Supply Quantitative Model à la Leontief *

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1 Moer Ecoomy, 2011, 2, o: /me Pubhe Oe September 2011 ( Suppy Quattatve Moe à a Leotef * Abtract Ezra Davar Iepeet Reearcher, Amo Vtamar, Netaya, Irae E-ma: ezra.avar@gma.com Receve May 13, 2011; reve Juy 6, 2011; accepte Juy 16, 2011 Th paper focue o the uppy quattatve moe ytem of put output, whch equvaet to the ema quattatve moe ytem of Leotef. Th moe aow u to efe the tota uppe quatte of commote for ay gve uppe quatty of prmary factor, a coequety eabe u to efe the fa ue of commote. The uppy quattatve moe bae o the rect output coeffcet of prmary factor. The Haamar Prouct ao ue. The quattatve uppy ytem moe mght be uefu too pag the ecoomc of coutre that have hgher uempoymet of prmary factor, epecay abour. Keywor: Leotef, Dema a Suppy Quattatve Moe, Output Coeffcet, the Haamar Prouct 1. Itroucto Leotef ue the term Iput-Output the tte of h frt a ema paper o Iput-Output Aay, Quattatve Iput a Output Reato the Ecoomc Sytem of the Ute State [1]. Th mea that every actvty ecoomc mutaeouy characterze by two e: come (reveue) expeture, ema uppy, put-output, export-mport, a o o. I other wor, the come of a certa ecoomc ut (houeho, frm, ttuto, coutry) cocurrety expeture for aother ecoomc ut; ema for ay commoty by ay vua ao uppy for aother vua or frm; put of ay commoty to a certa ector ao output for the ector proucg that commoty. Ug th potuate, Leotef ecrbe h put-output tabe a: Each row cota the reveue (output) tem of oe eparate bue (or houeho) If rea vertcay, coum by coum, the tabe how the expeture e of the ucceve accout [1]. Therefore, th aow u to ecrbe a aayze ecoome o two e o that f the ame coto ext, the reut mut be equvaet for both quatty a prce term. For exampe, the eveopmet of the whoe ecoomy mght be moee o ether the put e or the output e. Each recto ha t ow target a aow u to ove fferet type of probem of cotemporary eco- * The author thak Prof. A. Broy a Prof. E. Ey for uefu uggeto; Th paper ecate to Leotef 100 th brthay, a 70 year ce h frt paper o Iput-Output. omc. Sce that pero ecoomc terature o Iput- Output, there have bee attempt to formuate moe ecrbg the whoe ecoomy both e: ema (put) a uppy (output) for quatty a uppy (put) a ema (output) for prce. Ut toay, oy two type of ytem moe of put-output have bee formuate. Thee ytem moe were frt formuate by Leotef ther orga form, a the foowg year they were mprove upo: quattatve ema (put) a prce uppy (put) moe ([1-4]). The frt moe aow u to efe the ema (requre) quatty of the tota proucto (put) of commote for ay gve amout of fa ue a coequety ao for efg the ema (requre) quatty of the prmary factor. The eco moe aow u to efe the cot of proucto (uppy prce) of commote o the ba of prmary factor prce whch are eterme accorg to ther requre quatte by mea of ther tota uppy curve ([5,6]). Whe Ghoh ([7,8]) formuate the aocato moe, t wa ufortuatey abee to a output (uppy, uppy-rve) moe by h foower ([9-12]) 1. Moreover, Detzebacher ([10]; ee ao [13]) attempte to prove that Ghoh aocato moe equvaet to Leotef prce moe. However, the recet paper [6] 1 I ao, ufortuatey, fay cae Ghoh moe a output moe, epte the fact that my frt paper [19] the trbuto a output coeffcet (are) were ue equvaety, but my book [5] I metoe trbuto coeffcet oy oce a after that output coeffcet a output moe were ue. Copyrght 2011 ScRe.

