5. THE IOTs IN WAGES-VALUE AND THE FORMATION OF OPERATING SUPLUSES (OSs). EXAMPLE OF THE YEAR 2014.

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I have built the world IOTs in wages-value for the year 2000 to 2014, as well as the Excel tables explaining OSs formation of the 2,464 world productive units for the same years.

Data on labour compensation, I have used for building the IOTs in wages-value, come from the World Input-Output Database site too.

The world IOTs in prices of the WIOD describe 43 countries and the rest of the world, i.e. 44 areas, (list of areas) and 56 activities (list of activities), i.e. 2,464 productive units (56 x 44 = 2,464).

Taking the example of the year 2014, I present below in 5 steps the method I have followed to build the world IOTs in wages-value and the Excel tables explaining the OSs formation of the 2,464 productive units of the world economy. This method follows what I had exposed at the Bordeaux ADEK conference in July 2010, except that at Bordeaux I had described a process (cf. § 6, p.p. 19 et 20) to build a single world IOT in value from several national IOTs in prices. I have not used this process, since I have started from a single world IOT in prices.

I join the computer process I have used.

The 1^st step for building the IOT in wages-value is the determination of the vector $\overrightarrow{\mathrm{VA’}}$ of the year 2014.

This vector of 2,464 elements gives the incomes paid by each of the 2,464 productive units including the rest of the world. The "socio-economic accounts" of the WIOD give the se incomes (wages and labour-incomes of the independent workers ("labour compensation") paid by each of the 56 industries of the 43 countries, but not by the rest of the world. These figures being in local currency have to be converted in U.S. dollars, which is the common currency of the world IOT, using the exchange rates given by the WIOD. This gives the 2,408 (56 x 43) first elements of the vector $\overrightarrow{\mathrm{VA’}}$. For finding the 56 last elements of this vector relating to the rest of the world, I have supposed that, for each industry of the rest of the world, the ratio labour compensation/value added was equal to the average ratio of the 43 countries for the same industry. One can find the value added of the 2,464 productive units (including these of the rest of the world) in the world IOT in prices under the Excel or Rdata form.

One can see the vector giving the labour compensations of the 2,464 productive units thus obtained under a matrix form. Despite this matrix presentation, it is a true vector, the vector $\overrightarrow{\mathrm{VA’}}$, which can be got by writing end to end the different lines of the matrix.

2^nd step: building of the vector $\overrightarrow{\mathrm{K}}$ giving the ratios wages-value/price for each of the 2,465 productive units.

The IOT in prices gives the vector $\overrightarrow{\mathrm{\Sigma }}$ of the production of the 2,464 productive units and the matrix of intermediate consumption IC (2,464 x 2,464). One can get the diagonal matrix Σ and the square matrix (2,464 x 2,464) Σ - IC. But this matrix has no inverse, because some elements of $\overrightarrow{\mathrm{\Sigma }}$ are null, which implies that some column vectors of Σ - IC are also null. As a remedy to this problem, I have replaced all the null elements of $\overrightarrow{\mathrm{\Sigma }}$ by one. The only effect of that change is to replace a null production by a production of 1 million of USD, which is negligible on a world level.

After that, the solve function of R gives the vector $\overrightarrow{\mathrm{K}}$ of which I give the matrix form similar to that of $\overrightarrow{\mathrm{VA’}}$ .$$

3^rd step: synthesis and verification.

We know the vector $\overrightarrow{\mathrm{K}}$, thus we know the IOT in wages-value, but it is difficult to represent the whole of it, since each line of the 2,464 commodities is spread on 2,690 columns. Nevertheless I give a synthetic representation of the totals measured in wages-value and US dollars for the year 2014:

One can verify that the total of the rows of commodities equals the total of the columns of industries.

4^th step: analysis of the formation of accounting OSs (i.e. of the OSs as they are measured in the accounts).

I decompose the OSs of the different productive units following the method developped in my Bordeaux communication. The accounting OS of these units is given by the vector ${\overrightarrow{\mathrm{P}}}_{\mathrm{A}}$ of 2,464 elements (cf. paper, equation 13, bottom of page 15) with:
${\overrightarrow{\mathrm{P}}}_{\mathrm{A}}=\; (\overrightarrow{\mathrm{C}}-\overrightarrow{\mathrm{C\text{'}}})\; +\; (IC\; -\; IC\text{'})\overrightarrow{\mathrm{u}}-{}^{\mathrm{t}}\mathrm{(IC\; -\; IC\text{'})}\overrightarrow{\mathrm{u}}+\; (Inv\; -\; Inv\text{'})\overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{(\Delta S}}-\overrightarrow{\mathrm{\Delta S\text{'}}})$

The vector ${\overrightarrow{\mathrm{P}}}_{\mathrm{A}}$ of the world productive units accounting OSs may be written in 5 components I build below.

