1. THE CHAOTIC MODEL OF INFLATION AND UNEMPLOYMENT: EU

2. Vesna D. Jablanovic Faculty of Agriculture, University of Belgrade Nemanjina 6, 11081 Belgrade Serbia vesnajab@ptt.yu

3. Abstract The basic aims of this paper are : The basic aims of this paper are : firstly, to set up the chaotic model of inflation and unemployment in the European Union countries during the period 1970-2001 in the formal framework of the Rossler model (1976); firstly, to set up the chaotic model of inflation and unemployment in the European Union countries during the period 1970-2001 in the formal framework of the Rossler model (1976); secondly, to estimate private consumption deflator regression equation and unemployment regression equation ; secondly, to estimate private consumption deflator regression equation and unemployment regression equation ; and thirdly, to analyze the possibility of effective governmental economic policy. and thirdly, to analyze the possibility of effective governmental economic policy.

4. 1. Introduction Chaos theory attempts to reveal structure in aperiodic, unpredictable dynamic systems. Chaos theory shows the difficulty of predicting their long-range behavior. In this sense, it is important to construct deterministic, nonlinear economic dynamic models that elucidate irregular, unpredictable economic behavior. Chaos theory attempts to reveal structure in aperiodic, unpredictable dynamic systems. Chaos theory shows the difficulty of predicting their long-range behavior. In this sense, it is important to construct deterministic, nonlinear economic dynamic models that elucidate irregular, unpredictable economic behavior. Unlike a linear system, in which a small change in one variable produces a small and easily quantifiable systematic change, a nonlinear system exhibits a sensitive dependence on initial conditions: small or virtually unmeasurable differences in initial conditions can lead to wildly differing outcomes. This sensitive dependence is sometimes referred to as the” butterfly effect“, the assertion that the beating of a butterfly's wings in Tokyo can eventually change the weather in New York City. Nonlinear dynamics has shown that even systems governed by simple equations can exhibit complex behavior. Applied chaos theory is used to prove that erratic and chaotic fluctuations can indeed arise in completely deterministic models. Unlike a linear system, in which a small change in one variable produces a small and easily quantifiable systematic change, a nonlinear system exhibits a sensitive dependence on initial conditions: small or virtually unmeasurable differences in initial conditions can lead to wildly differing outcomes. This sensitive dependence is sometimes referred to as the” butterfly effect“, the assertion that the beating of a butterfly's wings in Tokyo can eventually change the weather in New York City. Nonlinear dynamics has shown that even systems governed by simple equations can exhibit complex behavior. Applied chaos theory is used to prove that erratic and chaotic fluctuations can indeed arise in completely deterministic models.

5. Chaos theory started with Lorenz's (1963) discovery of complex dynamics arising from three nonlinear differential equations leading to turbulence in the weather system. Li and Yorke (1975) discovered that the simple logistic curve can exibit very complex behaviour. Further, May (1976) described chaos in population biology. Chaos theory has been applied in economics by Benhabib and Day (1981,1982), Day (1982, 1983,1997, ), Grandmont (1985), Goodwin (1990), Gaunersdorfer, A. and Hommes, C.H. (2005), Hommes (2000, 2005, 2006 ), Medio (1993,1996), Medio, A. and Lines, M ( 2004), Lorenz (1993), Shone, R.(1999) among many others.Gaunersdorfer, A.

6. The Phillips curve Two closely related indicators of economic performance are inflation and unemployment. How are these two measures of economic performance related to each other? The Phillips curve describes the negative relationship between inflation and unemployment. Two closely related indicators of economic performance are inflation and unemployment. How are these two measures of economic performance related to each other? The Phillips curve describes the negative relationship between inflation and unemployment. The tradeoff between inflation and unemployment described by the Phillips curve holds only in the short run ( Phillips 1958; Samuelson and Solow 1960 ). The tradeoff between inflation and unemployment described by the Phillips curve holds only in the short run ( Phillips 1958; Samuelson and Solow 1960 ). In the long run expected inflation adjusts to changes in actual inflation and the short-run Phillips curve shifts. As the result, the long-run Phillips curve is vertical at the natural rate of unemployment (Phelps 1968; Friedman 1968 ). In the long run expected inflation adjusts to changes in actual inflation and the short-run Phillips curve shifts. As the result, the long-run Phillips curve is vertical at the natural rate of unemployment (Phelps 1968; Friedman 1968 ).