2 E. DAVAR 643 how that Leotef Iput-Output ytem moe ffer from Ghoh ytem, a therefore they caot be equvaet. Th paper focue o the uppy quattatve moe that aow u to efe the tota uppe quatte of commote for ay gve uppe quatty of prmary factor, a coequety to efe the fa ue of commote for both phyca a moetary put-output ytem. The uppy quattatve moe à a Leotef bae o the output coeffcet of prmary factor a put coeffcet of commote. The output coeffcet of prmary factor are the vere magtue of ther put coeffcet; therefore, f the put coeffcet are gve a cotat by aumpto, the the output coeffcet are ao gve a cotat. The Haamar Prouct ao ue. Hece, th moe aow u to efe the tota uppe quatte of commote for ay gve uppe quatty of prmary factor, a coequety to efe the fa ue of commote. Such approach aow u to mapuate by each compoet of prmary factor (type of abour or fxe capta). Whe Ghoh moe bae o the aocato coeffcet of commote, whch are ot verte of the put coeffcet, a o the put coeffcet of prmary factor, a therefore, aow mapuate geeray by a aggregate magtue of vaue ae. Th paper cot of two ecto. Foowg the troucto the frt ecto ecrbe uppy quattatve equbrum ytem moe for Iput-Output phyca term; a the eco ecto ea wth uppy quattatve equbrum ytem moe for Iput-Output moetary term. Fay cocuo are preete. 2. Suppy Quattatve Equbrum for I-O Phyca Term à a Leotef 2 Let u tart wth the ema quattatve moe of Leotef put-output ytem, before ecrbg the uppy quattatve moe, for two reao: (1) to coer atoa property of the ema moe; a (2) to compare a uerta charactertc of thee moe. The ema quattatve equbrum for I-O phyca term cot of two ytem [6]:,or x A x y x I A y,or x By (1.1) 1 ˆ 0 v V Cx Cx v or v Cx CBy v 0 (1.2) 2 Iput-Output phyca term, th paper, ffer from the phyca put-output tabe (PIOT), whch recety appeare put-output terature. I the former, each commoty ha t ow phyca meauremet: meter, to, ut, M 3 a o o, whe the atter, a commote have uform phyca meauremet, for exampe to. where X x j the quare matrx (*) of the quattatve fow of commote the proucto; Y y r the matrx (*R) of the quattatve fow of commote to the categore of fa ue; y the coum vector (*1) of commote quatte for fa ue; x the coum vector (*1) of the tota output quatty of commote; V v kj the matrx (m*) of the quattatve fow of prmary factor to the ector of proucto; v the coum vector (m*1) of the tota quatte of prmary factor requre the proucto; A a j the quare matrx (*) of the rect put coeffcet of commote rea (phyca) term the proucto a ˆ 1 xj A X x,.., e aj ; (1.3) x.e., the put coeffcet a j meaure quatty of commoty requre for the proucto of oe ut of commoty j phyca term; C c kj the matrx (m*) of the rect put coeffcet of factor rea phyca term the proucto a ˆ 1 vkj C V x,.., e ckj ; (1.4) x.e., the put coeffcet of prmary factor c kj meaure quatty of factor k requre for the proucto of oe ut of commoty j phyca term; B Leotef vere matrx, a bj the tota requre quatte (rect a rect put) of commoty to a atfe oe ut of ema of the commoty j; v 0 the vector of the avaabe quatte of prmary factor; a ut coum vector (*1); The ytem (1.1) aow u to obta the tota requre quatte of commote for ay gve quatte of fa ue for the certa coto of the rect put coeffcet of commote A. Coequety, by the ubttuto of the obtae requre output quatte the ytem (1.2), the requre quatte of prmary factor are efe a v. Therefore, f the requre quat- te of prmary factor are wth the mt quatte rawg from ther uppy curve,.e., f the requre quatte are e or equa to the avaabe quatte ( v v0 ), the there a quattatve equbrum a the a prce equbrum etabhg mght be coere. j j Copyrght 2011 ScRe.

3 644 E. DAVAR Coverey, whe, f at eat the requre quatty for oe factor arger tha t avaabe quatty, the the proce mut be carre out for the ew fferet quatte for fa ue, ut the above coto are atfe. Worthy to cu the character of chage of the tota requre quatte of prmary factor ue to chage of quatte of fa ue. We aume, for the mpfcato, that oy the quatty of fa ue for a oe of ector (commoty) chage (creae) ( y ), whe the fa ue for other ector (commote) tay uchage. Subttute th (1.1), a we have: 1 1 x By B y y x x (1.5) where y the coum vector (*1) a compoet of whch are zero except of the compoet that equa to y. So a j x B y (1.6) x b y, j, 1,2, (1.7) From (1.7) we ca cocue that creag the fa ue of the commoty of a certa ector ether creae the tota proucto of commote of the ector where accorg vere coeffcet of put are more tha zero (b j > 0, j = ) or uchage f accorg vere coeffcet of put equa zero (b j = 0, j = ). Coequety, the quatte of prmary factor are ether creae, f rect put coeffcet of prmary factor are more tha zero (c kj > 0), or uchage f rect put coeffcet of prmary factor are equa to zero (c kj = 0) the ector where the tota proucto creae. Th kj kj j v c b y, k 1,2,, m; j 1,2,, (1.8) Now, the tota creae of each prmary factor eterme a v j 1v j 1 c b y, k 1,2,, m (1.9) k kj kj j y eterme a y v k j1ckjbj From (1.9) (1.10) However, thee tota creae of each prmary factor mut be e or equa to t uempoye quatte, th : Therefore vk vk0 vk, k 1,2,, m (1.11) m 1km k0 k kj j max y v v c b, (1.12) Th prove the foowg theorem: j1 Theorem 1 If matrx A potve (A 0) a prouctve (x > xa), a f the quatty of fa ue of a certa ector y creae a fa ue for a other ector are uchage, the the requre quatte of prmary factor are ether creae f rect put coeffcet of prmary factor are more tha zero (c kj > 0) or uchage f rect put coeffcet of prmary factor are equa to zero (c kj = 0) for the ector where the tota proucto creae; ao the magtue of the creae of fa ue of a certa ector (commoty) mte by the uempoye uppy quatte of prmary factor (1.12). To um up, th theorem cate that creag of the quatty the fa ue of the commoty of a certa ector, creae the requre quatte of prmary factor amot a ector. O the other ha, carefu examato of the ema ytem moe how that they mght be ue for oppote purpoe (recto) too. Namey, the ytem (1.1) mght be ue to obta the tota quatte of fa ue for ay gve tota quatty of commote, rewrtg t a: y I A x (1.13) So, (1.13) aow u to obta the tota quatty of fa ue for a gve tota quatte of commote. Th mea that orer to eterme the tota ema of fa ue, the tota quatte of commote have to be kow, o that the atter have to be coecte wth prmary factor. For exampe, the tota quatte of commote mut be eterme o the ba of the gve quatte of prmary factor. I other wor, the oppote moe to (1.2) requre. The queto, therefore, whether the ytem (1.2) may be traforme to uch a moe whch may aow u to eterme the tota quatte of commote for ay gve quatte of prmary factor. Ut toay, the awer wa obvouy egatve. It aerte that the coum of prmary factor for a certa ector, for the putoutput ytem phyca term, heterogeeou a therefore, ot be umme. Thu the egatve awer bae o the orary aay of put-output ytem moe. Let u try aother approach. Let tart from the etermato of the fow of prmary factor to ector of proucto (V matrx). From (1.2) t eterme that: V Cxˆ (1.14) If we take to accout the fact that whe reguar matrx mutpe o a agoa matrx, t mea that the frt compoet of each row of reguar matrx mutpe o the eemet of the frt coum of the agoa Copyrght 2011 ScRe.

