Table Of ContentELEMENTS
OF DYNAMIC
OPTIMIZATION
AlphaC .C hiang
ProfeosfEs coorn omics
TheU niverosfCi otnyn ecticut
McGraw-Hill, Inc.
New York StL.o uisS anF ranciscoA ucklandB ogota
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NewDe lhi ParisS anJ uan SingaporSeyd neyT okyoT oronto
ELEMENTS OF DYNAMIC OPTIMIZATION
InternatEidointailo1 n9s9 2
Exclusriivgeh btysM cGraw-HBiololk C o-Singafpoomrra en ufactaunrde
exporTth.i sb ookc annobter e-expofrrtoemdt hec ounttroyw hiciht i s
consignbeyMd c Graw-Hill.
Copyrig©h t1992b yM cGraw-HiIlnlc,A. l lr ighrtess ervEexdc.e pats
permitutnedde trh eU niteSdt atCeosp yrigAhctto f1 976n,o p arotf t his
publicamtaiyob ne r eproduocred di stribiunta endyf ormo rb ya nym eans,
ors torienda datbaa soer r etriesvyaslt ewmi,t hotuhtep riowrr itten
permissoifot nh ep ublisher.
5 6 7 8 9 S0W K9H L6 5
LibraorfyC ongreCsast aloging-in-PDuabtlai caton
ChianAgl,p hCa. ,( date).
Elemenotfsd ynamiocp timizatAilopnh Ca. C hiang.
I
p. em.
Includbeisb liograrpehfiecraeln acnedis n dex.
ISBN0 -07-010911-7
1.M athematiocpatli mizatio2n.E. c onomicMsa,t hematical.
I.T itle.
HB143.7.C45 1992
330'.'0511 '-dc20 91-37803
Thibso okw ass etin C entuSrcyh oolboboySk c ienTcyep ographIenrcs.,
Thee ditwoars S cotDt. S tratford;
thep roductsiuopne rviwsaosDr e nisLe. P uryear.
Thec ovewra sd esignbeydJ ohnH ite.
Projescutp erviwsaisod no neb yS cienTcyep ographIenrcs.,
Artw asp repareelde ctronbicySa clileyn Tcyep ographIenrcs.,
Wheno rderitnhgit si tlues,eI SBN0 -07-112568-X.
PrintiendS ingapore.
CONTENTS
Preface
xi
Part Introduction 1
1
TheN aturoef D ynamiOcp timization 3
1
1.1S alient Features of Dynamic Optimization Problems 4
1.2V ariable Endpoints and Transversality Conditions 8
1.3T he Objective Functional 12
1.4A lternative Approaches to Dynamic Optimization 17
Par2t TheC alcuolfuV sa riations 25
TheF undamentParlo bleomf t heC alculus
2
ofV ariations 27
2.1T he Euler Equation 28
2.2S ome Special Cases 37
2.3T wo Generalizations of the Euler Equation 45
2.4D ynamic Optimization of Monopolist 49
2.5T rading Off Inflation and aU nemployment 54
TransversaClointdyi tions
3
forV ariable-EndPprooibnlte ms 59
3.1T he General Transversality Condition 59
3.2S pecialized Transversality Conditions 63
3.3T hree Generalizations 72
3.4T he Optimal Adjustment of Labor Demand 75
vii
viH CONTENTS
4 Second-OrCdoenrd itions 79
'·
4.1 Second-OrCdoenrd itions 79
4.2 TheC oncav;ictoyn vexSiutffiyc ieCnotn dition 81
4.3 TheL egendNreec essaCroyn dition 91
4.4 Firsatn dS econVda riations 95
5 InfiniPtlea nniHnogr izon 98
5.1 MethodoloIgsicsauolef sI nfinHiotrei zon 98
5.2 TheO ptimaIln vestmPeantth o fa Firm 103
5.3 TheO ptimal SSoacviianBlge havior 111
5.4 Phase-DiaAgnraalmy sis 117
5.5 TheC oncav/iCtoyn vexSiutffiyc ieCnotn dition 130
Again
6 ConstraiPnreodb lems 133
6.1 FourB asiTcy peosf C onstraints 134
6.2 SomeE conomAipcp licatRieofnosr mulated 144
6.3 TheE conomiocfEs x haustiRbelseo urces 148
Part OptimCaoln trTohle ory 159
3
OptimaClo ntroTlh:e M aximumP rinciple 161
7
7.1 TheS implePsrto bleomfO ptimaClon trol 162
7.2 TheM aximumP rinciple 167
7.3 TheR ationoaflt eh eM aximum Principle 177
7.4 AlternaTteirvmei nCaoln ditions 181
7.5 TheC alculoufVs a riatiaonndOs p timCaoln trol
TheorCyo mpared 191
7.6 TheP olitical BCuyscilnee ss 193
