1 The Dynamics of Random Economic Models Volker Böhm Universität Bielefeld Handout for MDEF2000 Workshop Modelli Dinamici in Economia e Finanza Urbino

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1 1 The Dynamics of Random Economic Models Volker Böhm Universität Bielefeld Handout for MDEF2000 Workshop Modelli Dinamici in Economia e Finanza Urbino September 28 30, 2000

2 CONTENTS 2 Contents 1 Introduction 3 2 Random Dynamical Systems 4 3 Affine Random Dynamical Systems 9 4 Aggregate Stochastic Growth Models 15

3 1 Introduction 3 1 Introduction ffl Random Difference Equations ffl F : X R m! X ffl x t+1 = F (x t ;w t ) ffl x t state at time t ffl w t realization of disturbance at time t =) Law of motion plus a model of exogenous noise ffl recent development of new mathematical techniques =) Arnold (1998) ffl combining dynamical systems theory with stochastic processes ffl allows dynamic analysis of stochastic phenomena ffl stability analysis of sample paths ffl bifurcation theory ffl new view of time series analysis in economics Economic examples ffl Linear rational expectations models ffl Stochastic Multiplier-accelerator model ffl stochastic growth models: Solow OLG RBC ffl sequential CAPM model ffl dynamic disequilibrium macro model

4 2 Random Dynamical Systems 4 2 Random Dynamical Systems Law of motion plus a model of exogenous noise werden für alle Zeitpunkte t beschrieben durch ffl x t+1 = F (x t ;w t ) ffl x t Zustand zum Zeitpunkt t ffl w t Störung zum Zeitpunkt t Topological dynamical system modeling the (deterministic) dynamics Definition 2.1 A (topological) dynamical system in discrete time N = f0; 1; 2;:::g on a set X ρ R d with parameter space R m is given by the time one map F : X R m! X : For given w and any initial state x 2 X, orbits are generated by 1. x t = F (x t 1 ;w) 8t 2 N 2. x t = F (:; w) ffi ffif (x; w) z } t mal 8t 2 N

5 2 Random Dynamical Systems 5 Ergodic dynamical systems modeling the noise process Definition 2.2 A metric dynamical system in discrete time consists of a probability space (Ω; F; P) and a measurable invertible time one map : Ω! Ω; which induces the family of iterated maps ( t ) given by (Ω; F; P,( t )) satisfies: t = ffi ffi z } t mal ffl P is ergodic with respect to, i.e. invariant sets have P measure zero or one ffl is measure preserving, i. e. P = P Examples ffl sequences of i.i.d. random variables ffl irreducible finite state Markov chains ffl "almost" deterministic processes with periodic motion ffl the tent map with appropriate measure ffl the logistic map #! = 4!(1!) with appropriate distribution P of! 2 [0; 1]

6 2 Random Dynamical Systems 6 The real noise process Definition 2.3 A real noise process for the parameter w associated with a metric dynamical system is a stochastic process fu t g; u t : Ω! R m, generated by the metric dynamical system, i.e. u t = u ffi t ; w t = u t (!) = u( t!) Definition 2.4 (Arnold 1998) A random dynamical system consists of a parametrised topological dynamical system, i.e. F : X R m! X and a metric dynamical system with real noise process (Ω; F; P,( t )); u t = u ffi t

7 2 Random Dynamical Systems 7 Random fixed points and stability Definition 2.5 A random fixed point of F is a random variable x? : Ω! R m on (Ω; F; P) such that almost surely: The definition implies that x? (!) = F (x? (!);u(!)) ffl If F independent of the perturbation!, then x? is a deterministic fixed point, ffl x Λ (# t+1!) = F (x Λ (# t!);u(# t!)) for all times t, ffl the orbit fx Λ (# t!)g t2n,! 2 Ω generated by x Λ solves the random difference equation x t+1 = F (x t ;u t (!)): ffl fx Λ (# t )g t2n is stationary and ergodic, since # is stationary and ergodic, ffl If, in addition, E jjx Λ jj < 1, then lim T!1 1 T TX t=0 for every B 2 B(X). 1 B (x Λ (# t!)) = x Λ P(B) := P f! 2 Ωjx Λ (!) 2 Bg ffl the empirical law of an orbit is well defined and is equal to the distribution x Λ P of x Λ. ffl if the noise process is i. i. d. =) the orbit is an ergodic Markov equilibrium in the sense of Duffie, Geanakoplos, Mas-Colell & McLennan (1994).

8 2 Random Dynamical Systems 8 The concept of stability to be used in the context of a random dynamical system is as follows: Definition 2.6 A random fixed point x Λ is called attracting on some set U ρ Ω X if lim t!1 jjx t(!) x Λ (# t!)jj = 0 for all (!; x 0 (!)) 2 U: Lra a random fixed point is attracting if nearby orbits converge to the orbit of the random fixed point.

