THE FINAL PROOF SUPPORTING THE TURNOVER FORMULA.

Transcription

1 THE FINAL PROOF SUPPORTING THE TURNOVER FORMULA. I would like to thnk Aris for his mthemticl contriutions nd his swet which hs enled deeper understnding of the turnover formul to emerge. His contriution needs to e recognized. In his first contriution he modelled the formul-in-generl where ll sles tended towrds common verge, proving its ccurcy. By eginning with the generl, we were following Mrx s method of investigtion which egins with the strct (the pure forms) efore proceeding to the concrete. Only through equl exchnge, necessry strction, ws Mrx le to prove the connection etween vlue nd price. Similrly, hd Aris s modelling proved the formul to e inccurte in its uncomplicted form, then clerly it would hve hd to e discrded. The rel question is how we move from the strct to the concrete therey proving the formul in rel life. Why it is the cse tht no mtter wht independent proof hs een provided, it lwys confirms the ccurcy of the formul, provided of course the ntionl dt on which it is sed is roust. To move to the concrete, I sked Aris to model the turnover formul using unverge sles. The results pper in the pper he presented elow. The importnce of this pper lies in prt rther thn prt 2 (in Tle rther thn Tle 2). The resons will ecome cler soon. Prt shows the effect on turnover of chnges to the weighting of the finl sle (sle 0). It shows tht the higher the weight, the slower the turnover. The numer of turnovers referred to here re highlighted in yellow in Tle. At first glnce this seems to contrdict the formul. How cn ten turnovers suddenly ecome 9 when the weight of the finl sle is 20% rther thn 0%? The nswer is ecuse concretely the turnover ws lwys 9. If the weight of intermedite sles is 80% nd tht of finl sles is 20%, then turnover from the outset would hve een 9 insted of hypotheticl 0. The confirmtion is provided y the mnner in which the sttisticl ureus construct their system of Ntionl Accounts (SNA). The SNA is orgnized y industry. Wht this mens is tht the sum of finl sles is in fct the finl sles for prticulr industry. Tke the Motor vehicles, odies nd trilers, nd prts industry listed under durle mnufcturing. The finl sles here dd up to the nnul sles (less inventory djustments) of motor vehicles nd prts to finl consumers (drivers), delers, lesing compnies, repir shops, nd so on. Wht is importnt is the weight of these sles reltive to the gross output for this industry. The smller the weight, the greter must e the weight of intermedite inputs. Over the lst 30 yers there hs een mjor restructuring of this industry. The ctul cr compnies hve divested much of their mnufcturing se nd reduced themselves more or less to ssemlers of crs. Much of wht they produced in-house they now purchse from the interntionl utomotive chins. The result is tht the weight of their finl sles s shre of the gross output of the cr industry hs shrunk. On the other hnd, the weight of intermedite sles hs grown. The result is tht turnover hs ccelerted from n nnul rte of 6.7 in 997 to 8.3 in 207, or conversely, the circulting period hs reduced from 54.5 dys to only 44 dys. These results re confirmed y ctul nd reported chnges in the industry itself. In other words, the turnover formul is cpturing the dynmic of this industry nd the chnging rtios etween intermedite sles nd finl sles.

