72 www.r-economy.ru R-ECOMONY, 2018, 4(2), 72–78 doi: 10.15826/recon.2018.4.2.011 Online ISSN 2412-0731 Original Paper FOR CITATION Petrenko, D. S. (2018) Inframarginal models of spatially allocated economic structures and the analysis of production processes. R-economy, 4(2), 72–78. doi: 10.15826/recon.2018.4.2.011 FOR CITATION Петренко, Д. С. (2018) Инфрамаргические модели пространственно разнесенных экономических структур и анализ производственных процессов. R-economy, 4(2), 72–78. doi: 10.15826/recon.2018.4.2.011 doi: 10.15826/recon.2018.4.2.011 Inframarginal models of spatially allocated economic structures and the analysis of production processes Dmitry S. Petrenko Regional Development Center Ekaterinburg, Central Bank of the Russian Federation, Ekaterinburg, Russia; e-mail: zlobec@gmail.com ABSTRACT The article discusses designing of labor division networks. Designing of the economic structure of labor division constitutes the main part of infram- aginal analysis. Inframaginal analysis normally uses predefined economic structures, which means that in certain cases some economic structures may be neglected. Such inaccuracies may be not important in the analy- sis of small enterprises but in the analysis of spatially allocated economic structures, some important aspects may be left unnoticed, which will lead to wrong decisions regarding labor allocation. To make an enterprise compet- itive it is essential to understand what is the optimal economic organization and the form of labor division in the given region. If some economic struc- tures are not taken into account in the analysis, the general equilibrium will be incorrect, which will negatively affect the decision-making. If we use inframarginal models to analyze the production process, it will allow us to take a fresh perspective on the problem. All possible structures of the divi- sion of labor are designed by using production factors and goods to reduce the risk of errors in the process of decision-making, which will make the production process of the enterprise more efficient. KEYWORDS inframarginal analysis, technology, division of labor, network effects, economic structures, regional economy Инфрамаргические модели пространственно разнесенных экономических структур и анализ производственных процессов Д. С. Петренко Региональный центр развития «Екатеринбург», Центральный Банк Российской Федерации, Екатеринбург, Россия; e-mail: zlobec@gmail.com РЕЗЮМЕ В статье обсуждается проектирование сетей разделения труда. Проек- тирование экономической структуры разделения труда составляет ос- новную часть инфрамагинального анализа. Инфрамагинальный анализ обычно использует предопределенные экономические структуры, а это означает, что в некоторых случаях некоторыми экономическими струк- турами можно пренебречь. Такие неточности могут быть не важны при анализе малых предприятий, но при анализе пространственно распре- деленных экономических структур некоторые важные аспекты могут остаться неучтенными, что приведет к неправильным решениям отно- сительно распределения рабочей силы. Чтобы сделать предприятие кон- курентоспособным, важно понять, что является оптимальной эконо- мической организацией и формой разделения труда в данном регионе. Если в анализе не учитываются некоторые экономические структуры, общее равновесие будет неверным, что негативно скажется на процессе принятия решений. Если мы используем инфрамаргинальные модели для анализа производственного процесса, это позволяет нам взглянуть на проблему с новой точки зрения. Все возможные структуры разде- ления труда разработаны с использованием факторов производства и товаров для снижения риска ошибок в процессе принятия решений, что сделает производственный процесс предприятия более эффективным. КЛЮЧЕВЫЕ СЛОВА инфрамаргинальный анализ, технологии, разделение труда, сетевые эффекты, экономические структуры, региональная экономика http://doi.org/10.15826/recon.2018.4.2.011 http://doi.org/10.15826/recon.2018.4.2.011 R-ECOMONY, 2018, 4(2), 72–78 doi: 10.15826/recon.2018.4.2.011 73 www.r-economy.ru Online ISSN 2412-0731 Introduction There are two types of business decisions: decisions associated with the choice of activity and decisions of resource allocation. Decisions of the first type can be illustrated by the choice of majors students make when entering the uni- versity. These are inframarginal decisions. Then students choose the courses they want to study and decide on the