Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy
The new book SRAFFA AND LEONTIEF REVISITED: Mathematical methods and models of a circular economy is dedicated to Wassiliy Leontief’s concepts of Input-Output Analysis and to the algebraic properties of Piero Sraffa's seminal models described consequently by matrix algebra and the Perron-Froben...
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irk-123456789-1731462020-11-24T01:26:28Z Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy Emmenegger, J.-F. Chable, D. Nour Eldin, H.A. Knolle, H. Нова книга The new book SRAFFA AND LEONTIEF REVISITED: Mathematical methods and models of a circular economy is dedicated to Wassiliy Leontief’s concepts of Input-Output Analysis and to the algebraic properties of Piero Sraffa's seminal models described consequently by matrix algebra and the Perron-Frobenius Theorem. The academic editor Walter de Gruyter-Oldenbourg, has published this monography in January 2020 in English language. Мета роботи – дати інформацію про нову книгу «Повернення до Сраффи та Леонтьєва. Математичні методи і моделі кругової економіки». Ця монографія була видана в січні 2020 англійською мовою німецьким науковим видавництвом Вальтер де Гройтер (Walter de Gruyter-Oldenbourg). Цель работы – дать представление о новой книге «Возвращение к Сраффе и Леонтьеву. Математические методы и модели круговой экономики». Эта монография была издана в январе 2020 на английском языке немецким научным издательством Вальтер де Гройтер (Walter de Gruyter-Oldenbourg). 2020 Article Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy / J.-F. Emmenegger, D. Chable, H.A. Nour Eldin, H. Knolle // Кібернетика та комп’ютерні технології: Зб. наук. пр. — 2020. — № 2. — С. 86-99. — Бібліогр.: 9 назв. — англ 2707-4501 DOI:10.34229/2707-451X.20.2.9 http://dspace.nbuv.gov.ua/handle/123456789/173146 519.86, 330.44 en Кібернетика та комп’ютерні технології Інститут кібернетики ім. В.М. Глушкова НАН України |
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The new book SRAFFA AND LEONTIEF REVISITED: Mathematical methods and models of a circular economy is dedicated to Wassiliy Leontief’s concepts of Input-Output Analysis and to the algebraic properties of Piero Sraffa's seminal models described consequently by matrix algebra and the Perron-Frobenius Theorem. The academic editor Walter de Gruyter-Oldenbourg, has published this monography in January 2020 in English language. |
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Emmenegger, J.-F. Chable, D. Nour Eldin, H.A. Knolle, H. |
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Emmenegger, J.-F. Chable, D. Nour Eldin, H.A. Knolle, H. |
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Emmenegger, J.-F. |
| title |
Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy |
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Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy |
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Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy |
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Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy |
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Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy |
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sraffa and leontief revisited: mathematical methods and models of a circular economy |
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Інститут кібернетики ім. В.М. Глушкова НАН України |
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2020 |
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Sraffa and Leontief Revisited: Mathematical Methods and Models of a Circular Economy / J.-F. Emmenegger, D. Chable, H.A. Nour Eldin, H. Knolle // Кібернетика та комп’ютерні технології: Зб. наук. пр. — 2020. — № 2. — С. 86-99. — Бібліогр.: 9 назв. — англ |
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Кібернетика та комп’ютерні технології |
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2025-07-15T09:40:53Z |
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2025-07-15T09:40:53Z |
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| fulltext |
NEW BOOKS
86 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
CYBERNETICS
and COMPUTER
TECHNOLOGIES
The new book SRAFFA AND LEONTIEF
REVISITED: Mathematical methods and mod-
els of a circular economy [1] is dedicated to
Wassiliy Leontief’s concepts of Input-Output
Analysis and to the algebraic properties of Piero
Sraffa's seminal models described consequently
by matrix algebra and the Perron-Frobenius
Theorem. The academic editor Walter de Gruy-
ter-Oldenbourg, has published this monography
in January 2020 in English language. 1
Keywords: Input-Output analysis, circular eco-
nomy, Perron-Frobenius Theorem, non-negative
matrix.
J.-F. Emmenegger, D. Chable †,
H.A. Nour Eldin, H. Knolle, 2020
UDC 519.86, 330.44 DOI:10.34229/2707-451X.20.2.9
J.-F. EMMENEGGER, D. CHABLE
†
, H.A. NOUR ELDIN, H. KNOLLE
SRAFFA AND LEONTIEF REVISITED:
MATHEMATICAL METHODS AND MODELS
OF A CIRCULAR ECONOMY
From natural circular processes to economic circular
economy. Life on earth depends on circular processes.
