1 Background

Keynesian macroeconomics inspired the seminal work of Samuelson (1939), who 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 (Chari 1994; Chow 1985; Dalla and Varelas 2015; Dalla et al. 2016; Dassios et al. 2014; Dassios and Kalogeropoulos 2014; Dassios and Zimbidis 2014; Day 1999; Karpetis and Varelas 2012; Kotsios and Leventidis 2004; Kotsios and Kostarakos 2015, 2016; Machado et al. 2015; Matsumoto and Szidarovszky 2015; Puu et al. 2004; Rosser 2000; Westerhoff 2006; Wincoop 1996). 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.

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.

Business cycle models are important as they are used to investigate economic behaviour in the study of optimal fiscal and monetary policies as well as examine the effects of economic shocks (Chari 1994). In addition, such models are also used to study investment, consumption and inventory cycles (Dalla and Varelas 2015; Dalla et al. 2016). The more realistic a business cycle model is, the more informed the analysis it provides. The model described in the present work is highly realistic as it incorporates multiple delays and trade factors for union of unlimited countries for an unlimited number of years and thus incorporates more of the factors that determine national incomes than any of the previous works seen in the literature. It also succeeds to provide a comprehensive explanation for the emergence of business cycles, while it also produces a stable trajectory for the expectation of the national income of each country which is part of the network.

The rest of this paper is organised as follows. In Sect. 2, we propose a new model for a national economy into a multi-country context where the interaction element is restricted to the annual governmental expenditure, which is determined according to the experience of the total system, and the trade relations of these countries (exports–imports). In addition, delayed variables are incorporated into the system of equations based on delayed information. Section 3 investigates the stability of the equilibrium state of the system and suggests a typical state feedback action for the different parameters involved, while it also proposes how to design the corresponding solution trajectories. Section 4 contains a practical application that provides further insight and better understanding as regards the control actions, system design and produced business cycles. Sections 5 and 6 conclude the entire paper.

2 Methods

The proposed model for a network of i countries, \(i=2, 3, \ldots , \omega\), is based on the following assumptions:

Assumption 1

National income \(T^i_k\) for the i country in year k equals to the summation of the following elements: consumption, \(C^i_k\), private investment, \(I^i_k\), governmental expenditure \(G^i_k\) and exports \(X^i_k\) less imports \(Y^i_k\)

$$T^i_k=C^i_k+I^i_k+G^i_k+X^i_k-Y^i_k.$$
(1)

Assumption 2

Consumption \(C^i_k\) in year k depends on past income (on more than one past year’s value) and on marginal tendencies to consume, modelled with \(a^i_1, a^i_2,\ldots , a^i_n\), the multiplier parameters, where \(0< a^i_1+a^i_2+\cdots +a^i_n <1\)

$$C^i_k=a^i_1T^i_{k-1}+a^i_2T^i_{k-2}+\cdots +a^i_nT^i_{k-n}.$$
(2)

Assumption 3

Private investment \(I^i_k\) in year k depends on consumption changes and on the positive accelerator factors \(b^i_1, b^i_2,\ldots , b^i_m\) . Consequently, \(I^i_k\) depends on the respective national income changes,

$$I^i_k=b^i_1\left( C^i_k-C^i_{k-1}\right) +b^i_2\left( C^i_{k-1}-C^i_{k-2}\right) +\cdots +b^i_m\left( C^i_{k-m+1}-C^i_{k-m}\right) .$$

By using (2) we get

$$I^i_k=b^i_1\left( \sum ^{n}_{q=1}a^i_q T^i_{k-q}-\sum ^{n}_{q=1}a^i_qT^i_{k-(q+1)}\right) +\cdots +\,b^i_m\left( \sum ^{n}_{q=1}a^i_qT^i_{k-(q+m-1)}-\sum ^{n}_{q=1}a^i_qT^i_{k-(q+m)}\right) ,$$

or, equivalently,

$$I^i_k=\sum ^{m}_{j=1}b^i_j \left( \sum ^{n}_{q=1}a^i_qT^i_{k-(q+j-1)}-\sum ^{n}_{q=1}a^i_qT^i_{k-(q+j)}\right) .$$
(3)

Assumption 4

The governmental expenditure depends not only on the country’s national income but also on the national income of the other countries participating into the multi-country union, i.e. governmental expenditure \(G^i_k\) in year k obeys the feedback law \(I^i_k\) in year k and depends on consumption changes and on the positive accelerator factors \(c^{i,d}_1, c^{i,d}_2,\ldots , c^{i,d}_p\) . Consequently, \(G^i_k\) is equal to,

$$G^i_k=\bar{G}^i+\sum ^{\omega }_{d=1}\left[ c^{i,d}_1T^d_{k-1}+c^{i,d}_2T^d_{k-2}+\cdots +c^{i,d}_pT^d_{k-p}\right] .$$
(4)

Note the \(G^i_k\), may be a fully controlled item by the governments, i.e.

$$G^i_k=\bar{G}^i_k+\sum ^{\omega }_{d=1}\left[ c^{i,d}_1T^d_{k-1}+c^{i,d}_2T^d_{k-2}+\cdots +c^{i,d}_pT^d_{k-p}\right] .$$

