Introduction to Supply and Use Tables, part 3 Input-Output Tables 1

Introduction to Supply and Use Tables, part 3 Input-Output Tables 1 Introduction This paper continues the series dedicated to extending the contents of the Handbook Essential SNA: Building the Basics 2. The aim of this paper is to provide some background to understand the compilation of supply and use tables (SUTs). Since this paper will exceed the intended length of papers in the series, we will distribute its content over three parts, of which the current document is the third 3. In the first part we introduced the basic terminology and structure of the SUT. In part 2 we reviewed the data sources needed for SUT compilation, looked briefly at the balancing process, explored the expression of the use table in basic prices, and reviewed the compilation of the SUT in constant prices. In this last part we will explore the links between SUT and input-output tables. Starting point: Supply and Use Tables In parts 1 and 2 we presented a version of the SNA 2008 example where we aggregated the activities into five industries, as outlined in table 1 below: Agri(culture), Man(ufacturing), Serv(ices) 1, Serv(ices) 2 and Serv(ices) 3 4. Products have been aggregated similarly. For the examples in this part, we will aggregate the 5x5 tables even further, into a 2x2 tables containing Prod(ucts) (consisting of Agri and Man ) and Serv(ices) (consisting of Serv1, Serv2 and Serv3 ), as outlined in table 1. Example part 3 Example parts 1,2 SNA 2008 Prod Agri Agriculture, forestry and fishing Man Manufacturing and other industry Construction Serv Serv 1 Trade, transport, accommodation and food Serv 2 Information and communication Finance and insurance Real estate activities Business services Serv 3 Public Administration Education, human health and social work Other services Table 1 Industries used in the examples The example supply table presented in parts 1 and 2 is as follows: 1 This paper was developed by DevStat Servicios de Consultoría Estadística in consortium with ICON Institute, under the project Essential SNA: Building the Basics, supported by EUROSTAT, for which information can be found at the following link: http://circa.europa.eu/irc/dsis/snabuildingthebasics/info/data/website/index.html 2 Henceforth called the Handbook ; this paper is based on the second (2012) edition; it can be found at the following link: http://epp.eurostat.ec.europa.eu/portal/page/portal/product_details/publication?p_product_code=ks-ra-12-001 3 The other parts are: Introduction to Supply and Use Tables, part 1 Structure; Introduction to Supply and Use Tables, part 2 Data Sources and Compilation 4 The example comes from chapter 14 of SNA 2008. The aggregation only serves to keep the tables small and hence easy to inspect. SNA also makes the distinction between market output, non-market output and output for own final use. Again, to keep the tables small, we will not follow SNA in this respect. 1

Supply Output by industries Output CifFob Imports (bp) Agri Man Serv 1 Serv 2 Serv 3 Total (4) (5) (6) (7) (8) (9) (10) (11) (12) (1) Agri 124 87 87 37 (2) Man 2498 2 2095 27 29 2153 345 (3) Serv 1 289 7 226 0 233-6 62 (4) Serv 2 615 3 7 587 597-4 22 (5) Serv 3 534 2 7 525 534 (6) CifFob 10-10 (7) DP.R 43 43 (8) Total 4103 89 2105 262 623 525 3604 0 499 Table 2 Supply table Total supply (column 4) consists of domestic output, valued in basic prices ( bp ), i.e. prices excluding taxes (column 10), and imports (column 12). Imports have to be Cif-Fob adjusted to make them comparable to output in basic prices (column 11). Also, direct purchases of residents abroad ( DP.R ) have been added to imports. Since we will express the use table also in basic prices there is no need to add columns for net taxes (i.e. taxes minus subsidies) and margins (consisting of trade margins and transport margins). Aggregating this table further into its 2x2 format gives (CifFob, DP.R and imports have been taken together): Prod output Serv output Total output Imports Total supply Prod 2184 56 2240 382 2622 Serv 10 1354 1364 117 1481 Total 2194 1410 3604 499 4103 Table 3 Supply table, aggregated The