The Main Theoretical Aspects of Input-Output Tables Model for Use in Macroeconomic Analysis
Constantin Anghelache
*, Mădălina-Gabriela Anghel
*** Prof. PhD, Faculty of Faculty of Cybernetics, Statistics and Economic Informatics, Bucharest University of Economic Studies / Faculty of Finance and Accounting, “Artifex”, University of Bucharest, Economu Cezarescu Street, no 47, Bucharest, Romania
e-mail: [email protected]
** Assoc. prof.PhD, Faculty of Finance and Accounting, “Artifex” University of Bucharest, Economu Cezarescu Street, no 47, Bucharest, Romania
e-mail: [email protected]
Abstract
The links between the branches of economy are realized in concrete terms, there being elements to be studied and analysed. Thus, within the national economy there is intermediate consumption that specifies the consumption of a branch of all other branches. For this, we calculate intermediate consumption technology coefficients that are used to determine the results of the national economy.
For this the input-output table was created and also the balance connection between branches which at the macroeconomic level is a complex model that provides the necessary elements of the calculation and analysis of economic performance.
In order to perform such an analysis it is important to determine how branches of national economy are aggregated. Aggregation of branches of national economy is the first step to study the links between them.
The number of branches and the concentration requirements are subordinated to identifying links and proportions that are achieved.
In this article, the meaning of a homogeneous branch is reduced to goods and services that meet certain criteria, according to international standards.
For the first time, the model was used in the United States, with the scientific support input-output table of Leontief. In Romania, this input-output system is used and complemented that gives an additional possibility to analyze and study the national economy.
Keywords:technological coefficient; model; pure branch; macroeconomic system; quadrant.
JEL Classification: D57, E00.
Introduction
In the economic modelling - mathematical models occupy a special place among branches based on input - output analysis, known in our country under the generic name of balance links between branches. Such models reflect existing flows and interdependencies between industries in the national economy in which economic activity is taking place. As part of the input branches there is a two-way flow of materials, products or services productive namely: on the
one hand entries in a particular branch or sub-branch, on the other hand outputs of each branch or sub-branch, completed production results and assigned destinations. These two categories of flows quantify the relationship between resources and their use, ensuring the material balance in the economy. Also in the branches of the input we distinguish technological relations, ie links that form the productive apparatus of the national economy in production or productive consumption. In the balance, each branch corresponds to one row and one column. On occasions the use falls production of each branch, ie intermediate consumption, ie consumption of own production and other industries, for accumulation, ie fixed capital, working capital and inventories, to fund public and private consumption, export and inevitable loss of domestic product or national product. Columns register for each branch structure of production material expenses, net output or value added items, ie net income, wages, net indirect taxes and import.
The rows in the table show essentially the distribution of each branch output, and a column- structure corresponding size. The main part of the input branches is the flow table products and services, to which we add production distribution coefficients table, table of technical coefficients or direct material consumption and total material consumption coefficients table.
The main table of the balance of branch links comprises four parts known as quadrants.
Literature Review
Alesina and Ardagna (2010) analyse the major modifications in the fiscal policies, Kaplan and Violante (2014) describe a consumption-oriented model as response to fiscal stimulation mechanisms. Mounford and Uhlig (2009), Parker (2011), Romer and Romer (2010) analyse the effects of fiscal policies under certain economic conditions. Anghelache (2008), Anghelache and Anghel (2016) present the basics of economic statistics, from the theoretical and practical viewpoint. Anghelache, Anghel (2014), Anghelache, Anghel, Manole, and Lilea (2016) develop on the economic modelling. Anghelache, Manole, Anghel, Diaconu (2016) characterize the correlations between macroeconomic variables, Anghelache, Manole, Anghel and Popovici (2016) study the links between external payment balance indicators and macroeconomic aggregates. Anghelache and Capanu (2004), Anghelache and Capanu (2003), Anghelache, Mitruţ and Voineagu (2013), Anghelache (coord., 2007). Anghelache, Isaic-Maniu, Mitruţ, and Voineagu (2007), Anghelache et.al. (2006), Anghelache (2006) are reference work on macroeconomic statistics, with a focus on the presentation and explanation of the system of national accounts of Romania. Barro and Redlick (2011) develop on the effects of public policies of taxation and acquisitions, effects measured at macroeconomic level, Ramey (2011) has developed on a similar topic. Bloom (2009) focuses on the uncertainty phenomenon and some of its effects. Chari, Kehoe and McGrattan (2007) develop on the accounting organized around business cycle. Christiano, Eichenbaum and Rebelo (2011), Woodford (2011) analyse the characteristics of the government spending multiplier. Ciccone (2002) presents some characteristics of the correlations between inputs and industrialization. Hendrickson, Lave and Matthews (2006) describe the evaluation of goods and services environmental lifecycle from an input-output perspective. Jones (2011) studies some correlations between economic growth, and input-output economics, Miller and Blair (2009) develop on the input-output topic. Pesaran, Pick and Pranovich (2013) approach some particular characteristics of forecasts. Saiz (2010) evaluates the housing supply from the geographic dimension angle.
