L N O Q. l q l q. I. A General Case. l q RANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY. Econ. 511b Spring 1998 C. Sims

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1 Econ. 511b Sprng 1998 C. Sm RAD AGRAGE UPERS AD RASVERSAY agrange mulpler mehod are andard fare n elemenary calculu coure, and hey play a cenral role n economc applcaon of calculu becaue hey ofen urn ou o have nerpreaon a prce or hadow prce. You have een hem generalzed o cover dynamc, non-ochac model a Hamlonan mehod, or a byproduc of ung Ponryagn maxmum prncple. n ac model agrangan mehod reduce a conraned maxmzaon problem o an equaon-olvng problem. n dynamc model hey reul n an ordnary dfferenal equaon problem. n he ochac model we are abou o conder hey reul n, for dcree me, an negral equaon problem or, n connuou me, a paral dfferenal equaon problem. negral equaon and paral dfferenal equaon are harder o olve han ordnary equaon or dfferenal equaon hey are boh le lkely o have an analycal oluon and more dffcul o handle numercally. he applcaon of agrangan mehod o ochac dynamc model herefore appear o be of le help n olvng he opmzaon problem han her applcaon o non-ochac problem. Conequenly many reference on dynamc ochac opmzaon gve lle aenon o agrange mulpler, nead emphazng more drec mehod for obanng oluon. he economc leraure ha o ome exen been guded by h paern of empha. h unforunae, becaue agrangan mehod are a helpful n economc nerpreaon of model n ochac a n non-ochac model. Alo, n general equlbrum model, ue of agrangan mehod urn ou omeme o mplfy he compuaonal problem, n comparon o approache ha ry o olve by more drec mehod all he eparae opmzaon embedded n he general equlbrum.. A General Cae Snce n h coure we are more nereed n ung hee reul han n provng hem, we preen hem backward. ha, we begn by wrng down he reul we are amng a, hen dcu lm on range of applcably, and hen only a he end kech ome argumen a o why he reul are rue. We conder a problem of he form ubec o max E U ( ), ( ) C ( lc q Z l q l ) q e = = P (1) = = l q l q e g C( ), Z( ), =,...,. (2) = = We aume ha he vecor Z an exogenou ochac proce, ha, ha canno be nfluenced by he vecor of varable ha we can chooe, C. or a dynamc, ochac eng, he nformaon rucure an eenal apec of any problem aemen. nformaon revealed over me, and decon made a a me can depend only on he nformaon ha ha been 1

2 revealed by me. Here, we aume ha wha known a lz( ) q =,.e. curren and pa value of he exogenou varable. f coure mplcly h mean ha alo lc( ) q = known a, nce choce of C( ) alway mu be a funcon of he nformaon avalable a. he cla of ochac procee C ha have h propery are ad o be adaped o he nformaon rucure. We can generae fr order condon for h problem by fr wrng down a Hamlonan expreon, el q= l q= el q= l q= = = E U C ( ), Z ( ) λ g C ( ), Z ( ) and hen dfferenang o form he C : oce ha: H U+ g+ = E λ C( ) C( ) C( ) = = + P, (3) P =, =,...,. (4) n conra o he deermnc cae, he Hamlonan n (3) and he C n (4) nvolve expecaon operaor. he expecaon operaor n he C E, condonal expecaon gven he nformaon e avalable a, he dae of he choce varable vecor C( ) wh repec o whch he C aken. Becaue U and g each depend only on C daed and earler, he nfne um n (4) nvolve only U and g daed and laer. he erm a he lef n (4) uperfluou and uually u omed. n fne-dmenonal problem, fr order condon are neceary and uffcen condon for an opmum n a problem wh concave obecve funcon and convex conran e. he condon n (4) are no a powerful, becaue h an nfne-horzon problem. r order condon here, a n mpler problem, are applcaon of he: Separang Hyperplane heorem: f x maxmze he connuou, concave funcon V ( ) over a convex conran e Γ n ome lnear pace, and f here an (nfeable) x wh V ( x ) > V ( x), hen here a connuou lnear funcon f ( ) and a number a uch ha f ( x) > a mple ha x le oude he conran e and f ( x) < a mple V ( x) < V( x). n a fne-dmenonal problem wh x n 1, we can alway wre any uch f a where he f are all real number. n f ( x) = f x, (5) f he problem ha dfferenable V and dfferenable conran of he form g x alo be rue ha we can alway pck = 1 b g, hen wll 2

