Official Statistics

Annex A

Published 19 September 2024

1. Introduction

For further transparency on the methodology used in the Regional Employment bulletin, Annex A has been designed to guide users through the process of estimating indirect jobs by a worked example.

2. Methodology in Practice

As a worked example, we can explore the indirect jobs in the Manufacturing of Metal Products (MOD SIC group 17) as a result of expenditure on Shipbuilding and Repairing (MOD SIC group 27).

From Table 6 of MOD’s Regional Expenditure bulletin’s supporting data tables we know that the direct spend on Shipbuilding and Repairing was £5,497 million in 2022/23. To see how many indirect jobs this sector alone supports we can take this figure as our MOD vector of final demand and set all other values to zero.

Let f be the MOD vector of final demand, where here we now have,

Description of mathematical notation: defines the input vector of final demand. In abridged format it shows f equals a fifty-two by one matrix where the input expenditure for Shipbuilding, SIC group 27, is £5,497 million. All other groups have input spend as zero.

We then need to construct the aggregated Input-Output Table (IOT) according to MOD’s own 52 SIC groupings from the 100+ SIC codes used in the Office for National Statistics’ (ONS) IOT tables starting with the outlined IOT in Table 1. Note that this is a subset of the whole table for illustrative purposes.

Table 1: 2019 Input-Output Table (Domestic Use, Basic Prices, Product by Product, £ million)

	01	02	03	..	94n	..	Total Demand for Products at Basic Prices
01	4,219	1	0	..	0	..	28,145
02	86	210	0	..	0	..	1,482
03	0	0	0	..	0	..	2,084
..	..	..	..	..	..	..	..
Total Consumption	16,295	699	1,012	..	0	..	3,824,433
Imports of Goods and Services	652	20	320	..	0	..	735,802
Taxes Less Subsidies on Products	437	13	28	..	0	..	238,191
Taxes Less Subsidies on Production	-1,833	-2	4	..	0	..	26,908
Compensation of Employees	3,338	391	370	..	0	..	1,090,157
Gross Operating Surplus	9,256	360	350	..	0	..	883,092
Total Output	28,145	1,482	2,084	..	0	..	3,824,433

The ONS IOT displays domestic use in Agriculture, Forestry and Fishing against the separate SIC codes of 01, 02 and 03 respectively. For MOD expenditure these are grouped together as MOD SIC group 1, so we simply sum each entry in row and column for SIC codes 01, 02 and 03 to get the aggregated domestic use for MOD SIC group 1. This is repeated across all ONS SIC codes to transform the IOT into the now aggregated IOT table according to MOD’s 52 SIC groups as shown in Table 2.

Table 2: Aggregated 2019 Input-Output Table (Domestic Use, Basic Prices, Product by Product, £ million)

	1	2	3	..	48	..	51	52
1	4,516	0	0	..	0	..	51	0
2	0	0	0	..	0	..	4	0
3	1	0	804	..	0	..	2	0
..	..	..	..	..	..	..	..	..
48	0	0	0	..	0	..	0	0
..	..	..	..	..	..	..	..	..
51	225	0	15	..	0	..	8,982	0
52	0	0	0	..	0	..	0	0
Total Consumption	17,994	192	5,937	..	0	..	44,306	0
Imports of Goods and Services	991	7	976	..	7,499	..	7,227	0
Taxes Less Subsidies on Products	477	4	61	..	6,785	..	3,967	0
Taxes Less Subsidies on Production	-1,832	-12	78	..	0	..	1,123	0
Compensation of Employees	4,100	79	1,541	..	67,602	..	39,553	3,917
Gross Operating Surplus	9,967	89	7,589	..	28,417	..	33,958	0
Total Output	31,699	358	16,182	..	110,303	..	130,133	3,917

Since we are only interested in jobs supported in industry, figures under SIC group 48 for public administration and defence are then nilled out to remove later interactions within the public sector. We then create the A matrix by dividing each cell by the column sum in Total Output, with the result shown in Table 3. We also drop any rows and columns that do not directly relate to SIC groups to get the A matrix in the desired format of 52 rows by 52 columns for each SIC crosstab interaction.

