GTAP Elvth Aual Cof, 28 "Futu of Global Eooy" lsk SAM laboato as a ultobtv austt pobl Casao Maqu Laa Pñat Dpto Aálss Eoóo Aplao Uvsa Las Palas Ga Caaa (Spa aqu@aa.ulpg.s Dolos R. Satos-Pñat Dpto Métoos Cuattatvos Eooía y Gstó Uvsa Las Palas Ga Caaa (Spa satos@.ulpg.s Abstat Wh laboatg a SAM ay ata austts hav to b a. I ou stuy w assu w to aust th houshol ata o o to th ata o labo payts v fo th IO Tabl, assug that th ata o houshols s th o that s to b aust. I ths austt both th stbuto of ah labo o aog th fft houshol typs a th stbuto of th fft labo os aog ah houshol typ hav to b sultaously tak to aout. W popos ultta optzato thqus to solv ths pobl a to f a st of fft solutos. W th show fft ways to ak o slt btw ths fft solutos. W lu vaous xapls to show th ffts of applyg fft so ta.
. Itouto. Wh laboatg a SAM ay ata austts hav to b a. Just to tak a xapl, ost of th ass th ata sous us to stbut labo payts btw labo atgos (.g. g, tag t a ot th sa as th o s to stbut labo o aog fft typs of houshols. Ths ata ssu xplas th ffs both shs. I ou stuy w laboat a xapl that oul b us lat o to solv th to aust th houshol ata o o to th ata o labo payts v fo th IO Tabl, assug that th ata o houshols s th o that s to b aust. I ths austt both th stbuto of ah labo o aog th fft houshol typs a th stbuto of th fft labo os aog ah houshol typ hav to b sultaously tak to aout. W popos ultta optzato thqus to solv ths pobl. Th obtvs a xpss ts of th olu a ow offts a w f a st of fft solutos. W foulat ths austt pobl as a ultta optzato pobl wh th obtvs a xpss ts of th olu a ow offts, a w us ultta optzato thqus suh as opos pogag to f a st of fft solutos. W us th u su of oss tops as spaato asu btw th tal a fal offts ats. W popos ultta optzato thqus to solv ths pobl a to f a st of fft solutos. W th show fft ways to ak o slt btw ths fft solutos. W us ultobtv pogag obg both th wghtg tho wth th ostat thos. W lu vaous xapls to show th ffts of applyg fft so ta. 2
2. Austt ta, ultobtv foulatos a play aalyss. Gv a o-gatv, atx, ( Μ ( X x,,,,, a t [ ] w wat to t a atx of postv + lts, α ( α,, Μ (,,, whh satsfs α x,,,, a optzs ta obtv. Ths obtv s f as a spaato asu btw th tal a fal ats, t X x X α x, o btw th tal a fal ow (,,,,,,,, a ( a olu offt ats. + Fo a atx of o-gatv lts, X ( x,, Μ (,, ot S {(, {,, } {,, } : x > + }. + offts, A ( a,, Μ (,,, + B ( b,, Μ (, a f as follows:,,, w Th atx of ow a that of olu offts, A a,,,, Lt ( offt ats. If b x α x wth α x k α x k k. x a,,,,,,, x k k k x b,,,,,,. x k a ( B b,,,, αx,,,, w hav b th tal ow a olu a α x α x α x k k k a 3
Th spaato asu hos s th su of oss tops. W apply ths spaato asu fo ow a olu offts lag to two fft obtvs: a x f ( X, X a l l α α (, S a (, S + + b α α fv( X, X bl x l (, S b (, S + + wh α x x, a x,,,. Patula ass. Cas : α α,,. I ths as, fo th total ow otos, t follows that α,,,,,,. Th A A, f ( X, X a x fv( X, X l. (, S k k + x k x k k k k k Cas 2: α β,,. t If a vto β [ β β ] wth β >,,,, suh that X x, B B, f ( X, X a β,,,, xsts, th fo ( V X β f x (, l. X X β β (, S + I o to t th sall tval of vaato of postv α olus whh guaats a soluto to th syst of la 4
quatos pobl: αx,,,, w solv th followg optzato st.. z α x,,,, l α u,,,,,,, u l z,,,, α ε,,,,,, ( wh ε s a vy sall valu whh s tou to avo zo z ax u l,,,, that s, w solutos. Th optu s { } z th axu ag u l. I patula, f z th a t vto β [ β β ] wth β >,,,, vfyg X β xsts. If z u k lk >, w a f postv valus α wth ay ow a olu, whh satsfy z fo α α z, α x,,,. Usg th optu z w a t o aust bous fo olus by solvg th pobl st.. ( u l α x,,,, l α u,,,,,,,,,,, u l z α ε,,,,,,. (2 If th optal soluto to ths pobl s gv by l, u,,,, w hav postv offts α wth l α u,,, whh satsfy α x,,,. 5