7.7 EnergyU sea ndE nvironmeQnutaalli ty 200
Moreo nO ptimaClo ntrol 205
8
8.1 An EconomIinct erpretoaftt hieoM na ximumP rinciple 205
8.2 TheC urrent-VHaalmuiel tonian 210
8.3 SufficieCnotn ditions 214
8.4 Problewmist She verSatla taen dC ontrVoalr iables 221
8.5 AntipolluPtoiloinc y 234
InfiniHtoer-izoPnr oblems 240
9
9.1 TransversCaolnidtiyt ions 240
9.2 SomeC ounterexamRpeleexsa mined 244
9.3 TheN eoclassTihceaolro yfO ptimaGlr owth 253
9.4 ExogenoaunsdE ndogenoTuecsh nologiPcraogresls 264
OptimaClo ntrwoilt hC onstraints 275
10
10.1 ConstraiInntvso lvCionngt rVoalr iables 275
ix
PRJ:P'A.CE
10.2T he Dynamics of a Revenue-Maximizing Firm 292
10.3St ate-Space Constraints 298
10.4E conomic Examples of State-Space Constraints 307
10.5L imitations of Dynamic Optimization 313
315
Answers to Selected Exercise Problems
323
Index
PREFACE
Inr ecyeenatrI hs a vree cemiavneryde quteoes xtpsa mnyFd undamental
MethoodfMs a tmhaetiEccooamnli tcois n cltuhdseeu bjoefdc ytn amic
optimiSzianttciheoee nx .i sstiioznfetg h abto owko uilmdp oass ee vere
spaccoen stIrd aeicnittdop,e r de stehtneot p oifdc y naompitci miiznaa tion
sepavroaltueSm eep.a rateness tnhopetr weisvteohnlstutc maaebnn ed ing,
consiadsea rc eodn tinoufaF tuinodna mMeetnhtooadfMlsa t hematical
Ecomniocs.
Ast htei tElelem teosnDyf n maiOcpt iamtiiizompnl tihebisos,o i ks
inteansda enid n trodtuecrxtatot rthyhe raa nne ncycltoopmeWedh.ii cl e
thbea soifct shc el ascsailccaoulfvl aursi aatniidot nmsso decronu sin,
opticmoanltt rhoeloa ryr,ee x platihnoerdo udgiehffrleyn,gt aimaealsn d
stochcaosnttairrcoe l i nnoctl uDdyenda.pm rirocag mmiinsegx plaiinn ed
thdei screftroem-I;te ixmcelt uhdceeo ntinuovuesr-sbtieioucmnsa eie t
needs dpiaeffrrteineatqliua alt aisao p nrse reqwuhiisciht ehw,ao vuel d
takuesfn a arfi eld.
Althotuhagedh v enotp toicfmo anltt rhoelho racysa utsheceda lculus
ofv ariattoi oobvnreess hadIod weeedimi,t n advtiods iabslmteiht seos p ic
ovfa riactaillocunFusao.lor n teh iank gn,o wloeftd hgleea titisen rd ispens
abfloeu rn derstmaanncdyli ansgs ic ecworniotmitintec hn ce pa alpceurs
luos-f-varimaotliBdoe.ns sit dhemese ,t hiousds eedv einnr ecwernitt ings.
Finaalb layc,k ngdr otuihnce a lcoufvl aursi aftcaiiolniastb aetteatsne dr
fluluenrd erstoafon pdtiincmgoa nltt rhoeloT rhyer. e adwehroi so nly
interienos ptteidm alt hceoomnryat yri,ofd l e sisrkeiPdpa, r 2 to ft he
present Bvuotlw uoImu esl.tdr ornegcloym mreenadda itln ega st the
flolowCihnag2p :(. t hEeu leeqru atSieo4cn.2. )( ,c hecckoinncg1ac vointy
vexiatnyd) ,5S .e(cm1.e dtohlooigsiscouafiel ns fi nhiotreir zeolne,av lasnot
too ptimalt hceoonrtyr)o.l
Certfaaeitnu orfte hsbi oso akrw eo rptohi ntiInndg e voeulto.p ing the
Euleeqru atIsi uopnpm,lo yr e dtehtaani olmtsoh sbeotro iknos r dtehra t
three acdaebnre tatpeprr etchibeae taeuo tfty hl eo giincv o(lSvee2.c1d)..
xi
xii PREFACE
Inc onnectwiiotinhn finite-phroorbilzeoImn as t,t emtpotc larsiofmye
commomni sconcepatbiooutnths ec onditfioocrno sn vergoefin mcper oper
integ(rSael5cs.. 1 I)a .l stor tyo a rguteh atth ea llecgoeudn terexamples in
optimcaoln trtohle oaryg ainisntfi nite-htorrainzsovne rcsoanldiittyi ons
mayb es pecisoiunstc,h ee iyn volav fea iltuorr ee cogntihzepe r eseonfc e
implificxiettd e rmisntaalt ients h oseex amp(lSese9 c..2 ).