9 3 Affine Random Dynamical Systems 9 3 Affine Random Dynamical Systems Iterated Function Systems (IFS) Affine models with discrete i. i. d. noise x t+1 = A i x t + b i i 2 I Consider a finite family fg i g of affine maps G i : R K! R K with associated probabilities fß i g, ß P i ßi = 1. > 0, i = 1;::: ;r, und If all mappings G i are contractions, then f(g i ); (ß);i= 1;::: ;rg is called an iterated function system Lemma 3.1 (Arnold (1998), Barnsley (1988)) Let f(g i ); (ß);i= 1;::: ;rg be an iterated function system. 1. f(g i ); (ß);i= 1;::: ;rg has a unique compact attractor A ρ k 1 ; k r Λ, 2. A is independent of the probabilities fß i g. 3. A is the limit of a decreasing sequence of finite unions of compact intervals (cubes), 4. there exists a unique invariant measure μ on A, 5. there exists a unique globally attracting random fixed point x? of the associated random dynamical system, whose empirical distribution is μ =) A is often a fractal set! Cantor set =) the measure μ may have a very complex density

10 3 Affine Random Dynamical Systems 10 Theorem 3.1 (Arnold (1998)) Consider G : X R 3m! X with real noise process (A; b) t := (A; b) ffi # t, A : Ω! R m R m and b : Ω! R m measurable, over the ergodic dynamical system (Ω; F; P; (# t )) where G (A; b) is the family of invertible affine difference equations x t+1 = A( t!)x t + b( t!): With some ("mild and natural") additional assumptions (hyperbolicity, integrability, contractivity) there exists a unique globally attracting random fixed point. Example 1: Sequential CAPM Consider the sequential CAPM model (cif. Böhm & Chiarella 2000) under unbiased prediction with price and expectations process (3.1) (3.2) p t = 1 R [D t( )+Rμ t 1 E t 1 D t ( )]! 1 μ t = fi a μx + Rμ t 1 E t 1 D t ( ): ψ X a and an AR(1) dividend process modeled as (3.3) D t+1 = ffd t + t ; where with 0 < ff < 1 and t ο uniform i.i.d. over [a; b] Theorem 3.2 The random dynamical system given by equations (3.2) 3.3 has a unique random fixed point μ Λ if and only if R 6= 1. μ Λ is globally attracting if and only if 0 < R < 1.

11 3 Affine Random Dynamical Systems p t t Figure 3.1: Convergence of prices to random fixed point 25 p t t Figure 3.2: Prices with AR(1) dividends p t t Figure 3.3: prices with AR(1) dividends

12 3 Affine Random Dynamical Systems 12 Example 2: Random Multiplier accelerator model (cif. Böhm & Jungeilges 2000) (3.4) (3.5) (3.6) (3.7) C = m 0 + my 1 I = v 0 + v(y 1 Y 2 ) Y = C + I + G Y = (m 0 + v 0 )+(m + v)y 1 vy 2 =) (3.8) Y = (m 0 + v 0 )+(m + v)y 1 vy 2 =) (3.9) (m 0 i ;v0 i ) 0; 0» (m i;v i )» 1 (3.10) X ß i = 1 ß i fl 0 i 2 I

13 3 Affine Random Dynamical Systems 13 Example 3: Productivity shocks in a linear growth model Consider ffl 0 < a 1 < ::: < a i < < a r ffl mit Wahrscheinlichkeiten fß i g, ß P i ßi = 1, > 0, i = 1;::: ;r, und ffl parametrised family of affine maps fg i g, G i : R! R ffl mit zugehörigen Fixpunkten G i (k t ) = (1 ffi + sb)k t + sa i ; (1 + n) k i := sa i n + ffi sb i = 1;::: ;r: Beispiel r = 2. a 1 = 0:5, a 2 = 0:7, ß 2 = :02 s = b = 0:5 ffi = 0:5 bzw. ffi = 0:85

14 3 Affine Random Dynamical Systems 14 Figure 3.4: Invariante Verteilung im linearen Modell auf einem Intervall Figure 3.5: Invariante Verteilung im linearen Modell auf einer Cantormenge

