Michael Baker - Thesis - Problems in Longterm Forecasting and Planning

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A3. Net Physical Volume of Production

A3.1 Introduction

As explained in Chapter 1, statistics on freight transport generally give details of the tonnes of goods lifted and the distance over which they are moved, expressed in terms of the product tonne-km. An average tonne of a commodity is usually moved more than once so that in general the tonnes of a commodity lifted in any year by all modes of transport exceeds the tonnage of the commodity produced. To find out how often a commodity is moved it is necessary to know how much is produced as well as how much is lifted.

There are several sources of data on the volume of production of commodities. However they do not cover all commodities. Also they tend to be at a more disaggregated level than the commodity classifications used in freight transport statistics. For example it may be possible to find the physical output of both sugar and confectionery. However these cannot simply be added together when finding the output of the commodity food, since some of the output of sugar will be consumed by the manufacture of confectionary. If they were added together directly some the Food commodity output would have been double counted since some of its output is used in its own production. The volume of output required for each commodity is the net output of that commodity.

When aggregating physical output data it is desirable to know how much of each commodity is used in the production of all other commodities within the new commodity group. The simplest way to find this would be from a physical input-output table. There are several methods which could be used to obtain physical input-output tables. Perhaps the best method would be to use original sources such as the Census of Production (COP) (Business Statistics Office annual) which have much physical output data. However, some of this data is in the wrong units (e.g. nos of motor cars rather than tonnes of motor cars).

Possibly a simpler method would be to convert existing monetary input-output tables to physical units using price data. Some prices can be obtained for the COP reports when physical volume is measured in tonnes or when the conversion factor from the physical unit (e.g. litres of fuel oil) to tonnes is known. However, it cannot be used where the conversion factor not known (i.e. for motor cars). There are other possible sources for some commodities, such as the physical volumes of output of commodities as recorded in Central Statistical Office (annual a). However some commodities are not quoted in terms of tonnes (e.g. cars, cutlery, clothing, tyres etc which are usually measured in terms of numbers of items).

Another possible source of price data is the Annual Statement of the Trade of the United Kingdom Volume 1 (Her Majesty’s Customs and Excise annual), but it is only useful for competitive imports. However, even within its very disaggregated commodity groupings those things which are imported are often complementary rather than competitive in that they could not be replaced by domestically produced items because of differences in quality, specification etc and consequently may well cost more (or less) than the domestic ''equivalent'. Where it is expected that both imports and the domestic "equivalents" have the same price this source could be used.

The method of constructing physical input-output tables described in this appendix is one which finds the prices of all commodities at the same time. However, it does entail a substantial degree of estimation and the prices derived should (wherever possible) be checked against any obtained from other sources. If they do not agree the items in the procedure which were estimated should be changed. How this could be done has not been determined.

The appendix contains an example of how the net physical production in 1968 of the eight commodities used in Chapter 1, was found. The method used is then explained mathematically. Details are then given on how a time series both financial and physical input-output tables could be built up.

A3.2 Example based on 1968

To find the net physical production of eight commodities in 1968. the 1968 input-output tables (Central Statistical Office 1973) were first reduced to 12 categories. The first eight categories correspond as closely as possible to the commodity classification used in Chapter 1. The last four categories were Construction; Gas, electricity and water; Transport; and Distribution and services. All four are either services or their output is not moved by what are considered to be freight transport modes. (The movement of such things as for example water, sewage and gas by pipeline are generally not considered as freight transport since they are only moved in their own pipeline systems). A table showing the correspondences used in reducing the tables is shown in Table A3.1.

Table A3.1 Correspoindence between freight Transport Categories and Input-Ouput Categories

Freight Transport Categories (as used in Chapter 1)		1968 Input-Output Categories
1	Food, Drink and Tobacco	1,2,6-14
2	Crude Minerals	4,5
3	Coal and Coke	3,15
4	Petroleum and Products	16
5	Chemicals and Fertilisers	17,18,22-25
6	Building Materials	68-71
7	Iron and Steel	26,27,0.5of54
8	Other	19-21,28-53 0.5of54,55-67 72-80
9	Construction	81
10	Gas, Electricity and Water	82-84
11	Transport	85-87
12	Distribution and Services	88-90

The reduced Make, Absorption and Imports tables are given in Tables A3.2 to A3.4.

