## 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
$Y_1=3/2, \: Y_2=25/16, \: Y_3=1$
.