Stochastic Versions of Turnpike Theorems in the Sense of Uniform Topology

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1 MPRA Munich Peronal RePEc Archive Sochaic Verion of urnie heorem in he Sene of Uniform oology Darong Dai School of Buine, Nanjing Univeriy 6. Aril Online a h://mra.ub.uni-muenchen.de/46/ MPRA Paer No. 46, oed. July : UC

2 Sochaic Verion of urnie heorem in he Sene of Uniform oology Darong Dai School of Buine, Nanjing Univeriy, Nanjing 93, P. R. China In hi aer, a ochaic endogenou aggregaive growh model i conruced and wo main reul are eablihed, baed on endogenou horizon of he economy and endogenou erminal caial oc, which i alo efficien caial accumulaion in ome ene. Fir, rong urnie heorem under uncerainy and in he ene of uniform oology are obained; econd, inefficacy of emorary fical olicy, which i choen o be caial income axaion, ha been demonraed in comaraively wea condiion differen from Yano (998). Key ord: Sochaic endogenou growh; Uniform-oology urnie heorem; Endogenou erminal oc; Endogenou welfare funcion; Caial income axaion. JEL Claificaion Number: C6, C6, E, E, E6. Correonding auhor.

3 . INRODUCION Our goal of hi aer i o udy urnie heorem and he effec of emorary fical olicy, which i ecifically choen o be caial income axaion, in a ochaic endogenou growh model, wih he ource of uncerainy i he oulaion ize of he rereenaive houehold. Comeiive equilibrium aumion are alo emloyed, ha i he firm, uing AK roducion echnology (e.g., Barro, 99; Rebelo, 99; urnovy, ; Aghion, 4), ha zero rofi in he equilibrium of he economy. In he a everal decade, he o-called urnie heorem have been exenively udied and well underood. Mo of hem (e.g., Morihima, 96, 965; uui, 966, 967; McKenzie, 963, 976; iner, 967; Cole, 985; Yano, 984a, 984b, 985; Bewley, 98; Gale, 967; Ganz, 98; Drandai, 966; and Araujo and Scheinman, 977), however, focued on he following four ye of ecificaion: fir, muli-ecor economie or general equilibrium model wih many conumer and roducer; econd, fical olicie are generally excluded in heir model; hird, hey ju concern he deerminiic cae, i.e., uncerainy i uually excluded in heir model; and fourh, he horizon of he abrac economy, fixed finie of infinie, and he erminal oc are all exogenouly given. here are cerainly ome exceion, for inance, fical olicy ha been conidered and carefully udied in Yano (998) model. Raher, Yano demonraed ha a emorary change in fical olicy ha almo no effec on reen and fuure conumion wih aing he general equilibrium rice effec ino accoun in a dynamic general equilibrium model, hinging on he following hree ye of aumion: fir, he exience of an inerior dynamic general equilibrium; econd, he moohne of uiliy and roducion funcion; and hird, he uniquene of a aionary equilibrium conumion vecor in he cae of undicouned fuure uiliie. Moreover, Johi (997) rovided a comrehenive develomen of urnie heory in a ochaic aggregaive growh model, exending he claical urnie heory o general non-convex and non-aionary environmen. Alhough he model in he aer i a ochaic aggregaive growh model wih he effec of emorary fical olicy being horoughly examined, unlie Yano (998), our concluion of he inefficacy of emorary fical olicy on equilibrium conumion ah hold rue in comaraively wea condiion, ay, given he iniial level of caial oc ufficienly low, in he cae of dicouned fuure uiliie, and in a more realiic

4 ochaic environmen. ha more, here he ource of uncerainy i uoed o be oulaion ize of he rereenaive houehold, hereby leading o a ochaic diffuion roce of caial accumulaion, while Johi (997) direcly and exogenouly inroducing he ochaic environmen a indeenden variable ino roducion funcion. Furhermore, when dicuing efficien caial accumulaion (e.g., Gong and Zou,,), efficiency i uually defined wih reference o he final ae (ee, Radner, 96; Kurz, 965) or he erminal oc (ee, McKenzie, 963, 976). In hi aer, alo, he erminal oc, equivalen o efficien caial accumulaion in ome ene, i endogenouly deermined a well a he oing ime of he economy, which i an oimal oing ime ha maximize he final-ae objecive funcion of he rereenaive houehold, i.e., chooing a minimum ime o a o maximize he dicouned uiliy funcion, which, o ome exen, reemble Kurz (965) ecificaion, ha i, minimizing he ime o economic mauriy. And hence i i argued ha one conribuion of hi aer i o how ha he horizon of he economy and he erminal caial oc, alo efficien caial accumulaion, can be imulaneouly and endogenouly deermined, hereby endogenouly generaing a ingle welfare funcion in an aggregaed model of oimal growh. And i i eay o ee ha our reul i a naural correondence o Bewley (98), which how ha he ocial welfare funcion i endogenouly deermined by he mare mechanim in decenralized model of oimal growh. Finally, noing ha exiing urnie heorem, in oimal growh heory, a Yano (985) argued ha, can be ummarized a he following wo ye, one i neighborhood urnie heorem (ee, Yano, 984b; McKenzie, 98) which aer ha an oimal ah in a growh model converge o a mall neighborhood of a aionary ah, he oher i aymoic urnie heorem (e.g., Araujo and Scheinman, 977; Bewley, 98; Yano, 985) which mean ha an oimal ah converge o a aionary ah. Here, we have roved much ronger urnie heorem in he ene of uniform oology, which we may call uniform-oology urnie heorem, and hi would aear o be he econd innovaion of he reen aer. he re of he aer i organized a follow: ecion reen he model and our ey heorem, ecion 3 give ome concluding remar and he aendix rovide he main mahemaical derivaion. 3

