## 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 |

\[A=\left( \begin{array}{ccc} 0.4 & 0.2 & 0.3 \\ 0.3 & 0.1 & 0.1 \\ 0.3 & 0.7 & 0.6 \end{array} \right)\]

. 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

\[\begin{pmatrix}G\\I\\H\end{pmatrix}=\begin{pmatrix}29/48\\5/16\\1\end{pmatrix}\]

.