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WS 2006/07 6. Chaos-Theory and the Business Cycle Endogenous and real business cycles Derived from natural sciences Flourished in 1980s and 1990s Meanwhile out of fashion again (too technical, lacking policy receipts) But interesting insights in dynamics, applicable to distribution-battle 1 U van Suntum, Vorlesung KuB 1

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WS 2006/07 KuB 7 Chaos-theory: An example*) *) cf. Ian Stewart, Spielt Gott Roulette? Chaos in der Mathematik, Basel u.a. 1990, S. 163 ff.; Heubes, Konjunktur u. Wachstum, a.a.O., S. 108 x t+1 = r x t (1 – x t ) x t+1 xtxt X max = 0,25 r 0,5 KuKuB 7 2 1,0 0 U van Suntum, Vorlesung KuB 2

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WS 2006/07 KuB 7 Derivation of maximum: x t+1 = r x t (1 – x t ) x t+1 xtxt X max = 0,25 r 0,5 3 U van Suntum, Vorlesung KuB 3

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WS 2006/07 KuB 7 x t+1 = r x t (1 – x t ) Derivation of fixed point (equlibrium): x t+1 xtxt X max = 0,25 r 0,5 45 o x t+1 = x t fixed point Existence of equilibrium is not sufficient to guarantee its feasibility and stability! 4 U van Suntum, Vorlesung KuB 4

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WS 2006/07 KuB 7 x t+1 = r x t (1 – x t ) with 0 convergence to fixed point with 3 fluctuations (bifurcations) with 3,58 chaos with occasional periodicity with r > 4 => exploding 45 o x t+1 xtxt Points which are feasible in principle Fixed point (stable if absolute slope of curve < 1) Startwert X max = 0,25 r Y t+1 = aY t (1 – Y t ) 5 Chaosgleichungbeispiel.xls U van Suntum, Vorlesung KuB 5

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WS 2006/07 KuB 7 a) convergence (0 fixed point = 0,6428 start: x = 0,4 fixed point: x = 0,6428 temporal behavior of x 6 U van Suntum, Vorlesung KuB 6

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WS 2006/07 KuB 7 b) bifurcations (3 fixed point = 0,6875 start: x = 0,4 points of bifurcation:*) x = 0, 7995 und x = 0,5130 *) numerically derived by Excel-Solver: conditions: x t+1 = x t+3 and x t+2 = x t+4 7 temporal behavior of x U van Suntum, Vorlesung KuB 7

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WS 2006/07 KuB 7 c) chaos (3,58 fixed point = 0,7368 start: x = 0,4 8 temporal behavior of x U van Suntum, Vorlesung KuB 8

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WS 2006/07 KuB 7 d) explosion ( r > 4); here: r = 4,2 start: x = 0,4 9 temporal behavior of x U van Suntum, Vorlesung KuB 9

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WS 2006/07 KuB 7 Economic application: Goodwin/Pohjola-model (1967/1981) (cf. Heubes, Konjunktur und Wachstum) wages w (and wage share u = W/Y) rise in Y grwoth rate g Y declines in wage share u (1 – u) g u g u 10 U van Suntum, Vorlesung KuB 10

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WS 2006/07 KuB 7 Application: Business Cycle Model of Goodwin/Pohjola (1967/1981) Assumptions: Leontief production function=> g denotes growth rate of Y, K and N Classical saving function: amount G is saved, amount W consumed Additional simplifications here: no technical progress, size of labor force fixed Wages increase in employment and labor productivity Variables: I = investment, K = capital stock, N= labor, k = capital coefficient K/Y, w = wage rate, d = wage adjustment parameter, N * = equilibrium employment (fixed point of chaos model) 11 U van Suntum, Vorlesung KuB 11

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WS 2006/07 KuB 7 Formal Derivation of Goodwin/Pohjola-Model chaos with k < 0,39 12 U van Suntum, Vorlesung KuB 12

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WS 2006/07 KuB 7 Explanation: investment proportional to profit share (1-u) because all profits are saved (eq.1) employment proportional to total income because of Leontief PF (eq. 2) high employment and high productivity increase wage rate (eq. 3) model culminates in a single difference equation (eq. 6, 7). thus with given starting point N and constant N * employment is determined at any t empirical estimation by autoregressive methods (using only N t, N t-1 etc.) KuKuB 7 13 Chaosgleichungbeispiel.xls U van Suntum, Vorlesung KuB 13

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WS 2006/07 KuB 7 Grafical exposition with k < 0,39 (here: k = 0,38) employment N(t) KuKuB 7 14 U van Suntum, Vorlesung KuB 14

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WS 2006/07 KuB 7 Damped fluctuations with k >> 0,4 (here: k) bifurcation with k > 0,4 (here: k = 0,6) employment N(t) Summary: anything goes… 15 U van Suntum, Vorlesung KuB 15

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WS 2006/07 KuB 7 Critique of Chaos-Theory Strengths: Shows dynamics of business cycle Easily transformable into econometrics Can explain erratic fluctuations Weaknesses: technical, relatively little economic content Poor policy relevance Very simple economic model 16 U van Suntum, Vorlesung KuB 16

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WS 2006/07 KuB 7 Lerning goals/Questions How do mathematicans and economists define chaos? What is the fixed point of a dynamic model? What are bifurcations? Where do the dynamics in the Goodwin/Pohjola model come from? How does the wage share behave over the business cycle? 17 U van Suntum, Vorlesung KuB 17

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WS 2006/07 KuB 7 Exercise: KuKuB 7 18 Assume the following difference equation for total demand in the business cycle: Let parameter a be 4.20 1)What is the maximum possible value of Y? 2)What is the fixed point of the model?

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