Leontief's Input-Output Model in R • leontief

Common variables

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"]

Input requirement matrix

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)

	agriculture_fishing	mining	manufacturing_industry	electricity_gas_water	construction	retail_hotels_restaurants	transport_communications_information	financial_services	real_state	business_services	personal_services	public_administration	consumption_over_demand
agriculture_fishing	0.147	0.000	0.125	0.003	0.000	0.007	0.000	0.000	0.000	0.001	0.002	0.002	0.110
mining	0.008	0.071	0.030	0.002	0.005	0.002	0.001	0.002	0.000	0.002	0.001	0.001	0.001
manufacturing_industry	0.185	0.052	0.137	0.050	0.194	0.077	0.061	0.010	0.002	0.026	0.047	0.027	0.261
electricity_gas_water	0.007	0.056	0.028	0.308	0.004	0.014	0.008	0.005	0.006	0.006	0.014	0.035	0.210
construction	0.002	0.001	0.001	0.009	0.123	0.008	0.005	0.002	0.129	0.004	0.011	0.028	0.001
retail_hotels_restaurants	0.049	0.027	0.037	0.023	0.053	0.070	0.057	0.015	0.004	0.031	0.040	0.018	0.522
transport_communications_information	0.036	0.034	0.057	0.027	0.019	0.099	0.144	0.041	0.004	0.044	0.019	0.037	0.311
financial_services	0.028	0.006	0.015	0.018	0.028	0.036	0.020	0.109	0.039	0.019	0.009	0.003	0.438
real_state	0.003	0.003	0.005	0.002	0.003	0.048	0.015	0.010	0.014	0.024	0.023	0.009	0.699
business_services	0.029	0.091	0.065	0.041	0.056	0.094	0.073	0.090	0.019	0.123	0.042	0.047	0.051
personal_services	0.001	0.001	0.003	0.001	0.001	0.004	0.005	0.003	0.001	0.003	0.034	0.003	0.451
public_administration	0.001	0.001	0.002	0.002	0.000	0.004	0.003	0.001	0.000	0.001	0.002	0.003	0.034
wage_over_demand	0.146	0.090	0.104	0.064	0.241	0.251	0.154	0.255	0.030	0.299	0.554	0.541	0.000

Output allocation matrix

Let B be the output allocation matrix.

B <- output_allocation(X, d)
rownames(B) <- rownames(X)
colnames(B) <- rownames(X)
kable(B)

	agriculture_fishing	mining	manufacturing_industry	electricity_gas_water	construction	retail_hotels_restaurants	transport_communications_information	financial_services	real_state	business_services	personal_services	public_administration
agriculture_fishing	0.147	0.000	0.524	0.002	0.001	0.020	0.000	0.000	0.000	0.002	0.003	0.002
mining	0.003	0.071	0.054	0.001	0.004	0.002	0.001	0.001	0.000	0.002	0.001	0.000
manufacturing_industry	0.044	0.029	0.137	0.010	0.086	0.050	0.035	0.002	0.000	0.012	0.021	0.005
electricity_gas_water	0.008	0.153	0.138	0.308	0.009	0.044	0.024	0.006	0.008	0.015	0.031	0.033
construction	0.001	0.001	0.002	0.004	0.123	0.012	0.007	0.001	0.077	0.004	0.011	0.012
retail_hotels_restaurants	0.018	0.024	0.058	0.007	0.037	0.070	0.050	0.006	0.001	0.022	0.028	0.005
transport_communications_information	0.015	0.034	0.100	0.010	0.015	0.113	0.144	0.017	0.002	0.035	0.015	0.013
financial_services	0.029	0.013	0.064	0.015	0.053	0.098	0.047	0.109	0.044	0.036	0.017	0.002
real_state	0.003	0.006	0.019	0.002	0.005	0.115	0.033	0.009	0.014	0.040	0.038	0.006
business_services	0.015	0.111	0.142	0.018	0.055	0.133	0.090	0.047	0.011	0.123	0.041	0.020
personal_services	0.000	0.002	0.006	0.000	0.001	0.005	0.007	0.002	0.000	0.003	0.034	0.001
public_administration	0.002	0.003	0.009	0.002	0.000	0.013	0.009	0.001	0.000	0.002	0.004	0.003

