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1、Mathemaaticall methhods ffor ecconomiic theeory: a tuttoriallby Marrtin JJ. OsbborneTable oof conntentss Introduuctionn and instrructioons 1. Reviiew off somee basiic loggic, mmatrixx algeebra, and ccalcullus o 1.1 Loggic o 1.2 Mattricess and soluttions of syystemss of ssimulttaneouus equuationns o
2、1.3 Inttervalls andd funcctionss o 1.4 Callculuss: onee variiable o 1.5 Callculuss: manny varriablees o 1.6 Graaphicaal reppresenntatioon of functtions 2. Topiics inn multtivariiate ccalcullus o 2.1 Inttroducction o 2.2 Thee chaiin rulle o 2.3 Derrivatiives oof funnctionns deffined impliicitlyy o 2.
3、4 Diffferenntialss and compaarativve staatics o 2.5 Hommogeneeous ffunctiions 3. Conccavityy and conveexity o 3.1 Conncave and cconvexx funcctionss of aa singgle vaariablle o 3.2 Quaadratiic forrms 3.2.1 DDefiniitionss 3.2.2 CCondittions for ddefiniitenesss 3.2.3 CCondittions for ssemideefinitteness
4、s o 3.3 Conncave and cconvexx funcctionss of mmany vvariabbles o 3.4 Quaasiconncavitty andd quassiconvvexityy 4. Optiimizattion o 4.1 Inttroducction o 4.2 Deffinitiions o 4.3 Exiistencce of an opptimumm 5. Optiimizattion: interrior ooptimaa o 5.1 Neccessarry connditioons foor an interrior ooptimuum
5、o 5.2 Suffficieent coonditiions ffor a locall optiimum o 5.3 Connditioons unnder wwhich a staationaary pooint iis a ggloball optiimum 6. Optiimizattion: equallity cconstrraintss o 6.1 Twoo variiabless, onee consstrainnt 6.1.1 NNecesssary ccondittions for aan opttimum 6.1.2 IInterppretattion oof Lagg
6、rangee multtiplieer 6.1.3 SSufficcient condiitionss for a loccal opptimumm 6.1.4 CCondittions underr whicch a sstatioonary pointt is aa globbal opptimumm o 6.2 n vvariabbles, m consstrainnts o 6.3 Envvelopee theoorem 7. Optiimizattion: the KKuhn-TTuckerr condditionns forr probblems with inequualityy
7、 consstrainnts o 7.1 Thee Kuhnn-Tuckker coonditiions o 7.2 Wheen aree the Kuhn-Tuckeer connditioons neecessaary? o 7.3 Wheen aree the Kuhn-Tuckeer connditioons suufficiient? o 7.4 Nonnnegattivityy consstrainnts o 7.5 Summmary of coonditiions uunder whichh firsst-ordder coonditiions aare neecessaary
8、annd suffficieent 8. Diffferenttial eequatiions o 8.1 Inttroducction o 8.2 Firrst-orrder ddifferrentiaal equuationns: exxistennce off a soolutioon o 8.3 Sepparablle firrst-orrder ddifferrentiaal equuationns o 8.4 Linnear ffirst-orderr diffferenttial eequatiions o 8.5 Phaase diiagramms forr autoonomo
9、uus equuationns o 8.6 Seccond-oorder diffeerentiial eqquatioons o 8.7 Sysstems of fiirst-oorder lineaar diffferenntial equattions 9. Diffferencce equuationns o 9.1 Firrst-orrder eequatiions o 9.2 Seccond-oorder equattions Mathemaaticall methhods ffor ecconomiic theeory: a tuttoriallby Marrtin JJ. Os
10、bborneCopyrigght 1997-2003 Martiin J. Osborrne. VVersioon: 20003/122/28. THIS TUUTORIAAL USEES CHAARACTEERS FRROM A SYMBOOL FONNT. Iff yourr operratingg systtem iss not Windoows orr you thinkk you may hhave ddeleteed youur symmbol ffont, pleasse givve youur sysstem aa charracterr checck beffore uusi
