Bifurcation Routes and Economic Stability Miloslav S. Vosvrda

Transcription

1 Bifucation Routes and Economic Stability Miloslav S. Vosvda Institute of Infomation Theoy and Automation, Academy of Sciences of the Czech Republic Institute of Economic Studies, Faculty of Social Sciences, Chales Univesity Pague Abstact: An economic model of cycles focused on the foeign capital investment phenomenon will be biefly intoduced. We conside a system of the fist ode nonlinea diffeential equations whee a feedback function is contolled by a capital/output atio paamete. A value of the potential goss domestic poduct YP will be consideed as a unit of the system. An influence of the paametes changes on a macoeconomic stability is analyzed. Convegence of this system eithe to stable state o limit cycle o chaotic state is demonstated. Keywods: macoeconomic stability, foeign investment phenomenon, system of the nonlinea diffeential equations, capital/output atio paamete feedback, the Hopf bifucation JEL: E, E. Intoduction Thee ae two basic tools to analyze fundamental issues in dynamical macoeconomics. One of them is a model of optimal gowth descibing savings behavio. The second one is the Solow-Swan model with a constant aggegate popensity to save out of income. A steady state of the dynamical economic system coesponds to a gowth path satisfying a stationay solution of a popely defined diffeential system and thus exhibiting cetain conditions of constant gowth ate in a single-secto model. One of the possible mechanisms fo egulating gowth path is a savings ate. A egulation though the savings ate is made possible by distinguishing two types of income, two social classes o by intoducing money and financial assets (see Henin (986)). The standad neoclassical gowth theoy, Kaldo (956), Pasinetti (96), Samuelson and Modigliani (966) wee investigating the question to what extent diffeent saving behavio of the two income goups (labo and capital) might influence the gowth path. The case that each agent is able to save by accumulating capital but not to boow fom the othe is solved by Bewley (998). Woodfod (99) went beyond Bewley s esults both showing that equilibium cycles ae possible and exhibiting conditions unde which equilibium dynamics ae chaotic. The aggegate savings function needs no longe be concave, so that multiple and unstable steady states can occu. The ole of diffeential simple savings behavio and distibution effects fo stability of stationay states was investigated in Böhm and Kaas (). They demonstated that the economy exhibits unstable steady states and fluctuations if the income distibution vaies sufficiently and if shaeholdes save moe than wokes. A cental question of this pape is whethe using and/o un-using of foeign investments change the qualitative popeties of a gowth path of the dynamical economic system. The analysis will be povided in the two steps: Fist, stability the economic system without a foeign financing will be analyzed by the behavio of the eigenvalues of the Jacobian matix; Second, the economic system with a foeign financing will be analyzed by the Hopf bifucation. This eseach was suppoted by the Gant Agency of the Czech Republic unde Gants 4/99/6, 4//49, 4//4.

2 . An economic system without a foeign financing The economic system without a foeign investment will be consideed as follows St!() = ayt () + pst () ( YPY ()) t (a) Yt! () = v St () Haodian (b) whee S - savings of households, Y - Goss Domestic Poduct, YP (=) - the potential GDP, a - the vaiation of the maginal popensity to savings, p - the atio of capitalized pofit, v - the capital/output atio. If Y(t) < YP epesents a emakable incease of pofit, the activities ae not constained. High pofits ae deived fom these sectos thei makets ae not yet satuated. A quality of a capitalization of the pofits, e-injecting its into the financing cicuit, is expessed by a non-lineaity in the equation (). If Y(t) > YP this epesents a possibility of the inflation appeaing. A lack of new investment oppotunities modifies using of savings. The saving deviation is acceleated by intoduced non-lineaity between the Y(t) and YP. The fundamental constaint in the development pocess is the capital financed hee by the savings only (Haodian)..a Stationay Equilibium This system possesses a unique equilibium as follows * T T = fx (, μ ) = ( S Y ) = ( ) whee T means a vecto tansposition. The following question is: Does this system possess a peiodic solution? Fo solving we use Bendixson's citeion fo the nonexistence of a peiodic solution (Dazin (994), p. 8). An application of this citeion we get the following poposition: We suppose that Y(t)!, and p!. Thee is no closed obit in the analyzed domain since ( ps( t) ( Y ( t)) + ay( t) ( v S( t)) + = p( Y ( t)) St () Yt ().b Necessay and sufficient stability conditions The Jacobian matix of ou system equation is p( YPY ) apsy p a J = = (a) / v / v S = with the chaacteistic equation χ ( x) = x w x+ u (b) whee w= t( J), u = J. We solve the chaacteistic equation λ w λ+ u = and we get the following eigenvalues, 4 p p a λ = ± +. v Subsequently, because the deteminant J has, in the equilibium point, negative value equal to a/v and two eal eigenvalues having the opposite sign. Thus the stationay point is an unstable saddle point. Example Let us conside the following values of paametes a=.45, p=.5, v= and fo initials of savings of households S()=.45, of output Y()=., and of the potential output YP= we obtain the following solution w=.5, u= -.5. The equation Y =