,, W a tst ats (,, α α suh that th ffs btw offts α a sall, both fo ah olu a ah ow. Exapl. Cos th followg saos: a 5 7 23 49 2 4 5 35 X a, a z. 2 7 3 2 65 5 9 2 8 55 b 5 7 23 63 2 4 5 42 X b, b z.3268696. 2 7 3 2 69 2 3 4 46 29 5 7 23 89 2 4 5 42 X, z.34844237. 2 7 3 2 69 2 3 4 46 78 5 7 23 63 X 2 4 5, 42 z. 2 7 3 2 69 5 7 23 89 X 2 4 5, 42 z.34844237. 2 7 3 2 69 6
β β β wth X β. I xapls b, a suh a t I saos a a th s a vto [ ] β >,,,, suh that vto β os ot xst, ths ass w a f postv valus α wth αx,,,, vfyg α αk z,, k, fo ay olu. Obsv that, fo kow algba sults, th soluto to th syst X β as a s uqu, as w hav a ft st of solutos. Multobtv austt foulatos. + W wat to f a atx α ( α,, Μ (,, whh povs a bala soluto spt to th hozotal a vtal obtvs, f( X, X a fv( X, X, wth oat ffs btw valus α fo ah olu a ow. t Wh z th s a vto β [ β β ] wth β >,,,, t suh that X β, X x w hav f ( X, X. W β,,,, a fo ( obtv f as { } { } a b tst atag qual offts fo ah olu ( β >,,,, a obta thos whh z f (, X X a th ff btw th, wth ths a w os a so ax β β a foulat th followg bobtv poga: V ( f X X u l (,, s.t. β x,,,, l β u,,,, β ε,,,. (3 7
Exapl 2. Cos th sao Exapl wh 5 7 23 63 X 2 4 5, 42 a z. If w optz th two 2 7 3 2 69 obtvs f (, a X X u l ptly, w obta th followg obtv valus a solutos: f ( X, X, u l.6374,2.945 β.666.58.429 2.238, ( ( wth [ ] ( f ( X, X, u l (.6434,.688 wth β [ ].5662.5246.5246 2.227. I o to obta altatv solutos to Pobl (3 w a apply ultta optzato thqus suh as ultobtv pogag, opos pogag, goal pogag a oths. I Exapl 2, usg th ostat tho fo ultobtv pogag, w obta a altatv soluto solvg th pobl: f ( X, X s.t. β x,,,, l β u,,,, u l γ β ε,,,, (4 wh γ s a valu ops btw.688 a 2.945; so, fo γ.75 w obta ( f ( X, X, u l (.6379,.75 a β [.595.5643.4563 2.263 ]. If t z > a vto β [ β β ] wth,,,, β X, β > suh that os ot xst a, fo ay olu, th u ag of vaato of α whh povs a postv soluto to th syst αx,,,, s. f (, X X a V (,, z If w a tst th obtvs f X X a sultaously w wat to kp th vaato ags fx to z, w solv th pobl 8
( f X X fv X X (,, (, s.t. α x,,,, z α α z,, k 2,,,,,, k α ε,,,,,,. (5 I ostats k z a z by (, ( z α α z th bous ay b lax plag g z h z (also wh z. Exapl 3. Th last two olus ota th valus ospog to th Pato optal soluto to Pobl (5 wh w z f X, X + f X, X. ( V ( Sao X X a 2 X 3 X 4 X 5 X a 6 X 7 X 8 X a h z f z g( z ( z.3484 z z.3484 z ( z + a.3484 2z ( z +.3484 2z Tabl f V z.77 z.729 2.7246 z.6374 z.539 4.3747 z +..5 2z.66.985 z +.6.89 2z.564.8 I so of ths saos, th solutos obta a ot usful baus, u to th ltato pos o th vaato ags fo α, th soluto otas valus los to zo (valus qual to th low 9
bou pos oα. I Tabl, w a obsv so saos wh ths ltato s oflt wth th hozotal a vtal, f X, X.. obtvs ( f ( X X a V ( I o to ob th zato of obtvs (, V (,. f X X a f X X wth th zato of th ffs btw offts α olus a ows, w f th followg obtvs: (, f X X a ax ax fv ( X, X { { α} { α} } ( { α} { α} ( { α} { α} ax whh a lu th followg foulato: { } ( ( f( X, X, fv( X, X, ax u l, u l, u l s.t. α x,,,, l α u, 2,,,,,, l α u, 2,,,,,, α ε,,,,,,. (6 To solv ths pobl w tou th of th obtvs th st of ostats lag to th poga:
( f X X fv X X (,, (,, s.t. ( u l ( u l α x,,,, l α u, 2,,,,,, l α u, 2,,,,,, ( u l γ z α ε,,,,,,, ρ ρ (7 wh ( z γ, ρ, a ρ, a fx. W a obta fft solutos assgg fft wghts th obtv futos fo fft γ z, ρ, a ρ, ths pou s a xtu of wghtg valus of ( a ostat thos fo ultobtv pogag. Exapl 4. 5 7 23 89 Cos X 2 4 5, 42. Tabl 2 shows th obtv 2 7 3 2 69 valus fo th solutos obta wh w z ah obtv sgag th oth os. Th agoal of th tabl otas th al valus.
