Theoretical and scientifical journal 36 No. 1 / 2019 EXPERIENCE OF THE INPUT-OUTPUT MODELS APPLICATION TO THE MOLDOVAN ECONOMY Elvira NAVAL1, Doctor of informatics, Coordinating Researcher, Institute of Mathematics and Computer Science “Vladimir Andrunachievici”, Republic of Moldova The main goal of this article is to present an overview of the input-output models which have been applied to Moldovan economy development study. We examined: static input-output model, dynamic input- output model restricted by limited energy resources, and the Markov chain approach based on the input- output tables. All these models have been examined using statistic data referring to the input-output table, constructed on the base of 19 and 16 aggregated branches of the national economy. Static and dynamic optimization models were formulated, simulation calculation was done and analysed. Input-output table balancing problem was solved using RAS method. For dynamic model matrix of the investment coefficients was constructed. The emphasis was put on the problem of applying the theory of Markov chain for examination of the 19 and 16 branches in the framework of the input-output model for Republic of Moldova. A square exchange matrix of order nn  has been constructed. Every branch was considered as one state of the Markov chain with n states. We introduced a new )1( n -th absorption state so that the examined matrix became of the order )1()1(  nn . The obtained transition matrix – probabilities matrix has been used for forecasting. Keywords: input-output models, static optimization model, dynamic optimization model, Markov chain, exchange matrix, investment matrix, transition matrix, forecasting. Obiectivul principal al acestui articol constă în prezentarea unei sinteze asupra modelelor interramurale utilizate în studierea dezvoltării economice a Moldovei. Au fost examinate modelul de optimizare static şi modelul de optimizare dinamic restricţionat de resurse energetic limitate, la fel şi abordarea stocastică bazată pe lanţurile Markov, obţinute în baza tabelelor intrări-ieşiri. Modelele menţionate au fost dotate cu date statistice în vederea construirii tabelelor intrări-ieşiri, având la bază 19 şi 16 ramuri agregate ale economiei naţionale. Modelul static şi modelul dinamic de optimizare au fost formulate, calculele de simulare în baza lor au fost efectuate şi analizate. Problema balansării tabelelor intrări-ieşiri a fost soluţionată prin aplicarea metodei RAS. Pentru modelul dinamic s-a construit matricea coeficienţilor investiţionali. Accentul a fost pus pe problema aplicării lanţurilor Marcov la examinarea a 19 şi 16 ramuri în cadrul modelului intrări-ieşiri pentru Republica Moldova. Matricea pătrată a cheltuielilor materiale directe de ordinul nn  a fost construită. Fiecare ramură fiind considerată ca o stare a unui lanţ Markov cu n stări. A fost adăugată o stare absorbantă încât matricea de tranziţie s-a transformat într-o matrice de ordinul )1()1(  nn . Matricea de tranziţie obţinută – matricea de probabilităţi, s-a folosit în scopuri de previziune. Cuvinte-cheie: modele intrări-ieşiri, model de optimizare static, model de optimizare dinamic, lanţuri Markov, matricea cheltuielilor materiale directe, matricea investiţională, matricea de tranziţie, previziune. Главная цель настоящей статьи заключается в предоставлении обзора межотраслевых моделей, используемых для изучения экономического развития Молдовы. Были рассмотрены оптимизационная статическая модель и динамическая модель с ограничениями на энергетические ресурсы, а также стохастическая модель, основывающаяся на цепях Маркова, построенных на основе таблиц затраты-выпуск. Рассмотренные модели были снабжены статистическими данными, необходимыми для построения таблиц затраты-выпуск по 19 и 16 агрегированным отраслям национальной экономики. Были сформулированы статическая и динамическая оптимизационные модели, по которым были проведены и проанализированы имитационные расчеты. Таблицы затраты-выпуск были сбалансированы с помощью метода RAS. Для динамической модели была построена матрица коэффициентов инвестиционных затрат. Основной акцент был поставлен на применение Марковских цепей для изучения 1 © Elvira NAVAL, elvira.naval@math.md mailto:elvira.naval@math.md ECONOMY AND SOCIOLOGY 37 No. 1/ 2019 19 и 16 агрегированных отраслей в рамках модели затраты-выпуск для Республики Молдова. Каждая отрасль ассоциировалась с одним из состояний цепи Маркова с n состояниями. К рассматриваемой цепи добавилось еще одно, абсорбированное состояние так, что переходная матрица приобрела порядок )1()1(  nn . Построенная переходная матрица – вероятностная матрица была использована в целях прогнозирования. Ключевые слова: модели затраты-выпуск, статическая оптимизационная модель, динамическая оптимизационная модель, цепи Маркова, технологическая матрица, инвестиционная матрица, переходная матрица, прогнозирование. DOI: https://doi.org/10.36004/nier.es.2019.1-03 JEL Classification: C61, C68 UDC: 330.45 Introduction. For empirical applications, the input-output table has the same importance as a mathematical model. Most analyses start from tables in money units and devote a great deal of effort to making sure that the total money value of each row is equal to that of the corresponding column before deriving a coefficient matrix. By contrast, Leontief [1-4] stressed the technological interpretation of each column of coefficients and urged the collaboration of economists with engineers and other technological experts to project, column by column, coefficient matrices representing hypothetical changes in technologies in different industries based on the information in physical units. Scientific approach is based on the modeling modern theory, relied on the input-output tables approach. Interbranch models are made up from matrices of direct material expenditure of one producing industry for other producing industries. Method for interbranch balances construction has been proposed and implemented by Russian scientist Vasilii Leontief [1-2]. In the year 1972 V. Leontief was awarded the Nobel Prize in economic science “for elaboration of the input-output method and its application for solving important economic problems”. In time interbranch models passed through spectacular evolution, being studied both deterministic and stochastic models; static and dynamic models with lagged capital investment (one or more years), simulation and optimization models. Interbranch models are largely applied to economic development examination taking in account environmental medium. Such models have been used for world and regional economy evaluation, as a forecasting and indicative planning tool for medium and long term. Now these models are used in many industrial developed countries, but also in some less developed countries. The purpose and scientific basis of the research The present research deals with the overview of author`s research experience in the interbranch modeling and the last research related to the Markov chain applications relied on input-output table. They studied deterministic static [6-8] and dynamic [9-10] optimization models, general equilibrium models, stochastic models [11]. Empirical researches, simulation