1 Introduction
Many authors have studied generalized discrete, see (Apostolopoulos and Ortega 2019; Dassios 2015a; Dassios and Kalogeropoulos 2013; Ogata 1987; Ortega and Apostolopoulos 2018), and continuous time systems, see (Dassios et al. 2019, 2020; Lewis 1986, 1987, 1992; Liu et al. 2019; Milano and Dassios 2016, 2017), and their applications especially in cases where the memory effect is needed including generalized discrete, see (Dassios and Baleanu 2015; Dassios and Kalogeropoulos 2014; Dassios and Szajowski 2016), and continuous time systems with delays, see (Liu et al. 2017, 2019, 2020; Tzounas et al. 2020). Many of these results have already been extended to systems of differential, see (Dassios and Baleanu (2018a, b; Klamka 2010; Klamka and Wyrwał 2008; Podlubny 1999) and difference equations with fractional operators, see (Dassios 2015c, 2018b).
Keynesian macroeconomics inspired the seminal work of (Samuelson 1939), who actually introduced the business cycle theory. Although primitive and using only the demand point of view, the Samuelson’s prospect still provides an excellent insight into the problem and justification of business cycles appearing in national economies. In the past decades, many more sophisticated models have been proposed by other researchers, see (Barros and Ortega 2019; Dassios and Devine 2016; Dassios and Zimbidis 2014; Dassios et al. 2014; Dorf 1983; Kuo 1996; Puu et al. 2004; Rosser 2000). All these models use superior and more delicate mechanisms involving monetary aspects, inventory issues, business expectation, borrowing constraints, welfare gains and multi-country consumption correlations.
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Some of the previous articles also contribute to the discussion for the inadequacies of Samuelson’s model. The basic shortcoming of the original model is the incapability to produce a stable path for the national income when realistic values for the different 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.
2 The original model
The original version of Samuelson’s multiplier–accelerator original model is based on the following assumptions:
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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 \(G_k\)
$$\begin{aligned} T_k=C_k+I_k+G_k. \end{aligned}$$
(1)
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 \(0< a < 1\),
$$\begin{aligned} C_k=aT_{k-1}. \end{aligned}$$
(2)
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,
$$\begin{aligned} I_k=b(C_k-C_{k-1})=ab(T_{k-1}-T_{k-2}). \end{aligned}$$
(3)
Assumption 2.4
Governmental expenditure \(G_k\) in year k remains constantHence, the national income is determined via the following second-order linear difference equationSee (Samuelson 1939) for the needed theory of difference equations that lead to the solution of the above equation.
$$\begin{aligned} G_k={\bar{G}}. \end{aligned}$$
$$\begin{aligned} T_{k+2}-a(1+b)T_{k+1}+abT_k={\bar{G}}. \end{aligned}$$
3 The reformulation: Singular Samuelson’s model
LetThen, (1) can be written asor, equivalently,Equation (2) can be written asor, equivalently,Finally (3) can be written asor, equivalently,or, equivalently,Hence the above expressions can be written in a matrix form. We proved the following theorem:
$$\begin{aligned} Y_k=\left[ \begin{array}{c}T_k\\ C_k\\ I_k\end{array}\right] . \end{aligned}$$
$$\begin{aligned} 0=-T_k+C_k+I_k+G_k, \end{aligned}$$
$$\begin{aligned} \left[ \begin{array}{ccc}0&0&0\end{array}\right] Y_{k+1} =\left[ \begin{array}{ccc}-1&1&1\end{array}\right] Y_k+G_k. \end{aligned}$$
$$\begin{aligned} C_{k+1}=aT_k \end{aligned}$$
$$\begin{aligned} \left[ \begin{array}{ccc}0&1&0\end{array}\right] Y_{k+1} =\left[ \begin{array}{ccc}a&0&0\end{array}\right] Y_k. \end{aligned}$$
$$\begin{aligned} I_{k+1}=b(C_{k+1}-C_k). \end{aligned}$$
$$\begin{aligned} -bC_{k+1}+I_{k+1}=-bC_k. \end{aligned}$$
$$\begin{aligned} \left[ \begin{array}{ccc}0&-b&1\end{array}\right] Y_{k+1} =\left[ \begin{array}{ccc}0&-b&0\end{array}\right] Y_k. \end{aligned}$$
Theorem 3.1
The difference equation (1) can be written in the form of the following singular discrete time system:whereNote thatFis singular (det\(F=0)\). 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 numbers, see (Dassios and Baleanu 2013).
