NUMERICAL ANALYSIS OF TWO COUPLED KALDOR-KALE KALDOR-KALECKI MODELS WITH DELAY

Transcription

1 Introduction NUMERICAL ANALYSIS OF TWO COUPLED KALDOR-KALECKI MODELS WITH DELAY Beata Zduniak Urszula Grzybowska Arkadiusz Orłowski Warsaw University of Life Sciences (SGGW) Faculty of Applied Informatics and Mathematics FENS 2014

2 Introduction Introduction to our problem This paper is concerned with two coupled Kaldor-Kalecki models of business cycles with delays in both the gross product and the capital stock. In our study we consider two types of investment functions that lead to different behavior of the system. We introduce the model with unidirectional coupling to investigate the influence of the European Union economy as a global one on the local Polish economy. We present the results of numerical analysis.

3 Introduction Great depression and theories of trade cycle The research on the reasons of trade cycles was stimulated by great depression ( ). The most important contributions were done by: John Maynard Keynes (1936) Michał Kalecki (1933) Nicholas Kaldor (1940)

4 Introduction Michał Kalecki ( )

5 Introduction Michał Kalecki ( ) Kalecki was born in Lodz He studied to become an engineer Due to difficult financial conditions he had to break the studies He earned a living writing articles to economic newspapers In 1933 he wrote An Essay on the Theory of the Business Cycle In 1935 he published his papers in Revue d Economie Politique and Econometrica In 1936 he obtained a scholarship, which enabled him to travel to Sweden

6 Introduction Michał Kalecki In 1937 Kalecki went to England He visited the London School of Economics and afterward went to Cambridge In 1939 he wrote one of his most important works: Essays in the Theory of Economic Fluctuations During the war he worked in the Oxford Institute of Statistics After the war Kalecki traveled due to his duties as an political or economic adviser In 1955 he returned to Poland and in 1957 he was appointed chairman of the Committee for the Perspective Plan

7 Introduction Kalecki s model of business cycle (1933) The most important contribution was incorporation of a new factor into the model: investment decisions B(τ) at the moment τ. According to Kalecki, the increase of capital K(t) at the moment t results form investment decisions made at the moment t τ, where τ denotes the average time of investment completion. Kalecki assumed that the investment decisions B(t) are increasing with respect to savings S and decreasing with respect to capital K: B(t) = gs(t) hk(t) I (t + τ) = gi (t) hk(t) I (t) = K(t)

8 Introduction Kalecki s model of business cycle (1933) The basic equation of the model is given by: K ak(t) + bk(t θ) = 0 where a = g θ and b = h + g θ

9 Introduction Nicholas Kaldor ( )

10 Introduction Nicholas Kaldor ( ) Miklos Kaldor was born in Budapest He studied in Berlin and at the London School of Economics He worked at the LSE till 1947 He worked as a political adviser for various governments and parties In 1952 he became a lecturer at Cambridge University, where he was appointed professor of economics in 1966

11 Introduction Kaldor model of business cycles (1940) Ẏ = α(i (Y, K) S(Y, K)) K = I (Y, K) δk where I (Y, K) is an investment function depending on income (or outcome) Y and capital K, S(Y, K) is a saving function, α is an adjustment coefficient and δ is the depreciation rate of capital stock (δ (0,1)).

12 Introduction Kaldor model of business cycles (1940) Major contribution of Kaldor was the assumption that the investment function and the saving function are nonlinear functions of income Y. The idea is that the rate of investment will be quite low at extreme output levels (very high, point C, and very low, point A).

13 Introduction Kaldor model of business cycles (1940) The Kaldor model is a prototype model of nonlinear dynamics. Its behavior can be examined either with help of the Poincare-Bendixson Theorem or Hopf Bifurcation Theorem.

14 Introduction Poincare Bendixson Theorem A non-empty compact limit set of a C 1 dynamical system in R 2, which contains no equilibrium point, is a closed orbit.

15 Introduction Application of the Poincare Bendixson Theorem We can obtain the phase diagram to detect the compact limit set. We first obtain the isoclines dy dt = 0 and dk dt = 0. Application of the implicit function theory and analysis of the relation between the slopes of investment and saving function leads us to the following phase diagram.

16 Introduction Kalecki Kaldor model of business cycles (1999) In 1999 A. Krawiec & M. Szydłowski applied the Kalecki s idea of investment gestation period to the Kaldor model. This resulted in a system of delay differential equations: dy dt dk dt = α(i (Y (t), K(t)) S(Y (t), K(t))) = I (Y ((t T ), K(t)) δk(t) where I (Y, K) is an investment function depending on income Y and capital K, S(Y, K) is a saving function, α is an adjustment coefficient, δ is the depreciation rate of capital stock (δ (0,1)) and T is the time delay.

17 Introduction Further contributions to the Kalecki-Kaldor model The analysis done by Krawiec & Szydłowski revealed that the limit cycle behavior of the system is independent of the assumption that the investment function is s-shaped. Krawiec & Szydłowski proposed the following form of the investment function: I (Y, K) = ηy + βk where η > 0 and β < 0. In the analysis of the model (6) the authors used Poincare Andronov Hopf Theorem to predict the occurrence of a bifurcation to a limit cycle for some value of parameter T.

18 Introduction Further contributions to the Kalecki-Kaldor model In the following years many researchers contributed to the Kalecki Kaldor model. The amendments done concerned delay and the form of investment and saving functions. dy dt dk dt = α(i (Y (t), K(t)) S(Y (t), K(t))) = I (Y ((t T ), K(t T )) δk(t) where I (Y, K) = I (Y ) + βk S(Y, K) = γy (see Krawiec & Szydłowski, Kodera, Wu for more details). Also investigation of the cyclic behavior of the Kalecki Kaldor model for s shaped investment function was done (see, e.g., Kodera).