Assumption 5

Imports of country i in year k depend on past income and on the multipliers \(m^{i}_{t}\):

$$Y^{i}_{k} = m^{i}_{1} T^{i}_{k-1} + m^{i}_{2} T^{i}_{k-2}+\cdots +m^{i}_{t} T^{i}_{k-t}$$

or, equivalently,

$$Y^{i}_{k} = \sum ^{t}_{u=1} m^{i}_{u} T^{i}_{k-u}.$$
(5)

Assumption 6

The imports are also a linear sum of the exports of all other countries in year k:

$$Y^{i}_{k}= \bar{e}_{i,1} X^{1}_{k} + \bar{e}_{i,2} X^{2}_{k} +\cdots +\bar{e}_{i,\omega } X^{\omega }_{k},$$

or, equivalently,

$$Y^{i}_{k}= \sum ^{\omega }_{q=1} \bar{e}_{i,q} X^{q}_{k}.$$

Now let \(\bar{Y}_{k}\), \(\bar{E}\) and \(\bar{X}_{k}\) be \(\omega \times 1\), \(\omega \times \omega\) and \(\omega \times 1\) matrices, respectively. The ith elements of \(\bar{Y_{k}}\) and \(\bar{X_{k}}\) are \(Y^{i}_{k}\) and \(X^{i}_{k}\), respectively, while \(\bar{e}_{i,q}\) is the entry in the ith row and qth column of \(\bar{E}\). As a result, the above expression can be written as

$$\bar{Y}_{k}= \bar{E} \bar{X}_{k},$$

Let

$$\begin{aligned} E=\left\{ \begin{array}{ll} \bar{E}^{-1},&\quad {\hbox {if}}\; {\hbox {det}}\bar{E}\ne 0\\ \bar{E}^{\dagger },&\quad {\hbox {if}}\; {\hbox {det}}\bar{E}=0\end{array}\right\} . \end{aligned}$$

Then

$$\bar{X}_{k}= E \bar{Y}_{k},$$

which means the ith row of \(\bar{X}_{k}\) may be written as

$$X^{i}_{k}= e_{i,1} Y^{1}_{k} + e_{i,2} Y^{2}_{k} +\cdots +e_{i,\omega } Y^{\omega }_{k},$$

or, equivalently,

$$X^{i}_{k}= \sum ^{\omega }_{q=1} e_{i,q} Y^{q}_{k},$$
(6)

where \(e_{i,d}\) is the entry in the ith row and qth column of E. We also assume

$$\begin{array}{ll}\sum \limits ^{\omega }_{q=1} e_{i,q}=1,&\quad \forall i=1,2,\ldots ,\omega ,\\ e_{i,i}=0,&\quad \forall i=1,2,\ldots ,\omega . \end{array}$$

Combining equations (6) and (5) gives

$$X^{i}_{k}= \sum ^{\omega }_{q=1}\left[ e_{i,q} \sum ^{t}_{u=1} m^{q}_{u} T^{q}_{k-u}\right] .$$
(7)

Hence, by replacing (2), (3), (4), (5) and (7) into (1), the national income is determined via the following high-order linear difference equation

$$\begin{aligned} T^i_k&=\bar{G}^i+\sum ^{p}_{q=1}\sum ^{\omega }_{d=1}c^{i,d}_qT^d_{k-q}+\sum ^{n}_{q=1}a^i_qT^i_{k-q}+\sum ^{m}_{j=1}b^i_j \left( \sum ^{n}_{q=1}a^i_qT^i_{k-(q+j-1)}-\sum ^{n}_{q=1}a^i_qT^i_{k-(q+j)}\right) \\&\quad +\sum ^{\omega }_{q=1}\left[ e_{i,q} \sum ^{t}_{u=1} m^{q}_{u} T^{q}_{k-u}\right] -\sum ^{t}_{u=1} m^{i}_{u} T^{i}_{k-u},\quad i=1,2,\ldots ,\omega . \end{aligned}$$
(8)

Although Eq. (8) describes the general case of our model, its form is quite difficult and probably not so informative for further studies. Hence, we restrict our attention to a more realistic case where consumption depends on p − 1 past year’s income values, private investments depend on consumption changes within the last year, and governmental expenditures and export–imports depend on p past year’s income values. Then, (8) takes the form

$$\begin{aligned} T^i_k&=\bar{G}^i+\sum ^{p}_{q=1}\sum ^{\omega }_{d=1}c^{i,d}_qT^d_{k-q}+\sum ^{p-1}_{q=1}a^i_qT^i_{k-q}+\sum ^{p-1}_{q=1}b^i_1a^i_q\left( T^i_{k-q}-T^i_{k-(q+1)}\right) \\&\quad +\sum ^{\omega }_{d=1}\left[ e_{i,d} \sum ^{p}_{q=1} m^{d}_{q} T^{d}_{k-q}\right] -\sum ^{p}_{q=1} m^{i}_{q} T^{i}_{k-q},\quad i=1,2,\ldots ,\omega . \end{aligned}$$
(9)

3 Results

In this section, we study the stability of the equilibrium state of (9). In addition, by using concepts and results of linear control theory for time-invariant linear discrete state equations (Azzo and Houpis 1995; Dorf 1983; Kuo 1996; Ogata 1987), we investigate the case in which the governmental expenditure \(\bar{G}^i_k\) of country i at time k, \(i=1,2,\ldots ,\omega\), is a fully controlled variable, as also mentioned in Assumption 4 in Sect. 2. We prove the following theorem.

