Planning Economy - An Art for Matrix Eigenvalue and Eigenvector

[NewMajor]

The Positive Eigenvector Method (PEM) proposed by Chinese mathematician Hua Luogeng(1984) in his later years, is a model used in National Economic Plan based on Input-Output Analysis and Perron-Frobenius Theorem. And I think that it could be also a reference model for investment planning in Parecon.

Don’t worry, I’ll use plain language to describe Input-Output Analysis and only use the mathematics of a system of bivariate linear equations.

PEM reveals that there is a mathematical upper limit to the economic growth rate which is depended on investment decision between different sectors, and its growth path is a knife edge. Once off the "knife edge" it will increasingly deviate from the normal track until there is a recession.

[NewMajor]

1. Fixed Input Rate and Technology of Production

To produce a unit of certain product, a certain proportion of inputs must be put into it, such as 2×[wheel] and 5kg×[steel pipe] to produce 1×[bicycle].

I call this quantity of another input consumed in the production of one unit of a product the direct consumption coefficient. So in the above example, “2×[wheel]/bicycle”, and “5kg×[steel pipe]/bicycle” are the two direct consumption coefficients of the bicycle production line, which could be denoted as a1 and a2.

But if in the future it is possible to save only one kilogram of steel to produce a bicycle, then the technology of producing a bicycle is progressive, it saves the required inputs. Thus the “direct consumption coefficient” reflects the technology of production.

Similarly, 1.58 kW of electricity is used to produce one ton of Steel, which is determined by the smelting technology used to make the steel. It is an objective parameter determined by factors other than the economic model, so it can be regarded as exogenous.

[NewMajor]

2. A simple example of a two-sector economy

Now we introduce the simplest “two-sector model” to discuss economic growth. Each sector produces one product, wheat an steel.

The two conditions are known as follows:

1) endowment (initial resources)

In the first year, there were 45 tons of wheat and 20 tons of steel.

2) technology (direct consumption coefficients)

To produce one ton of wheat, 0.25 tons of wheat + 0.14 tons of steel need to be invested;
To produce one ton of steel, 0.4 tons of wheat + 0.12 tons of steel need to be invested.

The above initial resource quantities and direct consumption coefficients form the initial conditions for solving for output growth.

To find the output of products at the end of the year, let the output of wheat and steel in the 1st year be x₁ and y₁, respectively.

Next, we will calculate how much wheat and steel will be consumed in the production process:

3) wheat consumption

To produce one ton of wheat, 0.25 tons of wheat are needed as input;
to produce two tons of wheat, 0.5 tons of wheat are needed as input,… …
Then, the output of x₁ ton of wheat requires an input of 0.25x₁ tons of wheat.

Similarly, to produce y₁ tons of steel, 0.4y₁ tons of wheat are also required.

From this, the first equation could be obtained, the wheat consumption equation:

4) steel consumption

Next, how much steel is consumed in total for the production of x₁ wheat and y₁ steel?
By following the above analysis, it can be concluded in the same way that,
Therefore, if the 20 tons of steel as the initial resource is invested in production, the steel consumption equation that can be obtained is:

When combined with the equation for wheat consumption, a system of growth equations is obtained:

So at the end of the first year, the output of wheat(x₁) and steel(y₁) was:

Similarly, in the second year, if the 100 tons of wheat and 50 tons of steel produced at the end of the first year are continued to be used for production, let the output of wheat and steel in the second year be and respectively, then the growth equation in the 2nd year can be obtained as follows:

And the output at the end of the 2nd year was:

However, something strange happened.
After going through the same process in the 3rd year, the output of wheat turned out to be negative.
In contrast, the increase in the output of steel was extremely significant (the growth equation for the 3rd year and its solution are as follows):

So where is the problem?