Table A3.2 Make Matrix 1968

£ million
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv
Comd'ty
Food	7236	-	-	-	4	-	-	5	-	-	-	-
Minerals	3	226	-	-	-	8	1	-	-	-	-	-
Coal	-	-	1004	-	2	-	-	-	-	5	-	-
Petrol'm	-	-	-	941	27	-	-	1	-	-	-	-
Chemical	12	-	7	5	2251	-	-	79	-	3	-	-
B. mats.	-	3	-	-	8	1025	1	7	15	-	-	-
Iron & S	-	-	-	-	-	1	2127	41	-	-	-	-
Other	1	-	-	1	61	6	54	21925	31	-	-	-
Const'n	10	2	10	-	-	11	8	55	5847	123	26	-
G.El.& W	-	-	24	3	15	-	12	1	-	2181	-	-
Transp't	176	13	2	-	23	29	10	189	-	-	4036	-
Dist.& S	119	-	-	-	47	10	3	276	12	55	7	15306

Table A3.3 Absorption Matrix 1968

£ million
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv
Comd'ty
Food	2299	-	-	1	26	1	-	60	-	1	22	41
Minerals	7	4	-	1	10	36	26	8	91	15	1	-
Coal	12	1	139	-	35	27	107	46	1	384	1	-
Petrol'm	61	11	4	51	88	36	43	110	29	91	61	45
Chemical	216	9	8	36	489	27	26	562	4	7	1	39
B. mats.	65	1	4	-	14	81	39	126	497	9	11	20
Iron & S	8	2	33	-	2	7	510	1146	156	22	14	2
Other	467	21	96	27	198	105	224	6837	925	121	256	1438
Const'n	59	8	28	-	5	2	3	124	946	35	6	64
G.El.& W	70	8	42	8	78	39	119	271	16	67	43	288
Transp't	194	3	44	168	97	123	102	379	103	66	314	500
Dist.& S	725	17	14	28	203	83	147	1588	232	139	146	1047

Table A3.4 Imports Matrix 1968

£ million
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv	Final Dem- and	Total
Comd'ty
Food	982	-	-	-	15	-	-	175	-	-	5	4	1078	2259
Minerals	1	-	-	584	41	24	91	134	1	13	-	-	15	904
Coal	-	-	-	-	-	-	-	-	-	-	-	-	-	-
Petrol'm	17	4	1	79	25	9	9	28	9	27	17	16	54	295
Chemical	30	2	1	6	161	6	5	145	-	2	-	11	59	428
B. mats.	1	-	-	-	1	5	2	12	12	-	-	4	10	47
Iron & S	1	-	2	-	-	1	29	69	21	1	-	-	13	137
Other	29	1	6	2	41	10	39	1637	79	4	22	94	1818	3782
Const'n	-	-	-	-	-	-	-	-	-	-	-	-	-	-
G.El.& W	-	-	-	-	-	-	-	-	-	2	-	-	-	2
Transp't	-	-	-	-	-	-	-	-	-	-	-	-	-	-
Dist.& S	-	-	-	-	-	-	-	-	-	-	-	8	450	2

As illustrated in Figure A3.1, all the material which appears in any commodity initially comes from old scrap (after being used within final demand), imports or primary inputs. Primary inputs are such things as domestically mined minerals and coal, wood and farm produce. The system boundary for imports and exports can conveniently be taken as the point at which they pass through Customs since they are well documented by Her Majesty’s Customs and Excise (annual). The system boundary for primary inputs could be the point at which they are removed from the natural environment. However it is more convenient to take it as the point at which they leave the industry which produces them since this is more likely to be reliably documented. The system foundry for waste is the point at which the waste returned to the natural environment.

Figure A3.1 Structure of the Industrial and Distribution System

As explained below construction of a physical imports table is possible using the same Customs and Excise data from which the original financial imports table was constructed. For simplicity exercise the physical imports table was constructed by distributing the total mass of each of the eight physical commodities over the industries in proportion to the financial entries in the table. This was done by dividing all the entries for each commodity by its price. All the prices are shown in Table A3.5. The one exception was that crude oil was removed from the imports of minerals and treated separately. The complete physical import table is shown in Table A3.6.

Table A3.5 Price of Imports 1968

Commodity	£ million		million tonnes		£/tonne
Food	2259		18.8		120.2
Minerals	904		103.5		-
crude oil		584		81.3	7.2
other minerals		320		22.2	14.4
Coal	-		-		-
Petroleum products	295		23.2		12.7
Chemicals	428		1.0		428.0
Building materials	47		8.6		5.5
Iron and Steel	137		3.5		39.1
Other manufactures	3782		9.5		398.1