calculations and obtained results analysis have been effectuated. Father, general theory of the input-output models will be followed [18]. Scientific basis of the research is the input-output Leontief model [4], which consists of n industries n,...,2,1 . The i - th industry requires an amount 10  ij a of goods (in physical units) from industry j to produce 1 dollar's worth of goods. The external demand for the industries goods, in physical units, is given by the vector ),..,,( 21 n yyyy  . Let A be the )( nxn matrix of inter-industry coefficients ij a , the industries produce total amounts given by the vector ),..,,( 21 n xxxx  and F is the matrix )( kxn of factor input per output (one row for each factor), and f is total factor use. Then the basic static input-output model looks as: yxAI  )( , or yAIx 1 )(   (1) .Fxf  (2) The amounts of goods which the industries will need just to meet their internal demands, is given by the vector xA . In order to meet the external demand y and the internal demands the industries must produce total amounts given by a vector .),..,,( 21 n xxxx  which satisfies the equation yxAx  . The inverse matrix 1 )(   AI has been called the Leontief inverse, also known as the multiplier matrix or matrix of multipliers. Theoretical and scientifical journal 38 No. 1 / 2019 Equations (1) and (2) are a quantity input-output model, and the coefficients in A and F matrices are ratios of physical units. If y is given, the solution vector x represents the quantities of sector`s outputs. Assume that each industry’s output is measured in a unit appropriate for that sector. A mixed-unit flow table accommodates variables measured in different units and can be constructed with no conceptual difficulty. In the coefficient matrix A derived from such a flow table, the thij  element is equal to the th ij element of the flow table divided by the th ij row total. So, mixed unit A matrix may be constructed as columns of coefficients. Equations (1) and (2) represent an abbreviation of the basic input-output model. The full model involves two additional equations (where )(.  indicates transposition): orFvAIp   )( 11 )()(   AIFAIvp  (3) Fxxvyp   (4) Here, p is the vector of goods unit prices, v is value-added, the total money value of factor inputs per unit of output, and  is the vector of factors prices. Equation (4), called the income equation, is derived from Equations (1) and (3), the GDP identity assures that the value of final deliveries is equal to total value-added The Leontief inverse is strictly positive, i.e., each element is positive. It follows from basic economic logic which requires that an increase 0 y in final demand in Equation (1) should result in an increase 0 x in total output. If the matrix 1 )(   AI was not strictly positive, this logic could be violated. In consequence, Equation (1) always has a solution 0x for 0y . There are a number of equivalent statements about A : 1. 0)( 1   AI . 2.   321 )( AAAIAI (The series  k A is convergent). 3. The successive principal minors of 1 )(   AI are positive. 4. There exists a choice of units such that all row sums or all column sums of A are smaller than unity. 5. The matrix A has a dominant eigenvalue 10,   . 6. A dominant eigenvalue  of A is larger if one element of A is increased, and  gets smaller if one element of A is decreased. Statement 2 is important for distinguishing the industries contributing output in different phases of production. It says that output  )( AyAAyyx So the quantity y must be produced, plus Ay which is the vector of input to produce y, etc. Statement 3 is the well-known Hawkins-Simon condition, which assures that each subsystem is productive; that is, each subgroup of industries ,...,, kji requires less input from the economic system than it produces in terms of outputs. According to statement 4, the Brauer-Solow condition, value-added in each sector is positive in coefficient matrices derived from input- output tables in (nominal) money values. Assuming that the matrix describes a viable economy, this property assures that if output is measured in any chosen physical units, there exists a set of prices such that each industry has a positive value-added (i.e., revenue left to pay for factor inputs). The dominant eigenvalue  is a measure of the size of the intermediate outputs produced in the economy relative to total production. That is,  indicates the net surplus of an economy in the sense that the larger  (within the bounds described by statement 5), the smaller the net output. The surplus so defined can be consumed, invested for growth, devoted to environmental protection, etc. Statement 6 is useful for interpreting the role of technological change. For example, a technological innovation that reduces the need for certain intermediate inputs results in a lower dominant eigenvalue for the new coefficient matrix, leaving more surplus. Innovations that are not cost-reducing, on the other hand, will result in a larger  . Input-output analysis can effectively identify those industries where increased technological efficiency would have a significant economy-wide impact. Thus,  is a kind of efficiency indicator in that of two matrices describing two different economies, the one with a larger dominant eigenvalue represents the economy that is less efficient economically although it may have other desirable features. Eigenvalues also play an important role in dynamic models, where they have an interpretation ECONOMY AND SOCIOLOGY 39 No. 1/ 2019 in terms of rates of growth or contraction and profit rates. If the economy does not produce a surplus (i.e., 0y in Equation (1)), we deal with a closed model of the following form .Axx  In this special case, A has a dominant eigenvalue equal to unity, and total output x is the Perron- Frobenius eigenvector of A . To solve this model for x means to solve it for this eigenvector. The solution provides only the production proportions; the scale has to be determined in other ways, such as external knowledge about the size of certain elements of x . Above we interpreted the th i column of the input coefficient matrices A and F as representing the technology to make good i and claimed that the coefficients represented an average technology. This interpretation allows for the existence of differences in technology among enterprises in the same industry. The use of an average avoids the complication of having to distinguish products and technologies where the distinction does not add much useful information for the purposes of the analysis. In terms of the model, it means that a one-to-one relation is established between the typical commodities and also average technology for producing it. Suppose a new technology becomes available to produce the thi  good. The th i columns of A and F will also be balanced if the new technology is adopted by this industry. If two technologies are available for producing the th i good, the model can determine which technology is the lower-cost choice in terms of the overall use of factors. Equation (3) shows that the cost of factor use is equal to xv  or Fx  . If the new technology is cheaper in terms of overall factor inputs, it is more efficient than the old one and in