2) Line tangent to function (i.e., 1st order Taylor Series approx) is above

Transcription

1 Optzato Theor Backroud Io Cove Fucto - three detos use whchever s easest Chord coect a two pots o ucto les below the ucto, ad,, e taet to ucto.e., st order Talor Seres appro s above the ucto: ' Based o secod de', reader ter o st order Talor Seres appro s eatve: '' ˆ ˆ, ; sucet to sa '' Cocave Fucto - eatve o cove ucto s cocave; lp equaltes ro cove de's Prcpal Subatr - k k subatr o atr A ored b delet - k colus ad sae respectve rows k-th Order Prcpal Mor - deterat o k k prcpal subatr ead Prcpal Subatr A k - k-th order prcple subatr created b delet the last - k rows ad last - k colus ro A; eaple, or atr, lead prcpals: a a a a a a,, ad a a a a a a a a ead Prcpal Mor - deterat o the lead prcple subatr Dervatve Shorthad - ca rewrte / as or ' ad / as or '' ' - Quadratc For - real-valued ucto o or: Q,, a whch ca be rewrtte Q T A A s setrc atr Postve Dete - T A > e.., Postve Sedete - T A e.., Neatve Dete - T A < e.., Neatve Sedete - T A e.., T Idete - A > or soe R ad < or soe other e.., o 5

2 Deter Deteess - A s postve dete all lead prcpal ors are > OR check all eevalues > A s eatve dete all lead prcpal ors alterate s: A <, A >, A <, etc..e., k A k > OR check all eevalues < A s postve eatve sedete sae as ad wth ad ; Note: ou have to check all k-th order prcpal ors choose k o the! OR check all eevalues postve sedete or eatve sedete 4 I k-th order lead prcpal or s ozero but does t t a o above, the A s dete OR have ed postve ad eatve eevalues Matr - e dete: < & >.e., a < & a a a > Hessa - secod dervatve atr; setrc b You's Theore H '' You's Theore - order o deretato does't atter or cotuous ucto; Bordered Hessa - secod dervatve atr, bordered b rst dervatves rst row ad colu BH Quascocave - two detos Chord coect two pots doa les above level curve o oe o the oral pots ' ad '' D ad,, ' '' [ ', '' ] Set o all pots reater tha a level curve s cove : ' ' D s a cove set { } Quascove -reverse the equaltes above ad chae to a evel curves ' '', '' ' Cove set evel curve:, ' ' ' evel curves '', '' Cove set evel curve:, ' NOT Quascocave '' ', '' o 5

3 Varable Case raphs do't eeralze to ultple varables Quascocave cove relates to a sle peaked sle trouh ucto so a local a s a lobal a Mootoc Fuctos - all ootoc uctos.e., creas or decreas o oe varable are both quascove ad quascocave ' ' {: '} Ths s a cove set ths s quascocave ' ' {: '} Ths s ot a cove set ths s ot quascocave WARNING - Quascocave s ot the sae as cocave; quascocave s talk about the level curves the doa space; cocave s talk about the ucto tsel - space Theore - a cocave ucto s also quascocave but quascocave cocave Proo: Deto o cocave:, That eas [, ] a cocave ucto s quascocave Rht sde s spl a wehted averae so [ ] Satato Pot - level curves or cocetrc "rs" do't have to be crcles approach a sle pot where the ucto s azed or zed eaple raph below Test or Quascocave/cove - ook at level curves or ucto raph o rht Red les are taets to level curve, c ad have decreas slopes ook at total deretal: d d d I we look at cha, but wat to sta o the level curve, we have a plct ucto h d That eas dh/d d /d - / Note ro the red les that as we crease, / s decreas,.e., d, h, h d dh d d dh d dvso rule & cha rule Multpl out ters ad substtute dh/d - /, the actor out / wll requre add a to soe ters ad realze that So order or ths to be, the deterat o the bordered hessa ust be Ths works or the blue les as well Suar: >.e., red les d / /d BH <.e., blue les d / /d BH Note: Ths s't talk about postve or eatve deteess lke we check or cocave/cove; we're ol cocered wth the s o the deterat Theore - BH >, ucto s quascocave; BH <, ucto s quascove Theore - a Cobb-Doulas.e., α β ucto s quascocave ' Quascocave level curve slope latte level curve slope steeper o 5