11、ng the ttutoriial. IIf youu systtem dooes noot passs thee testt, seee the page of teechniccal innformaation. (Notte, inn partticulaar, thhat iff yourr browwser iis Nettscapee Naviigatorr verssion 66 or llater, or MMozillla, yoou neeed to make a smaall chhange in thhe broowser setupp to aaccesss the
12、symbool fonnt: heeres how.) IntroduuctionnThis tuutoriaal is a hyppertexxt verrsion of myy lectture nnotes for aa secoond-yeear unndergrraduatte couurse. It coovers the bbasic matheematiccal toools uused iin ecoonomicc theoory. KKnowleedge oof eleementaary caalculuus is assummed; ssome oof thee prer
13、requissite mmateriial iss reviiewed in thhe firrst seectionn. Thee mainn topiics arre mulltivarriate calcuulus, concaavity and cconvexxity, optimmizatiion thheory, diffferenttial eequatiions, and ddifferrence equattions. The emphaasis tthrougghout is onn techhniquees ratther tthan aabstraact thheory
14、. Howeever, the ccondittions underr whicch eacch tecchniquue is appliicablee are stateed preeciselly. A guidiing prrincipple iss acccessibble prrecisiion. Severall bookks proovide addittionall exammples, disccussioon, annd prooofs. The llevel of Maathemaatics for eeconommic annalysiis by Knut Sysdaa
15、eter and PPeter J. Haammondd (Preenticee-Halll, 19995) iss rougghly tthe saame ass thatt of tthe tuutoriaal. Maathemaatics for eeconommists by Caarl P. Simoon andd Lawrrence Blumee is ppitcheed at a sliightlyy highher leevel, and FFoundaationss of mmathemmaticaal ecoonomiccs by Michaael Caarter is m
16、oore addvanceed stiill. The onlly wayy to llearn the mmateriial iss to ddo thee exerrcisess! I welcoome coommentts andd sugggestioons. PPleasee let me knnow off erroors annd connfusioons. The enttire ttutoriial iss copyyrightted, bbut yoou aree welccome tto proovide a linnk to the ttutoriial frrom y
17、oour siite. (If yoou wouuld liike too trannslatee the tutorrial, pleasse wriite too me.) Acknowlledgmeents: I havve connsulteed manny souurces, inclludingg the bookss by SSydsaeeter aand Haammondd, Simmon annd Bluume, aand Caarter mentiioned abovee, Matthemattical analyysis (2ed) by Toom M. Aposttol
18、, EElemenntary diffeerentiial eqquatioons annd bouundaryy valuue prooblemss (2edd) by Williiam E. Boycce andd Richhard CC. DiPPrima, and Diffeerentiial eqquatioons, ddynamiical ssystemms, annd linnear aalgebrra by Morriis W. Hirscch andd Stepphen SSmale. I haave taaken eexamplles annd exeercisees fr
19、oom sevveral of thhese ssourcees. Instrucctionss The tuttoriall is aa colllectioon of mainn pagges, wwith ccross-referrencess to eeach oother, and linkss to ppages of exxercisses (wwhich in tuurn haave crross-rrefereences and llinks to paages oof sollutionns). The maiin pagges arre lissted iin thee
20、tablle of conteents, whichh you can ggo to at anny poiint byy presssing the bbuttonn on tthe leeft maarked Conttents. Each paage haas navvigatiional buttoons onn the left-hand side, whicch youu can use tto makke youur wayy throough tthe maain paages. The mmeaninng of each buttoon dissplayss in yyour
21、 bbrowseers sstatuss box (at tthe boottom of thhe scrreen ffor Neetscappe Navvigatoor) whhen yoou putt the mousee overr thatt buttton. OOn mosst pagges thhere aare teen butttons (thouugh onn thiss inittial ppage tthere are oonly ssix), with the ffollowwing mmeaninngs. o Go to tthe neext maain paage.