principle will be installed. Because each price is the sum of the costs of the primary resources used directly and indirectly in its production, introducing the new technology will assure minimal cost. These considerations enable to formalize the economically most efficient choice of technology as a minimization problem. The non-substitution theorem formulated and proved in [7, 9] identifies the choice among several alternative technologies that minimizes the use of priced resources for each product. It can be shown that for a particular final demand y , there is a unique, cost-minimizing set of technologies, provided that the possibility of factor constraints is ignored. There are three types of impact, propagated through economy: direct, indirect, and induced that could be estimated by input-output models. Economy wide shocks deal with changing in the initial expenditure. Impact of the input changing between industries contributing to modifications in one or more specific technologies can be estimated. Data sources and utilized methods Direct material expenditure coefficients matrices were constructed on the National Accounts base in constant prices [8]. Direct investment expenditure coefficients matrix and limiting coefficients were constructed for the dynamic optimization model with one or more lags. Leontief inverse matrix was determined, Markov chain was constructed on the base of direct material expenditure coefficients matrix, transposing input-output matrices into transition one. Optimization methods, Solver application, RAS method software (proper elaboration) were utilized. 1. Own results and discussions 1.1. Static input-output model for the Republic of Moldova Input-Output table elaboration in the Republic of Moldova has been started since USSR by the State Planning Institute. Our country at that time has occupied leading position at the diverse dimensions input-output table elaboration. After tearing of the USSR, this task was incumbent on the Institute of the Market Problems and to the National Bureau of Statistics. At present, principal elements of the input- output tables could be extracted from National Accounts. Regretfully, from the year 2015 this data for input-output table elaborating was ceased. Researching of the interbranch models based on the input-output tables has been effectuated by the institutes of the Academy of Sciences of Moldova such as: Institute of Mathematics and Computer Science “Vladimir Andrunachievici”, National Institute of Economic Research and the Institute of Energetic, which shows a major interest in the interbranch models examination. Relying on the data in current prices for the 23 branches from the National Accounts of the Republic of Moldova, years 1996-2014, input-output table in constant prices in their classic form was constructed. Namely, for mentioned years, information was selected and processed. This data has formed I-III quadrant of the interbranch balances in constant prices. Theoretical and scientifical journal 40 No. 1 / 2019 Having input-output tables for year 2014 in the structure of the n (23) examined branches the model of the economic development will be formulated and solved. Four branches: Public administration and defense (L), Associate activities not included in other activities (O91), Recreation, culture and sport activities (O92) and Other activities and services (O93) are not productive branches. Having zero over the line, these branches consume only, so do not affect the matrix of direct material expenditure, thus may be excluded from the interbranch balance. The interbranch static and dynamic models have been studied in [6-11], based on it, much simulation was done. Further static optimization problem based on the input- output table for the year 2014 will be formulated. An input-output table (year 2014) for the 19 productive aggregate industries of the Moldovan economy is principalelement when static optimization problem is formulated. Our country being in profound scarcity of the proper sources of energy, is imposed to import it in sufficient large proportions for covering own necessities. As a result, both growing of the world prices of energy resources and internal tariffs growing, contribute to the domestic prices of the energy resources changing. This in its turn affects both production sector in total and households dramatically influencing country energy security and wellbeing of the population, being on the limit of poverty. So, the problem of energy tariffs growth studying, in this context has major importance. Interbranch balances can be very useful in the structure of 19 productive aggregate industries, from which aggregate industry of Electric energy, gase and water being one on the whole based on import. So, we will research the influence of the energy tariffs growing on the economy as a whole and on the population in special using interbranch static optimization models. Suppose that energy tariffs grow 1.5 times, then components of technology vector will grow at the same rate. In such circumstances, what will be the impact on Gross Domestic Product? Knowing output volume X in the one specified year t having modified electric industry technology vector (E), mentioned problem will be formulated as a problem of final consumption optimization. It can be mentioned that both elements of the input-output tables, output vector, and final consumption vector are measured in constant prices. So, the following static optimization problem needs to be solved: maximize final consumption when energy resources tariffs are increasing (changing vector-column (E) and fixing output volume vector value). Then formalized model is written as:   19 1 max i i y , (5) subject to the following restriction XYAI   1 )( , (6) here X is the output vector volume which is known and Y is the final demand vector, that must be maximized or   19 1 max i i x (7) subject to the following restriction YAXX  . (8) Let`s examined matrices of the direct material expenditure A in money values. Elements of these matrices are less than one and its sum along the column and line is strictly smaller than one. Matrix A , constructed from statistical data [8] for year 2014 satisfied partly the earlier exposed particularities, namely, the sum along the column is less than one, but the sum along the line for some industries for example (Processing industry) is greater than one. In such circumstances inverse matrix existence do not done. Such phenomena implies problem in satisfying restriction (6). This imposed to done simulations based on the problem (7), changing prices following restriction (8). 1.2. Simulation`s calculations Table 1 Expenditure changes in respect to price modification Expenditure E 28500717 2843392 31307171 30698998 30359116 29846954 29211061 28548610 29411015 28962435 Resources X 221303904 4298115 1836452 192390282 201135294 218661243 221303904 227370333 236115346 244860358 Prices 1 1,05 1,1 1,15 1,2 1,25 1,3 1,4 1,45 1,5 Iterations 1 2 3 4 5 6 7 8 9 10 Source: Author`s calculations. ECONOMY AND SOCIOLOGY 41 No. 1/ 2019 Simulation calculations effectuated in concordance with price modifications demonstrate that price growing can be only applied till 1,3 limit, after that some components of the Gross Domestic Product become negative, challenging economy collapse. Figure 1. Resource modification relative to price changing Source: Author`s calculations. Figure 2. Modifications in Gross Domestic Product relative to price changing Source: Author`s calculations. 