Research Methodology and Data
General aspectsMaking the input-output table is based on input-output method developed by Professor W.
Leontief. This method - as stated W. Leontief, applies both to study the national economy as a
whole, the level of its structural elements, and for studies of large firms. By this method, the highlight and analysed linkages between (branches) studied the economic system, connections reflected by the linear mathematically equation, the equations underlying the determination of the coefficient inputs and outputs shown in the table. In this table we can highlight the interdependence between branches, each branch of production is presented in two ways:
o as a result of inputs of product, respectively the deliveries from other branches and manpower consumption, looking at it like a delivery from a separate sector of the economy;
o the output, deliveries to other branches to supply intermediate consumption and final consumption (household consumption, government consumption, investment).
Table lines express the outputs and columns express the inputs. In a whole, the table shows flows of goods and services from all economic sectors (Anghelache and Anghel, 2016).
For compiling input-output table a special classification of branches is required, which is one of the basic problems of the table.
The basis of this classification are three main principles, namely: classification to reflect the state of the labour division and to take account of trends in the country for which the table is formed; classification to satisfy various indicators necessary for economic analysis;
classification made to take into account the particularities of branches classification used in practice of national and international statistics.
Application of these principles requires a thorough knowledge of the economy, the areas where to use the table, of how it will solve methodological issues, and the possibilities they offer evidence at the current stage of development and improvements that will brings to the making of the table.
Classification of branches for this table is intended to make branches more uniform, and try to organize the data collection and processing. Number of branches included in the classification determines the size of the table.
Statistical work of determining the content of each branch according to the type of product has to contain the following: compiling lists of goods and services that are produced, transport or sell, consume productive or accumulates in the national economy, as well as services provided or can be provided; grouping the products and services of manufacturing technology, the raw materials used or the destination of the products; development based on formed clusters, the product nomenclature and classification of branches (Anghelache, Mitruţ and Voineagu, 2013).
Starting from a firm (enterprise), the branch is conceived as all the companies producing similar products. But companies usually are multipurpose factories and this makes the branch not to have a homogeneous content. Branches, in this case, in addition to basic production corresponding to the profile, manufacture also goods that do not meet its extra-profile. It is necessary for the production of a branch to be identical because otherwise relations between the branches cannot be correctly characterized and the calculated coefficients are not real, being negatively influenced by costs that are not matched to the profile of the sector.
In this case, the branches obtained from companies or parts thereof, must undergo a process of elimination of the so-called production extra-profile, in order to obtain homogeneous branches, and branches called pure. By pure branch is meant production activity that results in products of the same kind, regardless of the form of subordination and organization of the company.
The purity problem is mainly linked to the branches of industries including companies performing industrial activity different from their basic profile. For example, companies producing sugar and also make alcoholic drinks or steel companies that produce besides steel, coke and electricity. In some companies, production not falling in their profile can have a specific higher weight and in others it may be insignificant. The degree of its size is a problem
with the composition of the input fields, as it distorts the calculated coefficients and technological ties between branches.
In order to obtain pure branches it is necessary to know the size of the production branch that does not match the profile of the so-called production extra-profile.