5 . HE MODEL e aume ha he economy admi a rereenaive houehold wih inananeou uiliy funcion u() = ln(), i.e., wih log reference. Our goal in he aer i o inveigae urnie heorem in a ochaic abrac economy, and here he ource of uncerainy i he oulaion ize L ( ) (e.g., Meron, 975), which grow in accordance wih he following ochaic differenial equaion (SDE), dl() = nl() d + L() db() () where Î i ome conan and B( ) i a andard Brownian moion on a given comlee robabiliy ace (,, { } ³, ) wih naural filraion{ } ³ and B () = -a... o reare for he houehold oimizaion, le u denoe he ae holding of he rereenaive houehold a ime by ( ), hen we ge he following law of moion for he oal ae of he houehold () = (- r ) r() () + w() L() -c() L() () where c ( ) i conumion er caia of he houehold, r ( ) i he inere rae on ae, w () L () i he flow of labor income earning of he houehold and r i uoed o be an effecive ax rae on he rae of reurn from caial income. Pu er caia ae a a () = () L (), hen i follow from (), () and Iô formula ha, da() = [(- r ) r() a() + w() -c() - na() + a()] d - a() db() (3) On he oher hand, we ecifically ado he following aggregae roducion funcion, Y () = AK () wih A >. Noice ha hi roducion funcion doe no deend on labor, hu wage earning, w ( ), in (3) will be equal o zero. Dividing boh ide of hi equaion by L ( ), and a uual, define () K () L () a he caial-labor raio, we obain er caia ouu a f ( ( )) = y ( ( )) º y ( ) Y ( ) L ( ) = A ( ) (4) 4

6 from which i i eaily een ha ouu i only a funcion of caial, and here are no diminihing reurn. ha more, he Inada condiion are no longer aified. In aricular, which i eenial for uained growh. lim f ( ( )) = A> () he condiion for rofi-maximizaion require ha he marginal roduc of caial be equal o he renal rice of caial, R ( ) = r ( ) + d, in whichd i he dereciaion rae. Since, a i obviou from equaion (4), he marginal roduc of caial i conan and equal o A, hu R( ) which imlie ha he ne rae of reurn on he aving i conan and equal o = Afor all, r () = A- d, " ³ (5) Nex uing he fac ha a ( ) = ( ), w () =, c () = (-r ) A () and equaion in (5), one can rewrie (3) a d() = [ r A-d-( A-d) r - n + ] () d - () db() (6) wih () and r denoing he aving rae. hen i follow ha, LEMMA. here i ome e (, ) < uch ha for " < and " < <. Proof. See Aendix A. u ( ) e(, Now, we conider he following ecial objecive funcion, ) ò ex( -r( - ))ln[ c()] d+ U (7) where < and i an -oing ime, which wih he ermu are deermined by he following oimal oing roblem g e r A (, ) -r (, ( )) u ln[( - ) ( )] Î æ ö (, ) -r ç= ç u e ln[ c( )] è Î ø 5

7 ln[( ) ( )] (7 ) (, ) - r = e -r A ubjec o he ochaic differenial equaion in (6), and { - oing ime}. In wha follow, we will calculae he oimal oing ime in a ochaic diffuion roce. Le Y ( ) ( +, ( )) and () (, ()) Y = (, ), hen he generaor of Yi, ( ) f f f f(, ) = + [ ra-d-( A- d) r+ - n ] + (8) If we ry a funcionf of he form -r l f(, ) = e for ome conanl Î we ge -r l (, ) = e { - + [ ra- -( A- ) r+ - n] + [ ( -)]} f r d d l l l = e -r l h( l) in which h( l) ( l ) + [ r A-d-( A- d) r + ( ) -n] l-r (9) Solving equaion h( l ) = give he unique oiive roo, ( ) d+ A- d r + n-r A- ( ) + D l = () where wih hi value ofl we u D= [ ra - -( A- ) r+ ( ) - n] + d d r -r l ìï e C, (, ) Î D f(, ) = ï í ï -r ïî e ln[( -r ) A], (, ) Ï D () -r for ome conanc, o be deermined. If we le (, ) g e ln[( -r ) A], we have g e r A ra A r n -r (, ) = {-rln[(- ) ] + [ -d-( - d) + - ]-( )} > < r A- - A- r - n+ - r A ex{[ d ( d) ( )] r} [( ) ] 6

8 herefore, U r A A r n r A = {(, ); < ex{[ -d-( -d) - + ( )] r} [(- ) ]} () hu, we gue ha he coninuaion region D ha he form for ome uch hau Í D, i.e., D={(, ); < < } (3) Hence, by (3) we can rewrie () a r A A r n r A ³ ex{[ -d-( -d) - + ( )] r} [(- ) ] (4) r l ìï e C, < < f(, ) = ï í ï - r ïî e ln[( - r ) A], ³ - (5) for ome conanc > (o be deermined). e gue ha he value funcionf ic a = and hi give he following high conac -condiion, and Combining (6) wih (7) one can ge l C ( ) = ln[( - r) A] (coninuiy a = ) (6) l Cl ( ) - = ( ) - (differeniabiliy a = ) (7) l C ( ) ln[( -r ) A] = l- - Cl( ) ( ) and o ummarize, hen we ge, = [ex( l)] [(- r ) A] (8) -l -l C= ( ) l= {[ex( l)] [(- r ) A]} l (9) HEOREM. (ENDOGENOUS EFFICIEN ERMINAL CAPIAL SOCK) Under above aumion and conrucion, if <, < r, and d d d d r + ( A- ) r + n+ ( ) < + r A + n+ ( A- ) r + -( ), hen we obain he oimal - oing ime = D inf{ ³ ; ( ) = }. In oher word, 7

9 - r -l l g (, ) = e ( ) l = U, which i a uermeanvalued majoran of g(, ) wih and l i given by (8) and (), reecively. Proof. See Aendix B. REMARK. he heorem how ha he horizon of he economy and he erminal oc, which i alo efficien caial accumulaion in he ene of maximizing he dicouned welfare funcion of he rereenaive houehold referring o (7 ), are endogenouly deermined. Nex we will udy he urnie heorem in he ochaic growh model. HEOREM. (LOCAL UNIFORM-OPOLOGY URNPIKE HEOREM) Given a comlee filered robabiliy ace { } ³ (,,, ).If + = d+ - d +, ra ( A ) r n hen () i uniformly bounded for Î [, ]( " > ) and for aaw..,and furhermore () uniformly converge o for Î[, D ] and for aaw..,where D i he oimal oing ime defined in heorem. Proof. See Aendix C. Now, we will rovide ome local characerizaion of he efficien erminal caial oc by he following heorem. HEOREM 3. (NEIGHBORHOOD PROPERIES OF HE EFFICIEN CAPIAL SOCK) If + ¹ d+ + - d, () ra n ( A ) r will ill be a local maringale on robabiliy ace (,, ) (" > ), where i equivalen o, and i () ochaically ulimaely bounded. Moreover, here exi a conan E > and a iener meaure n, defined on he canonical robabiliy ace for Brownian moion, on Borel igma algebra B ( C [, )) generaed by ( ( ); ³ ) uch ha di(, ) (i) ê B ( ) ú, a a - E 8