4 Multple Varables - Sucet or Quascocave - - k BH k > k ecessar cod: - k BH k Sucet or Quascove - BH k < k ecessar cod: BH k Note: o alterat ss or quascove Rules/Wars - k uber o varables BH k BH k s the kth prcple subatr o BH ever delete border row or colu 4 ever use ewer tha varables k ; does't sa uch because deterat wll alwas be zero varable ad border row/col BH - 5 start wth atr 6 ca use lead prcple ors ou have > ; ot strct, ou ust check all cobatos o varables Geeral Rule - ou have a eplct ucto, use H cocave/cove or BH quascocave/cove; ou do't have a eplct ucto use the basc detos as the proo above or cocave quascocave Optzato Bascs Two Questos. Does soluto est?. Fd/characterze the soluto Ma vs. M - M s sae as Ma - ocal Ma - pot * such that * ear * Strct Ma - < stead o Global Ma - d all local a/ ad copare the or pose stro assuptos o ucto to esure ol oe /a Case - Ucostraed wth Sle Varable Necessar - ' * ca also add cocave,.e., '' Sucet - ' * ad '' < * ear * eed that or specal cases lke - 4 Get results ro Talor Seres: st Order - * ' ˆ * ; or * to be a, reader ter ust be eatve; the derece - * ca be ether postve or eatve deped o where s relatve to *: ' ˆ * ad ' ˆ * ; thereore, ' * d Order - ˆ * ' * * '' * ; or * to be a, reader ter ust be eatve; the derece ter s squared so t's postve; that eas '' ˆ < 4 o 5

5 Case - Costraed wth Sle Varable Costrat - a b; resolves estece proble theores [we wo't prove] sa ou have cotuous ucto ad copact set [closed & bouded], there s a a ad Ier Pots - sae codtos are ucostraed case Boudar Pots - deret or lower & upper bouds: ower Boud - Necessar - ' a < or ' a ad '' a Sucet - sae, but chae to < last part Upper Boud - Necessar - ' b > or ' b ad '' b Sucet - sae, but chae to < last part Case - Ucostraed wth Ma Varables Necessar - st Order -,,, equatos & ukows d Order - Hessa s eatve sedete Sucet - sae as ecessar ecept Hessa s eatve dete Two Varable Case - ecessar codto: or sucet ake the strct equaltes,, st Order - d Order -,, ad -.e., H eatve sedete Case 4 - Costraed wth Ma Varables Two Varable Case - Ma, Subect to,.e., a level curve o Theor - use costrat to eerate plct ucto h ad solve ucostraed proble o oe varable: Ma,h or ew ucto Ma F df d st Order - d d d Note: Fro costrat, d d d Ecooc Iterpretatos - MB MC Maral / beet o / Maral opportut cost o creas Slope o Costrat Slope o Obectve / / / / / / '' does't atter a b '' does atter d /d ; how costrat orces to chae whe ou chae Phscal opportut cost o creas a b Slope o level curve o obectve ucto Slope o costrat 5 o 5

6 6 o 5 Sae Beet per Dollar Spet / / / / d Order - Ths ets ess! Wll use dervatve shorthad / Start wth st order: d dh Take dervatve o that, but realze these are uctos o two varables so rewrte t rst:,,,, h h h h Ad the dervatve s: d dh d dh d dh d dh Now substtute dh/d - / ad recoze that ad To spl urther, actor out a / [ ] Sucet Codto: [ ] < Necessar Codto: [ ] We ca ether check the whole th or the sucet ad ecessar codtos or we ca check the dvdual parts. For the latter, t helps to roup ore ters toether: Maral beet o Beet per dollar spet Maral cost o Dvso Rule Cha Rule Product Rule

