Sekerka, Bohuslav

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Theory and Methods RRI Input-Output Archive 4-1-1974 Problems of Relations Among Balances in Various Classifications Bohuslav Sekerka Follow this and additional works at: https://researchrepository.wvu.edu/rri_iotheorymethods Recommended Citation Sekerka, Bohuslav, "Problems of Relations Among Balances in Various Classifications" (1974). Theory and Methods. 44. https://researchrepository.wvu.edu/rri_iotheorymethods/44 This Article is brought to you for free and open access by the RRI Input-Output Archive at The Research Repository @ WVU. It has been accepted for inclusion in Theory and Methods by an authorized administrator of The Research Repository @ WVU. For more information, please contact beau.smith@mail.wvu.edu. □ id.74-242 csg SIXTH INTERNATIONAL CONFERENCE t ON input - output techniques ( Vienna. 22 - 26 April 1974) Session 3: Studies of national applications PROBLEM OF RELATIONS AMONG BALANCES IN VARIOUS CLASSIFICATIONS by Bohuslav Sekerka National Economy Computing Institute of the State Planning Commission Czechoslovakia Received on 7.1.1974 This document has been reproduced by the UNIDO. The views and opinions expressed in this paper are those of the author and do not necessarily reflect the views of the UNIDO. !' Problem of Relations Among Balances in Various Classifications 1. Introduction by Bohuslav Sekerka This paper deals with the problem of relations among balances in various classifications. We shall consider in?ut-output balances, first with rows and columns corres-ponding to commodities, second with rows corresponding to commodities and columns corresponding to producers (enter-prises) and finally with rows and columns corresponding to producers. ½ben investigating the relations among these ba-lances we shall use the methods of disaggregation, as descri-bed in paragraph 4. All these questions are important e.g. for the middle-term planning of socialist economy, in which orgnnization aspects for short-term planning and commodity aspects for long-term planning are considered. The descr~bed methods of disaggregation are important for analyses, when a certain phenomenon is investigated in various classifications. For example, when expressing quantities of a certain nomencla-ture in another nomenclature. We denote that the development in computation technique, construction of information systems and databanks for central - 2· -control enable us to introduce various classificstion aspects into the investi.e;ation of economic events. We shall consider the economy in which m enterprises pruduce n commodities. We denote st X• • 1. J quantity of commodity 1., produced by enterprises and used for the production of commodity j by enter-prise t during a certain period of time; yf quantity of commodity i, produced by enterprise s during a certain period and used as final product; z~ quantity of commodity i, produced by enterprises 1 during a certain period. We shall assume the variabl es in monetary expression, so that all required 3ggregatio~s are allowed. The length ,Jf time interval facilitates to assume that the produced commo-dity is spent during the same period or that initial and final stocks of a certain commodity approximately equal. The symbol . on the place of index expresses the values aggregated by this index. We shall demand the natural rela-tions be fulfilled, i.e. aggregation corresponds to addition. Then zi denotes both the total production of commodity i and ~ Lz~. In some cases the aggregation bias may appear. If we, s 1 for example, define for every enterprise the quantity of commo-dity which should be produced, and define the total production of commodity by input-output relations, then it is not necessary that f!:. z1.~ be equivalent to z1: • The balance method secures s=l - 3 -that these values equal. So we shall assume the aggregated value as the sum of non-aggregated values. When a bias appears, it is necessary to remove it by the change of dependences, structures and so on. We denote that for values describing the realized events, the condition of non-biased aggregation is always fulfilled. The ba!ic relation of distribution of commodity i produ-ced by the enterprises could be expressed ( 1. 0) x~· + y~ = z~ J.. J. 1 which could be also written n (l.Oa) L j=l ( 1. Ob) where i = l, ••. ,n, s y. 1 s = z. 1 s = l, ... ,m. If we aggregate the relations (l.Oa) in accordance with the enterprise~, we shall obtain (1.1) n 2-- x• • + y!. = z ! j=l i j ;i. 1 i = l, ..• ,n representing the balance of distribution of commodity i for - 4 -the proauction of otner co:nmodities. Let us assull'e the con-sumption of a certain c0n®odity f0r the prJduction of another com~odity be directly proportionate to the production of the certain commodity. It means that there exist the coeffi-cients aij independent uf the ent~rprise, so thqt • t t X• , = a . . Z.; l J l. .J oJ for any i , j = 1 , ... , n , t = l, ... ,rri. Thc~e coefficients correep0nd to the technical coefficients. From it follows (l.la) I i = l, .