The linear model of a diversified economy

 

Leontief based on an analysis of the US economy and the period before World War II was installed is an important fact: in the long-term value of change very little and can be considered as constants. This phenomenon is understandable in light of the fact that production technology remains at the same level for quite some time, and, therefore, consumption of the j-th branch of production i-th industry in the production of its product volume xj have the technological constant.

In view of this fact, you can make the following assumption: production for the j-th industry volume xj need to use the products of the i-th industry volume aijxi, where aij - constant. With this assumption, the linear production technology is adopted, and this very assumption is called the hypothesis of linearity. At the same time the number of coefficients Aij are called direct costs. According to the hypothesis of linearity, we have

 

Then equation (16.2) can be rewritten as a system of equations

We introduce the column vectors of the volume of output (gross output vector), the volume of production of final consumption (final consumption vector) and the matrix of coefficients of direct costs:

Then the system (16.4) in matrix form is

Typically, this ratio is called the equation of linear input-output balance. Together with the description of the matrix representation (16.5) this equation is called the Leontief model.

Interbranch balance equation can be used for two purposes. In the first, the most simple case, when the known vector of gross output , required to calculate the vector of final consumption and, and - a similar problem was discussed above (n. 16.1, Example 5).

In the second case, the equation of interbranch balance is used for planning purposes, the following formulation of the problem: for the time period T (for example, one year) known vector of final consumption and want to define the vector's gross output. It is necessary to solve a system of linear equations (16.6) with a known matrix A and the target vector . In the future we will deal precisely with such a task.

Meanwhile, the system (16.6) has a number of peculiarities arising from the applied nature of this problem; first of all, all the elements of the matrix A and the vectors and must be non-negative.

 

Lecture

Functions of one variable. The concept of the function. Limit function. The concept of continuity of a function. Points of discontinuity of functions.

 

Functions of one variable. The concept of the function. Limit function. The concept of continuity of a function. Points of discontinuity of functions.

 

 

 

 

 

 

 

 

Lecture