19 Introduction Coupled Kaldor-Kalecki models Ẏ 1 Ẏ 2 Ẏ 3 Ẏ 4 = α 1 (F 1 (t) δ 1 Y 2 (t) γ 1 Y 1 (t)) = F 1 (t τ) δ 1 Y 2 (t τ) δy 2 (t), = α 2 (F 2 (t) δ 2 Y 4 (t) γ 2 Y 3 (t)) = F 2 (t τ) δ 2 Y 4 (t τ) δy 4 (t) + s 2 (Y 1 (t) Y 3 (t)) where F 1 (t), F 2 (t) are investment functions, s 1 and s 2 coupling coefficients, α is the adjustment coefficient, δ is the depreciation rate of capital stock (δ (0, 1)), γ 1, γ 2, δ 1, δ 2 are constant and τ is a delay.

20 Investment functions In our study we consider two types of investment functions that lead to different behaviour of the system. 1 Logistic functions for local and global economy F 1,2 (t) = e Y (t) /(1 + e Y (t) ) 2 Logistic function for global and sinus function for local economy F 1 (t) = e Y (t) /(1 + e Y (t) ) and F 2 (t) = 0.8 sin(y (t)) Delay In our study delay which is used in investment functions is constant and equals 3.

21 Investment functions Logistic functions for local and global economy

22 Investment functions Global economy: European Union Ẏ 1 Ẏ 2 = α 1 (e Y1(t) /(1 + e Y1(t) ) δ 1 Y 2 (t) γ 1 Y 1 (t)) = e Y1(t τ) /(1 + e Y1(t τ) ) δ 1 Y 2 (t τ) δy 2 (t) Local economy: Poland Ẏ 3 = α 2 (e Y3(t) /(1 + e Y3(t) ) δ 2 Y 4 (t) γ 2 Y 3 (t)) Ẏ 4 = ey 3 (t τ) δ 1+e Y 3 (t τ) 2 Y 4 (t τ) δy 4 (t) + s 2 (Y 1 (t) Y 3 (t))

23 Investment functions 1 α is bigger for richer, stronger market (global economy). We consider α = 1.5, 3, 4 and 9. But always in our considerations α 1 > α 2. 2 Unidirectional coupling aims to show the impact of a stronger market for weaker one for s 2 > 0.

24 Investment functions for logistic functions for local and global economy Case without coupling with α 2 = 1.5 and reference parameters. Time series: y 1 (t), y 2 (t) and y 3 (t), y 4 (t) The system with small values of the α is not able to function properly, so small adjustment coefficient prevents us from obtaining the business cycle.

25 Investment functions for logistic functions for local and global economy Case without coupling with α 2 = 1.5 and reference parameters. Phase portraits: y 1 (t), y 2 (t), y 3 (t), y 4 (t)

26 Investment functions Case with coupling s2 = 0.1 and α2 = 1.5

27 Investment functions Case with coupling s 2 = 0.2 and α 1 = 4 Adding an unidirectional coupling s 2, causes that y 1, y 2 directs y 3, y 4. An attempt to fit the y 3, y 4 to the y 1, y 2 resulting in disorder oscillations, as well as changing the length of the local amplitude of the system.

28 Investment functions Logistic function for global but sinus function for local economy

29 Investment functions Global economy: European Union Ẏ 1 Ẏ 2 = α 1 (e Y1(t) /1 + e Y1(t) δ 1 Y 2 (t) γ 1 Y 1 (t)) = e Y1(t τ) /1 + e Y1(t τ) δ 1 Y 2 (t τ) δy 2 (t) Local economy: Poland Ẏ 3 Ẏ 4 = α 2 (0.8sin(Y 3 (t))) δ 2 Y 4 (t) γ 2 Y 3 (t)) = 0.8sin(Y 3 (t τ)) δ 2 Y 4 (t τ) δy 4 (t) + s 2 (Y 1 (t) Y 3 (t))

30 Investment functions for logistic function for global but sinus function for local economy Case without coupling with α 1 = 4 and reference parameters.

31 Investment functions Case without coupling with α 1 = 4 and reference parameters. Phase portraits:

32 Investment functions Case with coupling s 2 = 0.1 and α 1 = 4

33 Investment functions Case with coupling s 2 = 0.1 and α 1 = 4. Phase portraits:

34 Investment functions Case without coupling, α 2 = 1.5

35 Investment functions We add unidirectional coupling s 2 = 0.1. We demonstrate that the management system of the weak by the powerful global system is able to restore its oscillatory character. This is true even for a small value of the coupling coefficient.

36 Investment functions Other types of coupling:s2 = 1, α2 = 1.5

37 Investment functions Other types of coupling:s2 = 5, α2 = 1.5

38 Investment functions Other types of coupling:s 1 = 0.1, α 2 = 1.5

39 Investment functions Innovation idea: Kaldor Kalecki model has not been used as a system two coupling Kaldor Kalecki models. In our study we model the interaction of two economic systems. We have also demonstrated and modelled that the stronger global system directs the interdependent local system. Despite the influence the local system is able to restore its oscillatory behavior, even if it has very adverse economic parameters and has been blanked system. It is planned to examine the impact of the delay on the action of the local and global the original unidirectional coupling.

40 Investment functions The Kaldor-Kalecki model, although created many years ago, can be adopted to describe today s business cycle working in larger structures by modifying and taking into account coupling between the described structures. It is worth noting that most of the research of this model done so far is based on a very narrow range of parameters.