Table A3.6 Physical Imports Matrix 1968

million tonnes
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv	Final Dem- and	Total
Comd'ty
Food	8.2	-	-	-	0.1	-	-	1.5	-	-	0.1	-	9.0	18.8
Minerals	0.1	-	-	81.3	2.8	1.7	6.3	9.3	0.1	0.9	-	-	1.0	103.5
Coal	-	-	-	-	-	-	-	-	-	-	-	-	-	-
Petrol'm	1.3	0.3	0.1	6.2	2.0	0.7	0.7	2.2	0.7	2.1	1.3	1.3	4.2	23.2
Chemical	0.1	-	-	-	0.4	-	-	0.3	-	-	-	-	0.1	1.0
B. mats.	0.2	-	-	-	0.2	0.9	0.4	2.2	2.2	-	-	0.7	1.8	8.6
Iron & S	-	-	0.1	-	-	-	0.7	1.8	0.5	-	-	-	0.3	3.5
Other	0.1	-	-	-	0.1	-	0.1	4.1	0.2	-	0.1	0.2	4.6	9.5
Const'n	-	-	-	-	-	-	-	-	-	-	-	-	-	-
G.El.& W	-	-	-	-	-	-	-	-	-	-	-	-	-	-
Transp't	-	-	-	-	-	-	-	-	-	-	-	-	-	-
Dist.& S	-	-	-	-	-	-	-	-	-	-	-	-	-	-

The total of all imports used by each industry can be found by summing the columns in Table A3.6. However, this is of no great interest at present because not all inputs to an industry appear the output of that industry. Many, like coal or petroleum, are consumed by the industry and returned to the natural environment. Others suffer a degree of wastage within the manufacturing process.

A problem to be overcome is that caused by the non-principal products of several industries. These appear as off diagonal entries in the Make Matrix (e.g. £12 millions of chemicals produced by the Food, Drink and Tobacco Industry. see Table A3.2). The problem is caused by the possibility that the various outputs of an industry may require different mixes of inputs, so it is not possible to attribute the commodities absorbed by an industry (Table A3.3) directly to each of its outputs (Table A3.2). A way to overcome this is to construct Make and Absorption tables for industries producing single commodities.

With these highly aggregated tables there were only six elements off the leading diagonal in the make matrix (Table A3.2) which were larger than the corresponding elements in the absorption matrix (Table A3.3). Of these six only one was within the physical commodities section of the table. It seemed to me that a simple way of constructing an absorption table for single commodity industries would be to subtract the make matrix from the absorption matrix, and to leave the leading diagonal blank. The table would then describe the net quantities of each commodity absorbed by each industry. The corresponding net quantity of each commodity produced would be found by subtracting the elements of the leading diagonal in the absorption matrix from the corresponding elements in the make matrix. The resulting two net or modified matrices are shown in Tables A3.7 and A3.8.

Table A3.7 Modified Make Matrix 1968

£ million
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv
Comd'ty
Food	4937	-	-	-	-	-	-	-	-	-	-	-
Minerals	-	222	-	-	-	-	-	-	-	-	-	-
Coal	-	-	865	-	-	-	-	-	-	-	-	-
Petrol'm	-	-	-	889	-	-	-	-	-	-	-	-
Chemical	-	-	-	-	1762	-	-	-	-	-	-	-
B. mats.	-	-	-	-	-	944	-	-	-	-	-	-
Iron & S	-	-	-	-	-	-	1616	-	-	-	-	-
Other	-	-	-	-	-	-	-	15088	-	-	-	-
Const'n	-	-	-	-	-	-	-	-	4900	-	-	-
G.El.& W	-	-	-	-	-	-	-	-	-	2114	-	-
Transp't	-	-	-	-	-	-	-	-	-	-	3722	-
Dist.& S	-	-	-	-	-	-	-	-	-	-	-	14252

Table A3.8 Modified Absorption Matrix 1968

£ million
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv
Comd'ty
Food	-	-	-	1	22-	1	-	55	-	1	22	41
Minerals	4	-	-	1	10	28	25	8	91	15	1	-
Coal	12	1	-	-	33	27	107	46	1	379	1	-
Petrol'm	61	11	4	-	61	36	43	109	29	91	61	45
Chemical	204	9	1	31	-	27	26	483	4	4	1	39
B. mats.	65	-2	4	-	6	-	38	119	482	9	11	20
Iron & S	8	2	33	-	2	6	-	1105	156	22	14	2
Other	466	21	96	26	137	99	170	-	894	121	256	1438
Const'n	49	6	18	-	5	-9	-4	69	-	-88	-20	64
G.El.& W	70	8	18	5	63	39	107	270	16	-	43	288
Transp't	18	-10	42	168	74	94	92	190	103	66	-	500
Dist.& S	606	17	14	28	156	73	144	1312	220	84	139	-

The single commodity industry absorption table is in effect a commodity by commodity table. A more sophisticated method of constructing a commodity by commodity absorption matrix has been developed Clopper Almon (1970). It is based upon the commodity technology assumption. Almon's method could be used in any future analyses and is described below (see Section A3.4).

To convert the modified absorption matrix to physical units the elements for each commodity were divided by that commodity's price. However, before the price of each commodity could be found the net physical volume production of each commodity, corresponding to the financial elements in the modified make matrix, had to be found.

Estimates of the proportion of each commodity absorbed by an industry which appeared in the output of that industry are given in Table A3.9. These proportions may seem high but it must be remembered that any new scrap which was produced by an industry and returned to another industry was part of the output of the first industry. The difference (1 - use coefficient) is the proportion of the commodity absorbed by an industry, which it returned to the natural environment.

