Applications - A Guide to MATLAB for Beginners and Experienced Users - page 172

fact that the last column of T is the sum of columns 4 (total inter-industry
outputs) and 8 (total final demand) translates into the matrix equation
X = AX + Y ,or Y = (1 − A ) X . Let's check this:
Y = T(1:3, 8); X = T(1:3, 9); Y - (eye(3) - A)*X
ans =
0
0
0
Now one can do various numerical experiments. For example, what would
be the effect on output of an increase of £ 10 billion (10,000 in the units of
our problem) in final demand for industrial output, withno corresponding
increase in demand for services or for agricultural products? Since the
economy is assumed to be linear, the change X in X is obtained by solving
the linear equation Y = (1 − A ) X , and
deltaX = (eye(3) - A) \ [0; 10000; 0]
deltaX =
1.0e+004 *
0.0280
1.6265
0.1754
Thus agricultural output would increase by £ 280 million, industrial output
would increase by £ 16.265 billion, and service output would increase by
£ 1.754 billion. We can illustrate the result of doing this for similar increases
in demand for the other sectors with the following pie charts:
deltaX1 = (eye(3) - A) \ [10000; 0; 0];
deltaX2 = (eye(3) - A) \ [0; 0; 10000];
subplot(1, 3, 1), pie(deltaX1, {'Agr.', 'Ind.', 'Serv.'}),
subplot(1, 3, 2), pie(deltaX, {'Agr.', 'Ind.', 'Serv.'}),
title('Effect of increases in demand for each of the 3
sectors', 'FontSize',18),
subplot(1, 3, 3), pie(deltaX2, {'Agr.', 'Ind.', 'Serv.'}) ;