4 E. DAVAR 645 matrx a the eco eemet of each row mutpe o the eemet of the eco coum, a o o. Therefore, the agoa matrx may be repace by a matrx where a eemet of a certa coum are etca a equa to the accorg agoa magtue; a ew matrx meo efe accorg to the meo of matrx C,.e. (m*). Th, takg cae x ˆ uer cuo, mght be repace by the matrx X (m*) where a eemet of the frt coum wou be the tota output of the frt ector, a eemet of the eco coum the tota output of the eco ector, a o o: X x x x x x x m1 m2 m x x x (1.15) It eceary to emphaze that there mght be a oppote cae, amey, whe a agoa matrx mutpe by a reguar matrx, a, uch a cae, each eemet of the row of repacg matrx ha to be etca a the meo of the matrx mut be accorg to the reguar matrx (ve fra). Now, (1.14) mght be rewrtte a V Cxˆ C X (1.16) The g () mea the Haamar prouct of two matrce C a X whe matrx V forme by the eemetwe mutpcato of ther eemet. The matrce mut be the ame ze. So, every compoet of V obtae a the foowg: each compoet of matrx C mutpe o the accorg compoet of matrx X, for exampe, the eemet c 23 mutpe o the accorg eemet x 23. O the other ha, from (1.16) X mght be eterme a o X V C (1.17) o where C the matrx of rect output coeffcet of prmary factor, whch are verte of the rect put coeffcet of prmary factor a t the ame ze a o tructure of the matrx C, th, ckj 1 ckj f ckj 0 o a f c kj = 0 the c kj ao equa to 0; the output coeffcet cate the quatte of commoty j prouce by a ut of prmary factor k. If, by aumpto, the rect put coeffcet of prmary factor are gve a cotat, the the rect output coeffcet wou ao be gve a cotat. Therefore, accorg to (1.17) orer to eterme the tota quatte of commote, the fow of prmary factor to ector (matrx V) requre. A metoe above for the equbrum tate, whe t eterme from the ema e, the eemet of a certa coum of X are etca, a they are the ame quatty. But, whe the eemet of matrx V are eterme accetay a uppy (otate a V ), accorg to the avaabe quatte of prmary factor, a they have to ue for etermato of the tota output of commote, the the tota quatty of a certa commoty may be fferet for varou prmary factor. I uch a cae, t eceary to chooe oe amout from them (ve fra). The requre quatte of prmary factor (v -coum vector), whch are eterme by the requre fow of prmary factor to ector of proucto (V ), ha to be a ource for the etermato of the uppe vero of the atter matrx ( V ). If the requre quatte of prmary factor are far from ther avaabe quatte (v < v 0 ), the there are uempoye quatte of prmary factor (cug abour). Therefore, uch a tuato, the oppote proce erabe, amey, the proce ha to tart from the e of prmary factor tea of the e of fa ue a the prevou cae. Here, the begg, the amout of quatte of prmary factor (otate a v the tota uppy quatte of prmary factor) are eterme a the ther trbuto betwee the ector of proucto mut be eterme. So, the queto ow how the gve quatte of prmary factor have to be trbute betwee ector of proucto. There are fte way of trbuto of the gve uppy quatte of prmary factor betwee proucto ector, tartg from the occaoa trbuto a fhg wth the pag trbuto accorg to a certa crtero. Let u cu the type of trbuto where the tructure of ew trbuto etca to the tructure of the trbuto for the ema e. For the purpoe of efg the tructure of the ema e et u rewrte the equato ytem (1.2) a foow: or k k k k v v 1v 2 v, k 1,2,, m (1.18) vk vk vk vk vk1 vk2 vk, k 1,2,, m (1.19) v v v a k k k k k k k k k k v 1v 2 v v, k 1,2,, m (1.20) where vkj kj, k 1,2,, m; j 1,2,, v k kj 1, k 1, 2,, m (1.22), j1 (1.21) kj the hare of the ector j the tota requre quatte of prmary factor k. From (1.21) we ca efe Copyrght 2011 ScRe.