Leontief inverse matrix

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)

	agriculture_fishing	mining	manufacturing_industry	electricity_gas_water	construction	retail_hotels_restaurants	transport_communications_information	financial_services	real_state	business_services	personal_services	public_administration
agriculture_fishing	1.214	0.014	0.181	0.021	0.043	0.029	0.017	0.005	0.007	0.009	0.014	0.011
mining	0.020	1.080	0.042	0.007	0.016	0.007	0.005	0.003	0.003	0.004	0.004	0.003
manufacturing_industry	0.283	0.088	1.225	0.106	0.286	0.128	0.106	0.029	0.043	0.051	0.077	0.055
electricity_gas_water	0.030	0.094	0.059	1.454	0.025	0.033	0.023	0.013	0.013	0.016	0.027	0.055
construction	0.007	0.005	0.007	0.018	1.145	0.022	0.013	0.006	0.151	0.011	0.019	0.035
retail_hotels_restaurants	0.087	0.048	0.074	0.050	0.090	1.100	0.085	0.029	0.019	0.048	0.055	0.032
transport_communications_information	0.090	0.066	0.110	0.067	0.067	0.150	1.195	0.067	0.018	0.071	0.041	0.059
financial_services	0.052	0.018	0.037	0.039	0.051	0.058	0.037	1.129	0.053	0.031	0.019	0.011
real_state	0.014	0.011	0.016	0.010	0.014	0.061	0.027	0.017	1.018	0.032	0.030	0.014
business_services	0.087	0.137	0.126	0.094	0.120	0.152	0.124	0.129	0.045	1.162	0.070	0.073
personal_services	0.003	0.003	0.005	0.003	0.003	0.006	0.008	0.005	0.001	0.004	1.036	0.004
public_administration	0.003	0.002	0.003	0.003	0.001	0.005	0.004	0.001	0.001	0.002	0.002	1.004

Equilibrium output

The required output is given by L * d.

eq <- equilibrium_output(L, d)
rownames(eq) <- rownames(X)
colnames(eq) <- "output"
kable(eq)

	output
agriculture_fishing	25832
mining	31674
manufacturing_industry	81363
electricity_gas_water	23428
construction	28817
retail_hotels_restaurants	47168
transport_communications_information	50606
financial_services	21588
real_state	18686
business_services	51363
personal_services	23148
public_administration	9746

Multipliers

Output multiplier

The output multiplier is the column sum of L.

out <- output_multiplier(L)

Income multiplier

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)

Employment multiplier

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)

Summary of multipliers

sm <- round(cbind(out,inc,emp),4)
rownames(sm) <- rownames(X)
colnames(sm) <- c("output_multiplier", "income_multiplier", "employment_multiplier")
kable(sm)

	output_multiplier	income_multiplier	employment_multiplier
agriculture_fishing	1.89	0.291	94.3
mining	1.57	0.187	21.4
manufacturing_industry	1.88	0.250	46.2
electricity_gas_water	1.87	0.177	22.3
construction	1.86	0.400	55.3
retail_hotels_restaurants	1.75	0.393	78.3
transport_communications_information	1.65	0.278	41.3
financial_services	1.44	0.354	24.9
real_state	1.37	0.109	14.5
business_services	1.44	0.393	29.1
personal_services	1.40	0.639	90.8
public_administration	1.36	0.607	53.9