This statement is valid for plants, animals and the human
being. In early times the human being has found food in
nature, without influence on the circular processes. With
animal husbandry and agriculture, he has learnt to adapt
the life cycles of animals and plants to his own needs, but
this type of economy was still cyclical, and the human be-
ing has not intervened yet into the circularity of water, of
nitrogen and carbon. This has changed with the area of
industrial production and exploitation of not renewable
natural resources. The many disadvantages of this mode
of production, from which the climate change most wor-
ries, are now clearly visible. Actually, a new conscious-
ness awakes the interest for the concept of circular econ-
omy, which involves the fact that the ‘resources are avail-
able in a certain quantity and have an inherent capacity of
renewal’, see Aurez [2] (2019), p. 24. In the last century
there were two economists who understood and described
the economy as a circular process, Wassily Leontief
(1906 – 1999) [3] and Piero Sraffa (1898 – 1983).
Wassily Leontief (1906 – 1999) was a Russian Ameri-
can economist, who in 1928 has devoted his doctoral the-
ses to the circularity of economy. He conceived the Input-
Output tables (IOT), which today are used in the Statisti-
cal Offices of all the countries, in order to establish the
annual national production. These tables also contain the
results of national accounting.
With his book „Production of Commodities by Means
of Commodities“ [4], published in the year 1960, Piero
Sraffa (1898 – 1983) explained the basics to a modern
understanding of the economic processes of circularity.
He considered commodities as means of production and
end products. He supposed constant rate of profits and
constant wage rates and computed production prices, in a
system producing surplus, separated in profits for entre-
preneurs and wages for workers.
1 The authors donated a copy of the book to the V.I. Vernadsky
National Library of Ukraine (Eds.)
https://doi.org/10.34229/2707-451X.20.2.9
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 87
From agriculture of the ancient civilizations to the classics and neoclassic economies. The exist-
ing testimonies of the ancient civilizations of the Egyptians and Mesopotamians in the region of the Fer-
tile Crescent illustrate intensively, that the human beings from the very beginning of history observed the
processes of nature. They grasped the cycles of the seasons, the water flow of the Nile, the Euphrates and
the Tigris, the influence of weather, and finally the circularity of the life of the plants and the animals.
Setting up their agriculture, these early civilizations have observed the circularity of life processes and
imitated it. In this way the first economic circular processes have been created. The construction of basic
tools followed the same logic. The organisational and artisanal set up led to the flourishment of these ear-
ly civilizations. Especially the Egyptians have presented on graphic wall pictures the succession in time of
the different steps of work in the cultivation of cereals, ploughing, sowing, harvesting, storing, conserving
or consuming. The first tools have also been visually presented.
The knowledge of economic circular processes is for these reasons since the ancient civilizations part
of our history. For millenaries it has been transmitted from generation to generation and has always been
testified by great thinkers and scientists. Thus, the physician and physiocrat François Quesnay (1694 –
1774) has graphically presented the concept of circularity of economic production processes in his books
(1759) as tableaux économiques and illustrated with calculations.
There are two events which can be identified, leading to a deviation from the understanding of the
circularity of economic processes. As first event there is the statement of: Adam Smith (1723 – 1790)
who “asserts the apparently self-contradictory notion that capitalism transforms selfishness into its oppo-
site: regard and service for others”, see Foley ([5], p. 2). This mechanism was conducted by the invisible
hand. When every individual selfishly maximizes his personal utility, then following Smith the wealth of
nations is also guaranteed. Thus, “neither Smith nor any of his successors has been able to demonstrate
rigorously and robustly how private selfishness turns into public altruism“, see Foley [5], p. 3).
The concentration on the goal of personal utility leads away from holistic comprehension of econom-
ic processes, because the attention is focused on the end product, this means on the object which gives the
psychologically and individually justified utility. The unalterably necessary means of production, as
wheat, iron, wood, from which tools are constructed in the economy, as they are presented on the wall
pictures of the ancient Egyptian civilization, are put on the background.
The area of classical political economy (classics), whose representatives are Adam Smith, David Ri-
cardo and others, based on the comprehension of circularity of economic processes, lasted until 1870.