use table in basic prices was derived in part 2: Use Intermediate consumption by industries Total Exports FCE GCF (bp) Agri Man Serv 1 Serv 2 Serv 3 IC HH GG (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) Agri 124 3 70 3 6 5 87 7 25 2 3 (2) Man 2498 34 896 51 82 111 1174 409 548 3 364 (3) Serv 1 289 4 97 26 14 13 154 71 61 3 (4) Serv 2 615 4 142 40 91 97 374 12 206 23 (5) Serv 3 534 1 2 1 4 38 46 2 123 363 (6) DP.R 43 43 (7) DP.NR 29-29 (7a) Net Tx 133 1 40 2 2 3 48 10 54 0 21 (8) Total 4236 47 1247 123 199 267 1883 540 1031 368 414 Table 4 Use table in basic prices Total use in basic prices is built up from intermediate consumption (IC, column 7) and the final use components exports (column 8), final household consumption (FCE HH, column 9), final government consumption (FCE GG, column 10) and gross capital formation (GCF, column 11). Direct purchases of residents abroad ( DP.R ) are added to household consumption, direct purchases of non-residents in the country ( DP.NR ) are subtracted from household consumption and added to exports. Net product taxes ( Net Tx, subtracted from the use table in purchasers prices during the conversion into basic prices as outlined in part 2) have been added as row 7a. The value added (VA) components have not been presented here. Aggregating this table further into its 2x2 format gives (Exports, HFE and GCF merged into Final demand ; DP.R and DP.NR merged with final demand): 2

Prod IC Serv IC Total IC Final demand Total use Prod 1003 258 1261 1361 2622 Serv 250 324 574 907 1481 Total 1253 582 1835 2268 4103 VA + Net Tx 941 828 Table 5 Use table in basic prices, aggregated In this last table we have added total value added by industry to the net taxes. In the rest of this paper we will present the supply and use tables together in the following form, where imports have been subtracted from final demand, giving final demand net, henceforth abbreviated as FD net : Use table Supply table Prod IC Serv IC FD net Total use Prod output Serv output Total supply Prod 1003 258 979 2240 2184 56 2240 Serv 250 324 790 1364 10 1354 1364 VA + Net Tx 941 828 1769 Total 2194 1410 1769 2194 1410 Table 6 Supply and Use tables in basic prices, aggregated It is now easy to see that the tables are balanced: there is equality between supply and use both by columns (2194, 1410) and by rows (2240, 1364). From SUT to Input-Output Table The idea of an input-output table is that it allows one to construct total output in an industry by following its products as they are used either as inputs in other industries or as final demand. This is captured for our 2x2 case by the following equations: x 11 + x 12 + x 1d = x 1 x 21 + x 22 + x 2d = x 2 Here we use the following symbols: x ij = output from industry i used in industry j x id = (net) final demand for output of industry i x j = output of industry j The first equation says that x 11 units of output of the Prod industry are used by the Prod industry itself, x 12 units of output of the Prod industry are used by the Serv industry and x 1d units of output of the Prod industry are used up as final demand. The sum of these outputs equals total output of the Prod industry, denoted as x 1. We can define input-output coefficients by dividing the inputs by the total industry output as a ij = x ij /x j which can be rewritten as or x ij =a ij x j and substituted in the above equations, giving: a 11 x 1 + a 12 x 2 + x 1d = x 1 a 21 x 1 + a 22 x 2 + x 2d = x 2 These two equations can be combined by using matrix notation 5 as: Ax + Y = x 5 For our purposes, a matrix can be seen a table, so that A is a 2x2 table. 3