22、 o Go to tthe neext toop-levvel seectionn. o Go backk to tthe prreviouus maiin pagge. o Go backk to tthe prreviouus topp-leveel secction. o Go to tthe maain paage (text) forr thiss secttion. o Go to tthe exxercisses foor thiis secction. o Go to tthe soolutioons too the exerccises for tthis ssectioon
23、. o Go to tthe taable oof conntentss. o Search throuugh alll pagges off the tutorrial ffor a strinng. o View teechniccal innformaation aboutt viewwing aand prrintinng pagges. If youd likke to try uusing the bbuttonns noww, preess thhe blaack riight-ppointiing arrrow (on a yelloow bacckgrouund), whic
24、hh willl takee you to thhe nexxt maiin pagge; too comee backk heree afteerwardds, prress tthe bllack lleft-ppointiing arrrow oon thaat pagge. After yyou foollow a linnk on a maiin pagge, prress tthe whhite Text buttton too retuurn too the page if yoou wissh to do soo befoore gooing tto thee nextt ma
25、inn pagee. To help you kknow wwhere you aare, aan abbbreviaated ttitle for tthe maain paage too whicch thee butttons oon thee leftt corrresponnd is givenn at tthe toop of the llight yelloow pannel. (For tthis ppage, for eexamplle, thhe abbbreviaated ttitle is IIntrodductioon.) Pagess of eexamplles a
26、nnd sollutionns to exerccises have orangge bacckgrouunds tto makke it easieer to know wheree you are. If yoou gett lostt, preess thhe Teext bbuttonn or Conteents buttoon. Techniccalitiies The tuttoriall usess fraames extennsivelly. Iff yourr browwser ddoesnt suppport framees, Im nott suree whatt you
27、ll seee; I suggeest yoou gett a reecent versiion off Netsscape Naviggator. (Othher feeaturees thaat I uuse maay alsso nott be ssupporrted bby othher brrowserrs.) Some veery olld broowserss thatt suppport fframess do nnot haandle the Back and Forwward buttoons coorrecttly inn frammes. HTML haas no ta
28、gs to diisplayy mathh. I hhave fakedd thee mathh by uusing text italiic fonnts foor romman leetterss, thee Winddows ssymboll fontt for most symbools (ggifs ffor otthers), smaall foonts ffor suubscriipts aand suupersccriptss, andd tablles foor aliignmennts. TThe reesult is reeasonaable uusing Netscca
29、pe NNavigaator wwith aa 12 oor 14 pointt basee fontt and a rellativeely hiigh reesoluttion mmonitoor, buut mayy not be soo greaat undder otther ccircummstancces. IIf whaat youu see on yoour sccreen lookss awfuul, leet me know and IIll ssee iff I caan do anythhing aabout it. MathML, a vaariantt of HH
30、TML, has eextenssive ccapabiilitiees forr beauutifullly diisplayying mmath, but iis currrentlly suppporteed onlly by Netsccape NNavigaator 77.1 annd itss coussins (e.g. Mozillla). I am workiing onn a MaathML versiion off the tutorrial. 1. Reviiew off somee basiic loggic, mmatrixx algeebra, and ccalc
31、ullus1.1 LoggicBasicsWhen maaking preciise arrgumennts, wwe oftten neeed too makee condditionnal sttatemeents, like if the pricee of ooutputt incrreasess thenn a coompetiitive firm increeases its ooutputt or if the demannd forr a goood iss a deecreassing ffunctiion off the pricee of tthe goood annd
32、thee suppply off the good is ann incrreasinng funnctionn of tthe prrice tthen aan inccreasee in ssupplyy at eevery pricee decrreasess the equillibriuum priice. These sstatemments are iinstannces oof thee stattementt if A thhen B, where AA and B staand foor anyy stattementts. Wee alteernatiively writ
33、ee thiss geneeral sstatemment aas A impliies B, or, usiing a symbool, ass A B. Yet twoo moree wayss in wwhich we maay wriite thhe samme staatemennt aree A is a suffiicientt condditionn for B, and B is a necesssary condiition for AA. (Note tthat BB comees firrst inn the seconnd of thesee two stateeme
34、ntss!) Importaant noote: TThe sttatemeent A B dooes noot makke anyy claiim aboout whhetherr B iss truee if AA is NNOT trrue! IIt sayys onlly thaat if A iis truue, thhen B is trrue. WWhile this pointt may seem obvioous, iit is somettimes a souurce oof errror, ppartlyy becaause wwe do not aalwayss app
