7 The Rudiments of Input-Output Mathematics

Transcription

1 7 The Rudimets of Iput-Output Mthemtics The first si chpters of this volume, which costitute self-cotied uit, descrie the iput-output system without the use of mthemtics. The costructio of iput-output model d some of its pplictios were illustrted y rithmetic emples. With oe eceptio these rithmetic smples were sufficiet to demostrte how iput-output tle is put together, d how it c e used for vriety of purposes. I Chpter 2 the cocept of iverse mtri ws metioed d umericl emple of iverted mtri give. As oted i tht chpter, the meig of these terms ws deferred util the preset chpter. While the geerl solutio of iput-output system c e illustrted y umericl emple, the ctul process of ivertig mtri c oly e illustrted y mes of mtri lger. To roud out the epositio of iput-output system two techiques for ivertig mtri will e discussed here. This is the etet to which we will pursue the mthemtics of iput-output lysis. For this purpose we will eed to drw upo some of the more elemetry propositios of mtri lger, d these will e give without proof d without y ttempt t either mthemticl elegce or rigor. Before turig to discussio of some of the fudmetls of mtri lger, some prelimiry commets o ottio will e helpful, d it will lso e ecessry to discuss riefly the cocept of determit s prerequisite to lter discussio of mtri iversio. The Summtio Sig Mtri lger dels with systems of equtios, d whe delig with lrge system of equtios it is cumersome to write out every term ech time equtio is used. A compct ottio is eeded, d this is provided y the summtio sig. Some of the elemetry rules for usig the summtio sig re give elow: The symol for summtio is Σ, the Greek upper-cse letter sigm. It is used to show tht dditio hs tke plce. If, for emple, there re oservtios of vrile, the () i i The ide i shows where we strt coutig, d the letter where we stop. I this cse ll items from the first through the th re dded. It is lso possile to use this shorthd ottio to symolize the dditio of pirs of oservtios. For emple, (2) ( 3 + y3) + ( 4 + y4 ) + ( 5 + y5 ) i + y 5 i 3 5 i 3 Clerly this could e eteded to y umer of sets of oservtios. The ide shows tht i this cse we strt coutig the third pir of oservtios d go through the fifth. A set of products, for emple, costts times vriles, my e writte s: (3) i i Note, however, tht set of vriles times sigle costt is writte s: 6 i i

2 (4) i or 6 i 6 i i I this cse the costt c e tke outside the summtio sig sice (4) is equivlet to ( ) Cosider et the dditio of set of vriles mius costt: i ) i (5) ( ) + ( 2 ) + ( 3 ) ( ) ( This my lso e writte s i i The summtio sig sves oth time d spce. Becuse iput-output lysis dels with lrge umers of vriles d equtios it is coveiet to use this symol to summrize etire systems of equtios d their solutios. I redig equtios which coti oe or more summtio sigs, the reder should oserve the opertios tht hve ee performed efore the results re summed. A equtio which cotis umer of summtio sigs my pper formidle t first glce, ut the, oly idictes tht the simplest of rithmetic opertios dditio hs tke plce. Some simple illustrtios of the use of this shorthd symol i descriig iput-output tle will e give lter i this chpter. Determits The otio of determit my e itroduced y mes of emple. Cosider the followig system of lier equtios i which d y re the ukows. + y c + y c These equtios c e solved y "elimitig" etwee them, solvig for y, the sustitutig the vlue of y i oe of the equtios d solvig for. The system c lso e solved usig determits, however, s illustrted y the followig emple: We defie the determit D s 2 2, d the solutio to the ove equtios is give y:

3 c c c c, y The vlue of the determit is give y D ( ) d the vlues of the epressios i the umertors of d y re foud i the sme wy. This is illustrted y the followig umericl emple. Give the equtios: 3+ 4y 8 + 2y D [(3) (2) () (4)] (6 4) 2 2 To solve for the ukows, sustitutios re mde s i the geerl epressio ove, d the followig computtios re crried out: (36 32) 4 [( 8 ) (2 ) (4 ) (8 )] (24 8) 6 y [(3)(8) (8)()] Isertio of these vlues i the equtios shows tht they hve ee solved. The determit descried ove is of the secod order sice it hs two rows d two colums. Determits of higher order c e formed for the solutio of lrger systems of equtios. They re lso used i oe of the methods for ivertig mtri to e give i lter sectio of this chpter. This is the purpose of icludig discussio of determits i this ook, d o ttempt will e mde to give complete epositio. Further detils will e foud i most first-yer lger tets. A detiled discussio of the properties of determits d their use i ecoomic lysis hs ee give y R. G. D. Alle. 2 Some Properties of Determits A determit cosists of umer of qutities rrged i rows d colums to form squre. If there re four qutities, the determit will cosist of two rows d two colums; if there re ie, it will

