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  
\[C(C \mathbf{v})=C^2 \mathbf{v}={\left( \begin{array}{cccc} 0.1 & 0.3 & 0.0 & 0.0 \\ 0.7 & 0.6 & 0.0 & 0.0 \\ 0.1 & 0.0 & 1.0 & 0.0 \\ 0.1 & 0.1 & 0.0 & 1.0 \end{array} \right)}^2 \begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix}=\left( \begin{array}{cccc} 0.22 & 0.21 & 0.0 & 0.0 \\ 0.19 & 0.57 & 0.0 & 0.0 \\ 0.14 & 0.03 & 1.0 & 0.0 \\ 0.23 & 0.16 & 0.0 & 1.0 \end{array} \right) \begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix}\]
.
The amount in each country after  
\[n\]
  days is  
\[C^n \mathbf{v}\]
.