Compute environmental footprints.

Trace environmental extensions through the supply chain using the Leontief inverse, following the FABIO methodology (Bruckner et al., 2019). The footprint shows how much of an environmental pressure (e.g. land use, water, emissions) is embodied in the final consumption of each product in each country.

The multiplier matrix is computed as \(MP_{ij} = (e_i / X_i) \cdot L_{ij}\), where \(e_i\) is the extension for sector \(i\). For each demand category, the footprint is decomposed per target item using the FABIO diagonal approach: \(FP = MP \cdot \text{diag}(y)\), aggregated by item.

For large systems, pass z_mat and x_vec instead of l_inv. This solves \((I - A) x = Y\) directly using a sparse LU factorisation, avoiding the dense Leontief inverse entirely and reducing memory from \(O(n^2)\) to \(O(nnz)\).

Usage

compute_footprint(
  l_inv = NULL,
  x_vec,
  y_mat,
  extensions,
  labels,
  z_mat = NULL,
  fd_labels = NULL,
  output_tol = 1e-08,
  value_added_floor = 0.001,
  conserve_extensions = TRUE
)

Arguments

l_inv

Leontief inverse matrix from compute_leontief_inverse(). Ignored when z_mat is provided.

x_vec

Numeric vector of total output per sector.

y_mat

Final demand matrix from build_io_model().

extensions

Numeric vector of environmental extensions (e.g. hectares of land use) per sector. Must have the same length as x_vec.

labels

Tibble with area_code and item_cbs_code mapping row/column indices to their meaning. From build_io_model().

z_mat

Optional inter-industry flow matrix from build_io_model(). When provided, the system is solved directly (sparse LU), and l_inv is not needed.

fd_labels

Optional tibble labelling Y columns. Pass fd_labels[[i]] from build_io_model() output. When provided, footprints are decomposed per target item using the FABIO diagonal approach, and the result includes a target_fd column. When omitted, columns of Y are treated as sectors (appropriate only when Y is square).

output_tol

Minimum output considered valid when computing extension intensities. Sectors with x_vec <= output_tol get zero intensity to avoid infinite or numerically explosive footprints from zero-output residuals.

value_added_floor

Minimum share of each sector's output that is treated as non-intermediate leakage when constructing A from z_mat. Column sums larger than 1 - value_added_floor are rescaled to that maximum. Ignored when a precomputed l_inv is supplied without z_mat.

conserve_extensions

If TRUE, rescale positive footprint flows within each origin area/item so their sum does not exceed the corresponding positive extension total. This keeps footprint outputs conservative when capped coefficients or negative final demand columns would otherwise make positive-only paths larger than the source extension.

Value

A tibble with footprint results containing:

• origin_area: Country where the pressure occurs.

• origin_item: Item causing the pressure.

• target_area: Country consuming the product.

• target_item: Item consumed.

• target_fd: Demand category (e.g. "food"). Only present when fd_labels is provided.

• value: Footprint value in extension units.

Examples

z_mat <- matrix(c(0, 5, 10, 0), nrow = 2)
x_vec <- c(100, 200)
l_inv <- compute_leontief_inverse(z_mat, x_vec)
#> Computing Leontief inverse (2x2 matrix)...
#> Inverting (I - A)...
#> ✔ Leontief inverse computed.
y_mat <- matrix(c(85, 195), ncol = 1)
extensions <- c(50, 30)
labels <- tibble::tibble(
  area_code = c(1L, 1L),
  item_cbs_code = c(1L, 2L)
)

# Small system: pass pre-computed L
compute_footprint(l_inv, x_vec, y_mat, extensions, labels)
#> ℹ Computing footprint for 2 sectors.
#>   2 sectors have non-zero extensions.
#>   Final demand: 1 column.
#> Computing multiplier matrix...
#> Computing footprints...
#> ✔ Footprint complete: 2 non-zero flows.
#> # A tibble: 2 × 5
#>   origin_area origin_item target_area target_item value
#>         <int>       <int>       <int>       <int> <dbl>
#> 1           1           1           1          NA  47.5
#> 2           1           2           1          NA  30.0

# Using Z directly (computes L internally)
compute_footprint(
  x_vec = x_vec, y_mat = y_mat,
  extensions = extensions, labels = labels,
  z_mat = z_mat
)
#> ℹ Computing footprint for 2 sectors.
#>   2 sectors have non-zero extensions.
#>   Final demand: 1 column.
#> Sparse solve path (no dense Leontief inverse).
#> Computing footprints...
#> ✔ Footprint complete: 2 non-zero flows.
#> # A tibble: 2 × 5
#>   origin_area origin_item target_area target_item value
#>         <int>       <int>       <int>       <int> <dbl>
#> 1           1           1           1          NA  47.5
#> 2           1           2           1          NA  30.0