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Modelyuvannya Economy - Vіtlіnsky VV

11.3. Koefіtsієnti chat i Povny materіalnih vitrat

Zdіysnyuyuchi analіz modelі mіzhgaluzevogo balance, potrіbno rozglyanuti osnovnі vlastivostі matritsі koefіtsієntіv directly materіalnih vitrat A. Tsі koefіtsієnti for viznachennyam Je nevіd'єmnimi, otzhe, matrix In tsіlomu Je nevіd'єmnoyu: Huh? 0. Process vіdtvorennya not mozhna Bulo b zdіysniti, Yakscho used for Vlasnyi vіdtvorennya in the Branch vitrachavsya bіlshy obsyag product nіzh stvoryuvavsya. Zvіdsi obviously scho dіagonalnі Elements matritsі A menshі nіzh odinitsya: aii <1, i = 1, ..., n.

System rіvnyan mіzhgaluzevogo balance vіdobrazhaє realnі ekonomіchnі processes in Cauterets Sens mozhut mother deprivation nevіd'єmnі local Gross vipuskіv; this rank vector valovoї produktsії skladaєtsya s nevіd'єmnih Components Connection vector X, yaky Je nevіd'єmnim vector: X> 0. Postaє power for yakih drain system Ekonomichna zdatna zabezpechiti nevіd'єmny kіntsevy Key infrastructure in vsіh Galuzo? Vіdpovіd on tse power pov'yazana s ponyattyam produktivnostі matritsі koefіtsієntіv directly materіalnih vitrat.

Signification. Nazivatimemo nevіd'єmnu matrix A productive Yakscho іsnuє Taqiy nevіd'єmny vector X, scho

X> AX. (11.13)

Obviously, scho Umov (11.13) oznachaє іsnuvannya nevіd'єmnogo vector kіntsevoї produktsії Y> 0 for modelі mіzhgaluzevogo balance (11.6).

Matrix Direct dwellers koefіtsієntіv materіalnih vitrat A bula productivity neobhіdno i dostatno, Abi vikonuvalasya one perelіchenih nizhche s minds:

• matrix (E - A) Got Booty nevіd'єmno obernenoyu, tobto guilty іsnuvati Wrap matrix (E - A) -1? 0;
• matrix series Got zbіgatisya, Ak ® 0, k ®, and Yogo scrip dorіvnyuє obernenіy matritsі (E - A) -1?;
• naybіlshy per module l rozv'yazok (Vlasnyi value) of the characteristic rіvnyannya Got Booty strictly Mensch od odinitsі;
• usі golovnі mіnori matritsі (E - A) tobto viznachniki matrix, scho utvorenі Some of the elements i ryadkіv Perche Perche stovpchikіv tsієї matritsі order od 1 and n, toil Buti dodatnimi.

Bіlsh downtime, ale deprivation dostatnoyu Find our produktivnostі matritsі And Je obmezhennya її the value of the norm, the value of tobto naybіlshoї іz sumi elementіv matritsі And cutaneous stovpchiku. Yakscho rate matritsі A strictly Mensch od odinitsі then tsya matrix Je productivity. Nagolosimo scho given Umov deprivation dostatnoyu Yea, i matrix A Mauger viyavitisya productive minutes from razі, Yakscho bude bіlshoyu її rate per unit.

Naybіlshy for module korіn the characteristic rіvnyannya induced in tretіy umovі produktivnostі matritsі A (poznachimo Yogo through l *), Mauger sluguvati for otsіnku zagalnogo rіvnya koefіtsієntіv directly materіalnih vitrat and otzhe, the value of (1 - l *) harakterizuє zalishok pіslya vitrat, tobto produktivnіst . Chim bіlshim Je (1 - l *), note bіlshimi Je mozhlivostі dosyagnennya іnshih tsіley, okrіm virobnichogo stream processes. Іnshimi words denote vischim Je zagalny rіven koefіtsієntіv matritsі A note bіlshim - Maximum per module Vlasnyi values (l *) i nizhchim - rіven produktivnostі, i navpaki, denote nizhchy zagalny rіven koefіtsієntіv matritsі A Tim Mensch Je maximum modulus Vlasnyi value ( l *) i vischoyu produktivnіst.

Proanalіzuymo matrix koefіtsієntіv Povny materіalnih vitrat, tobto matrix B = (E - A) -1. Element tsієї matritsі bij pokazuє, skіlki vsogo neobhіdno virobiti produktsії i -ї the Branch, dwellers will win odinitsyu kіntsevoї produktsії j -ї the Branch.

Damo Lots Other aforesaid koefіtsієnta Povny materіalnih vitrat s look around to those scho okrіm directly vitrat іsnuyut oposeredkovanі vitrati tієї chi іnshoї produktsії for virobnitstva produktsії danoї the Branch. Rozglyanmo for butt formuvannya vitrat elektroenergії on Key infrastructure rolled steel, obmezhuyuchis tehnologіchnim lantsyuzhkom "ore-chavun-steel-rolling." Vitrati elektroenergії for otrimannya rental Zi stalі nazivatimemo directly vitratami, Ti samі vitrati for otrimannya stalі s chavunu - oposerednenimi vitratami 1st order and vitrati elektroenergії for otrimannya chavunu s Rudy - oposeredkovanimi vitratami elektroenergії on Key infrastructure Stalevo rental 2nd toscho order. Otzhe can dati TAKE aforesaid:

Koefіtsієntom kvazіpovnih materіalnih vitrat cij nazivayut the bag directly i oposeredkovanih vitrat produktsії i -ї the Branch for virobnitstva odinitsі produktsії j -ї the Branch through promіzhnі products on vsіh poperednіh stadіyah virobnitstva. Yakscho koefіtsієnti oposeredkovanih materіalnih vitrat k-th order through poznachati Then Location Got formula

(11.14)

Yakscho to enter a matrix rozglyadu koefіtsієntіv kvazіpovnih materіalnih vitrat C = (cij) is the matritsі koefіtsієntіv oposeredkovanih materіalnih vitrat rіznih poryadkіv Then poelementnu formula (11.14) can be tribute to matrichnіy formі:

(11.15)

W look around to zmіstovnu are koefіtsієntіv oposeredkovanih materіalnih vitrat mozhna zapisati takі matematichnі spіvvіdnoshennya:

for vikoristannya Cauterets matrix formula (11.15) nabiraє viglyadu

(11.16)

Yakscho matrix koefіtsієntіv directly materіalnih vitrat And Je productive, then Druha s minds produktivnostі іsnuє matrix B = (E - A) -1, yak Yea sumoyu zbіzhnogo matrix row

(11.17)

Porіvnyuyuchi virazi (11.16) that (11.17), dіstanemo:

B = E + C

abo in poelementnomu zapisі:

Tse viznachaє ekonomіchny Sens, scho poyasnyuє vіdmіnnіst mіzh koefіtsієntami (Some of the elements) of the matrix B is the C: on vіdmіnu od koefіtsієntіv matritsі C scho vrahovuyut deprivation vitrati on virobnitstvo produktsії, koefіtsієnti matritsі In vklyuchayut in itself, okrіm vitrat, takozh odinitsyu kіntsevoї produktsії, KOTRA vihodit outside the scope virobnitstva.