3 and nearly alway wre f V = b x g (6) g ( x) f ( x) = λ x. (7) he nearly neceary becaue of wha known a he conran qualfcaon. poble ha he fr-order propere of he conran near he opmum do no gve a good local characerzaon of he conran e 1. However, f we can fnd an x vecor and a e of nonnegave λ ha afy he conran and (6) and (7), we have found he eparang hyperplane and hence he opmum. he andard agrange mulpler equaon are herefore uffcen condon for an opmum, and hey are nearly uffcen: We know here wll alway be a eparang hyperplane, and uually we wll be able o wre n he form (7), bu here are ome knfe-edge (.e., rare) pecal cae n whch h wll no be rue. h ufe he common raegy of ryng o olve uch problem by lookng for oluon o (6) and (7). he uffcency par of hee reul can be ummarzed a: Kuhn-ucker heorem 2 : f ) V a connuou, concave funcon on a fne-dmenonal lnear pace, ) V dfferenable a x wh graden Vbxg, ) g k, = 1,..., are convex funcon, each dfferenable a x wh graden gbxg, v) here a e of non-negave number λ, = 1,..., k uch ha Vbxg gbxg = λ, and v) gbxg, = 1,..., k, hen x maxmze V b g = over he e of x afyng g x, 1,..., k. Bu n an nfne dmenonal pace may no be rue ha we can wre every connuou lnear funcon a an nfne um analogou o (5), and he poenally nfne um n (7) and n (5) wh f defned by (6) mgh no converge. hee complcaon provde addonal reaon ha here can be model n whch he agrange mulpler equaon are no neceary condon for an opmum, bu more mporanly hey mean ha hey are no longer uffcen condon, even for 1 f you wan an example of h, ry o ue agrange mulpler mehod o olve he problem of b g ubec o bx g y and bx g y. h problem 2 2 maxmzng x y afe all he dfferenably, concavy and convexy one mgh lke, ye doe no yeld o a drec agrange mulpler approach becaue a he opmum fr-order expanon of he conran do no characerze he conran e. he dffculy goe away f he rgh-hand-de of he conran are changed from 1 o 1.1, ay. 2 h veron of he Kuhn-ucker heorem no he mo general poble, even for fnedmenonal pace. 3

4 problem wh concave obecve funcon and convex conran e. o handle hee problem ha we mpoe on nfne horzon problem wha are called ranveraly condon. o apply he agrange mulpler dea o our curren problem, nerpre V a gven by he maxmand n (1), x a beng C, he opmal C equence, and x a beng a generc C equence. n our ochac problem, (5)-(7) become E HG = = U H Cv, Z v= v v= K C f C C J = e K = E l q= HG = λ = g H Cv, Z v= v v= K C C n order o ge from (8) wha are gven a C n (4) above, we nerchange he order of ummaon n he expreon on he lef and rgh de of (8), hen equae coeffcen of correpondngly ubcrped C. he veron of (8) wh order of ummaon nerchanged E HG = = U H Cv, Z g C Z v v v K v, = = v C E λ C J = G = = C K H H = v v= K C KJ KJ (8), (9) from whch eay o ee ha (4) follow, f we equae he coeffcen on C erm on he wo de of he equaon. Bu o ufy hee manpulaon, we mu be careful abou ue of convergence. Dealng wh convergence of hee um checkng ranveraly. oe ha mply equang coeffcen on he lef and rgh of (9) mgh eem o mply (4) eher whou he E operaor or wh an unubcrped E operaor. o underand why he E appear, remember ha C a random varable, a rule for choong a numercal value for C a a funcon of nformaon avalable a. coeffcen n (9) herefore he um of all he erm ha mulply, over boh dae and poble ae of he world gven nformaon a. he um over ae conen wh nformaon avalable a ha reul n he E operaor n he C. h ufcaon may be hard o underand a h pon. made explc n a mple pecal cae a he end of hee noe. n mo economc model, here are only fnely many lag a argumen o g and U, whch make many of he nfne um n (8) and (9) become fne. n fac mo commonly U ha no lag n argumen. o ge veron of ranveraly ha are cloer o wha commonly dcued n economc model and allow u o prove reul, we now pecalze o he cae where U = UbC, C 1, Zg and g = gbc, C 1, Zg. h allow u o wre a veron of he Kuhn- ucker heorem for nfne-dmenonal pace a: nfne-dmenonal Kuhn-ucker: Suppoe 4

