Alpha chiang 4th edition PDF
Preview Alpha chiang 4th edition
FUNDAMENTAL METHODS OF Mathematical Economics ALPHA C. CHIANG KEVIN WAINWRIGHT Preface This boos is writen for those students of economics inter: om Leaning tke basic mate- vaglcaebods Hh ave become indispensable for apuoperndonsaing the eueee cans Herture, Unfoturatc, sving watheieais is foe mas. something aia 1 laking biturasting rsivino—sbsohutely necessary. but extrntsly unplsasant. Sach emat- tide, refered to as “ena anxuet” hs its roats—aeelieve largely in che imag tate in which mathomotice offen proscntal ts sales fhe belief Un conciseness means sleganse, explanations olleed ae lequently to hie fr elaiy thus parang su des ané séving them an undeserved sense of inellzetniuksquney. Au oly dermal ate, he nul secompacied by sry inudveilusteaons oF demonetsa “an impart mottsation, A uneven progression inthe level af mace can rake cectain matheratisl topos appear mre ili Haw hey scully a. Final eerie pro 2 supbiatvaled may tend ta shatter sindents' cont deuce, athe tan stimulete thiskine a itendec evar: tha sme ean With chac ss mind, we avez sczigus ella \o minimize unxiery-causing Features, Ti the exten pose, patent rather thaa cryptic explanations arc offered. Lhe style is a- liberavely informal and “reaue Sriendly.” As a mastar af resine, we Iry W anlitizale em sever questions thal. ate key Ip aria in the snus ais choy rd, Vo underscore the relovane of aathesuatis io economies, we let the analvical needs of econm:sts mo viva he stay of che celated ezathematica techriquss and ter ‘usa he kale will ape Dregriete ccoamic mutes insite afterwad. Also, dhe wathertical too kits bulk yuna e-etelly graCuated schedule, with the aiemertary tools serving ay sein stoues te che noreadvanced inos discus les; Wacrover appropri, graphic visa reinforctounl to Uhe algebraic results, ind we have desigced the exorcise problems 48 drills 20 help soliddy grasp and bolster contidcnee, rather than caae alleges Ta right unwittingly ustate and inteiece the nice Tw this book, Te following mice types of economic analysis me eset sie fx Tibrium analysis, comparacive varies, opti tion poubkems (asa spovial type of sates, Atynaaie, andl dytasnie upiimtzzsion. Te tackle these, te Sllowine mathomutical methods se ntreduced in due course: mattis ager, fferenial and integral cafeulus ilenatial equations, difference ecuntions, and ein'al simul Aww. Bevause of the substanial nucher ob illustrunve evoneanie models—both maczo asd micte appeorang here, this ‘book should be eseful alse ce those who ate already ratvcmatically tesined stil in ne ‘of agate to-ushes them fom the real af mratheina i ihe ud o economies, For the sams 1eas0, the book shoulé not only serve at a text Zora course on mathematical mith 045, but elso as supplementary recding in s.ch mouse as mireceunomi Ihoery. msLT0 sconams they, and eesti srowdh smd development ‘We huve attempted te retain the princrpal objectives and style af the previous edit, owever. the present action contains sevecl significant charges, The auterial en mathe tmatiea’ prog atnining is now preseatel earlie: in a new Chap. 13 enticed “Further ‘igics in Opsimization.” This chapter has twa myer theres: ipl miation wilh inequality <u stia.n ant the envelope: dnooren” Loder the Set theme, the Kube-Tncker conditions ate vil et ped in mach the same manera in the previous edition, omever, the topic hay be