Coursenotes_ECON301

Notice that exce for the f and he e ept first

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online