1.3. Conclusions It must be mentioned that through this study, the utility of the proposed tool, based on the input-output tables constructed on the money flows in constant prices [8] for years 1996-2014, was demonstrated. Two static optimization models (1)-(2) based on the data from year 2014 have been solved. The first model, given satisfied interbranch balance restrictions, starts with known volume of production in order to maximize final consumption. The second model deals with production volume maximizing, given known final consumption and interbranch restrictions satisfied. Calculations results have demonstrated that upper limits of the price increasing is equal to 30%, after that some components of the final demand become negative, which ascertained that final demand can not be satisfied. Using interbranch balance for 2014 year, it could be possible to demonstrate that energy resources tariffs growth, have negative impact on the Gross Domestic Product, which influences both economy as a hole and population especially. Such analysis can be effectuated for every industry from X10 X9 X8 X7 X6 X5 X4 X3 X2 Y0 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Theoretical and scientifical journal 42 No. 1 / 2019 those 19 examined in accordance with proposed objective. It is very unfortunate to know that beginning from year 2015 interbranch tables elaboration in the classical manner was ceased. Another examined aspect is referred to the input-output tables unbalance namely that sum over line may be greater than one, as result it is not equal to the column sum, which is less than one. This industries utilized resources exceeding more production volume, using in plenty import resources (especially energy resources) surpassing domestic production. Input-output table balancing problem is possible to be solved by using RAS method. Next, RAS method description and application follows. 1.4. Input-output matrices balancing Table 2 Sum of the direct material expenditure matrices coefficients along the columns and lines Aggregate branches  i ij a  j ij a i x i x -  i iij xa i x -  j jij xa A Agriculture, hunting economy and forestry 0.65 0.47 8194443 4307985 3617782 B Fishing 0.01 0.45 27364.6 14925 26046 C Mining 0.44 0.52 303098.6 145482 1449980 D Processing industry 5.94 0.74 15657828 4022168 20667046 E Water distribution; waste administration 0.72 0.67 1985815 650721 1498246 F Construction 0.39 0.79 5660514 1185158 822897 G Wholesale and retail trading 0.07 0.46 8310259 4528646 237941 H Hotels and restaurants 0.16 0.57 892875.4 384848 287496 160-63 Transport and storing 0.63 0.70 5439666 1636009 2207758 164 Communications 0.45 0.36 2427941 1560084 862180 J Financial activity 0.35 0.26 2011364 1492015 934737 K70 Estate transactions 0.65 0.44 3103366 1735042 1474654 K71 Cars and equipments rent 0.02 0.39 38119.96 23372 16664 K72 Computers and adjoint activity 0.13 0.48 743423.5 387512 141666 K73 Investigations - elaborations 0.05 0.34 173374.3 113617 66855 K74 Other commercial activities 0.31 0.55 1611441 719303 862941 L Public administration and defense 0 0.33 2049550 1376683 73720 M Education 0.03 0.29 2680886 1884680 47213 N Health and social assistance 0.02 0.38 2241138 1391747 121516 O90 Recovering and waste removing 0.08 0.32 306535.1 208836 0 O91 Activities not included in other categories 0 0.55 558130.2 253929 0 O92 Recreation, cultural and sport activities 0 0.55 744108.9 336151 0 O93 Other activities and services 0 0.49 277459.9 141804 3617782 Source: Author`s calculations. Table 2 offers calculations of the sum direct material expenditure matrices coefficients along the columns and lines, effectuated in accordance with model data. Showed noticed from this table that column sum over the column is less than one for all examined branches but sum over line of the Processing industry exceeds much more one unit, in such a manner encroach upon productivity property of the direct material expenditure coefficients matrices. Second observation certified that double accounting rule don`t have to stand. To surpass this phenomena one will e appeal to the RAS [13] method, well known and wide utilized automation tool for balancing input-output matrices. Main objective of the RAS method consists in column and line balancing of the input-output matrices, in other words balancing demand and use through modification and revising of initial input-output table. Main equation which is cyclic dependent on the existing discrepancy can be written as: ECONOMY AND SOCIOLOGY 43 No. 1/ 2019 new n new Rn new R new R new Cn new Cn new C AtXtXtXAtXtXtX   )()(...)()(...)()( 1111 . Here )( 1 tX new C is the new volume production vector over column at the time 1 t , A is the direct material expenditure matrix, )( 1 tX new R is the new volume production vector over line at the time 1 t . These equations have been easy solved with special soft or by utilizing Excel applications. Being known precise sums over columns and lines in practice this method is reduced to revising existent totals over columns and lines so as it coincides with that precise. The RAS adjustments can be seen as one iterative process in which columns and lines (lines and columns) have been modified successively in order to obtain correct balanced totals over columns and lines so as differences among two consecutive lines be equal to zero. At the first step ( 1 t ) (first iteration), matrix A will be replaced by the line sum in concordance with the formula: AtXtA new R new R  )()( 11 Procedure is applied to all matrix A lines. At the second step (first iteration) all column sums are recalculated in concordance with formula: AtXtA new C new C  )()( 11 . First iteration is finalized with the matrix A recalculated over columns and lines as follow )()()( 111 tXAtXtA new R new C new  . Table 3 Input-output table initial data before applying RAS procedure mil lei Data in mil lei, constant prices A B C D E F G H 160-63 164 J A Agriculture, hunting economy and forestry 1452154 4436 0 2000419 0 28031 29847 81591 0 0 0 B Fishing 0 1 0 20161 0 0 1890 1219 0 0 0 C Mining 57053 33 59 393858 322219 421803 53898 14088 17250 3304 1193 D Processing industry 2225370 6147 109014 7755851 542075 3567832 1315381 251983 2141284 325315 101408 E Electric energy, gase, water 41203 653 9484 320939 242408 26013 197846 47139 94006 146554 15007 F Construction 0 0 4076 194858 33087 103635 51275 15649 82651 53258 51162 G Wholesale and retail trading 0 0 0 66789 11179 6319 38772 2985 48556 8163 11803 H Hotels and restaurants 1564 38 521 24229 2951 35096 82510 762 71752 4154 2861 160-63Transport and storing 48850 166 30891 322883 34823 63718 564646 4542 889936 78445 16272 164 Communications 12232 165 511 88005 36106 39813 113264 22217 114752 153710 33824 J Financial activities 7177 92 701 71932 56425 27614 345550 8845 93080 13043 221442 K70 Estate transactions 13681 258 1657 167510 12464 128751 617018 23065 169002 19204 16641 K71 