Table A3.9 Use Coefficients

(Proportion of commodity i absorbed by industry j  which appears in the output of industry j)
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Dist- rib. &Serv
Comd'ty
Food	1	-	-	-	-	-	-	-
Minerals	-	1	-	.9	.8	.8	.8	.8
Coal	-	-	1	-	-	-	-	-
Petrol'm	-	-	-	1	.5	-	-	-
Chemical	-	-	-	.5	1	.8	.4	.8
B. mats.	-	-	-	-	-	1	-	-
Iron & S	-	-	-	-	-	.9	1	.9
Other	-	-	-	.5	.9	.9	-	1

To find the quantity of imports appearing in the output of each industry it was necessary to multiply the elements of the physical imports matrix (Table A3.6) by the corresponding elements of the Use matrix (Table A3.9). The result of doing this is shown in Table A3.10. The total imports appearing in the output of each industry were found by summing each column and are also shown in Table A3.10.

Table A3.10 Imports which appear in the Outputs of Industries

million tonnes
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Dist- rib. &Serv
Comd'ty
Food	8.2	-	-	-	-	-	-	-
Minerals	-	-	-	73.2	2.2	1.4	5.0	7.4
Coal	-	-	-	-	-	-	-	-
Petrol'm	-	-	-	6.2	1.0	-	-	-
Chemical	-	-	-	-	0.4	0.9	-	0.2
B. mats.	-	-	-	-	-	-	-	-
Iron & S	-	-	-	-	0.1	-	-	4.1
Other	-	-	-	-	0.1	-	-	4.1
Total	8.2	-	-	79.4	3.7	2.3	5.7	13.3

Since the volume of production of primary inputs is generally measured at the point at which they leave their respective industries they were assumed to appear in the output of those industries without loss. In this analysis old scrap was ignored due to lack of data. The totals of primary inputs and imports which appeared in the outputs of industries are shown in Table A3.11.

Table A3.11 Primary Inputs and Imports appearing in the Output of Industries

million tonnes
Industry	Primary Inputs	Imports	Total
Food	72.1	8.2	80.3
Minerals	314.9	-	314.9
Coal	171.0	-	171.0
Petroleum	-	79.4	79.4
Chemicals	-	3.7	3.7
Building  materials	-	2.3	2.3
Iron and Steel	-	5.7	5.7
Other	1.2	13.3	14.5

The imports and primary inputs shown in Table A3.11 appeared in the outputs of these industries. However as shown in Figure A3.1, some of the output of other industries was also absorbed by each industry. The proportion of the output of each industry absorbed by the other industries was found for each commodity by dividing the elements in that row of the modified absorption matrix by the net output of that commodity given in the modified make matrix. The resulting matrix is shown in Table A3.12. Some of the commodities absorbed also appear in the outputs of the respective industries. The proportions absorbed which appeared in the olutputs are those given in the Use matrix (Table A3.9).

Table A3.12 Commodity i absorbed by Industy j per unit of Output of Industry i

Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Dist- rib. &Serv
Comd'ty
Food	-	-	-	-	.004	-	-	.011
Minerals	.018	-	-	.005	.045	.126	.117	.036
Coal	.014	.001	-	-	.038	.031	.124	.053
Petrol'm	.069	.012	.004	-	.069	.040	.048	.122
Chemical	.116	.005	.001	.018	-	.015	.015	.274
B. mats.	.069	-.002	.004	-	.006	-	.040	.126
Iron & S	.005	.001	.020	-	.001	.004	-	.684
Other	.030	.001	.006	.002	.009	.007	.011	-

As in the case of imports, the proportion of the net output of an industry which appeared in the output of another industry was found by multiplying the elements of othe use matrix by the elements of the matrix shown in Table A3.12. The result of doing this is shown in Table A3.13.

Table A3.13 Proportion of the Output of Industry i which appears in the Output of Industry j

App'rs in Output of Industry	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Dist- rib. &Serv
Proportion of Output of Ind'try
Food	-	-	-	-	-	-	-	-
Minerals	-	-	-	.004	.036	.101	.090	.029
Coal	-	-	-	-	-	-	-	-
Petrol'm	-	-	-	-	.034	-	-	-
Chemical	-	-	-	.009	-	.012	.006	.219
B. mats.	-	-	-	-	-	-	-	-
Iron & S	-	-	-	-	-	.003	-	.615
Other	-	-	-	.001	.008	.006	-	-

The flow of imports and primary inputs was then followed throught the productive process as shown in Table A3.14. The first material which appeared in the output of each industry is shown in the first row. However some of this output appeared in the output of other industries. The figures in the first row were transfered to the first column. These were then multiplied by the row of corresponding elements in Table A3.13. This led to further output by some industries. Again some of this further output also appeared in the output of other industries so the totals of the further production of each industry for this round were transfered to the first column again and the process was repeated. The total net production of each industry was then found by summing each column. The reason why this process stops is given below in the section on the mathematical derivation of this method (see Section A3.3).