Theorem 3.1

Assume a union of \(\omega\) countries and let \(T^i_k\), \(i=1,2,\ldots ,\omega\), be the national income for the ith country in year k. If for every country that participates into the multi-country union, consumption depends on p − 1 past year’s income values, private investments depend on consumption changes within the last year, and governmental expenditures and export–imports depend on p past year’s income values, then:

  1. 1.

    The national income is determined via the following linear matrix difference equation

    $$\begin{aligned} T_k&=\bar{G}+\left[ C^{(1)}+A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)}\right] T_{k-1}\\&\quad +\sum _{q=2}^{p-1}\left\{ \left[ C^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) +(E-I_\omega )M^{(q)}\right] T_{k-q}\right\} \\&\quad +\left[ C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)}\right] T_{k-p}, \quad k \ge 0,\quad p \ge 3. \end{aligned}$$
    (10)

    where \(I_\omega\) is the identity matrix and

    $$\begin{aligned} T_k&= \left[ T^i_k\right] _{i=1,2,\ldots ,\omega },\\ A^{(q)}&= {\text {diag}}\left\{ a^i_q\right\} _{i=1,2,\ldots ,\omega },\quad q=1,2,\ldots ,p-1,\\ B&= {\text {diag}}\{b^i\}_{i=1,2,\ldots ,\omega },\\ C^{(q)}&= \left[ c^{i,d}_q\right] _{1\le i,d\le \omega },\quad q=1,2,\ldots ,p,\\ E&= [e_{i,d}]_{1\le i,d\le \omega },\\ M^{(q)}&= {\text {diag}}\left\{ m^i_q\right\} _{i=1,2,\ldots ,\omega },\quad q=1,2,\ldots ,p,\\ G&= [\bar{G}^i]_{i=1,2,\ldots ,\omega }. \end{aligned}$$

    The parameters \(a^i_q\), \(b^i\), \(c^{i,d}_q\), \(e_{i,d}\), \(m^i_q\), are defined in (2), (3), (4), (5) and (6). In addition, if

    $${\text {det}}\left[ \sum _{q=1}^{p-1}\left\{ C^{(q)}+A^{(q)}+(E-I_\omega )M^{(q)}\right\} +C^{(p)}+(E-I_\omega )M^{(p)}+BA^{(1)}\right] \ne 0,$$

    then the equilibrium

    $$T^*=\left[ \sum _{q=1}^{p-1}\left\{ C^{(q)}+A^{(q)}+(E-I_\omega )M^{(q)}\right\} +C^{(p)}+(E-I_\omega )M^{(p)}+BA^{(1)}\right] ^{-1}G$$

    of (10) is asymptotically stable, i.e. \(\lim _{k\longrightarrow \infty }T_k=T^*\), if and only if \(\forall\) i, \(i=1,2,\ldots ,\omega\) and \(\phi (\lambda )=0\):

    $$\left| \lambda \right| <1,\quad \lambda \in \mathbb {C}.$$

    where

    $$\begin{aligned} \phi (\lambda )&={\text {det}}\left( -\lambda ^p+\left[ C^{(1)}+A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)}\right] \lambda ^{p-1}\right. \\&\quad +\sum _{q=2}^{p-1}\left\{ \left[ C^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) +(E-I_\omega )M^{(q)}\right] \lambda ^{p-q}\right\} \\&\quad +\left. \left[ C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)}\right] \right) . \end{aligned}$$
    (11)
  2. 2.

    If governmental expenditure is a fully controlled variable for each country, then the national income is determined via the following system

    $$Y_{k+1}=FY_k+QG_k.$$
    (12)

    where \(Y_k=[Y^i_k]_{i=1,2,\ldots ,\omega }\in \mathbb {R}^{p\omega \times 1}\), \(F\in \mathbb {R}^{p\omega \times p\omega }\), \(G_k=[G^i_k]_{i=1,2,\ldots ,\omega }\in \mathbb {R}^{\omega \times 1}\) and

    $$\begin{aligned} F=\left[ \begin{array}{llllll} 0_{\omega ,\omega }&\quad I_\omega &\quad 0_{\omega ,\omega }&\quad \cdots &\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }\\ 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad I_\omega &\quad \cdots &\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }\\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \vdots \\ 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad \cdots &\quad 0_{\omega ,\omega }&\quad I_\omega \\ v_1&\quad v_2&\quad v_3&\quad \cdots &\quad v_{p-1}&\quad v_p\end{array}\right] ,\quad Q=\left[ \begin{array}{c}0_{\omega ,\omega }\\ 0_{\omega ,\omega }\\ \vdots \\ 0_{\omega ,\omega }\\ I_\omega \end{array}\right] \in \mathbb {R}^{p\omega \times \omega }. \end{aligned}$$

    where

    $$\begin{aligned} v_1&= C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)} \\ v_2&= C^{(p-1)}+A^{(p-1)}+B\left( A^{(p-1)}-A^{(p-2)}\right) +(E-I_\omega )M^{(p-1)}\\ v_3&= C^{(p-2)}+A^{(p-2)}+B\left( A^{(p-2)}-A^{(p-3)}\right) +(E-I_\omega )M^{(p-2)} \\&\qquad \qquad \qquad \vdots \\ v_{p-1}&= C^{(2)}+A^{(2)}+B\left( A^{(2)}-A^{(1)}\right) +(E-I_\omega )M^{(2)}\\ v_p&= C^{(1)}+A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)} \end{aligned}$$
    (13)

