Leontief Model

From Open Risk Manual


A Leontief Model is an economic model following the proposals of economist Wassily Leontief who developed a system of economic analysis (Input-Output Model) in the 1930s and 1940s. It is based on a linear economy assumption and demand-style economic modeling[1]


Starting with the Market Balance or Total Output equation:

x & = A x + y \\
x_i  & = \sum_j A_{ij} x_j + y_i


  • x is the column vector of output of endogenous accounts
  • y the column vector of Final Demand and

A is the so-called Technical Coefficient Matrix, whose elements are the elements of the Transactions Matrix divided by the total of their corresponding output column:

        A & = Z\hat{x}^{-1} \\
        A_{ij} & = \frac{Z_{ij}}{x_j}

The Leontief inverse matrix L = (I-A)^{-1} is obtained, satisfying the condition x = L y.

        x = (\mathrm{I}- A)^{-1}y = L y 

with \mathrm{I} defined as the identity matrix (Kronecker delta) with dimension equal to the size of A.


In the demand-driven, upstream or Leontief model, a matrix of direct requirements A is defined as the inter-industrial flows a_{ij} from an industry i to an industry j per gross output of sector j.

An element l_{ij} of the total requirements matrix, or Leontief inverse, L represents the amount of gross output from sector i that was produced to satisfy a unit of final demand y from sector j.


An important goal of IO analysis is to examine the interdependencies between production and consumption within an economy. Such analysis includes the flow of goods of services between the economy and the rest of the world and can be expressed in monetary or other measurement units.

See Also

Further Resources


  1. Leontief 1970, JSTOR