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).
This vignette is therefore descriptive, documenting how the
iotables package uses Eurostat-style terminology
consistent with ESA 2010 and the System of National Accounts (SNA
2008).
Retrieve the demo dataset, the Germany 1995 Symmetric Input-Output Table:
An input–output table (IOT) describes the flow of products and services in an economy, linking industries (producers) with their inputs, outputs, and final uses. In a symmetric input–output table (SIOT), both rows and columns represent either products or industries.
Eurostat distinguishes four quadrants of a SIOT:
| Quadrant | Description | ESA 2010 / SNA 2008 Category |
|---|---|---|
| I | Intermediate consumption — flows between industries or products | Intermediate use |
| II | Final demand — purchases by households, government, investment, exports | Final use |
| III | Primary inputs — compensation of employees, net taxes, gross operating surplus | Primary inputs |
| IV | Output totals — column sums at basic prices | Output (P.1) |
Quadrants I–III correspond to the rows of the table, while the column totals in Quadrant IV represent the total output of each industry or product.
Input–output analysis is built on a small set of matrices and
vectors.
The iotables package follows the same notation used in
Beutel (2008).
| Symbol | Meaning | Definition | Function in iotables |
|---|---|---|---|
| Z | Intermediate-consumption matrix | Flows of products between industries | iotable_get() |
| A | Input (technical) coefficient matrix | \(A_{ij} = Z_{ij} /
X_{j}\)|input_coefficient_matrix_create()| | **(I − A)** | Leontief matrix | Identity minus A |leontief_matrix_create()| | **L = (I − A)⁻¹** | Leontief inverse | Total (direct + indirect) requirements |leontief_inverse_create()` |
|
| f | Final-demand vector | Exogenous demand by category | retrieved from IOT |
| x = Lf | Total output vector | Fundamental Leontief equation | – |
Eurostat equation references:
\(a_ij\) = \(x_ij\) / \(x_j\) — technical coefficient (19), (43) identical formulations for SIOTs
\(x\) = \((I − A)⁻¹\) — total output
Each element of the input coefficient matrix A shows the share of input i used in producing one unit of output of industry j:
\[ a_{ij} = \frac{x_{ij}}{x_j} \]
where
- \(a_{ij}\) = input coefficient for
domestic goods and services
- \(x_{ij}\) = flow of product
i to industry j (a cell in Quadrant I)
- \(x_j\) = total output of industry
j
These coefficients are sometimes called direct requirements or technical coefficients.
hey are computed in iotables by:
input_coefficient_matrix_create(
data_table = iotable_get(source = "germany_1995"),
digits = 4
)
#> iotables_row agriculture_group industry_group construction
#> 1 agriculture_group 0.0258 0.0236 0.0000
#> 2 industry_group 0.1806 0.2822 0.2613
#> 3 construction 0.0097 0.0068 0.0158
#> 4 trade_group 0.0811 0.0674 0.0578
#> 5 business_services_group 0.0828 0.0890 0.1263
#> 6 other_services_group 0.0353 0.0139 0.0071
#> trade_group business_services_group other_services_group
#> 1 0.0011 0.0010 0.0015
#> 2 0.0761 0.0173 0.0597
#> 3 0.0098 0.0339 0.0180
#> 4 0.1378 0.0156 0.0413
#> 5 0.1218 0.2790 0.0672
#> 6 0.0208 0.0217 0.0434The output coefficient matrix (Ghosh-type) expresses how the output of industry i is distributed across purchasing sectors j:
\[o_{ij} = \frac{x_{ij}}{x_i}\]
where
\(o_{ij}\) = output coefficient (distribution ratio)
\(x_{ij}\) = flow of product i to industry j
\(x_{i}\) = total output of product i
These ratios describe the supply-side distribution structure and are
computed by output_coefficient_matrix_create().
Two types of linkages are typically analysed:
Backward linkage – measures the strength of a
sector’s demand on its suppliers.
It is given by the column sum of input coefficients or,
when based on the Leontief inverse, includes all indirect effects
through the supply chain.
Forward linkage – measures the influence of a sector as a supplier to others. It is given by the row sum of output coefficients, describing how the sector’s output feeds into other industries.
An input indicator represents a specific row of the
coefficient matrix A,
for example value added (GVA), labour, or emissions per unit of
output.
The income, product, employment, or emission multipliers are calculated with the following generic formula:
\[ (63) Z = B(I-A)^{1} \] - \(B\) = matrix of input coefficients for primary input (income, employment, product, or pollutant) - \(I\) = unit matrix - \(A\) = matrix of input coefficients for intermediates - \(Y\) = Diagonal matrix for final demand by product - \(Z\) = matrix with results for direct and indirect requirements for primary inputs1.
Embodied emissions in final demand:
\[ (66) Z = B(I-A)^{1}Y \]
See Eurostat Manual of Supply, Use and Input–Output Tables (Beutel, 2008, pp.503-506.)
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.
\(o_{ij}\) = 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
| No. | Formula | Description |
|---|---|---|
| (5) | \(o_{ij}\) = \(x_{ij}\) / \(x_i\) | Output coefficient (Ghosh model) |
| (9) | \(a_{ij}\) = \(x_{ij}\) / \(x_j\) | Input (technical) coefficient |
| (19), (43) | – | Identical to (9) for SIOTs |
| (47) | \(x\) = \((I - A)^{-1}\) | Leontief model for total output |
| (L) | \(L\) = \((I - A)^{-1}\) | Leontief inverse |
| (m) | \(m\) = \(a \times L\) | Multiplier equation |
All analytical functions of iotables reproduce the
benchmark examples from Beutel (2008), Eurostat Manual of Supply,
Use and Input–Output Tables.
| Function | Concept | Beutel Table | Page |
|---|---|---|---|
input_coefficient_matrix_create() |
Direct requirements (A) | 15.6 | 485 |
leontief_matrix_create() |
Leontief matrix \((I − A)\) | 15.9 | 487 |
leontief_inverse_create() |
Total requirements (\(L\) = \((I − A)^{1}\)) | 15.10 | 488 |
supplementary_add() |
Environmental extension (CO₂, CH₄) | 15.13 | 494 |
input_indicator_create() |
Input indicators (GVA, Compensation) | 15.14 | 498 |
multiplier_create() |
Total multipliers (GVA, Employment) | 15.16 | 503–504 |
In the Eurostat Manual, the calculations are shown with (63) wages, (64) employment, (65) capital.↩︎