    In addition, there exists a state feedback law of the form

    $$G_k=-KY_k,$$

    where \(K=\left[ \begin{array}{ccccc} K_1&K_2&\cdots&K_p\end{array}\right] \in \mathbb {C}^{\omega \times p\omega }\), \(K_i\in \mathbb {C}^{\omega \times \omega }\), such that the eigenvalues \(\tilde{\lambda }\) of the closed-loop system can be assigned arbitrarily and be given as the roots of the polynomial \(\tilde{\phi }(\tilde{\lambda })\):

    $$\begin{aligned} \tilde{\phi }(\tilde{\lambda })&={\text {det}}\left( -\tilde{\lambda }^p+\left[ C^{(1)} +A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)}-K_p\right] \tilde{\lambda }^{p-1}\right. \\&\quad +\sum _{q=2}^{p-1}\left\{ \left[ C^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) +(E-I_\omega )M^{(q)}-K_{p-q}\right] \tilde{\lambda }^{p-q}\right\} \\&\quad +\left. \left[ C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)}-K_1\right] \right) . \end{aligned}$$

Proof

For the proof of (a), from (9), \(\forall i=1,2,\ldots ,\omega\):

$$\begin{aligned} T^i_k&=\bar{G}^i+\sum ^{p}_{q=1}\sum ^{\omega }_{d=1}c^{i,d}_qT^d_{k-q}+\sum ^{p-1}_{q=1}a^i_qT^i_{k-q}+\sum ^{p-1}_{q=1}b^i_1a^i_q\left( T^i_{k-q}-T^i_{k-(q+1)}\right) \\&\quad +\sum ^{\omega }_{d=1}\left[ e_{i,d} \sum ^{p}_{q=1} m^{d}_{q} T^{d}_{k-q}\right] -\sum ^{p}_{q=1} m^{i}_{q} T^{i}_{k-q}, \end{aligned}$$

or, equivalently,

$$\begin{aligned} T^i_k&=\bar{G}^i+\sum ^{p}_{q=1}\left[ \begin{array}{cccc}c^{i,1}_q&c^{i,2}_q&\cdots&c^{i,\omega }_q\end{array}\right] \left[ \begin{array}{c}T^1_{k-q}\\ T^2_{k-q}\\ \cdots \\ T^\omega _{k-q}\end{array}\right] + \sum ^{p-1}_{q=1}a^i_qT^i_{k-q}+\sum ^{p-1}_{q=1}b^i_1a^i_q\left( T^i_{k-q}-T^i_{k-(q+1)}\right) \\&\quad +[\begin{array}{cccc}e_{i,1}&e_{i,2}&\cdots&e_{i,\omega }\end{array}] \left[ \begin{array}{c}\sum \nolimits ^{p}_{q=1} m^{1}_{q} T^{1}_{k-q}\\ \sum \nolimits ^{p}_{q=1} m^{2}_{q} T^{2}_{k-q}\\ \cdots \\ \sum \nolimits ^{p}_{q=1} m^{\omega }_{q} T^{\omega }_{k-q}\end{array}\right] -\sum ^{p}_{q=1} m^{i}_{q} T^{i}_{k-q}, \end{aligned}$$

or, equivalently,

$$\begin{aligned} T^i_k&=\bar{G}^i+\sum ^{p}_{q=1}\left[ \begin{array}{cccc}c^{i,1}_q&c^{i,2}_q&\cdots&c^{i,\omega }_q\end{array}\right] \left[ \begin{array}{c}T^1_{k-q}\\ T^2_{k-q}\\ \cdots \\ T^\omega _{k-q}\end{array}\right] \\&\quad +\sum ^{p-1}_{q=1}a^i_qT^i_{k-q}+\sum ^{p-1}_{q=1}b^i_1a^i_qT^i_{k-q}-\sum ^{p}_{q=2}b^i_1a^i_{q-1}T^i_{k-q}\\&\quad + \sum ^{p}_{q=1}\left[ \begin{array}{cccc}e_{i,1}&e_{i,2}&\cdots&e_{i,\omega }\end{array}\right] {\text {diag}}\left\{ m^i_q\right\} _{i=1,2,\ldots ,\omega } \left[ \begin{array}{c}T^{1}_{k-q}\\ T^{2}_{k-q}\\ \cdots \\ T^{\omega }_{k-q}\end{array}\right] -\sum ^{p}_{q=1} m^{i}_{q} T^{i}_{k-q}, \end{aligned}$$

