SAM UPDATING USING MULTIOBJECTIVE OPTIMIZATION TECHNIQUES ABSTRACT

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1 6 H A NNUAL C ONFERENCE ON G LOBAL E CONOMIC A NALYSIS June 12-14, 23 Schevenngen, he Hgue, he Netherlnds SAM UPDAING USING MULIOBJECIVE OPIMIZAION ECHNIQUES D.R. Sntos Peñte 1, C. Mnrque de Lr Peñte 2 1 Deprtmento de Métodos Cuntttvos en E. y G. Unversdd de Ls Plms de G.C., 3517 Ls Plms de G.C., Spn E-ml: drsntos@empresrles.ulpgc.es 2 Deprtmento de Análss Económco Aplcdo Unversdd de Ls Plms de G.C., 3517 Ls Plms de G.C., Spn E-ml: csno@empresrles.ulpgc.es ABSRAC In ths work we present n dustment/updtng procedure of Socl Accountng Mtrces (SAM) usng multoectve optmzton technques. he usul settng concentrtes the dustment effort n ether the column or row coeffcents of the SAM. he norml tendency conssts n usng the column coeffcents ecuse they etter reflect the techncl coeffcents emedded n the SAM. Usng the enchmrk SAM s the ss, nd ncorportng the new nformton vlle we pose doule-oectve optmzton prolem nd pply the method of compromse progrmmng tht llow us to work together wth column nd row coeffcents. We present smples of the dustment procedure where we show nd compre the Preto optml solutons otned together wth other solutons otned wth dfferent dustment crter. We compre the results otned wth the ones otned wth other pproches lke RAS nd the mnmum sum of cross entropy together wth other methods proposed y the uthors. hs comprson llows us to evlute the dfferences exstng etween these dfferent methods nd the pure vertcl or horzontl dustment crter. We eleve tht the dustment/updtng crter used does not need to e the sme long the dfferent prts of the SAM. We my wnt to updte the nput-output mtrx ncluded n the SAM concentrtng n the vertcl/column/techncl coeffcents. But there my e other prts of the SAM where we thnk tht the use of row coeffcents, or together oth row nd column coeffcents, s more pproprte. he method we propose cn e lso ppled to stutons where the dustment crter s not the sme for dfferent prts of the SAM.

2 1 Introducton A Socl Accountng Mtrx (SAM) s squre mtrx whose rows nd columns represent the resources nd uses of seprte economc ccounts whch together summrze the sc flows wthn the dfferent groups of trnsctors or ctegores of trnsctons n n economy. he ncomngs (rows) nd outgongs (columns) re necessrly equlrted followng stndrd economc ccountng stndrds, therefore the sum of columns nd rows for ech ccount must e the sme. hs mtrx presents n ccountng system tht provdes nformton out spects of n economy lke the structure, composton nd level of producton, the vlue dded generted y the fctors of producton nd the dstruton of ncome mong dfferent groups of households. Normlly, SAM s constructed usng n Input-Output le s strtng pont nd ncludes fgures out consumpton, the structure of producton, nd the flows relted to foregn trde, svngs nd nvestment. he dustment of SAM conssts n determnng the SAM correspondng to perod t= strtng from known SAM, whch corresponds to moment t=, nd from certn vlle nformton out the SAM n moment t=. It s queston of fndng SAM X whose structure s, n certn sense, s close s possle to the structure of \ the SAM n moment t=, X. Normlly ths structure s defned n terms of the row (horzontl) coeffcents nd the column (vertcl) coeffcents, ssumng tht f element x of the ntl mtrx s null then the correspondng element x of the fnl mtrx s lso null. If X=(x ) 1, n s SAM nd S={(,) : x }, then the mtrx of row coeffcents, A = ( ) 1, n, nd tht of column coeffcents, B = ( ) 1, n, cn e defned s x = x x = x f (,) S, nd = f (,) S f (,) S, nd = f (,) S We consder tht the dustment oectve conssts n determnng SAM X such tht the rtos, or, for (,) S, s smlr s possle to unty, dstngushng etween two types of dustment, the horzontl nd the vertcl one, dependng on wether we pproxmte the rtos of horzontl or vertcl coeffcents to unty. he defnton of the concept of proxmty determnes the dustment crterum. In ths pper we use the followng crter for ech of the dustment types (horzontl nd vertcl): MEL1: MEL1H, MEL1V nd MEL1HV. FH ( A, A ) = 1 (, ) S 1.

