\[C_1,...,C_n\]
trade with each other. The fraction of the income of country \[C_j\]
spent on imports from country \[C_i\]
is \[a_{ij}\]
. The \[n \times n\]
matrix \[A\]
with entry \[a_{ij}\]
in the ith row and jth column is called the 'exchange matrix'.Yf the income
\[Y_1\]
of country 1 is given by\[Y_1=a_{11}Y_1+a_{12}Y_2+...+a_{1n}Y_n\]
\[Y_2=a_{21}Y_1+a_{22}Y_2+...+a_{2n}Y_n\]
\[\vdots \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \vdots \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \vdots \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \vdots \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \]
\[Y_n=a_{n1}Y_1+a_{n2}Y_2+...+a_{nn}Y_n\]
We can write this in matrix form as
\[\mathbf{P}=A\mathbf{Y} \rightarrow (A-I)\mathbf{Y}\]
where \[\mathbf{Y}=\begin{pmatrix}Y_1\\Y_2\\ \vdots\\Y_n\end{pmatrix}\]
is called the income matrix.Suppose
\[A=\left( \begin{array}{ccc} 1/4 & 2/5 & 1/2 \\ 1/2 & 1/5 & 1/2 \\ 1/4 & 2/5 & 0 \end{array} \right)\]
.Then
\[(\left( \begin{array}{ccc} -3/4 & 2/5 & 1/2 \\ 1/2 & -4/5 & 1/2 \\ 1/4 & 2/5 & -1 \end{array} \right) ) \begin{pmatrix}Y_1\\Y_2\\Y_3\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}\]
.Solving this system by Gaussian Elimination for example, gives
\[Y_1=3/2, \: Y_2=25/16, \: Y_3=1\]
.