Here A is the matrix (or table) with input coefficients (a 11, a 12, a 21, a 22 ), Y is the column with final demand values (x 1d, x 2d ) and x is the column with output values (x 1, x 2 ). We want to obtain these data from the SUT, but in the form given in table 6 this is not possible for several reasons. First, final demand is not given by industry but by product. Second, the use table gives the use of products in industries irrespective of the industry where these products come from. As can be seen in the supply table the Prod industry makes 10 units of services ( Serv products) as secondary output, next to its main output of products of 2184 units. So the use of outputs as inputs cannot be determined by industry. Note however that in case such secondary outputs do not exist the use table would be able to generate the input values by industry, because in that case products are uniquely identified by their industry. Hence, to enable us to construct the matrix A from the intermediate consumption (IC) part of the use table we have to: Remove secondary products from the supply table (with their associated inputs); Replace products by the industries of which they are main product. Carrying this out on the SUT as given in table 6 will give us the industry by industry input-output table (of which the coefficients form the matrix A above), as we will see in the next section. The Industry by Industry Input-output Table The objective is to move secondary output for a particular industry (column in the supply table) up or down the column to the corresponding position of main output for that industry. In the supply table this means that the value 10 (secondary output of the Prod industry) should move up, and the value 56 (secondary output of Serv industry) should move down. Keeping in mind the balances that exist between the rows of the supply table and those of the use table, it comes as no surprise that the corresponding rows in the use table should also change. How do we calculate these changes? The total change over the rows in the use table must be equal to the value of secondary output moved (10 or 56). We can proportionally allocate these values over the use rows in two ways: we can apply the structure of the row where the secondary output comes from or the structure of the row where the secondary output goes to. Let us investigate the first possibility: Use table Supply table Prod IC Serv IC FD net Total use Prod output Serv output Total supply Prod 1003 258 979 2240 2184 56 2240 Serv 250 324 790 1364 10 1354 1364 VA + Net Tx 941 828 1769 Total 2194 1410 1769 2194 1410 Table 7 Assumption of fixed product sales structures This is known as the assumption of fixed product sales structures. The idea is that each product has its own specific sales structure, irrespective of the industry where it is produced. The term "sales structure" indicates the proportions of the output of a product in which it is sold to the respective intermediate and final users. So for the 10 units of output we want to move up we must calculate the row with sales structures in the Serv row of the use table. There are three of these structures: 250/1364*10 = 1.83, 324/1364*10 = 2.38 and 790/1364*10 = 5.79. In the use table these numbers must be subtracted from the second row and added to the first row. A similar exercise can be done to move the 56 secondary output for the Serv industry down. These two sets of changes and the resulting tables are as follows: 4

M4 Use table Supply table Prod Serv FD net Total use Prod Serv Total supply (1) (2) (3) (4) (5) (6) (7) Original Prod (1) 1003 258 979 2240 2184 56 2240 Serv (2) 250 324 790 1364 10 1354 1364 VA + Net Tx (3) 941 828 1769 Total (4) 2194 1410 1769 2194 1410 Change 1 Prod (5) 1.83 2.38 5.79 10 Serv (6) -1.83-2.38-5.79-10 VA + Net Tx (7) Change 2 Prod (8) -25.08-6.45-24.48-56 Serv, (9) 25.08 6.45 24.48 56 VA + Net Tx (10) Final Prod (11) 980 254 960 2194 2194 0 2194 Serv (12) 273 328 809 1410 0 1410 1410 VA + Net Tx (13) 941 828 1769 Total (14) 2194 1410 1769 2194 1410 Table 8 Industry by industry Input-output table (assumption of fixed product sales structures) The final table (rows 11,..,14) is the input-output table and is derived by adding the two change tables (5,..,7 and 8,..,10) to the original table. Note the following: The rows are now interpreted as industries, giving an industry by industry input-output table (IOT). Since we have as many rows as columns (both are broken down by industry) the table is symmetric and we now have a symmetric input-output table (SIOT) in rows 11,..,14. We will label this table M4. The output table is now diagonal. Instead of an output table an output row (or column) is