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1 econstor Make Your Publications Visible A Service of Wirtschaft Centre zbwleibniz-informationszentrum Economics Dassios, Ioannis K; Devine, Mel T Article A macroeconomic mathematical model for the national income of a union of countries with interaction and trade Journal of Economic Structures Provided in Cooperation with: Pan-Pacific Association of Input-Output Studies PAPAIOS) Suggested Citation: Dassios, Ioannis K; Devine, Mel T 206) : A macroeconomic mathematical model for the national income of a union of countries with interaction and trade, Journal of Economic Structures, ISSN , Springer, Heidelberg, Vol 5, Iss 8, pp -5, This Version is available at: Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public If the documents have been made available under an Open Content Licence especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence wwweconstoreu

2 DOI 086/s RESEARCH Open Access A macroeconomic mathematical model for the national income of a union of countries with interaction and trade Ioannis K Dassios * and Mel T Devine,2,3 *Correspondence: jdasios@mathuoagr Mathematics Applications Consortium for Science and Industry MACSI), Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland Full list of author information is available at the end of the article Abstract In this article, we assume a union of countries where each national economy interacts with the others We propose a new model where a) delayed variables are incorporated into the system of equations and b) the interaction element is restricted into the annual governmental expenditure that is determined according to the experience of the total system and the trade relations of these countries exports imports) In addition, we consider the equilibriums) of the model a discrete-time system) and study properties for stability, the appropriate control actions as well as the total system design in order to obtain a stable situation Finally, a practical application is also investigated that provides further insight and better understanding as regards the system design and produced business cycles Keywords: National economy, Network, Difference equations, Stability, Control, Trade Background Keynesian macroeconomics inspired the seminal work of Samuelson 939), 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 994; Chow 985; Dalla and Varelas 205; Dalla et al 206; Dassios et al 204; Dassios and Kalogeropoulos 204; Dassios and Zimbidis 204; Day 999; Karpetis and Varelas 202; Kotsios and Leventidis 2004; Kotsios and Kostarakos 205, 206; Machado et al 205; Matsumoto and Szidarovszky 205; Puu et al 2004; Rosser 2000; Westerhoff 2006; Wincoop 996) 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 206 The Authors) This article is distributed under the terms of the Creative Commons Attribution 40 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made

3 Page 2 of 5 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 994) In addition, such models are also used to study investment, consumption and inventory cycles Dalla and Varelas 205; Dalla et al 206) 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,, ω, is based on the following assumptions: Assumption National income Tk i for the i country in year k equals to the summation of the following elements: consumption, Ck i, private investment, Ii k, governmental expenditure Gk i and exports Xi k less imports Y k i Tk i = Ci k Ii k Gi k Xi k Y k i ) Assumption 2 Consumption Ck i 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, ai 2,, ai n, the multiplier parameters, where 0 < a i ai 2 ai n < Ck i = ai T k i ai 2 T k 2 i ai n T k n i 2) Assumption 3 Private investment Ik i in year k depends on consumption changes and on the positive accelerator factors b i, bi 2,, bi m Consequently, Ii k depends on the respective national income changes, ) ) ) Ik i = bi Ck i Ci k b2 i Ck i Ci k 2 bm i Ck m i Ci k m

4 Page 3 of 5 By using 2) we get Ik i = bi a i q T k q i a i q T k q) i bm i a i q T k qm ) i a i q T k qm) i, m Ik i = a i q T k qj ) i a i q T k qj) i bj i j= 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, ie governmental expenditure Gk i in year k obeys the feedback law Ik i in year k and depends on consumption changes and on the positive accelerator factors c i,d, ci,d 2,, ci,d p Consequently, Gi k is equal to, G i k = Ḡi ω d= Note the Gk i, may be a fully controlled item by the governments, ie G i k = Ḡi k ω d= [ c i,d T k d ci,d 2 T k 2 d ci,d p T k p d [ c i,d T k d ci,d 2 T k 2 d ci,d p T k p d 4) Imports of country i in year k depend on past income and on the mul- Assumption 5 tipliers m i t : Y i k = mi T i k mi 2 T i k 2 mi t T i k t t Yk i = m i u T k u i u= 5) Assumption 6 The imports are also a linear sum of the exports of all other countries in year k: Y i k =ē i,x k ē i,2x 2 k ē i,ωx ω k, ω Yk i = ē i,q X q k

