3) Suppose that three industries are interrelated so that their outputs are used
ID: 3405820 • Letter: 3
Question
3) Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3 × 3 consumption matrix
where aij is the fraction of the output of industry j consumed (purchased) by industry i. Let pi be the price charged by industry i for its total output. A problem is to find prices so that for each industry, total expenditures equal total income. This leads to the system of linear equations Ap = p, where p = [p1, p2, p3]T . Find a solution p with nonnegative p1, p2 and p3.
.1 .5 0 .8 0 .4 .1 .5 .6Explanation / Answer
Given matrix is
we need to find p such that Ap=p, where p=[p1, p2, p3]
Ap=p implies [A-I]p=O, where O is a zero matrix o=[0,0,0]. Now A-I=
determterminant of A-I = (-0.9)(0.4-0.2)-(.05)(-0.32-.014)
= -0.9*0.2-0.5*(-0.18)
=-0.18+0.90
=0.72, therefore the matrix A-I is invertible
which implies that that the system of equation [A-I]p=O, has only trivial solution i.e,
P=[0,0,0] i.e p1=0, p2=0, p3=0.
.1 .5 0 .8 0 .4 .1 .5 .6Related Questions
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