10 (ii) in which And ì ü limu ï ( ) dï í - E ò, ïî ï E ³ -, a (iii) n { Ba ( )} { - < a " ³ }, { Î a } B( ) ( ); ( ), a di(, ) log( ), inf ; ( ) B( ), B ( ) which i he Kullbac-Leibler diance beween and wih E < a, " a >, " () = > and i defined in (8). Proof. See Aendix D. Moreover, we can obain he following urnie heorem abou caial accumulaion, hereby exending he concluion in heorem. HEOREM 4. (UNIFORM-OPOLOGY URNPIKE HEOREM ) here exi ome C(, ) > uch ha a Paricularly, if, hen we have Proof. See Aendix E. u ( )- C(, ) for " > and " > lim u ( )-. ê REMARK. hi urnie heorem imlie ha he ah of caial accumulaion will uniformly converge o he efficien caial oc, alo he erminal caial oc, if he ochaic effec i ufficienly cloe o zero. And hu hi heorem rovide condiion under which he erminal caial oc i uniformly reachable, which i obviouly much ronger han Johi (997) argumen. Now we conider he following ochaic oimal conrol roblem facing he rereenaive houehold, 9

11 ubjec o max ex( -r( -))ln( c( )) d c () ò d r A n c d db () = {[(- )( -d)- + ] ()- ()} - () () e rove ha here exi a coninuouly differenial funcion ( ( )), aifying he following Bellman-Iaac-Fleming differenial equaion, r( ( )) - ( ) ( ( )) = max( ln( c ()) + ( ()){[( )( ) -r A-d - n+ ] ()- c ()})() c () Alying he maximizaion oeraor, yield he following condiion for a maximum a Subiuing () ino () roduce ry c () = ( ()) () r - =- + -r A-d - n+ - ( ( )) ( ) ( ( )) ln[ ( ( ))] ( ( ))[( )( ) ] ( ) ( ( )) = C + C ln( ( )) for ome conanc, C o be deermined. hen i i eay o ge, And And hence by () C r r r r A d n r - - = {ln( )-( ) + [(- )( - )- + ]- } C = r - c ( ) = r( ) = r()ex{[ r A-d-( A- d) r + ( ) -n] - B( )} () hu, in order o udy he effec of emorary fical olicy, i.e., caial income axaion, on equilibrium conumion ah, we now define c () c (, r ) = r() r(, r ), c () c (, r) = r () r(, r), () º () where r and r are wo differen emorary fical olicie. hen we ge he following heorem, HEOREM 5. (INEFFICACY OF EMPORARY FISCAL POLICY) If we chooe () uch ha

12 () ( + ) ( ) - ex( ) - e e 3, " < <, " e > where d d, ra- -( A- ) r + -n d d, ra- -( A- ) r + -n ra d A d r n, - -( - ) ¹, hen we obain, Proof. See Aendix F. limu c () -c () a e. ê REMARK. hi heorem how ha, given wo differen emorary caial income axaion olicie r and r, he diance beween he correonding equilibrium ah of conumion allocaion, c () and c ( ), i arbirarily mall in he ene of mean-quare uniform oology if he iniial level of caial oc i ufficienly low, which differ from Yano (998) requiremen ha he dicoun facor i ufficienly cloe o. By heorem, one can u, And, by (), Iô formula and (6), we ge hen we ge he following heorem, c (- r ) A dc () = r[ r A-d-( A-d) r - n + ] () d - r() db() (3) HEOREM 6. (UNIFORM-OPOLOGY URNPIKE HEOREM ) here exi ome C(, ) > uch ha u c ( ) - c C(, ) " ³ Moreover, if r or, hen we ge

13 lim u c ( ) -c " ³. ê Proof. See Aendix G. REMARK. hi urnie heorem how ha he equilibrium conumion ah will uniformly converge o he efficien conumion allocaion of he dynamic equilibrium economy, condiioned on ufficienly mall dicoun facor or ochaic effec. And i i eay o find ou he difference beween hi urnie heorem and hoe in Yano (984a, 984b, 985), which, in aionary environmen, require he dicoun facor ufficienly cloe o one. Now we will rove he urnie heorem for equilibrium allocaion vecor ah of he dynamic economy, and we define (- ) Φ() () y() c () (4) (- y c ) Φ (5) where i defined in (8), and c (- r ) A = (-r ) y. hen by (4), (6) and (3), we u æ d() ö æ [ r A d ( A d) r n ] ö æ ö dφ () = dy() A[ r A d ( A d) r n ] () d A = () db() (6) çdc () r[ r A d ( A d) r n ] ç ç-r è ø è ø è ø hen we obain he following heorem, æv ö æ ö V d () A +- db () () çv ç- r è ø è ø HEOREM 7. (UNIFORM-OPOLOGY URNPIKE HEOREM 3) here exi ome C(, ) > uch ha 3 Moreover, if, hen we have u ( ) Φ - Φ C (, ) " ³, ê lim u ( ) Φ -Φ " ³, ê where denoe L - norm.