The calculus results are given in a unique Excel file here with the following sheets:

1. vector $\overrightarrow{\mathrm{C}}$
2. vector $\overrightarrow{\mathrm{C\text{'}}}$
3. primitive OS (1^st component) $\overrightarrow{\mathrm{\Pi }}=\overrightarrow{\mathrm{C}}-\overrightarrow{\mathrm{C\text{'}}}$, this sheet being the difference between sheets 1 and 2
4. earnings coming from intermediate consumption's sales (2^nd component) $\mathrm{(IC\; -\; IC\text{'})}\overrightarrow{\mathrm{u}}$
5. losses coming from intermediate consumption's purchases (3^rd component) ^t(IC - IC')  $\overrightarrow{\mathrm{u}}$
6. net earnings coming from intermediate consumption exchanges, i.e. earnings minus losses of both precedent sheets
7. net OS after intermediate consumption exchanges. This sheet is the sum of sheets 3 and 6
8. earnings coming from GFCF's sales (4^th component)
9. OSs coming from the production of commodities sold to be stored  $\overrightarrow{\mathrm{\Delta S}}-\overrightarrow{\mathrm{\Delta S\text{'}}}$ (5^th component)
10. accounting OSs ${\overrightarrow{\mathrm{P}}}_{\mathrm{A}}$ got by addition of sheets 7, 8 and 9

At last the same Excel file presents three more sheets whose object is wages-value OSs calculus (see 5^th step below).

Each sheet is a matrix (56 industry rows, 44 area columns) giving the values of each productive unit. If one considers end to end the columns of each of these matrices, one gets vectors of 2,464 elements equivalent to the n elements vectors of the theorical model.

a) building of the 1^st component $\overrightarrow{\mathbf{\Pi }}=\overrightarrow{\mathbf{C}}-\overrightarrow{\mathbf{C\text{'}}}$.

The primitive OS, realized only by the sales of consumption commodities before any exchange between firms, is given by the vector $\overrightarrow{\mathrm{\Pi }}=\overrightarrow{\mathrm{C}}-\overrightarrow{\mathrm{C\text{'}.}}$. This vector, and the two vectors which compound it, have, for each of them, 2,464 elements and can be represented with a matrix of 44 rows of countries x 56 columns of activities.

Each element of these vectors represents the sales of one of the 2,464 commodities as consumption good to the 44 countries (including the rest of the world and the country making the good). On a IOT the final uses appear on a matrix  located on the "right" of the matrix IC of the intermediate consumptions. With the software R, I have extracted from the IOT in prices published on the WIOD site, the matrix efprix of the final uses in prices. After that, one need to extract from this last matrix the sales of consumption goods from each productive unit to the different countries.

The final purchases of each country are divided on five successive columns (purchases by the consumers, purchases by non-profit institutions serving households, purchases by government, purchases of GFCF, variations of stocks). I have considered all the sales of consumption, therefore the sales given by the first three columnns. I have given a double index to each productive unit, an index b for the activity varying from 1 to 56 (cf. list of activities), and the index p for the area varying from 1 to 44 (cf. list of countries).

Thus one can represent the vector $\overrightarrow{\mathrm{C}}$ by building the matrix Ctotprix (44 x 56) with the program lines in R language:

Aefprix <- array(t(efprix),dim=c(5,44,56,44))
# 5 types of final use, 44 countries purchasers, 56 activities producers, 44 countries producers
Ctotprix[b,p]=sum(Aefprix[1:3,,b,p]) 

The last program line means that only the first three types of final use have been selected, that all the coutries are purchasers (no figure between both commas), but that only the activity b of the country p is producer.

The matrix giving the final uses measured in value is efval = K efprix and from program lines similar to the above lines, we can build the matrix Ctotval (44 x 56), which represents the vector $\overrightarrow{\mathrm{C\text{'}}}$.

Results are written on the two firsts sheets of the Excel file for $\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{C\text{'}}}$. The initial OS $\overrightarrow{\mathrm{\Pi }}$ is written on the 3^rd sheet by difference between the two first.

b) building of the 2^nd component, gains realized by the sale of intermediate consumption (IC - IC') $\overrightarrow{\mathrm{u}}$ and of the 3^rd component, losses ^t $\overrightarrow{\mathrm{u}}$ (IC -IC') realized by purchase of intermediate consumption.

The calculus of gains and losses made by each of the 2,464 productive units is done easily from the matrices of intermediate consumption in prices IC and in value K IC.

One can verify that the total of the earnings got by the whole of the 2,464 productive units, that are written on the sheet 4 of the Excel file, equals the total of the losses got by the whole of the same productive units written on the sheet 5. Therefore the total of the net earnings of the whole of the productive units written on sheet 6 is null.

