Compute the Leontief inverse from a technology-coefficient matrix.
Usage
leontief_inverse_create(technology_coefficients_matrix, digits = NULL)
leontieff_inverse_create(technology_coefficients_matrix, digits = NULL)Arguments
- technology_coefficients_matrix
A technology-coefficient matrix created by
input_coefficient_matrix_create().- digits
Optional integer. Precision for rounding. Default
NULL(no rounding).
Value
A data.frame with the original key column and the Leontief inverse in the
remaining columns. If digits is provided, values are rounded.
Details
The Leontief inverse is defined as
\(L = (I - A)^{-1}\),
where \(A\) is the input-coefficient matrix created by
input_coefficient_matrix_create().
In the Eurostat Manual of Supply, Use and Input–Output Tables (Beutel, 2008), this formulation appears in Chapter 15 (see equations (19), (43), etc.). The UN Handbook on Supply and Use Tables and Input–Output Tables with Extensions and Applications (2018, Rev. 1) also uses this standard definition (see pp. 619–621).
For the analogous inverse from output coefficients, see ghosh_inverse_create().
See also
Other analytic object functions:
ghosh_inverse_create(),
input_flow_get(),
leontief_matrix_create(),
output_coefficient_matrix_create()
Examples
# A tiny 2x2 system with hand-calculable inverse
minimal_matrix <- data.frame(
sector = c("A", "B"),
A = c(0.2, 0.4),
B = c(0.1, 0.2)
)
leontief_inverse_create(minimal_matrix, digits = 3)
#> sector A B
#> 1 A 1.333 0.167
#> 2 B 0.667 1.333
# With a full example from the package
tm <- input_flow_get(
data_table = iotable_get(),
households = FALSE
)
leontief_inverse_create(technology_coefficients_matrix = tm)
#> iotables_row agriculture_group industry_group construction
#> 1 industry_group 0.0158664274 -0.0351941783 1.122398e-02
#> 2 construction -0.0006395334 0.0015456369 -4.406410e-04
#> 3 trade_group 0.0015287258 -0.0040074561 1.047752e-03
#> 4 business_services_group -0.0002338040 0.0005257947 -1.849282e-04
#> 5 other_services_group -0.0000288592 0.0001102479 -1.428857e-05
#> 6 total -0.0006639034 0.0013999740 -4.692178e-04
#> trade_group business_services_group other_services_group
#> 1 1.997032e-02 3.051518e-02 -0.0149026568
#> 2 -8.150269e-04 -1.265044e-03 0.0005982047
#> 3 1.991114e-03 3.150908e-03 -0.0014409066
#> 4 -2.988295e-04 -4.446801e-04 0.0002229223
#> 5 -4.776073e-05 -6.789378e-05 0.0000280663
#> 6 -8.239659e-04 -1.309127e-03 0.0006233165