Table A3.14 Industrial Output

million tonnes
	Outp. prev. round	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf
Primary Inputs & Imports	-	80.3	314.9	171	79.4	3.7	2.3	5.7	14.5
Food	80.3	-	-	-	-	-	-	-	-
Minerals	314.9	-	-	-	1.3	11.3	31.8	28.3	9.1
Coal	171.0	-	-	-	-	-	-	-	-
Petrol'm	79.4	-	-	-	-	2.7	-	-	-
Chemical	3.7	-	-	-	-	-	-	-	0.8
B. mats.	2.3	-	-	-	-	-	-	-	-
Iron & S	5.7	-	-	-	-	-	-	-	3.5
Other	14.5	-	-	-	-	0.1	0.1	-	-
Petrol'm	1.3	-	-	-	-	-	-	-	-
Chemical	14.1	-	-	-	0.1	-	0.2	0.1	3.1
B. mats.	31.9	-	-	-	-	-	-	-	-
Iron & S	28.3	-	-	-	-	-	0.1	-	17.4
Other	13.4	-	-	-	-	0.1	0.1	-	-
Petrol'm	0.1	-	-	-	-	-	-	-	-
Chemical	0.1	-	-	-	-	-	-	-	-
B. mats.	0.4	-	-	-	-	-	-	-	-
Iron & S	0.1	-	-	-	-	-	-	-	0.1
Other	20.5	-	-	-	-	0.2	0.2	-	-
Chemical	0.2	-	-	-	-	-	-	-	-
B. mats.	0.1	-	-	-	-	-	-	-	-
Other	0.1	-	-	-	-	-	-	-	-
Total	-	80.3	314.9	171	80.8	18.1	34.7	34.1	48.5

The prices of domestically produced commodities were found by dividing the value of net production from the modified make matrix (Table A3.7) by the net quantities procuded (Table A3.14). The result of doing this is shown in Table A3.15. For comparison the prices of imports are also shown. As previously mentioned the modified make matrix was found in physical units by dividing the elements of the matrix in financial units (Table A3.8) by the corresponding commodities' prices (Table A3.15). The result is shown in Table A3.16.

Table A3.15 Price of Domestically Produced Commodities 1968

Commodity	£ million	million tonnes	£/tonne	Imports £/tonne
Food	4937	80.3	61.5	120.2
Minerals	222	314.9	0.7	14.4
Coal	865	171.0	5.1	-
Petroleum	889	80.8	11.0	7.2	crude
				12.7	products
Chemicals	1762	18.1	97.3	428.0
Building  materials	944	34.7	27.2	5.5
Iron and Steel	1616	34.1	47.4	39.1
Other	15088	48.5	311.1	398.1

Table A3.16 Modified Absorption Matrix 1968

million tonnes
Ind'y	Food Drink &Tob.	Crude Min.	Coal and Coke	Pet. and Prods	Chem. and Fert.	Build Mats.	Iron and Steel	Other Manuf	Cons- truc- tion	Gas El.& Water	Trans -port	Dist- rib. &Serv
Comd'ty
Food	-	-	-	-	0.4	-	-	0.9	-	-	0.4	0.7
Minerals	5.7	-	-	1.4	14.2	39.7	35.5	11.3	129.1	21.3	1.4	-
Coal	2.4	0.2	-	-	6.5	5.3	21.3	9.1	0.2	74.9	0.2	-
Petrol'm	5.5	1.0	0.4	-	5.5	3.3	3.9	9.9	2.6	8.3	5.5	4.1
Chemical	2.1	-0.1	0.1	-	0.2	-	0.3	5.0	-	-	-	0.4
B. mats.	0.2	-	0.7	-	-	0.1	-	23.3	3.3	0.5	0.3	-

To find the supply of each commodity within the economy it was necessary to add to the net domestic production of each commodity the amount of that commodity imported. However to avoid double counting the amount of commodity imported by the industry which produced that commodity was first subtracted. The total supply of the eight commodities in 1968 is shown in Table A3.17.