14 Proof. See Aendix H. REMARK. he economic inuiion of hi urnie heorem i ha he equilibrium allocaion vecor ah of he dynamic economy will uniformly converge o he efficien allocaion vecor including caial, ouu and conumion, when he ochaic effec i ufficienly mall. And, wha more, we can eaily ee ha hi urnie heorem doe no deend on he conrain of dicoun facor lie hoe urnie heorem roved in Yano (984a, 984b, 985). 3. CONCLUDING REMARKS In he aer, ochaic verion of urnie heorem have been eablihed in a ochaic endogenou growh model and he inefficacy of emorary fical olicy which i ecifically choen o be caial income axaion ha alo been demonraed under relaively wea condiion. o ummarize, here are hree novelie in he aer: fir, we rovide a oible way maing he horizon of he economy and he erminal caial oc, alo efficien caial accumulaion in ome ene, all endogenouly deermined; econd, we rove ha a ingle welfare funcion in an aggregaed model of oimal growh can alo be endogenouly defined a i hown in decenralized model; hird, we rove much ronger urnie heorem under uncerainy and in he ene of uniform oology, which we call uniform-oology urnie heorem. Obviouly, he reen udy can be eaily exended a lea from he following hree direcion: fir, jum diffuion roce lie Iô-Lvy roce can be inroduced ino ochaic oimal growh model; econd, more comlicaed and more comrehenive mehod, ay, inegro-variaional inequaliie for oimal oing roblem (ee, Øendal and Sulem, 7) in ochaic analyi, maing he horizon, he erminal oc and furher he welfare funcion of he abrac economy endogenouly deermined, can be reaonably emloyed; hird, he mehodology of udie on urnie heorem can be naurally exended o inveigae he diance and he convergence beween differen economical yem, when heir evoluionary or develomen ah are abracly deermined by differen differenial equaion, ordinary or ochaic, of caial, including hyical caial, environmenal caial and alo human caial. APPENDIX 3

15 APPENDIX A: Proof of Lemma Since by (6) where d( ) = f ( ( )) d + g( ( )) db( ) f ra A r n ( ( )) [ -d-( - d) + - ] ( ) v ( ) g ( ( )) - ( ) hen by he Iô formula, ò ò ò () = () + áf(()), () ñ d+ g (()) d+ á (), g (()) db () ñ where á, ñdenoe andard inner roduc. Chooe ome g uch ha, hu for ome e= e( ) and Î, ], ìï áf ( ( )), ( ) ñ g ( ( )) ( g+ ) ( ) æ u ( ) eï í () + ( + g) ( ) d + u á ( ), g ( ( )) db ( ) ñ ï ç èò ø ò ïî I follow from Cauchy-Schwarz inequaliy ha / æ ( ) ö - u ( ) e () g ( ) d u ( ), g ( ( )) db ( ) ç + + ò + ò á ñ çè ö ü ï ï ø (-) aing execaion and for + g >, we have ìï üï ïî ï ( -) u ( ) ú e+ g ï í () + ( ) d+ u á ( ), g ( ( )) db ( ) ñ ï ê ú ï ò ê ò úï Alying he Burholder-Davi-Gundy inequaliy (ee, Karaza and Shreve, 99,.66), and for ome e= e ( ), /4 ì ( -) ü u ( ) ú e + g ï í () + ( ) d+ ( ) g ( ( )) d ï ê ú ò êò (A-) ú ïî ï Nex, by he Young inequaliy (ee, Higham e al, 3) and Hölder inequaliy, æ ö () g (()) d u () ç g (()) d êò ú ê èò ø /4 /4 / ú 4

16 ( -) / e + g u ( ) + g ( ( )) d ( -) e + g ú êò e - u ( ) + + g ( ) d ( -) e + g ú êò Subiuing hi ino (A-) yield, - ì ( ) e g ü - u ( ) e g ï + ú + () ( ) d ( ) d í + + ï ê ú ò êò ïî ï - If ( e + g ) ³, hen for ome e = e( ), 3( -) æ ö u ( ) + g e ç () + ( ) d ê ú è ò ø æ 3( ) () ö - + g e( ) () [ex( e ˆ ) ] + - ç eˆ çè ø in which ˆ ˆ( ) [ ( ) ( ) ] ( ) e= e r A-d- A- d r + - n + Given (), here i ome e(, ) < uch ha u ( ) e(, ) APPENDIX B: Proof of heorem By he heorem in Øendal (3),.4-6, i i eay o ee ha we ju need o rove he following cae, (i) e need o rove haf ³ g on D, i.e., ha C l ³ ln[( - r ) A] for < < Define l ( ) C l -ln[(-r ) A]. By our choen value ofc and we have l ( ) = l ( ) =. l- - Moreover, incel ( ) = Cll ( - ) +, if we ul >, hen l ( ) > for < < and hu we have l ( ) > for all < <. By (), 5

17 l > [ ra - -( A- ) r+ ( ) - n] + d d r > + -d- - d + - [ ra ( A ) r ( ) n] If henl > alway hold. Oherwie, u hen, ra (3 ) d ( A d) r n , ra+ > d+ n+ ( A-d) r- ( ) l > [ ra - + ( ) -( A- ) r- n] + d d r > { + [ ra-d-( A- d) r+ ( ) - n]} r+ d+ ( A- d) r + n> + r A hu, l > when ra (3 ) d n ( A d) r (B-) or l > when r d d d d + + ( A- ) r + n> + r A> + n+ ( A- ) r -( ) (B-) o um u, eiher (B-) or (B-) can mae (i) hold rue. -r (ii) Ouide D we have f(, ) = e ln[( -r) A] and herefore f r d d -r (, ) = e {- ln[( - r) A] + [ ra- -( A- ) r+ ( ) -n]} for " ³ ³ " ³ ex{[ ( ) ( )] } [( r A d A d r n r r ) A], which hold by (4). ³ r A- - A- r - n+ - r A ex{[ d ( d) ( )] r} [( ) ] (iii) o chec if D < a,.. we conider he oluion () of (6). Fir, we define 6

18 G () ln[ ()] hen by Iô formula, dg r A A r n d db () = [ -d-( - d) + ( ) - ] - () Hence, And hi give he oluion e ee ha if And hen, G G ra A r n B () = () + [ -d-( - d) + ( ) - ]- () ( ) = ex{[ ra-d-( A- d) r+ ( ) -n ] - B ( )} (B-3) ra ( ) d ( A d) r n + > (B-4) < (B-5) lim ( ) = a.. by he law of he ieraed logarihm of Brownian moion. And in aricular D < a..,a required. Remar: A comarion of (B-) and (B-4) how ha we mu u r > ( ) (B-6) (iv) Sincef i bounded on[, ], i uffice o chec ha -r { e ln[( r ) A( )]} - Î i uniformly inegrable on[, ) For hi o hold i uffice ha here exi a conan M uch ha -r { e [ln((- r ) A( ))] } M for all Î and ( ) Since < ln[( - r ) A( )] < (- r ) A( ) on[, ) Hence by (B-3) we have -r { e [ln((- r ) A( ))] } - -r { e [( r ) A( )] } ³ 7