Gnhanced with severed aew economic apphications, Zackading poak-load pricing and 09 sunet rationing, The second theme is relia to fe develope the emveiope thantom, the maximumevalue fipclion, amd the natinn of cvaity. By applying the envclope theur=m tw vacious economic models. ve derive important results such we Roy's ier, Shephard lenunia, and Hitctling’s fomma, ‘The sevond zusjur ation to this catition isa naw Chap, 20/60 optimal conten taeory, ‘he pucpose of tis chapter i to fnlruduve te wader tothe base if opinal centre and euwasirale how it nay he applied in economies, including examples fiver natural r= source esonomics anu april grow: fheory. The eucovnl in this chapter rca in ere part from the disenssion of optimal contol theory in Elements of Dynanie Opubmizaion hy Alpbui © Chiang, (MoCraw-Hill 1992, now puslshed by Waveland Press lc, which pee seots a thorough teataunt of lb upli wuntl ai ils precser,caeulus of vations. ‘Aide from the two ew chapters, there are several siziticamt adits an reteset lu this eicion, Ie Chap. 3 we bars cxpanded the discusiot of solving iugher-onder poly- somal equations by factoring (See. 3.3). n Chap. 4, a new seetion ont Merk ulin has ‘Wn hfe! (See. 47). And. in Chap. 5, we have introduced the checking ofthe rank of tmatrix via am schelun cute (See 5.11, wn the Hawkins-Siman condition in connection with the Leontet input-output model sSec. 5.7). Wilh respeet te wear applicatne ian nev examples Rave boon added and sovne o€ the existing applications have been ene Trance. A linear vetsiu of the I-T Mf mort has hoon included in Sou, 5.6, and amore gen cal kvm ofthe model in Sec. 8.6 bas boon expanded la socoaspass bath a ced al open seonomy, Gurehy demowstzling » mich richer application of esmpacaive statics eneral-fuaction models. Other additions include a discussion of expected uty and risk proferences (Nec. 4:2}, a proft staxinhization tacéel that inearporates the Cabb-Duulas production fimelion (Soe. 11.8), ants woperind infecte-pora, choise problem (See. 12.3), Finally, the exercise problems tave been cevised an! augneicl,giviag ste dems a gecaler opportuni to har heir sill Contents PART ONE INTRODUCTION (1 Chapter 1 (he Nature of Mathematical Economies 2 aw 12 (Mathecnatical versus Nenmmathematical Reaworniss 2 ‘Mathermaticat Economics versus oonometrios 4 Chapter 2 Economic Models 3 24 22 23 24 2S 26 a7 Ingrediems of a Matbematical Model § Variables Const and Parameters 3 Exstivas and Bdentiioy Tw Real-Nurubcr System 7 The Conceptaf Seis & Set Nation Relotionships berseen Sess 9 Operations wm Se UF Eaves of 31 Operations broxice 23 14 Relations and Funecions 15 ndered Pats 15 Peiations ani Fuvctions 16 Exenive24 19 Types of Fucvetion 20 Comins Putin, 0 Posmamal Bwerent 20 ettongi unsoas 22 romalgetn Fuectoms 28 1 Digmession on Btgonanis 24 Buanie 2S Fusions ut Twa ue Mose Independent Variables 2 Levels ef Gereralsty 27 PART TWO STATIC (OR EQUILIBRIUM) ANALYSIS 20 Chapter 3 Faquilibrium Analysis in Economics 3 4.1 The Mesning of Egeiibriam 30 3.2. Parsi Maskst quikirium—A Linear Model 31 Consircing 33 Madel 38 wnetain The Quadhats Farmala 66 Mother Graphical Sebats ¥ 34 General Market Lqui:bcium 4) Teantomeman Marker Model 44 Mamerica Example 42 Commits Cae 44 Soleo fet Goneraiguacon Seems 26 Exerine M448 3.5 equilibrium in Nationateinccnn:Anidysis 4h Chapter 4 Linear Mudels and Matrix Algebra 48 41 Vaarces ont iets 4 Mouriecayatrrays 32 Berto a Specie! Manes 30 Eeencive.n! 31 42. Mais Operations ‘Alina snd Suomen gf edocs $2 Seale Ssdinication $2 ai Contents The esto of evan 36 Ei rere 42 4.3. Noteson Veclor Operations 53 Haale ovion uf teers 59 eames interpuctscn Peter oe aperane, 2 i Nectar Space 84 Frenne 538 44 Commutacive, Aswusiulve, urd Disrbutive Laws 8 ian ddan italic Attain 8 pacar 4.5 [dontcy Mavicesarat Null Varies 70 Mersin Waser sea Maasces. 