## The Closed Leontief Model

Suppose the economy of a country is described by the following simple model. There are three sectors: the Government, industry and households. The table below shows the fraction of the output of each sector which is consumed by each sector.
 Consumed By\Fraction Of Government Budget Industrial Output Household Budget Government 0.4 0.2 0.3 Industry 0.3 0.1 0.1 Households 0.3 0.7 0.6
Let
$A=\left( \begin{array}{ccc} 0.4 & 0.2 & 0.3 \\ 0.3 & 0.1 & 0.1 \\ 0.3 & 0.7 & 0.6 \end{array} \right)$
. The 'closed Leontief mode;' is one in which 'what goes into the economy is what comes out', so that if
$\mathbf{x} = \begin{pmatrix}Government \: Budget\\Industrial \: Output\\Household \: Budget\end{pmatrix}$
then
$A \mathbf{x}=\mathbf{x}$
.
$\left( \begin{array}{ccc} 0.4 & 0.2 & 0.3 \\ 0.3 & 0.1 & 0.1 \\ 0.3 & 0.7 & 0.6 \end{array} \right) \begin{pmatrix}G\\I\\H\end{pmatrix}=\begin{pmatrix}G\\I\\H\end{pmatrix}$
.
The solution to this, using Gaussian Elimination for example, is
$\begin{pmatrix}G\\I\\H\end{pmatrix}=\begin{pmatrix}29/48\\5/16\\1\end{pmatrix}$
.