Answer to Question #183922 in Linear Algebra for Bhakti

Question

The following data is the input-output tables for different sectors in 

an economy. Find the technology matrix and also the total output against the 

changes in the final demand given:

INDUSTRY X Y FINAL 

DEMAND

X 15 25 40

Y 20 30 50

D=(

50

75)

Answer 1

The input output coefficient matrix for X and Y is given by-

$A=\begin{bmatrix} 0.66 &0.04 \\ 0.2 & 0.33 \end{bmatrix}$

Then the Technology matrix( leontiff matrix) is given by-

$I-A=\begin{bmatrix} 0.34 &-0.04 \\ -0.2 & 0.67 \end{bmatrix}$

Imagine that the vector of final demands becomes $y = [y_1, y_2] =[50,70]$ .Then, to find the corresponding activity levels in $x = [x_1, x_2, x_3]$ we must solve the system $(I − A)x = y.$ We have-

$\begin{bmatrix} 0.34 &-0.04 \\ -0.2 & 0.67 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}=\begin{bmatrix} 50 \\ 70 \end{bmatrix}$

Solving above matrix for x and y and we get-

$x=165.15,y=153.71$