A NOTE ON THE EXISTENCE OF AN OPTIMAL SOLUTION FOR CONCAVE INFINITE HORIZON ECONOMIC MODELS

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1 A NOTE ON THE EXISTENCE OF AN OPTIMAL SOLUTION FOR CONCAVE INFINITE HORIZON ECONOMIC MODELS by Smdeb Lahiri Discussin Paper N. 221, Sepember 1985 Cener fr Ecnmic Research Deparmen f Ecnmics Universiy f Minnesa Minneaplis, Minn ABSTRACT lhe purpse f his paper is esablish cndiins fr he exisence f an pimal sluin in cncave maximizain prblems In infinie hrizn ecnmic mdels.

2 A Ne On: The Exisence f an Opimal Sluin fr Cncave Infinie Hrizn Ecnmic Mdels Smdeb Lahiri Deparmen f Ecnmics, Universiy f Minnesa (Twin Ciies) 1035 Managemen & Ecnmics h Avenue Suh Minneaplis, MN This prblem was suggesed me by Prf. J. Jrdan. While his paper was in prgress i was brugh he auhr's nice by Prf. R. Marimn ha an pimizain resul such as he abve exiss in a bk by I. Ekeland and T. Turnbull: 'Infinie Dimensinal Opimizain and Cnvexiy'. Our se up differs frm heirs in w respecs. There he pimizain 'prblem is assumed be valid eiher 1) in a finie hrizn ecnmy r 2) in an infinie hrizn ecnmy wih uiliy being discuned a a psiive rae. Furher, uiliies are assumed bunded in he L nrm alng any feasible pah as als he ime derivaive f he pah iself. Our pimizain prblem makes n such assumpins. Hwever, ur pimizain prblem is a cncave pimizain prblem. Fr helpful suggesins I am graeful Prfessrs L. Hurwicz, M.K. Richer, D. Kahn and P. Rej.

3 Inrducin The purpse f his paper is esablish cndiins fr he exisence f an pimal sluin in cncave maximizain prblems in infinie hrizn ecnmic mdels. Such mdels have been used exensively in he sudy f pimal ecnmic grwh and planning. The general ype f prblem we rea is: ( S ) Sup 6 OI(x, ~, )d s..(x, ~) e:ac::r 2m x() = x where A is cnvex wih in (A)~~ and I a cncave upper semi-cninuus funcin which is bunded abve by an inegrable funcin (see Secin 1 fr full deails). We ne ha here is a cnnecin beween his ype f pimizain prblem and general equilibrium hery wih an infinie number f cmmdiies as reaed fr insance in Bewley [1972]. We culd cnsider (S) as an equilibrium prblem f ne cnsumer where his chice variable is he cnrl pah and given he iniial endwmen he chses he pimal cnrl pah. Fr a specific versin.f ur prblem, Peleg and Ryder (1972) ga exisence wihu a resricive bundedness assumpin n A, which appears in ur analysis. Secin 1: Le AeR 2m be a cnvex se wih In. (A)~. Suppse I:A x R + R saisfies: + Assumpin (*): (a) I is upper semi-cninuus (hereafer referred

4 as u.s.c.) and fr each ER, 1(,) is cncave. l:a x R + R + + is upper-semicninuus if and nly if lim sup l(',y') ~ (',y')+ (,y) (b) There exiss an inegrable funcin S:R + R such + ha l(y,) ~ s() fr all yea. 2 l(,y) (c) If YEaA (he plgical bundary f A) and y/a and if limy = y wih Y EA hen lim sup 1 (y,) = _00 n n+ n+ n fr every Remark 1: - If l(y,) = e m(y) and m is a bunded cninuus funcin n a cmpac se A hen Assumpin (*) is saisfied. Since A is n necessarily clsed Assumpin (*) is als saisfied if m: (0,1] x R + R is given by m(y1,y2) lg y1 Sme nain will help us sae and sudy prblem (S). Le L 00 dene he space f (equivalence classes f) measurable funcins w:r+ + R m such ha ess sup Iw() 1<+00, endwed wih he M>O and nrm Iwl = ess sup Iw() I. Le E ={zel 00 / here exiss such ha sup 16 x(s)ds I <M}. E is a :Linear subspace f L 00 may hus be endwed wih he sup nrm. Le us cnsider he prblem:. (p) Sup ll(x + 6z(s)ds, z(), )d subjec (x + Iz(s)ds, z(»ea 0 where he sup is aken amng all measurable funcins z:r + + m. R, such ha ess sup Iz() I<+~ and ess sup Ib z()d) 1<+00. Le v:r2m x R+ + [_00, +00) be he exended real funcin given by:

5 3 v(y,) v(y,) I (y, ) if yea - 00 if Y I- A Assumpins (*)(a) and (c) imply ha v(,) is a cncave upper-semicninuus funcin, i.e. limy = y, implies lim n n+ Fr every WfL ' JOO zee, le J(w,z) = 0 v(w(),z(),)d. Since v is u.s.c. {(y',')ea x R /v(',y' )<a} is clsed + - vaer and hence v is a measurable funcin. Assumpin (*) implies ha J:L x E + 00 R is a well defined cncave funcinal. In fac since v(y,) is u.s.c. in y and In. (A) f ~, -v is a nrmal cnvex inegrand in he language f Rckafellar (1968). Furhermre if we define fr each 2m per,v*(p,) = Inf {- <p,y> - v(y,)} (he cncave cnjugae y f v) hen -S()2 v*(,)2 -v(x + 6u(s)ds,u(),). I fllws ha J is a well defined cncave funcinal (Rckafellar (1968) Therem 1 page 532). Thus we may rewrie (P) as sup (W,Z)E L 00 xe J(w,z) s.. w() = X + 6z(s)ds We sar by shwing ha J is an u.s.c. funcinal. Here all limis are aken wih respec he srng plgy. Lemma 1: If lim (w,z ) = n n (w,z)el x E hen 00 n+ I i III sup J ( w,z ) <J ( w, Z ) n n- n+

6 4 Prf: The prf will parallel a prf f an analgus herem in Arauj and Scheinkman [1980]. We may wihu lss f generaliy assume ha any subsequence n such ha li~j(w, k k+ n k Z ) = lidlsup J(w,Z ) is such ha (x + (z (s)ds, z (» n n+ n n 0 n k k n k EA a.e., and w () = x + OI z (s)ds, fr herwise he n 0 k n k inequaliy is rivial. Since v(y,)~s() i fllws frm Fau's lemma ha lin) sup J(w,z ) k+ n k n k ~01lirnsup v(x + 01zn (s)ds, z (),) d k+ 0 k nk ~6OOv(x + 6z(s)ds,z(),)d Since v is u.s.c. and fr each <, li~z () k+ n:a, and limx + 6z (s)ds = x + 01z(s)ds. k+ 0 n 0 k Therem 1: Nw we prceed ur main herem. z() Q.E.D. If A is a bunded subse f R 2m, and here exiss an inegrable funcin S:R + R such ha l(y,)~s() ~yea, + hen S admis an pimum sluin in L X E. Prf: If A is a bunded subse f R 2m, hen /I (w n ' znll +00 implies max {~w /I, ~ z /I}+ +00 n n max {ess sup /lwn() /I, ess S P IIzn() IP+ +00 here exiss NE~, such ha fr all n>n meas ER I(w (),z (»ER 2m A}>O + n n v(wn(),zn(),)d 1V(Wn(),zn(),)d {:R /(w (),z (» EA} + n n + 1v(wn(),zn(),)d {:R+/(wn(),zn(» ER2m_.A}