7. The basic aims The basic aims of this paper are: The basic aims of this paper are: firstly, to set up an endogenous, chaotic model of the private consumption deflator and the unemployment rate in the European Union during the period 1970-2001 in the formal framework of the Rössler (1976) model ; firstly, to set up an endogenous, chaotic model of the private consumption deflator and the unemployment rate in the European Union during the period 1970-2001 in the formal framework of the Rössler (1976) model ; secondly, to estimate the private consumption deflator regression equation and unemployment regression equation, secondly, to estimate the private consumption deflator regression equation and unemployment regression equation, and thirdly, to suggests that lacking prior knowledge of the system's parameters and initial values, it is difficult to see how one could determine them from a disturbed time series. In this sense, it would be impossibe to predict future movement of inflation and unemployment from its past behaviour. and thirdly, to suggests that lacking prior knowledge of the system's parameters and initial values, it is difficult to see how one could determine them from a disturbed time series. In this sense, it would be impossibe to predict future movement of inflation and unemployment from its past behaviour.

8. 2. The model of inflation and unemployment

9. The short-run Phillips curve The essence of the model can be simply formulated in differential form. The rate of change of private consumption deflator, p ', is a proportion, , of the private consumption deflator, p. Further, when unemployment rate,u, grows then private consumption deflator, p, declines, at the rate . The short-run Phillips curve depicts the negative relationship between the private consumption deflator and unemployment rate. Then : The essence of the model can be simply formulated in differential form. The rate of change of private consumption deflator, p ', is a proportion, , of the private consumption deflator, p. Further, when unemployment rate,u, grows then private consumption deflator, p, declines, at the rate . The short-run Phillips curve depicts the negative relationship between the private consumption deflator and unemployment rate. Then : p ' =  +  p -  u (1) p ' =  +  p -  u (1)

10. The long-run Phillips curve Further, it is assumed that the rate of change of unemployment rate, u', is a proportion, , of private consumption defltor, p. According to the natural-rate theory, the long-run Phillips curve is vertical at the natural unemployment rate; as long as unemployment is below the natural rate, inflation will tend to increase and unemployment moves back to its natural rate. When unemployment rate, u, grows then the rate of unemployment rate change, u', increases at the rate . Thus Further, it is assumed that the rate of change of unemployment rate, u', is a proportion, , of private consumption defltor, p. According to the natural-rate theory, the long-run Phillips curve is vertical at the natural unemployment rate; as long as unemployment is below the natural rate, inflation will tend to increase and unemployment moves back to its natural rate. When unemployment rate, u, grows then the rate of unemployment rate change, u', increases at the rate . Thus u' = +  p +  u (2) u' = +  p +  u (2)

11. The system (1)-(2) is linear because it contains only linear differential equations which, as usual, means that p ' and u' are not raised to any power than one. The system is autnomous because the coefficients, , , , , , and the terms, , are constants. The complete solutions are the sum of the homogenous and particular solutions: (i) Real and distinct roots:

12. (ii) Real and equal roots: where the roots r1 and r2 are the solution to the characteristics equation; (iii) Complex roots: p (t) = e ht  A1 cos (v t ) + A2 sin ( v t )  +p*

13. 3. Empirical evidence We use and normalise data on the private consumption deflator, p, and unemployment rate, u, during the period 1970-2001 (OECD 1999) to estimate our system (1)-(2) (see Fig. 1.). We use and normalise data on the private consumption deflator, p, and unemployment rate, u, during the period 1970-2001 (OECD 1999) to estimate our system (1)-(2) (see Fig. 1.).