Linkages

Backward and forward linkage

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)

	backward_linkage	forward_linkage
agriculture_fishing	0.496	0.288
mining	0.343	0.123
manufacturing_industry	0.506	0.869
electricity_gas_water	0.486	0.491
construction	0.487	0.322
retail_hotels_restaurants	0.462	0.424
transport_communications_information	0.393	0.561
financial_services	0.288	0.330
real_state	0.217	0.159
business_services	0.283	0.770
personal_services	0.243	0.059
public_administration	0.211	0.019

Power of dispersion

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)

	power_dispersion	power_dispersion_cv
agriculture_fishing	1.165	2.14
mining	0.965	2.40
manufacturing_industry	1.161	2.10
electricity_gas_water	1.154	2.59
construction	1.147	2.07
retail_hotels_restaurants	1.079	2.05
transport_communications_information	1.014	2.38
financial_services	0.884	2.68
real_state	0.845	2.60
business_services	0.888	2.64
personal_services	0.860	2.65
public_administration	0.836	2.66

Sensitivity of dispersion

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)

	power_dispersion	power_dispersion_cv
agriculture_fishing	0.965	2.60
mining	0.736	3.12
manufacturing_industry	1.526	1.54
electricity_gas_water	1.136	2.63
construction	0.887	2.68
retail_hotels_restaurants	1.058	2.09
transport_communications_information	1.234	1.94
financial_services	0.947	2.52
real_state	0.779	2.84
business_services	1.430	1.61
personal_services	0.667	3.39
public_administration	0.636	3.46

Multiplier product matrix

mp <- multiplier_product_matrix(L)
rownames(mp) <- rownames(X)
colnames(mp) <- rownames(X)
kable(mp)

	agriculture_fishing	mining	manufacturing_industry	electricity_gas_water	construction	retail_hotels_restaurants	transport_communications_information	financial_services	real_state	business_services	personal_services	public_administration
agriculture_fishing	0.152	0.116	0.240	0.179	0.140	0.167	0.194	0.149	0.123	0.225	0.105	0.100
mining	0.126	0.096	0.199	0.148	0.116	0.138	0.161	0.124	0.102	0.187	0.087	0.083
manufacturing_industry	0.151	0.116	0.240	0.178	0.139	0.166	0.194	0.149	0.122	0.225	0.105	0.100
electricity_gas_water	0.151	0.115	0.238	0.177	0.138	0.165	0.192	0.148	0.122	0.223	0.104	0.099
construction	0.150	0.114	0.237	0.176	0.138	0.164	0.191	0.147	0.121	0.222	0.103	0.099
retail_hotels_restaurants	0.141	0.107	0.223	0.166	0.129	0.154	0.180	0.138	0.114	0.209	0.097	0.093
transport_communications_information	0.132	0.101	0.209	0.156	0.122	0.145	0.169	0.130	0.107	0.196	0.091	0.087
financial_services	0.115	0.088	0.182	0.136	0.106	0.126	0.148	0.113	0.093	0.171	0.080	0.076
real_state	0.110	0.084	0.174	0.130	0.101	0.121	0.141	0.108	0.089	0.163	0.076	0.073
business_services	0.116	0.088	0.183	0.136	0.107	0.127	0.148	0.114	0.094	0.172	0.080	0.076
personal_services	0.112	0.086	0.177	0.132	0.103	0.123	0.143	0.110	0.091	0.166	0.078	0.074
public_administration	0.109	0.083	0.172	0.128	0.100	0.120	0.139	0.107	0.088	0.162	0.075	0.072

Induced effects (labor/consumption)

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)

	backward_linkage_wie	forward_linkage_wie
agriculture_fishing	0.643	0.398
mining	0.433	0.123
manufacturing_industry	0.609	1.130
electricity_gas_water	0.550	0.702
construction	0.728	0.323
retail_hotels_restaurants	0.713	0.947
transport_communications_information	0.547	0.872
financial_services	0.543	0.768
real_state	0.247	0.858
business_services	0.582	0.821
personal_services	0.797	0.510
public_administration	0.752	0.053
wage	3.090	2.729

References

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.

Leontief’s Input-Output Model in R