7 of Maris Abgche inence and Soe Properies 75 Mies aud Sotuine imear Eguatcs Spee Exerive 46 AT Finite Marky Chains 7% Sovedal Cveedbverbing Mace Chains 8 Reena? 8 Chapter 5 Linear Models and Matrix Algebra (Continued) $2 5.1 Contino or Nonsinguiarity of2 Mamnix 82 Newer frsns Sion Cembicams 2 at Even 5 5.2. Test of Noos:eyclarly hy Lee of Determinant 88 ued Newser Frabasing an et-Cndee Deter by mo Laglocs Epa Feene £2 9 53 34 55 56 57 58 Basie Proparties of Detecminants 1 Paicrimontad Crtrion for Nonamguleriy 96 Regi ofa Marie Ned Ye Trevvive 3498 Piudingthe Inverse Matric) Deseret be Aton agus Cafacrne 99 Maire wrsicn it) Brewiw $4102 Crumars Rats WU tie Rae Prervation 6 si Seta #05 on 16 Buvvise 13107 Applivatin: to Masket and Najioral-Income Models 107 Moré Maal 107 aime Mole BBLH Merit Cased Seaman 109 Sri Aig wens Btinination of Bivens 134 proms 36D Leona! pulutant Models 12 Sr she Oram Medel SF 4 Kasterioat Exapie Tie Exige of evapo Sota 8 Exetonis Mesming of tie HawbrseSinan cea tga Chipur Mae 1 Cumin 18 Ihe Clara Evence$? 12) Limetal ans of static Analysis 120 PART THREE COMPARATIVE-STATIC ANALYSIS 123 Chapter 6 Comparative St s and the Concept of Derivative 124 61 62 The XeLav ot Comparative Statics 124 Rate af Change au Ike Duivalive 128 Tae Dighsonee Queniont 125 ie Daria 226 Fava td 63 The Derivaiveandthe Shope ofaCave 12 64 ‘he Coneen of Lint 129 Kaede biden ag See 128 rahi Bronte of Porno Me fae Lint Cones Frise 139 6.5. Digrossion on Inoqualitis ae Absolu: Valin 136 Auderof leet 136 we Hien cd eg siratiuey 136 nit 840 ae on ofan Press Boers hiS 1 66 Lani Teorey 138 Shears tending a Sing Panation 129 Toor Ieleing Tua Fuacane 1h Lint fs Pobmenial apiion 44 Exercise 56 347 6.7 Contumity ond Duteerebilly als Function 111 Coniinste P ts Ewa? 36 Chapter 7 Rules of Differentiation and Their Use in Comparative Statics 148 m1 2s of Differentiation fara Funct a wed #5 Exerie 21 182 7.2. Rules oF Divan abe valving Taw ot Mon: Funetions of che Same Variable Sue-iermice Bue 182 lie 2 Haat Sanna Nowcrte Sane aareayerRosee Eonerine 6 Quecvem Bide 188 eta fwecge-Can Prewtions 189 yereine 226 p Betaven urscna-Cis an CConten:s ali 9.3. Rules of bifferentiation Levatvmg unctians wDilfnsat Variables 18) Chain Rule 152 Furcivm Rade 9h ce 7.4 Vartisl Vigleremation 105 vovches E85 Netiies 167 Geen Bes 10 7.5. Applicadonste Compan Anaysis 170 ‘ric Mat 170 tsb Output Met 3 sxeribe 25075 7.6 Noten dacobar. De-ern:inams 175 Chapter 8 Camparative-Sratic Analysis of Gencral-Funetion Models 178 81 Dillereatials 179 inter Peviaives 179 Pagloventals and You whats 14 Piewie 2M 8.2 Total Differentiate “34 Bremen? ia 83 Rules of Ditferentials Brense hd 16 8.4 Talal Devivsises 189 Breny te Paal Dertnve 4 Bandar ane Tene 88 Avon eatin Some enol Remacts 192 Frewaea #398 85. Derivativosof Implicit Functions. 