# Terminology

## Introduction

This vignette reviews the terminology used in input–output analysis, with reference to the Eurostat Manual of Supply, Use and Input–Output Tables (Beutel, 2008).

Work in progress: a more comprehensive review will follow, comparing terminology with the UN Handbook (2018), OECD input–output matrices, and recent economics literature. This vignette should therefore be read as descriptive, showing how the iotables package currently uses Eurostat-style terminology.

    library(iotables)

Retrieve the demo dataset, the Germany 1995 Symmetric Input-Output Table:

    germany_siot <- iotable_get()

### First quadrant

The first quadrant of the symmetric input-output tables, or in other words, the domestic intermediate matrix, input-output flows between a simplified, 6 sectors of the German economy in 1995.

    germany_siot[c(1:7), c(1:7)]
    #>              iotables_row agriculture_group industry_group construction
    #> 1       agriculture_group              1131          25480            1
    #> 2          industry_group              7930         304584        64167
    #> 3            construction               426           7334         3875
    #> 4             trade_group              3559          72717        14190
    #> 5 business_services_group              3637          96115        31027
    #> 6    other_services_group              1552          14986         1747
    #> 7                   total             18235         521216       115007
    #>   trade_group business_services_group other_services_group
    #> 1         607                     710                  762
    #> 2       41082                   11981                30360
    #> 3        5296                   23457                 9155
    #> 4       74399                   10835                21008
    #> 5       65755                  193176                34223
    #> 6       11225                   15058                22070
    #> 7      198364                  255217               117578

Row-wise, we see the flow of inputs. For example, at basic prices, out of the total demand for business services (423 933), 193 176 of the production of business services is used by other business service providers, 65 755 by trade enterprises, and 34 223 by other service industries.

    germany_siot[5, c(1, 5:8)]
    #>              iotables_row trade_group business_services_group
    #> 5 business_services_group       65755                  193176
    #>   other_services_group  total
    #> 5                34223 423933

The columns describe the structure of the input of the corresponding sector. For example, the business service sectors needed 15058 input from the other services, and 13371 imports from abroad (shown in the table below), to meet the final domestic demand of 423933 for business services (shown in the table above):

    germany_siot[c(5:9), c(1, 6)]
    #>               iotables_row business_services_group
    #> 5  business_services_group                  193176
    #> 6     other_services_group                   15058
    #> 7                    total                  255217
    #> 8                  imports                   13371
    #> 9 intermediate_consumption                  277061

## Coefficients

### Input coefficients for domestic intermediates

The `input_coefficient_matrix_create()` function relies on the following equation. The numbering of the equations is the numbering of the Eurostat Manual.

9. [recap: (43) is the same]

\(a_{ij}\): input coefficient for domestic goods and services \(x_{ij}\): flow of domestic commodity i to sector j, i.e., a cell in the first quadrant. \(x_j\): output of j

    input_coefficient_matrix_create_2(iotable_get(), digits = 2)

### Output coefficients

The output coefficients are ratios derived from quadrant I (intermediates) and quadrant II (final demand) of a sector.

The `output_coefficient_matrix_create()` function creates these coefficients based on equation (5) in the Eurostat Manual.

5. \(o_{ij}\) = \(x_{ij}\) / \(x_i\)

\(o_{ij}\) = output coefficient for domestic goods and services (i = 1, …, 6; j = 1, …, 6) \(x_{ij}\) = flow of commodity i to sector j \(x_j\) = output of sector i

## Inter-industry analysis

The term forward linkage refers to the interconnection of a sector to those to which it sells its output. In its simplest form, the strength of the forward linkage of a sector i is given by the row-wise sum of the direct output coefficients. This considers the direct supply-side economic effects of sector i on the entire economy.

The backward linkage of a sector j is given by the column sum of the direct input coefficients. This captures the direct demand-side effects of sector j on the economy. When calculated from the Leontief inverse, backward linkages also include the indirect effects propagated through the supply chain.

## Next steps

This terminology vignette is intended as a descriptive overview. Future releases of `iotables` will expand this section to align with the UN Handbook on Supply and Use Tables and Input–Output Tables with Extensions and Applications (2018), OECD input–output matrices, and recent economics literature. A systematic review of terminology and extended validation tests are planned for the next major release.