7 [ ] Fro the thrd terpretato o the rst order codto beet per dollar spet, realze:... that eas ad Plu these to the rst set o parethess above ad et Factor out the ad ultpl out the : Obvousl, the ters the parethess are deterats o bordered hessas ust a lttle ccs there. Us atrces: det det > Wehted Derece - ths s a wehted derece o deterats o bordered hessas BH BH - s stateet o curvature o level curves o obectve ucto : Quascocave - BH > Quascove - BH < *** Ths ol works or varable case *** S cases - - are a; 4-6 are, c, quascocave & quascove c & quascocave, d order stll holds c & quascove, d order stll holds,, c 4 quascove & quascocave c Case - s quascocave & s quascove; ths assures the secod order codto wll hold ad s a stroer sucet codto tha ecessar; eerall ths s the codto we'll use because we wo't be work wth eplct uctos Case - & are quascocave, but wehted derece s stll > ; ou ove alo costrat ucto, ou caot reach a hher level curve o, so ou are at a au at the taec pot Case - slar to ecept & are quascove 5 & quascocave, d order does't hold c 6 & quascove, d order does't hold 7 o 5

8 Case 4 - s quascove & s quascocave; ths volates the secod order codto ad akes the wehted derece < so the taec pot s a u Case 5 - slar to ecept & are reversed; wehted derece < so taec pot s a u; ou ca ove alo costrat ad reach hher level curves o Case 6 - slar to 4 ecept & are reversed; wehted derece < so taec pot s a u; ou ca ove alo costrat ad reach hher level curves o Stroer d Order - sats d order codto BH > quascocave ad BH < quascove; ths s Case above; t's stroer ore restrctve tha the sucet d order codto lsted above; ths s what we'll usuall assue whe we et to cosuer theor or Cases ad, we eed to kow the eplct ucto to check the wehted d. Practce - use araa ucto Multple Varables - st Order - Opto - - /,,...,- ad h,,... - Opto - ad,,..., d Order - s quascocave ad s quascove Optzato wth araa Fucto Tpcal proble: Decso varables "Subect to" Ma s.t. : Dee ew varable arae ultpler arae Multpler - atheatcal ter or varable troduced to the optzato proble Ecooc Iterpretato - Shadow Prce; how uch t would be worth to rela the costrat.e., how uch ou would be wll to pa or a addtoal ut o resource araa Epresso - - "Prc Out the Costrat" - subtract the costrat s stadard otato althouh ou could add t; subtract esures whch s ood or error check ad better or terpretato as a shadow prce; proo o s otes or evelope theore Optz - ote that Ma Ma because st Order - treat as a decso varable ust lke ; results equatos ad ukows /... sae as we had beore "ba or the buck" /... costrat coes back st order codto; optal to sats the costrat 8 o 5

9 9 o 5 d Order - look at hessa or atr H Pla wth ths to ake t look lke a bordered hessa; rst use so ad ro st order cod. so Use ths o to rewrte the hessa: H Note we ultpl a row or colu b a scalar, ou ultpl the deterat b the sae value, so we ca ultpl row ad colu b - ad ot chae the s o the deterat. I we ultpl row ad colu b, we ultpl the deterat b whch s postve so t does't chae the s o the deterat; we ow have: I order or a pot that satses the rst order codtos to be a au, ths atr ust sats the rules or quascocavt.e., - k M k > k ; a spler check would be to requre to be quascocave ad to be quascove Multple Costrats - wth equalt costrats we're lted to the uber o costrats be less tha the uber o varables ; wll have oe larae ultpler or each costrat araa Epresso - k k k H - H a wehted derece o hessas Ma s.t. : : :

10 st Order - wll have rst dervatves k k,,..., k aral beet o,,..., aral cost o rela costrats because o creas Note ew otato: k k k ad k d Order - s coveto to check quascocave depeds o ; "t's Cha" M k k s.t. k,,..., Iequalt Costrats Matheatcall the sae as Case 4 costraed wth equaltes; wo't solve t as a equalt, but wll show that t's theoretcall the sae Noeatvt - ; usuall cluded or odel to prevet osescal soluto e.., cosupto ust be oeatve Greater Tha - ca be wrtte as we ultpl b - Ma s.t.,,..., Ma s.t. Ma s.t. - Slack Varable - troduce varable to take up the "slack" ad orce the equalt to be a equalt; square t to ake sure t's oeatve wthout us a equalt; ote costrat s tht.e.,, the the slack s zero s Ma s.t. Costrat t - wth equalt costrats, we eed varables costrats, but otce slacks add oe varable or each costrat so we do t have that proble Geeral Case - show o rht; ths case wll be used to derve the Kuh-Tucker codtos Ma,s,v u β v s.t. s : Ma,s s.t. s [ ] [ ] st Order - o the!, β, v have each; s, have each s : - v : β,,..., o 5