or, equivalently,

$$\begin{aligned} T^i_k&=\bar{G}^i+\sum ^{p}_{q=1}\left\{ \left( \left[ \begin{array}{ccc}c^{i,1}_q&\cdots&c^{i,\omega }_q\end{array}\right] +[\begin{array}{cccc}e_{i,1}&\cdots&e_{i,\omega }\end{array}]{\text {diag}}\left\{ m^i_q\right\} _{i=1,\ldots ,\omega }\right) \left[ \begin{array}{c}T^{1}_{k-q}\\ \cdots \\ T^{\omega }_{k-q}\end{array}\right] -m^{i}_{q} T^{i}_{k-q}\right\} \\ &\quad +\sum ^{p-1}_{q=1}\left\{ a^i_qT^i_{k-q}+b^i_1a^i_qT^i_{k-q}\right\} -\sum ^{p}_{q=2}b^i_1a^i_{q-1}T^i_{k-q}, \end{aligned}$$

or, equivalently, by denoting \(b^i_1\) as \(b^i\)

$$\begin{aligned}T^i_k&=\bar{G}^i+\left( \left[ \begin{array}{ccc}c^{i,1}_1&\cdots&c^{i,\omega }_1\end{array}\right] +[\begin{array}{cccc}e_{i,1}&\cdots&e_{i,\omega }\end{array}]{\text {diag}}\left\{ m^i_1\right\} _{i=1,\ldots ,\omega }\right) \left[ \begin{array}{c}T^{1}_{k-1}\\ \cdots \\ T^{\omega }_{k-1}\end{array}\right] \\&\quad +\left( a^i_1+b^i_1a^i_1-m^{i}_{1}\right) T^i_{k-1}+\sum ^{p-1}_{q=2}\left\{ \left( \left[ \begin{array}{ccc}c^{i,1}_q&\cdots&c^{i,\omega }_q\end{array}\right] \right. \right. \\&\quad \left. +\,[\begin{array}{cccc}e_{i,1}&\cdots&e_{i,\omega }\end{array}]{\text {diag}}\left\{ m^i_q\right\} _{i=1,\ldots ,\omega }\right) \left[ \begin{array}{c}T^{1}_{k-q}\\ \cdots \\ T^{\omega }_{k-q}\end{array}\right] \\&\quad + \left. \left( a^i_q+b^i\left( a^i_q-a^i_{q-1}\right) -m^{i}_{q}\right) T^i_{k-q}\right\} +\left( \left[ \begin{array}{ccc}c^{i,1}_p&\cdots&c^{i,\omega }_p\end{array}\right] \right. \\&\quad \left. +\, [\begin{array}{cccc}e_{i,1}&\cdots&e_{i,\omega }\end{array}]{\text {diag}}\left\{ m^i_p\right\} _{i=1,\ldots ,\omega }\right) \left[ \begin{array}{c}T^{1}_{k-p}\\ \cdots \\ T^{\omega }_{k-p}\end{array}\right] +\left( b^i_1a^i_p-m^{i}_{p}\right) T^i_{k-p}, \end{aligned}$$

or, equivalently, in matrix form,

$$\begin{aligned}T_k&=\bar{G}+\left( C^{(1)}+EM^{(1)}+A^{(1)}+BA^{(1)}-M^{(1)}\right) T^i_{k-1}\\&\quad + \sum ^{p-1}_{q=2}\left\{ \left( C^{(q)}+EM^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) -M^{(q)}\right) T_{k-q}\right\} \\&\quad + \left( C^{(p)}+EM^{(p)}+BA^{(p)}-M^{(p)}\right) T^i_{k-p}, \end{aligned}$$

which leads to (10). Let \(T^*\) be the equilibrium of (10). Then, for \({\text {det}}[\sum _{q=1}^{p-1}\{C^{(q)}+A^{(q)}+(E-I_\omega )M^{(q)}\}+C^{(p)}+(E-I_\omega )M^{(p)}+BA^{(1)}]\ne 0\) we have

$$\begin{aligned} T^*&=\bar{G}+\left\{ C^{(1)}+A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)}\right. \\&\quad + \sum _{q=2}^{p-1}\left[ C^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) + (E-I_\omega )M^{(q)}\right] \\&\quad \left. +\, C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)}\right\} T^* \end{aligned}$$

or, equivalently,

$$T^*=\left[ \sum _{q=1}^{p-1}\left\{ C^{(q)}+A^{(q)}+(E-I_\omega )M^{(q)}\right\} +C^{(p)}+(E-I_\omega )M^{(p)}+BA^{(1)}\right] ^{-1}\bar{G}.$$

We are interested in the asymptotic stability of the equilibrium, i.e. when \(lim_{k\longrightarrow \infty }T_k=T^*\). By adopting the notation

$$\begin{array}{c} T_{k-p}=Y^1_k,\\ T_{k-p+1}=Y^2_k,\\ \vdots \\ T_{k-2}=Y^{p-1}_k\\ T_{k-1}=Y^p_k \end{array}$$
(14)

and

$$\begin{aligned} \begin{array}{cc} &T_{k-p+1}=Y^1_{k+1}=Y^2_k,\\ &T_{k-p+2}=Y^2_{k+1}=Y^3_k,\\ &\vdots \\ &T_{k-1}=Y^{p-1}_{k+1}=Y^p_k\\ \end{array}\end{aligned}$$
$$\begin{aligned} T_{k}&=Y^p_{k+1}=\bar{G}+\left( C^{(1)}+A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)}\right) Y^p_k\\ &\quad +\sum \limits _{q=2}^{p-1}\left\{ (C^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) +(E-I_\omega )M^{(q)})Y^q_k\right\} \\ &\quad +\left( C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)}\right) Y^1_k. \end{aligned}$$