Table A3.17 Total Supply of Commodities 1968

million tonnes
Commodity	Imports	Import used by same industry	Remaining Imports	Domestic Production	Total
Food	18.1	8.2	10.6	80.3	90.5
Minerals	22.2	-	22.2	314.9	337.1
Coal	-	-	-	171.0	171.0
Petroleum	104.5	87.5	17.0	80.8	97.8
Chemicals	1.0	0.4	0.6	18.1	18.7
Building  materials	8.6	0.9	7.7	34.7	42.4
Iron and Steel	3.5	0.7	2.8	34.1	36.9
Other	9.5	4.1	5.4	48.5	53.9

One of the sources of error in this estimate of the net domestic production commodities was the very high level of aggregation. Consequently the implied assumption of a constant price for each commodity to every industry doubtful. In future work the level of disaggregation used should be as low as feasible, at which the constant price assumption would be more valid. Any aggregation required to commodity categories corresponding to those used in freights transport statistics would only be made after the physical make and absorption tables had been produced. In the process of aggregation non-zero elements will appear off the leading diagonal of the absorption matrix. Both the make and absorption matrices will be converted to net tables by subtracting the leading diagonal of the absorption matrix from both matrices.

A3.3 Mathematics of the method used to find net physical production

To greatly simplify the analysis we will assume single product industries. (Section A3.4 below explains a method for producing a commodity by commodity absorption table Which is effectively a single product industry absorption table. As an aid to comprehension this section is written in terms of single product industries.) One of the implications of the single product assumption is that no industry produces re-eyeleable new scrap. Any scrap which is produced is waste which is returned natural environment.

The material for all commodities produced initially comes from old scrap, imports and primary inputs (see Figure A3.1). The total of imports going into industry are

n Σ i=1 Cij

j=1,n

where C_ij is the import of commoditiy i by industry j for all i and j. However not all of the imports going into the j^th industry will appear in its output. The remainder is wasted and returns to the environment.

If we let

Uij = proportion of commodity i absorbed by industry j and which appears in the output of industry j (the"use" coefficient)

i=1,n j=1,n

Then the total of all imports by the j^th industry which appear in its output will be

n Σ i=1 UijCij

j=1,n

A conventional assumption can be made that primary inputs first appear in the industry producing that commodity without any loss.

Let:

Pi = primary input of commodity i (to industry i)

i=1,n

and

hi = imports and primary input appearing in the output of industry j

j=1,n

Then

hj = Pj + n Σ i=1 UijDCij

j=1,n

(A3.1)

So far old scrap has been excluded from the analysis but it could be included as a further term in equation A3.1.

The total of domestically produced commodities appearing in the output of industry j is

n Σ i=1 Uijbij

j=1,n

where U_ij is as before and

bij = absorption of cfommodity i by industry j

i=1,n j=1,n

However U_ijb_ij can be expressed in terms of the proportion of the output of commodity i which appears in the output of industry j for all i and j.

Let

ai = net output of commodity i

i=1,n

Wij = the proportion of the output of commodity i which appears in the output of industry j

i=1,n j=1,n

then

Uijbij = Wijai

i=1,n j=1,n

(A3.2)

∴

Wij = Uijbij/ai

i=1,n j=1,n

(A3.3)

So far all quantities (a_i, b_ij, C_ij, P_i, h_i) are in terms of mass. However make and absorption tables are usually expressed in monetary terms. It is possible (see example above and/or the next section) to construct in monetary terms a make matrix with elements a'_ij and an absorption matrix with elements b'_ij,

where

a'ij = 0

i=1,n i≠j j=1,n

and

b'ij = 0

i=1,n

That is all industries only make one commodity and do not absorb any of that commodity. It is also possible to assume that

bij/ai = b'ij/a'ii

i=1,n j=1,n

then from equation A3.3

Wij = Uijb'ij/a'ii

(A3.4)

(As will be seen later this implies that the price paid by all industries for a commodity is the same). From equation A3.2

n Σ i=1 Uijbij = n Σ i=1 Wijai

j=1,n

The output of each industry consists of imports, primary inputs and other commodities.

aj = n Σ k=1 UkjCkj + Pj + n Σ k=1 Ukjbkj

= hj + n Σ k=1 akWkj

j=1,n

or

aT = hT + aTW

where a^T and h^T are row vectors whose elements are

ai and bi

i=1,n

respectively, and W is the matrix whose elements are

Wij

i=1,n j=1,n

aT (I - W) = hT

(A3.5)

or

aT = hT (I - W)-1

Alternatively, continuing from equation A3.1, for all j some of h_ij appears in the output of other industries. That is

hiWij appears in the output of industry j

i=1,n j=1,n.

Over all commodities there will be an extra

n Σ i=1 hiWij in the output of industy j, or for all

industries the extra output will be h^TW.

However some of this extra output will appear in yet more industries output and this will amount to

hTW2

This process can be repeated until we have

aT = hT (I + W + W2 + W3 + … )

= hT (I - W)-1

(A3.6)

This second derivation is the same as the method used in the previous numerical example. The row sums in W all sum to less than one. Consequently the series

I + W + W2 + W3 + … will converge.