3 FV ( B, B ) = 1 (, ) S 1. FHV1 = FH1( A, A ) + FV1 ( B, B ) MEL2: MEL2H, MEL2V nd MEL2HV. (, ) 1 2 FH A A = (, ) S (, ) 1 2 FV B B = (, ) S FHV2 = FH 2 ( A, A ) + FV2 ( B, B ) MMAX: MMAXH, MMAXV nd MMAXHV. FH ( A, A ) = 1+ S mx 1 3 (, ) S (, ) S FV ( B, B ) = 1+ S mx 1 3 (, ) S (, ) S FHV = FH ( A, A ) + FV ( B, B ) RAS: RASH, RASV nd RASHV. 4(, ) = ln (, ) S FH A A 4(, ) = ln (, ) S FV B B FHV = FH ( A, A ) + FV ( B, B ) 4 4 4

4 MSCE: MSCEH, MSCEV nd MSCEHV. 5(, ) = ln (, ) S FH A A 5(, ) = ln (, ) S FV B B FHV = FH ( A, A ) + FV ( B, B ) Where = x nd =. herefore s the sum of the elements of row, whch s equl to the sum of the elements of column, nd s the sum of ll the elements of mtrx X. he dustment prolem s formulted s n optmzton prolem where the oectve functon s one of the prevously mentoned nd the restrctons re determned y the column nd row totls, whch re supposed to e known, n ddton to other gven vlues. he vertcl crterum FV 1 hs een used y Mtuszewsk, Ptts nd Swyer (1964). Crterum FV 4 corresponds to the well known RAS lgorthm nd FV 5 s sed on n entrophy mesure whch hs een used y Goln, Judge nd Ronson (1994), nd Ronson, Cttneo nd El-Sd(2), mong others. Crter FV 4 nd FV 5 re prtculr cses of the dustment crterum of the mnmzton of the sum of weghted crossed entropes (McDougll, 1999). In ths pper we comne oth the vertcl nd the horzontl dustment crter pplyng multoectve optmzton technques. We wll develop n dustment method usng crter MEL1H nd MEL1V (FH 1 nd FV 1 ), whch defne oectve prolem. For the exmples we put forwrd, we wll clculte dfferent Preto optml solutons nd the compromse set. We wll represent ths solutons, together wth the ones otned wth other dustment crter n the spce of the 3 oectves nd nlyze the results otned. he rest of the pper s orgnzed s follows. In secton 2 we present the compromse progrmmng pproch ppled to the dustment of Socl Accountng Mtrces. In secton 3 we llustrte ths dustment method wth dfferent exmples. In secton 4 we dscuss the dustment prolem when the crter re not the sme for ll sectors. Fnlly, n secton 5 we nclude some conclusons nd possle extensons of our work. 2 Adustment of SAM usng compromse progrmmng In most of the SAM dustment efforts, only vertcl coeffcents re used dsregrdng the horzontl dustment. On the other hnd, the dustment crter commonly used cn e clssfed nto two groups. In the frst group we cn fnd the crter sed n the mnmzton of certn metrc L p, whle we cn fnd n the second group those defned strtng wth n entropy mesure. Crter MEL1, MEL2 nd MMAX, defned n secton 1, elong to the frst group (nvolve metrcs L 1, L 2 nd L ). he RAS nd MSCE crter elong to the second group. he crter MEL1, MEL2 nd MMAX cn e expressed n the form 1(, ), 2(, ) L ( UC) L ( UC) L U C L U C nd 1, + S, respectvely, eng U the squre mtrx of order n whose elements re ll