Then, the matrix difference equation (10) takes the form

$$Y_{k+1}=FY_k+\left[ \begin{array}{c}0_{\omega ,1}\\ \vdots \\ 0_{\omega ,1}\\ G\end{array}\right] .$$

The stability of the above system depends on the eigenvalues of F. Let \(\lambda\) be an eigenvalue of F. Then, the equilibrium of (10) is asymptotic stable if and only if

$$\begin{array}{cc}\left| \lambda \right| <1,&\lambda \in {\mathbb {C}}.\end{array}$$

In addition, if \(U=\left[ \begin{array}{c}U_1\\ U_2\\ \vdots \\ U_p\end{array}\right]\) is an eigenvector of the eigenvalue \(\lambda\), then

$$\begin{aligned} \left[ \begin{array}{cccccc} 0_{\omega ,\omega }&\quad I_\omega &\quad 0_{\omega ,\omega }&\quad \cdots &\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }\\ 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad I_\omega &\quad \cdots &\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }\\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \vdots \\ 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad \cdots &\quad 0_{\omega ,\omega }&\quad I_\omega \\ v_1&\quad v_2&\quad v_3&\quad \cdots &\quad v_{p-1}&\quad v_p\end{array}\right] U=\lambda U, \end{aligned}$$

or, equivalently,

$$\begin{array}{c} U_2=\lambda U_1,\\ U_3=\lambda U_2,\\ \vdots \\ U_p=\lambda U_{p-1},\\ \sum \nolimits _{i=1}^pv_iU_i=\lambda U_p, \end{array}$$

or, equivalently,

$$\begin{array}{c} U_2=\lambda U_1,\\ U_3=\lambda ^2 U_1,\\ \vdots \\ U_p=\lambda ^{p-1} U_1,\\ v_1U_1+v_2U_2+\cdots +v_pU_p=\lambda U_p. \end{array}$$

By replacing the first \(p-1\) equations into the pth equation of the above expression, we get

$$v_1U_1+\lambda v_2 U_1+\cdots +\lambda ^{p-1}v_pU_1=\lambda ^p U_1,$$

or, equivalently,

$$\left[ v_1+\lambda v_2+\cdots +\lambda ^{p-1}v_p-\lambda ^pI_\omega \right] U_1=0.$$

Hence, the eigenvalues \(\lambda\) are the roots of the polynomial \(\phi (\lambda )={\hbox{det}}[v_1+\lambda v_2+\cdots +\lambda ^{p-1}v_p-\lambda ^pI_\omega ]\), which by replacing (13) is polynomial (11). Thus, the equilibrium \(T^*\) of (10) is asymptotically stable if and only if \(\forall i=1,2,\ldots ,\omega\) and for \(\phi (\lambda )=0\)

$$\begin{array}{cc}\left| \lambda \right| <1,&\lambda \in \mathbb {C}.\end{array}$$

For the proof of (b), since governmental expenditure is a fully controlled variable for each country, equation (10) will take the form

$$\begin{aligned} T_k&=\bar{G}_k+\left[ C^{(1)}+A^{(1)}+BA^{(1)}+(E-I_\omega )M^{(1)}\right] T_{k-1}\\&\quad + \sum _{q=2}^{p-1}\left\{ \left[ C^{(q)}+A^{(q)}+B\left( A^{(q)}-A^{(q-1)}\right) +(E-I_\omega )M^{(q)}\right] T_{k-q}\right\} \\&\quad + \left[ C^{(p)}-BA^{(p-1)}+(E-I_\omega )M^{(p)}\right] T_{k-p}, \quad k \ge 0,\quad p \ge 3. \end{aligned}$$

By adopting (14) and applying it into the above expression, we arrive at (13) which is a linear discrete-time control system with input vector \(G_k\). Linear control involves modification of the behaviour of a given system by applying state feedback. It is known Azzo and Houpis (1995), Dorf (1983), Kuo (1996), Ogata (1987) that this is possible if and only if the system is completely controllable. The necessary and sufficient condition for complete controllability is that

$${\text {rank}}\left[ \begin{array}{ccccc}Q&FQ&F^2Q&\cdots&F^{p-1}Q\end{array}\right] =p\omega .$$