4 cosist of three rows d three colums. The order of determit depeds upo the umer of rows d colums; secod order determit hs two rows d two colums, third order determit hs three rows d three colums, d so o. The qutities withi the determit re clled its elemets. These elemets my represet umers, costts, vriles, or ythig which c tke o sigle umericl vlue. The result of evlutig the determits tht will e used i this chpter will lso e sigle umer. It will e importt to rememer this whe we tur to discussio of mtrices i lter sectio. Determits of the secod d third order re esy to evlute d to work with. Determits of higher order ecome somewht cumersome, ut everythig tht hs ee or will e sid out secod d third order determits i this chpter lso holds for higher-order determits. Miors d Cofctors The elemets of third or higher-order determit c e epressed i terms of miors d cofctors. I defiig these terms we will itroduce somewht differet ottio of the determit, s follows: This ottio will e useful i epliig the meig of miors d cofctors, d lso i our lter discussio of mtrices. The suscripts i the ove determit idetify the row d colum of ech of its elemets. The first umer idetifies the row, d the secod idetifies the colum. For emple, the elemet 23 idictes tht it elogs i the secod row d third colum; the elemet 2 goes i row oe d colum two. The mior of y elemet of third order determit cosists of the secod order determit which remis whe the row d colum of the give elemet re deleted or igored. Miors will e idicted y the symol Δ, which is the uppercse Greek letter delt. Approprite suscripts will idicte the mior of give elemet. For emple, the mior of elemet will e writte s: Δ i.e. the rows d colums which remi fter row d colum I re deleted. Similrly, the mior of 22 will cosist of the elemets i the rows d colums remiig fter row 2 d colum 2 re struck out. It is writte s: Δ The cofctor of elemet cosists of tht elemet's mior with the pproprite sig ttched. This is where the ottio which hs ee used i this sectio comes i hdy sice the sig of the cofctor c e determied from its suscripts. We will use the symol A to represet cofctors, s distict from miors. If the sum of the suscripts is eve umer, such s A, the cofctor will hve plus sig; if the sum of the suscripts is odd umer, for emple A 2, the cofctor will hve mius sig. The

5 cofctors of the ove determit my e writte s follows: A , 33 A , 3 33 A d so o. Ech of the cofctors is evluted s follows: A ( ) 32, A ), d A ) ( ( Oly three of the cofctors hve ee writte out ove, to illustrte the rule of sigs, ut similr cofctors c e writte for ech of the ie elemets of the third order determit. Whe ivertig three-ythree mtri, ll ie cofctors re eeded. To evlute third order determit y mes of epsio, however, oly three of the cofctors re eeded. Both of these processes will e illustrted lter i this chpter whe determits re used to ivert third order mtri. Mtrices At first glce mtri resemles determit. But there is importt differece. It will e reclled tht whe determit is evluted the result is sigle umer. This is ot true of mtri, which is defied s rectgulr rry of umers. We will use the symol [ ij ] to idicte mtri. I this ottio, i refers to the rows of mtri d j to the colums. To distiguish the mtri from determit we eclose the former i squre rckets, d cotiue the covetio of usig stright lies to idetify determit. A third order mtri d third order determit will thus e idetified s follows: [ ij ] D Before proceedig to discussio of the iversio of mtri, it will e coveiet to itroduce some defiitios d some of the compct ottio of mtri lger. We will lso give the rules of mtri lger eeded for uderstdig of mtri iversio. Ulike determits, mtri eed ot e squre, i.e. it is ot ecessry for the umer of rows to equl the umer of colums. Iput-output lysis dels with squre mtrices, however, d this is the oly kid which will e cosidered i detil i this chpter. Oe other type of mtri, which hs specil me, will e cosidered sice it ws used i Chpter 3 d plys itegrl prt i iput-output lysis. A specil kid of mtri cosists of sigle colum d y umer of rows. Such mtri is referred to s colum vector. I Chpter 3, whe the severl colums i the fil demd sector were collpsed ito sigle colum, the result ws referred to s colum vector. Similrly, we spek of row vector, which is ctully mtri cosistig of sigle row d y umer of colums. Filly, mtri c cosist of sigle row d sigle colum oly, i.e. sigle elemet. The ltter is typiclly referred to s sclr. The two types of vectors d sclr re illustrted elow:

6 [ ] [ ] Colum Vector Row Vector Sclr Returig to the otio of squre mtri, this c e writte i its most geerl form s K M O M ij m L m To simplify ottio it is coveiet to use cpitl letters to represet complete mtri. Ideed, oe of the gret dvtges of mtri lger is tht we c write comple systems of equtios i terms of sigle mtri equtio, d opertios c e performed with these mtrices s though they were sigle umers (which, it is worth repetig, they re ot!). For emple, if we hve the followig system of equtios: h h h We c epress the etire system s squre mtri d two colum vectors, h h2.,.. h

7 d this system my the e writte s the followig mtri equtio: A h I this compct ottio, A the squre mtri with 2 coefficiets ( ij ); is the colum vector of elemets, d h is secod colum vector of elemets. I ordiry lger if A d h were umers d "ukow," the solutio of (2) would e h. I mtri lger if ll the coefficiets ( ) A ij of A were kow, s well s the elemets of the colum vector h, we could solve for ll the ukow 's y logous (ut ot ideticl) procedure. Some Mtri Defiitios We hve lredy defied squre mtri, row d colum vectors, d sclr. As is true of determit, the order of squre mtri is give y the umer of rows (or colums). The pricipl (or mi) digol of squre mtri cosists of the elemets ruig from the upper left to the lower right corers, i.e. ll of the elemets i which the row suscript is equl to the colum suscript. A squre mtri is osigulr if the determit of tht mtri is ot equl to zero. This is importt property to rememer sice if mtri is sigulr (i.e. if its determit ) its iverse cot e defied. A mtri which cosists of 's log the mi digol with ll other elemets equl to zero is clled idetity mtri. Such mtri, which is geerlly symolized y I, plys essetilly the sme role i mtri lger s the umer does i ordiry lger. Two mtrices re equl if d oly if they re of the sme order, d if ech elemet of oe is equl to the correspodig elemet of the other. Tht is, two mtrices re equl if d oly if oe is duplicte of the other. Oe other defiitio is required efore turig to some of the sic lws of mtri lger. If the rows d the colums of mtri re iterchged the result is trsposed mtri. We idetify the trspose of give mtri s follows: the trspose 3 of A A T For emple, if 5 A 4 2 3, the 7 6 T A Bsic Mtri Opertios Mtri dditio d sutrctio. If two mtrices A d B re of the sme order, we my defie ew mtri C s A + B. Mtri dditio simply ivolves the ddig of correspodig elemets i the two mtrices A d B to oti the elemets of C. This is illustrted i the followig emple: A 3 5 2, d B the 7 3 C A+ B 2 4

8 We could lso hve writte C B + A to oti the sme result; tht is, the commuttive lw of dditio holds (for mtrices of the sme order), d A + B B +A. While it will ot e demostrted here, the ssocitive lw of dditio lso holds, i.e. (A + B)+C A + (B + C) for mtrices of the sme order. This is so ecuse i mtri dditio correspodig elemets re dded, d the order of dditio of these elemets does ot mtter. Sutrctio my e cosidered s iverse dditio; tht is, if we hve the umers +5 d -5, their sum is. Thus if A d B re two mtrices of the sme order, sutrctio my e cosidered s tkig the differece of A d B. For emple, if 5 2 A 4 3, d 3 2 B, the 8 A B 3 4 I geerl, the dditio d sutrctio of mtrices is like the dditio d sutrctio of ordiry umers sice these opertios re performed o the correspodig elemets of mtrices of the sme order. As oted ove, oth the ssocite d commuttive lws hold for mtri dditio. This is ot true of mtri sutrctio, however. The ssocitive lw does ot hold sice, for emple, 4 - (5-2) is ot the sme s (4-5) - 2. Similrly, the commuttive lw does ot hold sice, for emple, is ot the sme s Usig the origil ottio for the geerl elemets of two mtrices, we my summrize mtri dditio d sutrctio for mtrices of the sme order y: Sclr multiplictio my e defied s: A + B [ ij + ij ], d A B [ ] ka [ k ij ], tht is, ij ij ech elemet of A is multiplied y k. If we hve, for emple, Mtri Multiplictio A, d k 3, the ka 3 Mtri multiplictio is restricted to mtrices which re coformle. A mtri A is coformle to other mtri B oly whe the umer of colums of A is equl to the umer of rows of B. The the product AB hs the sme umer of rows s A d the sme umer of colums s B. It will e coveiet, t lest iitilly, to defie mtri multiplictio usig letters isted of umers. If we hve two mtrices A d B defied s follows: A , d B , the AB is defied s