5 646 E. DAVAR v v, k 1,2,, m; j 1,2,, (1.23) kj Therefore kj k V V, (1.24) where a g of the Haamar prouct; [ kj ] the matrx (m*) of trbuto of prmary factor betwee ector of proucto; V the matrx (m*) where a eemet of a certa row are etca (ve upra) a equa to the requre quatty of the accorg factor. So, aumg that cotat (1.21) aow u to eterme V whe V gve, that, eterme V whe V (v ) gve. To um up, the proce compete. If the tota uppy quatte of prmary factor are gve the (1.24) aow u to eterme ther trbuto betwee brache of proucto; ubttutg the obta reut to (1.17), the tota uppy quatte of commote are obtae; thu, ubttutg the atter to (1.13), accorg quatte of the fa ue of commote are eterme. Therefore, aumg that the ew tota quatty of prmary factor v 3, the matrx V compe where a eemet of each row are the ame accorg to v. Subttutg t (1.24), the matrx V obtae. Namey: Subttute the atter to (1.17) we have: V V (1.25). o X V C (1.26). Becaue of that the tota quatte of varou prmary factor are epeety eterme from the put tructure of ector, coum of the matrx X mght be heterogec, a that, compoet of a certa coum mght be fferet. So, there mght be the foowg x x x x x x X (1.27) m1 m2 m x x x where x kj the tota quatte of commoty j eterme accorg to the uppy quatte of prmary factor k. I uch a tuato, t eceary to chooe oe compoet from each coum accorg to the foowg crtero: j 1km kj x m x, j 1, 2,,, (1.28) 3 Where () expree the fact that thee quatte are eterme from the uppy e. Such otato ue orer ot to cofue t wth v the tota requre put of a prmary factor for a certa ector, ug for the put-output ytem moetary term. Th mea that for each coum the owet tota quatty choe to guaratee extece of requre quatte of a prmary factor. Subttute thee tota quatte of commote x to Equato (1.13) a the tota quatte of fa ue y are obtae. To um up, the uppy quattatve equbrum for Iput-Output phyca term ca be pace to the foowg ytem: j 1km o X V C, (1.29) kj x m x, j 1, 2,,, (1.30) y I A x (1.31) where V, C o, A are gve. The Equato (1.29) efe the matrx of pobe tota prouct of commote for each prmary factor X a the Haamar prouct of the matrx of the fow of prmary factor to ector (V ) a the matrx of rect output coeffcet of prmary factor (C o ). Here, there mght be m fferet tota quatte of commoty for each ector (commoty). Coequety, the equato ytem (1.30) aow for the choog of oe tota quatty for each ector o that t mght be pobe from the pot of a prmary factor. Fay, the equato (1.31) aow obtag the fa ue of commote for the choog of tota quatte of proucto. From the pot of ug the uppy quattatve moe practce worthy to coer the character of chage of the tota quatte of fa ue accorg to chagg of prmary factor. To mpfy, aume that oy the quatty of prmary factor for oe ector of proucto chage, whe other ector are uchage. Th mea that the tota proucto of the atter ector are ao uchage. Aume that the quatty of the prmary factor k (a bour) for the ector j creae by v kj 0 >. Subttute th (1.26) we have 1 o j kj kj kj j j j x v v c x x x (1.32) Aumg that uch a creae of the tota proucto of the commoty j ao pobe from the e of other prmary factor th ector,.e., there ext uempoye quatte of the ret prmary factor. Th mea that quatte of a prmary factor are accorgy creae. The, whe we ubttute the atter the Equato (1.31) we have: 1 1 y I A x I A x x I A x I A x y y (1.33), Copyrght 2011 ScRe.

6 E. DAVAR 647 where x the row vector (1*) a compoet of whch are zero except of the compoet (= j) that equa to. So x j Therefore whe j y I A x (1.34). 1 1 o j kj c kj y a x a v o, (1.35), y aj x j aj vkj ckj, whe j, 1,, j1, j1,, (1.36). From th we ca cocue the foowg: (1) ce a 1 the fa ue of ector ( y, whe = j) creae by 1a x j ; a (2) ce a j (whe I j) mght be ether a j > 0 or a j = 0 the fa ue of ector ( y, whe j) ether ecreae by a x j or ot chage. Yet, the creae of the fa ue of the commoty queto caot be more tha t uatfe quatty, that : 0 y y y, whe j (1.37). where y the maxmum quatty of ema of the 0 commoty. A the ecreae of the fa ue of the other commote caot be more tha ther quatte of fa ue, that y y, whe ( j) (1.38) Therefore, the arget magtue of the creae of the prmary factor equa to the maet magtue betwee the creae of fa ue of the ector queto (ee 1.35) a the ecreae of fa ue of other ector (ee 1.36): max v 1 kj o o y y a ckj j y a ckj m 1 ; j 0 j (1.39) By th the foowg theorem proofe: Theorem 2 If matrx A potve (A 0) a prouc tve (x > xa), a f quatte of a prmary factor v j of a certa ector j are creae by the ame rate a prmary factor for a other ector are uchage, the the fa ue of the ector queto y ( = j) creae a the fa ue of other ector y ( j) are ether ecreae whe aj > 0 or uchage whe a j = 0 (whe j); a the magtue of the certa prmary factor (factor ) creae a certa ector mte by the uatfe fa ue of the ector queto a fa ue of other ector (1.39). From the above we cocue that creag the qua- tty of ay prmary factor for a certa ector, creae the fa ue of th ector a ecreae or o t chage the fa ue of a other ector. To utrate the uggete uppy quattatve moe of put-output, et u ue Leotef mpfe putoutput moe ([14]; ee ao [15), whe makg two chage: frt, tea of two type of Capta Stock, oy oe type coere; a eco, Capta Stock meaure moetary term tea of phyca term: From th Tabe we ca efe the rect put coeff- of commote a prmary cet factor: A X x ˆ P 20.0P P 0, (1.40) 14.0Y 6.0Y Y 0.25 PP 0.4 PY 0.14Y P 0.12Y Y C V x ˆ $ 350.0$ P MH 135.0MH Y 2.5$ P 7.0$ Y 0.55MH P 2.7MH Y (1.41) where: P-Pou, Y-Yar, MH-Ma-Hour. Aumg that the quatty of the fa ue of the eco commoty creae (by 10.0 Y) a the quatty of the fa ue of the frt commoty uchage a they are equa to (y 1 ) = (55.0P 40.0Y); the ug (1.1a) we obta accorg tota output of commote: The a x I A y By 1.45 PP PY55.0P 0.232Y P 1.242Y Y 40.0Y P Y $ $ V Cxˆ, 58.4MH 168.5MH $ v V 226.9MH (1.42). (1.43). Aother aumpto that the avaabe quatte of prmary factor are (v 0 ) = (800$ 300MH). So, comparo to requre quatte v 1 a the avaabe quatte, we ca cocue that th quattatve equbrum a there are uempoye amout of both prmary factor. Yet, by comparg the ew matrce of fow of prmary factor to ector V 1 wth accorg matrx from Copyrght 2011 ScRe.

