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Unformatted text preview: hat is prod ds te e, duced is produced only for the sake of satisfyin the inpu requireme f e ng ut ents of the (n+1) indus stries in the model. e At fir glance, the convers rst t sion of the o open sector into an ad dditional ind dustry would d not s seem to cre eate any sig gnificant cha ange to the analysis. Actually, ho e owever, 229 since the new in e ndustry is a assumed to have a fixe input rat as does every other ed tio indus stry, the supply of wha used to b the prima input (la at be ary abour) mus now bear st a fixe proportio to what used to be called the f ed on final deman nd. More concretely this may mean, for e e y, example, th househo hat olds will consume each commodity in a fixed proportion to the labo services they provi h our s ide. This certa ainly constit tutes a sign nificant theo oretical change in the a analytical fr ramework comp pared to the open mod e del. Math hematically speaking, t disappe the earance of final deman means that we will nds now have a hom mogeneous s-equation s system. As ssuming fou industries only ur (inclu uding the newly create one de ed esignated b the subsc by cript 0), the "correct" e outpu levels will be those that satisfy the followi system of equation ut y ing ns: Beca ause this eq quation sys stem is hom mogeneous, it can have a non-triv solution e vial n if and only if the 4 4 tech d e hnology ma atrix (I A) has a vanis shing deter rminant. Of co ourse, this condition is always sa c s atisfied. In a closed mo odel, there is no more e prima input (la ary abour) and hence, eac column s ch sum in the input outp put coeff ficient matr A must n rix now be exac equal to (instead o less than 1. That ctly of n) mean that ns a0j + a1 + a2j + a3j = 1 1j or a0j = 1 a1j a2j a3j Yet, this implies that in eve column of the matr (I A) above, the to element s ery rix op t ways equal to the nega ative sum o the other three elem of r ments. As a is alw cons sequence, the four row are linea dependent and we must find t ws arly e that I A= 0. This guarantees that the system poss s sesses non n-trivial solu utions. In fa it has an act, al s. h infinite number of non-trivia solutions This implies that in the closed model with a hom mogeneous equation syst...
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