15 4 Aggregate Stochastic Growth Models 15 4 Aggregate Stochastic Growth Models Literatur: Mirman (1972, 1973),, Ramsey (1928) ) RBC Models ffl `local' linearized statistical description near "noisy" steady state ffl no stability analysis ffl general statistical description of random behavior as Markov equilibria Becker & Zilcha (1997) ffl no stability of sample paths ffl no global dynamic analysis Stability in the stochastic Solow model ffl production function f : R +! R +, with parameters ffl A > 0 multiplicative scale factor of f, ffl n > 1 rate of population growth ffl 0» ffi» 1 rate of depreciation ffl 0» s» 1 aggregate propensity to save. =) difference equation parametrized in (ffi;n;s;a) k t+1 = (1 ffi)k t + saf(k t ) ; (1 + n)

16 4 Aggregate Stochastic Growth Models 16 with ffl ffl ffl k t+1 = (1 ffi( t!))k t + ο( t!)f(k t ) 1+n( t!) ffi(!) 2 [ffi min ;ffi max ] ρ [0; 1] n(!) 2 [n min ;n max ] ρ ] 1; 1] ο(!) 2 [ο min ; 1[ ρ ]0; 1[ mit E ο < 1 ffl if the Inada conditions hold =) for every given vector of parameters (ffi;n;ο) there exist a unique nontrivial positive fixed point, =) strictly decreasing in n + ffi and strictly increasing in ο. =) Let ffl k := k(ffi max ;n max ;ο min ) denote the smallest and ffl k := k(ffi min ;n min ;ο max ) the largest possible fixed points of the deterministic model. =) eventually Λ all sample paths stay in the compact interval k; k =) the long run behavior occurs in the interval whichisaforward invariant set of the random dynamical system.

17 4 Aggregate Stochastic Growth Models 17 Theorem 4.1 (Schenk-Hoppé & Schmalfuß (1998)) Let f be strictly monotonically increasing, strictly concave, and continuously differentiable. If i) ii) iii) ffi max + n max > 0 0» lim k!1 f(k)=k < (ffi max + n max )=ο min < lim k!0 f(k)=k» 1 E log 1 ffi(!)+ο(!)f 0 (k) 1+n(!) < 0; there exists a unique nontrivial globally attracting random fixed point k?. Corollary 4.1 If the perturbations (ffi(!);n(!);ο(!)) are i.i.d., the random fixed point is given by a unique ergodic Markov equilibrium.

18 4 Aggregate Stochastic Growth Models 18 k 14 k t t Figure 4.1: Stable random fixed points in the Solow Swan model. Left coloumn: tent map. Right coloumn: i.i.d. shock.

19 REFERENCES 19 References Arnold, L. (1998): Random Dynamical Systems. Springer-Verlag, Berlin a.o. Arnold, L. & X. Kedai (1994): Invariant Measures for Random Dynamical Systems, and a necessary Condition for Stochastic Bifurcation from a Fixed Point", Random and Computational Dynamics, 2(2), Barnsley, M. (1988): Fractals Everywhere. Academic Press, New York a.o. Becker, R. & I. Zilcha (1997): Stationary Ramsey Equilibria under Uncertainty", Journal of Economic Theory, 75(1), Böhm, V. (1999): Stochastische Wachstumszyklen aus dynamischer Sicht", Wirtschaftswissenschaftliches Seminar Ottobeuren, 28, Böhm, V. & C. Chiarella (2000): Mean Variance Preferences, Expectations Formations, and the Dynamics of Random Asset Prices", Discussion Paper No. 448, University of Bielefeld. Böhm, V. & J. Jungeilges (2000): The Dynamics of Stochastic Affine Systems.", in preparation. Böhm, V. & J. Wenzelburger (1997): Expectational Leads in Economic Dynamical Systems", Discussion Paper 373, Department of Economics, University of Bielefeld. (1999): Perfect Predictions in Economic Dynamical Systems with Random Perturbations", Revised Version September 1999, Discussion Paper No. 356, University of Bielefeld. Duffie, D., J. Geanakoplos, A. Mas-Colell & A. McLennan (1994): Stationary Markov Equilibria", Econometrica, 62(4), Mirman, L. J. (1972): On the Existence of Steady State Measures for One Sector Growth Models with Uncertain Technology", International Economic Review, 13(2), (1973): The Steady State Behavior of a Class of One Sector Growth Models with Uncertain Technology", Journal of Economic Theory, 6, Ramsey, F. (1928): A Mathematical Theory of Saving", The Economic Journal, 38, Schenk-Hoppé, K. R. & B. Schmalfuß (1998): Random Fixed Points in a Stochastic Solow Growth Model", Discussion Paper 387, Department of Economics, University of Bielefeld. Schmalfuß, B. (1996): A Random Fixed Point Theorem Based on Lyapunov Exponents", Random and Computational Dynamics, 4(4), Schmalfuß, B. (1998): A Random Fixed Point Theorem and the Random Graph Transformation", Journal of Mathematical Analysis and Application,, 225(1),