3 λ w λ+ u = has two eal oots with diffeent signs = -.5, and =.45. This case is demonstated in the following figues. a=.45 p=.5 v= YP=.967 Savings of households Dynamics.555 Output Dynamics y y tt tt a) b) Savings of households to Output Output to Savings of households.555 y y 5. 5 y y.967 c) d) Figue. An economic system with a foeign financing Now we will analyze the economic system whee Y (t) is detemined by the level of the investment financed by both the savings and the foeign capital inflows. Theefoe the continuous- system is consideed by the following way (, ), n m x! = f x μ x R, μ R () Specifications on ou investigated system appeaing the following fom of equations St!() = ayt () + pst () ( YPY ()) t Yt! () = v ( St () + Ft ()) (4) Ft! () = mst () Yt () whee F the foeign capital inflow, m the capital inflow/savings atio, the debt efund/output av atio, and we assume that > and a >..a Stationay Equilibium This system possesses a unique equilibium as follows * = fx (, μ ) = ( S Y F ) T whee am a m a m Y = + YP S = + YP F = + YP p m p m p Fo paametes fom ou example a=.5, p=.7, v=.4, m=., =.4, and fo initials of savings of households S()=.45, of output Y()=, of a foeign investment F()=.5, and of the potential output YP= we obtain the following solution ( ) T

4 .b Necessay and sufficient stability conditions (Routh-Huwitz (see Gandolfo 997, pp. )) The Jacobian matix of ou system equation is am p p( YPY ) apsy a m J = / v / v = / v / v m m am Y= + YP (5) p am S= + YP m p am F= + YP m p with the chaacteistic equation χλ ( ) = λ + g λ + h λ+ q (6) whee g =t( J) q =J ( apsy) h = am = m p v = ( a + ) v m p = a + (+ ) v m The coefficient h epesents the pincipal minos of the Jacobian J. We solve the chaacteistic equation and we get the following eigenvalues. The equation λ + g λ + h λ+ q= has one eal and two complex conjugate oots if the disciminant g g h q h g = + + is positive (see Loenz pp. 86). An estimating of a sign of the disciminant is, a little bit cumbesome, intoduced in the appendix A. It is shown that the sign of this disciminant is, fo the system (5), positive. In ou case with the following values of the paametes a=.5, p=.7, v=.4, m=., =.4 ae the g=.5, q=.975, h=., and the value of the is equal.5. By computing we eally get one eal and two complex conjugate oots λ =.6, λ = i, λ =.46.9i. The Routh-Huwitz theoem states that necessay and sufficient conditions fo all oots of the equation λ + g λ + h λ+ q= to have negative eal pats ae given by the following inequalities (see Gandolfo 997, pp. ): The leading pincipal minos of the matix J g q g q M =g M = M = h h g q must be positive, i.e., that a m M = > a p M =. am + ( a ) > v v av M = ( a m+ p) am + p( a ) > Fom these inequalities follow that <<a, <m/<p<. In ou case we again get M =.5, M =., and M =.. Because these values of paametes ae positive the necessay and sufficient stability conditions ae fulfilled. 4

5 In the following examples a behavio of the stable system unde changes paametes p, a,, m espectively is demonstated. Example Changing of the paamete p Let us conside the following values of paametes a=.5, v=.4, p=., m=p-., =a-., YP=, and we solve the system of diffeential equations with the following initials Y () = ( ) T. The solution in the gaphical fom is shown in the following figues output to savings of households.5 foeign capital inflow to output y 4 y y y savings of households to output output to a foeign capital inflow y y y y Figue 4 In the D view we can see inteactive behaviou of these thee dimensions.5 Figue 5 The paamete p will be changed fom the value. until.85 and next fom.5 till.885 and the othe paametes will be unchanged. Fo low values of this paamete p is this system in unstable state. Fo inceasing values of the paamete p this system conveges in the stable state. Theefoe this system unde changing of the paamete p has so-called supecitical Hopf bifucation. It means that when the paamete p is inceasing fom p =. till p=.885, the single fixed point changes its 5

