The Leontief Economic Model

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.