Employing Input-Output Model to Assess …
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3.6 Mathematical Models for Input-Output Analysis
3.6.1 Mathematical Model
Using linear algebra theory according to the elements recorded in the IO table can
establish IO mathematical models and study the quantitative relationship between
various departments. IO models mainly include a row model based on the balanced
relationship between the input and output tables and a column model based on the
balanced relationship between the input and output tables [7]. Another form of IO
analysis technology is the mathematical model of IO analysis, which is based on
the IO table. It is a way of expressing quantitative relationships and connections
using mathematical operation symbols. The IO mathematical model is based on the
quantitative relationship between the economic indicators in the IO table, and the
mathematical form of linear equations is used to reflect the input in the economic
system. And the relationship between output. Therefore, it is necessary to define and
calculate certain key coefficients in order to convert the form of the IO table into a
mathematical model. After the model is built, various calculations can be carried out
according to the needs of economic analysis, and then the calculation results can be
obtained, which can be used for empirical analysis, planning, forecasting, and other
tasks.
3.6.2 Parameter Settings
(1) Calculate the direct consumption coefficient matrix and the Leontief inverse
matrix
The direct consumption coefficient is one of the basic concepts in the IO analysis
technology. It is the coefficient value obtained by dividing the total input of the jth door in the IO table by the value of the product directly consumed by the i-th
department. The formula is as follows:
a i j =
x i j
X j
where a i j represents the input of department i that department j needs to increase
unit output; set A be the direct consumption coefficient square matrix.
For the n products in the IO table of the national economy, there must be n × n
direct consumption coefficients. The matrix of n × n direct consumption coefficients
arranged in the order of the standard is called the direct consumption coefficient
matrix and is denoted as A as the following:
165
3.6 Mathematical Models for Input-Output Analysis
3.6.1 Mathematical Model
Using linear algebra theory according to the elements recorded in the IO table can
establish IO mathematical models and study the quantitative relationship between
various departments. IO models mainly include a row model based on the balanced
relationship between the input and output tables and a column model based on the
balanced relationship between the input and output tables [7]. Another form of IO
analysis technology is the mathematical model of IO analysis, which is based on
the IO table. It is a way of expressing quantitative relationships and connections
using mathematical operation symbols. The IO mathematical model is based on the
quantitative relationship between the economic indicators in the IO table, and the
mathematical form of linear equations is used to reflect the input in the economic
system. And the relationship between output. Therefore, it is necessary to define and
calculate certain key coefficients in order to convert the form of the IO table into a
mathematical model. After the model is built, various calculations can be carried out
according to the needs of economic analysis, and then the calculation results can be
obtained, which can be used for empirical analysis, planning, forecasting, and other
tasks.
3.6.2 Parameter Settings
(1) Calculate the direct consumption coefficient matrix and the Leontief inverse
matrix
The direct consumption coefficient is one of the basic concepts in the IO analysis
technology. It is the coefficient value obtained by dividing the total input of the jth door in the IO table by the value of the product directly consumed by the i-th
department. The formula is as follows:
a i j =
x i j
X j
where a i j represents the input of department i that department j needs to increase
unit output; set A be the direct consumption coefficient square matrix.
For the n products in the IO table of the national economy, there must be n × n
direct consumption coefficients. The matrix of n × n direct consumption coefficients
arranged in the order of the standard is called the direct consumption coefficient
matrix and is denoted as A as the following: