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SOME NOTE ABOUT THE EXISTENCE OF CYCLES AND CHAOTIC SOLUTIONS IN ECONOMIC MODELS

SOME NOTE ABOUT THE EXISTENCE OF CYCLES AND CHAOTIC SOLUTIONS IN ECONOMIC MODELS. by Beatrice Venturi Department of Economics University of Cagliari Italy. EXISTENCE OF CYCLE AND CAOTIC SOLUTIONS IN ECONOMIC-FINAN C IAL MODELS. We analyze the global structure of

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SOME NOTE ABOUT THE EXISTENCE OF CYCLES AND CHAOTIC SOLUTIONS IN ECONOMIC MODELS

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  1. SOME NOTE ABOUT THE EXISTENCE OF CYCLES AND CHAOTIC SOLUTIONS IN ECONOMIC MODELS by Beatrice Venturi Department of Economics University of Cagliari Italy MDEF 2008 BEATRICE VENTURI

  2. EXISTENCE OF CYCLE AND CAOTIC SOLUTIONS IN ECONOMIC-FINANCIAL MODELS We analyze the global structure of a tree-dimensional abstract continuous time stationary economic model that includes some determinates parameters. MDEF 2008 BEATRICE VENTURI

  3. THE MODEL We shall consider a generic non-linear first order system with some structural parameters: MDEF 2008 BEATRICE VENTURI

  4. Where f, g and h are complicate non-linear functions of class C2 (twice continuously differentiable) in all their arguments. The parameters: are real and positive. MDEF 2008 BEATRICE VENTURI

  5. A stationary (equilibrium) point of our system is any solution of : Assuming the existence of such a solution at some point MDEF 2008 BEATRICE VENTURI

  6. THE JACOBIAN MATRIX • The local dynamical properties of system from (2.1) to (2.3), at a hyperbolic equilibrium point P*, can be described in terms of the Jacobian matrix , for brevity. In fact, the nature of the eigenvalues of J*, plays a key role. MDEF 2008 BEATRICE VENTURI

  7. We consider the system as a one-parameter family of differential equations dependent of the parameter .We fixe the other parameters . We assume that in our model exists a parameters set where the Jacobian has two eigenvalues complex conjugate : MDEF 2008 BEATRICE VENTURI

  8. EMERGENCE OF STABLE CYCLES First we use rigorous arguments to show as an equilibrium of the model could be destabilized into a stable cycle in the dynamic of R3 under the following two alternative assumptions: • A steady state has three stable roots. b) A closed orbit has a two dimensional manifolds in which it is asymptotically stable . MDEF 2008 BEATRICE VENTURI

  9. APPLICATIONS • Next we apply these results to a general non-linear fixed-price disequilibrium IS-LM model as formulated by Neri U. and Venturi B. (2007). MDEF 2008 BEATRICE VENTURI

  10. APPLICATIONS • Neri U. and Venturi B. (2007) discuss the effect of a change of the adjustment parameter in the money market, via the Hopf bifurcation’s approach in a three dimensional fixed-price disequilibrium IS-LM model. MDEF 2008 BEATRICE VENTURI

  11. THE ECONOMIC MODEL (1) • S = savings, • T= tax collections, • G=government expenditure, • B = interests payment on perpetuities, • L = liquidity preference function • m = is the real money supply. • I = investment, • r =interest rate, • y = output (income) • w =wealth • α >0 and µ>0 are the adjustment parameters in their respective markets (2) (3) Specifically, our “generalized model” is a non-linear systemin the independent state variablesr, y and m. MDEF 2008 BEATRICE VENTURI

  12. THE ECONOMIC MODEL (1) describes the traditional disequilibrium of dynamic adjustment in the product market; (2) describes the corresponding disequilibrium in the money market; (3) represents the governmental budget constraint. MDEF 2008 BEATRICE VENTURI

  13. THE ECONOMIC MODEL • Next, we define the disposable incomey and the wealthw as follows: MDEF 2008 BEATRICE VENTURI

  14. THE ECONOMIC MODEL • We assume that the functions: I, S, T are of class C2 • Recall that a stationary (equilibrium) point of our system is any solution of. Assuming the existence of such a solution at some point P*(y*,r*,m*), we want to analyze its local properties (e.g. stability, etc.) around P*. MDEF 2008 BEATRICE VENTURI

  15. THE ECONOMIC MODEL We shall rewrite our system in the equivalent form: with h: MDEF 2008 BEATRICE VENTURI

  16. THE ECONOMIC MODEL Since G and B are fixed a choice of the policies, implies: Remark1: Unlike Schinasi G.J. ,1982 , we allow the functions L and S (liquidity and savings) to depend on wealth w. because all deficit must be financed either by creation of money or by creation of new debt. MDEF 2008 BEATRICE VENTURI

  17. THE ECONOMIC MODEL • The monotonicity, of the restrictions: , , , is assumed and these assumptions imply the (economic) conditions: . MDEF 2008 BEATRICE VENTURI

  18. Dynamical analysis • Theorem 1. A hyperbolic stationarypoint of the system (1), (2), (3) is locally asymptotically stable if the following assumptionshold at: with N given by (*) at P* : N = the first integer such that N a2> a3 , at P*, MDEF 2008 BEATRICE VENTURI

  19. Dynamical analysis i) the marginal propensity to invest out of income is greater than unity ii) iii) MDEF 2008 BEATRICE VENTURI

  20. Dynamical analysis • Corollary 1. Let the hypotheses of Theorem 1 hold at a hyperbolic stationary point of the our system. Then the steady state for each α >0 and all is locally asymptotically stable. MDEF 2008 BEATRICE VENTURI

  21. Dynamical analysis • Corollary 2. Assume Corollary 1 and let J*=J(P*) as before. Then, there exists a valueµ>0for which J* has a pair of purely imaginaryeigenvalues. MDEF 2008 BEATRICE VENTURI

  22. is now replaced by : Theorem 2.Assume the hypotheses of Theorem1 except that Then, there exists a continuous function µ(δ) with µ(0)=μ’ and for all δsmall enough , there exists a continuous family of non-constant positive periodic solutions [r*(t ,), y*(t,), m*(t,)] for the dynamical system (1), (2), (3) which collapse to the stationary pointP * . MDEF 2008 BEATRICE VENTURI

  23. THE ECONOMIC MODEL REMARK For a three-dimensional non linear dynamical model the general version of the Hopf bifurcation theorem, ensures the existence of a small amplitude periodic solutions bifurcating from the steady state only in the center manifold(a two-dimensional subspace of 3). MDEF 2008 BEATRICE VENTURI

  24. Dynamical analysis • From an economic point of view, sub-critical or super-critical orbits are both reasonable. • Since the third real root of the Jacobian matrix is negative the existence of a super-critical Hopfbifurcations becomes very interesting in the analysis of macroeconomic fluctuations for a IS-LM model • A stable economy, by the increase or decrease of its control parameters, could be destabilized into a stable cycle in the dynamic of R3 MDEF 2008 BEATRICE VENTURI

  25. CONCLUSIONS • The sub-criticalHopf bifurcations may correspond to the Keynesian corridor (Leijonhufvud, 1973): the economy has stabilityinside the corridorwhile it will loose the stability outside the corridor. • In such a case the dynamics are either converging to an equilibrium point or the trajectories go somewhere else, and it is also possible that another attracting set exists, but often the alternative is diverging trajectories. MDEF 2008 BEATRICE VENTURI

  26. CONCLUSIONS • We have seen that fluctuations derive from the mechanisms through which money markets reflect and respond to the developments in the real economy. • Our analysis provides an example of the classical thesis concerning endogenous explanations to the existence of fluctuations in some real world economic variables MDEF 2008 BEATRICE VENTURI

  27. The general version of the Hopf bifurcation theorem For a three-dimensional non linear dynamical model the general version of the Hopf bifurcation theorem, ensures the existence ofa small amplitude periodic solutions bifurcating from the steady state onlyin the center manifold(a two-dimensional subspace of 3). MDEF 2008 BEATRICE VENTURI

  28. Shil’nikov showed that if the real eigenvalues has large magnitude than the real part of the complex eigenvalues of the Jacobian of system, then there are horseshoes present in return maps near the homoclinicorbit of the model. Shil’nikov Theorem MDEF 2008 BEATRICE VENTURI

  29. The orbits, in the dynamics of the center manifold, can generally be either attracting or repelling.In the case of an attracting orbit (the so- called sub-critical case) trajectory on the center manifold are locally attracted by this orbit, which becomes a limit set.. MDEF 2008 BEATRICE VENTURI

  30. In this situation the stationary pointis an unstable solutionmeanness from an economic point of view(unless the initial conditions happen to coincide with the stationary value).Conversely, if the cycle is unstable(the so- called super-critical case) the steady state is attracting. MDEF 2008 BEATRICE VENTURI