Extra-profile production record is pretty hard to be kept for companies. In this case, to obtain pure branches, selective research statistics determined based on the specific weight of the total output of the production of each branch extra-profile. Depending on these indicators the extra- profile sector is deleted from the production sector and it is includes in the corresponding branches of her profile.
Aggregation of branches
Aggregation of the branches is a basic problem for studying links between branches, closely related to classification of branches of national economy in general and of industries in particular. If in the table it cannot appear a large number of branches, then we proceed to the union of two or more branches, the process is called aggregation (Anghelache, 2008).
Aggregation determines the number of branches in the breakdown, using criteria that allow accurate characterization of technical and economic links between branches. The number of branches and their level of aggregation needs are determined by the characterization and analysis of the economy needs to characterize the most important links and proportions made in economic activity.
Aggregations production branches must be made according to the specific features of each sector. Branches must be homogeneous. Homogeneity of production of a branch can be determined by identity criterion of a manufactured product, where it is intended for consumer production, by the raw materials used. These criteria are used to determine the homogeneity branches and are the criteria underlying their aggregation. So when aggregating branches we used five criteria, namely: the identity of products manufactured; common destination of products; similarity raw materials consumed; similarity of the processes used; quantitative structure similarity with the cost of production.
Choosing aggregation criterion is based on the principle of classification adopted for classification of production branches. When branches classification was made according to the purpose of the products the aggregation should be based on the same criteria.
Aggregation is of two types: horizontal and vertical (Anghelache, Isaic-Maniu, Mitruţ and Voineagu, 2007).
Horizontal aggregation means aggregating products parallel to production processes, which is one and the same stage of production. It is based on the assumption that products whose manufacture consumes the same kind of raw material and production at the same stage and usually have the same functions. This kind of aggregation finds application of the destination criterion for final products. As an example, branches can be given: the production of artificial silk; production of silk; production of wool fabrics; production of cotton fabrics; the production of fabrics. All these branches have a homogeneous character, used as feed yarn and finished products (fabric) have the same destination. They can be aggregated into a single branch of industry - production of fabrics.
Aggregation aims to bring together the vertical state production of successive, independent parties in the production of the various branches of industry. This is exemplified in figure branches: the production of cotton, production of cotton yarn; production of cotton fabrics.
These three branches can aggregate a branch namely cotton industry (Anghelache, 2006).
Using horizontal or vertical branches aggregation is determined by the purpose which it pursues economic analysis. Developing classification branches has a great influence on indicators
calculated based on the table. The degree of homogeneity of branches influences the character coefficients calculated as the number of branches, so far as this number corresponds to the degree of development of the division of labour influence the amount and quality of information to be provided to the economic analysis, economic forecasting calculations.
Activity branch
It is envisaged that branches exercise the same type of productive activity, irrespective of whether the institutional units to which they belong generate or not production market or non- market. Industries may be classified in three categories:
o industries producing market goods and services (market branches activity) or goods and services for own final use;
o industries of government producing non-market goods and services (industry with non- market public administrations);
o Branches of activities of non-profit institutions serving households producing non-market goods and services (non-market branches of non-profit institutions serving households).
Local economic activity units are only liable to some extent for the demands required for production process analysis. For this type of analysis (i.e. for the analysis of input-output) unit that suits best are the ones that have homogeneous production unit. Homogeneous production unit (HPU) is characterized by a single activity, namely the inputs, a production process and outputs.
Products constituting inputs and outputs are themselves characterized both by their nature, stage of development and production technique used and by the reference to the nomenclature of products. If an institutional unit producing goods and services contains a principal activity and one or more secondary activities, they will be separated in the same number of homogeneous production units. On the contrary, auxiliary activities are not separated from the main activities secondary or the ones they serve (Anghelache and Capanu, 2004).
The homogeneous branch consists of a group of units of homogeneous production. The set of activities retained for a homogeneous branch is identified by reference to a product classification. The homogeneous branch produces only goods and services specified in the classification. Homogeneous branches may be classified in three categories:
o homogeneous branches producing market goods and services or goods and services for own final use;
o homogeneous branches of general government producing non-market goods and services;
o homogeneous branches of non-profit institutions serving households producing non-market goods and services.
