Product balance model A x+y=x, where a is the direct consumption coefficient matrix; X is the column vector of the total output value of each department; Y is the final product column vector.
The inverse of the term is: (I-A)- 1y=x, where I is identity matrix. The value composition model ATx+v+m =x, where AT is the transposed matrix of a; V is labor remuneration; M is the product of the remainder. After the shift term is inverted, (I-at)- 1 (v+m) = 10. Consumption coefficient In the input-output principle, consumption coefficient can be divided into direct consumption coefficient and complete consumption coefficient. The former is also called input coefficient, process coefficient or technical coefficient, which is used to reflect the production technology structure of the national economy. Generally, it is represented by the symbol a ij, that is, pure department I products consumed by unit products produced by pure department J, such as pig iron consumed by smelting one ton of steel. The calculation formula is that x ij is the consumption of products in department I when department J produces products, also called intermediate flow; X j is the output of j department.
The direct consumption coefficient is basically the same as the consumption quota widely used in planning statistics, but there are some differences. The differences are as follows: ① Consumption quota refers to the process consumption of producing unit products, and the direct consumption coefficient includes the corresponding consumption of workshops, factories and companies in addition to this consumption; ② Consumption quota is generally only measured in kind, while direct consumption coefficient is measured in currency except in kind; (3) the consumption quota is generally determined according to the specific varieties and models of a product, such as the specific varieties and models of steel, while the direct consumption coefficient is generally determined according to the categories of products (such as steel).
On the basis of direct consumption coefficient, the complete consumption coefficient can be calculated, that is, the sum of direct consumption and indirect consumption of a total product or intermediate product by the final product of a production unit. For example, the production of a machine not only consumes steel directly, but also consumes electricity, while power generation requires equipment, and production equipment consumes steel. The consumption of steel by production machines through power generation equipment is called indirect consumption.
The formula for calculating the complete consumption coefficient of K kinds of final products per production unit for I kinds of products (denoted as b ik) is (i, j, k= 1, 2, 3, …, n).
The above formula is written as matrix b = ab+i. from this
B=(I-A)- 1
There is also a formula for calculating the total consumption coefficient (i, j, k= 1, 2, 3, …, n).
Where c ik is the complete consumption coefficient of K kinds of final products to I kinds of products per production unit. The above formula is written as matrix c = a+a C, thus:
C=(I-A)- 1A
The relationship between the two complete consumption coefficients is as follows:
b-C =(I-A)- 1-(I-A)- 1A =(I-A)- 1(I-A)= I
In this way, the difference between the two complete consumption coefficients is a identity matrix, and the element on its main diagonal is 1, and the other elements are 0. Economically speaking, the final product is a product separated from the production process and should not be included in production and consumption. Coefficient C should be taken as the complete consumption coefficient, but coefficient B is the basis of calculating C, which can reflect the dependence between the final product and the total product.