[Background knowledge-Keynesian economic growth model]
John M. Keynes established the famous national economic growth model. Let Y represent gross national income, C represent total consumption, E represent total expenditure, I represent investment, and G represent government investment (such as infrastructure, etc.). Then there is
【6.1】
Among them, c0 represents the minimum consumption, which is determined by the savings rate, etc.; c (0≤c≤1) is called marginal consumption, which reflects the coefficient of increase in consumption as income increases.
In [6.1], let Y=E, that is, 
[6.2]
It can be seen from [6.2] that the larger I and G are, the larger Y is; the larger c is, the larger Y is. That is, expanding consumption, increasing investment and state input can promote the increase of gross national income. Which 
is called the multiplier.
【Questions raised】
Expanding consumption promotes investment, thereby increasing national income. Samuelson (PA Samuelson), winner of the Nobel Prize in Economics in 1970, established a very simple multiplier-accelerator model, which can be transformed into a second-order difference equation. By solving this equation, we can explain the changes in economic growth. some important phenomena.
【Symbol Description】
【Model construction】
The gross national income in year t is equal to the sum of total consumption and total investment in that year, that is,
The total consumption in year t is equal to the sum of basic consumption c0 and the total income and marginal consumption c of the previous year. 
The total investment in year t is equal to the sum of spontaneous investment and induced investment, that is 
Among them, β (0≤β≤1) describes the stimulating effect of consumption growth on investment.
Summarizing the above analysis results, we get the following mathematical model
【6.3】
After iterating the three equations, 
[6.4] is sorted out
This is a second-order linear non-homogeneous difference equation with constant coefficients about national income. The characteristic equation corresponding to its corresponding homogeneous equation is
The discriminant about the root is
The two roots of the characteristic equation are
According to Vedic theorem, 
we have
That is, there are only two situations for characteristic roots:
Substituting 
[6.4] into the solution, 
the necessary and sufficient condition for the stability of this equilibrium point is that the characteristic roots are all located within the unit circle, that is, 
[6.5]
That is, the proportion c of national income Y used for consumption and the proportion β of consumption increment used for investment satisfy [6.5].