Classification of industries
Defining branches depends on the classification of activities is directly related to the products, because each activity is characterized by the products they carry. Consequently, activities and products are two complementary visions to define the contours of actual production.
System of National Accounts (SNA 95) proposes a classification of activities in accordance with the "International Standard Classification of activities" (ISIC-International Standard Industrial Classification) conducted by the Statistical Office of the UN, the third version, published in 1990.
This classification is the basis used in each country, thus ensuring international comparability.
European Union uses NACE (statistical classification of economic activities of the European Community defined by Regulation no.3037/90). Under this classification, industries are broken down into four levels named in ascending order: classes, groups, subdivisions and divisions (Anghelache and Capanu, 2003).
In Romania using classification by economic activity (CAEN-2), the classification is therefore used by all traders. CAEN-2 is harmonized with NACE and ISIC.
Classification ensures identification of all activities and consolidates them into a single system.
It lets you organize, streamline and computerize social and economic information flows, creating processing facilities to integrate into national and international presentation and analysis of information.
In CAEN-2, socio-economic activities are grouped into five steps (sections, subsections, groups and classes), established the principle of homogeneity as a whole of activities that have common characteristics, namely:
o nature of goods and services (their physical composition, the manufacturing stage and the needs they can meet);
o Usage of goods and services by businesses (intermediate consumption, final consumption, capital formation, etc.);
o Raw material and technological processes used, the organization and financing of production.
Content of input-output tables
The models developed by W. Leontief include the so-called accounting system or statistical input-output table and the analytical system. Statistical table comprises a number of rows and columns that highlight the volume of inputs and outputs (Chari, Kehoe and McGrattan, 2007).
The main section of the table shows flows of goods and services produced and consumed in the production of various branches. The consumer products are intermediate consumption. It does not include consumption of fixed capital (depreciation).
The second section of the table shows flows of goods and services out of production and the final consumption, the third section highlights the manpower and fixed capital used (depreciation), elements that represent the so-called primary inputs and expressed by added value indicator (primary income plus depreciation). The second part is the material content of the third section (II = III). From a graphical point of view these sections are as follows:
I II
Intermediate consumption by Final consumption by industry origin industry origin III IV
Added value by industry provenance Reflects the redistribution process Presentation of the statistical table figures are based on the following principles (Anghelache, Isaic-Maniu, Mitruţ and Voineagu, 2007):
o The production of each branch can be consumed in the analyzed branch, from other branches of production, other economic activities and export system. Mathematically, assuming n branches, xi is expressed as follows:
X x x ⋯ x y
X x x ⋯ x y
⋮ ⋮ ⋮ ⋮ ⋮
X x x ⋯ x y
(1)
where x x ⋯ x represent intermediate consumption and yn is final consumption.
For all the branches, the ecuation will be:
Xi = ∑ xij + yi ; i 1, n. (2)
These equations show that the output balance of each branch is equal to the intermediate consumption plus the final consumption.
o Each branch in order to be able to carry out the production process must consume products from other branches of production, labour and fixed capital. These inputs (primary and intermediate) give the total costs of the branch, respectively the intermediate costs and consumption of primary inputs. Mathematical equation for one branch will be:
X x x ⋯ x T F (3) where:
xi1 = value of material inputs; i = 1, 2, ..., n.
T1 = consumption of labour;
F1 = consumption of fixed capital (depreciation);
T1 + F1 = added value.
The value equation of expenditures will be expressed, mathematically modeled as:
∑ j
j j
V F
T (4) Between the inputs and outputs the two equations must be equal:
∑ ∑ j
j j
V F
T (5) Table 1. The synthetic input-output table
Intermediate consumption Final consumption
Total production Consuming
branches Components Total 1…..j…..n 1…..f…..F Producing
Branches 1
⋮
i
⋮
nI quadrant
xij
II quadrant
yif yi Xi
Added Value
1
⋮
k Components
⋮
K ____
Total
III quadrant
Vkj
Vi
IV quadrant
Total production
Xi
Source: Anghelache, C., Isaic-Maniu, A., Mitruţ, C. and Voineagu, V., 2007. Sistemul Conturilor Naţionale – Ediţia a II-a. Bucharest: Economic Publishing House.