5 equl to one, nd C the squre mtrx of order n whose element (,), c, s equl to the rto of coeffcents (horzontl or vertcl) n moment t= nd t=, f (,) S s non null, nd c =1 otherwse. he entropy mnmzton crterum (WMSCE) conssts n the mnmzton of the functon w( ) = ln (, ) S c E C w c c where w w re postve weghts. RAS results from mkng = n the vertcl dustment nd w = n the horzontl one, whle the MSCE crterum cn e otned y mkng w =1, (,). Consderng tht the preservton of the horzontl structure, together wth the vertcl one, cn tke prt n the dustment oectves, eng for some sectors even the most mportnt of oth crter, nd strtng from the orgnl pproch of Mtuszewsk, Ptts nd Swyer, who ppled the vertcl dustment wth metrc L 1, we pose oectve prolem where we comne oth the vertcl nd horzontl dustment. hs oectve cn e expressed s follows mn f ( X) = ( f ( X); f ( X)) 1 2 suect to: x =, = 1,..., n (1) x =, = 1,..., n (2) x =, (,) S x, (,) eng f ( X) = 1 1 (, ) S f ( X) = 1 2 (, ) S x x =, =, (, ) S

6 he lst restrcton n (2) cn e elmnted ecuse t s lnel comnton of restrctons (1) nd the rest n (2). Introducng new vrles, the prolem cn e stted s lnel progrm resultng the followng equvlent prolem: mn g( X) = ( g ( X, Y, Z, Y, Z ), g ( X, Y, Z, Y, Z )) h h v v h h v v 1 2 suect to: x =, = 1,..., n x =, = 1,..., n 1 h h 1 y, (, ) z S = v v 1 y, (, ) z S = x =, (,) S h h v v x, y, z, y, z, (,) S eng h h v v h h g ( X, Y, Z, Y, Z ) = ( y + z ) 1 2 (, ) S h h v v v v g ( X, Y, Z, Y, Z ) = ( y + z ) (, ) S x x =,, (, ) S = where X = ( x ), Y h = ( y h ), Z h = ( z h ), Y v = ( y v ), Z h = ( z v ) A wy to otn Preto optml solutons conssts n mnmzng weghted sum of the oectves wth strctly postve weghts, ddng the efore mentoned restrctons. he del pont results from mnmzng ech oectve seprtely, tkng ech of the optml solutons otned. Applyng the method of the restrctons we cn otn the pyment mtrx wth the Preto optml solutons correspondng to the del vlues. he del pont, whose components re the mnml vlues of ech of the oectves, s normlly not ttnle (f not there would e no conflct etween the dfferent oectves). One of the pproches of the multoectve optmzton s sed on the de tht the est optons re those

7 fesle solutons whch le ner to the del pont n the oectve spce. hs de s the se of compromse progrmmng (Yu (1973) nd Zeleny (1973)). he compromse set s formed y the optml solutons(optml vlues n the oectve spce) of the prolems of the mnmzton of the L p (normlzed) dstnces to the del pont. he optml solutons of the prolems of mnmzton of dstnces L p wth 1 p re Preto optml. For p = these solutons re only wek Preto optml nd there lwys exst t lest one Preto optml. For lnel progrms, the solutons to the prolems for 1 < p < re locted etween the solutons otned for p = 1 nd p =. 3 Exmples In order to show n pplcton of the compromse progrmmng ppled to the dustment of Socl Accountng Mtrces, we present three exmples where the enchmrk mtrx, X, s mtrx of order 12x12 generted y the ggregton of certn sectors n the SAM of Mozmque of yer 1994 s ppers n Ronson, S., A. Ctneo nd M. El-Sd (21). We consder three cses, cses, nd c, whch result from tkng dfferent vlues for the sums of the elements n the columns of X. he vlue of these three new column totls nd the ncrese the new vlues represent from the enchmrk elements of mtrx X, pper n tle 1. Whle n cse the ncrements le etween 2.4% nd 9.41%, n cses nd c these ncrements vry etween.89% nd 53.6%, nd.12% nd 58.36% respectvely. he equltes: =, (, ) S (3) relte the horzontl wth the vertcl coeffcents. hese reltons seem to show tht f the rtos whch pper n (3) re very dfferent the conflct etween the horzontl nd vertcl dustment s hgher. In cse the mnmum nd mxmum vlues of rtos for the ntl mtrx re.7 nd respectvely, whle for the fnl mtrx these vlues re.6 nd he dfferences, n relton to these extreme vlues, etween mtrces X nd X re hgher n cses nd c. In cse the extreme vlues for the fnl mtrx re.1 nd 35, nd n cse c result to e.7 nd he quotent of rtos tht fgure n (3) vry etween.948 nd n cse,.6592 nd n cse, nd.6447 nd n cse c.