7 648 E. DAVAR Tabe 1, we ca ao ee that each eemet of the frt arger tha the accorg eemet of the eco, whch accorg to Theorem 1. Th becaue, th cae, a vere put coeffcet of commote a a rect put coeffcet of prmary put are trcty potve (>0). Now aumg that the goa of ecoomc to acheve fu empoymet for both prmary factor (the avaabe quatty mu 3% for reerve), th the ew vector of uggete quatte of prmary factor w be (v 1 ) = (776.0$ 281.0MH). For the foowg we ee matrx a V. The frt mght be compute o the ba of Tabe 1, a t (1.44) a the eco A V V V 776.0$ 776.0$ 281.0MH 281.0MH 324.0$ 452.0$ 81.0MH 200MH (1.45) (1.46) o A, C the matrx of rect output coeffcet of prmary factor 0.4 $ $ o 1 P Y C C (1.47) 1.82 PMH 0.37YMH Subttute (1.48) a (1.49) to (1.29) we obta o X V C 324.0$ 452.0$ 0.4 P $ Y $ 81.0MH 200MH 1.82 P MH 0.37Y MH 129.6P 64.6Y 147.4P 74.0Y (1.48). We ca ee that the tota output ffer for varou prmary factor both ector a therefore, ug the crtero of choce (1.30) we obta (that) x = (129.6 P 64.6 Y). Th mea that the uppy quatte of the frt factor, Capta Stock, fuy empoye, whe the eco factor, Labour, ot fuy empoye, here t uempoye part. Therefore, orer to creae empoymet of the eco factor t eceary to creae the frt factor,.e. vetmet mut be creae. Fay, accorg quatty of fa ue eterme by mea of (1.31), amey y I A x 0.75 PP 0.4 PY129.6P (1.49) 0.14Y P 0.88Y Y 64.6Y 71.4P 38.8Y Depte the fact that the tota proucto are creae both ector, (129.6P > 106.2P, a 64.6Y > 62.4Y), the fa ue of the frt ector creae (71.4P > 55.0P), however, the fa ue of the eco ector ecreae (38.8Y < 40.0Y). Thee reut are accorg to Theorem 2, becaue the rate of creae of the frt ector greater tha the eco ector (0.22 > 0.035) a therefore, the creag of the fa ue of the eco ector ervg from the creag t tota proucto ( = 2.1) e tha the ecreag ervg from the creag of the tota proucto of the frt ector ( = 3.3). Whe, the creag of the fa ue of the frt ector ervg from the creag t tota proucto ( = 17.55) greater tha the ecreag ervg from the creag of the tota proucto of the eco ector ( = 0.96). For the ceary emotrato properte of the Theorem 2, aume that the whoe uempoye quatty of the frt factor ue the eco ector,.e., v 12 = o 526.0$; a therefore, x2 v12c $ Y $. Sce, the tota proucto of the frt ector uchage (= 100.0P, ee Tabe 1)), the the accorg fa ue are: y I A x 0.75 PP 0.4 PY100.0P (1.50) 0.14Y P 0.88Y Y 75.2Y 45.0P 52.2Y The fa ue of the frt ector ecreae by 10.0P {= ( ) or (45.0P 55.0P)}, a the fa ue of the eco ector creae 22.2Y {= ( ) or (52.2Y 30.0Y)}. Tabe 1. Hypothetca put-output phyca term Agrcuture Maufacturg Houeho Tota Agrcuture 25.0 Pou 20.0 Pou 55.0 Pou Pou M aufacturg 14.0 Yar 6.0 Yar 30.0 Yar 50.0 Yar Capta Stock $ $ $ Labor 55.0 Ma-Hour Ma-Hour 40.0 Ma-Hour Ma-Hour Copyrght 2011 ScRe.