19 = ( r ) A [ex{[r A d ( A d) r n r] B( )}] = ( r ) A [ex{[r A d ( A d) r 3 n r] }] e conclude ha if ra (3 ) d n r ( A d) r he deired reul i hen immediae. Remar: A comarion of (B-4) and (B-7) how ha we mu u (B-7) r > (B-8) APPENDIX C: Proof of heorem By (6), we have { d d } () = ex [ ra- -( A- ) r+ ( ) -n ]- B () Le Pu hen -, we have { d d } () = ex [ ra- -( A- ) r+ ( ) - n ] + B () ra-d-( A- d) r+ ( ) - n=- ra d ( A d) r n + = (C-) Hence, wih B B (), we have Le³ ³, one can find { } () = ex B- ( ) [ ] = { B- } () ex ( ) ê { } { } = ex B -( ) ex ( B B) - { } { } = ex B -( ) ex ( B -B ) 8

20 { x- x - } ex [ ( )] = ex { B -( ) } ò dm( x) ( -) { } { } = ex B -( ) ex ( )( - ) ò {- x- - - } ex [ ( )] ( ) ( -) { } = ex B -( ) = () dm( x) wihm he canonical Lebegue-Sielje meaure. Hence, () i an -maringale wr....on he oher hand, noing ha by (C-) { d d } () = êex [ ra- -( A- ) r+ ( ) -n ]- B () { d d } = ex [ r A- -( A- ) r + - n] = () < hu, by he Doob maringale inequaliy, { l } u ( ) ³ ( ) =," l >, " > l l ihou lo of generaliy, we ul = m for m Î, hen, m { } u ( ) ³, " mî m By he Borel-Canelli lemma, m for infiniely many { m} u ( ) ³ = So for aaw.. here exi m( w) uch ha hu, u ( ) < m for m³ m( w) limu u ( ) < m for m³ m( w) (C-) Conequenly, ( ) = w (, ) i uniformly bounded for Î [, ]( " > ) and for aaw... Moreover, i 9

21 i eaily een ha ( )- i alo an -maringale. So, alying Doob maringale inequaliy again, we obain, ( )- u ( )- ³ ê ú, " e >, " > e { e} Uing he definiion of D in heorem, we ee ha here exi a > uch ha he above maringale inequaliy ill hold for " Î B ( ) { ; - < } a a D D.ihou lo of generaliy, we - ea = m, " m Î. Hence, " ÎB a ( ) and according o he coninuiy of maringale wr.. m (given w ), condiion (C-) and Lebegue bounded convergence heorem, we have which yield - Leing e = i, "Î i, we ge D limu ( m) ê - limu u ( ) e m - ³ { } =. a.. m m e { u ( ) e} limu - < ³ a.. m m { u ( ) - i } limu - < = " iî a.. m I follow from Faou lemma ha, hu, by he Borel-Canelli lemma, So for aaw.. here exi i ( w) uch ha m -i { } u ( )- < = " iî a.. D -i for infiniely many i { } u ( )- < = D -i u ( )- < for i³ i( w) D herefore, ( ) uniformly converge o for Î[, D ] and for aaw... APPENDIX D: Proof of heorem 3 Noe from heorem ha () will no be a maringale on robabiliy ace (,, ) for " > when ra+ ¹ d+ n+ ( A- d) r. Since,

22 d() = b(, w) d + (, w) db() where b r A A r n (, w) [ -d-( - d) + - ] () (, w) -() B() -a.. e now u b (, w) d+ n+ ( A-d) r -r A- q() = q, for aa..(, w) Î, ] (, w) hen, ì ü Z() ex ï í- q( ) db() - q () dï ò ò ïî ï = - - ex( qb ( ) ( q )) Define a meaure on by, d( w) = Z( ) d ( w) i.e., Z( ) i he o-called Radon-Niodym derivaive. Since, [ Z ( )] = [ex{ -qb ( )-( q )}] = ex{ q -( q )} = which how, according o Giranov heorem, ha i a robabiliy meaure on equivalen o and ( ) i a local maringale wr.... Moreover,, i æ ö ex ç ( ) q ( d ) = ex( q ) < ê è ò for < ø which aifie he Noviov condiion. Uing Giranov heorem again, we conclude ha he following roce ˆ( ) () () (), ò q q B d+ B = + B i a Brownian moion wr... wih B ˆ() = B() = aand.. exreed in erm of B ˆ( ) we can ge

23 =- ˆ d() () db(), hu, i i eaily een ha = -B ˆ - ( ) ()ex{ ( ) ( )} which i defined on he meaure ace (,, ). hen, ˆ () = [ex{ -B ()-( )}] = (D-) and = - B ˆ ê ú - ( ) [ex{ ( ( ) ) ( 4)}] = - ex( 8) hu, lim ( ) = Now for any e > and any conan H >, by he Chebyhev inequaliy, Hence, () ú > H { () H} { H} limu ( ) > which imlie { H} limu ( ) = herefore, ( ) i ochaically ulimaely bounded. Now we define M ( ) ( ) -, alo a -local maringale, aifying Hence, M () = () - () + M -Bˆ - + lim ( ) lim[ ex{ ( ) ( )}] = Bˆ lim{ ex[( ( ) )( ) ( )]} = + = a.. by he rong law of large number for maringale and he fac =. Hence,

24 lim M ( ) <+ a.. For any inegeri ³, define he oing ime (or Marov ime), Clearly, i a,.. and ( )=, where Noe ha for any ³, i inf{ ³ ; M( ) ³ i} i= { w; ( w) } i = Leing and uing Faou lemma, we obain hu, M ( i ) i lim u M ( i) lim u M( i) M( i) = i ( i ) ê - ú i< Since ( )- i a - maringale, hu by (D-) and he Doob maringale inequaliy, i ( i ) ì ü - ïw;u ( i ) lï ê ú í - ³ ïî ï l ( i ) + + = l l " l, > On he oher hand, by Kolmogorov inequaliy, we have Hence, we have var ( i ) - ïì w;u ( i ) lï ü ê ú í - ³, " l, > ïî ï l var ( i ) ê - + l l " l, > var ( i )- ( + ) l " l, > (D-) Since by he Minowi inequaliy, var ( i) ( i) ê - ú ê - ú-( - ) " > 3