One of its pillars was the theory of labour value, which affirms that the price of a commodity depends es-
sentially on the working time necessary for its elaboration. After 1870, and this is the second event, the
theory of marginal utility, which at the beginning only was a theory of the behaviour of consumers, puts
the theory of labour value in the background. Then, at the eve of the next century some economists con-
centrated their efforts on the analytical apparatus of the theory of marginal utility. They put besides the
marginal utility of the commodities the marginal productivity of the „ factors of production “, which are
labour and capital. This way of arguing is since 1900 the dominating school of thought and is called neo-
classical economy, even if it has nearly nothing to do with classics. Generally, one considers the decisive
difference between classics and neoclassical economy in its respective price theories, which is also the
theory of labour value versus the theory of marginal utility. For our purposes another difference is more
important. In the neoclassical economy the production process is a one sense road, leading from the pro-
duction factors labour (L) and capital (K) to an end product (Y) which is generally not precisely defined,
thus the process is shortly written as “exploit – produce – consume – draw away”, see also Aurez [2],
(2019), p. 24 . This relation is described by the equation Y = f (L, K), where f is the „production function“.
What undesired waste goods are generated, from where the labour force comes, and if and how many
times the process can be repeated, nobody asks for that, see Knolle [6]. Aurez [2], (2016), p. XIV, speaks
of «linear industrial economy».
J.-F. EMMENEGGER, D. CHABLE †, H.A. NOUR ELDIN, H. KNOLLE
88 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
Sraffa’s formal representation of economic production as circular production processes.
A detailed word-based presentation of circular production processes, starting from any number of indus-
tries and all corresponding parts of means of production tends to the limit of linguistic possibilities. Some
authors trying it have been confronted to this difficulty. These linguistic limits are here illustrated with
the most elementary wheat example of Sraffa ([4], paragraph 1, p. 21), just comprising two industries and
two products.
The next explanations lean on those of Helmut Knolle [6]. Sraffa starts his oeuvre Production of
commodities by means of commodities [4] with the consideration of a very elementary national economy,
consisting of two industries, producing only wheat and iron. With iron we understand simple iron tools,
which are unusable after one year and must be replaced. The productivity is so small, that no surplus is
generated. There is also no profit. The workers are not paid with money but get wheat as subsistence
wages. In the cadre of the chosen technology the specific quantities of wheat and iron in both sectors have
to be in a fixed proportion: every miner needs a pickaxe and each agricultural worker needs a scythe, etc.
The annual activity is described by following scheme: The first industry produces from 8 tons (t) of iron
and 120 quarters (qr) of wheat the quantity of 20 t of iron and, secondly, agriculture produces from 12 t of
iron and 280 qr of wheat 400 qr of wheat.
The system uses also in both industries 8 + 12 = 20 t of iron and produces the same amount of iron,
respectively, uses 120 + 280 = 400 qr of wheat and produces the same amount of wheat. The iron industry
produces over the own use a surplus of 12 t of iron, which are exchanged against 120 qr of wheat in order
to pay subsistence wages to the workers and entrepreneurs. In agriculture one needs just 12 t of iron and
can exchange its surplus of 120 qr of wheat with the iron industry.
With this scheme Sraffa has developed a price model, in order to get the equilibrium between the to-
tal quantity of means of production and the total final use, occurring over the determination of production
prices (exchange values). One observes that 1 t of iron has just the value of 10 qr of wheat, in order that
the whole system can be reproduced year after year. The currency of this economy is wheat. The workers
and entrepreneurs get subsistence wages in wheat.
For agriculture the value of the means of production is equal to the value of the total wheat produc-
tion: 12 10 + 280 1 = 400 1 = 400 qr of wheat, for the iron industry the value of the means of pro-
duction is equal to the value of the produced iron: 8 10 + 120 1 = 20 10 = 200 qr of wheat. The total
production has a value of 600 qr of wheat. One can formulate this equivalence in another way: Agriculture
has to deliver at the end of the year the quantity of 120 qr of wheat of own production against 12 t of iron
from the iron industry, which needs itself 120 qr of wheat and can thus deliver 12 t of iron to agriculture.
Agriculture needs the remaining 280 qr of wheat partly as seeds, partly to pay its workers and entre-
preneurs, and the 8 t of iron as iron tools for the next annual production. The iron industry needs the ob-
tained 120 qr of wheat, in order to pay subsistence wages for its workers and entrepreneurs. The remain-
ing 8 t of iron are used as iron tools for the own production process. The system is in equilibrium, when
the inputs and outputs are equal, thus, 10 qr of wheat have to be exchanged against 1 t of iron. This was a
verbal description of the most elementary example of Sraffa. Here, there is no production of surplus, no
profit for entrepreneurs, no wages for workers, there are exclusively subsistence wages for all the active
persons. The complexity of a verbal description of a circularity process has become visible throughout this
example. This seems indeed to be a reason why the detailed concept of circularity is apparently uncom-
mon to a larger public. For economists who exclusively use word-based descriptions and not number-
based matrix algebra, the accessibility is seemingly also restricted or difficult.
