Introduction

    The mathematics model for the economy of a country or a region is based on the various sectors of this economy.  The Leontief model is addresses this problem.  Assume that each industry in the economy has two types of demands: an external demand (from outside the system) and an internal demand (demand placed on one industry by another in the same system), the Leontief model is a system of linear equations. The Leontief model was invented in the 30's by Wassily Leontief who developed an economic model of the United States economy by dividing it into 500 economic sectors.  Wassily Leontief received the economics Nobel Prize on October 18, 1973.  

The Closed Leontief Model

    Assume that an economy consists of  n  interdependent industries (or sectors)  .  Each industry will consume some of the goods produced by the other industries, including itself (for example, a power-generating plant uses some of its own power for production).  An economy is called closed if it satisfies its own needs; that is, no goods leave or enter the system.  We make the following conventions:

(i)          is  the production level of industry  ,  
    
(ii)         is the number of units produced by industry that is necessary to produce one unit by industry ,  

(iii)         is the number of units produced by industry    and consumed by industry  ,  

(iv)      is the total number of units produced by industry  .  

Since the economy is closed, the total production for industry equals it total consumption and we have the equations

           

for  .

If the economy is balanced, the total production of each industry must be equal to its total consumption. This results in the linear system:

                

which can be written in matrix form  
        
        .

The matrix    is called the input-output matrix, and    is the production vector.  

Proof Leontief Model  Leontief Model  

Example 1.  Consider the closed three sector economy   consisting of say:  Energy, Manufacturing, and Services where the input-output matrix is given by

.   Find the production vector  .   

Solution 1.

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.

The Open Leontief Model

    The closed Leontief model describes the case when no goods leave or enter the economy.  But often times, an economy has to satisfy an outside demand.  In this case, let    be the demand from the outside industry.  Let    and    be as in the closed model, then  

          

for  .

This results in the linear system:

        

which can be written in matrix form  
        
        ,

where   and   are the same as in the closed model and   is the demand vector.

One way to solve this linear system is

        
        
          
        
        .
    
Of course, the matrix    must be invertible, which might not be always the case.  If, in addition, has nonnegative entries, then the components of the vector   are nonnegative and therefore they are acceptable as solutions for this model.  We say in this case that the matrix   is productive.

Proof Leontief Model  Leontief Model  

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.

Example 4.  Consider the open five sector economy   consisting of say: Agriculture, Construction, Energy, Manufacturing, and Services where the input-output matrix is given by

  and the demand vector is  .  Find the production vector  .   

Solution 4.

Research Experience for Undergraduates

Leontief Model  Leontief Model  Internet hyperlinks to web sites and a bibliography of articles.  

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(c) John H. Mathews 2004