25 Hence, by (D-) we ge ( i ) ê - ú ( + ) l+ ( - ) < " l, > (D-3) hu, ( )- ( " > ) i quare-inegrable maringale. Define i x i ( )- " iî i And le x ( ) - u ( ) - " i Î i i i { } ê i ú ( i) - ( )- " iî denoe he L - norm andl - norm, reecively. Lez > be ome conan, hen by Doob maringale inequaliy and Fubini heorem, we have xi z ê ú= l { w; xi ( w) z ³ l} dl ò æ ö ò ç xi ( w)d ( w) dl çèò { wx ; i ( w) z³ l} ø æ x ( w) c { ; ( ) } d ( w) ö = ò ç dl i èò wxi w z³ l ø æ xi z xi ( w) ö = ò ç dl d ( w) çèò ø I follow from Hölder inequaliy and = ò xi( w)( xi ( w) z)d ( w) = xi( xi z) ( x z) z i < ha, i = ( ) ê i ú i i x z x z x x z x z x i i Hence, alying Lebegue dominaed convergence heorem, x = lim x z x i z i i i.e., 4

26 { i } ì ï ü u ( i) ( ) í ï - ê - ú " iî ïî ï u ( i) 4 ( i) - ê - ê ú 4 l( + ) + 4( - ) " l>, " iî by (D-3). Leingi and by Lebegue bounded convergence heorem, hu u ( )- 4 l( + ) + 4( - ) " l> herefore, here exi ome conan F uch ha lim u ( )- < a.. ( )- F " ³ (D-4) almo urely. Moreover, ince on he robabiliy ace (,, ) we have d() =- () dbˆ () Hence, we can define he following characeriic oeraor of ( ), ˆ g ( ) for any >. e define he Kullbac-Leibler diance (ee, Bomze, 99; Imhof, 5) beween and a follow g g ( ) di(, ) log( ) ³ hen, ˆ ( ) g =, for any > hu, by (D-4), ˆ g ( ) + F E (D-5) where E ( ) + F> i ome conan. Define, 5

27 { > - < a ³ } B( ) ( ) ; ( ), a { Î a } inf ; ( ) B( ) B ( ) a where B ( ) denoe B ( C ) cloure. Suoe haa > E, for every Ï B ( ), i.e., Î B ( ), a we have by (D-5). hen by Dynin formula, a ˆ g ( ) - a + ò { g [ ]} g ˆg d g E a { } ( ) = ( ) + ( ( )) ( ) + ( - ) Since a.hen by Lebegue monoone convergence heorem, we have, E { } g ( ) ( E a ) ( w) + - which yield, g ( ) di(, ) ( w ) ê [ ( w )] B ( ) ú = = (D-6) a a -E a -E a required in (i). Furhermore, for ome conan > g( ), e u inf{ ³ ; g( ( )) = } hen, by Dynin formula and inequaliy (D-5), { [ ( )]} ( ) ˆ g = g + ò g ( ( )) d ò g ( )- ( )- d+ E ( ) a a If, hen which yield,, and by Lebegue bounded convergence heorem, ò g ( )- ( )- d+ E g ( ) ò ()- d + E hu, limu ( )- d E ê ò (D-7) 6

28 hen he required aerion in (ii) follow. If we le c denoe he indicaor funcion of C ( C B ) ( ) a, and len, induced by Brownian moion B ˆ( ), ³, denoe he iener meaure (ee, Karaza and Shreve, 99,.7) on Borel igma algebra B ( C [, )) generaed by ( ), ³, hen we ge Ba n c B C ( ) ê limu C ( ( )) d a = Ba ( ) ê ò ú () - lim u d ò E a a Hence we have, E n Ba ( ) ³ - a which give he deired reul in (iii). (D-8) APPENDIX E: Proof of heorem 4 By (6), we have where d( ) = f ( ( )) d + g( ( )) db( ) f ra A r n ( ( )) [ -d-( - d) + - ] ( ) v ( ) g ( ( )) - ( ) Now, by Iô formula, ()- = () - + á ()-, f(()) ñd ò ò ò + á ( )-, g ( ( )) db ( ) ñ+ g ( ( )) d where á, ñdenoe he andard inner roduc. For Î [, ], and h= h( ), we ge ìï - ï - +ê ( ) d ï ïî u ( ) hí () ò ( ) / / + u á ( )-, f( ( )) ñ d + u á ( )-, g ( ( )) db ( ) ñ üï ò ò ïï I follow from Cauchy-Schwarz inequaliy ha 7

29 { ( ) u ( )- h ()- + - ò ( ) d / / + u á ( )-, f( ( )) ñ d + u á ( )-, g ( ( )) db ( ) ñ üï ò ò ïï aing execaion and for ome h= h( ), we have ( ) u ( )- - h () - + ( ) d ê ú êò ú { / / ü + u á ( )-, f( ( )) ñ d + u á ( )-, g ( ( )) db ( ) ñ ï ú ï ê ò ú ê ò úï ï Alying Burholder-Davi-Gundy inequaliy (ee, Karaza and Shreve, 99,.66), for ome h= h ( ) ( ) u ( )- - h () - + ( ) d ê ú ò { /4 /4 üï + ()- f(()) d + ()- g (()) d ï ê ò ú ê ò ú (E-) ï ï Nex, by he Young inequaliy (ee, Higham e al, 3) and Hölder inequaliy, ò ()- f(()) d /4 / æ ö u ( )- f( ( )) d ê ç èò ø /4 ú ( -) / h ( ) u ( )- + ( ( )) f d ( -) ( h) ê ú êò ( -) ( h ) ( -) êu ( ) ú v ( ) ( -) ò - + d ( h ) ê ú (E-) Similarly, we ge /4 ()- g (()) d u ()- êò ( -) ( h ) ú Hence, ubiuing (E-) and (E-3) ino (E-) yield, ( -) ( h ) ( -) + () d (E-3) ò 8

30 ( ) u ( )- - h () - ê ê ú ú here mu be ome h = h( ) uch ha, { ( )} - + ò () d+ h () d + v êò 3( ) æ ö u ( ) - - ( + v ) h( ) () - + ( ) d ê ú ç è ò ø Since by (6), hu where { d d } ( ) = ()ex [ ra- -( A- ) r+ ( ) -n ] - B ( ) { d d } ( ) = () ex ra [ - -( A- ) r + ( ) -n ] - B ( ) = () ex ( ) ( h ) h d d ( ) [ r A- -( A- ) r + ( ) - n] + ( ) Hence herefore, Pu hen we have ò () ( ) d = ê ú [ex( h) ] h - æ 3( ) () ö - u ( ) ( v ) h( ) - + () - + [ex( h) -] ê ú ç h è ø æ 3( ) () ö - (, ) ( ) C v h( ) + () - + [ex( h) - ] ç h çè ø u ( )- C(, ) In aricular, when, by Levi lemma we have 9