Cars and equipments rent 0 0 0 312 0 0 0 218 1089 65 41 K72 Computers and adjoint activity 4749 35 0 10232 5757 5489 0 4026 6928 8372 6614 K73 Investigations - elaborations 7879 0 0 30759 1819 636 0 520 2072 1408 698 K74 Other commercial activities 13756 398 545 132954 24649 6646 332874 16425 57195 40887 33512 M Education 0 6 129 8726 2396 1178 10600 1687 7788 8630 2332 N Health and social assistance 793 14 0 0 0 0 1643 0 0 0 0 O90 Recovering and waste removing 0 0 28 25243 6735 12782 24595 11066 6316 3346 4540 A Agriculture, hunting economy and forestry 3886462 12440 157617 11635660 1335093 4475356 3781612 508028 3803658 867859 519349 B Fishing 3886462 12440 157617 11635660 1335093 4475356 3781612 508028 3803658 867859 519349 C Mining 0 0 0 0 0 0 0 0 0 0 0 Theoretical and scientifical journal 44 No. 1 / 2019 Data in mil lei, constant prices K70 K71 K72 K73 K74 M N O90 Line`s sum Correct sum A 0 0 0 3762 0 7245 10298 0 3617782 3886462 B 0 0 0 0 0 626 2149 0 26046 12440 C 52932 50 1994 962 2208 57387 46117 3571 1449980 157617 D 666487 5071 153753 20905 494036 399234 538800 47098 20667046 11635660 E 171257 733 6897 3456 10361 105962 53869 4458 1498246 1335093 F 72239 345 795 11799 27980 70995 45663 3428 822897 4475356 G 14161 0 329 531 22631 1603 3974 147 237941 3781612 H 7707 44 3939 1084 20080 15227 4929 8050 287496 508028 160-63 25519 390 18008 3826 71576 15163 15754 2348 2207758 3803658 164 74832 157 57778 1416 79611 13721 13184 6883 862180 867859 J 28956 535 17736 893 12878 6272 13377 8188 934737 519349 K70 163450 6353 29377 1916 69923 23517 7954 2914 1474654 1368324 K71 127 0 0 62 77 4966 9600 108 16664 14748 K72 10639 299 51233 1032 4290 7479 12509 1984 141666 355942 K73 0 0 3078 5777 2730 4189 5291 0 66855 59757 K74 70470 483 8518 1799 60164 49925 8903 2839 862941 892138 M 1290 12 1240 107 9560 8896 7825 1317 73720 796206 N Health 0 0 0 0 0 0 44763 0 47213 849392 O90 8259 273 1267 429 4034 3799 4434 4368 121516 97699 Coloan`s sum 1368324 14748 355942 59757 892138 796206 849392 97699 Correct sum 1368324 14748 355942 59757 892138 796206 849392 97699 Balansing 0 0 0 0 0 0 0 0 Source: Author`s calculations. Then equality of the new sums over columns and lines is verified. If the equality of the new sums over columns or lines doesn’t exist, then matrix )( 1 tA new is copied and replaced instead of the matrix A . Iterations are repeated till both column sums and line sums coincide with that correct one, and namely with differences along the sum between last two iterations being equal to zero. In conclusion, for RAS procedure application it is necessary to indicate data for direct material expenditure coefficients matrix A (with n lines and n columns) and correct sums over respective columns and lines. Then the balancing procedure realized in Excel application can be applied. Table 4 Input-output table initial data after applying RAS procedure (mil lei) Iteration 63. Line transformations A B C D E F G H 160-63 164 J A 1785404 4510 0 1972238 0 33450 18718 64735 0 0 0 B 0 0 0 10872 0 0 648 529 0 0 0 C 7926 4 6 43874 33193 56872 3819 1263 1196 195 61 D 1680428 3838 54992 4696367 303547 2614945 506628 122790 807227 104328 28268 E 59883 785 9208 374035 261259 36695 146663 44211 68208 90459 8051 F 0 0 24377 1398725 219636 900423 234114 90401 369360 202468 169062 G 0 0 0 1441519 223125 165072 532281 51853 652450 93314 117271 H 4785 96 1064 59454 6697 104238 128782 1505 109614 5398 3232 160-63 147949 416 62501 784174 78210 187309 872266 8878 1345598 100901 18192 164 22464 251 627 129601 49171 70965 106096 26330 105208 119885 22931 J 7609 80 496 61149 44358 28414 186846 6051 49262 5873 86660 ECONOMY AND SOCIOLOGY 45 No. 1/ 2019 Iteration 63. Line transformations A B C D E F G H 160-63 164 J K70 21687 338 1755 212928 14652 198093 498878 23594 133744 12928 9738 K71 0 0 0 771 0 0 0 433 1675 85 46 K72 22177 134 0 38315 19935 24879 0 12132 16151 16603 11402 K73 10685 0 0 33451 1829 837 0 455 1403 811 350 K74 25595 612 677 198367 34010 12002 315903 19722 53127 32309 23017 M 0 102 1891 153604 39001 25095 118695 23898 85350 80461 18897 N 89869 1275 0 0 0 0 95028 0 0 0 0 O90 0 0 24 26217 6469 16069 16248 9250 4084 1841 2171 Coloan`s sum 3886462 12440 157617 11635660 1335093 4475356 3781612 508028 3803658 867859 519349 Correct sum 3886462 12440 157617 11635660 1335093 4475356 3781612 508028 3803658 867859 519349 Balansing 0 0 0 0 0 0 0 0 0 0 0 Iteration 63. Line transformations K70 K71 K72 K73 K74 M N O90 Line`s sum Correct sum A 0 0 0 1485 0 4267 1655 0 3886462 3886462 B 0 0 0 0 0 202 189 0 12440 12440 C 3968 4 147 43 127 3819 837 263 157617 157617 D 271589 2439 61812 5070 154948 144417 53176 18852 11635660 11635660 E 134315 679 5337 1613 6254 73773 10233 3434 1335093 1335093 F 348957 1969 3791 33920 104029 304437 53423 16264 4475356 4475356 G 205682 0 4712 4590 252999 20668 13981 2097 3781612 3781612 H 12726 86 6417 1065 25521 22321 1971 13057 508028 508028 160-63 41708 753 29037 3722 90039 21999 6236 3770 3803658 3803658 164 74160 184 56491 835 60725 12071 3165 6700 867859 867859 J 16565 361 10010 304 5670 3185 1853 4601 519349 519349 K70 139818 6413 24792 975 46037 17858 1648 2448 1368324 1368324 K71 211 0 0 61 98 7327 3865 176 14748 14748 K72 26810 889 127371 1548 8320 16731 7634 4910 355942 355942 K73 0 0 2223 2516 1538 2721 938 0 59757 59757 K74 70756 573 8437 1075 46494 44498 2165 2800 892138 892138 M 15287 174 14491 757 87169 93553 22452 15329 796206 796206 N Health 0 0 0 0 0 0 663220 0 849392 849392 O90 5773 225 874 179 2170 2357 751 2999 97699 97699 Coloan`s sum 1368324 14748 355942 59757 892138 796206 849392 97699 Correct sum 1368324 14748 355942 59757 892138 796206 849392 97699 Balansing 0 0 0 0 0 0 0 0 Source: Author`s calculations. 2. Dynamic input-output model for the Republic of Moldova 2.1. Model structure Optimization models present special interest in its application for economic problems solving. Peculiarity of the optimization models consists in the possibility to elaborate methods for efficient using of the limited resources for maintaining sustainable economic development. Further, dynamic optimization model for Republic of Moldova will be presented. Input-output table`s statistic data for Republic of Moldova have been collected from National Accounts (years 2000-2014) in the structure of 19 industrial branches) in concordance with European Union Standards. Objective function deals with maximizing the Gross Domestic Product restricted by limited energy consumption for the period of [1,T]: Theoretical and scientifical journal 46 No. 1 / 2019     T t tttt xAxexf 1 )(max Here, t x is the vector of the production volume, t A is the matrix of the direct material expenditures and e is the unity vector. Objective maximization function is followed to meet following restrictions: ttt xcC    , 211 tttttttt yyxxBxAx      ,)( 1 ttttttftt xxBexAxess     ,)1( 1 ttttt yexAxes    , 2 ttttft yexAxes  , 1211   ttt xqxxq .0, 1  tt yx In previous relationships t c is the vector of energy consumption needed for production of the one unit of output in industries examined at the moment t ; t B is the matrix of the investment coefficients; t y 1 is the final consumption vector; t y 2 is the vector of net export; t s is the rate of internal savings, equal to (0.62), ft s is the weight of the foreign capital in GDP, equal to 0.67; 1 q is the fast growth limited coefficient, equal to 1.127; 2 q is the coefficient of economic recession limit, equal to 0.873. The investment coefficients of the matrix t B were calculated according to the following formula: . 