8 E. DAVAR Suppy Quattatve E qubrum for I-O Moetary Term à a Leotef I practce t ot away pobe to eparate quatte a prce wth objectve a ubjectve reao [16]. Hece, the reut of ecoomc actvte are uuay preete moetary term. Therefore, amot a extg emprca I-O are compe moetary term ce Leotef frt put-output ytem [1]. Emprca (Marxa-Leoteva) I-O characterze by quatty moetary term [17]. Th mea that thee cae, prce a quatte are ot eparate a they are amagamate to oe eemet. Each eemet cue a quatty a prce. Therefore, emprca I-O ha a uform meauremet for a part: commote, factor a categore of fa ue, amey, moey meaure. O the oe ha, th create ome probem whe t ue for pag a aay. O the other ha th aow exteg a cope of aay by the formuato of atoa moe. For exampe, a t wa metoe above, Ghoh formuate the aocato moe whch, ufortuatey, wa abee to a output (uppy, uppy-rve) moe by h foower ([9-12]). It mportat to tre that t mpobe to formuate uch moe for the I-O phyca term. Th ue to the heterogeeou character of both the tructure of the ue of factor for the proucto of certa prouct a the tructure of commote for a certa category of fa ue. Moreover, Detzebacher [10] ha attempte to prove that Ghoh aocato moe equvaet to Leotef prce moe. But, the recet paper [6] how that Leotef Iput-Output ytem moe ffer from Ghoh ytem, therefore they caot be equvaet. At th pot et u tart from the ema quattatve equbrum moe moetary term, whch etca to quattatve equbrum for I-O phyca term a cot of two ytem: x,or, A x y x I A y 1 or x By 0 (2.1) v V ˆ Cx Cx v,, (2.2) or v Cx CBy v A otato, etermato a exe here are etca to ytem (1.1) a (1.2), except that they are moetary term. Here a we a for I-O phyca term, by mea of ytem (2.1), the tota requre output of commote are obtae for the gve quatte of fa ue the certa coto for the matrx of the rect put coeffcet (A); (a) coequety, by the ubttuto of the obtae requre output quatte the ytem (2.2) 0 the requre quatty of prmary fa ctor are efe a v. If requre quatte are e or equa to the avaabe quatte (v v 0 ), the there a quattatve equbrum a the prce equbrum mght be coere. Coverey, whe at eat the requre quatty for oe factor arger tha t avaabe quatty, the the proce mut be carre out for the ew fferet quatte for fa ue, ut the above coto w be atfe. The ema quattatve moe ytem moetary term wey ue practce, a t' worth whe to coer the character of chage of the tota requre quatte of prmary factor ue to chage of quatte of fa ue mar to the ema quattatve moe phyca term (ve upra). To carfy the matter, et' aume that oy the quatte of fa ue for a oe of ector (commoty) chage (creae) ( y ), whe fa ue for other ector (commote) tay uchage. Subttute th (2.1) a we phyca put-output (ee (1.5), (1.6), (1.7)) a we have: x b y, j, 1,2,, (2.3) j From (2.3) we ca cocue that creag the fa ue of commoty of a certa ector ether creae the tota proucto of commote of part of ector whe accorg vere coeffcet of put more tha zero or oe't chage f accorg vere coeffcet of put equa zero. Therefore, the quatte of prmary factor are ether creae f rect put coeffcet of prmary factor are more tha zero (c kj > 0) or uchage f rect put coeffcet of prmary factor are equa to zero (c kj = 0) ector where the tota proucto creae. I ato, put-output moetary term the equbrum tate characterze by the baace betwee the tota vaue ae for a ector a the tota fa ue for a ector too ([18]; [5]): vj y (2.4) j1 1 Th ao true for the partcuar cae whch cue. Chage (creag) of vaue ae a ector (a prmary factor ue each ector) mut be equa to the chage the fa ue of the ector queto, that : m vkj vj y j j1 k1 j1 (2.5) If we take to accout (2.5), we ca cocue that creag the fa ue the ector queto geeray more tha creag of vaue ae th ector. By th we prove the foowg theorem: Theorem 3 If matrx A potve (A 0) a prouctve (x > xa), a f quatte of fa ue of a certa Copyrght 2011 ScRe.