The table consists of four quadrants, formed by a number of rows and columns (Anghelache, 2006).
Quadrant I is the balance that reflects interdependencies that are formed between the branches and sub-branches of the national economy as a result of mutual deliveries of items of work meaning the technological dependencies between industries. Rows of the quadrant show the distribution of production of each branch between all branches of material production, so the material flows entering the power delivery and the columns show the expenditure structure material nature of the objects of labour for growing in every branch. The quadrant has the shape of a square matrix in which rows are entered the branches where the national economy is divided. Each branch is analysed in a double capacity: producing and consuming. Material flows are measured using the coefficients of intermediate consumption, which means the referenced xij amount should be consumed in the production and manufacturing industry to achieve the projected output branch j consuming. In quadrant I should be n variables Xi. In reality the number of these variables is lower because there is no direct link between absolutely all national economic branches: some variables xij are void. Each variable xij has a double meaning: the output of the i input branch and the j producing and consuming branch. The line totals are not equal to the column totals for all lines, these shows that the production of each branch producing for consumption in the national economy and all columns show the size of material expenses made in the branches consuming to achieve projected yields.
Quadrant II includes flows of products and services emerging from the production sphere and entering in the end-use sphere. The quadrant reflects the product or the final consumption, the final demand in the economy by branches providing origin and destination. The final product represent a part of production of a branch that is meant for satisfaction of the final demand symbolized with Y. This is structured around the following elements: public final consumption, private final consumption, gross fixed capital formation, change or growth of stocks, export;
There are branches of national economy whose end product does not cover all these needs (Anghelache, Mitruţ and Voineagu, 2013).
This means that the global product of branch (X) is divided into two parts: intermediate product and final product. The intermediate product assigned for the productive consumption is subject to a new manufacturing process and enters the material costs of branches. The final product is not subjected to further processing but it is used for social or personal consumption, getting out permanently from the production. Intermediate overall product distribution to the final consumer is characterized by a system of equations for the distribution of production:
⋯
⋯
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋯
⋯
(5.1)
The system of equations can be written also as:
; where i 1, n
(5.2) The III quadrant includes branches of the national economy following indicators: depreciation of fixed capital (Z), Net value added (VAN) structured in indirect taxes net (value added tax - VAT, customs duties - TV, wages salary - S, operating surplus, profit - P), gross value added
(GVA) (depreciation + net value added), total costs of production (CM) (labour costs + depreciation nature objectives), and sometimes imports (Im). Summing elements in quadrants I and III is characterized by the equation of balance of the costs of production which have the following form:
⋯
⋯
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋯ ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋯
⋯
(5.3)
In a reduced form the equation system can be written also as:
, 1,
(5.4) The balance equation has the following form:
(5.5) The branches of the input consist of differential equations system in three types, namely:
distribution of the production equation by summing the elements of each branch constructed in quadrants I and II
formation of the production costs equations constructed by summing the elements in quadrants I and III
balance equations and branch of the national economy.
The IV quadrant reflects the redistribution linking the primary income of the III quadrant and final consumption in the II quadrant.
In the IV quadrant we must include income units and works in the field considered unproductive.
Usually this dial is not completed due to theoretical and methodological difficulties and in terms of content is not significant enough analysis and macroeconomic forecasting.
The quadrant lines I and II give the elements the total proportion of branch data (∑xij + yi = Xi) from the point of view of the material structure.
Columns I and III quadrant elements give the total output of data, in terms of the structure value:
∑xij + vj = Xj (6) The sum of quadrant I and II elements are equal with the sum of quadrant I and II elements:
∑xij + yi = ∑xij + vj (7) or:
∑xi = ∑xj (8)
Also, the sum of the elements of the quadrant II shows the structure of the final use is equal to the amount added (primary income + depreciation).
