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1、Evaluation Warning:The document was created with Spire.Doc for.NET.Mathe emati ical meth hodsfor econ nomic c the eory:at tutor rialb b byby M Ma a artiartin n n n J.J.OsbOsbo o orneorneTable e of cont tents sIntro oduct tion and inst truct tions s1.Re eview w of some e bas sic l logic c,ma atrix x
2、alg gebra a,an nd ca alcul luso1.1 L Logic co1.2 MMatri ices and solu ution ns of f sys stems s of simu ultan neous s equ uatio onso1.3 I Inter rvals s and d fun nctio onso1.4 C Calcu ulus:one e var riabl leo1.5 C Calcu ulus:man ny va ariab bleso1.6 G Graph hical l rep prese entat tion of f funct ti
3、ons s2.To opics s in mult tivar riate e cal lculu uso2.1 I Intro oduct tiono2.2 T The c chain n rul leo2.3 D Deriv vativ ves o of fu uncti ions defi ined impl licit tlyo2.4 D Diffe erent tials s and d com mpara ative e sta atics so2.5 H Homog geneo ous f funct tions s3.Co oncav vity and conv vexit t
4、yo3.1 C Conca ave a and c conve ex fu uncti ions of a a sin ngle vari iable eo3.2 Q Quadr ratic c for rms3.2.1 1 Def finit tions s3.2.2 2 Con nditi ions for defi inite eness s3.2.3 3 Con nditi ions for semi idefi inite eness so3.3 C Conca ave a and c conve ex fu uncti ions of m many vari iable eso3.
5、4 Q Quasi iconc cavit ty an nd qu uasic conve exity y4.Op ptimi izati iono4.1 I Intro oduct tiono4.2 D Defin nitio onso4.3 E Exist tence e of an o optim mum5.Op ptimi izati ion:inte erior r opt timao5.1 N Neces ssary y con nditi ions for an i inter rior opti imumo5.2 S Suffi icien nt co ondit tions
6、s for r a l local l opt timum mo5.3 C Condi ition ns un nder whic ch a stat tiona ary p point t is a gl lobal l opt timum m6.Op ptimi izati ion:equa ality y con nstra aints so6.1 T Two v varia ables s,on ne co onstr raint t6.1.1 1 Nec cessa ary c condi ition ns fo or an n opt timum m6.1.2 2 Int terp
7、r retat tion of L Lagra ange mult tipli ier6.1.3 3 Suf ffici ient cond ditio ons f for a a loc cal o optim mum6.1.4 4 Con nditi ions unde er wh hich a st tatio onary y poi int i is a glob balo optim mumo6.2 n n var riabl les,m con nstra aints so6.3 E Envel lope theo orem7.Op ptimi izati ion:the Kuhn
8、 n-Tuc cker cond ditio ons f for p probl lems with hine equal lity cons strai intso7.1 T The K Kuhn-Tuck ker c condi ition nso7.2 WWhen are the Kuhn n-Tuc cker cond ditio ons n neces ssary y?o7.3 WWhen are the Kuhn n-Tuc cker cond ditio ons s suffi icien nt?o7.4 N Nonne egati ivity y con nstra aints
9、 so7.5 S Summa ary o of co ondit tions s und der w which h fir rst-o order r con nditi ions arenece essar ry an nd su uffic cient t8.Di iffer renti ial e equat tions so8.1 I Intro oduct tiono8.2 F First t-ord der d diffe erent tial equa ation ns:e exist tence e of a so oluti iono8.3 S Separ rable e
10、fir rst-o order r dif ffere entia al eq quati ionso8.4 L Linea ar fi irst-orde er di iffer renti ial e equat tions so8.5 P Phase e dia agram ms fo or au utono omous s equ uatio onso8.6 S Secon nd-or rder diff feren ntial l equ uatio onso8.7 S Syste ems o of fi irst-orde er li inear r dif ffere entia
11、 al eq quati ions9.Di iffer rence e equ uatio onso9.1 F First t-ord der e equat tions so9.2 S Secon nd-or rder equa ation nsMathe emati ical meth hods for econ nomic c the eory:at tutor rialb by Ma artin n J.Osbo orneCopyr right t 1 1997-2003 3 Mar rtin J.O Osbor rne.Vers sion:200 03/12 2/28.THIS TU
12、TORIAL USES CHARACTERS FROM A SYMBOL FONT.If your operatingsystem is not Windows or you think you may have deleted your symbol font,please give yoursystem a character check before using the tutorial.If you system does not pass the test,see thepage of technical information.(Note,in particular,that if