1 )1(       n j ijttj ijtit ijt ax as b Optimization model formulated earlier, endowed with determined ijt b coefficients, have been solved using Solver application and the following conclusions were done. 2.2. Calculating model parameters and forecasting Table 6 Macroeconomic indicators values for st and stf Indices Year 2010 Year 2011 Year 2012 Year 2013 Year 2014 Mean st -0.6714 -0.6961 -0.6818 -0.4089 -0.6243 -0.6165 sft 0.6714 0.6961 0.6818 0.6478 0.6243 0.6643 Source: Author`s calculations. Table 7 Energy`s Industry intermediate consumption Years Year 2010 Year 2011 Year 2012 Year 2013 Year 2014 Max Ct Ct 3664845.609 4152344 4421277 4867877 5373075 5373075 Source: Author`s calculations. Table 8 GDP forecasted values in (mil lei) Objective function Year 2015 Year 2016 Year 2017 Year 2018 GDP 104512319.9 110708443.8 117496102.2 127693108.6 Source: Author`s calculations. 2.3. Conclusions ECONOMY AND SOCIOLOGY 47 No. 1/ 2019 Economic development scenario restricted by energy saving polices have been elaborated dealing with small changes, in the limits of existing possibilities. So, following results presented in Table 9, small diminishing of the energy industry volume in the Gross Domestic Product structure and increasing weight in the services industries were confirmed. In conclusion, to reach energy saving it is necessary to develop such services industries which consume a small volume of energy resources: only human resources and energy preserving technologies. Table 9 Forecasting results (year2018) Aggregate branches Year 2015 Year 2016 Year 2017 Year 2018 A Agriculture, forestry and fishing 12,9178 14,0059 15,9075 15,9075 B Mining 0,4762 0,5163 0,5562 0,5864 C Processing industry 24,6010 20,6616 17,2410 14,0811 D Production and supplying of electric energy 3,1200 2,6404 2,1866 1,7858 E Water distribution; waste administration 1,,3585 1,4730 1,5867 1,6729 F Construction 8,8936 9,6427 10,3874 10,9519 G Wholesale and retail trading 12,7075 13,7779 14,8419 15,6485 H Transport and storing 8,5466 9,6427 9,9821 10,5246 1 Accommodation and public feed 1,4029 1,5210 1,6385 1,7275 J Information and communications 3,8597 2,5360 0,7761 0,8183 K Financial activity and insurance 2,4480 2,6541 2,8591 3,0145 L Estate transactions 4,8759 5,2866 5,6949 5,2798 M Professional, research and technical activities 0,2724 0,2953 0,3182 0,3354 N Professional services offer and support activities 1,9612 2,1264 2,2906 2,4151 O Public administration and defense 3,2202 3,4914 3,7611 3,9655 P Education 4,2121 4,5669 4,9196 5,1870 Q Health and social assistance 3,5212 3,8178 4,1126 4,3361 R Art, research and pleasure activity 1,1691 1,2676 1,3655 1,4397 S Other activities and services 0,4359 0,4727 0,3944 0,3221 N Professional services offer and support activities 1,9612 2,1264 2,2906 2,4151 M Professional, research and technical activities 0,2724 0,2953 0,3182 0,3354 O Public administration and defense 3,2202 3,4914 3,7611 3,9655 P Education 4,2121 4,5669 4,9196 5,1870 Q Health and social assistance 3,5212 3,8178 4,1126 4,3361 R Art, research and pleasure activity 1,1691 1,2676 1,3655 1,4397 S Other activities and services 0,4359 0,4727 0,3944 0,3221 Source: Author`s calculations. Table 10 Direct investment expenditure coefficients matrix Investment matrix Bt A B C D E F G H I A Agriculture, hunting economy and forestry 0,00015 0 2E-05 0 0 2E-06 9,2E-07 0 0,000168 B Mining 5,4E-10 2E-09 3E-10 2,56E-08 7,8E-09 2,8E-09 1,5E-10 7E-11 2,69E-09 C IProcessing industry 5,4E-05 0,008 2E-05 0,000109 0,00044 6E-05 9,2E-06 2E-05 0,000122 D Electric energy, thermic energy, gas, hot water, and conditional air production and supplying 1,3E-06 8E-04 9E-07 6,22E-05 7,6E-05 5,6E-07 1,8E-06 1E-06 2,91E-05 E Water distribution; sanitation, waste administration, decontamination activities 0 1E-05 3E-07 8,24E-06 5,9E-05 1,3E-06 1E-06 4E-07 3,25E-05 F Construction 0 0,005 7E-06 0,000111 0,00023 2,9E-05 6E-06 1E-05 0,000126 Theoretical and scientifical journal 48 No. 1 / 2019 Investment matrix Bt A B C D E F G H I G Wholesale and retail trading; îkeeping and repairing of cars and motocars 0 0 8E-06 0,000121 3,3E-05 5,7E-06 1,4E-05 3E-05 8,33E-05 H Transport and storing 5,2E-06 0,01 3E-06 3,09E-05 5,8E-05 4,7E-06 1,7E-05 4E-05 9,69E-06 I Accomodation and public nourisment activities 1,9E-08 2E-05 2E-08 2,87E-07 1,2E-05 2,9E-07 2,8E-07 4E-07 1,78E-07 J Information and communication 4E-07 3E-05 2E-07 8,22E-06 2,3E-05 7,5E-07 7,7E-07 1E-06 1,24E-05 K Financial and asurance activities 7,2E-08 2E-05 6E-08 4,65E-06 7,5E-06 1,9E-07 9,8E-07 4E-07 1,75E-06 L Estate tranzactions 1,5E-06 5E-04 2E-06 1,13E-05 6,9E-05 9,8E-06 2E-05 8E-06 5,02E-05 M Professional, scientific and technical activities 1,6E-07 0 6E-08 3,17E-07 0 9,3E-09 0 2E-08 2,17E-07 N Activități de servicii Administration servivies and support servicies 2,1E-07 2E-05 2E-07 3,05E-06 8,5E-06 6,9E-08 1,4E-06 4E-07 4,88E-06 P Education 6,2E-09 4E-04 8E-07 2,22E-05 0,00011 9,1E-07 3,4E-06 4E-06 3,75E-05 Q Health and social assistance 6,4E-06 0 0 0 0 0 3,7E-06 0 0 Investment matrix Bt J K L M N P Q A Agriculture, hunting economy and forestry 0 0 0 2E-05 0 3,5E-06 1,6E-06 B Mining 9,09E-11 1,3E-11 2E-09 4E-10 1,05E-10 2,4E-09 5,5E-10 C IProcessing industry 2,083E-05 2,7E-06 6E-05 2E-05 5,96E-05 4,2E-05 1,6E-05 D Electric energy, thermic energy, gas, hot water, and conditional air production and supplying 8,512E-06 5,1E-07 2E-05 5E-06 1,6E-06 1,4E-05 2,1E-06 E Water distribution; sanitation, waste administration, decontamination activities 1,219E-06 7,3E-07 4E-06 3E-06 2,96E-06 2,4E-06 8,2E-07 F Construction 3,929E-05 2,3E-05 1E-04 0,0002 5,65E-05 0,00013 2,3E-05 G Wholesale and retail trading; keeping and repairing of cars and motocars 2,002E-05 1,7E-05 7E-05 4E-05 0,000148 3,7E-05 2,2E-05 H Transport and storing 1,852E-05 1,9E-06 1E-05 2E-05 3,82E-05 7,1E-06 2,1E-06 I Accomodation and public nourisment activities 1,7E-07 3,7E-08 3E-07 6E-07 1,17E-06 7,8E-07 7,2E-08 J Information and communication 1,151E-05 1E-06 7E-06 3E-06 9,89E-06 2,2E-06 7,6E-07 K Financial and asurance activities 5,477E-07 2,4E-06 1E-06 4E-07 6,36E-07 2,7E-07 1,7E-07 L Estate tranzactions 9,518E-06 2E-06 6E-05 1E-05 3,8E-05 1,1E-05 1,1E-06 M Professional, scientific and technical activities 1,69E-07 1,6E-08 0 6E-06 2,85E-07 3,8E-07 1,4E-07 N Activități de servicii Administration servivies and support servicies 1,32E-06 5,5E-07 4E-06 1E-06 4,46E-06 3,2E-06 1,7E-07 P Education 1,972E-05 2,8E-06 5E-06 6E-06 5,3E-05 4,3E-05 1,1E-05 Q Health and social assistance 0 0 0 0 0 0 0,00045 Source: Author`s calculations. Calculations results, depicted in the considered table, also demonstrate considerable GDP growth during the examined years under sufficient decreasing of the energy resources weight of the energy industry. For the dynamic model as well as for the static model, necessary data were gathered and processed. So the data from Statistic Yearbooks for the years 2000-2015 were selected, and were completed by specific features for dynamic model. Because since year 2015, Moldovan statistics has been changed in accordance with industrial structure of the European Union, the 19 industries structure instead of 23 was adopted. The new industry structure also looks like in table 9. Examined model offered many opportunities for diverse scenario elaboration and solving both in order to modify rates of growth of the industries produce volumes, and to modify technology matrix structure, and also to increase profitability of the productive industries. Simulations calculations results obtained can be useful for decision making person’s in formulating development polices in ECONOMY AND SOCIOLOGY 49 No. 1/ 2019 particular at economy as a whole level and at the sector level in special. 