31 lim u ( )- = lim u ( )- APPENDIX F: Proof of heorem 5 Noing ha Hence, we now rove ha u c () - c () = r u () - () ê ú ê ú (F-) limu () - () ê From lemma and for " < < here i ome conan uch ha where u ( ) u ( ) (F-) ê ú ê ú () = () + ò [ -d-( - d) + - ] () + (-) () () ò r A A r n d db () = () + ò [ -d-( - d) + - ]() + (-) () () ò ra A r nd db Suoe () ()," ³, oherwie we ju conider () and () inead of () and ( ), reecively, for ome <. In wha follow, we firly define he following oing ime, inf{ ³ ; ( ) ³ }, inf{ ³ ; ( ) ³ }, By he Young inequaliy (ee, Higham e al, 3) and for any u >, u ( ) - ( ) = u () - () c{, } u () () > > + - c { or, } ú u ( )- ( ) c ê > { } u -( ) + u ( ) - ( ) + {, or } (F-3) ( -) u 3

32 I follow from (F-) ha, { } c ( ) = ê { } ú u ( ) And imilarly, { } ( ). So, { or, } { } + { } hu we obain, Hence (F-3) become, Define ( ) - u ( )- ( ) u ( ) + ( ) + u ( -) u ( ) - ( ) u ( )-( ) + + / - ê ú ê ú (F-4) u d d ra- -( A- ) r + -n d d ra- -( A- ) r + -n hu by he Cauchy-Schwarz inequaliy and he riangle inequaliy, ò ò ( )- ( ) = [ ( )- ( )] d+ [ ( )- ( )] db ( ) () () d [() ()] db() ê ò ò ()- () d+ - () d+ [() - ()] db () ò ò ò So for any, by Iô iomery, we have u ( )- ( ) ( ) + ( ) - ( ) d + - ( ) d êò ú ò ( + ) ò u ( )-( ) d+ - ( ) d ê ú ò Since by (6), ú 3

33 { d d } () = ()ex êra- -( A- ) r + ( ) -n - B () hu here ê ( ) ú = () ex( ) ra d A d r n - -( - ) hen ò () ( ) d= ex( ) - ( ) Accordingly, u ( )- ( ) ( ) ê ú So he Gronwall inequaliy (ee, Higham e al, 3) yield u ( )- ( ) + ò u ( )- ( ) d ( )( ) + - () ex( ) - () - ex( ) - ex ( + ) ( ) Inering hi ino (F-4) give () ( + ) ( ) u ( ) - ( ) - ex( ) - e + u ( -) ( -) u + + Hence, for " e >, we can chooe ome u and uch ha, + u e ( -) and ( -) 3 u e 3 And for any given, we u () uch ha So, for " e >, herefore, we have () ( + ) ( ) - ex( ) - e e 3 u ( ) - ( ) e 3

34 lim u ( ) - ( ), a e By Levi lemma, we obain limu () - () ê which yield, limu c () -c () a e ê by (F-). APPENDIX G: Proof of heorem 6 By (3), we have, dc ( ) = f ( ( )) d + g( ( )) db( ) here f ra A r n ( ( )) r[ -d-( - d) + - ] ( ) w ( ) g ( ()) -r () Now, by Iô formula, c () - c = c ()- c + ác () -c, f( ()) ñd ò ò ò + ác( ) -c, g ( ( )) db ( ) ñ+ g ( ( )) d where á, ñdenoe he andard inner roduc. For Î [, ], and z = z( ), " ³ we ge ìï c - c ï c - c + ( r ) d + ïî ï ê ú u ( ) zí () êò ( ) u ác ( ) -c, f( ( )) ñ d + u ác ( ) -c, g( ( )) db( ) ñ üï ò ò ïï I follow from Cauchy-Schwarz inequaliy ha { ( ) u ( ) z () r - ò ( ) c - c c - c + d+ 33

35 u ác ( ) -c, f( ( )) ñ d + u ác ( ) -c, g( ( )) db( ) ñ üï ò ò ïï aing execaion and for ome z = z( ), we have { ( ) u c ( ) - c r - z c ()- c + ( ) d + ê ú êò ú ü u ác ( ) -c, f( ( )) ñ d + u ác ( ) -c, g( ( )) db( ) ñ ï ú ï ê ò ú ê ò úï ï Alying Burholder-Davi-Gundy inequaliy (ee, Karaza and Shreve, 99,.66), for ome z = z ( ), ( ) u c ( ) - c r - z c ()- c + ( ) d+ ê ú ò { /4 /4 üï c ( ) - c f( ( )) d + c ( ) -c g( ( )) d ï ê ò ú ê ò ú (G-) ï ï Nex, by he Young inequaliy (ee, Higham e al, 3) and Hölder inequaliy, ò c () -c f(()) d /4 / æ ö u c ( ) - c f( ( )) d ê ç èò ø ( r ) u c ( ) - c + f( ( )) d ( -) ( r z ) ê ú êò /4 ( -) / z ( -) ( r z ) ( -) u c ( ) - c + w ( ) d ( -) ( r z ) ê ú êò (G-) Similarly, we ge /4 c ( ) - c g( ( )) d u c ( ) -c êò ( -) ( r z ) ú ( -) ( r z ) ( -) + r () d êò (G-3) Hence, ubiuing (G-) and (G-3) ino (G-) yield, ( ) u c ( ) - c r - z c ()-c ê ê ú ú { ú 34