13、 your browser is Netscape Navigatorversion 6 or later,or Mozilla,you need to make a small change in the browser setup to access thesymbol font:heres how.)Intro oduct tionThis tuto orial l is a hy ypert text vers sion of m my le ectur re no otes for a se econd d-yea ar un nderg gradu uate cour rse.It
14、 c cover rs th he ba asicmath hemat tical l too ols u usedin e econo omictheo ory.Know wledg ge of f ele ement tarycalc culus s is assu umed;som me of f the e pre erequ uisit te ma ateri ial i is re eview wed i in th he fi irst sect tion.The e mai into opics s are e mul ltiva ariat te ca alcul lus,c
15、onc cavit ty an nd co onvex xity,opt timiz zatio on th heory y,di iffer renti iale equat tions s,an nd di iffer rence e equ uatio ons.Theemph hasis s thr rough houtis o on te echni iques s rat therthan nabs strac ct th heory y.Ho oweve er,t the c condi ition ns un nderwhic ch ea ach t techn nique e
16、isappl licab ble a are s state edpr recis sely.Ag guidi ing p princ ciple e is acc cessi ible prec cisio on.Sever ral b books s pro ovide e add ditio onal exam mples s,di iscus ssion n,an nd pr roofs s.Th he le evel ofMMathe emati ics f for e econo omic anal lysis s by Knut t Sys sdaet ter a and P P
17、eter r J.Hamm mond(Pre entic ce-Ha all,1995 5)is s rou ughly y the e sam me as s tha at of f the e tut toria al.MMathe emati ics f fore econo omist ts by y Car rl P.Sim mon a and L Lawre ence Blum me is s pit tched d at a sl light tly h highe er le evel,and d Fou undat tions s of math hemat tical l
18、eco onomi ics b by Mi ichae el Ca arter r is more e adv vance edst till.The o only way to l learn n the e mat teria al is s to do t the e exerc cises s!I wel lcome e com mment ts an nd su ugges stion ns.P Pleas se le et me e kno ow of f err rors and conf fusio ons.The e entir re tu utori ial i is co
19、 opyri ighte ed,b but y you a are w welco ome t to pr rovid de a link k to the tuto orial lfro om yo our s site.(If f you u wou uld l like to t trans slate e the e tut toria al,p pleas se wr rite to m me.)Ackno owled dgmen nts:I ha ave c consu ulted d man ny so ource es,i inclu uding g the e boo oks
20、 b by Sy ydsae eter andHamm mond,Sim mon a and B Blume e,an nd Ca arter r men ntion ned a above e,Ma athem matic cal a analy ysis(2ed d)by y Tom m M.Apos stol,Ele ement tary diff feren ntial l equ uatio ons a and b bound dary valu uepr roble ems(2ed)by Will liam E.B Boyce e and d Ric chard d C.DiPr
21、rima,and d Dif ffere entia aleq quati ions,dyn namic cal s syste ems,and line ear a algeb bra b by Mo orris s W.Hirs sch a and S Steph henS Smale e.I have e tak ken e examp ples and exer rcise es fr rom s sever ral o of th hese sour rces.Instr ructi ionsThe t tutor rial is a a col llect tion of main
22、 n pa ages,wit th cr ross-refe erenc ces t to ea ach o other r,an ndli inks to p pages s of exer rcise es(w which h in turn n hav ve cr ross-refe erenc ces a and l links s to page es of fsol lutio ons).The m main page es ar re li isted d in the tabl le of f con ntent ts,w which h you u can n go to a
23、 at an ny po oint byp press sing the butt ton o on th he le eft m marke ed C Conte ents.Each page e has s nav vigat tiona al bu utton ns on n the e lef ft-ha and s side,whi ich y you c can u use t to ma akey your way thro ough the main n pag ges.The mean ning of e each butt ton d displ lays in y you
24、rbrow wsers st tatus s box x(at t the e bot ttom of t the s scree en fo or Ne etsca ape N Navig gator r)wh hen y youp put t the m mouse e ove er th hat b butto on.O On mo ost p pages s the ere a are t ten b butto ons(thou ugh o on th hisi initi ial p page ther re ar re on nly s six),wit th th he fo
25、ollow wing mean nings s.oGo to o the e nex xt ma ain p page.oGo to o the e nex xt to op-le evel sect tion.oGo ba ack t to th he pr revio ous m main page e.oGo ba ack t to th he pr revio ous t top-l level l sec ction n.oGo to o the e mai in pa age(tex xt)for this s sec ction n.oGo to o the e exe erci