9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Cosider ow the followig umericl emple which lso gives the rule for multiplyig 2 X 2 mtrices: Let 3 A 2 4 2, d B, the 3 ( ) ( ) 5 3 AB (2 2 + ) ( ) 4 8 Notice, however, the result of reversig the order of multiplictio. ( ) BA ( + 3 2) ( ) ( ) The mtri product BA does ot equl the product AB. Tht is, i geerl, mtri multiplictio is ot commuttive. 4 The ocommuttive ture of mtri multiplictio c lso e illustrted y multiplyig row vector times colum vector. If, for emple, we hve the followig row d colum vectors: FG 2 F [ 2 3] d G 4, the [( 2 ) + ( 2 4 ) (3 )] [ ] 7 2 (2 ) (2 2) (2 3) (4 ) (4 2) (4 3) ( ) ( 2) ( 3) 2 3 But, GF [ ] A row vector times colum vector, multiplied i tht order, equls sclr. But colum vector times row vector yields mtri.

10 The ssocitive lw holds i mtri multiplictio. Tht is, if we hve three mtrices A, B, d C, the (AB)C A(BC). But s the ove emples hve show, the order of mtri multiplictio cot e reversed. There is oe importt eceptio to this geerliztio. I the et sectio we will defie the iverse of mtri which is symolized s A -. The order of multiplictio of mtri times its ow iverse does ot mtter, i.e. AA - A - A. I this cse it is immteril whether A or A - is o the left; i oth cses the result is I, the idetity mtri. Tht is: AA A A I Ivertig Mtri I erlier sectios we discussed the cocept of determit, d the miors d cofctors of determit. We lso covered mtri dditio d sutrctio, sclr multiplictio, d mtri multiplictio. Most of these will ow e used i our discussio of mtri iversio, the mjor gol of this chpter. The iverse of specil kid of mtri, to e discussed lter, gives us geerl solutio to the equtios i iput-output system. It will e reclled from our erlier discussio tht mtri A times its iverse A - equls I, the idetity mtri. Thus fter mtri hs ee iverted it c e multiplied y the origil mtri. If the result is mtri with 's log the mi digol d zeros everywhere else we hve check o our procedure d re ssured tht A - is ideed the iverse of the origil mtri. The emple chose to illustrte the process of mtri iversio is etremely simple oe. I prticulr, it hs ee chose to give us determit with vlue of. The sole purpose of this is to keep the rithmetic s simple s possile so tht ttetio c e focused o the process of mtri iversio rther th o the computtios themselves. The prolem is to fid A - of the mtri A The first step is to evlute the determit of this mtri y epdig log the cofctors of row s follows: 23 D ( 2 6) 2(4 3) + 3(2 3) The vlue of the determit, s metioed ove, is uity. The et step ivolves idetifictio of ll the cofctors of the determit. These re give elow:

11 Cofctors of D (6) ( ) ( ) A, A2, A ( 2 ) () () A2, A22, A ( 3) () () A3, A32, A The umers i pretheses ove ech of the cofctors represet the vlues of the cofctors with pproprite sigs tke ito ccout. The vlues of the cofctors re the rrged i mtri form, d this mtri is trsposed. It will e reclled tht to trspose mtri we covert ech colum ito row (or vice vers). To void cofusio with trsposed mtri s such, the trsposed mtri of cofctors is clled the djoit mtri. These steps re illustrted elow: Mtri of cofctors Adjoit Mtri Oly oe step remis to oti the iverse of the origil mtri. This is to divide ech elemet i the djoit mtri y the vlue of the origil determit. Sice i our emple the vlue of the determit is, the umers i the djoit mtri re ot chged it is A -, the iverted mtri we re seekig. To e sure of this, however, we will multiply the origil mtri y the iverse mtri. If the result is idetity mtri we re sure there hve ee o errors i the clcultio of A -. Tht is, we must fid out if A. A I The detils of the multiplictio re give elow: {( 6) + (2 ) + (3 )} {( 2) + (2 ) + (3 )} {( 3) + (2 ) + (3 )} {( 6) + (3 ) + (3 )} {( 2) + (3 ) + (3 )} {( 3) + (3 ) + (3 )} {( 6) + (2 ) + (4 )} {( 2) + (2 ) + (4 )} {( 3) + (2 ) + (4 )} Ech of the epressios withi the rckets { } will ecome elemet i the mtri which results from this multiplictio.

12 Crryig out the ove rithmetic opertios we oti: I This is the idetity mtri, d it proves tht A - is i fct the iverse of A. It will e reclled tht mtri multiplictio is ot commuttive i geerl. I this specil cse, however, the order of multiplictio does ot mtter. We could hve reversed the order of multiplictio, d the result would hve ee the idetity mtri. Ivertig Mtri y Mes of Power Series The iverse of the ove mtri is ect. The method employed is lso strightforwrd d esy to use for ivertig 3 3 mtri eve if the determit is positive umer lrger th. All this ivolves is dividig ech elemet of the trsposed mtri of cofctors y the vlue of the determit. The method is ot efficiet oe, however, for ivertig lrge mtri, sy 4 4. The computtiol procedure followed whe lrge mtri is iverted y computer is quite comple d will ot e illustrted here. Aother techique for otiig the pproimte iverse of mtri will e descried (ut ot illustrted) sice this techique rigs out the "multiplier" effect of epdig iput-output mtri to oti tle of direct d idirect requiremets per dollr of fil demd (Tle 2-3). This is the method of epsio y power series, d it will e compred with ect method for otiig the iverse of Leotief iput-output mtri. The mtri tht is iverted to oti tle of direct d idirect requiremets per dollr of fil demd is kow s the Leotief iput-output mtri. It is defied s (I A), d its iverse is the (I A) -. I these epressios, I is the idetity mtri d A is the mtri of direct coefficiets such s Tle 2-2. Thus the tle of direct d idirect requiremets per dollr of fil demd is the trsposed iverse of the differece etwee the idetity mtri d mtri of direct iput coefficiets. The mtri (I A) - c lso e pproimted y the followig epsio: I + A +A 2 + A A Tht is, the tle of direct iput coefficiets is dded to the idetity mtri. This is how we show the iitil effect of icresig the output of ech idustry y oe dollr. The the successive "rouds" of trsctios re give y ddig the squre of A to (I +A), d to this result ddig A to the third power, d so o util the ecessry degree of pproimtio is chieved. 5 Sice ll of the iitil vlues i the tle of direct coefficiets re less th oe, ech of the mtrices cosistig of higher powers of A will coti smller d smller umers. As A is crried to successively higher powers the coefficiets will get closer d closer to zero. This is other wy of syig tht t some poit the direct d idirect effects of icresig the output of ech idustry i the iput-output model y oe dollr will ecome egligile. I prctice, if the A mtri is crried to the twelfth power, workle pproimtio of the tle of direct d idirect requiremets per dollr of fil demd will e otied. Tle 7- o pge 46 shows the ect iverse of the Leotief mtri used i Chpters 2 d 3, d i pretheses elow ech cell etry is the pproimtio otied y crryig the A mtri to the twelfth power d ddig the result to the idetity mtri. Trsposed iverse (I A) T - Power series pproimtio 6 [I + A + A A 2 ] T