sufficient. The supply table has effectively disappeared and the use table has been transformed into a SIOT. To obtain input-output coefficients from the SIOT we divide the column entries by their column totals; we will come back to this issue in the last section. The table with coefficients is the matrix A above. We can calculate this SIOT (and other SIOTs below) with matrix algebra for any number of rows and columns. We will not give the mathematical details, but the formulas can be found in the UN Handbook Of Input-Output Table Compilation And Analysis (paragraphs 4.40 4.74). These formulas can be implemented in EXCEL 6 for small SUTs (up to 60x60). The second way we can allocate the secondary output is if we can use the structure of the row where the secondary output goes to: Use table Supply table Prod IC Serv IC FD net Total use Prod output Serv output Total supply Prod 1003 258 979 2240 2184 56 2240 Serv 250 324 790 1364 10 1354 1364 VA + Net Tx 941 828 1769 Total 2194 1410 1769 2194 1410 Table 9 Assumption of fixed industry sales structures 6 Using the matrix functions MINVERSE and MMULT; these should be set up as array formulas; this is done as follows: Set up the function in a single cell; Highlight the range of cells where the array will need to come, with the cellpointer in the single cell with the function; Press the edit function key F2; Press <ctrl><shift><enter> 5

This is known as the assumption of fixed industry sales structures. In this case each industry has its own specific sales structure, irrespective of its product mix. So for the 10 units of output we want to move up we must calculate the row with sales structures in the Prod row of the use table (this fixed row is used for all secondary outputs). There are again three of these structures: 1003/2240*10 = 4.48, 258/2240*10 = 1.15 and 979/2240*10 = 4.37. A similar exercise can again be done to move the 56 secondary output for the Serv industry down. These two sets of changes and the resulting tables are as follows: M3 Use table Supply table Prod Serv FD net Total use Prod Serv Total supply (1) (2) (3) (4) (5) (6) (7) Original Prod (1) 1003 258 979 2240 2184 56 2240 Serv (2) 250 324 790 1364 10 1354 1364 VA + Net Tx (3) 941 828 1769 Total (4) 2194 1410 1769 2194 1410 Change 1 Prod (5) 4.48 1.15 4.37 10 Serv (6) -4.48-1.15-4.37-10 VA + Net Tx (7) Change 2 Prod (8) -10.26-13.30-32.43-56 Serv (9) 10.26 13.30 32.43 56 VA + Net Tx (10) Final Prod (11) 997 246 951 2194 2194 0 2194 Serv (12) 256 336 818 1410 0 1410 1410 VA + Net Tx (13) 941 828 1769 Total (14) 2194 1410 1769 2194 1410 Table 10 Industry by industry Input-output table (assumption of fixed industry sales structures) The final table (rows 11,..,14) is again a SIOT, now labeled M3. This SIOT is of course different from M4. Although this may not be evident in the above small 2x2 setup, this SIOT could have negative elements, for example in a more disaggregated form when there would not have been final demand in a particular row where secondary output exists. The Product by Product Input-output Table In the previous section we converted the SUT into SIOT from the industry perspective, by moving secondary output up or down the industry columns of the supply table, with similar movements in the use table. We can switch perspective and move secondary output left or right across the product row of the supply table, again with similar movements in the use table. As before we have two possibilities: we can use the structure of the column where the secondary output comes from or the structure of the column where the secondary output goes to. Let us investigate the first possibility: Use table Supply table Prod IC Serv IC FD net Total use Prod output Serv output Total supply Prod 1003 258 979 2240 2184 56 2240 Serv 250 324 790 1364 10 1354 1364 VA + Net Tx 941 828 1769 Total 2194 1410 1769 2194 1410 Table 11 Industry technology assumption The actual calculations are similar as before but the use ratios are now calculated by column. In this case each industry has its own input structure and if we move the secondary output for the Prod 6