26、s ses f for t this sect tion.oGo to o the e sol lutio ons t to th he ex xerci ises for this s sec ction n.oGo to o the e tab ble o of co onten nts.oSearc ch th hroug gh al ll pa ages of t the t tutor rial for a st tring g.oView tech hnica al in nform matio on ab bout view wing and prin nting g pag g
27、es.If yo oud like e to try usin ng th he bu utton ns no ow,p press s the e bla ack r right t-poi intin ng ar rrow(on aye ellow w bac ckgro ound),wh hich will l tak ke yo ou to o the e nex xt ma ain p page;to come e bac ck he erea after rward ds,p press s the e bla ack l left-poin nting g arr row o o
28、n th hat p page.After r you u fol llow a li ink o on a main n pag ge,p press s the e whi ite Text t bu utton n to retu urn t to th hepa age i if yo ou wi ish t to do o so befo ore g going g to the next t mai in pa age.To h help you know wwhe ere y you a are,an a abbre eviat ted t title e for r the e
29、 mai in pa age t to wh hich the butt tons on t the l leftcorr respo ond i is gi iven at t the t top o of th he li ight yell low p panel l.(F For t this page e,fo or ex xampl le,t the a abbre eviat ted t title e is Int trodu uctio on.)Pag ges o of ex xampl les a and s solut tions s to exer rcise esha
30、 ave o orang ge ba ackgr round ds to o mak ke it t eas sier to k know wher re yo ou ar re.I If yo ou ge et lo ost,pres ss th he T Text but tton or Cont tents s bu utton n.Techn nical litie esThe t tutor rial uses s fr rames s ex xtens sivel ly.I If yo our b brows ser d doesn nt s suppo ort f frame e
31、s,I Im n nots sure what t you ull see;I s sugge est y you g get a a rec cent vers sion of N Netsc cape Navi igato or.(Othe er fe eatur res t that I us se ma ay al lso n not b be su uppor rted by o other r bro owser rs.)Some very y old d bro owser rs th hat s suppo ort f frame es do o not t han ndle
32、the Bac ck a and Forw ward but ttons s cor rrect tly i in fr rames s.HTML has no t tags to d displ lay m math.I h have fak ked the math h by usin ng te ext i itali ic fo ontsfor roma an le etter rs,t the WWindo ows s symbo ol fo ont f for m most symb bols(gif fs fo or ot thers s),s small lfon nts f
33、for s subsc cript ts an nd su upers scrip pts,and tabl les f for a align nment ts.T The r resul lt is srea asona able usin ng Ne etsca ape N Navig gator r wit th a 12 o or 14 4 poi int b base font t and d a r relat tivel lyhi igh r resol lutio on mo onito or,b but m may n not b be so o gre eat u und
34、er r oth her c circu umsta ances s.If f wha atyo ou se ee on n you ur sc creen n loo oks a awful l,le et me e kno ow an nd Ill s see i if I can do a anyth hingabou ut it t.MathMML,a a var riant t of HTML L,ha as ex xtens sive capa abili ities s for r bea autif fully ydis splay ying math h,bu ut is s
35、 cur rrent tly s suppo orted d onl ly by y Net tscap pe Na aviga ator 7.1and its cous sins(e.g g.Mo ozill la).I am m wor rking g on a Ma athML L ver rsion n of thetuto orial l.1.Re eview w of some e bas sic l logic c,ma atrix x alg gebra a,an nd ca alcul lus1.1 L Logic cBasic csWhen maki ing p preci
36、 ise a argum ments s,we e oft ten n need to m make cond ditio onal stat temen nts,like eif th he pr rice of o outpu ut in ncrea ases then n a c compe etiti ive f firm incr rease es it ts ou utput torif th he de emand d for r a g good is a a dec creas sing func ction n of the pric ce of f the e goo o
37、d an nd th he su upply y of thegood d is an i incre easin ng fu uncti ion o of th he pr rice then n an incr rease e in supp ply a at ev very pric ce de ecrea asesthe equi ilibr rium pric ce.These e sta ateme ents are inst tance es of f the e sta ateme entifA then n B,where eA a and B B sta and f for