•. ,n well known from the classic conceptic i1 of intercommodi ty 1·e-lations. In practice it is ass~med that for any j = 1, ••• ,n lS i=l a .. , 1 ]. J and so (l.la) is the one to one relation among i = l, ... ,n. From (l.Ob) follows, that for any i = 1, ••. ,n could be written ~ t = L__ z. t=l 1 • - 5 -The left side of the last relation represents the sum of con-sumption for further production of a certain commodity classi-fied in accordance with the enterprises and the final consump-tion of a certain commodity. The right side represents the pro-duction of a certain commodity classified in accordance with the enterprises. Thus the commodity-producer model may be obtai-ned. Analogically from (l.Oc) follows the balance (1.3) s=l, ••• ,m, which corresponds to the organization classification of the production. The value zs may characterize the total produc-tion of the enterprises. The last relation describes relations among different enterprises in the consumption for production. There appearethe question of transformation among rela-tions (1.1), (1.2) and (1.3) or the methods of transformation of some of these relations to another ones. This problem will be solved in the next paragraphs. Let us denote that besides the relations mentioned before, some other balance classification could be used, which however, is not the subject of this paper. 2. B9lance in Commodity-Prcd~cer Classification From the previous ~aragrapb we easily see, that zt can • characterize the total production of any enterprise t. When the enterprise t produces comreoJity i = l, ••• ,~ in the quan-tity z! , then we put 1. (2.1) From it follows n (2.2) ;· i=l t b. l = 1 t = l, ... ,n We can see that the production programme of enterprise t could be characterized by b~ , i = l, .•. ,n and by zt • To 1 • a certain extent the proe;rafilf!le of r:i.0 0duction coulJ be defined in advance e.g. by allocating production tasks or by the deci-sions of the enterprise. It comes out that the coefficients bf are frequently dependent on given decisions and so they cannot be considered invariable which means they represent only technological relations. In many a case some "inertia" exists in the production of enterprises and so these coeffi-cients cannot be supposed a subject of short-term essential changes or rather their substantial changes are conditional on the methods of decision. Problems of the methods of deci-sion are not the subject of this paper. Further we shall assu-th bt . . h . . me at i is given.Wen a disproportion apperars in calcula--7-tion, these coefficier.ts may be altered. In accordance with the previous .t - a zt xi j - ij j for any i , j = 1 , ••• , n , which gives t = 1, ••. ,m, n .t X· n ~ -= '-- a •. z~ = L a .. b~ zt J j=l i J J • = t zt r. 1 1. j=l 1J where n (2.3) r~ = > a •. b~ i = 1, ••. , n 1 j=l 1J J t = 1, ••• ,m • So (1.2) may be written as m m (2.4) "5--r~ zt + y! = ~ b~ t i = l, ... , n '-- z t=l 1 • 1 t=l 1 • which are the basic relations for the next analysie. ' Our task is - for given yi , i = l, .•• ,n - to find such proauction programmes of enterprises that the relation (2.4) is fulfilled. Generally this task needn't have any solution if the coefficients rl and bi , i = l, ••• ,n, t = l, ••• ,m are given before. The existence of solution depends on values of YI• Further on we shall show necessary and sufficient condi-tion that the task should have a solution with given values yi. Let y. i = l, ••• >n correspond to the values Z; l · - 1 n ~" th"t ·,·,.:f.l) ;s v~l:i_d. Let ,1 2.i_), - i2,]) tJ'.? l - . ' • • • ' . ,. ·., ·"" - - - - ' m <., b! t (2.5) .. = 17" l = l . . • , n ·- . .. ' t=l ,_ • ~ Proof: m ; r n m "' -- t t \ - •·-••· ht, t ,- - ' b~ t ·, ' r.; ~ = H; z = a, . z .:. t= l_ 7 , • - · lJ <l • - • 5= J .. V IJ ,j= l t=1 t= 1 t=l, ... ,n 9 and fr0!T! it immedi8tely foll ows ~2.4). Contrarywise, from (2.4) f ollows m L m b~ zt + y: = :::_ J • . 1 t=l i=l, ••. ,n. j=l Because to any Yi 1 = l, ••• , n bJ (J .la ) there eYistsonly one z: i = l, ••• ,n we can easily see th&t (2.5) must be valid. 1. That is the proof of our nsserti on. Let us show the appl i.cat ion of this statement. We assume the coefficient s in (l.la) and (2.4) are given. The zt which fulfill the (2.4) can be specified by (2.5) • • The right sides of (2.5) we obtai n from (l.la). - 9 -If the relations (2.5) have no solution, we have to alter the coefficients in (2.4) because the production pro-grammes can teoretically be chosen as desired. In that case t we specify such zi , i = l, ••• ,n , t = l, ••• ,m that i = l, ... ,n From t easi1J sp e cify t and b! and with the use z . we can z l 1 of and t specify the coefficients r! and obtain a .