Obtv z ax{ u l} ( u l ( u l f f V { } ax.3484.3938 8.27.54 z u l 4.3889 ( u l ( u l.6969.6969 8.896.69 9.64.8395 3.3582.3 f.8395 3.3582.3 f V.8356.8584 8.695.267 6 Tabl 2.9 4 α α,, Th optal ats (,, ospog to ass Tabl 2 a show Tabl 3 wh th last ll lus th soluto wf X, X + wf X, X obta wh w z th futo ( 2 V ( subt to ostats of Pobl (7 wth w w2 a γ ( z.7, ρ.8, a ρ 4. ( f fv γ ρ ρ ( Th obtv valus obta a,,,,.3263,.467,.7,.8, 4. 2
ax{ } ( u l z u l 2 3 4.3484.667.3484 3.527 2.E-9.783.E-9 2.742 3.E-9.783.E-9 2.742 2 3 4.E-9.783.E-9 3.4 2.E-9.783.E-9 2.742 3.E-9.783.E-9 2.742 ( u l 2 3 4.9348.9348.9348.9348 2.3548.3548.3548.3548 3.952.952.952.952 f 2 3 4.9348.9348.9348.9348 2.3548.3548.3548.3548 3.952.952.952.952 f V f ( X, X + fv ( X, X 2 3 4.649.634.65 3.4 2.643.689.643 2.5655 3.643.698.643 2.5965 2 3 4.687 2.986.687 2.986 2.5523 2.74.5877 2.74 3.476.4986.4928.4986 Tabl 3 Colusos Ths xapls show how th ultta optzato thqus allow us to ob vaous austt ta a to obta o qulbat solutos. I th last ol, wh w ob th fv obtvs, w a obsv how w obta o alst solutos. Ths sults off a tool to solv th SAM austt pobls to th touto of ths wok. 3
Rfs. Ballsto, E. a C. Roo (998: Multpl ta so akg a ts applatos to oo pobls. Kluw Aa Publshs. F, M. a P.L. Yu (976: So w sults o opos solutos fo goup so pobls. Maagt S, 22,6, 688-693. Gola, Jug a Mll (996: Maxu topy oots, obust stato wth lt ata. Joh Wly & Sos. Gola, Jug a Robso (994: Rovg foato fo oplt o patal ultstoal oo ata. Th Rvw of Eoos a Statsts, 76, 54-549. Igzo, J.P. (983: Galz goal pogag: A ovvw. Coputs a Opatos Rsah,, 4, 277-289. Igzo, J.P. (985: Itouto to la goal pogag. Sag Publatos, Sag Uvsty Pss, Bvly lls. Magll, S.M. (977: Thotal popts of bpopotoal atx aust. Evot a Plag A, 9, 687-7. Magll, S.M. (978: Covg a lat popts fo a of bptopotoal atx pobl. Evot a Plag A,, 499-56. Maqu Laa, C. a D.R. Satos Pñat (24 Nw ola appoahs fo th austt a upatg of a SAM. Eoos of Plag, 36, 259-272. MDougall, R.A. (999: Etopy thoy a RAS a fs. http: //www.sf.k /gtap /paps /MDougall.pf. Mtt, K.M. (999. Nola ultobtv optzato. Kluw. Robso, S., A. Catao a M. El-Sa (2: Upatg a statg a soal aoutg atx usg oss tophy thos. Eoo Systs Rsah, 3,, 47-64. Satos-Pñat, D.R. a C. Maqu Laa Pñat (27 SAM upatg usg ultobtv thqus. Wokg Pap. Yu, P.L. (973: A lass of solutos fo goup so pobls. Maagt S, 9, 8, 936-946. 4