Since

$${\text {det}}\left[ \begin{array}{ccccc}Q&FQ&F^2Q&\cdots&F^{p-1}Q\end{array}\right] =(-1)^{p\omega }\ne 0,$$

which means system (13) is controllable and the state feedback replaces the input \(G_k\) by

$$\bar{G}_k=-KY_k,$$

where

$$K=\left[ \begin{array}{cccc}K_1&K_2&\cdots&K_p\end{array}\right]$$

and \(K_1,K_2,\ldots ,K_p\in \mathbb {C^{\omega \times \omega }}\). The system then takes the form

$$Y_{k+1}=(F-QK)Y_k,$$

where \(QK=\left[ \begin{array}{c}0_{\omega ,\omega }\\ \vdots \\ 0_{\omega ,\omega }\\ K \end{array}\right]\). The basic problem is that of choosing a state feedback K such that the resulting (closed-loop equation) is stable. The stabilisation in the time-invariant case is via results on eigenvalue placement in the complex plane. In our situation, eigenvalues of the closed-loop system are specified to have modulus less than unity for stability. Thus, we have

$$\begin{aligned} F-QK=\left[ \begin{array}{cccccc} 0_{\omega ,\omega }&\quad I_\omega &\quad 0_{\omega ,\omega }&\quad \cdots &\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }\\ 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad I_\omega &\quad \cdots &\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }\\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \vdots \\ 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad 0_{\omega ,\omega }&\quad \cdots &\quad 0_{\omega ,\omega }&\quad I_\omega \\ v_1-K_1&\quad v_2-K_2&\quad v_3-K_3&\quad \cdots &\quad v_{p-1}-K_{p-1}&\quad v_p-K_p\end{array}\right] , \end{aligned}$$

where \(v_i\), \(i=1,2,\ldots ,\omega\) are given by (14). Then, the characteristic equation of \(F-QK\) will be \(\tilde{\phi }(\tilde{\lambda })\), with \(\tilde{\lambda }\) the desired eigenvalues. The proof is completed. \(\square\)

4 Numerical examples

In this section, we present two numerical examples in order to provide further insight and better understanding regarding the system design and produced business cycles. For both examples, we model \(\omega =5\) countries that each look back p = 3 time steps. Let \(A^{(1)}\), \(A^{(2)}\), B and \(C^{(3)}\) take the following values:

$$\begin{aligned} \begin{array}{l}A^{(1)}={\text {diag}}\{0.3,0.5,0.2,0.35,0.4\},\\ A^{(2)}={\text {diag}}\{0.3,0.1,0.25,0.25,0.35\},\\ B={\text {diag}}\{0.5, 0.7, 0.6,0.4, 0.3\},\end{array}\quad C^{(3)}=\left( \begin{array}{lllll} 0.26&\quad 0.07& \quad 0.07&\quad 0.04& \quad 0.03\\ 0.39&\quad 0.30&\quad 0.27&\quad 0.21&\quad 0.10\\ 0.39&\quad 0.35&\quad 0.33& \quad 0.29&\quad 0.02\\ 0.29&\quad 0.26&\quad 0.20&\quad 0.03&\quad 0.01\\ 0.32&\quad 0.21&\quad 0.20&\quad 0.19&\quad 0.06\\ \end{array} \right) . \end{aligned}$$

while \(\bar{E}\) and \(M^{(i)}\), \(i=1,2,3\), are given by:

$$\begin{aligned} \begin{array}{l} M^{(1)}={\text {diag}}\{0.9,0.95,0.925,0.9,0.8\},\\ M^{(2)}={\text {diag}}\{0.82,0.85,0.825,0.8,0.775\},\\ M^{(3)}={\text {diag}}\{0.76,0.75,0.725,0.7,0.75\}, \end{array}\quad \bar{E}=\left( \begin{array}{lllll} 0&\quad 0.6 &\quad 0.2 &\quad 0.02&\quad 0.03\\ 0.25&\quad 0 &\quad 0.2 &\quad 0.03&\quad 0.02\\ 0.25&\quad 0.1 &\quad 0 &\quad 0.05&\quad 0.05\\ 0.25&\quad 0.1 &\quad 0.2 &\quad 0 &\quad 0.9\\ 0.25&\quad 0.2 &\quad 0.4 &\quad 0.90&\quad 0 \\ \end{array} \right) . \end{aligned}$$

In addition let \(C^{1}\), \(C^{2}\) be given by:

$$\begin{aligned} C^{(1)}=\left( \begin{array}{lllll} c^{1,1}_{1} &\quad 0.32 &\quad 0.28 &\quad 0.24 & \quad 0.20\\ 0.32 & \quad 0.28 &\quad 0.24 & \quad 0.20& \quad 0.16 \\ 0.28 & \quad 0.24 & \quad 0.20& \quad 0.12 & \quad 0.08 \\ 0.24 &\quad 0.20& \quad 0.12 & \quad 0.08 & \quad 0.04 \\ 0.20& \quad 0.12 &\quad 0.08 & \quad 0.04 & \quad 0.01\\ \end{array} \right) ,\quad C^{(2)}=\left( \begin{array}{lllll} c^{1,1}_{2}& \quad 0.24& \quad 0.10& \quad 0.04& \quad 0.01\\ 0.36&\quad 0.30&\quad 0.26&\quad 0.25&\quad 0.13\\ 0.34&\quad 0.27&\quad 0.20&\quad 0.20&\quad 0.13\\ 0.32&\quad 0.31&\quad 0.28&\quad 0.19&\quad 0.05\\ 0.36&\quad 0.29&\quad 0.29&\quad 0.23&\quad 0.08\\ \end{array} \right) . \end{aligned}$$

In the matrices \(C^{(1)}\) and \(C^{(2)}\), the first elements (i.e. \(c^{1,1}_{1}\) and \(c^{1,1}_{2}\)) are not assigned specific numerical values as these entries are varied in Examples 4.1 and 4.2.