To find the size of the elements b_ij we can use equation A3.2 to get

bij = Wijai/Uij

i=1,n j=1,n

but from equation A3.4

Wij = Uijb'ij/a'ii

i=1,n j=1,n

∴

bij = (Uijb'ij/a'ii)(ai/Uij) = b'ij(ai/a'ii)

i=1,n j=1,n

To find b_ij we divide b'_ij by the price a'_ii/a_i for all i and j. As mentioned above this imples that the price of a commodity is the same for all industries.

A3.4 Construction of a commodity by commodity absorption table

The following method of constructing a commodity by commodity a absorption matrix was developed by Clopper Almon (1970). The method is based upon the commodity technology assumption. That is that all the industries which produce a Commodity use the same technology to do so. Consequently all industries require the same mix of inputs for a given commodity. If

B is the absorption matrix, where element bij is the input of commodity i to industry j

A is the market share matrix, where element aij is the fraction of commodity i made by industry j so that

n Σ j=1 aij = 1

i=1,n

(A3.6)

and B' is the modified or pure absorption matrix where element

b'ij is the input of commodity i into indsutry j.

Then B = B'A

or

bij = n Σ l=1 b'ilalj

i=1,n j=1,n

(A3.7)

so

B' = BA-1

(A3.8)

However equation A3.8 leads to some small elements (some of which are negative) in B' which were zero in B. It is possible to use a modified form of iterative procedure for the solution of equation A3.8 which avoids negative elements in B'.

From equation A3.7

0 = B - B'A

and adding B' to both sides

B' = B + B' (I - A)

or

b'ij = bij + b'ij - n Σ l=1 b'ilalj

i=1,n j=1,n

(A3.9)

b'ij = bij - n Σ l=1 l≠j b'ilalj + b'ij(1 - ajj)

i=1,n j=1,n

(A3.10)

Or in words, the quantiry of commodity i used in the manufacture of commodity j is equal to the quantity of commodity i used by industry j less the use of commodity i by industry j for commodities other than j plus commodity i used for commodity j in industries other than j, for all i and j.

Both sides of equation A3.6 can be multiplied by b'_ij to get

b'ij (ajj + n Σ l=1 l≠j ajl) = b'ij

i=1,n j=1,n

b'ij n Σ l=1 l≠j ajl = b'ij (1 - ajj)

i=1,n j=1,n

(A3.11)

Equation A3.11 says that

b'ij n Σ l=1 l≠j ajl

the amount of commodity i removed via the second terms in the set of equations A3.10 from other industries due to their production of commodity j is exactly equal to that added via the third term to industry j's consumption of commodity i (b'_ij(1-a_jj)), for all i and j.

Substituting equation A3.11 in equation A3.10

b'ij = bij n Σ l=1 l≠j b'ilalj + b'ij n Σ l=1 l≠j ajl

i=1,n j=1,n

(A3.12)

It is unlikely that half or more of a commodity will be produced in industries other than that for which the commodity is the principal product. Consequently the row sums of the absolute value of (I - A) will be less than one and the iterative process for the solution of equation A3.12 will converge. The first approximation to b'_ij

is

b(0)'ij = bij

i=1,n j=1,n

then

b(k+1)'ij = bij - n Σ l=1 l≠j b(k)'ilalj + b(k)'ij n Σ l=1 l≠j ajl

i=1,n

(A3.13)

To ensure that

b(k+1)'ij ≥ 0

i=1,n j=1,n

we can modify equation A3.13 such that

bij - fij n Σ l=1 l≠j b(k)'ijalj ≥ 0

i=1,n j=1,n

where

fij = 1, if bij ≥ n Σ l=1 l≠j b(k)'ilalj

i=1,n j=1,n

and

fij = bij / n Σ l=1 l≠j b(k)'ilalj, if bij < n Σ l=1 l≠j b(k)'ilalj

i=1,n j=1,n

or

fij = minimum of (bij and n Σ l=1 l≠j b(k)'ilalj) / n Σ l=1 l≠j b(k)'ilalj

i=1,n j=1,n

(A3.14)

The effect of f_ij (when f_ij < 1) is to reduce, from the otherwise expected level, the quantity of commodity i used in industry j for the production of all commodities except j, by equal amounts.

The amount of commodity i used by industry q for the production of commodity p is

fiqb(k)'ipapq

i=1,n p=1,n q=1,n

The amount of commodity i used by industry q for the production of all commodities except q is

n Σ p=1 p≠q fiqb(k)'ipapq = fiq n Σ p=1 p≠q b(k)'ipapq

i=1,n q=1,n

(A3.15)

The amount of commodity i used by industries other than p for the production of commodity p is

n Σ q=1 q≠p fiqb(k)'ipapq = b(k)'ip n Σ q=1 q≠p fiqaqp

i=1,n p=1,n

(A3.16)

We can replace the second and third terms in A3.13 by A3.15 and A3.16 if we exchange j for q in A3.15 and j for p in A3.16. Then

b(k+1)'ij = bij - fij n Σ p=1 p≠j b(k)'ipapj + b(k)'ij n Σ q=1 q≠j jiqajq

i=1,n j=1,n

(A3.17)

The calculationn of b(k+1)' is now a two pass process involving first the calculation of all f_ij using equation A3.14 and then the use of equation A3.17.