14. Fig.1. The private consumption deflator, p, and unemployment rate, u, in the European Union during the period 1970-2001.

15. It is important to choose the regression lines of the private consumption deflator and the unemployment rate that best fit the data points ( in the sense that the sum of the squared vertical deviations of each observed point form the line is a minimum). We obtain p(t)^ =e -0.117874 t[ -0.128414cos(0.119351 t)+ 2.086188 sin (0.119351t)] + 0.17066 R= 0.91784 Variance explained 84.243% (3) A D E Estimate-.128422.08618.170661 Std.Err..08481.18436.028924 t(29)-1.5141711.315705.900234 p-level.14081.00000.000002 A low p- values 0.00000 is smaller than 0.005 and implies that we are »safe« in rejecting the null hypothesis that the coefficients D and E are zero and concluding that they are significantly different from zero. A high p- values of the coefficient A is greater than 0.005 and implies that we are not »safe« in rejecting the null hypothesis that the coefficients A are zero and concluding that A are statistically insignificant.

16. u(t)^= e-0.117874 t[-0.671958cos(0.119351 t ) - 0.929128 sin (0.119351 t)]+0.873093 R=0.95684 Variance explained 91.554% (4) A D E Estimate-.6720-.92913.87309 Std.Err..0641.13931.02186 t(29)-10.4852-6.6692939.94574 p-level.0000.00000.00000 A low p- values 0.00000 is smaller than 0.005 and implies that we are »safe« in rejecting the null hypothesis that the coefficients A, D and E are zero and concluding that they are significantly different from zero.

17. Fig. 2. The regression line of the private consumption deflator

18. Fig. 3.The regression line of the unemployment rate

19. Or p' = 0.4286992369 - 0.2631864004 p - 0.34276798 u (5) u' = - 0.0483271691 + 0.0869769 p + 0.0274384 u (6) Now, we can see that h = - 0.117874 or h<0. In this sense we conclude that the private consumption deflator, p(t), and unemployment rate, u(t), in the European Union converged in ever-decreasing oscillations to its steady state values p* (=0.170660 or 2.269778%) and/or u* (=0.873093 or 10.3024974%) because the real part of the complex root is negative. Using p0=0 and u0=0 as inital values, the system (5)-(6) is shown in Figure 4.

20. Fig. 4. Estimated model of the private consumption deflator, p, and unemployment rate, u, (5)-(6)

21. 4.Governmental behavior as the stabilizator of endogenous business cycles ? Instead of defining an upper and lower bound to a varibale, it is possible to posit a control governmental parameter which provides a growing downward pressure beyond a given high (positive) value of the private consumption deflator, and a growing upward one for low (negative) values of the private consumption deflator. Therefore, define a third variable, g, as control parameter, i.e. parameter of economic policy, i.e., indirect taxes, to the other two variables, with which it is dynamically coupled, thus g ' =  + g ( p -  ) (7) where  is equilibrium value of the private consumption deflator or the estimated steady state value of private consumption deflator, g ' is the change of control parameter, i.e., indirect taxes. The control parameter, g, (indirect taxes ), has been treated as simply device to produce version of the irregularity characteristic of the private consumption deflator evolution, p, and its time- series.

22. If g'>0 then we obtain p ' =  +  p -  u -  g (8) Now, following the Rössler's ideas (1976) we find a particularly simple system of equation (8), (2) and (7) which is probably the most elementary geometric construction of chaos in continuous system p ' =  +  p -  u -  g u ' = +  p +  u (9) g ' =  + g ( p -  ) where : p - the private consumption deflator, u- unemployment rate, g - control parameter, i.e., indirect taxes,  - equilibrium value of the private consumption deflator, and , , , ,, , ,  - adjustable constants.