36 hen here mu be ome z = z( ) uch ha, ( )} - + ò () d+ z () d r + r w êò 3( ) æ ö u c ( ) - c r - ( r + w ) z ( ) c () - c + ( ) d ê ú ç è ò ø Since by (6), hu where { d d } ( ) = ()ex [ ra- -( A- ) r+ ( ) -n ] - B ( ) { d d } ê ( ) ú = () êex ra [ - -( A- ) r + ( ) -n ] - B ( ) = () ex z ( ) ( ) ( ) [ z r A-d-( A- d) r + ( ) - n] + ( ). Hence ò () ( ) d = ê ú [ex( z) ] z - herefore, u c ( ) - c r r + w ( ) 3( -) æ () ö z( ) c ()- c + [ex( z) - ] ç z çè ø Pu æ 3( ) () ö - (, ) ( ) C r r + w z ( ) c ()- c + [ex( z) - ] ç z çè ø hen we have u c ( ) - c C(, ) hu, if r or, hen by Levi lemma we ge lim u c ( ) - c = lim u c ( ) -c 35

37 APPENDIX H: Proof of heorem 7 By (6), we ee ha, æ d() ö æ [ r A d ( A d) r n ] ö æ ö dφ = dy A r A A r n d A = db çdc () r[ r A d ( A d) r n ] ç ç-r è ø è ø è ø () () [ d ( d) ] () () () ( ) + ( - - ) V V V3 d () A r db () () Vd () + -A -r db () () ( ) f ( ( )) d+ g ( ( )) db ( ) Now, by Iô formula, Φ() - Φ = Φ() - Φ + á () -, f(()) ñd ò Φ Φ + ò áφ( )-Φ, ( ( )) ( ) ñ+ ( ( )) ò g db g d where á, ñdenoe he andard inner roduc. For Î [, ], and z = z( ), " ³ we ge ï ìï ( A ) d ï ïî u Φ( )- Φ zí Φ() - Φ + ê ( + r + ) ( ) ò / / + u á ( ) -, f( ( )) ñ d + u á ( ) -, g ( ( )) db ( ) ñ üï ò Φ Φ ò Φ Φ ïï I follow from Cauchy-Schwarz inequaliy ha { ( ) u Φ( )- Φ z Φ() - Φ + ( + r + ) - ( ) ò A d / / + u á ( ) -, f( ( )) ñ d + u á ( ) -, g ( ( )) db ( ) ñ üï ò Φ Φ ò Φ Φ ïï aing execaion and for ome z = z( ), we have ( ) u Φ( ) - Φ ( + r + A ) - z Φ() - Φ + ( ) d ê ú êò ú / / ü u ( ), f ( ( )) d + á - ñ + u á ( ) -, g( ( )) db( ) ñ ï ú ï ê ò Φ Φ ú ê ò Φ Φ úï ï { 36

38 Alying Burholder-Davi-Gundy inequaliy, for ome z = z ( ), ( - ) u ( ) - ( + r + A ) z ()- ê Φ Φ ú Φ Φ { + d + - f d ò () ê () (()) ò Φ Φ /4 üï + () - g( ( )) d ï ê ò Φ Φ (H-) ú ï ï Nex, by he Young inequaliy (ee, Higham e al, 3) and Hölder inequaliy, f d ò Φ ()- Φ ( ()) /4 æ ö u Φ( ) - Φ f( ( )) d ê ç èò ø u Φ( ) -Φ ( -) (( ) ) + r + A z ú (( + r + A ) z ) + f ( ( )) d êò ( -) / u Φ( ) -Φ ( -) (( ) ) + r + A z ú ( -) (( + r + A ) z ) ( -) + V () d êò (H-) Similarly, we ge /4 /4 ú g d ò Φ ()- Φ ( ( )) /4 u Φ( ) -Φ ( -) (( ) ) + r + A z ú ( -) (( + r + A ) z ) ( -) + ( + r + A ) ( ) d êò (H-3) Hence, ubiuing (H-) and (H-3) ino (H-) yield, ( - ) u ( ) - ( + r + A ) z ()- + ê Φ Φ ú Φ Φ { 37

39 ò ( )} () ( ) - d A () + + r + z d V + ( + r + A ) êò hen here exi ome z = z( ) uch ha, u Φ( ) -Φ ú ( ) 3( - ) ( ( + + A A ) ) r V r z æ ö () ç ê Φ - Φ ú + ( ) d è ò ø ( ) 3( -) ( + r + A ) V + ( + r + A ) z( ) æ ö () ç ê Φ - Φ ú + ( ) d è ò ø Since by (6), hu where { d d } ( ) = ()ex [ ra- -( A- ) r+ ( ) -n ] - B ( ) { d d } ê ( ) ú = () êex ra [ - -( A- ) r + ( ) -n ] - B ( ) = () ex z ( ) ( ) z d d ( ) [ r A- -( A- ) r + ( ) - n] + ( ) Hence ò () ( ) d = ê ú [ex( z) ] z - herefore, u Φ( ) -Φ ú ( ) 3( - ) ( ( + r + A V + + r + A ) ) æ () ö z( ) () [ex( z) ] Φ - Φ + - ç z çè ø Pu ( ) 3( -) C(, ) ( + r + A ) V + ( + r + A ) 38

40 æ () ö z( ) () [ex( z) ] Φ - Φ + - ç z çè ø hen we have u ( ) Φ - Φ C (, ê ) hu, if, by Levi lemma we ge he following deired reul, lim u ( ) - = lim u ( ) - ê ú ê Φ Φ Φ Φ REFERENCES Aghion, P., 4. Growh and Develomen: A Schumeerian Aroach. Annal of Economic and Finance 5, -5. Araujo, A. and J.A. Scheinman, 977. Smoohne, Comaraive Dynamic, and he urnie Proery. Economerica 45, 6-6. Barro, R.J., 99. Governmen Sending in a Simle Model of Endogenou Growh. Journal of Poliical Economy 98, 3-5. Bewley,., 98. An Inegraion of Equilibrium heory and urnie heory. Journal of Mahemaical Economic, Bomze, I.M., 99. Cro enroy minimizaion in uninvadable ae of comlex oulaion. Journal of Mahemaical Biology 3, Cole, J.L., 985. Equilibrium urnie heory wih Conan Reurn o Scale and Poibly Heerogeneou Dicoun Facor. Inernaional Economic Review 6, Drandai, E.M., 966. On Efficien Accumulaion Pah in he Cloed Producion Model. Economerica 34, Gale, D., 967. On Oimal Develomen in a Muli-Secor Economy. Review of Economic Sudie 34, -8. Ganz, D.., 98. A Srong urnie heorem for a Nonaionary von Neumann-Gale Producion Model. Economerica 48,

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