Chapter 3, Problem 59RE

Economy models A look at the industrial sector of an economy can be simplified to include three industries: the mining industry, the manufacturing industry, and the fuels industry. The technology matrix for this sector of the economy is given by

$\begin{array}{l}\begin{array}{ccc}\text{M}& \text{Mf}& \text{F}\end{array}\\ A=\left[\begin{array}{ccc}0.4& 0.2& 0.2\\ 0.2& 0.4& 0.2\\ 0.1& 0.2& 0.2\end{array}\right]\begin{array}{l}\text{Mining}\hfill \\ \text{Manufacturing}\hfill \\ \text{Fuels}\hfill \end{array}\end{array}$

Find the gross production of each industry if surpluses of $72$ units of mined goods,

$40$ units of manufactured goods, and $220$ units of fuels are desired.