4.1 Example 4.1

For this example, we assume that \(c^{1,1}_{2}\) takes the value 0.36. Then, we may use Theorem 3.1 and determine the value of parameter \(c^{1,1}_{1}\) such that the system is stable. Note that through \(c^{1,1}_{1}\) the government of country i = 1 controls the relevant governmental expenditure in order to obtain asymptotic stability in the system of national economy. The speed of the system’s response is basically characterised by the maximum value r of the following set

$$r=\max \left\{ \begin{array}{ccc}\left| \lambda \right| ,&\forall \lambda \in {\mathbb {C}},&\phi (\lambda )=0\end{array}\right\} ,$$

where \(\lambda\) are the roots of (11). High-speed response coincides with minimising the value of r. According to Theorem 3.1, we require to have \(r<1\), which is satisfied for \(0<c^{1,1}_{1}<6.1\), see Figs. 1 and 2, \(c^{1,1}_{1}\).

Fig. 1
figure 1

Illustration of 2D diagram between r and \(c^{1,1}_{1}\)

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Fig. 2
figure 2

Illustration of 2D diagram between r and \(c^{1,1}_{1}\)

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4.2 Example 4.2

For this example, we may use Theorem 3.1 and determine the value of parameters \(c^{1,1}_{1}\) and \(c^{1,1}_{2}\) such that the system is stable. Note that through \(c^{1,1}_{1}\) and \(c^{1,1}_{2}\), the government of country i = 1 controls the relevant governmental expenditure in order to obtain asymptotic stability in the system of national economy. The speed of the system’s response is again characterised by the maximum value r, defined in Example 4.1. As before, high-speed response coincides with minimising the value of r. Again, according to Theorem 3.1, we require to have \(r<1\). Figures 3, 4 and 5 show the regions for which this is true. Figure 3 is a three-dimensional graph showing the values of r for different values of \(c^{1,1}_{1}\) and \(c^{1,1}_{2}\). In Fig. 4, the different colours represent the different values of r. Figure 5 displays when the system is stable (blue region) and unstable (red region), i.e. when \(r<1\) and \(r\ge 1\), respectively.

Fig. 3
figure 3

Illustration of 3D diagram between r, \(c^{1,1}_{1}\) and \(c^{1,1}_{2}\)

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Fig. 4
figure 4

Illustration diagram between \(c^{1,1}_{1}\) and \(c^{1,1}_{2}\). The bar on the right represents the values of r

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Fig. 5
figure 5

Illustration diagram which displays when the system is stable (blue region) and unstable (red region)

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5 Discussion

A further extension of this paper is to apply (incorporate) fractional operators into the model. The fractional nabla operator is a very interesting tool for this, since it succeeds to provide information from a specific year in the past until the current year, see Dassios and Baleanu (2013), Dassios et al. (2014), Dassios (2015)

  • The nabla operator of nth order, n Natural, applied to a vector of sequences \(Y_k:\mathbb {N}_\alpha \rightarrow \mathbb {C}^{m}\) is defined by:

    $$\nabla ^nY_k=\sum ^{n}_{j=0}a_jY_{k-j}=a_0Y_k+a_1Y_{k-1}+\cdots +a_nY_{k-n},$$

    where \(a_j=(-1)^j\frac{1}{\Gamma (n+1)\Gamma (j+1)\Gamma (n-j+1)}\).

  • The nabla fractional operator of nth order, n Fractional, applied to a vector of sequences \(Y_k:\mathbb {N}_\alpha \rightarrow \mathbb {C}^{m}\) is defined by

    $$\nabla _\alpha ^nY_k=\sum ^{k}_{j=\alpha }b_jY_j=b_kY_k+b_{k-1}Y_{k-1}+\cdots +b_\alpha Y_\alpha ,$$

    where \(b_j=\frac{1}{\Gamma (-n)}(k-j+1)^{\overline{-n-1}}\) and \((k-j+1)^{\overline{-n-1}}=\frac{\Gamma (k-j-n)}{\Gamma (k-j+1)}\).

Results on stability, robustness, duality, etc., of this operator, see Dassios (2016), Dassios and Baleanu (2015), Dassios (2015), may also be used successfully on the suggested macroeconomic models.

Furthermore, the coefficients used in the model (i.e. \(a^{i}_{q}\), \(b^{i}\) and \(c^{i,d}_{q}\)) are all deterministic and assumed exactly known. In reality, it may be difficult to determine these exactly, particularly coefficients that represent predicted future values. Future work will examine the effects and stability of making the coefficients in the model stochastic. For all these, there is some research in progress.

6 Conclusions

In this article, we assumed a union of countries where each national economy interacts with the others according to past year's experience and trade relations (exports–imports). We considered the equilibrium of this model (a discrete-time system) and provided a theorem for its stability, the appropriate control actions and the total system design in order to obtain a stable situation. Finally, numerical examples were given in order to provide further insight and better understanding as regards the system design and produced business cycles.