38、 a any s state ement ts.WWe al ltern nativ vely writ te th his g gener ral s state ement t asAimp plies s B,or,u using g a s symbo ol,a asA B.Yet t two m more ways s in whic ch we e may y wri ite t the s same stat temen nt ar reAis a su uffic cient t con nditi ion f for B B,andB is a ne ecess sary c
39、ond ditio on fo orA.(Note e tha at B come es fi irst in t the s secon nd of f the ese t two s state ement ts!)Impor rtant t not te:T The s state ement t A B do oes n not m make any clai im ab bout whet ther Bis s tru ue if f A i is NO OT tr rue!It s says only y tha at if f A i is tr rue,then n B i i
40、s tr rue.Whil le th hisp point t may y see em ob bviou us,i it is s som metim mes a a sou urce of e error r,pa artly y bec cause e we do n nota alway ys ap pply the rule es of f log gic i in ev veryd day c commu unica ation n.Fo or ex xampl le,w when wes say if it ts f fine tomo orrow w the en le et
41、s play y ten nnis we prob bably y mea an bo oth if it ts f finetomo orrow w the en le ets play y ten nnis and d if it ts n not f fine tomo orrow w the en le ets not play yten nnis (an nd ma aybe also o if f its no ot cl lear whet ther the weat ther is g good enou ugh t topl lay t tenni is to omorr r
42、ow t then Ill l cal ll yo ou).Whe en we e say y if f you u lis sten to t the r radio o at 8ocloc ck th hen y youl ll kn now t the w weath her f forec cast,on n the e oth her h hand,we do n not m meanalso o if f you u don nt l liste en to o the e rad dio a at 8 ocl lock then n you u won nt k know the
43、 weat therfore ecast t,b becau use y you m might t lis sten to t the r radio o at 9 ocloc ck or r che eck o on th he we eb,f fore examp ple.The poin nt is s tha at th he ru ules we u use t to at ttach h mea aning g to stat temen nts i inev veryd day l langu uage are very y sub btle,whi ile t the r r
44、ules s we use in l logic cal a argum ments s are eabs solut tely clea ar:w when we m make the logi ical stat temen nt i if A then n B,tha ats exac ctlywhat t we mean n-n no mo ore,no l less.We ma ay al lso u use t the s symbo ol to mean n on nly i if o or i is im mplie ed by y.T ThusB Ais eq quiva a
45、lent t toA B.Final lly,the symb bol me eans imp plies s and d is impl lied by,or if and o only if.Thu usA Bis eq quiva alent t toA B and B B A.IfA is a a sta ateme ent,we w write e the e cla aim t that Ais s not t tru ue as snot(A A).IfA and B ar re st tatem ments s,an nd bo oth a are t true,we writ
46、 teAand d B,and i if at t lea ast o one o of th hem i is tr rue w we wr riteAor B.Note,in part ticul lar,that t wri iting g A or B B in nclud des t the p possi ibili ity t that both h sta ateme ents are true e.Two r rules sRule 1If th he st tatem mentA Bis tr rue,then n so too is t the s state ement
47、 t(not B)(not t A).The f first t sta ateme ent s says that t whe eneve er A is t true,B i is tr rue.Thus s if B is s fal lse,A mu ust b befa alse-he ence the seco ond s state ement t.Rule 2The s state ement tnot(A Aand d B)is eq quiva alent t to the stat temen nt(not A)o or(n not B B).Note the or in
48、 the seco ond s state ement t!If f it is n not t the c case that t bot th A is t true and B is stru ue(t the f first t sta ateme ent),the en ei ither rAi is no ot tr rue o or B is n not t true.Quant tifie ersWe so ometi imes wish h to make e a s state ement t tha at is s tru ue fo or al ll va alues
49、s of a va ariab ble.For exam mple,let tting g D(p)be e the e tot tal d deman nd fo or to omato oes a at th he pr rice p,it t mig ght b be tr rue t thatD(p)10 00 fo or ev very pric ce p in the set S.In th his s state ement t,f for e every y pri ice is a a qua antif fier.Impor rtant t not te:WWe ma ay
50、 us se an ny sy ymbol l for r the e pri ice i in th his s state ement t:p p is s ad dummy y var riabl le.A After r hav ving defi ined D(p)to o be the tota al de emand d for r tom matoe es at tthe e pri ice p p,fo or ex xampl le,w we co ould writ teD(z)10 00 fo or ev very pric ce z in the set S.Given