capa ncome, 3% flucuang from VND.5 - mllon, 8.76% wh per capa ncome from VND -.5 mllon,.9% remanng ncome per capa from VND.5 o 3 mllon, and 46.5% wh hgher VND 3 mllon per capa ncome durng perod 8-. Consder process conssng of one year from 3 o 7, here remans 4*59 observaons. We jus consder lowes sae (lower VND mllon and hghes sae (hgher VND 3.5 mllon for enormous ncome per capa durng hs perod. We cu our economy n 5 followng saes:
N. K. MINH, P. VAN KHANH 495 Sae : Per capa GDP VND mllon. Sae : VND mllon per capa GDP VND.5 mllon. Sae 3: VND.5 mllon per capa GDP VND 3 mllon. Sae 4: VND 3 mllon < per capa GDP VND 3.5 mllon. Sae 5: Per capa GDP VND 3.5 mllon. We oban ranson probably marx as:.663.5.857.535.4599.67.343.6587.4663.5337.96.96.988 Sascs Ch-square for esng he nvarance of probably marx over me s: n p p p 3.33595. χ, j Sascs χ n p p p, j has dsrbuon Ch-square wh free order ( n( n. Here, 5 s he number of perods, and n 5 serves as he number of sae layers n Marov chans. Lmaon probably wh sgnfcance level α 5% s:.5 χ4 5 4.879478. herefore, we can affrm esed ranson probably marx and use for forecasng and researchng consdered ergodc Marov process. In hs case, Marov chan s ergodc because all posve elemens belong o ranson probably marx level 5.36.6.65.8.344.5.398.876.8.5877.3.5.5.9.8764.9.57.8.374.9359.39.96.45.9.947 And lmaon ranson probably represens a ranson marx behnd 8 or 3 seps. We have ranson probably marx afer 3 years as:.63.39.97.9.9.63.39.97.9.9.63.39.97.9.9.63.39.97.9.9.63.39.97.9.9 Based on hs resul, Sae 5 s lvng bu ohers are nearly gone ou. Ye, Sae remans.63%. Hence, we have dsrbuon forecas for 8, 9, and as follow: F8 (.384.394.697.3.63 F9 (.36.33.49.558.745 F (.8.38.8.79.8 F.6.37.96.9.95 Esmaed resuls show ha alhough here are sgns of convergence, hs process aes place n a very long fuure, abou 3 years more. Predced average value n 9 was approxmaely calculaed as follows: 5 y ( a + a F 689( vnd In, s 6397 (vnd and wll be 87 (vnd n. 4. Comparson o Classc Barro Models Aachng he same daa se N 59, 7 of Venam s provnces whn 99-997, we found ha he resuls calculaed under he algorhm. share he general pon wh Marov chan algorhm n he fac ha he resuls ndcae he common characersc, he convergence of he model. However, f usng classc Barro models for he same daa se, we have no ye found common characerscs menoned above. hs s also explaned by heorem.3 relaed o he lac of accuracy n classc Barro models compared o correspondng exended models. 5. Conclusons hs research has used dfferen mehods o sudy he ncome convergence of Venam s provnces whn 99-7. he esmaon resuls from Barro regresson models show ha Barro model doesn have sasc meanng. Esmaed resuls from expanson Barro regresson models based on cross daa ndcae sgns of convergence over he whole sudy perod. Resuls from exended Barro model are conssen wh he resuls under approachng mehod based on Marov chans and has been proved ha he error from exended models s smaller han from classc Barro models. Wh curren suaon and no change n polcy and regme, accordng o expandng Barro model, Venam s ncome per capa s abou 8 USD/year whch s very low compared o he developed economes oday. Marov chan model s used o llusrae long-erm flucuaons whn average growh n ncome per capa among 59 provnces across Venam n 99-7. Esmaon resul from Marov chan model ndcaes convergence
496 N. K. MINH, P. VAN KHANH sgns n dsrbuon bu should be n over 3 decades me. he reason mgh be obsacles n regme whch slows down echnology ranson process leadng o lmed mobly. Moreover, lac of necessary nfrasrucure and unfar dsrbuon causes echnology dffuson slowdown. he speed of hs process may vary among provnces. 6. Acnowledgemens hs research s funded by Venam Naonal Foundaon for Scence and echnology Developmen (NAFOSED under gran number II.--8. REFERENCES [] R. J. Barro and X. Sala- Marn, Convergence across Saes and Regons, Broongs Papers on Economc Acvy, Vol., No. 99, 99, pp. 7-58. [] R. J. Barro and X. Sala- Marn, Economc Growh, Mc Graw-Hll, New Yor, 995. [3] X, Sala- Marn, he Classcal Approach o Convergence Analyss, he Economc Journal, Vol. 6, No. 437, 996, pp. 9-36. hp://dx.do.org/.37/35375 [4] D.. Quah, Emprcal Cross-Secon Dynamcs n Economc Growh, European Economc Revew, Vol. 37, No. -3, 993, pp. 46-434. hp://dx.do.org/.6/4-9(9393-5 [5] A. B. Bernard and C. I., Jones, Comparng Apples o Oranges: Producvy Convergence and Measuremen across Indusres and Counres, Amercan Economc Revew, Vol. 86, No. 5, 996, pp. 6-38.