11 Costrat Slack Varable - u u Copletar Slackess - each costrat s assocated wth a slack varable u ad a larae ultpler ; at least oe o these ust be zero; ths ollows drectl ro the rst order codto above Ecooc Iterpretato - s the shadow prce; <.e., the slack u >, the costrat s ot bd so we would't pa to rela t... the shadow prce s zero; look at t the other wa, >.e., we'd be wll to pa to rela the costrat, the costrat ust be bd.e., u ad... otce also that -/ Specal Case - u ; costrat s tht, but we do't a b rela t; could be because there's a satato pot let or ultple costrats. Rewrte st Order - there are several was to wrte ths, but ths wa s ood because t's utuall eclusve ad ehaustve; ote that t corporates the rst order codto or the larae ultpler a > / b / Noeatvt Slack Varable - β v v Iterpretato - sae as costrat slack varable above; ether β or v or both; β, the oeatvt costrat s ot tht.e., > ; ths s the optal soluto we ust have / or creas would prove the soluto; v, the oeatvt costrat s tht; or ths to be a optal soluto we ust have / ; that s, we'd lke to decrease to prove the soluto, but we ca't because we ht the boudar o the oeatvt costrat Rewrte st Order - ust lke beore, we ca rewrte ths to corporate other rst order codtos a utuall eclusve ad ehaustve wa a > / b / Rela costrat does't chae optal soluto Feasble reo beore ad ater Feasble reo beore ad ater Kuh-Tucker Codtos - suarzed boes above; these rst order codtos capture all rst order codtos; ote that the slack varables ad larae ultplers assocated wth oeatvt are't eeded Cobatos o Codtos - there are deret was to cobe the K-T codtos; each cobato represets oe o our possble tpes o solutos: Corer Soluto - All Costrat Bd - Soe Costrats Not Bd - o 5

12 4 No Costrat Bd - Us The wth a Eplct Fucto - look at proble to see we ca lt possble solutos e.., ad Wrte out laraa Wrte out Kuh-Tucker Codtos 4 st all possble rea cases... up to e.., a, b, a 5 For each case, solve the equatos or the ukows; these are the crtcal or statoar pots 6 Check the equaltes the K-T codtos or the soluto 7 Substtute each soluto to the obectve ucto ad pck the best oe Well Behaved - ou ver beore had that the obectve ucto s loball quascocave ad the costrats are loball quascove, there wll be ol oe soluto, the lobal a; that allows us to od the steps above... choose the easest cases to solve rst; ol oe case wll sats the K-T codtos so ou ca stop as soo as ou d t Us The wth a Geeral Fucto - we usuall assue the d order codtos No Noeatvt - ths replaces the lower bo above wth / Deret ower Boud - costrat s b, ust replace ro the codtos above wth b.e., > b ad b Costrat Qualcato - K-T codtos b theselves are't ecessar or a optal soluto; there are restrctos that ust be placed o the costrat ucto ad the costrat set order or the K-T codtos to be ecessar; orall wrt the costrat qualcatos s't trval the're coplcated, but here are two eaples: Costrat Fucto - ca't have lecto pot costrat at boudar o the easble reo.e., rst dervatve o costrat ; ths ves the wro sal ad trcks the proble to thk the costrat s ot bd whe t s Costrat Set - ust have oept teror; ept set or sle pot or easble reo wll ot work wth K-T codtos Slater's Codto - sucet or costrat qualcatos to hold Aother Sucet Codto - hav all lear costrats ad a oept easble reo s sucet or costrat qualcatos to hold ote, ths s uch ore restrctve, but lots easer to reeber Specc Eaple - ed o otes has a specc eaple o proble that does't sats costrat qualcato Trasoratos - ca chae soe ths ad stll et the sae soluto Obectve - Ma /4 /4 Ma / 4 l... ves sae soluto althouh obectve value wll obvousl be deret Costrats s sae as - 7 ; ths could be portat achev costrat qualcato Eaple p. o hadout Ma /4 /4, s.t. - 5 : - 7 :, o 5