• b. we can 1J 1 ]. the values suitable to (2.4). ). Balance in Producer Classification and Disaggregation We have seen in the previous text, that the balances in commodity classification and in commodit.),·-producer classi-fication are closel;y connected interme ,1iarJ· the structure for production of different enterprises and intermediary the technical coefficients. Let us deal with the problem of relations between the balances in commodity classifica-tion and the balance in producer classification. For this reason let us assume the values xst , z 8 and y 8 , t,s=l, ••• ,m • • • • to fulfill (1.3). Let us also assume that we know the balance in commodi-ty classification i.e. the values x: •. or a .. , z: and y ! 1J 1J 1 1 i,j = l, ••• ,n which~ulfill the relations (1.1) or (l.la). The question is, whether it is possible to specify the va-- 10 -lues x~~, z~ and y~ as well as to fulfill the relations 1 J 1 l. ( 1 • Oa) and ( 1 • Ob) • From the formal point of view there exists an infinite number of possible solutions of this question. So it is clear that we have to use further conditions. Let us see to the quantities describing production and final consumption. In many cases the values zf and yf are know and that's why it is possible to leave the question of their obtaining out. Let only the values z~ and z 8 be known. In that case l. • TI such qut:intities z~ that ~ z~ = z 5 and i i=l l. • we are searching n ~ s z: j_ z, = • s=l l 1. From the previous text comes out that these values can be specified optionally with regard to the technological dependance of commodity. However, it is clear that the pro-duction usage or any further conditions must be taken into account. If these conditions are formulated as the values realized before, then it is poseible to use any disaggrega-tion method (see paragraph 4). If we accept the assumption that the structure of pro-duction of commodity of different enterprises is relatively stead;y then it is convenient to use the first disaggregation method. In this case the weight will be b~. l r - 11 -Thus we obtain m L s=l b ~ ZS = o z 1.· 1. • i = 1, •.. , n The bias will appear if the second relation is not fulfilled. In that case we have to change either zs and z! l or the coefficients bi. Then zf can be extimated by disaggre-gation if they are not given. Analogically the values of final consumption can be treated. Let us study now the consumption for production. From zf and yI it is possible to obtain xf: by (1.0). If x~· and x:~ are known, we can specify the value 1.. lJ of x~•. by their lJ disaggregation. If x~• and 1.. xst are known, we can specify the value . . of x~t by their 1.. disaggregation. By disaggregation st lues of x . .• lJ st s. of x. and x . . l • 1. J we can specify the va-When all these dis~aggregations are non-biasing, we ob-tain the balanced systems. We disclose further possibilities of specifying st X, • l • We shall make use of technical coefficients. Those are assu-med to be independent of the enterprises, and so .t x .. = :LJ t a .. z. lJ J 1 s val ia for i, j = 1, ••• , n and t = 1, •.• , m • - 1. 2 -B t . .p .t ·t . ;/ '36 g re: g 8. . l () D O ,. iC i j ]. :LS • i .1 b+ · o t .:a ~OSSl.LJ e tn O .,91.I: X.: anu .. L • a. t . f' • t d st by isaggregn .1cri 0 .. x. an x 1 • • • we ht . st can o _ 2 1 r, x. . !. • We de-st note, that we c~rn modify th~ meth -J,} Jf oht3ir..i.ng xi. • We can consider t~G relat:on st _ st x· t X• - w. . 1. • 1. 1. • , i. = l, •.. ,n s,t = 1, ... ,m where the weight w~t s = l, ••• ,~ indicates the structure 1 of deliveries of commodity i fl' or!'l different producers to the enterprise t, which m~s~s that is valid. l!l .- st · -· w. = .1 s=l 1 l : , y ~- ' ••• '.I,, , t = 1, ••• ,m This modification assumes that the structure of deli-veries of a certain corr.moji t,y from different enterprises to a certain enterprise is relatively steady. This is similar to the case, when we consider the enterprises as consumers and not as suppliers. The described method can also be modified so that we d · t 8 • • th th ~ t ~ · ht v 9• t and so 1s aggrega · e x. w1 e use 01 1ie we1g s 1. l where st X· 1. m i = i = l, ... ,n, s,t = 1, ••• , m 1, ... , n s = 1, ... , m In this case we prefer the idea that the enterprise (producer of commodit;y) keeps to the structure of distribu-tion of this commodit;y to different enterprises. - 13 -Now we shall study the transformation of (1.1) to (1.)) and otherwise. Let us assume we knew the technical coefficients (aij) and the structure of production of commodity at different enterprises (bi) and the structure of deliveries of commodi-ty i from different enterpris•s to the enterprise t (wft). If we know the values of i = l, ••• ,n, then it is possible to obtain the values of zi and xi~, i,j = l, ••• ,n by using (1.1) and (l.la). By the relation (2.5) we can e3ti-s mate the values of z , s = l, •.