9 650 E. DAVAR ector y creae a fa ue for a other ec- tor are uchage, the the quatte of prmary factor are ether creae f rect put coeffcet are more tha zero (c kj > 0) or uchage f they are equa zero (c kj = 0) ector where the tota proucto wa creae; a the magtue of the creae of fa ue of a certa ector (commoty) mte by the uempoye uppy quatte of prmary factor; ao the erve creae of vaue ae of the ector queto e tha the creae of the fa ue th ector v v y (2.6) m j k1 kj j f at eat oe of c kj > 0 whe (j ). Smar to Theorem 1, th theorem cate that creag of quatte the fa ue of commoty of a certa ector, creae requre quatte of prmary factor amot a ector; ato, a creae of the tota requre quatte of prmary factor for the ector queto e tha the creae of the fa ue th ector. O the ba of the above, we ca ao cocue that the uppy quattatve equbrum for I-O moey term etca to the uppy quattatve equbrum for I-O phyca term a cot the foowg ytem: o X V C, (2.7) kj j x m x, j 1, 2,, (2.8) 1km y I A x (2.9) where V, C o, A are gve. A otato a etermato here are etca to ytem (1.29), (1.30) a (1.31), except that they are moetary term. The Equato (2.7) efe matrx of pobe tota proucto of commote for each prmary factor by orary mutpcato matrx of the fow of prmary factor to brache (V ) a matrx of rect output coeffcet of prmary factor (C 0 ). Here, there mght be m fferet tota quatty of commoty for a certa commoty. Coequety, the Equato ytem (2.8) aow u to chooe oe tota quatty o that t mght be pobe from the pot of a prmary factor. Fay, the equato (2.9) aow u to obta the fa ue of commote for choog tota quatte of proucto. The character of chage of the tota quatte of fa ue for the uppy quattatve moe moetary term ha atoa ecoomc ee becaue of the homogeety of meauremet of the moetary put-output. I th cae, the vaue of fferet prmary factor ue for a certa brach a the vaue of fferet commote emae for a certa category of fa ue mght be ummarze. Becaue the uppy quattatve equbrum for I-O moey term etca to the uppy quattatve equbrum for I-O phyca term we ca cocue that the above coere properte (ee Theorem 2) for the atter have to be correct ao for the former the ame framework. We ee that creag the quatty of ay prmary factor for a certa ector creae the fa ue of th ector a ecreae or o t chage the fa ue of a other ector: y 1 a x, whe j (2.10), y a x j j j j whe j, 1, 2,, j1, j1,, (2.11) Thee chage are erve from the chage (creae) of prmary factor for brach j. Becaue of the baace betwee the tota vaue ae for a ector a the tota fa ue for a ector (2.4), chage (creag) of vaue ae oe ector (a prmary factor ue th ector) mut be equa to chage fa ue of a ector, that : m vkj vj y y y j y j k1 1 1 (2.12) If we take to accout (2.12), we ca cocue that creag of fa ue the ector queto ervg from the creag of vaue ae th ector ge- theorem: eray more tha the atter. By th we prove the foowg Theorem 4 If matrx A potve (A 0) a prouctve (x > xa), a f quatte of a prmary factor v kj of a certa ector j creae by the ame rate, a prmary factor for a other ector are uchage, the the fa ue of the ector queto y ( = j) creae a the fa ue of other ector y (I j) ether ecreae whe a j > 0 or uchage whe a j = 0; a therefore, the erve creae of the fa ue of the ector queto more tha the creae of vaue ae th ector v y,whe j (2.13) j f at eat oe of a j > 0 ( j); a the magtue of the certa prmary factor (factor ) creae a certa ector mte by the uatfe fa ue of the ector queto a fa ue of other ector. Here ao, mar to Theorem 2, the creae of the quatty of ay prmary factor for a certa ector, creae the fa ue of th ector a ecreae or oe t chage the fa ue of a other ector; a, ato, erve creae of the fa ue th ector queto more tha the creae of the tota quatte of prmary factor for the ector. Copyrght 2011 ScRe.

10 E. DAVAR 651 The properte of Theorem 3 a 4 mght be utrate by mea of hypothetca put-output moetary term (for exampe $, whch o ot appear the Tabe 2): From the Tabe 2 we ca efe the rect put coeffcet of commote (A) a prmary factor (C) a coequety Leotef vere coeffcet (B) a output coeffcet of prmary factor (C o ): a A X x ˆ C V x ˆ (2.14) (2.15) B I A (2.16) o C 1 C (2.17). Frty, et u coer Theorem 3. For th purpoe, et aume that the fa ue of ector 2 creae by 85.0 $ (y 2 = 85.0$); the the tota proucto of ector 1 creae by 22.6 $ (x 1 = b 12 y 2 = = 22.6$), a of ector 2 creae by $ (x 2 = b 22 y2 = = 105.9$). From th creag of vaue ae wou be V A (2.18) v v1v (2.19) So, v2 = 76.01$ a t e tha y 2 = 85.0$, what acco rg to the Theorem 3. It worthy to tre that the tota amout of creag of vaue ae both ector equa to the amout of creag of the fa ue of the eco ector (= 85.0$). Now, aume that the quatty of the frt prmary factor for the eco ector creae by 8.66$ ( v12 = 8.66$); the the tota output of the ector 2 creae by ( o x2 c12v $ 105.7$ ), mar to the prevou exampe ( 105.9). Whe the tota output of the frt ector ot chage. From th for the fa ue we have 1 2 y y y I A x (2.20) So, the fa ue of the eco ector creae by 92.7$, whch more tha the tota creae of the vaue ae of th ector 75.8$, whch accorg to Theorem 4. At the ame tme the fa ue of the frt ector ecreae by 16.9$. But, the tota creae of fa ue both ector (92.7$ 16.9$ = 75.8$) equa to the tota creae of vaue ae the eco ector. It teretg to ote that epte the fact that both cae the tota vaue ae of the eco ector wa creae the ame magtue (75.8$), the fa ue of th ector creae by fferet magtue. Th ue to the fact that the eco cae oy the tota proucto of the eco ector wa chage (creae), whe the frt cae the tota proucto of both ector wa chage (creae). Therefore, orer to the fa ue of the frt ector ot chage t eceary to creae t proucto the magtue whch cover (equa) requremet to prouce atoa quatty of the eco ector a the frt ector tef. Namey, the put of the frt prmary factor Tabe 2. Hypothetca put-output moetary term Agrcuture Maufacturg Itermeate Tota Output Fa Ue y Tota Output x Agrcuture Maufacturg Itermeate Tota Iput Capta Stock Labour Tota vaue ae v Tota Iput x Copyrght 2011 ScRe.