13 All etries here re crried to four plces. There is greemet to the first two deciml plces i ll ut four of the cells. Ad whe rouded to the erest cet, more th two-thirds of the pproimtios y power series re ideticl to the etries i Tle 2-3. Thus the pproimtio y power series yields completely workle results. TABLE 7- Trsposed Iverse of Leotief Mtri d Approimtio y Power Series A B C D E F A (.3767) (.248) (.2795) (.44) (.274) (.2259) B (.448) (.244) (.66) (.845) (.82) (.2354) C (.263) (.3834) (.3788) (.23) (.649) (.392) D (.3424) (.25) (.2477) (.5266) (.644) (.434) E (.352) (.2559) (.352) (.3842) (.2798) (.2524) F (.3763) (.3529) (.2225) (.2933) (.296) (.327) As prcticl mtter, there is little poit i epdig mtri y mes of power series. With tody's high-speed electroic computers d efficiet computtiol methods, it is possile to oti ect iverse s rpidly, d t o higher cost, th to estimte the iverse y epsio of power series. The reso for metioig the power series pproimtio is tht it coveys more clerly th the mechicl process of iversio the step y step, or icremetl, wy i which the idirect effects of iteridustry trsctios re propgted throughout the system. Moore d Peterse hve lso suggested tht ech of the terms i the power series c e used to represet the iterctio etwee chges i fil demd, over time, d the direct d idirect trsctios required to stisfy the successive chges i fil demd. 7 A third method of pproimtig tle of direct d idirect effects will e metioed, ut will ot e descried here. This is the itertive method of computig successive "rouds" of productio eeded to stisfy give level of fil demd. Like the pproimtio y power series, this method hs the dvtge of showig clerly the icremetl ture of idirect effects. It lso shows how the idirect effects coverge towrd zero s successive "rouds" of trsctios re completed. 8 The Iput-Output System A Symolic Summry We re ow i positio to summrize the sttic, ope iput-output system i symolic lguge. Bsiclly, the iput-output model is geerl theory of productio. All compoets of fil demd re cosidered to e dt. The prolem is to determie the levels of productio i ech sector which re required to stisfy the give level of fil demd. The sttic, ope model is sed upo three fudmetl ssumptios. These re tht:

14 . Ech group of commodities is supplied y sigle productio sector. 2. The iputs to ech sector re uique fuctio of the level of output of tht sector. 3. There re o eterl ecoomies or disecoomies. The ecoomy cosists of + sectors. Of these, oe sector tht represetig fil demd is utoomous. The remiig sectors re outoomous, d structurl iterreltioships c e estlished mog them. 9 Totl productio i y oe sector durig the period selected for study my e represeted y the symol X i. Some of this productio will e used to stisfy the requiremets of other o-utoomous sectors. The remider will e cosumed y the utoomous sector. This situtio my e represeted y the followig lce equtio: () X i X i + X i X i + X f (i... ) where X f is the utoomous sector, d the remiig terms o the right-hd side of the equtio re the outoomous sectors i the system. Assumptio (2) ove sttes tht the demd for prt of the output of oe outoomous sector X i, y other outoomous sector X, is uique fuctio of the level of productio i X j Tht is: (2) X ij ij X j Sustitutig (2) i equtio () yields (3) X i i (X ) + i2 (X 2 ) +... i (X ) + X f (i... ) This my e writte more compctly s i ij j f j (4) X ( X ) + X ( i... ) where X ij is the mout demded y the j th sector from the i th sector, d X f represets the ed-product (fil) demd for the output of this sector. The model c e illustrted schemticlly i Figure 7-. From the trsctios tle (Tle 2-) the techicl coefficiets re computed (Tle 2-2). These coefficiets show the direct purchses y ech sector from every other sector per dollr of output. They re give i equtio (2) ove, which my e rewritte s: (5) ij X X ij j The coefficiets re computed for the processig sector oly i two steps:

15 () Ivetory depletio durig the se period is sutrcted from totl gross output to oti djusted gross output. (2) The etry i ech colum of the processig sector is divided y djusted gross output to oti the ij show i (5). This gives the followig mtri of techicl coefficiets. (6) A L j L M M M L L i ij i M M M L j L As oted i the precedig sectio, the tle of direct d idirect requiremets per dollr of fil demd is otied y ivertig Leotief mtri, which is defied s (I A). The ew mtri of coefficiets showig direct d idirect effects (Tle 2-3) is geerlly trsposed to oti (I A) T -. This mtri my e desigted s R.