7 5 Furher, implies ha = -"" vn>n J + "" v(w (),z (),)d n n + -"" 0 (x,x) E L x E 0 "" (x,x) E L x E "",..,,.., 0 (x,x) + (l-a)(x,x) E L,..,,-J "" x E and J"" [AV(X,X,) O 0 + (l-a)v(x,x,) 0 ]d,." - ~6""V(AX + (l-a)~, A~ + (l-a)~,)d (by cncaviy f v).' J(x,~) J"" 0 v(x(),x(),)d is a cncave funcin defined n L"" x E. Le (x,x ) be a maximizing sequence f J; ha is a n sequence f elemens f L"" n x E such ha 0 J(x,x ) + sup J(x,x) ex: n n 0 (x,x)el xe "" Ne ha ex: belngs a priri (-00,+00]; we will see frm wha fllws ha ex: +"". The sequence (x,x ) is bunded in n n 0 {( x, x) / J ( x, x ).2 ex:}. This is because he sequence J(x,~ ) is n n bunded belw. Thus we can exrac frm (xn'x n ) a subsequence. 0 (x,x ), wh~ch n. n. cnverges weakly in 100 x E an elemen (x,x) ~ ~ belnging L"" x E. (See Appendix) By Crllary f Ekeland and Temam [1976] r Crllary f Berberian [1974] and he fac {(x.~)el. - "" x E/ J(~,~)~a} is cnvex vaer, J is u.s.c. n L x E fr he weak 00 plgy f L "" x E and hence J(x,~)~lim sup J(x,~ ) n. n. n.+"" ~ ~ ~ ex:

8 6 (x,~) is a sluin f he pinal cnrl prblem and c I +00. Ne: (1) The abve lemma and herem are easily seen be valid if insead f Assumpin (*) ~ we had he fllwing assumpin (*)a': Assumpin (*) a': 1(,):A+R is u.s.c. fr all ER + and I:A x R + + R is a measurable funcin (2) The space E may be idenified wih he space f bunded, finiely-addiive se funcins h R~m which are absluely cninuus wih respec he lebesgue measure. This is he dual f L (R,R m) Appendix: Prpsiin 1: Le K be a bunded se in E. If lim. meas(d)+o I f()d = 0 D v f in K, hen K is weakly sequenially cmpac. Prf: Suppse K is bunded and ha he inegrals If()d saisfy he abve cndiin v f in K. D Le fnek and suppse ha Ifn~~C fr n=1,2,... c E where II f II =- I If () d I n EOn

9 7 By he Canr diagnal prcess, chse a subsequence {g } n f {fn} such ha he limi A (D) = exiss fr every lim Ig ()d n-+ D n D in B( [0,00]), (i.e. he Brel a-algebra n [0,00]). This limi exiss since B([O,]) is cunably generaed, and s he Canr diagnal prcess is peraive. L '. gn is weakly cnvergen in E. ll E.D. (The prf f his herem clsely parallels he prf f Therem 9 in Dunfrd and Schwarz [1957], v. 1, page 292). Prpsiin 2: (x.) is weakly cnvergen and A bunded J jen implies (x.,x~) J J jen is weakly cnvergen. Prf: x.() = x + Ix.(s)ds v >O J 0 0 J and he fac ha x.( ) -+ f(') (weakly), where J fee, implies x. () = x + I x. ( s ).ds + x + If(s)ds J. 0 0 J 0 Define x() = f(). Then x() = x + Ix(s)ds 0. x.( ).-+x( ) J a.e. Since A is a bunded subse f R 2m, prjl A exiss xer m wih (x,x)ea} is bunded. a.e.

10 8 sup {~xl / x E prjl A} < +00, where chis nrm is he Euclidean nrm. {II x.11 } is bunded abve. J L j EN x.( ) + x( ) a.e. implies ha J x.( ) + J x( ) weakly.

11 9 References 1. A. Arauj and Scheinkman, J.A. (1980): "Maximum Principle and Transversaliy Cndiin fr Cncave Infinie Hrizn Ecnmic Mdels", Repr 8019, Universiy f Chicag. 2. S.K. Berberian (1974): "Lecures in Funcinal Analysis and Operar Thery", Springer-Verlag, New Yrk Inc. 3. T. Bewley (1972): "Exisence f Equilibria in Ecnmies wih Infiniely Many Cmmdiies", Jurnal f Ecnmic Thery 4(3), N. Dunfrd and Schwarz, J. (1957): "Linear Operars", Par 1, Inerscience, Wiley. 5. I. Ekeland and Temam, R. (1976): "Cnvex Analysis and Variainal Prblems", Nrh-Hlland American Elsevier, New Yrk. 6. B. Peleg and Ryder, H.E. (1972): "On Opimal Cnsumpin Plans in a Muli-secr Ecnmy", Review f Ecnmic Sudies 39, R.T. Rckefellar (1968): "Inegrals Which Are Cnvex Funcinals", Pacific Jurnal f Mahemaics 24(3),