6 stability which is monitoed in figues 6, 7, 8. We will now change piecemeal values of the paametes p in the following way p=., p=.5, p=.6, p=.85, p4=.5, p5=.65, p6=.76, p7= Z Z4 Tajectoies of Savings Z4 p=. p=.5.5 Tajectoies of foeign capital inflows Z Z Z4 p=. p= Z Z4 6 4 Tajectoies of Output Z4 p=. p=.5 Figue 6 6

7 Figue 7 Figue 8 Example Changing of the paamete a Let us conside the following values of paametes a=.5, v=.4, p=., m=p-., =a-., YP=, and we solve the system of diffeential equations with the following 7

8 initials Y () = ( ) T. The paamete a will be changed fom the value.5 until.95 and the othe paametes will be unchanged. Fo low a values is this system in an unstable state. Fo inceasing a values is this system in a stable state. Theefoe this system unde changing of the paamete a has so-called supecitical Hopf bifucation. It means that when the paamete a is inceasing fom a =.5 till a =.95, the single fixed point changes its stability which is monitoed in figues 9,,. We will now change piecemeal values of the paametes a in the following way a=.5, a=.75, a=., a=.45, a4=.55, a5=.675, a6=.8, a7= Tajectoies of Savings Z Z Z4 a=.5 a= Z Z Z Z Z4 a=.5 a=.55 Tajectoies of foieng capital inflows Tajectoies of Output Z4 a=.5 a=.55 Figue 9 8

9 Figue In figue it is possible to see the tajectoies fo the values of the paamete a fo In figue ae shown the situations fo the values of the paamete a fo Figue 9

10 Example 4 Changing of the paamete Let us conside the following values of paametes a=.5, v=.4, p=., m=p-., =a-., YP=, and we solve the system of diffeential equation with the following initials Y () = ( ) T. The paamete will be changed fom the value.7 until.4 and next fom.455 till.79 and the othe paametes will be unchanged. Fo values less than.5 is this system in a stable state by the stability condition fo <a. Fo values lage than.5 is this system in an unstable state. Theefoe this system unde changing of the paamete has the subcitical Hopf bifucation again. It means that when the paamete is inceasing fom <.5 to >.5, the single fixed point changes its stability which is monitoed in figue. We will now change piecemeal values of the paametes in the following way =.7, =.9, =., =.4, 4=.455, 5=.567, 6=.679, 7= Tajectoies of Savings Z Z Z4 =.7 = Tajectoies of foeign capital inflows Z Z Z4 =.7 = Z Z4 Tajectoies of Output Z4 =.7 =.455 Figue

11 Figue In figue it is possible to see the tajectoies fo the values of the paamete fo In figue 4 is shown the situation fo the values of the paamete fo Figue 4

12 Example 5 Changing of the paamete m Let us conside the following values of paametes a=.5, v=.4, p=., =a-.5, m=., YP=, and we solve the system of diffeential equation with the following initials Y () = ( ) T. The paamete m will be changed fom the value.7 until.8 and next fom.57 till.88 and the othe paametes will be unchanged. Fo values m less than.57 is this system in a stable state. Fo values m lage than.57 this system is in an unstable state. Theefoe this system unde changing of the paamete m has the subcitical Hopf bifucation again. It means that when the paamete m is inceasing fom m <.57 to m >.57, the single fixed point changes its stability which is monitoed in figue 5. We will now change piecemeal values of the paametes m in the following way m=.7, m=., m=.57, m=.8, m4=.57, m5=.6, m6=.757, m7= Tajectoies of Savings Z Z Z4 m=.7 m= Tajectoies of foeign capital inflows Z Z Z4 m=.7 m= Z Z4 4 Tajectoies of Output Z4 m=.7 m=.57 Figue 5

13 Figue 6 In figue 6 it is possible to see the tajectoies fo the values of the paamete m fo In figue 7 is shown the situation fo the values of the paamete m fo Figue 7 The paametes v the capital/output atio is consideed as constant in the analyzed peiod.