8 le 2 contns the del pont nd the pyments mtrx for ech cse. he vlues of ths pyment mtrx correspond wth Preto optml solutons where one of the oectves cheves the del vlue. Fgures 1 to 6 show the del pont, the compromse set nd solutons correspondng to crter MEL2, MMAX, RAS nd MSCE, n the spce of the oectves. les 3, 4 nd 5, contn the coordntes of the Preto optml ponts correspondng to the del vlues of ech oectve, the extreme ponts of the compromse set nd the solutons correspondng to the dfferent crter consdered. he sme tles show the reltve dstnce mesures etween the ntl mtrces, for oth the vertcl nd horzontl coeffcents. If we nlyze the results we cn oserve tht crter RASH, RASV nd RASHV cheve the sme soluton. On the other hnd, n the spce of the oectves, whle n cses nd crter MEL2 nd MMAX gve solutons tht re nerer to the solutons of MEL1 thn the crter sed on n entrophy mesure, n cse c the soluton MMAX s the most dstnt, proly due to the weght ssgned to the mnmx component tht produces hgher nfluence on t thn the L 1 component. It cn lso e perceved tht the vertcl nd horzontl dustments do not concde for ll the crter, n the sense tht the vertcl soluton for crterum cn cheve vertcl vlue for MEL1 tht s worse tn the horzontl soluton. hus, for exmple, n cse the soluton of the MSCEV hs vlue for the vertcl oectve tht s worse thn the soluton of MSCEH nd MSCEHV. Somethng smlr occurs n cse c wth the MMAXH crterum nd the horzontl oectve vlue. 4 Adustment crter y sectors here my exst stutons where we my need to put forwrd vertcl dustment for certn sectors whle for others we should pursue horzontl dustment. In ths cse, the methods tht use only ether vertcl or horzontl coeffcents my result ndequte nd t seems necessry to comne oth types of dustment. hese stutons cn e undertken consderng two oectves correspondng to the horzontl nd vertcl dustment, n such wy tht the sectors tht hve to e dusted horzontlly pper n frst oectve whle those needng vertcl oectve re ntroduced n the second one. he sectors for whch the most sutle dustment s not settled could fgure n oth oectves. If H} nd V represent, respectvely, the sectors to whch horzontl nd vertcl dustment hs to e ppled, the oectves of the prolem for dustment crterum MEL1 cn e formulted n the followng wy: f ( X) = 1, f ( X) = 1, 1 2 (, ) S, Η R (, ) S, V R donde R = S ( H V )