16 (7) R r L r j L r M M M r L r L r i ij i M M M r L rj L r Alyticlly, the iput-output prolem is tht of determiig the iteridustry trsctios which re required to susti give level of fil demd. After trsctios tle hs ee costructed, we c compute the A d (I A) T - mtrices. For y ew fil demd vector iserted ito the system, we use these to compute ew tle of iteridustry trsctios s follows: (8) j X r X, the fi ij ' i (9) X T ij ' i ' Equtio (8) shows tht we multiply ech colum of ( ) T y the ew fil demd ssocited with the correspodig row. Ech colum is the summed to oti the ew totl gross output (X i '). Filly, i equtio (9), ech colum of the tle of direct iput coefficiets is multiplied y the ew totl gross output (X i ') for the correspodig row. The result is the ew trsctios Tle T' which c e descried y the followig ew lce equtio: I A () ' ' ' i ij j f i X ( X ) + X, ( i... ) Whe the "dymic" model discussed i Chpter 6 is used i mkig log-rge projectios, the fied techicl coefficiets the ij, of the origil A mtri re replced y ew coefficiets computed from smple of "est prctice" estlishmets i ech sector. All of the computtiol procedures descried ove remi uchged, however. This could e symolized y sustitutig ij for ij i () idictig tht ll compoets of the lce equtio re chged i the "dymic" model.

17 Refereces ALBERT, A. ADRIAN, Itroductio to Algeric Theories (Chicgo: The Uiversity of Chicgo Press, 94). ALLEN, R. G. D., Mthemticl Alysis for Ecoomists (Lodo: Mcmill d Compy, Ltd., 949). AYRES, FRANK, JR., Theory d Prolems of Mtrices (New York: Schum Pulishig Co., 962). CHENERY, HOLLIS B. d PAUL G. CLARK, Iteridustry Ecoomics (New York: Joh Wiley & Sos, Ic., 959). JOHNSTON, J., Ecoometric Methods (New York: McGrw-Hill Book Compy, Ic., 963). MACDUFFEE, CYRUS COLTON, Vectors d Mtrices, The Mthemticl Associtio of Americ (L Slle, Ill.: Ope Court Pulishig Co., 943). MOOD, ALEXANDER M., Itroductio to the Theory of Sttistics (New York: McGrw-Hill Book Compy, Ic., 95). School Mthemtics Study Group, Itroductio to Mtri Alger, Uit 23 (New Hve: Yle Uiversity Press, 96). U. S. Deprtmet of Agriculture, Computtiol Methods for Hdlig Systems of Simulteous Equtios, Agriculture Hdook No. 94, Agriculturl Mrketig Service (Wshigto, D.C.: U. S. Govermet Pritig Office, Novemer 955). U. S. Deprtmet of Commerce, Bsic Theorems i Mtri Theory, Ntiol Bureu of Stdrds, Applied Mthemtics Series 57 (Wshigto, D.C.: U. S. Govermet Pritig Office, Jury 96). Edotes For lucid d compct itroductio to mtri lger see Dvid W. Mrti, "Mtrices," Itertiol Sciece d Techology No. 33 (Octoer 964), While this rticle dels with the pplictio of mtri lger to vrious egieerig prolems, it lso serves s ecellet geerl itroductio to mtrices. 2 Mthemticl Alysis for Ecoomists (Lodo: Mcmill d Co., Ltd., 949), pp If A is iverted d trsposed, the result my e writte A T. 4 If three mtrices, A, B, d C, re coformle, the ssocitive lw of multiplictio holds. Tht is, A (BC) (AB) C. It should e oted, however, tht AB AC does ot ecessrily imply tht B C. 5 As cosequece of the ssocitive lw, powers of the sme mtri lwys commute. Thus the order of multiplictio of A d the higher powers of A does ot mtter. 6 After the power series pproimtio ws completed the resultig mtri ws trsposed to mke it comprle with Tle 2-3. It will e reclled tht trspositio of the iverse mtri is ot essetil prt of iput-output lysis; it is doe to mke the tle of direct d idirect requiremets esier to red. 7 Frederick T. Moore d Jmes W. Peterse, "Regiol Alysis: A Iteridustry Model of Uth," The Review of Ecoomics d Sttistics, XXXVII (Novemer 955), 38-8.

18 8 A detiled emple of the icremetl method is give i Hollis B. Cheery d Pul G. Clrk, Iteridustry Ecoomics (New York: Joh Wiley & Sos, Ic., 959), pp Otherwise stted fil demd, for ech sector, is eogeous vrile, d the iteridustry trsctios re edogeous vriles. To simplify the epositio we igore certi ivetory djustmets here which hve to e mde i prctice.