  31. The study of stability of emerging orbitson the centre manifold can be performed by calculating the sign up of-up-third order derivative of the nonlinear part of the system, when written in normal form. This case is particularly relevant from an economic point of view only if the initial conditions imply that the economy fluctuates right from the beginning. MDEF 2008 BEATRICE VENTURI

  32. In fact, the Hopf bifurcation theorem proves the existence of closed orbits but it gives no information on their number and their stability. Using the non linear parts of an equation system, a stability coefficient (as formulate for example by Guckenheimer J.- Holmes P.,1983) may be calculated in order to determine the stability properties of the closed orbits (see Foley , 1989, Feichtinger ,1992, Mattana - Venturi,1999, Anedda C. -Venturi B. ,2003). MDEF 2008 BEATRICE VENTURI

  33. Mattana P.- Venturi B. , 1999, analyzed a simplified three-dimensional version of Lucas’s model, a two sector endogenous growth model with externality. Considering the externality as a bifurcation parameter, they proved, the existence of small amplitude periodic solutions, Hopf bifurcating from the steady state in the center manifold. Venturi B., 2002, have elaborated a numerical simulation of this model. . MDEF 2008 BEATRICE VENTURI

  34. In Fig.1 is plotted the dynamics of one orbit Hopf bifurcating from the steady state of the reduced version of the Lucas model (see Venturi B., 2002) The orbits is super-critical MDEF 2008 BEATRICE VENTURI

  35. Figure 1 MDEF 2008 BEATRICE VENTURI

  36. THE ECONOMIC MODELS • We review a generalized two sector models of endogenous growth, with externalities, as formulated by Mulligan B.- Sala-I-Martin X.,1993. • We show that in this class of economic models, considering the externality as bifurcation parameter, the conditions of existence of periodic orbitsand chaotic solutionscome true. MDEF 2008 BEATRICE VENTURI

  37. The Mulligan B. - Sala-I-Martin X. model deal with the maximization of a standard utility function: • where • c= is per-capita consumption • = is a positive discount factor • = is the inverse of the intertemporal elasticity of substitution. MDEF 2008 BEATRICE VENTURI

  38. The constraints to the growth process are represented by the following equations k=is the physical capital, h=is thehuman capital MDEF 2008 BEATRICE VENTURI

  39. k, ,hare the private share of physicaland the humancapitalin the output sector k , hare the private share of human capital in the education sector MDEF 2008 BEATRICE VENTURI

  40. uandv= are the fraction of aggregate human and physical capital used in the final output sector at instant t (1- u) and (1- v) are the fractions used in the education sector, Aand B are the level of the technology in each sector, MDEF 2008 BEATRICE VENTURI

  41. is a positive externality, parameter in the production of physicalcapital is a positive externality parameter in the production of human capital All the parameters: MDEF 2008 BEATRICE VENTURI

  42. live inside the following set: • =(0, 1)(0, 1) (0, 1) (0, 1) (0, 1) (0,1) (01) (0 1) (0 1)  4. MDEF 2008 BEATRICE VENTURI

  43. The representative agent’s problem is solved by defining the current value Hamiltonian: i, i = 1, 2 =Lagrange multipliers (co-state variables).  = is a depreciation factor MDEF 2008 BEATRICE VENTURI

  44. Plus the usual two transversality conditions: and obtaining the first-order necessary conditions for an interior solutions. MDEF 2008 BEATRICE VENTURI

  45. The general model just presented collapses to Uzawa-Lucas’ when depreciation is neglected and the following restrictions are imposed: MDEF 2008 BEATRICE VENTURI

  46. According to the strategy used byMulligan B.C. - Sala-I-Martin X.,1993,we express the two multipliers 1 and 2 in terms of their corresponding control variables c and u and obtain an autonomous system of four differential equations in the four variables k, h, c, u. A solution of this autonomous system is called a Balanced Growth Path (BGP) if it entails a set of functions of time solving the optimal control problem presented such that k, h and c grow at a constant rate and u is a constant. MDEF 2008 BEATRICE VENTURI

  47. We choose a standard combination of the original variables that is stationary on BGP. We get a first order autonomous system of three differential equations MDEF 2008 BEATRICE VENTURI

  48. B.G.P that does not include positive externalities admits only the saddle-path stable solutions (see Mulligan B.C. - Sala-I-Martin X.,1993). We get a first order autonomous system of three differential equations MDEF 2008 BEATRICE VENTURI

  49. Where: MDEF 2008 BEATRICE VENTURI

  50. We put our model into normal form(see Guckenheim - Holmes 1983, pp. 573). The system becomes: MDEF 2008 BEATRICE VENTURI

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