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 89
With the means of matrix algebra, one succeeds on the other hand to describe the circular processes
in a very elegant way.
In the next section we show the contribution of the book “Sraffa and Leontief revisited”.
From Sraffa’s Elementary Example to the SWISS-IOT 2014 (49 sectors/product groups)
Rigorous matrix algebra representation of Sraffa’s circular economy. Sraffa’s oeuvre «Produc-
tion of commodities by means of commodities» [4] (1960) has met great interest soon after its publication
among famous economists (Pasinetti, Schefold) and mathematicians (Newman). Since, a great deal of
work has been accomplished to complete Sraffa’s book. Thus, Peter Newman and Ann Arbor [7] have ex-
plicitly shown in their article, that the theorem of Perron-Frobenius is the algebraic basis of the Sraffa
price model with surplus, which is transformed in profits and where the workers obtain subsistence wages.
Indeed, Newman has correctly shown here that the calculation of throughout positive prices leads to an
eigenvalue problem. Further, both authors have also solved the general Sraffa price model with wages for
workers and formulated the conditions for the existence of the appearing inverse matrices, referring to
theorems in Felix R. Gantmacher [8] and others, in order to obtain also here throughout correctly positive
prices as solutions.
In the year 1976 Bertram Schefold has once more reformulated and mathematically completed Sraf-
fa’s basic theses in an appendix of the German edition of Sraffa’s book [4].
Our intention is to complete the existing literature (Pasinetti, Schefold, Kurz and Selvadori) by num-
ber-based and application-oriented presentations, using throughout modern matrix notations, referring to
the specific theorems of matrix algebra. For the calculation of numerical examples and applications on the
Input-Output Tables of Germany, Switzerland and other countries, the software packages
MATHEMATICA and MATLAB have been used, relying on the calculus and graphic facilities of these
tools. In this sense, there is a line starting from Sraffa’s circularity models to their formal description by
matrices, referring to the needed Theorems, to the calculations with the help of modern software and the
interpretation of the numerical results.
Our experience shows that without a rigorous application of matrix algebra it is impossible to de-
scribe completely the circular economy as a closed system in any dimension. The above verbal descrip-
tion of the most elementary Sraffa price model illustrates precisely this fact. It is essential to explain what
domains of mathematics are on the basis of the calculations that are realized here: It is the group of the
Perron-Frobenius theorems. In our book we completed the variants of the different Lemmas and Theo-
rems belonging to this group of propositions, and we associated the different proves, when this seemed to
be necessary. We have been inspired by the standard oeuvre of Gantmacher [8].
The following model concept has been introduced to underline the applicability on free markets. Eve-
ry commodity, for example wheat or iron, is characterized in a free economy by four attributes: the quan-
tity, the price, the value and the object. The term object has to be understood in such a way that there ex-
ists in a free economy for each commodity more than one realisation, so that the buyer is free to choose
one of the possible item / industry / branch. This typically happens with the acquisition of a bicycle. Fi-
nally, the buyer decides to acquire a precise type of bicycle, an individually realized item or object from a
specific trade mark, a specimen. For each purchase and sale of a commodity there are therefore four de-
grees of freedom: the choice of the object e, the price p, the quantity q and the value x. This fundamental
concept leads to a renewal of the algebraic presentation.
This concept of four attributes has been applied to symmetric national Input-Output Tables (IOT),
which are composed of sectors (homogeneous CPA branches), producing CPA product groups. Sraffa’s
industries correspond to these sectors, Sraffa’s products to the CPA-product groups. The abbreviation
CPA means: Statistical Classification of Products by Activity in the European Economic Community.