5 P ; b, 1, Zg convex n C and C 1 for each Z ; m r uch ha each C a funcon only of dm r, l q fne wh he paral um defnng on he rgh hand 1,..., gcc, C 1, Zh ; cl q l qh = b g = ) V C, Z lm nf E U C, C 1, Z ) U concave and each elemen of g C C ) here a equence of random varable C nformaon avalable a, V C Z de of () convergng o a lm, and for each = v) U and g are boh dfferenable n C and C 1 for each Z and he dervave have fne expecaon; v) here a equence of non-negave random vecor lλ q, wh each λ n he correpondng nformaon e a, and afyng λ g C, C 1, Z = v) U C C c h wh probably one for all ; c + P = +, 1, Zh UcC + 1, C, Z+ 1h gcc, C 1, Zh gcc + 1, C, Zh E E + 1 C C C C P all (.e., he Euler equaon hold). λ λ for n U C,,,, C Z g C C Z lm up E C C H G c C C K J 1 h c λ 1 h d P. m r maxmze V ubec o gbc, C 1, Zg for all =,...,. n a feable equence of conumpon choce rule ha acheve a hgher m r, depe mc r afyng he condon of he heorem. We mplfy c, 1, Zh and ung g for gcc, C 1, Zh. By v) (ranveraly) for every feable C equence C hen C Proof: Suppoe C value of V han doe C noaon from h pon on by ung U for U C C dfferenably and by concavy of U and convexy of g, we know ha for each and mlarly d d d D U C C + D U C C U C, C, Z U (1) D1 g C C + D2 g C d d 1 C 1 gdc, C 1, Z g (11) n gve a hgher value of V m r, we conclude ha Ung (1), he defnon of V, and our workng hypohe ha C han doe C 5

6 lm E D1U C C + D2U C 1 C 1 = e d d P >. (12) Bu our Euler equaon a gven n (v) aure u ha (12) equae erm by erm, excep for a lefover erm on he end, o he expeced um of he graden of g, weghed by he he λ equence. n parcular, (12) exacly lm R S e d d U P V W λ = + E bdu 1 λ D1 gg dc C E D g C C D g C C. (13) Snce he C by hypohe feable, nce λ, and nce λ g zero wh probably one, e d. he fr expecaon whn curly bracke n (13) herefore le λ g C, C 1, Z g han or equal o zero for every, by convexy of g. hu he fr erm ha a lm up le han or equal o zero. he non-povy of he lm up of he econd erm n he curly bracke exacly wha we aumed n our ranveraly condon (v). h complee he proof by conradcon: whle (12) ha o exceed zero f C ha equal o (13), whch ha o be non-pove. n mprove on mc r, he condon of he heorem guaranee oe ha condon (v), ranveraly, no n que he uual form. he uual form would mply aer E λ c, 1, Zh CP. (14) C g C C fen n economc model he U erm n he rue ranveraly condon a gven n (v) drop ou or converge o zero auomacally. (14) hen guaranee ranveraly a one parcular pon, nc = l q ; hough n mo economc model he zero equence n he feable e, h need no alway be rue. he convenonal ranveraly condon alo oo rong n ha requre acual convergence, raher han only ha he lm nf be non-negave. oo weak n ha check only one pon n he feable e. here are model n whch, f we replaced our condon (v) by (14), here would be C equence ha afy all he condon of he modfed heorem ha are no n fac opma. A leadng example of uch a model he lnear-quadrac permanen ncome model wh a borrowng conran replacng he uual bound on he rae of growh of wealh. he andard lnear decon rule no opmal n uch a cae, bu afe he andard ranveraly condon (14), whle falng our condon (v). REERECES Kuhner, Harold J. [1965], n Sochac Exremum Problem: Calculu, Journal of ahemacal Analy and Applcaon 1, [1965], n he Sochac axmum Prncple: xed me of Conrol, Journal of ahemacal Analy and Applcaon 11, P 6