35、lly thee rulees of logicc in eeverydday coommuniicatioon. Foor exaample, whenn we ssay iif its finne tommorroww thenn lets plaay tennnis we prrobablly meaan botth if its finne tommorroww thenn lets plaay tennnis and if its nott finee tomoorrow then llets not pplay ttenniss (annd mayybe allso iif its
36、 nott cleaar wheether the wweatheer is good enouggh to play tenniis tommorroww thenn Illl calll you). Whhen wee say if youu listten too the radioo at 88 ocllock tthen yyoulll knoww the weathher foorecasst, oon thee otheer hannd, wee do nnot meean allso iif youu dont lissten tto thee radiio at 8 occl
37、ock then yyou woont kknow tthe weeatherr foreecast, beccause you mmight listeen to the rradio at 9 ocloock orr checck on the wweb, ffor exxamplee. Thee poinnt is that the rrules we usse to attacch meaaning to sttatemeents iin eveerydayy langguage are vvery ssubtlee, whiile thhe rulles wee use in loo
38、gicall arguumentss are absollutelyy cleaar: whhen wee makee the logiccal sttatemeent iif A theen B, thatts exxactlyy whatt we mmean-no mmore, no leess. We may also use tthe syymbol too meann onlly if or is immpliedd by. Thuss B A is equiivalennt to A B. Finallyy, thee symbbol meanns immpliess and is
39、s impllied bby, oor iff and onnly iff. Thhus A B is equiivalennt to A B annd B A. If A iss a sttatemeent, wwe wriite thhe claaim thhat A is noot truue as not(A). If A annd B aare sttatemeents, and bboth aare trrue, wwe wriite A and BB, and if at leeast oone off themm is ttrue wwe wriite A or B. Note
40、, iin parrticullar, tthat wwritinng A or B inclludes the ppossibbilityy thatt bothh stattementts aree truee. Two rullesRule 1 If the stateement A B is truee, theen so too iis thee stattementt (not B) (noot A). The firrst sttatemeent saays thhat whheneveer A iis truue, B is trrue. TThus iif B iis fal
41、lse, AA mustt be ffalse-hennce thhe seccond sstatemment. Rule 2 The staatemennt not(A aand B) is equiivalennt to the sstatemment (not A) or (not BB). Note thhe orr in the ssecondd stattementt! If it iss not the ccase tthat bboth AA is ttrue aand B is trrue (tthe fiirst sstatemment), thenn eithher A
42、is noot truue or B iis nott truee. QuantiffiersWe someetimess wishh to mmake aa stattementt thatt is ttrue ffor alll vallues oof a vvariabble. FFor exxamplee, lettting D(p) be the ttotal demannd forr tomaatoes at thhe priice p, it mightt be ttrue tthat D(p) 100 ffor evvery pprice p in tthe seet S. I
43、n thiss stattementt, foor eveery prrice is a quanttifierr. Importaant noote: WWe mayy use any ssymboll for the pprice in thhis sttatemeent: p is a dummmy vaariablle. Affter hhavingg defiined DD(p) to be thhe tottal deemand for ttomatooes att the pricee p, forr exammple, we coould wwrite D(z) 100 ffo
44、r evvery pprice z in tthe seet S. Given tthat wwe jusst useed thee notaation p for a priice, sswitchhing tto z in tthis sstatemment iis a llittlee odd, BUT theree is aabsoluutely nothiing wrrong wwith ddoing so! IIn thiis simmple eexamplle, thhere iis no reasoon to switcch nottationn, butt someetime
45、ss in mmore ccompliicatedd casees a sswitchh is uunavoiidablee (beccause of a clashh withh otheer nottationn) or conveenientt. Thee poinnt is that in anny staatemennt of the fform A(x) foor eveery x in tthe seet Y we may legittimateely usse anyy symbbol innsteadd of x. Anotherr typee of sstatemment
46、wwe sommetimees neeed to make is A(x) foor somme x in tthe seet Y, or, equuivaleently, there eexistss x in tthe seet Y suchh thatt A(x). For soome x (allternaativelly thhere eexistss x) iss anotther qquantiifier, likee forr everry x; myy commments aboutt notaation applyy to iit. Exercisses 1.1 Exeercisees on logicc1. A, B, aand C are stateementss. Thee folllowingg theoorem iis truue: if A iss truee and B is nnot trrue thhen C is ttrue.Which oof thee folllowingg stattementts folllow ffro