$$\begin{aligned} \begin{array}{cc} FY_{k+1}=GY_k+V_k,\quad k= 2, 3,\ldots , \end{array} \end{aligned}$$
(4)
$$\begin{aligned} F=\left[ \begin{array}{ccc} 0&{}0&{}0\\ 0&{}1&{}0\\ 0&{}-b&{}1\end{array}\right] ,\quad G=\left[ \begin{array}{ccc} -1&{}1&{}1\\ a&{}0&{}0\\ 0&{}-b&{}0\end{array}\right] ,\quad V_k=\left[ \begin{array}{c} G_k\\ 0\\ 0\end{array}\right] . \end{aligned}$$
Definition 3.1
Given \(F,G\in {\mathbb {R}}^{r \times m}\) and an arbitrary \(s\in {\mathbb {C}}\), the matrix pencil \(sF-G\) is called
1.
Regular when \(r=m\) and det\((sF-G)\ne 0\);
2.
Singular when \(r\ne m\) or \(r=m\) and det\((sF-G)\equiv 0\).
Corollary 3.1
Proof
The determinant det\((sF-G)=s^2-a(b+1)s+ab\ne 0\). 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 \(p_j\) (\(1\le j \le \nu\)), and the set of elementary divisors of type \({\hat{s}}^q=\frac{1}{s^q}\), called infinite elementary divisors, where q is the algebraic multiplicity of the infinite eigenvalue. \(\sum _{j =1}^\nu p_j = p\) and \(p+q=m\).
From the regularity of \(sF-G\), there exist non-singular matrices P, Q\(\in {\mathbb {R}}^{m \times m}\) 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 \(q\times p\). The matrix Q can be written as\(Q_p\in {\mathbb {R}}^{m \times p}\) and \(Q_q\in {\mathbb {R}}^{m \times q}\). The matrix P can be written as\(P_1\in {\mathbb {R}}^{p \times r}\) and \(P_2\in {\mathbb {R}}^{q \times r}\).
$$\begin{aligned} \begin{array}{c}PFQ=\left[ \begin{array}{cc} I_p&{}0_{p,q}\\ 0_{q,p}&{}H_q\end{array}\right] , \\ PGQ=\left[ \begin{array}{cc} J_p&{}0_{p,q}\\ 0_{q,p}&{}I_q\end{array}\right] .\end{array} \end{aligned}$$
(5)
$$\begin{aligned} Q=\left[ \begin{array}{cc}Q_p&Q_q\end{array}\right] . \end{aligned}$$
(6)
$$\begin{aligned} P=\left[ \begin{array}{c}P_1 \\ P_2\end{array}\right] . \end{aligned}$$
(7)
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\(k\ge 0\), is given by the formulaHere,\(D_k=\left[ \begin{array}{c} \sum ^{k-1}_{i=0}J_p^{k-i-1}P_1V_i\\ sum^{q_{*}-1}_{i=0}H_q^iP_2V_{k+i} \end{array}\right]\)and\(C\in {\mathbb {R}}^p\)is a constant vector. The matrices\(Q_p\), \(Q_q\), \(P_1\), \(P_2\), \(J_p\), \(H_q\)are defined by (5), (6) and (7).
$$\begin{aligned} Y_k=Q_pJ_p^kC+QD_k. \end{aligned}$$
4 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.
Theorem 4.1
We consider the system (4). Then in yeark, national income\(T_k\), consumption\(C_k\)and private investment\(I_k\)are given by
$$\begin{aligned} &T_k=s_1^{k+1}c_1+s_2^{k+1}c_2+a\sum ^{k-1}_{i=0}[(s_1^{k-1}+s_2^{k-1})]G_i,\\& C_k=a(s_1^kc_1+s_2^kc_2)+a^2\sum ^{k-1}_{i=0}[(s_1^{k-i-1}+s_2^{k-i-1})]G_i,\\& I_k=s_1^{k}(s_1-a)c_1+s_2^{k}(s_2-a)c_2+a\sum ^{k-1}_{i=0}[((s_1-a)s_1^{k-1} +(s_2-a)s_2^{k-1})]G_i. \end{aligned}$$
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 bySince we have one infinite eigenvalue, we haveand \(J_p\) 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:Finally, here \(P_1\) is the matrix which contains the right eigenvectors of the finite eigenvaluesHence,or, equivalently,or, equivalently,The proof is completed.