3. Markov chains and exchange matrix for the Republic of Moldova Let come back to the input-output model with 16 aggregate industries which described economy of the Republic of Moldova. Industry i necessitates the quantity 10  ij a of goods (in money value) from the industry j in one lei value. Let A be the direct material expenditure coefficients matrix with its elements .16,16,  jia ij Let the demand for final consumption be vector ),..,,( 21 n yyyy  . We will make up Markov chain taking as the states technology vectors in correspondence with examined industries, and as a transition probabilities ij a elements. It is well known that the direct expenditure coefficients matrix (technology coefficients matrix) has satisfied conditions njia ij  ,1,10 and line sum is equal to nia n j ij   1,1 1 and is strictly less than one. 1, then Further transition – probabilities matrix will be constructed for the exchange matrix in the input-output model. For this purposes 16x16 exchange matrixes for Republic of Moldova (year 2014) were considered. We examine stochastic matrix, which has nonnegative elements, each column of it sums to one. Economically, the system can be interpreted either as one in which there is no demand or one in which demand is considered as an industry which consumes all its own output. Such matrices also arise in the analysis of finite Markov chains: stochastic processes in which the probability of being in a particular state at any step depends only on the state occupied at the previous step. More exactly, consider repetitive trials of an experiment with a finite number of possible outcomes n SSS ,...,, 21 . The sequence of outcomes is a Markov chain if there is a set of 2n numbers ij p such that the conditional probability of outcome j S on any trial, given outcome i S on the previous trial, is ij p ; that is ,2,1,,1),1Pr(  knjiktrialonSktrialonSp ijij The transition probabilities ij p can be arranged in a stochastic matrix as follows:                nnnn n n ppp ppp ppp P     21 22212 12111 . Let )( k i p be the probability that the outcome on the thk trial i S and Tk n kkk pppp ),,( )()( 2 )( 1 )(  be the associated probability distribution vector. Then, for Markov processes, it is true that ,2,1, )()1(   kPpp kk Thus a Markov chain is completely characterized by its transition matrix P and an initial probability distribution )0( p in the sense that ,2,1, )0()(  kpPp kk In many applications of Markov processes, one is interested in the existence of equilibrium probability distributions; that is, vector p such that Ppp  . For example, if some positive power of P is strictly positive then it can shown that there is a unique strictly positive distribution p so that ppP k k   )0( lim for any initial distribution )0( p . The existence of equilibrium distribution vectors can be proven using the Brower Fixed Point Theorem. If S is the set of all probability distributions, then S forms the standard )1( n simplex in n R and for p in S , we have:      j j i ji j j i j i j ijiiij i i pppppppPp ,1)()( Theoretical and scientifical journal 50 No. 1 / 2019 so that Pp also lies in S . Thus, we can view P as a linear transformation from S to itself. Since P is continuous. We can consider exchange matrix A as the transition matrix of a Markov chain provided only normalized price vector s productive whose components sum to one. In the open model for an economy some output is accounted for by consumer demand. Every closed model may be considered as an open model too. In terms of matrix A , this means that some columns may sum to less then 1. The system, or the matrix A , is productive if there is a nonnegative vector X (output)such that AXX  . The vector X is a production vector. Theorem. A substochastic matrix A is productive if and only if AI  is nonsingular. Corollary If A is substochastic matrix, then the equation DXAI  )( has a nonnegative solution if 0D ; a positive solution if 0D ; a strictly positive solution if .0D In the transition matrix of the Markov chain, sum over its columns is equal to one. In order to satisfy this condition by the exchange matrix A , we proceed in the following manner. An absorbent state 0, denoted as  iji aa 10 is added to exchange matrix A , obtaining the new matrix A  , line sum over every line be equal to 1. A                               1000 1 1 1 121 222221 111211      j njnnn j jn j jn aaana aaaa aaaa . For columns sum to be equal to 1, it is needed to transpose the matrix A  and so transition matrix P , is obtained, in which column sum is equal to one. P                           j j j njjj nnnn n n aaa aaa aaa aaa 1111 0 0 0 21 21 22212 12111      Principal cofactor of the matrix P is equal to ),()1( )( nnAP nn nn   , and APnnP  *1 . Suppose that X is the production volume vector in the year 2015, and Y is the final demand vector in the same year (both normalized to one), then expression for the Gross Domestic Product is YAXX  ,while vector t is absorbtion expectation time, Nct  , where 1 )(   AIN and )1,...,1,1(c is the n – dimensional vector )1,2,2,2,1,1,1,2,1,2,4,1,3,4,4,2,9,1,1,3(t .   097185947,0det  AI , then inverse matrix   1  AI exists, and matrix A is productive. There is not any positive power of P which is strictly positive for the matrix A . Right now we calculated a power of the matrix A equal to 15, for which ,0 )0(15)15(  pPp then there is a unique strictly positive distribution 0p so that ppP k k   )0( lim for the initial distribution onetonormalisedXp , )0(  . So, beginning from the initial state 2015 X and technology matrix 2015 A examined economy has been tendedto collapse. ECONOMY AND SOCIOLOGY 51 No. 1/ 2019 3.1. Conclusions Markov chain approach to the input-output models study was examined. Input-output table for year 2015 in 16 structure has been transformed into transition matrix by adding one additional column and then transposing it. After transposing transformed matrix became transition one with columns sum equal to 1. Applying known theory about Markov chain to the considered matrix, it was possible to study stability problem and time of transition from one state to another. So, effectuated calculations demonstrates that, it is strictly necessary to modernize the economy by implementing new contemporary technologies, the faster the better, because economy collapse can be happened in the following ten years. General conclusions