14 4. Conclusions a) An inceasing of the capitalization of the pofits demonstates well-known esults in economics that the capitalization of pofits causes the stabilization of the system. Ou system can show a elation between a speed of convegence to stabilization and a level of capitalized pofits. b) The paamete a has vey simila behavio as the paamete p. Known esult is the savings means stability. c) If the debt/output atio is less than the vaiation of the maginal popensity to savings then the system is in a stable state. If the debt/output atio is geate than the vaiation of the maginal popensity to savings then this system conveges to limit cycle position. d) If the capital inflow/savings atio is less than double the atio of capitalized pofit then the system is in a stable state. If the capital inflow/savings atio is geate than double the atio of capitalized pofit then the system conveges to limit cycle. Fo highe diffeence between the capital inflow/savings atio and double the atio of capitalized pofit this system conveges to a chaotic state. Acknowledgement I am much indebted to Jan Kodea and Kael Sladky fo stimulating ideas to Economic Dynamics. I would like to thank my colleagues at the Institute of Theoy Infomation and Automation fo poviding a stimulating envionment to pusue the ideas eflected in this pape. Refeences Bewley, T.F.(986): Dynamic implications of the fom of the budget constaint, in Models of Economic Dynamics, ed. H.F.Sonnenschein, N.Y., Spinge Böhm V, Kaas L.(): Diffeential savings, facto shaes, and endogenous gowth cycles, Jounal of Economic Dynamics&Contol, 4, pp Bouli S.(999): Feedback Loop in Extended Van de Pol s equation Applied to an Economic Model of Cycles, Intenational Jounal of Bifucation and Chaos, vol. 9, no.4, pp Chiaella C., Flaschel P.(): The Dynamics of Keynesian Monetay Gowth, Cambidge, Dazin P.G.(994): Nonlinea System, Cambidge, Gandolfo G.(997): Economic Dynamics, Spinge, Henin P.Y.(986): Macodynamics: Fluctuation and gowth, Routledge&Kegan Paul, Kaldo N. (956): Altenative theoies of distibution. Review of Economic Studies, pp. 8- Loenz H.W.(997): Nonlinea Dynamical Economic and Chaotic Motion, Spinge, Pasinetti L.L.(96): Rate of pofit and income distibution in elation to the ate of economic gowth. Review of Economic Studies, 9, pp Puu T.(): Attactos, Bifucations, and Chaos, Spinge, Samuelson and Mondigliani(966): The Pasinetti paadox in neoclassical and moe geneal models. Review of Economic Studies,, pp. 69- Woodfod M.(989): Impefect financial intemediation and complex dynamics, in Economic Complexity: Chaos, sunspots, Bubles, and Nonlineaity, ed. by Banett et al.(eds.), Cambidge Univesity Pess, Cambidge. 4

15 Appendix A We estimate the sign of the disciminant value whee am m p p g = q = ( a+ ) h= a+ ( + ) v m v m By algebaic opeations we get the following expessions g q + g h 7 6 h g 9 expand expand, h 7 h 79 g 7 gq 8 g + h + 4 q 6 qgh + 6 g h 7 g h + 8 h g4 79 g6 Now we estimate this value fo usual values of the paametes a,m,v,,p. The paametes a,m,,p usually belong to the inteval (,) and the paamete v is usually geate than 5. Fo these values of paametes we get 79 g expand, a 79 a m 7 gq expand, g 7 a m v 7 a m p v 79 g6 expand, a 79 a6 m g h expand, a 8 a m 8 a m v 8 a m p v 4 q m expand, m a v m + a v p p + v v v v 6 qgh expand, a a m m a + v v a m + p v a m m + p + a v v p a + p v 8 g h 7 h expand, a 8 h g4 expand, a expand, a 8 a4 m 54 a m v 7 a m m p a v 8 v 7 a m a p v p 7 v 9 a + v 9 a p + v m 9 a 4 + v 9 a p 4 + v m 9 a p p 4 p 8 p + v m + 7 v + 9 v + m 9 v m + 7 v m 8 a5 m a4 m a4 m + p v v 7 a The behavio of the disciminant value fo typical values of the paametes is depicted in figue A. Thus the value is positive. a :=.. m :=. p :=. v := 6.5 :=. Behavio of the disciminat value...5 Figue A 5