As shown in the balance scheme, which highlights the main indicators are: intermediate consumption, final consumption, value added and total production or gross production (Miller and Blair, 2009).
Intermediate consumption includes goods and services used to produce materials of different branches. Production which is used for this purpose is called the intermediate output (Pi).
Final consumption includes goods and services that are used for: investments (replacement of fixed capital consumed and growth), increase stocks of materials, household consumption and export.
Added value includes: deferred revenues of the population and of the state; depreciation, of the net product (net added value) and depreciation.
These indicators are calculated by branch and the entire economy.
Total production (Pt) can be calculated in two ways:
Pt = Pi + Pf (9) Pt = Ci + VAB (10) The notations for the two computational relationships are specified in the text.
These two ways of calculating respond to different information needs. In the first case the total production is characterized by its material structure and branches, and the second way characterizes income structure factors of production, intermediate consumption and depreciation of total production.
In addition to the input-output statistical table, as has been shown, the table model comprises the so-called analytical system. Unlike statistical system (statistical table) which aims to present the facts in figures, the analytical system consists of mathematical models used to investigate the relationships between branches (sectors), models underlying the cyber analysis of the national economy.
So the input-output table is not only a statistical table, but an economic-mathematical model expressing the relationship between branches in a system of mathematical relationships. The number of equations included in the system is equal to the number of industries included in the classification branches used for compiling the table (Miller and Blair, 2009).
Assuming that the table includes n branches and the output of each branch is noted with a xi (i = 1, 2, ... n), we obtain the following table:
Table 2. Contents of the input-output sinthetic table
X1
X2
X3
⋮ Xn
I x11 x12 x13 … x1n
x21 x22 x23 … x2n
x31 x32 x33 … x3n
⋮
Xn1 xn2 xn3 … xnn
II y1
y2
y3
⋮ yn
v1 v2 v3 … vn III IV X1 X2 X3 … Xn
As result, your table presents arranged in a square format of its values arranged in rows and in columns containing mutual deliveries between branches of production, so the flow between branches. If in the output branches includes their own consumption, inter-flow matrix is called the matrix of the gross production, and if the own consumption (diagonal values) are not included, the matrix is called the matrix of the net production. The gross output matrix structure shows a better more complete structure of production, it has a greater practical value than the net output matrix.
Outside the square, in the table we have the yi column expressing elements of final consumption, and rows v, expressing elements of added value. Column yi express deliveries of each sector for investment, growth of circulating capital and stocks, unproductive consumption and for export. In the table, X11 express the production of branch 1 consumed in X12 express the delivery of production branch 2, and the X1n expresses the production of branch 1 delivered to branch n. The production of branch 1 is consumed in the system and in other branches (X11, X12, ... .X1n) it can be obtained as the sum of these intermediate consumption. Assuming that the number of branches is 74, the intermediate consumption will be obtained by using the following formula:
∑ (11) In order to obtain the total production of branch i (Xi) at the intermediate consumption we will add the final consumption (yi). In this case:
∑ (12) In order to produce each branch receives products from other branches. So branch 1 receives product from branch 2 in the amount of X21, and from branch 3 in the amount of X31, from branch 4 in the amount of X41, from branch 5 in the amount of X51 and from branch 6 in the amount the X61 etc. These entries represent the productive branches consumption. The amount of entries of product in each branch and of the added value is the total output of the branch.
Calculating supplies and products for all branches entries will get a table of inputs and outputs.
Rows of the table show delivery (outputs) production of each branch and columns, inputs, production consumption of each branch from the branches for which was made table.
Table diagonal marked with x11, x22, x33, x144, x155, etc. characterize the productive consumption of each branch, from the production of the branch, meaning the production of the branch consumed inside it (Anghelache and Anghel, 2016).
The production of each branch in the table has a double expression. The distribution of output for intermediate and final consumption is highlighted horizontally. Vertical expenses are highlighted with different materials and services consumed and the added value elements. From the table shows two sets of equations: equation of production allocation and equation making the output value of each branch:
Xi = ∑xij + yi; (13) Xj = ∑xij+Yj; (14) where:
i represents the production branches (1 to n);
j represents the consumption branches (1 to n).