5 Page 4 of 5 Now let Ȳ k, Ē and X k be ω, ω ω and ω matrices, respectively The ith elements of Y k and X k are Yk i and Xi k, respectively, while ē i,q is the entry in the ith row and qth column of Ē As a result, the above expression can be written as Ȳ k = Ē X k, Let E = { } Ē, if detē =0 Ē, if detē = 0 Then X k = EȲ k, which means the ith row of X k may be written as Xk i = e i,yk e i,2yk 2 e i,ωyk ω, ω Xk i = 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 ω e i,q =, i =, 2,, ω, e i,i = 0, i =, 2,, ω Combining equations 6) and 5) gives ω [ Xk i = e i,q u= t m q u T q k u 7) Hence, by replacing 2), 3), 4), 5) and 7) into ), the national income is determined via the following high-order linear difference equation T i k = Ḡi ω d= [ ω c i,d e i,q u= m q T k q d a i q T k q i t m q u T q k u t u= j= b i j a i q T k qj ) i a i q T k qj) i m i u T k u i, i =, 2,, ω 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 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

6 Page 5 of 5 T i k = Ḡi ω c i,d d= ω e i,d d= q T k q d a i q T k q i m d q T k q d b i ai q ) Tk q i T k q) i m i q T k q i, i =, 2,, ω 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 995; Dorf 983; Kuo 996; Ogata 987), we investigate the case in which the governmental expenditure Ḡk i of country i at time k, i =, 2,, ω, is a fully controlled variable, as also mentioned in Assumption 4 in Sect 2 We prove the following theorem Theorem 3 Assume a union of ω countries and let Tk i, i =, 2,, ω, be the national income for the ith country in year k If for every country that participates into the multicountry union, consumption depends on p 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: The national income is determined via the following linear matrix difference equation [ T k = Ḡ C ) A ) BA ) E I ω )M ) T k {[ C q) A q) B A q) A q )) E I ω )M q) } T k q [ C p) BA ) E I ω )M p) T k p, k 0, p 3 where I ω is the identity matrix and [ T k = Tk i, i=,2,,ω { } A q) = diag, q =, 2,, p, a i q The parameters a i q, bi, cq i,d, e i,d, m i q, are defined in 2), 3), 4), 5) and 6) In addition, if { det C q) A q) E I ω )M q)} C p) E I ω )M p) BA ) =0, then the equilibrium i=,2,,ω B = diag{b i } i=,2,,ω, [ C q) =, q =, 2,, p, c i,d q i,d ω E =[e i,d i,d ω, { } M q) = diag m i q G =[Ḡ i i=,2,,ω i=,2,,ω, q =, 2,, p, 0)

7 Page 6 of 5 { T = C q) A q) E I ω )M q)} C p) E I ω )M p) BA ) of 0) is asymptotically stable, ie lim k T k = T, if and only if i, i =, 2,, ω and φλ) = 0: G λ <, λ C where [ φλ) = det λ p C ) A ) BA ) E I ω )M ) λ {[ C q) A q) B A q) A q )) E I ω )M q) λ p q} [ C p) BA ) E I ω )M p)) ) 2 If governmental expenditure is a fully controlled variable for each country, then the national income is determined via the following system Y k = FY k QG k where Y k =[Yk i i=,2,,ω R pω, F R pω pω, G k =[Gk i i=,2,,ω R ω and 0 ω,ω I ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω I ω 0 ω,ω 0 ω,ω 0 ω,ω F = 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω I, Q = ω 0 Rpω ω ω,ω v v 2 v 3 v v p I ω where v = C p) BA ) E I ω )M p) v 2 = C ) A ) B A ) A p 2)) E I ω )M ) v 3 = C p 2) A p 2) B A p 2) A p 3)) E I ω )M p 2) v = C 2) A 2) B A 2) A )) E I ω )M 2) v p = C ) A ) BA ) E I ω )M ) In addition, there exists a state feedback law of the form G k = KY k, where K = [ K K 2 K p C ω pω, K i C ω ω, such that the eigenvalues λ of the closed-loop system can be assigned arbitrarily and be given as the roots of the polynomial φ λ): [ φ λ) = det λ p C ) A ) BA ) E I ω )M ) λ K p {[ C q) A q) B A q) A q )) E I ω )M q) λp q } K p q [C p) BA ) E I ω )M p) K ) 2) 3)