11 652 E. DAVAR of the frt ector mut be creae by 1.65$, or, atera- tvey, the put of the eco prm ary factor of the frt ector mut be cre ae by 7.3$. Both cae the prouc- to of the frt ector creae by 22.6$ ( $ = 22.34, or $= 22.6$). To compete the emotrato of the above metoe tatemet that the uppy quattatve moe à a Leotef equvaet to the ema quattatve moe of Leotef aume that the vaue ae of both ector are creae by 8.95$ a 76.0$ repectvey. The, the quatty of the frt ector creae by 22.6$ ( $ = 22.6$) a the quatty of the eco ector creae by 105.8$ ( $= 105.8$). Fay, the fa ue of ector w be creae whch are etca wth the reut of the ema quattatve moe (ve upra). 4. Cocuo Th paper exame the uppy quattatve moe ytem of put output for both phyca a moetary term, whch equvaet to the ema quattatve moe ytem of Leotef. The uppy quattatve moe bae o the rect output coeffcet of prmary fac- tor whch are the vero of the rect put coeffcet. Th paper ao took to coerato the properte of both ema a uppy quattatve ytem moe. It wa how that: (1) the creae of the fa ue of a certa ector geeray creae requre (ema) quatte of prmary factor for a ector a magtue of the creae of fa ue of a certa ector (commoty) mte by the uempoye uppy quatte of prmary factor the quattatve ema moe ytem phyca term (ee Theorem 1); (2) the creae of the prmary factor of a certa ector creae fa ue of th ector a geeray ecreae fa ue of the ret ector a the magtue of the creae of a certa prmary factor (factor) a certa ector mte by the uatfe fa ue of the ector queto a by the fa ue of other ector the quattatve uppy moe ytem phyca term (ee Theorem 2); (3) the quattatve ema moe moetary term the creae of the tota vaue ae of a certa ector, ervg from the creag quatty of fa ue of the ector queto, geeray e tha the atter a magtue of the creae of fa ue of a certa ector (commoty) mte by the uempoye uppy quatte of prmary factor (ee Theorem 3); a (4) the quattatve uppy moe ytem moetary term the creae of fa ue of a certa ector, ervg from the creag of the vaue ae of the ector queto, greate r tha the creae of the tota vaue ae of th ector a the magtue of the creae of a certa prmary factor (factor) a certa ector mte by the uatfe fa ue of the ector queto a by the fa ue of other ector (ee Theorem 4). Fay, the quattatve uppy ytem moe mght be uefu too pag the ecoomc of coutre that have hgher uempoymet of prmary factor, epecay abour. 5. Referece [1] W. Leotef, Quattatve Iput a Output Reato the Ecoomc Sytem of the Ute State, The Revew of Ecoomc Stattc, Vo. 18, 1936, pp o:org/ / [2] W. Leotef, The Structure of Amerca Ecoomy, Cambrge, (Mor): Harvar Uverty Pre, Seco Eto 1951, Oxfor Uverty Pre, New York, [3] R. Stoe, Iput-Output a Natoa Accout, Raom Houe, Par, DECD, [4] H. B. Cheery a P. G. Cark, Iterutry Ecoomc, Joh Wey& So, New York, [5] E. Davar, The Reewa of Caca Geera Equbrum Theory a Compete Iput-Output Sytem Moe, Hog Kog, Sgapore, Syey, Avebury, Aerhot, Brookfe USA, [6] E. Davar, Iput-Output Sytem Moe: Leotef veru Ghoh, 13 th Iteratoa Coferece o Iput-Output Techque, 27 Jue - 1 Juy Bejg, Cha, [7] A. Ghoh, Iput-output Approach a Aocato Sytem, Ecoomca, Vo. 25, 1958, pp [8] A. Ghoh, Expermet wth Iput-Output Moe, At the Uverty Pre, Cambrge, [9] M. Augutovc, Metho of Iteratoa a Itertempora Compoto of Structure, Cotrbuto to Iput-Output Aay, Vo. I, North-Hoa, Loo, [10] E. Detzebacher, I Vcato of the Ghoh Moe: A Reterpretato a a Prce Moe, Joura of Regoa Scece, Vo. 37, No. 4, 1997, pp o:org/ / [11] J. Ooterhave, O the Paubty of the Suppy- Drve Moe, Joura of Regoa Scece, Vo. 28, No. 2, 1988, pp o:org / /j tb01208.x [12] J. Ooterhave, Leotef veru Ghoha Prce a Quatty Moe, Souther Ecoomc Joura, Vo. 62, 1996, pp [13] L. e Mear, I the Ghoh Moe Iteretg? Joura of Regoa Scece, Vo. 49, No 2, 2009, pp Copyrght 2011 ScRe.

12 E. DAVAR 653 [14] W. Leotef, Techoogca Chage, Prce, Wage, a Rate of Retur o Capta the USA Ecoomy, Iput-Output Ecoomc, Seco Eto, Oxfor Uverty Pre, New York, [15] W. Leotef, The Choce of Techoogy, Scetfc Amerca, Vo. 252, No. 6, pp , o:org/ /cetfcamerca [16] E. Davar, Iput-Output Mxe Meauremet, 13 th Iteratoa Coferece o Iput-Output Techque, Augut, Macerate, Itay, [17] R. Dorfma, Samueo, P. a R. Soow: Lear Programmg a Ecoomc Aay, McGraw-H, New York, [18] W. Leotef, Iput-Output Ecoomc, Oxfor UP, New York, [19] E. Davar, Iput-Output a Geera Equbrum, Ecoomc Sytem Reearch, Vo. 1, No. 3, pp , Copyrght 2011 ScRe.