These two equations represent two different expressions for the production of branches:
∑xij + yi = ∑xij + Yj. (15)
This equation indicates equality between the output of a branch, seen from the perspective of its distribution, and the same branch of production, seen through the prism of social production costs.
In order to highlight the technical and economic relations between the branches the coefficients aij are calculated, called coefficients of direct expenses. They show the production of branch i and necessary to produce a value of the output branch unit j:
a i= l, 2, ... , n. (16) results in:
xij = aijXi (17) Aij coefficients can be calculated also in natural expression, when the balance is formed in physical units and not only in terms of value.
The coefficients in natural expression are called technological coefficients which express quantity, from product “i” consumed for obtaining a quantity from product “j”.
Aij coefficients are smaller than 1. They have a great significance for economic analysis, planning and forecasting calculations. For this purpose they have to be determined. Their stability depends on the economic structure of a country. Mainly the changes that occur in the production structure in its technical level, affects the stability of the coefficients. In this case it is necessary to correct them according to factors influencing their stability. Introducing the coefficients aij in the balance equations of the input branches (its model) can be represented as follows:
X a X a X ⋯ a X y
X a X a X ⋯ a X y
⋮ ⋮ ⋮ ⋮ ⋮
X a X a X ⋯ a X y
(18)
This system of equations can be written in a form of matrix as follows:
(E – A) X = y; (19) Square matrix of direct expense coeficients
A
… …
⋮ ⋮ ⋮
…
(20)
Matrix unit n
E
1 0
…
00 1
…
0⋮ ⋮ ⋮
0 0
…
1(21)
Vector of global product
X X X
⋮ X
(22)
Vector of final consumption
y y y
⋮ y
(23)
From the matrix system the following equation is established:
X = (E – A)-1y. (24) This formula allows direct computation of each branch of production if the growth rate is determined of final product for each branch. Also, they may be developed based on macroeconomic forecasting calculations (Anghelache, Anghel, Manole și Lilea, 2016).
Matrix (E - A)-1 matrix is called the total expenditure and it is noted as Aij, it is obtained by inversing the direct cost matrix. Knowledge of the inverse matrix elements (E - A)-1 provides useful data for economic analysis, planning and forecasting.
Conclusions
From the research presented in this article some theoretical conclusions highlight the complexity and usefulness of input-output tables in the calculation of macroeconomic indicators, analysis and forecast. The macroeconomic system is complex, structured on economic sectors and/or pure branches. The input-output method highlights and analyses the relationship between the branches of the economic system, as well as the mathematically reflected relations through linear equations, underlying the determination of the coefficients presented in a synthetic input- output table. The aggregation required to obtain the branches for the input-output tables can be:
products identification, common destination of the finished product, similarity raw materials consumed or processes used. The aggregate branches are used to establish the branch-to-branch balance or the input-output table. Currently, the system used is CAEN 2 which assures a classification of the branches and national economic activities. The input-output system includes a number of tables and a synthetic table of the model structured in four quadrants, each with its significance, which helps the economic analysis of production, economic results, calculation of domestic product per branch and total national economy. The input-output model involves solving important issues such as aggregation of industries, these industries achieve homogeneity and identifying connections established between them. Of course, for synthetic input-output table there are a sufficient number of scales expressing links to be established between these branches of the national economy. We insisted on building the synthetic model of input-output in order to reveal the need for reliable data, of correlations that are established and then on this basis by the algorithm agreed we can proceed to calculate all theological coefficients, of intermediate consumption, the final results, the import role and export role in achieving the macroeconomic indicators and many other aspects. Being a complex model of the economy at the macro level for example we might refer to a limited number of branches to emphasize the scholastic indicators that are calculated and the expressiveness of their analysis. It is also possible to establish the use of input-output table in microeconomic analyses, the results being relevant. It will be used to adapt the calculated and used indicators at microeconomic level - firm. Also, we could deepen the role of macroeconomic forecasts using these tables. It merely summarizes theoretical possible to suggest the possibility of using this model to those interested in macroeconomic analysiswith microeconomic extension.
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