J.-F. EMMENEGGER, D. CHABLE †, H.A. NOUR ELDIN, H. KNOLLE
90 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
TABLE 1. Designations of the sectors of the SWISS IOT 2014
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 91
Actually, one had in general only two levels of matrices, one for the commodity flow matrix and one
for the coefficient’s matrix. This structure has to be enlarged to four levels (4 Input Output matrices Z, T,
S, D, 4 state matrices A, B, C, D, 4 market activity vectors x, p, q, e). This mathematical completion and
rounding off are developed in our book and presented in graphical form as follows (Fig. 1):
FIG. 1
Sraffa`s production of commodities by means of commodities (PCMC) with the production cycles in
physical terms and in monetary terms. The row-sums of the matrices Z and S, and the stochastic produc-
tion matrix D of the inter industrial production determine q and x. The components of the vector x repre-
sent the values of inter industrial production, while the components of the vector q represent the quantity
of each product group. The elements of the technology matrix D represent the influence of every sector on
the technology of the other sectors.
J.-F. EMMENEGGER, D. CHABLE †, H.A. NOUR ELDIN, H. KNOLLE
92 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
The matrix and vector relationships of the inter industrial market (see Tabl. 2). The elements of the
Input-Output matrix Z indicate how the value vector x is composed, while the elements of the matrix T
determine how the price vector p is composed. The same statement is valid for the elements of the matrix
S determining the dependence of the production quantity vector q on the production of other sectors. The
positive vectors x, p, q and the technological object vector e are the Perron-Frobenius eigenvectors
(PF-eigenvector in the Table) of the corresponding state matrices A, B, C, and D.
TABLE 2. The Stochastic Similarity Table of Interindustrial Production
Row-sum PF-Eigenvector Stochastic similarity I/O matrices
Value x Z e = x A x = x ˆ ˆ -1
A = xDx ˆZ = xD
quantity q S e = q C q = q ˆ ˆ -1B = pDp ˆT = pD
price p T e = p B p = p ˆ ˆ -1C = qDq ˆS = qD
Object e D e = e D e = e ˆ ˆ-1
D = eDe ˆD = eD
Graphical illustration of the algebraic modelling of the Sraffa-Leontief economy and its relationships
with the inter industrial and the consumption markets. The state matrices A, B, C and D of the markets
are illustrated together with the corresponding Perron-Frobenius eigenvectors of value x, price p, quantity
q and item technology vector e (Fig. 2).
FIG. 2
Graphical illustration of the Input-Output Matrix Z (44 sectors / each sector produces one CPA
product group of exactly one price) of the Swiss Inter industrial production 2008 (510.79 Billion CHF),
(Fig. 3).
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 93
FIG. 3
Graphical illustration of the Input Output matrices Z8 and D8 - in table form - of the 8 largest sectors
of the SWISS IOT 2008, normalized by the Perron-Frobenius (PF) eigenvalue of the official Input Output
table SWISS-IOT-2008 (Fig. 4).
FIG. 4
J.-F. EMMENEGGER, D. CHABLE †, H.A. NOUR ELDIN, H. KNOLLE
94 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
FIG. 5
The matrices Z6, A6, D6 of the six largest sectors. The value vector x6 is the Perron-Frobenius ei-
genvector of the state matrix A6. It is simultaneously the row-sum of the matrix Z6. The matrix D6 is a
right stochastic matrix. Its eigenvector is the object vector e. The elements of the technology matrix D6
indicate how the technology of one sector is participating to the production technology of the other
5 sectors (Fig. 6).
FIG. 6
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 95
Sraffa`s basic industries, respectively basic products. Sraffa [4] distinguished in circular econo-
mies of n industries and n products, the case of single product industries, where each industry produces
only one product, and the case of multiple-product industries (joint production), where each industry
produces one or more products. In both cases, there is the fundamental economic notion of basic indus-
tries, respectively basic products, which are those indispensables for a given economy, as the branch of
agriculture, producing, wheat, wood, vegetables, and guarantee the existence of the whole system. On the
opposite, there are the non-basic industries, which may – historically – produce not indispensable luxuries,
as jewellery and race horses. An important question is the determination of the number m of non-basic in-
dustries among the total number n of industries, leading to the number n-m of basic industries. Sraffa ([4],
Par. 6, 60) has given in the case of single product industries and joint production, specific methods to
determine the number m. Schefold [9] has reformulated Sraffa’s method for joint production in terms of
matrix algebra.
In our book we have also picked up Schefold’s matrix method and formulated a matrix rank criterion
which allows to determine the number m. We have also put together other methods to determine the num-
ber m, as the calculation of the matrix proposed by Pasinetti, and illustrated the methods with many nu-
merical examples. We have also identified the origin of this matrix methodology as an idea, going back to
Lev Semyonovich Pontryagin (1908 – 1988) and is applied extensively in the system control theory as the
controllability-observability conditions. This is a highly mathematical concept which we present in the
Fig. 7 with subsequent explanations.
