Balance input-output flows so the footprint conserves.

RAS-balance an inter-industry flow matrix toward the product-balance margin. The row target is intermediate use, \(u = X - \text{rowSums}(Y)\); the column target keeps the original column composition, rescaled so its total matches \(\sum u\).

When the balanced system is productive (the spectral radius of \(A\) is below 1), \((I - A) X = Y\) holds and the footprint conserves – the case for the productive systems in the examples and tests.

Caveat for physical agriculture: livestock feed conversion makes some columns of \(A\) sum well above 1 (many tonnes of feed per tonne of product), so the balanced system is not productive. Balancing to faithful margins does not remove that, and the traced footprint then over-traces rather than conserving (on the real 2010 model, by about 15%). RAS therefore does not, on its own, fix the grassland under-tracing; always verify with check_footprint_conservation().

Pass the result as z_mat to compute_footprint() with a large max_column_sum (e.g. 1e12) and conserve_extensions = FALSE.

Usage

balance_io_flows(z_mat, x_vec, y_mat, max_iter = 1000L, tol = 1e-06)

Arguments

z_mat

Inter-industry flow matrix (dense or Matrix).

x_vec

Total output per sector.

y_mat

Final demand vector, or matrix whose row sums give total final demand per product.

max_iter, tol

Passed to balance_ras().

Value

The balanced flow matrix.

Examples

z_mat <- matrix(c(1, 4, 2, 3), nrow = 2)
x_vec <- c(10, 10)
y_mat <- c(3, 3)
balance_io_flows(z_mat, x_vec, y_mat)
#>         [,1]     [,2]
#> [1,] 2.65857 4.341428
#> [2,] 4.34143 2.658572