4b ftps merions 195 ithe Shes anvina-Seuuiin Cine 700 uerise £5 206 iy Contents 8.6. Comupstlive Staley of General-Funetion Models 205 Market Made! 2 ee of Taal Devtctives 28 fxcmee Movdh EM 240 oxen ae Sted Open Eewinay 28 Soman ofthe Proveire NG ea TP 8.7 Lamwauiens of Comparative Statics 218 PART FOUR OPTIMIZATION PROBLEMS | 219 Chapter 9 Optimization: A Speclal Variety of Kquilibriam Analysis 220 9.1 Optimum Values und Exieme Vdues 221 9.2. Rollie Maxintin and Minima Fra-Devivrive Tat 222 elie atom 2 De Eoeeise #2 9.3 Sovoya aa Higher Derivatves Doriatise 27 ‘a Deviate ukrprcaitn ofthe Sacond Deron lucia 290 Anite toward isk 282 Excise 23. 9.4 Socond-Rerivative Tes 233 sensu Sficion Condinane 23 Conley far Prt Maaimizaiea, 235 Cagitctems efo Crbie Psat. pros Soa Marina Renown Came 240 Feerive 94 24d 9.5 Muchvuria unl TaylurSeries 242 Macinarin Sos of SoSpuomial Funvisn 242 fete Sevis af Hb Espaucion of Lagrane Horne c'the Reminder 248 Premise 05 2500 n Anite Hamer 285 9.6 Mth-Derivative Test for Retatve Extrercurn of 4Funeliva of Owe ¥inivble | 250 ardor bxpansion and el Shee Spe Bk Deva Reeve 5:6 38 Chapter 10 Exponential and Lagariliumie Funetivms 255 ire kxtrorman 250 fe Caney 253 Yer 10.1 The Vature of Fxponentis! Functions 256 Sipe Bypanestio mamctian 756 Graphical Fe eneraized Exgrmenial Fm Sine aPreer ese 208 Berne U0! 260 1O.Z Nauru Fe penta Fucotinns and the Problem of Growth 260 The Menine e240 tnt 22 ning em De hat it, Ierpeenation nserens Compr ae Ingrantsneoas Rate of Growth 268 Coatinas sees Dacre Growth 205 Pemsting and Negae mnth 266 Exercise 12257 10.8 | ogwittuns 267 ‘ie Hearing of Logavthee 267 Comer Lay wed Sutera So Fes of Fagurthinn 209 An Spolication 77! Exercise £03272 10.4 Logarithmic Functions Log Eincvons and ie Tit Graphical Pos 273 Bare Comersion 774 Exercise 104 276 ivatives of Frontal and Logarthan:e Functions 277 Toone Rie 277 Eepanenta-areton Rate 7 “ie Res hen Th Case of uve > 259 igor Hervartses 2 An Anplicasin 282 Exercise 163° 242 ws 10.6 Optimal lining 282 A Prnblem of Bixe Yonnee IKE Maxinsiton Contos: 280 A Prolene Finbar Casing 28 Brercise 166 286 1.7 Further Applicatancof kapsonanr al and T uguilnie Derivatives 286 Finding le Base of Groah 28% ate af Gow Cambinain of cme 3H mabe Pun Elatig 288 ay Chapter 11 The Case of More than One Choite Variable 291 1941 Toe Ditlerersie Version of Option Conditions 280 Resende Conse 282 Second Outer Conon 292 Dari Cai ian of 180 Fire Order 0 iveaihCach Partial Devivacres 2658 Secand Oster Bool OiBerenat SoconsbOnder Condition 285 Exe lig 3% 11.3 Qualleatics Forum — Aa Taeursion 3d oF iam 44 Sex anode Yoon 30 rea 9 0 Qh Pitre snd Biggie tin Detroit Te So Digiienn se 302 Costa Fame 3S Kirtbde Onan Earn, 307 Dano Kia c Deioiieesas 307 Premise 193 V4 Objective Functions with More aa Two Variables 313 Freee Sosone Une Coton 1S etre Cae, 326 Bee 14 7 wernt Test for Sigh conten ae 11.8 Sounnnl-Drder Qorltings in Rulstin Is Coneaviey and Comvenity 218 !Comenty 320 Checking Concavin fe sion 4 er Fueinas maa Chane See eid 11.6 Eeonuniv Applications Proiee ut Price Uicrinicnion sat Deckivasefe Mm 138 ioe i6 at 117 Coupara.ivo-Static Aspects of Optimizesion 342 Raskin Fae Santions 2 Gove! Peetion Sted 45 Bemie lit 33 Chapter 12 Optimization with Equality Constraints 347 12.1 EAfecte ofa Chnsuane “MT V2.2 Findiag We Satiunary Values 29 ier Mehul 380 ns fovie und Medtontraint Cases Erewvite 22 1S adsOrder Cendlitions 356 conn OnerFeal Vievemat 356 Sevant-Onin Condens 387 ‘The Seadened Horsian AIP 1238 iarighie Case 362 Mtstowesraint Case 