8 Page 7 of 5 Proof For the proof of a), from 9), i =, 2,, ω: T i k = Ḡi ω d= ω q T k q d a i q T k q i c i,d e i,d d= m d q T k q d m i q T k q i, b i ai q ) Tk q i T k q) i T i k = Ḡi [ c i, q cq i,2 cq i,ω T k q T 2 k q T ω k q p m q T k q p [e i, e i,2 e i,ω m2 q T k q 2 p mω q T k q ω a i q T k q i m i q T k q i, b i ai q ) Tk q i T k q) i T i k = Ḡi [ c i, q cq i,2 cq i,ω T k q T 2 k q T ω k q a i q T k q i b i ai q T k q i b i ai q T k q i [ { } ei, e i,2 e i,ω diag m i q i=,2,,ω T k q T 2 k q T ω k q m i q T i k q, T i k = Ḡi [ c i, q c i,ω q { } a i q T k q i bi ai q T k q i } ) T [ei, e i,ω diag{ m i k q q i=,,ω Tk q ω b i ai q T k q i, m i q T k q i by denoting b i as bi

9 Page 8 of 5 [ Tk i = Ḡi c i, c i,ω } ) [ei, e i,ω diag{ m i T k i=,,ω a i bi ai mi )Tk i {[ c i, q c i,ω q { } ) T [e i, e i,ω diag m i k q q i=,,ω Tk q ω ) ) } a i q bi a i q ai q m i q Tk q i [ cp i, { } ) T [e i, e i,ω diag m i k p p i=,,ω Tk p ω c i,ω p T ω k ) b i ai p mi p Tk p i, in matrix form, T k = Ḡ C ) EM ) A ) BA ) M )) Tk i which leads to 0) Let T be the equilibrium of 0) Then, for det[ {Cq) A q) E I ω )M q) }C p) E I ω )M p) BA ) =0 we have { C q) EM q) A q) B A q) A q )) M q)) } T k q C p) EM p) BA p) M p)) T i k p, { T = Ḡ C ) A ) BA ) E I ω )M ) [ C q) A q) B A q) A q )) E I ω )M q) C p) BA ) E I ω )M p)} T { T = C q) A q) E I ω )M q)} C p) E I ω )M p) BA ) We are interested in the asymptotic stability of the equilibrium, ie when lim k T k = T By adopting the notation Ḡ and T k p = Y k, T k p = Y 2 k, T k 2 = Y k T k = Y p k 4)

10 Page 9 of 5 T k p = Yk = Y k 2, T k p2 = Yk 2 = Y k 3, T k = Y k = Y p k T k = Y p k = Ḡ C ) A ) BA ) E I ω )M )) Y p k { C q) A q) B A q) A q )) } E I ω )M q) )Y q k Then, the matrix difference equation 0) takes the form The stability of the above system depends on the eigenvalues of F Let λ be an eigenvalue of F Then, the equilibrium of 0) is asymptotic stable if and only if In addition, if U = C p) BA ) E I ω )M p)) Y k Y k = FY k λ <, λ C 0 ω, 0 ω, G U U 2 U p is an eigenvector of the eigenvalue λ, then 0 ω,ω I ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω I ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω I U = λu, ω v v 2 v 3 v v p U 2 = λu, U 3 = λu 2, U p = λu, p i= v iu i = λu p, U 2 = λu, U 3 = λ 2 U, U p = λ U, v U v 2 U 2 v p U p = λu p

