Leontiev Input-Output Model

Author :  Thomas Shores
University of Nebraska
Send comments to :  tshores@math . unl . edu

Introduction

Most administrative systems  in industry or government produce two kinds of goods:  those intended for external consumption, such as a salable product, public regulation, customer support, etc., and those intended for internal consumption, such as payroll accounting, worker training, facilities maintenance,etc.  With respect to the latter goods, the system can be likened to a closed economy.  That  is, everything produced by the system is consumed by the system.  Here is a simplified model of such a situation: suppose an administrative unit has four divisions serving the internal needs of the unit, which we label as (A)ccounting, (M)aintenance, (S)upplies and (T)raining.  Each unit produces the ``commodity'' its name suggests, and charges the other divisions for its services.  A study shows that the fraction of commodities consumed by each division is given by the following table, called an input-output  matrix:

        Produced by:        A     M     S    T 
                            A     0.2   0.1   0.4  0.4 
 Consumed by:    M    0.3   0.4   0.2  0.1 
                            S     0.3   0.4   0.2   0.3 
                            T     0.2   0.1   0.2   0.2

Note that the sum of entries in each column is  one.  This reflects the fact that the system is closed, i.e., everything produced by a given division is consumed by the divisions of the unit.  If the sum were less than one, there would be a surplus of the commodity represented by the 
column -- if greater than one, there would be a shortage of it!

Here is the problem: what price should each division charge for its commodity so that each division earns exactly as much as it spends?  Such a pricing scheme is called an equilibrium price structure  in the parlance of economics; it assures that no division will earn too little to do its  job.  Let  x , y , z  and  w be the price per unit commodity charged by A,  M,  S and  T, respectively.  The  requirement  of expenditures equalling earnings for each division results  in the four  equations following.  Click inside the next cell and press <Enter> to activate it....

{ 0.2x + 0.1y + 0.4z + 0.4w == x,
0.3x + 0.4y + 0.2z + 0.1w == y,
0.3x + 0.4y + 0.2z + 0.3w == z,
0.2x + 0.1y + 0.2z + 0.2w == w }

We can put this system into simpler form by multiplying each equation by  -10 and bringing all terms to the left hand side to obtain the following.  Click inside the next cell and press  <Enter> to activate it.

equations = {8x - y - 4z - 4w == 0,
-3x + 6y - 2z - 1w == 0,
-3x - 4y + 8z - 3w == 0,
-2x - 1y - 2z + 8w == 0 }

Clearly, a trivial solution to the system is  x = y = z = w = 0 .  This solution is  not informative.  Nonzero solutions give information about the relative value of each service.  Are there any nontrivial solutions?  Let's ask Mathematica for some help.  Activate the next cell.

rules = Solve[equations, {x,y,z,w}]

Very interesting ... what do you make of this ?  Try the next cell...

equations /. rules

Exercise

Suppose there are exactly five companies  (Ace, Brown, Collegiate, Drake, and Excalibur) that publish textbooks, and that universities have the option of buying any book from any publisher.  Also suppose that  a constant proportion of universities that buy books from a given company (say, Ace) will purchase books from another company (Drake) the  next year, and that  the empirical statistics for this are:

         current   year:             Ace    Brown     Collegiate    Drake    Excalibur
         
         next year:  Ace           0.30      0.20           0.15           0.13        0.12
                           Brown       0.15     0.35           0.30            0.10       0.22
                           Collegiate  0.25     0.05           0.40           0.20        0.25
                           Drake         0.10     0.12           0.02           0.33        0.10
                           Excalibur   0.20     0.28           0.13           0.24        0.31
                           
 Here 12% of the universities that bought from Brown in 1991 bought from Drake in 1992, for example.  Let a,b,c,d,e be the proportion of universities buying books from Ace, Brown,
 Collegiate, Drake, and Excalibur, respectively.
 QUESTION: Does this  situation have an equilibrium state? If so, find it.     
 (Note that  in order to represent the state as proportions of the market, we also need the equation  a + b + c + d + e = 1. )