342 124 Quasiconcaviny aed Quasivomesity sos set x sa a 12.S Tilly Meximiation axl Com Deru 37 Frew Order Cs SeconOnter Cr xi Gontanss Comparative Sate dnsysic ATE Prigsoriumate Changer in Prt Evecibe 125 382 126 Harageaecas hunctians 383 Limon Homegeniy 36S Coir Dexygla: Protertica Function 288 Exensions of te Renal 28% bromine 12.680 127 Least Cast Combinatinn of Taps 390 Frececie Condon 306 Sesona-Cher Coniaen 292 The Scene Ris 292 ianowher oettons 18H Eke of Setalacton 6 CES Priest Sunes 57 Cobb Denies | tae CES Fie: sian es a Special Cate ot 39 Chapter 13 Further Topics Optimization 402 13.1 Nonlinea: Progracining anc kutn-Lacker Conditions 482 Siep 4 Bec fons Aestirins 03 Step 2 Efe of. Inception the & 408 fatto am Consatiesase 10¥ 13.2 The Cousieein! Quulliestign 412 pers ty HE a5 cowatiy Constaines 2 dn Tucker Sraguisz ea Toe Cans (a 1 Ces Bsereiee 112 aU 13,3 Evonoeriy Appliesiors a8 iy ene Rat 13.4 Satticiency Theorems in Nonlinear Progamming 424 ia Tae 4 ComarainsQualgemina Tse Beereise 12.927 13.5 Maximucn-Value Functions and tae Favelope Thonrsra 38 ihe Havel Teun fr Cec iemaives Opie The Prof Punccan 479 Imerprotion ofthe Logvenye Mestntier 46 13.6 Quality and the Eavclape Theorsin The Fama! Polen The Pa Probie 436 ae 13.7 Some Concluding Remarks 12 PART FIVE DYNAMICANATYSIS. 443 Chapter 14 Reonomie Dynami Calentus 444 14.1 Pynamiss and Imegration 44d 14.2 Indelicite Integrals 446 Theis ef Tatra 44 Basic Pale of bg and Integral 14.3 Nefinite Incegrsls 494 Meaning of Donne Inegrois #89 te tear es am ti vader 9 aS Dok at the dette ste tages Beerioe 14 18) 6 ince atv Exaivo 64 368 14.5 Some Eoonomic Applications of Incegrals 464 Frame Margi tumesion 09 3 oat uetion 45 Iivestmens ad Capit Romcieon 345 Present Kos oft Cash Vow $0 rose Cate uf Pervenal Fon 470 Exercce 148 496 14.6 Dunas Growlt Model 471 Tie Framewirk 37) Fad the Sitio 422 Ihe Razors Big $72 Bier sie A ATH Chapter 15 Continuous Tine: First-Order Differential Equations 475 15.1 First-Order Lines Differential Equations ith Constaal Cetliien od Constant Tau 475 The Homayenenas Cave 476 The Runomogenscus Case 476 rion te Seon Rucci tot 48) 35.2 Dynamics uf Market Price 4% The Fearne St The Fie Path 8 he Liane Si Ar Abeomate B.of he Mk Bwoens 152 188 15.3 Variable Coefficient and Varahle‘lerm 485 Fhe Homcgenncnn Cas 86 Hie Rontionupenevan Case 385 Ever {3.4 4R6 15.4 Veal Dither Equarions 486 tout Hieron Sigusionn a Movil of Seaton 18? Imagreing Fctcr 89 Salaton of Pra-Ondet Linear Diferentiat Eguuions 15% Browse 14d of Beir 43 se Contents xi 15.5 Nonlinear Differential Fxnatians ol thy First Outer anal Fist Degree 292 eves Dioxin igaosions BIE Sacbie arabia 183 Naren Mop tbi Bvecine 153 0 15.6 The Qualitaive-Ciraphic Agprauch 495 The Phave bingrom 395 Tipe wbene Pa 96 Reerwn (5.8 8 ne Linear Fon 42 Guay Aeoysis S00 sro 290 Chapter 16 Higher-Order Differential Equations 303 16.1 Second Onder Linear Diflvontin! Heuations vile Constant Coublisenls and Constant Tem 3 The Revéadar eiggeltet Fe Comgometrs Basen {The Dy numir Sobilip of gustan 516 Praise 164 SH 16.2 Comples: Sembers and Circular Functions $11 Iruaguar amc Coen Sun Comps on Cneulon Fs Proportion of ss Fctiams 5 Ral Atternste Renees Numbers 519 Motaons niCampk Beenie 142300 16.3. saalsis ofthe Camplex-teot (se 52 Me Cierplementars in anf 8 Tho Fine Path 38 He Dic Sat ar ce 162 32 Eat 5
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