$$\begin{aligned} s_1=\frac{a(1+b)+\sqrt{a^2(1+b)^2-4ab}}{2},\quad s_2=\frac{a(1+b)-\sqrt{a^2(1+b)^2-4ab}}{2}. \end{aligned}$$
$$\begin{aligned} Y_k=Q_pJ_p^kC+Q\left[ \begin{array}{c} \sum ^{k-1}_{i=0}J_p^{k-i-1}P_1V_i\\ sum^{q_{*}-1}_{i=0}H_q^iP_2V_{k+i} \end{array}\right] . \end{aligned}$$
$$\begin{aligned} H_q=0 \end{aligned}$$
$$\begin{aligned} Y_k=Q_p \left[ \begin{array}{cc} s_1^k&{}0\\ 0&{}s_2^k \end{array}\right] C+Q\left[ \begin{array}{c} \sum ^{k-1}_{i=0}J_p^{k-i-1}P_1V_i\\ 0 \end{array}\right] . \end{aligned}$$
$$\begin{aligned} Q_p= \left[ \begin{array}{cc} s_1&{}s_2\\ a&{}a\\ s_1-a&{}s_2-a \end{array}\right] , \end{aligned}$$
$$\begin{aligned} Q_q= \left[ \begin{array}{c} 1\\ 0\\ 0 \end{array}\right] . \end{aligned}$$
$$\begin{aligned} Q=\left[ \begin{array}{ccc} s_1&{}s_2&{}1\\ a&{}a&{}0\\ s_1-a&{}s_2-a&{}0 \end{array}\right] \end{aligned}$$
$$\begin{aligned} Y_k= & {} \left[ \begin{array}{cc} s_1&{}s_2\\ a&{}a\\ s_1-a&{}s_2-a \end{array}\right] \left[ \begin{array}{cc} s_1^k&{}0\\ 0&{}s_2^k \end{array}\right] C \\& + \left[ \begin{array}{ccc} s_1&{}s_2&{}1\\ a&{}a&{}0\\ s_1-a&{}s_2-a&{}0 \end{array}\right] \left[ \begin{array}{c} \sum ^{k-1}_{i=0} \left[ \begin{array}{cc} s_1^{k-i-1}&{}0\\ 0&{}s_2^{k-i-1} \end{array}\right] P_1V_i\\ 0 \end{array}\right] . \end{aligned}$$
$$\begin{aligned} P_1= \left[ \begin{array}{ccc} a&{}1&{}\frac{a}{s_1}\\ a&{}1&{}\frac{a}{s_2} \end{array}\right] . \end{aligned}$$
$$\begin{aligned} Y_k= & {} \left[ \begin{array}{cc} s_1&{}s_2\\ a&{}a\\ s_1-a&{}s_2-a \end{array}\right] \left[ \begin{array}{cc} s_1^k&{}0\\ 0&{}s_2^k \end{array}\right] C \\& + \left[ \begin{array}{ccc} s_1&{}s_2&{}1\\ a&{}a&{}0\\ s_1-a&{}s_2-a&{}0 \end{array}\right] \left[ \begin{array}{c} \sum ^{k-1}_{i=0}\left[ \begin{array}{cc} s_1^{k-i-1}&{}0\\ 0&{}s_2^{k-i-1} \end{array}\right] \left[ \begin{array}{ccc} a&{}1&{}\frac{a}{s_1}\\ a&{}1&{}\frac{a}{s_2} \end{array}\right] \left[ \begin{array}{c} G_i\\ 0\\ 0 \end{array}\right] \\ 0 \end{array}\right] , \end{aligned}$$
$$\begin{aligned} Y_k= \left[ \begin{array}{l} s_1^{k+1}c_1+s_2^{k+1}c_2+a\sum ^{k-1}_{i=0}[(s_1^{k-1}+s_2^{k-1})]G_i\\ a(s_1^kc_1+s_2^kc_2)+a^2\sum ^{k-1}_{i=0}[(s_1^{k-i-1}+s_2^{k-i-1})]G_i\\ s_1^{k}(s_1-a)c_1+s_2^{k}(s_2-a)c_2+a\sum ^{k-1}_{i=0}[((s_1-a)s_1^{k-1} +(s_2-a)s_2^{k-1})]G_i \end{array}\right] , \end{aligned}$$
$$\begin{aligned} \left[ \begin{array}{c}T_k\\ C_k\\ I_k\end{array}\right] = \left[ \begin{array}{l} s_1^{k+1}c_1+s_2^{k+1}c_2+a\sum ^{k-1}_{i=0}[(s_1^{k-1}+s_2^{k-1})]G_i\\ a(s_1^kc_1+s_2^kc_2)+a^2\sum ^{k-1}_{i=0}[(s_1^{k-i-1}+s_2^{k-i-1})]G_i\\ s_1^{k}(s_1-a)c_1+s_2^{k}(s_2-a)c_2+a\sum ^{k-1}_{i=0}[((s_1-a)s_1^{k-1} +(s_2-a)s_2^{k-1})]G_i \end{array}\right] . \end{aligned}$$
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 (Dassios et al. 2017; 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 2015b, 2018a).