Table 3: The A Matrix (Matrix of Coefficients, Product by Product, Nilled Public Administration)

	1	2	3	..	48	..	51	52
1	0.1425	0.0000	0.0000	..	0.0000	..	0.0004	0.0000
2	0.0000	0.0000	0.0000	..	0.0000	..	0.0000	0.0000
3	0.0000	0.0000	0.0497	..	0.0000	..	0.0000	0.0000
..	..	..	..	..	..	..	..	..
48	0.0000	0.0000	0.0000	..	0.0000	..	0.0000	0.0000
..	..	..	..	..	..	..	..	..
51	0.0071	0.0000	0.0009	..	0.0000	..	0.0690	0.0000
52	0.0000	0.0000	0.0000	..	0.0000	..	0.0000	0.0000

A MOD version of the Leontief Inverse matrix according to MOD’s own SIC groups is then generated using the equation (I – A)-1, where I is the identity matrix.

Calculate the Leontief Inverse matrix,

Description of mathematical notation: calculates the Leontief Inverse matrix. The A matrix is subtracted from the identity matrix, and the inverse taken. This returns the Leontief Inverse matrix as a fifty-two by fifty-two matrix.

This constructs the Leontief Inverse matrix shown above. Since in this worked example we are looking at interactions on Manufacturing of Metal Products from Shipbuilding expenditure it has been expanded to highlight figures against SIC groups 17 and 27.

MOD total demand across all SIC groups is then calculated by multiplying the Leontief matrix by the MOD vector of final demand, i.e. (I – A)-1 f, where f is defined by MOD expenditure on Shipbuilding and Repairing.

Calculate MOD Total Demand,

Description of mathematical notation: calculates total demand. The calculated Leontief Inverse matrix is multiplied by the initial vector of final demand to produce a fifty-two by one matrix for total demand.

To remove the impact of direct expenditure on demand we subtract the MOD vector of final demand from total demand to return intermediate demand, i.e. that which occurs throughout the supply chain.

Calculate Intermediate Demand,

Description of mathematical notation: calculates intermediate demand. Subtracting final demand from total demand yields intermediate demand. In our example, this is a fifty-two by one matrix with values for intermediate demand now calculated for each SIC group.

Then, if we focus on only SIC group 17 for Manufacturing of Metal Products, from the ONS SUT Supply of Products table, published for 2019, we see that total domestic output of products in basic prices for this industry was £49,372,000,000. This total output figure includes products in iron and steel, other metals and casting, fabricated metal products, and further elements arising from the installation and repair of metal products.

Using employment data from the BRES, we can adjust full-time and part-time employees into FTE employment equivalents. For SIC group 17 this equates to 323,884 FTE employees in this industry. Dividing output by employment here provides the output at basic prices per FTE employment as £152,436 per FTE employment.

The estimate of indirect jobs supported in the Manufacturing of Metal Products solely as a result of MOD expenditure on Shipbuilding and Repairing is then given by dividing the respective intermediate demand for metal work, i.e. SIC group 17, by the industry’s output per FTE employment.

174,874,413 / 152,436 = 1,147 indirect jobs

In this case we therefore estimate that MOD’s £5,497 million of direct expenditure with UK industry in the Shipbuilding and Repairing sector supported a further 1,147 indirect jobs in the Manufacturing of Metal Products industry group.

3. Contact Details

Analysis-Expenditure Head of Branch

Telephone: 030 015 86554

Email: Analysis-Expenditure-PQ-FOI@mod.gov.uk

Analysis Directorate (Analysis-Expenditure)
Ministry of Defence
Oak 0 West, #6028
MOD Abbey Wood North
Bristol
BS34 8QW

For general MOD enquiries, please call: 020 7218 9000

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