23. In this section we use the value of parameters from system (5)-(6) and voluntary fix parameters :  =0.07 and  =0.00000005 p' = 0.4286992369 - 0.2631864004 p - 0.34276798 u-0.07 g u' = - 0.0483271691 + 0.0869769 p + 0.0274384 u. (10) g' = 0.00000005 + g ( p - 0.17066 ) where  =0.4286992369,  =- 0.2631864004,  =0.34276798, =-0.0483271691,  = 0.0869769,  =0.0274384,  =0.17066 with initial values p0=0, u0 = 0, g0=0. The system (10) is shown in Figure 5.

24. Fig.5. The model (10) with initial values p0=0, u0 = 0, g0=0.

25. The highly erratic behaviour is generated by an undistributed, entirely endogenous system (10). Without prior specification of the model, it would be impossible to predict future from past behavior. Lacking prior knowledge of the system's parameters, it is difficult to know how one could determine them from a distributed time-series. As an example, the identical system is shown in Figure 6. with an unchanged set of the system's parameter:  = 0.4286992369,  = - 0.2631864004,  = 0.34276798, =- 0.0483271691,  = 0.0869769,  = 0.0274384,  = 0.17066,  =0.07 and  =0.00000005 and a new set of initial values p0=0.3, u0=0.2, g0=0, which can generate quite striking alteration of system behavior.

26. Fig. 6.The model (10) with new initial values p0=0.3, u0=0.2, g0=0

27. Similarly, the system (10) is depicted in Figure 7. with an unchanged set of initial values: p0=0, u0 = 0, g0=0 but with a new set of the following system's parameters, i.e.: β= 0.2, γ=0.3. In this case system (10) is characterized by erratic dynamics

28. Fig.7.The system (10)with an unchanged set of initial values but with a new set of the system's parameters (β= 0.2, γ=0.3)

29. 5. Conclusion Empirical content of the model (9) confirms the fact that movement of the private consumption deflator and the unemployment rate were been stable in the European Union during the period 1970-2001. Empirical content of the model (9) confirms the fact that movement of the private consumption deflator and the unemployment rate were been stable in the European Union during the period 1970-2001. On the other hand, chaotic model (9) can give an endogenous, alternative explanation of irregular behaviour of the private consumption deflator and the unemployment rate. Such model introduce shocks in a novel way : i.e., a shock constitutes a new set of initial conditions and/or the system's parameters. However, without prior specification of the model (9) it would be impossible to predict future inflation and unemployment rate from its past behaviour. In this sense, the fact of repeated exogenous shocks poses a difficult problem for econometric analysis. Erratic character of the private consumption deflator and the unemployment rate can create the problems and difficulties faced by government. On the other hand, chaotic model (9) can give an endogenous, alternative explanation of irregular behaviour of the private consumption deflator and the unemployment rate. Such model introduce shocks in a novel way : i.e., a shock constitutes a new set of initial conditions and/or the system's parameters. However, without prior specification of the model (9) it would be impossible to predict future inflation and unemployment rate from its past behaviour. In this sense, the fact of repeated exogenous shocks poses a difficult problem for econometric analysis. Erratic character of the private consumption deflator and the unemployment rate can create the problems and difficulties faced by government. The model importance lies in the endogeneity of its erraticism, but this in no way denies the importance of the exogenous disturbances. The model importance lies in the endogeneity of its erraticism, but this in no way denies the importance of the exogenous disturbances. This paper gives hope that complicated dynamic economic behavior of inflation and unemployment can be formulated by low-dimensional mathematical models (i.e., a relatively small number of equations). The possibility of low-dimensional economic modeling of complicated economic behavior has tremendous practical applications. Low dimension economic models allow us to simulate economic processes inexpensively. This paper gives hope that complicated dynamic economic behavior of inflation and unemployment can be formulated by low-dimensional mathematical models (i.e., a relatively small number of equations). The possibility of low-dimensional economic modeling of complicated economic behavior has tremendous practical applications. Low dimension economic models allow us to simulate economic processes inexpensively.