Example 3.  Consider the open three sector economy   consisting of say:  Energy, Manufacturing, and Services where the input-output matrix is

  and the demand vector is  .  Find the production vector  .   

Solution 3.

We are done!

Aside.  We can verify the solution by direct substitution.

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  .  This is reminiscent of Jacobi iteration and Gauss-Seidel iteration for solving a linear system.  For the example above   and the following iteration will converge (slowly).

Aside.  Any demand vector can be used, it is your choice.  For example:

(c) John H. Mathews 2004