The Exchange and Income Matrices

Suppose that countries  
\[C_1,...,C_n\]
  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  
\[Y_1=3/2, \: Y_2=25/16, \: Y_3=1\]
.