FIG. 7
Graphical illustration of basic industries (The Sraffa notion industry can be replaced by sector, if the
application concerns IOTs): The industry i generates through its corresponding vectors xi, pi, qi, ei a cyclic
space of influence represented through its characteristic polynomial Pi (λ). The same is valid for industry j
with the characteristic polynomial Pj (λ). If the two industries influence other industries, then, the dimen-
sion of all influenced industries- including their own cyclic spaces – is given by the product P (λ) of three
polynomials. The two polynomials, associated to industries, are therefore not relatively prime. The blue
polynomial Ψij (λ) or Ψji (λ) represents the space influenced by both industries. If the dimension of P (λ) is
n, then, the two industries are basic. If the dimension of Pi (λ) is n, then the corresponding state matrix has
a simple (cyclic) structure and can be transformed to a Frobenius form (phase state form). In such a case
the polynomial coefficients are directly the coefficients of the Frobenius form. The origin of these condi-
tions goes back to L. S. Pontryagin. There are already efficient algorithms that determine not only the
influence space, but also determine the coefficients of influence over the state matrices A, B, C and D.
Further any appropriate convex combination of suggested influence vectors can be tested about their
influence.
J.-F. EMMENEGGER, D. CHABLE †, H.A. NOUR ELDIN, H. KNOLLE
96 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
The monthly operating/circulating capital K* (normalized according to the corresponding Perron-
Frobenius-(PF) eigenvalue of the flow commodity matrix of the Input-Output table of the country) for the
USA, GBR, DE, Euro-Zone over 17 years (1995–2011), (Fig. 8).
FIG. 8
With this book Sraffa and Leontief revisited we want to close the gap in the literature just described
above.
The book contains many different interesting results, some of which will be summarized here.
a) Hassan A. Nour Eldin achieved a complete analysis of the algorithmic properties of the inter in-
dustrial market. The inter industrial economy is formulated as a boundary value problem, where the vec-
tors of value x, price p, quantity q and object e are correspondingly the Perron-Frobenius eigenvectors of
the state matrices A, B, C and D. They are simultaneously the row-sum of the commodity flow matrix Z,
the matrices T, S and D. The technology matrix D is a right stochastic matrix. It is the stochastic similarity
matrix of the state matrices A, B and C. Its elements influence therefore all other matrices and vectors.
b) There is the fundamental economic notion of basic industries, respectively basic products, which
describe those industries indispensables for an economy, as the branch of agriculture, producing, wheat,
wood, vegetables, and guarantee that the whole system holds. On the opposite, there are the non-basic in-
dustries, respectively non-basic products, which may produce not indispensable luxuries, as jewellery and
race horses. Our book presents a comprehensive matrix rank criterion to determine the number of non-
basics. Thus, one can easily determine the number of basics in a Sraffa price model.
c) The Sraffa price model, where the surplus is not divided into profit and wages, offers the possibil-
ity to compute a measure of productivity R of the described economy. The commodity flow matrix of a
symmetric national Input-Output Table (IOT) has been taken as basis of such a Sraffa price model. Has-
san A. Nour Eldin could show that the mentioned productivity measure R can be computed, without need
to determine the prices of the Sraffa model. It also shows that we are here in presence of a boundary value
problem, as they are known from physics. Thus, one has shown that the Sraffa price model can be com-
bined with Input-Output Tables to a dynamic system.
d) Helmut Knolle extended the Sraffa price model in such a way that recycling of waste products
can be involved and nevertheless the economy turns out generating profits.
e) Jean-François Emmenegger has shown that the Sraffa price model can be extended in a way that
every industry has its own rate of profits and its own wage rate, thus, one can work with real distributions
of wage rates and profit rates.
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 97
f) Daniel Chable has proposed the extensive use of graphs and directed graphs to represent the pro-
duction schemes of Sraffa price models, as shown below, see Fig. 9. They are called Sraffa networks. We
show such a network for the third elementary example of Sraffa ([4], paragraph 5), developed form the
above presented elementary example, but with a total production of 575 qr of wheat. Circles represent the
industries and squares the products. We have also adapted graph-theoretical criteria to determine the
presence of basics and non-basics in graphs and directed graphs, representing production schemes of
Sraffa price models.