time they want to spend on each of the learning courses. These are decisions of the second type – marginal decisions of time allocation. In the context of the division of la- bor, inframarginal decisions are more important than marginal decisions. In most cases of inframarginal analysis, a set of economic activities which can be chosen by individuals is set exogenously and infram- arginalists are concerned with the problem of mathematical optimization of utility functions [4,  p.  14]. The set of economic activities which can be used in the division of labor is usually limited and well known. In real life, however, managers have enough practical experience to determine the optimality of particular structures of the division of labor in various cases. Complex and specific results of inframarginal articles are not practically useful for the decision-making process, which leads to a situation when “infra- marginalists write papers mainly for inframar- ginalists” [6, pp. 177]. The technology-oriented theory of produc- tion can be divided into function analysis and ac- tivity analysis depending on the object of analysis [1, p. 1055]. Inframarginal analysis is based on activity analysis, proposed by Koopmans. Func- tion analysis was introduced by Fandel [7, p. 41] to find the types of possible economic structures in the process of inframarginal analysis. Activity was defined by Koopmans as “the combination of certain qualitatively defined commodities in fixed quantitative ratios as ‘inputs’ to produce as ‘out- puts’ certain other commodities in fixed quantita- tive ratios to the inputs” [9]. Method and model Let us now consider the asymmetric mod- el with trading activities and heterogeneous pa- rameters introduced by Yang [13, pp. 111]. In the model of specialization, there are three types of goods x, y, and z. The number of goods which are sold on the market have index s. The number of goods which are purchased on the market have index d. The self-provided goods have index c. The transaction cost coefficient is 1− k, k is viewed as a transaction service and depends on the quan- tity of labor used in transactions. As a service, it can be self-provided or purchased on the market: k = rc + rd. In this case, rc and rd as transaction services relate to the distance between a pair of trade partners and their location problems. All indi- viduals are evenly spaced and the distance be- tween each pair of neighbors is a constant. The distance between a pair of trade partners may differ from the distance between a pair of neigh- bors. For example, they can be engaged in rural or urban relations. The utility function is identical for all individ- uals and has a form of the Cobb-Douglas utility function [5, p. 337]: [ ( ) ] [ ( ) [ ( ) . ] ] c c d d c c d d c c d d u x r r x y r r y z r r z α β γ = + + + + + + × × The set of activities known to an enterprise describes the technical opportunities of this en- terprise. This set is called technology and is desig- nated by symbol T [7, p. 43]. Therefore, technology can be written the fol- lowing way: | 1, , , 0 . l x yT l x y z z r −       = = ≥         Labor restrictions are equal for all economic agents and can be written as: [ 1, 0 ], 1 , , , , . x y z r i l l l l l i x y z r + + = ∈ + = Using the theorem of optimum configuration ‘the optimum decision does not involve selling more than one good, does not involve selling and buying the same good, and does not involve buy- ing and producing the same good’ [11], we can find vectors of activities for technology T. The producer-consumer uses only one pro- duction factor l (labor) in the production pro- cesses. The economic agent can produce a good only for their own consumption xc or produce an additional part of the good for sale in order http://doi.org/10.15826/recon.2018.4.2.011 74 www.r-economy.ru R-ECOMONY, 2018, 4(2), 72–78 doi: 10.15826/recon.2018.4.2.011 Online ISSN 2412-0731 to purchase other types of goods that the eco- nomic agent does not produce on their own xs. If the economic agent does not produce a good and purchases the good on the market, we put 0 (zero) in the activity vector. All the possible activities vectors can be writ- ten the following way: , 0 0 0 , , , , 0 0 , 0 c c c c c c c c s c s c s c c c c s c s c c c c c c c l l x x y y z z r l l l l l x x x x x x x y y y y y y z z z r r r r T − −                               − − − − −               + + +         + +                                                   ′  = 0 0 0 0 0, , , , , 0 0 c c s c c c c c c c c c c s c s c s c c c c c s c s c s l x y y z l l l l l l x x x x y y y y z z z z z z z z r r r r r r r r −      +         − − − − − −                                                           + + +                      + + +            0 0 0 0, , , , , 0 0 0 0 0 0 0 0 0 0 , 0 c s c s c s c c c s c s c s c c c c c c s l l l l l l x x x x x x x y y y y y y y z z r r l l x y z z  − − − − − −                       + + +             + + +                                                           − −         +     0 0 0 0 0 0, , , , 0 0 0 0 0 0 0, , 0 0 0 0 0 c c c c s c s c c c s c s c s c s c s c s l l l l x y z z z z z r r r r r r r l l l x x y y z z − − − −                                                 + +                   + + +          − − −        +     +        +           . , 0 c c c s l x y r r                                                                                −                          +      Each element in matrix T represents the pro- duction function of an economic agent. The eco- nomic agent can choose any production function. The agent’s choice represents their production process, and it will be an inframarginal choice. For each activities vector in matrix T we will find cases in which the utility functions of the eco- nomic agent will be positive. Combinations of activities vectors will give different types of eco- nomic structures. Some of these were reviewed earlier [13,  p.  115] and we will use them to show the method of construction of economic structures from technology matrix T. Results The simplest case is autarky: an individual self-provides three goods. Therefore, the num- ber of goods sold and purchased and the number of transaction services are 0. The technology has only one activity vector . 0 c c c l x yT z  −         =              The utility function can be written as: 0.c c cu x y z α β γ= > In this case only one activity vector is sufficient to achieve a positive value of the utility function and there is no network of labor division. The pattern of labor division is shown as a graph [2] in Figure 1. A[y] [x] [z] Figure 1. Autarky Activities 0 0 0 0, , , , , 0 0 c s c s c c c c s c s c c c c s c s c c c c c c l l l l l l x x x x x x y y y y y y z z z z z z r r r r r r  − − − − − −                        + +             + +                         + +                                     exist in cases of partial division of labor. In this case an individual sells one of the produced goods and purchases one of the goods for consumption. The utility function should be positive. For an in- dividual with the activity vector 0 c s c c l x x z r −   +            there should exist an individual with the activity vector 0 c s c c l y y z r −      +         http://doi.org/10.15826/recon.2018.4.2.011 R-ECOMONY, 2018, 4(2), 72–78 doi: 10.15826/recon.2018.4.2.011 75 www.r-economy.ru Online ISSN 2412-0731 and so on for activities 0 , . 0 c s c c s c c c l l x x y y y z r r − −       +     +                   Individuals form an economic structure with pro- duction processes described by technology 0 0 , c s c s c c c c l l x x y yT z z r r  − −        +     +=                        (see Figure 2). x/y [y][x] [z] y/x [z] [r] [r][xs] [xd] [yd] [ys] Figure 2. Partial division of labor The complete division of labor (Figure 3) is represented by technology 0 0 0 0, , 0 0 c s c s c s c c c l l l x x y yT z z r r r  − − −            +       +=              +                   with three activities vectors. In this case, individ- uals produce only one of the goods and purchase two on the market. The transaction service is self-provided. y/xz [y] [x] [z][r][r] [xs] [xd] [y d][y s] z/xy [r] Market [yd][zd] [zd] [xd] [zs] x/yz Figure 3. Complete division of labor For a complete production process, an indi- vidual with the activity vector 0 0 c s c l x x r −   +            needs two other activities: 0 0 c s c l y y r −      +         and 0 0 . c s c l z z r −         +     If all of these activities vectors are present, the utility functions for all individuals are positive and there is division of labor. Partial division of labor and transaction ser- vices can be represented by the combination of the following activities vectors: 0 0 0 0, , ; 0 0 c s c s c c c c s l l l x x y yT z z z r r  − − −            +       +=                          +        0 0 , , ; 0 0 0 0 c s c c c c s c s l l l x x y y yT z z r r  − − −            +        =              +           +        0 0, , . 