微观经济学的数学方法(1)hmuw.docx

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 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 tions 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 tyo3.1 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 eso3.4 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 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 terpr 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 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 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 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 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 TUTORIAL 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 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 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,conc 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 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 Peter 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 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 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 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 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 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 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 yourbrow 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 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 ercis 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 ges.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 on 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 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 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 es,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 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 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 under 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 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 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 od 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 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 cond 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 is 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 ets 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 row 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 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 rules 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 alent 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 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 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 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 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 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