13 Ths has 4 pars o K-T codtos whch traslates to 4 6 cases; we ca elate soe b look at the proble; t's obvous that there est soe easble solutos wth ad wth,, we ca ore all cases wth ad ; aother th to otce s that the obectve s ootoc both varables; that eas at least oe o the costrats ust be bd.e., ca't have ; we're ow dow to cases to cosder d Order - based o work below, obectve s quascocave ad costrats are quascove so there s a uque lobal au Costrat - t's lear whch s both quascove ad quascocave; we're terested costrats be quascove so ths s ood Costrat - look at deterat o bordered hessa < quascove Obectve - look at deterat o bordered hessa do't reall eed to because t's a Cobb-Doulas ucto whch s quascocave, but ths shows ou how to do t the hard wa 4 4 / 4 / 4 araa - / 4 / / 4 7 / 4 / 4 / 4 / 4 / / 4 / 4 / 4 / 4 / 4 7 / 4 Sce all we're worred about s the s, ot the atude, we ca ultpl ad dvde rows ad colus b scalars to ake ths easer to work wth Multpl row b 4, row b 6, ad row b 6 / 4 / 4 / 4 / / 4 / 4 / 4 / 4 7 / 4 / 4 Multpl colu b /4 / 4 / / 4 / 4 / 4 / 4 / 4 7 / 4 / 4 / 4 4 / 4 / 4 / 4 / 4 / 4 / 4 / 4 / 4 / 4 / 4 7 / 4 7 / 4 Dvde row b -/4 ad -/4 ; do the sae wth colu 7 / 4 / 4 / 4 / 4 / 4 / 4 / 4 7 / / / / / 8 / / > / 4 / 4 / 4 / 4 obectve s quascocave /4 / K-T Codtos - a > / /4 --/4 /4 - - b / /4 --/4 /4 - - kow we wo't have ths o 5

14 a > / /4 /4 --/4 - - b / /4 /4 --/4 - - kow we wo't have ths a > -/ - 5 b -/ - 5 4a > -/ - 7 4b -/ - 7 Case - a, a, a, 4b; solve the sste: /4 --/4 /4 - - /4 /4 --/ v Eq v sples ad because drops out That eas we ca rewrte Eq ad as /8 --7/4 /4 /8 -/4-7/4 Whch we ca solve or Substtute ths to Eq 5 5 / Now we have to check the equaltes or each case & are obvous because ad are both > Skp to 4 because we a ot have to solve or s 4 does't hold; / ths s ot the soluto to the K-T codtos Case - a, a, b, 4a; solve the sste: /4 --/4 /4 - - /4 /4 --/4 - - v - 7 Eq sples ad because drops out That eas we ca rewrte Eq ad as /4 --/4 /4 /4 -/4 -/4 Whch we ca solve or or spl Substtute that to Eq v 7 7/ Now we have to check the equaltes or each case & are obvous because ad are both > For we have 49/4 49/ satsed For 4 we have to solve or /4 --/4 /4 /47/ -/4 7/ /4.4 > satsed Ths soluto satses all our equaltes so t satses the K-T codtos ad s the optal soluto to the proble; we do't have to cosder case a, a, a, 4a, but ou do the work ou'll d a cotradcto the equaltes lke we dd case 4 o 5

15 Eaple does't sats Costrat Qualcato Ma, s.t :, Optal Soluto c Obectve s strctl quascocave. Costrat ca be rewrtte, whch s lear, thereore, t's quascove. Ths should ea a soluto to the K-T codtos wll be optal. A raph o the easble reo ad soluto s show above. araa K-T Codtos - a > / b / a > / b / a > / b / Optal Soluto /, /... ths does't sats the K-T codtos. Sce >, codto a ust hold, but / - - / - / /. Sae holds or codto a. ook at ths aother wa. I a holds >, the - -, but the look at a, we ust have or b so. ook at a, we ust have or b so. Ether wa we ed up wth ad thereore The uque soluto to the K-T codtos s,; case b, b, b. But ths s't the optal soluto because /, / s better. Proble - costrat has a lecto pot. 5 o 5