On sheet 7, I write the OSs got by each productive unit after the intermediate consumption exchanges have occured. That can be done by summing sheets 3 and 6. Thus it is a redistribution of the OSs $\overrightarrow{\mathrm{\Pi }}$ got in selling consumption commodities. One can verify that the total of these last OSs is equal to the total of the initial OSs.

c) building of the 4^th component, gains made by each productive unit when selling GFCF.

We have no matrices (2,464 x 2,464) Inv and Inv' as defined in my communication, in page 11, but efprix and efval give in 4^th position the sales of GFCF from each of the 2,464 productive units to the 44 areas respectively in prices and values. To get the sales of GFCF measured in prices by the productive unit whose indexes are b and p, one must apply the following program line: FBCFprix[b,p]=sum(Aefprix[4,,b,p]).

A similar program line gives the measure in value. Therefore one can calculate the gain got by each produtive unit. Results are given on sheet 8.

d) building of the 5^th and last component: vector of the OSs earned by the sale of commodities to be stored. $\overrightarrow{\mathbf{\Delta S}}-\overrightarrow{\mathbf{\Delta S\text{'}}}$.

The commodities produced by a productive unit, then stored by the same or an other productive unit, are given by the 5^th position of the matrices efprix and efval. Therefore the earnings made by the productive unit whose indices are b and p when producing commodities that are eventually stored are given by gainsStocksProd[b,p] = sum(Aefprix[5,,b,p]) - sum(Aefval[5,,b,p]). The results are recorded on the 9^th sheet of the Excel file.

e) displaying of the accounting OS ${\overrightarrow{\mathbf{P}}}_{\mathbf{A}}$ by summation on the 10^th sheet

5^th step: building of the final OSs vector in wages-value.

As I explain in my Bordeaux communication, §4.3, pp 18 and 19, to get the final OSs in value, one has to omit the 5^th component relative to the gains done with stock variations and to deduct a component I have called "4^th A" giving the losses done when buying GFCF. But there are no data relative to this new component in firms accounts whose goal is the calculus of accounting OS, and not a OS in value. Therefore the IOT, which is built on the same principles, cannot be used to calculate the losses done by each productive unit when buying GFCF.

But we can, through the world IOT's, either in prices or in wages-value, calculate the losses done by the whole of the productive units of a given area (country or rest of the world) when buying GFCF. Therefore I will begin to calculate these losses. Then I will estimate the losses done by each productive unit in sharing the losses of each area between the productive units of the same area.

Results are displayed on sheets 11 (estimation of losses by purchasing GFCF), 12 (estimation of the net earnings done through GFCF exchanges) and 13 (OSs in wages-value) of the Excel file.

a) building of the losses made by each area when purchasing GFCF

The purchases of GFCF by the area from all the productive units are given by the program line: FBCFprix[p]=sum(Aefprix[4,p,,]).

A similar program line gives their measure in value. Therefore one can calculate the losses realized by all the productive units from a same country when purchasing GFCF.

The result, i.e. the vector of 44 elements giving the losses coming from the GFCF purchases by each of the 44 areas, is written on line 63 of the sheet 11 of the Excel file. One can verify that the total of the gains made by the whole of the units (sheet 8) is equal to the total of the losses made by the whole of the areas.

b) estimation of the losses of each productive unit when buying GFCF

We have no matrix Inv, as defined in my communication at Bordeaux. This matrix would have given the purchases of GFCF by each of the 2,464 productive units from the different productive units.

I have estimated these purchases through a matrix of GFCF distribution between the activities for France and for the year 2009, which I have got unofficially and with courtesy from the French statistical office. This matrix gives the distribution of purchases of GFCF by France between the French productive units.

I have supposed that this distribution applied without any difference to all areas for the year 2014.

These estimations are given on the sheet 11 "pertes achats FBCF" of the Excel file. Their totals by area appear on line 58. To allow a comparison I have copied on line 63 the totals of the losses by area got directly (cf. § d). The figures are slighty different. Thus the total of the world losses got by summation of the losses of the 2,464 world productive units is inferior of 2,700 millions of US dollars to the total got by summation of the losses by area. This difference, inferior to 35 hundred thousandths, is absolutely negligible.

Then I introduce the sheet 12 giving the gain of each productive unit when selling GFCF net of its losses when buying GFCF. The total of these net gains should be null at the world level, but as the losses when buying GFCF have been underestimated of 2,700 millions, the sum of these net gains are overestimated of the same figure.

c) displaying of the final OSs vector in wages-value

Finally, on the same file Excel, I add the sheet 13 "OS final en valeur" which gives the vector of the final OSs in wages-value of the different productive units of the world, which I get by adding to the initial OSs vector $\overrightarrow{\mathrm{\Pi }}$ (sheet 3) the net gains coming from the sale of intermediate consumption (sheet 6) and of GFCF (sheet 12). One can verify that the sum of the OSs thus obtained differs from the sum of the initial OSs for the negligible amount of 2,700 millions of US dollars.