11 Page 0 of 5 By replacing the first p equations into the pth equation of the above expression, we get [v λv 2 λ v p λ p I ω U = 0 Hence, the eigenvalues λ are the roots of the polynomial φλ) = det[v λv 2 λ v p λ p I ω, which by replacing 3) is polynomial ) Thus, the equilibrium T of 0) is asymptotically stable if and only if i =, 2,, ω and for φλ) = 0 For the proof of b), since governmental expenditure is a fully controlled variable for each country, equation 0) will take the form By adopting 4) and applying it into the above expression, we arrive at 3) 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 995), Dorf 983), Kuo 996), Ogata 987) that this is possible if and only if the system is completely controllable The necessary and sufficient condition for complete controllability is that Since which means system 3) is controllable and the state feedback replaces the input G k by where v U λv 2 U λ v p U = λ p U, λ <, λ C [ T k = Ḡ k C ) A ) BA ) E I ω )M ) T k {[ C q) A q) B A q) A q )) E I ω )M q) } T k q [ C p) BA ) E I ω )M p) T k p, k 0, p 3 rank [ Q FQ F 2 Q F Q = pω det [ QFQF 2 Q F Q = ) pω =0, Ḡ k = KY k, K = [ K K 2 K p and K, K 2,, K p C ω ω The system then takes the form Y k = F QK )Y k, 0 ω,ω where QK = The basic problem is that of choosing a state feedback K such that 0 ω,ω K the resulting closed-loop equation) is stable The stabilisation in the time-invariant case

12 Page of 5 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 0 ω,ω I ω 0 ω,ω 0 ω,ω 0 ω,ω F QK = 0 ω,ω 0 ω,ω I ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω 0 ω,ω I ω v K v 2 K 2 v 3 K 3 v K v p K p, where v i, i =, 2,, ω are given by 4) Then, the characteristic equation of F QK will be φ λ), with λ the desired eigenvalues The proof is completed 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 ω = 5 countries that each look back p = 3 time steps Let A ), A 2), B and C 3) take the following values: A ) = diag{03, 05, 02, 035, 04}, A 2) = diag{03, 0, 025, 025, 035}, B = diag{05, 07, 06, 04, 03}, while Ē and M i), i =, 2, 3, are given by: M ) = diag{09, 095, 0925, 09, 08}, M 2) = diag{082, 085, 0825, 08, 0775}, M 3) = diag{076, 075, 0725, 07, 075}, In addition let C, C 2 be given by: C 3) = Ē = c, c, C ) = , C 2) = In the matrices C ) and C 2), the first elements ie c, and c, 2 ) are not assigned specific numerical values as these entries are varied in Examples 4 and 42 4 Example 4 For this example, we assume that c, 2 takes the value 036 Then, we may use Theorem 3 and determine the value of parameter c, such that the system is stable Note that through c, the government of country i = 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

13 Page 2 of 5 r = max { λ, λ C, φλ) = 0 }, where λ are the roots of ) High-speed response coincides with minimising the value of r According to Theorem 3, we require to have r <, which is satisfied for 0 < c, < 6, see Figs and 2, c, 42 Example 42 For this example, we may use Theorem 3 and determine the value of parameters c, and c, 2 such that the system is stable Note that through c, and c, 2, the government of country i = 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 As before, Fig Illustration of 2D diagram between r and c, Fig 2 Illustration of 2D diagram between r and c,

14 Page 3 of 5 high-speed response coincides with minimising the value of r Again, according to Theorem 3, we require to have r < 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, and c, 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), ie when r < and r, respectively 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 203), Dassios et al 204), Dassios 205) Fig 3 Illustration of 3D diagram between r, c, and c, 2 Fig 4 Illustration diagram between c, and c, 2 The bar on the right represents the values of r

15 Page 4 of 5 Fig 5 Illustration diagram which displays when the system is stable blue region) and unstable red region) The nabla operator of nth order, n Natural, applied to a vector of sequences Y k : N α C m is defined by: n Y k = a j Y k j = a 0 Y k a Y k a n Y k n, j=0 where a j = ) j Ŵn)Ŵj)Ŵn j) The nabla fractional operator of nth order, n Fractional, applied to a vector of sequences Y k : N α C m is defined by n α Y k = k b j Y j = b k Y k b k Y k b α Y α, j=α where b j = Ŵ n) k j ) n and k j ) n = Ŵk j n) Ŵk j) Results on stability, robustness, duality, etc, of this operator, see Dassios 206), Dassios and Baleanu 205), Dassios 205), may also be used successfully on the suggested macroeconomic models Furthermore, the coefficients used in the model ie a i q, bi and cq i,d ) 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