An exactly analogous procedure can also be used to calculate a commodity by commodity imports matrix.

The pure make matrix corresponding to the pure absorption matrix will consist of a matrix whose only elements are a leading diagonal of the total production of each commodity. The pure absorption table will contain non-zero elements on the leading diagonal. To obtain pure make and absorption tables for the net production of each commodity the leading diagonal of the absorption table can be subtracted from both tables.

A problem which I have not resolved with this procedure is that some industries produce new scrap. For example the motor vehicle industry produces scrap steel. However it does not use the same inputs to do so as the iron and steel industry uses to produce steel. A possible solution to this problem is that the factors f_ij might reduce the quantity of inputs to steel to those used by the steel industry.

A3.5 Finding input-ouput tables for years intermediate to those for which firm tables exist

There are input-output tables the years 1954 (Central Statistical Office 1961), 1963 (Central Statistical Office 1970), 1968 (Central Statistical Office 1973) and 1974 (Central Statistical Office 1981). The three tables of interest are the Make, Absorption and Imports tables. Of these the imports tables will be dealt with separately later in this section.

For years other than those indicated above the row and column totals are known or can be inferred form the Blue Book (Central Statistical Office annual b) and from 1969 from Annual Census of Production (Business Statistics Office annual). To obtain updated tables it would be possible to use the RAS method with the nearest adjacent firmly based table. However as Johnson and Lynch (1975) found when applying the RAS method to update the 1963 tables to 1968 there would be serious discontinuities in the time series of the elements of the tables at the points at which the change over from one firm table to the next were made. To overcome this difficulty a process of linear interpolation between the two adjacent firm tables could be used to construct base matrices to which RAS could be applied.

The RAS procedure is reversible. That is from one matrix as base, another set of row and column totals can be used to produce a second matrix. Then using the second matrix with row and column totals for the first matrix it is possible to return to the first matrix. The effect of this reversibility is that there are families of similar matrices from any of which the same matrix will be produced using a given set of row and column totals. Specifically any linear multiple of a matrix is in the same family of similar matrices and can be used instead of the original matrix as base matrix in the RAS procedure.

Consequently, to obtain compatible tables between which to make interpolations, the tables can all be reduced a common basis. This reduction can be achieved by dividing every element in a table by the sum of all elements. In every reduced table the sum of all elements will be equal to one.

To obtain base matrices for the intermediate years linear interpolations could be made for every element between the corresponding elements in the two adjacent reduced firm tables. The sum of all elements in these interpolated base matrices will also be equal to one.

As a test of this procedure tables for 1963 could be constructed using the tables for 1954 and 1968 for interpolation of the base tables and the actual row and column sums from the 1963 tables. The elements of the resulting tables would then be compared with the actual elements in the firm tables for 1963. A similar comparison would also be possible 1968.

Imports table

The basic sources of data for constructing imports tables are the statistics collected by Her Majesty’s Customs and Excise (annual) These statistics have a very fine breakdown into different commodities and usually both the value and quantity are recorded. For most categories of import it is possible to specify which industry was responsible for its import.

Consequently correspondence table between the Customs and Excise Commodity classification and the individual elements of the imports matrix can be constructed. Using correspondence table and the import statistics it would be possible to construct both monetary and physical imports tables.

A3.6 Sources of Data

As explained above the basic source of data for the construction of a time series of financial make and absorption tables would be the firmly based tables produced by CSO (1961, 1970, 1973, 1981). These could be further augmented by data on industrial production for intermediate years from the Blue Book (Central Statistical Office annual b) and the Annual Census of Production (Business Statistics Office annual). The basic source of data for the construction of imports tables would be the Annual Statement of the Trade of the United Kingdom (Her Majesty’s Customs and Excise annual).

Primary inputs comprise the products of agriculture, forestry, fishing and mining. Details in physical quantities of all these are given in the Annual Abstract of Statistics (Central Statistical Office annual a). The figures are also available in Agricultural Statistics for the United Kingdom (Ministry of Agriculture Fisheries and Food annual a), Output and Utilisation of Farm Produce in the United Kingdom (Ministry of Agriculture Fisheries and Food annual b). Sea fisheries Statistical Tables (Ministry of Agriculture Fisheries and Food annual c), United Kingdom Mineral Statistics (Institute of Geological Sciences annual) and Digest of United Kingdom Energy Statistics (Department of Energy annual). I have found no sources for such primary inputs as water and gases abstracted from the atmosphere. Nor do I know of any sources of data for the consumption of old scrap for final demand.

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