4.1 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
Proposition 4.1
Proposition 4.2
(See Dassios et al. 2014) Consider the system (4) with given IC. Then, the solution for the initial value problem is unique if and only if the IC are consistent. Then, the unique solution is given by the formulawhere\(D_k=\left[ \begin{array}{c} \sum ^{k-1}_{i=0}J_p^{k-i-1}P_1V_i\\ sum^{q_{*}-1}_{i=0} H_q^iP_2V_{k+i} \end{array}\right]\)and\(Z^p_2\)is the unique solution of the algebraic system\(Y_2=Q_pZ^p_2+D_2\).
$$\begin{aligned} Y_k=Q_pJ_p^kZ^p_{2}+QD_k, \end{aligned}$$
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 aswhereby for \(k=2,\) we getor, equivalently, using (2), (3)or, equivalently,But \(T_2=a(1+b)T_1-abT_0+{\bar{G}}\). ThusHoweverHenceor, 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.
$$\begin{aligned} Y_k= \left[ \begin{array}{c}T_k\\ C_k\\ I_k\end{array}\right] , \end{aligned}$$
$$\begin{aligned} Y_2= \left[ \begin{array}{c}T_2\\ C_2\\ I_2\end{array}\right] , \end{aligned}$$
$$\begin{aligned} Y_2 = \left[ \begin{array}{c}T_2\\ aT_1\\ ab(T_1-T_0)\end{array}\right] . \end{aligned}$$
$$\begin{aligned} Y_2 = \left[ \begin{array}{c}1\\ 0\\ 0 \end{array}\right] T_2+\left[ \begin{array}{c} 0 \\ 1 \\ b\end{array}\right] aT_{1}+\left[ \begin{array}{c} 0\\ 0\\ -b \end{array}\right] aT_0. \end{aligned}$$
$$\begin{aligned} Y_2 =\left[ \begin{array}{c}1+b\\ 1\\ b\end{array}\right] aT_1 +\left[ \begin{array}{c}-b\\ 0\\ b\end{array}\right] aT_0 +\left[ \begin{array}{c}1\\ 0\\ 0\end{array}\right] {\bar{G}}. \end{aligned}$$
$$\begin{aligned} colspanQ_p=\left\langle\left[ \begin{array}{c}1+b\\ 1\\ b\end{array}\right] , \left[ \begin{array}{c}-b\\ 0\\ b\end{array}\right] \right\rangle,\quad QD_2= \left[ \begin{array}{c}1\\ 0\\ 0\end{array}\right] . \end{aligned}$$
$$\begin{aligned} Y_2\in \left\langle\left[ \begin{array}{c}1+b\\ 1\\ b\end{array}\right] , \left[ \begin{array}{c}-b\\ 0\\ b\end{array}\right]\right\rangle +\left[ \begin{array}{c}1\\ 0\\ 0\end{array}\right] , \end{aligned}$$
$$\begin{aligned} Y_2\in colspanQ_p +QD_2. \end{aligned}$$
5 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.
Acknowledgements
We would like to express our sincere gratitude to Dr. Sung Cho for his helpful and fruitful discussion that improved this article.
Competing Interests
The authors declare that they have no competing interests.
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