FIG. 9
Our book offers a new perspective for the understanding of the inner nature of circular economy. Ma-
trix algebra reveals to be the ideal tool to describe circular economy of commodities, which appear ones
as means of production and ones as final products. There are also numerous didactical examples and ap-
plications to the Input-Output Tables (IOT) of different countries set up with complete calculations of the
solutions. The Theorems and Lemmas with the associated proves, necessary to understand the different
aspects and variants of Sraffa price models, respectively to grasp the mathematical basis, are presented in
completed form.
Acknowledgements The authors are grateful to T.A. Bardadym for help in preparation of this paper.
References
1. Emmenegger J.-F., Chable D., Nour Eldin H.A., Knolle H. Sraffa and Leontief Revisited: Mathematical Methods and
Models of a Circular Economy. Berlin, Boston. De Gruyter. 2020. 572 p. https://doi.org/10.1515/9783110635096
2. Aurez V., Georgeault L. Economie circulaire, Système économique et finitude des ressources. De Boeck Supérieur
s. a. Louvain-la-Neuve. 2016 and 2019.
3. Leontief W.W. Die Wirtschaft als Kreislauf. Archiv für Sozialwissenschaft und Sozialpolitik. 1928. 60. P. 577–623.
4. Sraffa P. Production of commodities by means of commodities. Cambridge. Cambridge University Press. 1960 (and the
German edition, Verlag Suhrkamp. 780. 1976, with an appendix of Bertram Schefold).
5. Foley D.K. Adam’s Fallacy, A guide to economic Theology. The Belknap Press of Harvard University Press. 2006.
https://doi.org/10.4159/9780674027077
6. Knolle H. Die Wachstumsgesellschaft, Aufstieg, Niedergang und Veränderung. Köln. PapyRossa. 2016.
7. Newman P., Arbor A. Production of commodities by means of commodities. Schweizerische Zeitschrift für Volks-
wirtschaftslehre und Statistik. 1962. 98. Jg. P. 58–75.
8. Gantmacher F.R. Matrizentheorie. Berlin. Springer-Verlag. 1986. https://doi.org/10.1007/978-3-642-71243-2
9. Schefold B. “Mr. Sraffa on Joint Production and other Essays”. London: Routledge. 1989.
https://doi.org/10.4324/9780203168721
Received 03.05.2020
https://doi.org/10.1515/9783110635096
https://doi.org/10.4159/9780674027077
https://doi.org/10.1007/978-3-642-71243-2
https://doi.org/10.4324/9780203168721
J.-F. EMMENEGGER, D. CHABLE †, H.A. NOUR ELDIN, H. KNOLLE
98 ISSN 2707-4501. Кібернетика та комп'ютерні технології. 2020, № 2
Emmenegger Jean-François,
Dr. math., lecturer in mathematics and economic statistics at the Department of Quantitative Economics,
University of Fribourg, Switzerland, emeritus, Marly, Switzerland,
https://orcid.org/0000-0001-9581-2455
Jean-Francois.Emmenegger@unifr.ch
Chable † Daniel L., (1942 – 2018)
dipl. math. ETHZ, Vice-Director and Chief actuarial mathematician, Nestlé S. A., Vevey, Switzerland,
Nour Eldin Hassan Ahmed,
Prof. Dr. sc. tech., Chair of automatic control and technical cybernetics University of Wuppertal, Germany, emeritus,
Dübendorf, Switzerland,
Knolle Helmut,
PD Dr., master of conference in Columbia, bio mathematician, expert at the Federal Office of Health in Bern, emeritus,
Wohlen near Bern.
УДК 519.86, 330.44
Ж.-Ф. Емменеггер, Д. Шабль†, Г.А. Нур Ельдін, Г. Кнолле
Повертаючись до Сраффи та Леонтьєва:
математичні моделі та методи кругової економіки
Листування: Jean-Francois.Emmenegger@unifr.ch
Вступ. Іноді нові результати в одній науковій області можуть допомогти у вивченні зовсім інших
областей. Нова книга – чудовий приклад застосування різних математичних методів до вивчення круго-
вої економіки.
Мета роботи – дати інформацію про нову книгу «Повернення до Сраффи та Леонтьєва. Матема-
тичні методи і моделі кругової економіки». Ця монографія була видана в січні 2020 англійською мовою
німецьким науковим видавництвом Вальтер де Гройтер (Walter de Gruyter-Oldenbourg).
