knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) library(knitr) options(scipen=1, digits=3)
Let X
be the input-output matrix, w
the wage vector, c
the household
consumption vector, d
the total final demand vector, and e
the employment coefficient.
library(leontief) X <- transaction_matrix w <- wage_demand_matrix[, "wage"] c <- wage_demand_matrix[, "household_consumption"] d <- wage_demand_matrix[, "final_total_demand"] e <- employment_matrix[, "employees"]
Let A
be the direct coefficients matrix.
A <- input_requirement(X, d) A_aug <- augmented_input_requirement(X,w,c,d) rownames(A_aug) <- c(rownames(X), "wage_over_demand") colnames(A_aug) <- c(rownames(X), "consumption_over_demand") kable(A_aug)
Let B
be the output allocation matrix.
B <- output_allocation(X, d) rownames(B) <- rownames(X) colnames(B) <- rownames(X) kable(B)
Let I
be the identity matrix. Leontief inverse is the same as solving I - A
.
L <- leontief_inverse(A); rownames(L) <- rownames(X) colnames(L) <- rownames(X) kable(L)
The required output is given by L * d
.
eq <- equilibrium_output(L, d) rownames(eq) <- rownames(X) colnames(eq) <- "output" kable(eq)
The output multiplier is the column sum of L
.
out <- output_multiplier(L)
Let W
be a matrix where each column is w
with the same dimension as L
. The income multiplier is the column sum of the element-wise multiplication of L
and W
element-wise divided by w
.
inc <- income_multiplier(L, w/d)
Let E
be a matrix where each column is e
with the same dimension as L
. The employment multiplier is the column sum of the element-wise multiplication of L
and E
element-wise divided by e
.
emp <- employment_multiplier(L, e/d)
sm <- round(cbind(out,inc,emp),4) rownames(sm) <- rownames(X) colnames(sm) <- c("output_multiplier", "income_multiplier", "employment_multiplier") kable(sm)
bl <- backward_linkage(A) fl <- forward_linkage(A) bfl <- cbind(bl,fl) rownames(bfl) <- rownames(X) colnames(bfl) <- c("backward_linkage", "forward_linkage") kable(bfl)
bl <- power_dispersion(L) bl_cv <- power_dispersion_cv(L) bl_t <- cbind(bl,bl_cv) rownames(bl_t) <- rownames(X) colnames(bl_t) <- c("power_dispersion", "power_dispersion_cv") kable(bl_t)
sl <- sensitivity_dispersion(L) sl_cv <- sensitivity_dispersion_cv(L) sl_t <- cbind(sl,sl_cv) rownames(sl_t) <- rownames(X) colnames(sl_t) <- c("power_dispersion", "power_dispersion_cv") kable(sl_t)
mp <- multiplier_product_matrix(L) rownames(mp) <- rownames(X) colnames(mp) <- rownames(X) kable(mp)
bli <- backward_linkage(A_aug) fli <- forward_linkage(A_aug) bfli <- cbind(bli,fli) rownames(bfli) <- c(rownames(X), "wage") # wie = with induced effect colnames(bfli) <- c("backward_linkage_wie", "forward_linkage_wie") kable(bfli)
Schuschny, Andres Ricardo. Topicos sobre el modelo de insumo-producto: teoria y aplicaciones. Cepal, 2005.
Pino Arriagada, Andres y Fuentes Navarro, Silvia. Derivacion y analisis de los multiplicadores de empleo para la economia nacional. Universidad del Bio-Bio, 2018.
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