## Simulating Trade in a Single Commodity

Four countries are
$C_1, \: C_2, \: C_3, \: C_4$
engaged in mutual trade of a single nonrenewable commodity, represented per day by the matrix
$C= \left( \begin{array}{cccc} 0.1 & 0.3 & 0.0 & 0.0 \\ 0.7 & 0.6 & 0.0 & 0.0 \\ 0.1 & 0.0 & 1.0 & 0.0 \\ 0.1 & 0.1 & 0.0 & 1.0 \end{array} \right)$
.
If the amounts of the commodity in each country are
$v_1, \: v_2, \: v_3, \: v_4$
respectively, what will the distribution of the commodity after
$n$
days>
The entry in the ith row and jth column, labelled
$c_{ij}$
represents the exports of country j to country i.
$c_{ij}v_j$
will be the the fraction of the commodity held by country j exported to country i on the first day, so the imports of country i on day 1 will be
$\sum_j c_{ij}v_j$
. The vector formed by these numbers represents the total imports of each country after one day.
The amount of the commodity in each country after two days is
\begin{aligned} C(C \mathbf{v}) &= C^2 \mathbf{v} \\ &= {\left( \begin{array}{cccc} 0.1 & 0.3 & 0.0 & 0.0 \\ 0.7 & 0.6 & 0.0 & 0.0 \\ 0.1 & 0.0 & 1.0 & 0.0 \\ 0.1 & 0.1 & 0.0 & 1.0 \end{array} \right)}^2 \begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix} \\ &= \left( \begin{array}{cccc} 0.22 & 0.21 & 0.0 & 0.0 \\ 0.19 & 0.57 & 0.0 & 0.0 \\ 0.14 & 0.03 & 1.0 & 0.0 \\ 0.23 & 0.16 & 0.0 & 1.0 \end{array} \right) \begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix} \end{aligned}
.
The amount in each country after
$n$
days is
$C^n \mathbf{v}$
.