Результати. Ця книга – свого роду відповідь на зростаючий запит щодо всестороннього розгляду
процесу економічного виробництва. У ній розглянуто підходи Василя Леонтьєва до аналізу таблиць
«витрати-випуск» і матеріали видатної книги П'єро Сраффа «Виробництво товарів за допомогою това-
рів». Однопродуктові галузі та спільне виробництво у круговій економіці описуються за допомогою
апарату матричної алгебри. Загальною основою для дослідження підходів Сраффи і Леонтьєва служить
теорема Перрона-Фробеніуса для невід'ємних матриць, а також їхніх власних значень і власних
векторів, що використовується для виявлення основних економічних передумов, притаманних процесу
економічного виробництва.
Висновки. Книга адресована як молодим дослідникам, що бажають вивчати основи економіки, так
і досвідченим працівникам, що мають намір вивчити потенціал цінових моделей Сраффи, пов'язаних
з аналізом таблиць „витрати-випуск” Леонтьєва. Студенти старших курсів та випускники, аспіранти
та їх викладачі економіки, політології чи прикладної математики, що прагнуть зрозуміти надбання
Сраффи та результати сучасних досліджень кругових процесів у міжгалузевих обмінах та економіці
в цілому, знайдуть численні приклади із повними розв’язками та багату формальну математичну мето-
дологію, що розкриває економічний зміст результатів. Застосування різних математичних методів
супроводжується великою кількістю прикладів та ілюстративного матеріалу.
Ключові слова: модель «витрати-випуск», кругова економіка, теорема Перрона-Фробеніуса,
невід’ємна матриця.
https://orcid.org/0000-0001-9581-2455
mailto:Jean-Francois.Emmenegger@unifr.ch
mailto:Jean-Francois.Emmenegger@unifr.ch
mailto:Jean-Francois.Emmenegger@unifr.ch
mailto:Jean-Francois.Emmenegger@unifr.ch
SRAFFA AND LEONTIEF REVISITED: MATHEMATICAL METHODS ...
ISSN 2707-4501. Cybernetics and Computer Technologies. 2020, No.2 99
УДК 519.86, 330.44
Ж.-Ф. Эмменеггер, Д. Шабль †, Х.А. Нур Эльдин, Г. Кнолле
Возвращение к Сраффе и Леонтьеву:
математические модели и методы круговой экономики
Переписка: Jean-Francois.Emmenegger@unifr.ch
Введение. Иногда новые результаты в одной научной области могут помочь в изучении совсем
других областей. Новая книга – замечательный пример применения различных математических методов
к изучению круговой экономики.
Цель работы – дать представление о новой книге «Возвращение к Сраффе и Леонтьеву. Матема-
тические методы и модели круговой экономики». Эта монография была издана в январе 2020 на англий-
ском языке немецким научным издательством Вальтер де Гройтер (Walter de Gruyter-Oldenbourg).
Результаты. Эта книга – своего рода ответ на растущий запрос относительно всестороннего
рассмотрения процесса экономического производства. В ней рассматририваются подходы Василия
Леонтьева к анализу таблиц "затраты-выпуск" и материалы выдающейся книги Пьеро Сраффа «Произ-
водство товаров с помощью товаров». Однопродуктовые отрасли и совместное производство в круговой
экономике описываются с помощью аппарата матричной алгебры. Общей основой для исследования
подходов Сраффы и Леонтьева служит теорема Перрона-Фробениуса для неотрицательных матриц и их
собственных значений и собственных векторов, что используется для выяснения основных экономиче-
ских предпосылок, присущих процессу экономического производства.
Выводы. Книга адресована как молодым исследователям, желающих изучать основы экономики,
так и опытным работникам, которые намерены изучить потенциал ценовых моделей Сраффы, связан-
ных с анализом таблиц «затраты выпуск» Леонтьева. Студенты старших курсов и выпускники, аспиран-
ты и их преподаватели экономики, политологии или прикладной математики, стремящиеся понять до-
стижения Сраффы и результаты современных исследований круговых процессов в межотраслевых об-
менах и экономике в целом, найдут многочисленные примеры с полными решениями и богатую фор-
мальную математическую методологию, раскрывающий экономическое содержание результатов. При-
менение различных математических методов сопровождается большим количеством примеров
и иллюстративного материала.
Ключевые слова: модель «затраты-выпуск», круговая экономика, теорема Перрона-Фробениуса,
неотрицательная матрица.
mailto:Jean-Francois.Emmenegger@unifr.ch
mailto:Jean-Francois.Emmenegger@unifr.ch
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