Presented overview of the input-output models application to the concrete economy in transition like Moldovan economy, has demonstrated usefulness of this tool for empirical study. RAS method for input-output table balancing has been applied and realized as software program. Both static and dynamic optimization models can be implicated to elaboration of the diverse scenarios and their analysis. New technology proposed for implementation could be tested before it. Direct expenditure coefficients investment matrix has been constructed starting with settled down industries growth rate investment figures and then based on it, dynamic optimization model was solved using Solver procedure. The same procedure was also used for solving static and dynamic optimization models. Assumption about constancy of both technological and investment matrixes was made in order to effectuate calculation in time. This assumption was justified actually by small changes in time of technological and investment coefficients. Markov chain application to the input-output model treatment, appear to be very useful from the stability problem point of view. All together, examined methods are recommended for policy makers in order to consult alternative opinion when decisions are taking. REFERENCES 1. LEONTIEF, Wassily. Input-Output Economics. Oxford: Oxford University Press, 1966. 257 p. 2. LEONTIEF, Wassily, FORD, Daniel. Air Pollution and Economic Structure: Empirical Results of Input-Output Computations. In: Input-output techniques 1971: Proceedings of the Fifth International Conference on Input-Output Techniques, january 1971. Amsterdam: North-Holland Publ. Co., 1972, pp. 9-30. ISBN 072043064X. 3. FONTILA, Emilio. Leontief and the Future of the World Economy. 2000, august. [Accesat 18.11.2018]. Disponibil: https://www.iioa.org/conferences/13th/files/Fontela_Leontief.pdf 4. PETERSON, Bruce, OLINICK, Michael. Leontief models, Markov chains, substochastic matrices, and positive solutions of matrix equations. In: Mathematical modeling. 1982, vol. 3, pp. 221-239. [Accesat 21.02.2018]. Disponibil: https://core.ac.uk/download/pdf/81156007.pdf 5. TRINH,Bui, PHONG, N.V. A Short Note on RAS Method. In: Advances in Management & Applied Economics. 2013, vol. 3, no. 4, pp. 133-137. ISSN 1792-7544, ISSN 1792-7552. [Accesat 16.12.2018]. Disponibil: https://www.researchgate.net/publication/308018908_A_Short_Note_on_RAS_Method 6. DIACONOVA, M., NAVAL, Elvira. Modelul interramural de prognoză a dezvoltării economiei naţionale. In: Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica. 1999, nr. 3 (31), 1999, pp. 51-60. ISSN 1024-7696. 7. NAVAL, Elvira, GHEREG, Veronica. Input-Output model for Republic of Moldova. In: Proceedings of The 4th Conference of Mathematical Society of the Republic of Moldova, june 28-july 2 2017. [Accesat 15.11.2018]. Disponibil: http://dspace.usm.md:8080/xmlui/bitstream/handle/123456789/1665/425_428_Input- Output%20model%20for%20Republic%20of%20Moldova.pdf?sequence=1&isAllowed=y 8. Biroul Naţional de Statistică. [Accesat 21.02.2018]. Disponibil: http://www.statistica.md 9. NAVAL, Elvira. Modelul interramural static de optimizare în condiţiile modificărilor structurale. In: Creşterea Economică în condiţiile globalizării: modele de dezvoltare durabilă: conferinţa internaţonală ştiinţifico-practică, 12-13 octombrie 2017. Ediţia a XII-a. Chişinău: INCE, 2017, vol. 1, pp. 47-52. ISBN 978-9975-3171-1-5. https://www.econbiz.de/Record/input-output-techniques-proceedings-of-the-fifth-international-conference-on-input-output-techniques-geneva-january-1971-br%C3%B3dy-andr%C3%A1s/10000056315 https://www.econbiz.de/Record/input-output-techniques-proceedings-of-the-fifth-international-conference-on-input-output-techniques-geneva-january-1971-br%C3%B3dy-andr%C3%A1s/10000056315 https://www.iioa.org/conferences/13th/files/Fontela_Leontief.pdf https://core.ac.uk/download/pdf/81156007.pdf https://www.researchgate.net/publication/308018908_A_Short_Note_on_RAS_Method http://cmsm4.math.md/ http://dspace.usm.md:8080/xmlui/bitstream/handle/123456789/1665/425_428_Input-Output%20model%20for%20Republic%20of%20Moldova.pdf?sequence=1&isAllowed=y http://dspace.usm.md:8080/xmlui/bitstream/handle/123456789/1665/425_428_Input-Output%20model%20for%20Republic%20of%20Moldova.pdf?sequence=1&isAllowed=y http://www.statistica.md/ Theoretical and scientifical journal 52 No. 1 / 2019 10. NAVAL, Elvira. Elaborarea modelelor economice în baza balanţelor interramurale. In: Akademos. 2017, nr. 2 (45), pp. 27-32. ISSN 1857-0461. 11. NAVAL, Elvira, GHEREG, Veronica. Modelul dinamic de optimizare pentru Republica Moldova. In: Modelare matematică, optimizare şi tehnologii informaţionale: conferinţa internaţională, 12-16 noiembriee 2018. Ediţia a V-a. Chişinău, 2018, pp. 146-151. ISBN 978-9975-62-421-3. 12. KUN, Dong. The dynamic Optimization Model of Industrial Structure with Energy-saving and Emission-reducing Constraint. In: Journal of Sustainable Development. 2008, vol.1, no. 2, july. [Accesat 17.12.2018]. Disponibil: https://pdfs.semanticscholar.org/e11d/95545f6b9702d6dc235ff0a697e88fdceb96.pdf 13. BUI, Trinh, NGUYEN VIET, P.A Short Note on RAS Method. In: Advances in Management & Applied Economics. 2013, vol. 3, no. 4, pp. 133-137. ISSN 1792-7544. 14. PESCHEK, W. Input-Output Models and Markov Chain. [Accesat 26.10.2018]. Disponibil: https://businessdocbox.com/amp/80612155-Marketing/Input-output-models-and-markov- chains.html 15. KEMENY, Laurie Snell, JOHN, G. Finite Markov Chains. Management Mathematics for European Schools. 1963. [Accesat 12.11.2018]. Disponibil: https://www.amazon.com/Finite-Markov- Chains-Laurie-Kemeny/dp/B000KYES0O 16. GRINSTEAD, Charles, SNELL, J. Laurie. Introduction to Probability. Chapter 11. American Mathematical Society. 1997. [Accesat 26.10.2018]. Disponibil: http://mathsdemo.cf.ac.uk/maths/resources/Probability_Answers.pdf 17. KAKUTANI, Shizuo. A generalization of Brower’s fixed point theorem. In: Duke Mathematical Journal. 1941, vol. 8 (3), pp. 413-589. 18. DUCHIN, Faye, STEENGE, Albert E. Mathematical Models in Input-Output Economics. Rensselaer Working Papers in Economics. 2007, april. [Accesat 21.02.2019]. Disponibil: https://www.researchgate.net/publication/24125153_Mathematical_Models_in_Input- Output_Economics 19. ARROW, K.J. Alternative Proof of the Substitution Theorem for Leontief Models in the General Case. In: T.C. KOOPMANS (ed.) Activity Analysis of Production and Allocation. New York, 1951, pp. 155-164. Recommended for publication: 10. 06.2019 https://pdfs.semanticscholar.org/e11d/95545f6b9702d6dc235ff0a697e88fdceb96.pdf https://businessdocbox.com/amp/80612155-Marketing/Input-output-models-and-markov-chains.html https://businessdocbox.com/amp/80612155-Marketing/Input-output-models-and-markov-chains.html https://www.amazon.com/Finite-Markov-Chains-Laurie-Kemeny/dp/B000KYES0O https://www.amazon.com/Finite-Markov-Chains-Laurie-Kemeny/dp/B000KYES0O http://mathsdemo.cf.ac.uk/maths/resources/Probability_Answers.pdf https://en.wikipedia.org/wiki/Shizuo_Kakutani https://www.researchgate.net/publication/24125153_Mathematical_Models_in_Input-Output_Economics https://www.researchgate.net/publication/24125153_Mathematical_Models_in_Input-Output_Economics