Find the ratio of the three commodities in the closed model

Question

suppose that production of 1 unit of mining require 5/7 unit of mining, 1/7 unit of manufacturing, and 1/7 unit of communication. To produce 1 unit of manufacturing requires 3/7 unit of mining, 2/7 unit of manufacturing, and 2/7 unit of communication. To produce 1 unit of communication requires 0 units of mining, 1/7 unit of manufacturing, and 6/7 unit of communication. Find the ratio of the three commodities in the closed model.

Answer 1

Let x = units of mining, y = units of manufacturing and z = units of communication.  Then you have the following 3 equations with 3 unknowns: 

 

5/7 x + 1/7 y + 1/7 z = 1 x                      or rewriting       -2/7 x + 1/7 y + 1/7 z = 0

3/7 x -  5/7 y + 2/7 z = 1 y                                                3/7 x - 5/7 y + 2/7 z = 0

    0x + 1/7 y + 6/7 z = 1 z                                                    0 x + 1/7 y - 1/7 z = 0

 

Notice the constant terms on the right hand side of the re-written equations are all zero.  When this occurs the equations are said to be homogenious and then there is either one solution (the trirvial solution or x = y = z = 0 ) or infinitely many solutions.  It turns out the latter is the case.  

 

Note the last line of the rewriten equation:  1/7 y - 1/7z = 0 or y = z.  Plugging y = z into the second equation yields 3/7 x - 5/7 z + 2/7 z =  or 3/7 x =  3/7 z  or x = z  so we have infinitely many solutions where x = y = z.  For example let x = y = z = 7  and you will see all equations are satisfied.  Similarly for any set of numbers where x = y = z.  Try it for yourself.

 

What do you conclude about the proportions?   Don't feel bad I fooled with this for some time.  When you get to determinants, you will see that the determinant of the matrix formed by these equations is 0.  Hence no inverse and no unique solution.  Hope this helps you.