Study of Samuelson ’ s General Model of National Economy and Definition of Asymptotic-Stability Conditions

In this particular paper we firstly deal with Samuelson’s model of national economy. We create a difference equation which reflects Samuelson’s model for the national income of a country taking into consideration the expenditure and the investments of the two previous years and not only the immediately previous one. Later we find the saddle-point and deal with its stability giving conditions concerning the coefficient of the difference equation and which are able (sufficient) and necessary in order for the saddle-point to be stable.


Introduction
First we present Samuelson's model of the national income of a country.
We make the following assumptions: If the consumption increases, more industries are needed for the production of consumer goods; thus, investments take place.The bigger the rise in consumer expenditure the more investments will take place.On the other hand if consumption decreases, investors block funds which had been invested; therefore, investments are reduced.
In this particular paper we considered that the investments depend on the change in consumption of the last two years.
Assumption 4: We assume that government expenditure k G is stable.
Using the above assumptions we are led to a difference equation which reflects Samuelson's model.

Finding a Difference Equation Which Reflects Samuelson's Model
As we mentioned in the introduction, national income We consider that the expenditures depend on the national income of the previous two periods of time.We also assume that there is stable government spending P on both previous periods of time.The reason we expanded the Samuelson's model from one to two years is the assumption that a country's national income does not depend solely on factors of the previous year but for many year.The former assumption is expressed mathematically by the equation: (2) We also consider that the investments depend on the change in consumption of both previous periods of time as following: During the time period κ an investor compares the expenditure Cκ of that period with the expenditure C k-1 of the previous period taking the difference Cκ -C k-1 .He also compares the expenditure C k-1 with the expenditure C k-2 of the time period k-2 using the difference C k-1 -C k-2 .
All the above are expressed mathematically by the relationship: and b 2 >0 (3) We continue combining the above equations ( 1), ( 2), (3) in order to create an equation which expresses Samuelson's model.
Specifically, in equation ( 2) we use κ -1 instead of κ.Then we have: Also, in equation ( 2) we use κ -2 instead of κ.Then we have: From equations ( 1) and (3) we have: We replace C κ , C κ-1 and C κ-2 from (2), ( 4), ( 5) and we have: We define: Therefore the equation becomes: We replace κ with κ + 4. So the previous equation becomes: which is a difference equation and which expresses Samuelson's model for the national income of a country.

Specifying the Saddle-Points of Samuelson's Equation (6)
Let's assume that we have the difference equation: A real number α is called saddle-point of equation ( 7) if assuming that g 0 = α it also applies that g κ+i = α where κ = 1, 2, … and i = 0, 1, …, n.This means that g κ+i = α is a stable solution of the equations set.
Thus the saddle-point of ( 6) is S =

Definition of the Conditions That Make Stable the Saddle-Point
Equation ( 6) is a 4 th order non homogeneous difference equation.
The equivalent homogeneous equation of this one is: which has a characteristic equation: According to the methods of solving difference equations (see "Discrete Dynamic Systems" Gr. H. Kalogeropoulos, "An Introduction to Difference Equations" S. N. Elaydi and "Introduction to Difference Equations" S. Coldberg) the saddle-point S = 1 2 1 P cc  will be asymptotically stable if i r <1 i=1,2,3,4 ,where r 1 , r 2 , r 3 , r 4 are the roots of equation ( 9) not necessarily discrete From the above results we have the deduction: For a saddle-point to be asymptotically stable all the roots of the characteristic equation must have magnitude less than one.

Definition 2.3.1
Let V be a vector space over a field F (= R or C).
A function x ≥ 0 for all , x V x o  and 00  2) For all , a F x V  we have a x a x    3) For all , x y V  we have x y x y    There are a lot of vector norms but we will use here the maximum norm or infinity norm (  -norm) on R n (or C n ).
We consider a function In this particular paper we will use  row sum norm which has the following form: Now we will prove the next statement which we will need for the proof of the theorem 2.3.1.
We are going now to theorem 3.2.1 which will play an important role.
Theorem 2.3.1 Let P(x) = α n x n +……….+α 1 x + α 0 be a polynomial, α i ∈C ∀ i=1,2,….n a n ≠0 n∈N The n roots of the polynomial are in a circular disk of center (0, 0) and radius p = max{1, 1 0 x n be the roots of the polynomial P(x) not necessarily different among them.Let x be one solution of: be a n x n matrix and vector We use the infinity norm  for vectors.Then from From the statement(2.3.1)we have A u A u because all rows of matrix A contain only the value 1, except for the last one row where the sum of its elements is So the solution of equation P(x) = 0 is in the interior and in the disk of center (0, 0) and radius We now return to the check of the stability of the saddle-point.
We will prove that |r| ≠ 1.We notice that z = 1 is not a solution of f(r) = 0 because if as it arises from equation (2).
Therefore, concerning the roots of polynomial f(r), if |r| = 1 our options are: a)It has the solution z = -1 with multiplicity of 4.
b)It has the solution z = -1 with multiplicity of 2 and two complex conjugate roots (let them be 00 , zz having 0 z = 1 = 0 z ).c)It has two complex conjugate roots, each one with multiplicity of 2 (let them be 00 , zz , having | 0 z = 1 = 0 z ).
If we have option (b), then:  from the properties of conjugate complex numbers.Therefore we have: So it has to be D = 1, which is a contradiction because |D| < 1.
In conclusion, if r is the solution of f(z) = 0, then |r| ≤ 1 and since we have shown that |r| ≠ 1, then |r| < 1.This means that all the roots of the characteristic equation ( 9) have absolute value less than one.
So far we have seen that the saddle-point given by the equation which expresses Samuelson's model of national economy is asymptotically stable when: |A| + |B| + |C| + |D| < 1 (10) where: More specifically, condition (11) becomes:

Conclusions
In this paper we studied Samuelson's model of national economy taking into account the investments and expenditure of the previous two years.We discovered the saddle-point and gave the condition to be asymptotically stable.
As a future paper we will generalize Samuelson's model for more than two past years and we will look for conditions that make asymptotically stable saddle-point.
We will also study conditions (11) geometrically in the plane so that we can find sectors consisting of points which if we replace its ordinates in Samuelson's equation, the saddle-point will be asymptotically stable.

Assumption 1 :
The national income k T at time step κ is the composition of three elements: consumer expenditure The expenditure (according to Samuelson's usual model) is equivalent to the national income k T .This income is consumed just after the acquisition time which means that the expenditure 1 k C  of time period κ + 1 is equivalent to the national income k T of time period κ.In this particular paper we assume that expenditure depends on the national income of the previous two years.Assumption 3: Investments depend on the change in consumption as follows: During time period κ + 1 an investor compares the consumer expenditures 1 k C  of that time period with the expenditure k If |A| + |B| + |C| + |D| < 1, then the roots of the polynomial equation r 4 + Ar 3 + Br 2 + Cr + D = 0 have absolute value less than one.Proof If |A| + |B| + |C| + |D| < 1, then p = max {1, |A| + |B| + |C| + |D|} = 1.
That is if A  Mn, then A is a n x n matrix and Mn =R nxn (or C nxn )