industry, we must use the input structure for that industry. The three ratios are: 1003/2194*10 = 4.57, 250/2194*10 = 1.14 and 941/2194*10 = 4.29. The calculations are as follows: M2 Use table Supply table Prod Serv FD net Total use Prod Serv Total supply (1) (2) (3) (4) (5) (6) (7) Original Prod (1) 1003 258 979 2240 2184 56 2240 Serv (2) 250 324 790 1364 10 1354 1364 VA + Net Tx (3) 941 828 1769 Total (4) 2194 1410 1769 2194 1410 Change 1 Prod (5) -4.57 4.57 Serv (6) -1.14 1.14-10 10 VA + Net Tx (7) -4.29 4.29 Change 2 Prod (8) 10.25-10.25 56-56 Serv (9) 12.87-12.87 VA + Net Tx (10) 32.89-32.89 Final Prod (11) 1009 252 979 2240 2240 0 2240 Serv (12) 262 312 790 1364 0 1364 1364 VA + Net Tx (13) 970 799 1769 Total (14) 2240 1364 1769 2240 1364 Table 12 Product by product Input-output table (industry technology assumption) The final table (rows 11,..,14) is now a product by product SIOT labeled M2. Note that no negatives can arise in this case since one will never subtract more of an input then there is. The other possibility is that we use the structure of the column where the secondary output goes to: Use table Supply table Prod IC Serv IC FD net Total use Prod output Serv output Total supply Prod 1003 258 979 2240 2184 56 2240 Serv 250 324 790 1364 10 1354 1364 VA + Net Tx 941 828 1769 Total 2194 1410 1769 2194 1410 Table 13 Product technology assumption M1 Use table Supply table Prod Serv FD net Total use Prod Serv Total supply (1) (2) (3) (4) (5) (6) (7) Original Prod (1) 1003 258 979 2240 2184 56 2240 Serv (2) 250 324 790 1364 10 1354 1364 VA + Net Tx (3) 941 828 1769 Total (4) 2194 1410 1769 2194 1410 Change 1 Prod (5) -1.83 1.83 Serv (6) -2.30 2.30-10 10 VA + Net Tx (7) -5.87 5.87 Change 2 Prod (8) 25.60-25.60 56-56 Serv (9) 6.38-6.38 VA + Net Tx (10) 24.02-24.02 Final Prod (11) 1027 234 979 2240 2240 0 2240 Serv (12) 254 320 790 1364 0 1364 1364 VA + Net Tx (13) 959 810 1769 Total (14) 2240 1364 1769 2240 1364 Table 14 Product by product Input-output table (product technology assumption) 7

The ratios are now calculated for the column where the secondary output goes to: 258/1410*10 = 1.83, 324/1410*10 = 2.30 and 828/1410*10 = 5.87. Unlike the previous case negative numbers may appear. We have now reviewed the four available methods to reallocate secondary output and come to a SIOT. The input-output values are different for the various approaches, as can be seen in the following table: SIOT 7 Type Assumption Prod,Prod Prod,Serv Serv,Prod Serv,Serv M1 Product Product technology assumption 1027 234 254 320 M2 Product Industry technology assumption 1009 252 262 312 M3 Industry Assumption of fixed industry sales structures 997 246 256 336 M4 Industry Assumption of fixed product sales structures 980 254 273 328 Table 15 Input-output coefficients by method (Prod,Prod = left upper cell, Prod,Serv = right upper cell, etc.) We see from the above analysis that there is no single unique way to derive an input-output table from a given SUT. A typical SUT has many more product rows than industry columns. The first choice is at what level of aggregation the SIOT needs to be prepared. Typically, this is at ISIC 2-digit division level. Although this is not obvious from the above, for some SIOTs (such as M1 and M3) we have to start out with a symmetric SUT, in other cases (M2, M4) we can leave the SUT asymmetric. One also needs to decide whether to compile a product or an industry SIOT. As for the choice between tables, one does not want negative values in the SIOT since these negative inputs have no meaning. As we saw, for some methods negatives can occur. There are in practice other technical methods than the above four to come to a SIOT without negatives. But the best thing is to go back to the original data on output and IC and to make sure that the statistical units for which data is collected are as homogeneous as possible, i.e. with little or no secondary output. In this case the adjustments for secondary output when converting SUT into SIOT will be relatively minor and the likelihood of negatives small. The recommended statistical unit for SUT compilation is the establishment, typically with one activity, rather than the enterprise, with