9 5 Conclusons In ths work we propose n dustment method for the dustment of Socl Accountng Mtrces tht comne horzontl dustment wth vertcl one. We formulte the dustment prolem s multoectve optmzton prolem tht we solve pplyng compromse progrmmng, usng the dustment crter of the mnmzton of dstnce L 1 (MEL1). Ech oectve corresponds to one of the types of dustment, vertcl nd horzontl. o llustrte the dustment method nd n order to show the poston of the solutons otned e other crter n the spce on oectves, we consder other crter sed n metrcs Lp (MEL2 nd MMAX) nd the entrophy concept (RAS nd MSCE). kng s the ntl mtrx the mtrx of order 12x12 generted y the ggregton of certn sectors n the SAM of Mozmque of yer 1994 s ppers n Ronson, S., A. Ctneo nd M. El-Sd (21), we present severl exmples creted y consderng dfferent vlues for the sum of the elements of the columns of the serched new mtrx. It cn e oserved tht the solutons otned y RASV, RASH nd RASHV concde, nd tht the soluton of the vertcl dustment for crterum MSCE does not necessrly correspond wth the est vlue for the vertcl oectve wthn those otned y the solutons cheved y ths crterum. he poston of the MMAX solutons n cse 3 proly owes to the fct tht there s domnnt nfluence of the mnmx component wth respect of the L 1 component, produced y the weghts used. hs work cn e extended n dfferent wys. Between other spects, one could study the usefulness of the multoectve optmzton technques n stutons where the dustment crterum s not the sme for ll sectors. he results of pplyng the proposed dustment method to other crter could e lso nlyzed. References Goln, Judge nd Mller(1996). Mxmum entrophy econometrcs, roust estmton wth lmted dt. John Wley & Sons. Goln, Judge nd Ronson (1994): Recoverng nformton from ncomplete or prtl multsectorl economc dt. he Revew of Economcs nd Sttstcs, 76, Hrrgn, F.J. nd I. Buchnn (1984): A qudrtc progrmmng pproch to IO estmton nd smulton. Journl of Regonl Scence, 24, 3, Lhr, M.L. (1998): A strtegy for producng hyrd regonl nput-output tles. Mcgll, S.M. (1977): heoretcl propertes of proportonl mtrx dusment. Envronment nd Plnnng A, 9, Mtuszewsk,.I., P.R. Ptts nd J.A. Swyer (1964): Lner progrmmng estmtes of chnges n nput coeffcents. Cndn Journl of Economcs nd Poltcl Scence 2, McDougll, R.A. (1999): Entropy theory nd RAS re frends. http: // /gtp /ppers /McDougll.pdf}. Mettnen, K.M. (1999). Nonlner multoectve optmzton. Kluwer. Ronson, S., A. Ctneo nd M. El-Sd (21): Updtng nd estmtng socl ccountng mtrx usng cross entrophy methods. Economc Systems Reserch,, 13,1,

10 Yu, P.L. (1973): A clss of solutons for group decson prolems. Mngement Scence, 19, 8, Zeleny,M. (1973): Compromse progrmmng. Multple crter decson mkng. Edted y J.L. Cochrne, M. Zeleny, Unversty of South Croln Press, Colum, South Croln,

11 COLUMNS otl X otl X (%) Increse otl X (%) Increse otl X -c (%) Increse le 1: Column totls n cses, nd c Cse c Oectve Horzontl Vertcl Idel Pont Pyments Mtrx Idel Pont Pyments Mtrx Idel Pont Pyments Mtrx le 2: Idel ponts nd pyments mtrces

12 Oectves Mesures (for vertcl nd horzontl coeffcents) Solutons Horzontl Vertcl L 1 L 2 MSCE RAS L pymm pymm compr compr met12v met12h met12vh Mmxv Mmxh Mmxvh Rsv Rsh Rsvh Mscev Msceh Mscevh le 3: Solutons nd mesures for cse

13 Oectves Mesures (for vertcl nd horzontl coeffcents) Solutons Horzontl Vertcl L 1 L 2 MSCE RAS L pymm pymm compr compr met12v met12h met12vh mmxv mmxh mmxvh rsv rsh rsvh mscev msceh mscevh le 4 : Solutons nd mesures for cse

14 Oectves Mesures (for vertcl nd horzontl coeffcents) Solutons Horzontl Vertcl L 1 L 2 MSCE RAS L pymm pymm compr compr met12v met12h met12vh mmxv mmxh mmxvh rsv rsh rsvh mscev msceh mscevh le 5: Solutons nd mesures for cse c

15 Fgure 1: Spce of oectves. Cse Fgure 2: Spce of oectves. Cse (enlrgement)

16 Fgure 3: Spce of oectves. Cse Fgure 4: Spce of oectves. Cse (enlrgement)

17 Fgure 5: Spce of oectves. Cse c Fgure 6: Spce of oectves. Cse c (enlrgement)