The Samuelson macroeconomic model as a singular linear matrix difference equation

In this paper, we revisit the famous classical Samuelson’s multiplier–accelerator model for national economy. We reform this model into a singular discrete time system and study its solutions. The advantage of this study gives a better understanding of the structure of the model and more deep and elegant results.

parameters (multiplier and accelerator parameters) are entered into the system of equations. Of course, this statement contradicts with the empirical evidence which supports temporary or long-lasting business cycles.
In this article, we propose an alternative view of the model by reforming it into a singular discrete time system.
The paper theory of difference equations is organized as follows. Section 2 provides a short review for the organization of the original model and in Sect. 3, we introduce the proposed reformulation into a system of difference equations. Section 4 investigates the solutions of the proposed system.

The original model
The original version of Samuelson's multiplier-accelerator original model is based on the following assumptions: Assumption 2.1 National income T k in year k equals to the summation of three elements: consumption, C k , private investment, I k , and governmental expenditure G k Assumption 2.2 Consumption C k in year k depends on past income (only on last year's value) and on marginal tendency to consume, modeled with a, the multiplier parameter, where 0 < a < 1, Assumption 2.3 Private investment I k in year k depends on consumption changes and on the accelerator factor b, where b > 0 . Consequently, I k depends on national income changes, Assumption 2.4 Governmental expenditure G k in year k remains constant Hence, the national income is determined via the following second-order linear difference equation See (Samuelson 1939) for the needed theory of difference equations that lead to the solution of the above equation.
(3) where Note that F is singular (detF = 0) . Throughout the paper, we will use in several parts matrix pencil theory to establish our results. A matrix pencil is a family of matrices sF − G , parametrized by a complex number s, see (Dassios and Baleanu 2013).
Definition 3.1 Given F , G ∈ R r×m and an arbitrary s ∈ C , the matrix pencil sF − G is called 1. Regular when r = m and det(sF − G) � = 0; 2. Singular when r = m or r = m and det(sF − G) ≡ 0.
Corollary 3.1 The system (4) has always a regular pencil ∀a, b.
Proof The determinant det(sF − G) = s 2 − a(b + 1)s + ab � = 0 . Hence from Definition 2.1, the pencil is regular. The proof is completed.
The class of sF − G is characterized by a uniquely defined element, known as the Weierstrass canonical form, see (Dassios 2017), specified by the complete set of invariants of sF − G . This is the set of elementary divisors of type (s − a j ) p j , called finite elementary divisors, where a j is a finite eigenvalue of algebraic multiplicity p j ( 1 ≤ j ≤ ν ), and the set of elementary divisors of type ŝ q = 1 s q , called infinite elementary divisors, where q is the algebraic multiplicity of the infinite eigenvalue. ν j=1 p j = p and p + q = m. From the regularity of sF − G , there exist non-singular matrices P, Q ∈ R m×m such that J p , H q are appropriate matrices with H q a nilpotent matrix with index q * , J p a Jordan matrix and p + q = m . With 0 q,p we denote the zero matrix of q × p . The matrix Q can be written as Q p ∈ R m×p and Q q ∈ R m×q . The matrix P can be written as P 1 ∈ R p×r and P 2 ∈ R q×r . The solution of system (4) is given by the following Theorem: Theorem 3.2 (See Dassios 2012) We consider the system (4). Since its pencil is always regular, its solution exists and for k ≥ 0 , is given by the formula The matrices Q p , Q q , P 1 , P 2 , J p , H q are defined by (5), (6) and (7).

Results and discussion
In this section we will present our main results. We will provide the solution to the system (4) and consequently we will derive the sequence for the national income, the consumption and the private investment. (5) Theorem 4.1 We consider the system (4). Then in year k, national income T k , consumption C k and private investment I k are given by Proof From Corollary 3.1, the pencil sF − G is always regular. Furthermore, the pencil has one infinite eigenvalue and two finite: From Theorem 3.2, the solution of (4) is given by Since we have one infinite eigenvalue, we have and J p is the Jordan matrix of the two finite eigenvalues: The matrix Q p has the two eigenvectors of the two finite eigenvalues: while Q q is the eigenvector of the infinite eigenvalue: Hence, and the solution Y k takes the form: a 0 s 1 − a s 2 − a 0   Finally, here P 1 is the matrix which contains the right eigenvectors of the finite eigenvalues Hence, or, equivalently, or, equivalently, The proof is completed.
The way this method in this theorem reconstructs the Samuelson's model can be also used in models of similar nature. For example, it can be used into other macroeconomic models, or models where the memory effect appears, and models with delays, see Moaaz et al. 2020a, b). In addition, this updated form of Samuelson's model can provide new alternative methods to prove stability of similar dynamical systems, see (Apostolopoulos and Ortega 2018;Boutarfa and Dassios 2017;Dassios 2015bDassios , 2018a.

Initial conditions
We assume system (4) and the known initial conditions (IC): Y 2 . Note that it is a necessity the initial condition to be Y 2 because Y k is defined from T k−1 , T k−2 , and for k = 2 , T 2 is defined by T 1 ,T 0 .
Definition 4.1 Consider the system (4) with known IC. Then, the IC are called consistent if there exists a solution for the system (4) which satisfies the given conditions.
is the unique solution of the algebraic system Proposition 4.2 The singular reformulated Samuelson's model (4) has always a unique solution for any given initial conditions.

Proof
The column Y k in (4) is defined as whereby for k = 2, we get or, equivalently, using (2), (3) or, equivalently, However Hence or, equivalently, Hence from Proposition 4.1, the IC of the singular reformulated Samuelson's model (4) are always consistent and from Proposition 4.1, the singular reformulated Samuelson's model (4) has a unique solution for given IC. The proof is completed.

Conclusions
In this article, we focused and provided a new alternative formulation of the famous Samuelson macroeconomic model. We proved that this model can be studied via an equivalent singular system of difference equations using pencil theory. We provided properties for existence of solutions. As a future research, it is interesting to study stability and stabilization properties for non-consistent initial conditions. For this case optimization methods, see (Dassios et al. 2015), and concepts from graph theory, see (Cuffe et al. 2016;Dassios et al. 2019), will be required. For this idea, there is already some ongoing research in progress.