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Unformatted text preview: th products a1 x1 + a12x2 + a13x3 +...+ a1nxn on the d The he 11 other hand, does have an econo h omic meaning it represents the total am g; mount of x1 ne eeded as an input for all the n in ndustries. 223 (1 a11) x1 a21 x1 a12x2 (1 a22) x2 2 a13x3 - ... a23x3 - ... - a1nxn = d1 a2nxn = d2 an1 x1 an2 x2 2 an3x3 - ... - (1 ann) xn = dn In ma atrix notatio this can be written as: on, n e p agonal of th matrix on the left ar ignored, then the he n re If the 1s in the principal dia matrix is simply A = [-aij]. As it is, o the other hand, the matrix is th sum of y on r he the id dentity matrix In (with 1 in the pr 1s rincipal diag gonal and z zeros every ywhere else e) and t matrix A. the Ther refore, we can write the above se of matrice as: c et es (I A)x = d re a variable vec and the final dema ctor e and wher x and d are, respectively, the v (cons stant-term) vector. Th matrix (I A) is called the tech he hnology ma atrix, and we e can d denote it by T. y Ther refore, we can also write this syst c tem as: Tx = d As lo as T is non-singula and the is no re ong ar ere eason why i should no be we it ot -1 shall be able to find its inve erse T , an obtain th unique solution of th system nd he he from the equatio on: _ 1 x = T-1 d = (I A)-1 d ) 224 del merical Exa ample The Open Mod A Num For t purpose of illustra the es ation, suppo that the are only three indu ose ere y ustries in the econ nomy and th the inpu co-efficient matrix is as follows: hat ut s e e n ss as Note that in A each column sum is les than 1, a it should be for the open mode Further, if we deno by a0j th dollar am el. ote he mount of the primary in e nput (labour r) th used in producing a dollar' worth of t j comm d 's the modity, we can write (by subtr racting each column s sum in the a above matri from 1): ix a01 = 0.3 0 a02 = 0.3 a03 = 0.4 With the matrix A above, th open inp output system ca be expre he put an essed in the e form we derived earlier...T = (I A)x = d as fol d Tx x llows: e d any c emands d1, Here we have deliberately not given a specific values to the final de d2, and d3. In th way, by keeping th vector d in parametric form, ou solution his he ur appear as a "formula" into which w can feed various s we specific d ve ectors to will a obtai various corresponding solution (i.e. like a demand f in c ns function in consumer theor ry). 225 Now, by invertin the 3 3 tech...
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