Tom aintaai sne nsoefc ontinwuiitthFy u ndamenMteatlh odosf
MathematEiccoanlo miIch sa,v we ritttehniv so lume aw ciotmhp arable
levoefle xpository apnadtI,ih eonpcece,l, a raintdyr eadabiTlhiedt iys.
cussioofmn a themattieccahln iqiusae lsw ays reiwniftonhru cmeedr ical
illustrations, ecoannodem xiecr cpeirxsoaebm lpelImensts .h, en umerical
illustraptuiropnoss,pe rleys etnhtes hortest-dpirsotbalnecmse-i am ple
I
problweimt ahw ell-knsoowlnu tiosne-viendr iaffle raelntte rnfaotrimvue
latioannsdu, s ei ta sa r unnitnhgr etahdr outghheb ook.
Int hceh oiocfee c onoemxiacm plmeysm ,a jocrr iteirsti hoesnu itabil
itoyf t hee conommiocd elsi llustroaftt ihoepn asr ticmualtahre matical
as
techniuqnudeessr t udAyl.t hourgehc eecnotn omaicp plicaatrineoa ntsu ral
candidfaotiren sc lusIih oanvn,eo ts hieadw ayf rocml asasritci cSloemse.
clasasritci calrneeo sto nlwyo rtsht udyiinnt gh eoiwrn r ighbtu,ta lstou rn
outt ob ee xcelfloeirnl tl ustpruartpiovbseee csa utshee miord eslt ructures
aruen cluttweirtsehed c ondcaorym plicaastsiunmgp tiAso an bsy.- product,
thjeu xtaposoifot liadon ndn ewe conommoicd eallss por oviadnei sn terest
inggl imposfte h dee velopomfee ncto nomtihco ugFhotri. n stafnrcoetm,h e
clasRsaimcs egyr owtmho de(lS ec5.. 3t)h routghhen eoclasgsriocwatlh
modeolf C as(sS ec9.. 3t)o t heR omegrr owtmho dewli tehn dogenous
technolopgroigcra(elSs es9c ..4 o)n,es eeas p rogresrseifivnee meinnt th e
analytfircaamle work. Sfirmoimtl haecr llays,Hs oitce llmiondgeo lf e x
haustible reso6u.r3tc)oe t sh eF( oSresctm.eo rd elosfe nergyu sea nd
pollu(tSieocn. a ndS ec8.. o5n)es, e etsh es hiifntt hef ocuosfs ocietal
7. 7
concefrrnosrm e source-exthoea nuvsitrioonnm qeunatlaiAlt cyo.m parison
oft hec lasEsviacn mso deolfd ynammiocn opo(lSies2ct.. 4 w)i tthh em ore
recemnotd eolfL elaonndt hed ynamiocfas r evenue-maxfiirmmi( zSiencg.
10.a2l)s iol lustornaeto efts h em anyd evelopmiennm tisc roeconomic
reorientation.
Inl inwei tmhy pedagogpihciallo soIp ahtyt,e mtpote xplaeianc h i·
'
economic imno dsetle p-bym-asntneepfr r omi tisn itcioanls truction
a
throutghhei ntricoafmc aitehse mataincaallyt soii tsfs i nsaoll utEivoenn.
thougthh er esullteinnggt htireera tmreenqtu ilriemsi ttihneng u mbeorf
econommoicd eplrse <;enItb eedl,i evet htdehe atta iled giusdi edsainrcaeb le
becauistse e rvteoms i nimitzheet repidaatnidfo rnu stroafttieoanns soci
atewdi tthh el earnoifmn agt hematics.
Int hew ritionftg h ibso okI, h avbee nefiitmemde nseflryo mt he
numerocuosm menatnsds uggestoifPo rnosf esBsrourcA e.F orstoeftr h e
UniverosfiW tyyo minwgh,o skee eeny ecsa ugmhatn ys inosfc ommission
ando missiiontn h eo rigimnaanlu scrSiipntc.Ie d idn ota cceaplthl i s
PREFACE xiii
suggoenshsto,iw eIva elnroe,s hobuelhd e lrde spofnostrih rbeel mea ining
impeecrtfiMoannsoy.fm ys tudoevnettrshy ee aornsw ,h oImt ritehde
eardlireaorff tt hsbi oso akl,hs eol pmeewd i tthh eqiure stainordne sa c
tioSncso.t Stt rDa.tm fyeo dridet,xo err,tt hreeid g ahmto uonfet n courage
menatn pdr esastcu rrietm iocmaelnt tok se empeg oiAnngdt.h ceo opera
tievffoer otsfJ oseMpuhr pahtMy c Grawa-nHSdia lrla,h Roesser, Cheryl
KraannzEd,l lSiiema otSn c ieTnycpeao pghreIrnsmc,a. d,te h per oduction
procsemsoso twhe lalsp leasTahnatna.kr sae l sdou teo E dwaTr.d
as
Dowlfirnf gre retoiunstgo mtey pohgircaeaprlr tohrsal tu rkientd h e
o
initpriianlot fit nhgbe o oFki.n amlylw yie,fE mialgya oiffenr emde
unstianstsiinsgot nam nacneu scprrieppta. rT aota iloolftn heIm ,
am
deegprlayut le.f
AplhCa.C higa n
ELEMENTSO F DYNAMIC OPTIMIZATION