• ,m and put . where all rl are defined by (2.)). When the structure of deliveries of commodity i from different enterprises to the enterprise t (w~t) is known, l then it is possible to put and obtain • From the difference between z1~ and x~• we l.. obtain Yi. It is easy to see that by this method we can obtain the values thot fulfill (l.Ob), from which (1.3) can be derived. This description of the solution consists of transfer-ring the balance in commoa ity clac.sificat ion to producer classification. - 14 -Let us deal with the so:ution of an opposite problem: ~□w to obtain the balance in commodity classification out of the given bal9nce in producer classification. We shall start with the values of z 5 , y 6 and xst which • • •• fulfill the relstions (1.)). If the coefficients aij and bi are known then no proble!J1 with the solution, because it is easy to specif;y z~ = b~ zt and from them. to obtain z•1. o l. l. • .• From zi an::l technical coefficients we can obtain xij and we can nut v: = z: - x:· or we can find y1: by (l.la). • "'l l l. All the pievious subject-matters are the methodical instructions for practical application. We must realize that all the aspects were not taken into consideration and that the questions connected with economic interpretation were not the subject of the study. 4. Supplement - Disaggregation Let us consider a quantity f, that resulted form aggre-g~tion of quantities fi. From value of f along with some further information we want to derive f. (disaggregation). l The most common disaggregation depends on the knowledge of hypothetical structure of quantity f, i.e. we know wi such, that ..__-,'-·- w. = 1 nnd we .Put for anv i . f. = w.f . i l. " 1 1 .. - 15 -This method we can modify in following way: from indi-ces • .-)f'. d . h th h 1 f 1 i we create certain classes 1 an wit e e po Known or assumed laws we specify fi* such, that their amount e1uals f. ThP-se quantities may be disaggregated the sa~e method as descri-bed before. The di.saggre5 ation is done with the use of hypothe-i ~ i tical wi~ fulfilling the relation wi* = 1. Thus it is pos-ifi~ sible to write i f. = W•:,t.. f.* 1 1 l i (:_ ·* l. I • Certain laws and relations can be used in disaggregation and so the disaggregated quantities are obtained by substitu-tion off or f.* into these relations. 1 _,..s -'- , Let us assume, now, that we know some quantities f. and 1 which are the amounts of certain ff. From this follows ~- f~ = fs 5-- f~ = f. • 1 , 1 1 i s We want to estimate f~ 1 on the basis of the known quanti-ties. It is easy to see that the condition for the existence of s:..1cl1 quantities is the rel at ion L f 8 = L f. • Anyhow, • 1 8 1 it ie clear there exists an infinite number of possibilities to svecify the st~died quantities. In some cases it is possible to assume the relative sta-bility in relations ( 1 a) s w. 1 0 .. ln .. If these relations are knov,n, then ( 2a) pS S .c-S 1. • : W • .• l 1 From basic conditions and definition f ollows ,, s 7 ~ s fs f' w. - .... w. = .. ' -;- 'l .. _ J. ). J. s s the values w. l by (2a) is biasing from th8 rrqth s ~oticsl poirt cf view. In weight8 so th,cit tht? rel.stionn should b;'.) :'.\i1.filled . bility in relations ..pS (lb) .... v~ = l 1 --r:- 0 J. When the relations are known then disaggregation can be do~e acccrdi~g ta (2b) = s v. l +' J. • l From the conditi'.:H:"1 fur .9dd5tio:-i '.'lfld f:r::-r1, the definition of vP follows l ·, f.'- = 1 ~ v~ li fs v. = '-·- l 1 ~ s , from disngJ~~~ ?t~cn desc~ibe~ by (2b) appears. - 17 -We shall describe another method of disaggregation. This method assumes that fi, fi and f 8 are given and that they fulfill the conditions for addition. If the new values fi and f 8 are given, then it is possible to define the indices pi and p 8 with the help of formulas f. = -L Pi - ' f. 1 fs PS= - -s f and disasgregate according to the formula f s. = r•s s p. • p • 1 1 1 We easily see that along with this method of disaggrega-tion goes that L_ f~ ::- p S l_ P1· f~ = i 1 i 1 -s fs w. 1 where vi and wi represent the weights defined by (la) and (lb) with ff , f 9 and fi • To prevent the bias at this type of dis-aggregation, the new given values f. and f 8 must fulfill the 1 relation [ -s = -s f. v. f l 1 1 ' • - 18 -For disaggregation we can use any of the mentioned methods, or choose the most well-founded one from the sub-ject point of view. It is evident that all these methods can be modified like the first example mentioned in this paper. We shan't deal with these problems. Let us denote at the end that when some values are given in advance, which are to arise from disaggregation, it is advisable to exclude them from disaggregation or to designate propriete weights. 

Problems of Relations Among Balances in Various Classifications

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Problems of Relations Among Balances in Various Classifications

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