16 Page 5 of 5 order to provide further insight and better understanding as regards the system design and produced business cycles Author details Mathematics Applications Consortium for Science and Industry MACSI), Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland 2 Economic and Social Research Institute ESRI), Whitaker Square, Dublin 2, Ireland 3 Department of Economics, Trinity College Dublin, Dublin, Ireland Acknowledgements Both authors acknowledge Science Foundation Ireland award 09/SRC/E780, while M T Devine also acknowledges the ESRI s Energy Policy Research Centre Competing interests The authors declare that they have no competing interests Received: 5 December 205 Accepted: 7 May 206 References Azzo JD, Houpis CH 995) Linear control system analysis and design, 4th edn Mc Graw-Hill, New York Chari VV 994) Optimal fiscal policy in a business cycle model J Polit Econ 024):52 6 Chow GC 985) A model of Chinese national income determination J Polit Econ 934): Dalla E, Varelas E 205) Second-order accelerator of investment: the case of discrete time Int Rev Econ Educ 2:48 60 Dalla E, Karpetis C, Varelas E 206) Modeling investment cycles: a theoretical analysis Mod Econ 703):336 Dassios I, Zimbidis A, Kontzalis C 204) The delay effect in a stochastic multiplier accelerator model J Econ Struct 3):7 Dassios I, Kalogeropoulos G 204) On the stability of equilibrium for a reformulated foreign trade model of three countries J Ind Eng Int 0:7 doi:0007/s Dassios I, Zimbidis A 204) The classical Samuelson s model in a multi-country context under a delayed framework with interaction Dyn Contin Discrete Impuls Syst Ser B Appl Algorithms 24 5b): Dassios IK, Baleanu D 203) On a singular system of fractional nabla difference equations with boundary conditions Bound Value Probl 203:48 Dassios I, Baleanu D, Kalogeropoulos G 204) On non-homogeneous singular systems of fractional nabla difference equations Appl Math Comput 227:2 3 Dassios I 206) Stability and robustness of singular systems of fractional nabla difference equations Circuits Syst Signal Process doi:0007/s x Dassios I, Baleanu D 205) Duality of singular linear systems of fractional nabla difference equations Appl Math Model 394): Dassios IK 205) Optimal solutions for non-consistent singular linear systems of fractional nabla difference equations Circuits Syst Signal Process 346): Dassios I 205) Geometric relation between two different types of initial conditions of singular systems of fractional nabla difference equations Methods Appl Sci Math doi:0002/mma377 Day R 999) Complex economic dynamics: an introduction to macroeconomic dynamics, vol 2 MIT Press, Cambridge Dorf RC 983) Modern control systems, 3rd edn Addison-Wesley, Boston Karpetis C, Varelas E 202) Fiscal and monetary policy interaction in a simple accelerator model Int Adv Econ Res 82):99 24 Kotsios S, Leventidis J 2004) A feedback policy for a modified Samuelson Hicks model Int J Syst Sci 356):33 34 Kotsios I, Kostarakos S 205) Fiscal policy design in Greece in the aftermath of the crisis: an algorithmic approach Kotsios I, Kostarakos S 206) Feedback policy rules for government spending: an algorithmic approach researchgatenet/ Kuo BC 996) Automatic control systems, 5th edn Prentice Hall, Upper Saddle River Machado JA, Mata ME, Lopes AM 205) Fractional state space analysis of economic systems Entropy 78): Matsumoto A, Szidarovszky F 205) Nonlinear multiplier? Accelerator model with investment and consumption delays Struct Change Econ Dyn 33205): 9 Ogata K 987) Discrete time control systems Prentice Hall, Upper Saddle River Puu T, Gardini L, Sushko I 2004) A Hicksian multiplier accelerator model with floor determined by capital stock J Econ Behav Org 563): Rosser JB 2000) From catastrophe to chaos: a general theory of economic discontinuities Academic Publishers, Boston Samuelson P 939) Interactions between the multiplier analysis and the principle of acceleration Rev Econ Stat 22):75 78 Westerhoff FH 2006) Samuelson s multiplier accelerator model revisited Appl Econ Lett 56:86 92 Wincoop E 996) A multi-country real business cycle model Scand J Econ 23:233 25