2 If we exmine GM s lnce sheet for the 9 months ended 30 th Septemer 207, nd, use its revenue of $98,983 million s gross output nd the sum of deprecition, sg costs (for wges) plus operting income to rrive t GVA we end up with turnover of 8.. (( The closeness in turnover occurs despite the fct tht GM does not relese ctul compenstion figures forcing the use of generl, dministrtive nd selling expenses s sustitute or proxy. Furthermore, this ccelertion in turnover is lwys ssocited with reduction in working cpitl. A glnce t GM s lnce sheet shows tht it not only hs reduced its working cpitl ut offloded prt of it onto suppliers, s credit received exceeds credit given y 300%. Which rings me to the second prt of Aris s modelling. Here he exmines vritions in sles numer 9 (in Tle 2). Now it is importnt to note tht sles 9 dd up to the sum of intermedite sles. These sles generlly elong to the output of other industries. Some of them will constitute finl sles in those industries. For exmple, the power used in the Mnufcturing industry would constitute finl sle tken in the Utilities industry. It is for this reson tht the vrition found in inputs is unimportnt. Wht is importnt is the ggregtion of these inputs to determine the rtio of these inputs reltive to the finl sles of the industry under investigtion. We remin unconcerned with the individul vritions within this ggregte totl. To repet, s fr s the formul is concerned, wht is importnt is the shre of intermedite sles nd the shre of finl sles in Gross Output. The greter the shre of intermedite sles the greter will e the distnce etween gross output nd gross vlue dded, the fster turnover nd vice vers. This is the generl lw. It is wht the turnover formul cptures. And tht is why it is ccurte. Tht is why it cn e verified y independent proofs ecuse they re mesuring the sme thing - the turnover tht is chrcteristic of tht industry. Hd it not een for Aris s hrd work nd the communictions etween us I my hve missed the essentil point tht finl sles were industry sles to other industries or end users. So the formul is mesuring sles coming into the industry versus sles leving the industry, together with the difference in the vlue of these sles ecuse of the vlue dded y the industry itself. This is reflected in the two-prt nture of the formul. Brin Green. 2

3 The Empiricl Turnover Eqution s vrince in one term D Cunh D Cost Dis Aris In this rticle we will exmine Green s empiricl turnover eqution s i vrince in one term. We hve: GO GO GVA 2GO GVA I ; + = (), GVA GVA GVA where I is the pproximte numer of turnovers per time unit, GO is gross output nd GVA is gross vlue dded. An Exmple Following Green ( ii ), we consider 0 industries, ech one s sles used s input of the next one. For industries # to #9, vlue dded is nd for industry #0 vlue dded is. Input equls ( i ) for industry #i, while the vlue of sles is i for industries # to #9 nd 9 + for industry #0. Input plus vlue dded equls vlue of sles. The totl sum of vlue dded is GVA nd totl sles re GO. Input +Vlue dded =Vlue of Sles st industry 0 2 nd industry 2 3 rd industry th industry th industry th industry th industry th industry th industry th industry 9 9+ Totl (Sums) From eqution () we hve: 2GO GVA I = I = I = GVA Let us vry, ttriuting to it k% of the totl vlue dded. 9k = k GVA GVA = = = = k k k 3

4 9k ( ) So we hve k k + k k I = = I = 9k 9 + 9( k) + 9k 9 k We vry, nd clculte the numer of turnovers I: Tle. (in units I (%GVA) of ) 5% 0,47 0,5 0% 0 5%,59 9,5 20% 2, % 3 8,5 30% 3, % 4,85 7,5 40% % 7,36 6,5 50% % 5,5 60% 3,5 5 65% 6,7 4,5 70% % 27 3,5 80% % 5 2,5 90% % 7,5 We lso hve for 00%GVA tht I nd for 0%GVA tht I. Vrince in one term Let s consider the generl cse where only one term vries. So, let every industry s vlue dded e, except the ith for which vlue dded is. Let the numer of industries preceding it e n nd following it m. Input +Vlue dded =Vlue of Sles st industry 0 2 nd industry 2 M M M M n n n th industry ( ) (n+) th industry n n + (n+2) th industry n + n + + M M M M (n+m+) th industry n + + m Totl (Sums) n + m + 4

5 n m Totl vlue of sles (equling GO) is n + mn + ( m + ) + i + j i= j= So we get n m 2 n + mn + m + + i + j n + m + 2 GO GVA i= j= I = I = GVA n + m + n m n + m( 2n ) + ( 2m + ) + 2 i + j i= j= I = ( n + m) + Thus for? we cn see tht: As m 0 (nd 0 ) lim I = m 0 0 As n 0 (nd 0 ) lim I = 2m+ n 0 0 ( ) ( ) As m n (nd 0 ) lim I = 2n + = 2m + = m + n + m n 0 From the ove we cn discern the turnover eqution s dependence on the position of. If the dominnt fctor of vlue production is the lst industry, the turnovers tend to. If it is the first industry then the turnovers re nerly doule tht of totlly homogenous distriution of vlue dded ( = ). Finlly, if the predominnt industry is in the middle of the distriution then the turnover chnges little from the totlly homogenous cse. Vrince in one term for ten industries. So we repet the previous clcultions, vrying y 5% of GVA intervls, nd chnging its industry s position in the set. Tle 2. 5

6 s position (%GVA) (units of ) 5% 0% 5% 20% 25% 30% 35% 40% 45% 50% 0,47,59 2,25 3 3,86 4,85 6 7,36 9 9,50 0,00 0,50,00,50 2,00 2,50 3,00 3,50 4,00 2 9,6 0,00 0,39 0,78,7,56,94 2,33 2,72 3, 3 9,72 0,00 0,28 0,56 0,83,,39,67,94 2,22 4 9,83 0,00 0,7 0,33 0,50 0,67 0,83,00,7,33 5 I 9,94 0,00 0,06 0, 0,7 0,22 0,28 0,33 0,39 0,44 6 0,06 0,00 9,94 9,89 9,83 9,78 9,72 9,67 9,6 9,56 7 0,7 0,00 9,83 9,67 9,50 9,33 9,7 9,00 8,83 8,67 8 0,28 0,00 9,72 9,44 9,7 8,89 8,6 8,33 8,06 7,78 9 0,39 0,00 9,6 9,22 8,83 8,44 8,06 7,67 7,28 6,89 0 0,50 0,00 9,50 9,00 8,50 8,00 7,50 7,00 6,50 6,00 s position (%GVA) (units of ) 55% 60% 65% 70% 75% 80% 85% 90% 95% 3,5 6, ,50 5,00 5,50 6,00 6,50 7,00 7,50 8,00 8,50 2 3,50 3,89 4,28 4,67 5,06 5,44 5,83 6,22 6,6 3 2,50 2,78 3,06 3,33 3,6 3,89 4,7 4,44 4,72 4,50,67,83 2,00 2,7 2,33 2,50 2,67 2,83 5 0,50 0,56 0,6 0,67 0,72 0,78 0,83 0,89 0,94 I 6 9,50 9,44 9,39 9,33 9,28 9,22 9,7 9, 9,06 7 8,50 8,33 8,7 8,00 7,83 7,67 7,50 7,33 7,7 8 7,50 7,22 6,94 6,67 6,39 6, 5,83 5,56 5,28 9 6,50 6, 5,72 5,33 4,94 4,56 4,7 3,78 3,39 0 5,50 5,00 4,50 4,00 3,50 3,00 2,50 2,00,50 6

7 We cn oserve the previously mentioned generl chrcteristics in this exmple of ten industries. When =, the numer of turnovers is 0 (the homogenous cse). If the predominnt industry in terms of vlue dded is t the eginning of the production chin, then the numer of turnovers tends to the doule of the homogenous cse. If the predominnt industry is t the middle of the production chin then the numer of turnovers chnges only little. And, finlly, if it is t the end of the production chin, the turnover ecome less thn 0. i APPLYING THE TURNOVER FORMULA TO THE SYSTEM OF NATIONAL ACCOUNTS TO DETERMINE BOTH THE AMOUNT OF WORKING CAPITAL AND ITS ANNUAL RATE OF TURNOVER, Brin Green, 207. ii APPLYING THE TURNOVER FORMULA TO THE SYSTEM OF NATIONAL ACCOUNTS TO DETERMINE BOTH THE AMOUNT OF WORKING CAPITAL AND ITS ANNUAL RATE OF TURNOVER, Brin Green,