Example 2.  Consider the closed five sector economy   consisting of say: Agriculture, Construction, Energy, Manufacturing, and Services where the input-output matrix is given by

.   Find the production vector  .   

Solution 2.

Enter the matrix  .  

If   is an eigenvalue of    and    an eigen-pair, then the solution to   will be a multiple of the eigenvector  .
Let us verify that    is an eigenvalue of  .  

The eigen-pair     is easily computed using Mathematica's subroutines  Eigenvalues and  Eigenvectors.  

Since    is a solution for any constant  c, we are permitted to choose any multiple of   we desire for the solution.  For illustration:

Aside.  Any value of  c  is permitted, it is your choice.  When we get

We are done.

How did we get that "nice" vector ?
By using exact arithmetic and the following calculations.

Form the homogeneous system .   
and the augmented matrix     and find its reduced row echelon form.

The equations for this augmented matrix are  

      

There is one free variables which we choose to be  .   It is used in computing  .  
Use and solve the previous equations for    

      

Get    

    

The solution vector    is

We are done.

Aside.  We can verify that this is the solution by direct multiplication A X.  This is just for fun !

We are really done.

Aside.  Iteration can be used to solve for the solution of the closed Leontief model.  If the sum of the production levels is known and a starting vector is given   then the simple iteration    will converge to  .  Because the largest eigenvalue is  ,  this is a simplified version of the "power method" for  finding the dominant eigen-pair.  For the example above   and the following iteration will converge.

(c) John H. Mathews 2004