more activities. However, in practice some secondary output is likely to remain. From SIOT to Input-output Analysis An important application of input-output tables is input-output analysis, of which we will give a brief impression here. Starting point are the input-output coefficients which are obtained from the SIOT, e.g. for M3 by dividing by column totals: IO (M3) IO coefficient Prod Serv Prod Serv Prod 997 246 0,45 0,17 Serv 256 336 0,12 0,24 VA 941 828 0,43 0,59 Tot 2194 1410 1 1 Table 16 From input-output value to input-output coefficient The input-output coefficients are contained in the matrix A given earlier. Recall the basic equation linking inputs and final demand to output: 7 Note that our order of presentation was the reverse of our numbering scheme, with M4 being reviewed first and M1 last; we kept the numbering M1,..,M4 in line with the usual presentation in the literature. 8

Ax + Y = x A = matrix of input-output coefficients Y = column of final demand x = column of output We can manipulate the above matrix equation as if it contained single numbers: x-ax = Y (I-A)x = Y I = unit matrix (I-A) = Leontief matrix The solution of this linear equation system is: x = (I-A) -1 Y (I-A) -1 is the inverse of the matrix (I-A) and is called the Leontief inverse. For our example we have: I-IO Leontief Inverse = (I-IO) -1 Prod Serv Prod Serv Prod 0,55-0,17 1,93 0,44 Serv -0,12 0,76 0,30 1,38 Table 17 From input-output coefficients to Leontief inverse Multiplying this inverse with the column of final demand for M3 (column 1 in the table below) gives us output (column 2) 8 : FD Output Δ FD Δ Output Δ FD Δ Output (1) (2) (3) (4) (5) (6) Prod 951 2194 10 19 0 4 Serv 818 1410 0 3 10 14 Table 18 Applying the Leontief inverse Of course the calculated output numbers 2194 and 1410 are not new, these were the total outputs of the Prod and Serv industries in the original SUT. However, once these statistical data on output have been encapsulated in the coefficients of matrix A, the Leontief inverse can be used to calculate other output configurations given other values for final demand. For example, in column 3 in the table above we change final demand for products with 10 units, and leave the demand for services the same (e.g. by lowering a product tax or providing a product subsidy). Applying the Leontief inverse as before will give us the change in output in column 4: 19 units of products more produced (rather than the additional 10 for which there is extra final demand) and 3 units of services, for which there was no extra demand at all. Columns 5 and 6 present the reverse case, with extra demand for services only. The reason for these unexpected results are the inter-linkages in the economy as modeled by the input-output relationships, with extra demand for products inducing extra demand for input services as well. These extra services in turn require additional products for the deliverance, so an additional cycle of extra product demand starts, and so on ad infinitum. It 8 This is matrix multiplication, as done with the MMULT function in EXCEL 9

would be impossible to derive these predications directly from the SUT, and it is for this reason that the SIOT is likely to be more important for economic analysis than the SUT. Concluding remarks This paper set out to provide some background on the derivation of an input-output table based on a balanced SUT. This can be done only after converting the use table to basic prices. In our example we used the total use table, including imports. Alternatively, we can derive the domestic input-output table from the domestic use table in basic prices, as also derived in part 2. Using similar procedures as those outlined in the last section we can then model the impact on domestic output of exogenous changes in final demand. In the next paper we will review another matrix-based extension of the SUT: the Social Accounting Matrix (SAM). To find out more, The 2008 SNA, European Commission, IMF, OECD, UN, World Bank, 2009, Chapter 14 The supply and use tables and goods and services account Handbook of Input-Output Table Compilation and analysis, Series F, No. 74, UN 1999 10