Abstract
In classic economics the interrelations in production are vital to understand the laws of production and distribution, and therefore to understand how an economic system works. Wassily Leontief1, a Russian born American economist made the greatest contribution in this line of thought by developing the input/output analysis2.
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- 1.
Wassily Leontief (1906–1999) was born and educated in St. Petersburg. He received his Ph. D. from Berlin University in 1929. He joined the famous Kiel Institute of World Economics in 1927 and worked there until 1930. He moved to the USA in 1931; worked at the Harvard University (1932–1975) and New York University (1975–1991). Leontief was awarded with Nobel Prize in Economics in 1973 for his contribution to input-output analysis.
- 2.
For a historical survey and review of the contributions of economists to input-output analysis see [17, 18].
- 3.
Notice that, in contrast to its row sums, no meaning can be attributed to column sums of Table 10.1, since columns consists of elements with different units of measurement (for example, coal is measured by tons, electricity by Kwh and trucks by numbers).
- 4.
See [16, 22] for the discussion of joint production.
- 5.
In [21], O’Neill & Wood dropped the continuity assumption and proved theorem by assuming column sums of B are not greater than one. The latter assumption can be fulfilled in practice by appropriately choosing the measurement units for each commodity.
- 6.
Another term for reducible and irreducible matrices are decomposable and indecomposable matrices.
- 7.
In terms of graph theory, the part (b) means that a directed graph G(A) with vertices \(\{1,\ldots, n\}\) and edges {j → k|a jk ≠0} is strongly connected, that is, for each two vertices j≠k there is a path from j to k.
- 8.
For the sake of simplicity, in the remainder of the book, this distinction will be omitted and the input coefficients will be denoted by a ij .
- 9.
The alternative assumption is that wages are paid in advance.
- 10.
Pierro Sraffa (1898–1983) was an Italian economist, who spent most of his life in Cambridge, England. In his famous book [28] he launched a strong critique of marginalist theory and laid the foundations of Neo-Ricardian school in economics.
- 11.
Sraffa analyzed the determination of prices at a given moment of time, given the prevailing technology. Therefore, he did not make any assumptions concerning the returns to scale and did not use input coefficients.
- 12.
Within the framework of [28] this is not a deficiency of the system. On the contrary it makes clear that the distribution of income between wage and profit earners can not be solved within the price system. It requires a much broader framework that may include politico-economic as well as financial variables.
- 13.
For r ≥ 1 the matrix (1 + r)A is not productive (again by Hawkins–Simon conditions), so, the problem has no economical sense.
- 14.
Oskar Perron (1880–1975) was a German mathematician who made a significant contribution in algebra, geometry, analysis, differential equations, and number theory.
- 15.
In 1907, Perron proved Theorem 10.12 under the additional condition that the matrix A is positive (or at least some its power is positive). In 1912, Frobenius extended this result to its complete form. For the survey of these articles of Perron and Frobenius and for the history of the theorem, we refer the reader to [13]. Theorem 10.10 (which is also sometimes referred as Perron–Frobenius theorem) can be obtained from either Perron or Frobenius results by limit argument.
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Aleskerov, F., Ersel, H., Piontkovski, D. (2011). Linear Model of Production in a Classical Setting. In: Linear Algebra for Economists. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20570-5_10
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DOI: https://doi.org/10.1007/978-3-642-20570-5_10
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