0 0 0 0 c c c c s c s c s l l l x x x y yT z z r r  − − −                    +=              +           +        y/xz [y] [x] [z]r/xy[r] x/yr [z] [z] [r][r] [y] [y] [x] [x] Figure 4. Partial division of labor http://doi.org/10.15826/recon.2018.4.2.011 76 www.r-economy.ru R-ECOMONY, 2018, 4(2), 72–78 doi: 10.15826/recon.2018.4.2.011 Online ISSN 2412-0731 The complete division of labor and transac- tion services can be represented by the combina- tion of the following activities vectors: 0 0 0 0 0 0, , , . 0 0 0 0 0 0 c s c s c s c s l l l l x x y yT z z r r  − − − −                +         +=                  +               +          y/xz [y] [x] [z][r][r] [x] [x] [y] [y] z/xy [r] r/xyz [y][z] [z] [x] [z] x/yz [r][r] [r] Figure 5. Complete division of labor and transaction service Discussion In the case of a complete production process, there should be four individuals who produce one type of goods or transactional service. It is easy to show that the following activity vectors , , 0 0 0 c s c c c c s c c c c s l l l x x x x y y y y z z z z  − − −            +       +             +                   cannot be a part of the production process and a part of labor division because it is impossible to find individuals with the corresponding activity vectors with the positive utility function for these cases. These four basic forms of the division of labor (autarky, partial division of labor, complete division of labor, and complete division of labor and transac- tion service) were discussed by X. Yang and Wai-Man Liu [13, p. 115], but the following activities vectors 0 0, , 0 c c c c c c c s c s c s l l l x x y y z z r r r r r r  − − −                                              + + +        were not considered. Activity vector 0 c c c s l y z r r −             +  can exist in the following production process: 0 0 0, , . 0 0 P c s c c sx c c c c s l l l x x y y yT z z z r r  − − −            +       +=                          +        The economic structure for this technology is showed in Figure 6. Technology Txp is character- ized by the production process with an intermedi- ate product. We can see that y is the intermediate product and x is the final product because all indi- viduals consume x and y is used for production x. x/yr [y] [x] y/xr [r] Market r/x [x] [r] [x] [r] Figure 6. Division of labor with the intermediate product The utility function for configuration x/yr can be written as: / ( ) .x yr c d d cu x r y z α β γ= The utility function for configuration y/xr can be written as: / ( ) .x yr d d c cu r x y z βα γ= The utility function for configuration r/x can be written as: ( ) .c d c cu r x y z α β γ= Another production process with activity vector 0 c c c s l y z r r −             +  http://doi.org/10.15826/recon.2018.4.2.011 R-ECOMONY, 2018, 4(2), 72–78 doi: 10.15826/recon.2018.4.2.011 77 www.r-economy.ru Online ISSN 2412-0731 can be written the following way: 0 0 0 0 0, , , . 0 0 0 0 0 F c s c c sx c c s c s l l l l x x y y yT z z z r r  − − − −                +         +=                  +               +          In this case, all individuals decide to spe- cialize in production of final goods. An individ- ual who provided a transactional service makes a decision of partial specialization and purchas- es final product x. [x] y/xzr [y] [z] [r] Market [x] [x] [x] [y] [y] [r] [r] [z] [z] z/xyr x/yzr r/x Figure 7. Division of labor This economic structure can exist if the trans- action service is different for other types of goods. Conclusion Analysis of the technological matrix makes it possible to find all economic structures for a giv- en set of production factors and goods. We can see that all types of predefined economic struc- tures can be found with the help of the technology matrix. We have also considered two economic structures with nonsymmetrical abilities, which were not considered in the initial formulation of the problem. 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Petrenko – Regional Development Center Ekaterinburg, Central Bank of the Russian Federation (Ekaterinburg, Russia); e-mail: zlobec@gmail.com. http://doi.org/10.15826/recon.2018.4.2.011 http://doi.org/10.1016/j.ejor.2016.02.053 http://doi.org/10.1109/TST.2014.6787371 http://doi.org/10.1109/TST.2014.6787371 http://doi.org/10.1111/j.1749-124X.2014.12086.x mailto:zlobec@gmail.com