Input–output model

In economics, an input–output model is a quantitative economic model that represents the interdependencies between different sectors of a national economy or different regional economies. Wassily Leontief (1906–1999) is credited with developing this type of analysis and earned the Nobel Prize in Economics for his development of this model. Francois Quesnay had developed a cruder version of this technique called Tableau économique, and Léon Walras's work Elements of Pure Economics on general equilibrium theory also was a forerunner and made a generalization of Leontief's seminal concept. Alexander Bogdanov has been credited with originating the concept in a report delivered to the All Russia Conference on the Scientific Organisation of Labour and Production Processes, in January 1921. This approach was also developed by L. N. Kritsman and T. F. Remington, who has argued that their work provided a link between Quesnay's tableau économique and the subsequent contributions by Vladimir Groman and Vladimir Bazarov to Gosplan's method of material balance planning. Wassily Leontief's work in the input-output model was influenced by the works of the classical economists Karl Marx and Jean Charles Léonard de Sismondi. Karl Marx's economics provided an early outline involving a set of tables where the economy consisted of two interlinked departments. Leontief was the first to use a matrix representation of a national (or regional) economy. The model depicts inter-industry relationships within an economy, showing how output from one industrial sector may become an input to another industrial sector. In the inter-industry matrix, column entries typically represent inputs to an industrial sector, while row entries represent outputs from a given sector. This format therefore shows how dependent each sector is on every other sector, both as a customer of outputs from other sectors and as a supplier of inputs. Each column of the input–output matrix shows the monetary value of inputs to each sector and each row represents the value of each sector's outputs. Say that we have an economy with n {displaystyle n} sectors. Each sector produces x i {displaystyle x_{i}} units of a single homogeneous good. Assume that the j {displaystyle j} th sector, in order to produce 1 unit, must use a i j {displaystyle a_{ij}} units from sector i {displaystyle i} . Furthermore, assume that each sector sells some of its output to other sectors (intermediate output) and some of its output to consumers (final output, or final demand). Call final demand in the i {displaystyle i} th sector d i {displaystyle d_{i}} . Then we might write or total output equals intermediate output plus final output. If we let A {displaystyle A} be the matrix of coefficients a i j {displaystyle a_{ij}} , x {displaystyle x} be the vector of total output, and d {displaystyle d} be the vector of final demand, then our expression for the economy becomes