RANDOM DYNAMICAL SYSTEMS IN ECONOMICS MUKUL MAJUMDAR Abstract. Random dynamical systems are useful in modeling the evolution of economic processes subject to exogenous shocks. One obtains strong results on the existence, uniqueness, stability of the invariant distribution of such systems when an appropriate splitting condition is satisfied. Also of importance has been the study of random iterates of maps from the quadratic family. Applications to economic growth models are reviewed. Key words. dynamical systems, Markov processes, iterated random maps, invari- ant distributions, splitting, quadratic family, estimation, economic growth 1. Introduction. Consider a random dynamical system (S, Γ, Q) where S is the state space (for example, a metric space), Γ an appropriate family of maps on S into itself (interpreted as the set of all possible laws of motion) and Q is a probability measure on (some σ-field of) Γ. The evolution of the system can be described as follows: initially, the system is in some state x; an element α1 of Γ is chosen randomly according to the probability measure Q and the system moves to the state X1 = α1(x) in period one. Again, independently of α1, an element α2 of Γ is chosen according to the probability measure Q and the state of the system in period two is obtained as X2 = α2(α1(x)). In general, starting from some x in S, one has Xn+1(x) = αn+1(Xn(x)), (1.1) where the maps (αn) are indepenent with the common distribution Q. The initial point x can also be chosen (independently of (αn)) as a random vari- able X0. The sequence Xn of states obtained in this manner is a Markov process and has been of particular interest in dynamic economics. It may be noted that every Markov process (with an arbitrary given transition probability) may be constructed in this manner provided S is a Borel sub- set of a complete separable metric space, although such a construction is not unique [Bhattacharya and Waymire [1], p. 228]. Hence, random it- erates of affine, quadratic or monotone maps provide examples of Markov processes with specific structures that have engaged the attention of prob- ability theorists. Random dynamical systems have been studied in many contexts in economics, particularly in modeling long run evolution of economic sys- tems subject to exogenous random shocks. The framework (1.1) can be interpreted as a descriptive model; but one may also start with a dis- counted (stochastic) dynamic programming problem, and directly arrive at a stationary optimal policy function, which together with the given law of transition describes the optimal evolution of the states in the form (1.1). 1 2 MUKUL MAJUMDAR Of particular significance are recent results on the “inverse optimal prob- lem under uncertainty” due to Mitra [10] which assert that a very broad class of random systems (1.1) can be so interpreted. To begin with, in order to provide the motivation, I present two ex- amples of deterministic dynamical systems arising in economics. The first is a descriptive growth model that leads to a dynamical system with an increasing law of motion. The second shows how laws of motion belonging to the quadratic family can be generated in dynamic optimization theory. In Section 3 we review some results on random dynamical systems that satisfy a splitting condition, first introduced by Dubins and Freedman [7] in their study of Markov processes. This condition has been recast in more general state spaces (see (3.10)). The results deal with: (i) The existence, uniqueness and global stability of a steady state (an invariant distribution): a general theorem proved in Bhattacharya and Majumdar [3] is first recalled (Theorem 3.1). The proof relies on a contrac- tion mapping argument that yields an estimate of the speed of convergence [see (3.11) and (3.13)]. Corollary 3.1 deals with “split” dynamical systems in which the admissible laws of motion are all monotone. (ii) Applications of the theoretical results to a few topics: (a) turnpike theorems in the literature on descriptive and op- timal growth under uncertainty: when each admissible law of motion is monotone increasing, and satisfies the appropriate Inada-type ‘end point’ condition, Corollary 3.1 can be applied directly: see Sections 3.2.1 - 3.2.2. (b) estimation of the invariant distribution: as noted above, an important implication of the splitting condition is an estimate of the speed of convergence. This estimate is used in Section 3.2.3 to prove a result on √n-consistency of the sample mean as an estimator of the expected long run equilibrium value (i.e., the value of the state variable with respect to the invariant distribution). Next, in Section 4 we briefly turn to qualitative properties of random iterates of quadratic maps: a growing literature has focused on this theme, in view of the discussion in Section 1.2 and of the privileged status of the quadratic family in understanding complex or chaotic behavior of dynam- ical systems. 1.1. The Solow Model: A Dynamical System with an Increas- ing Law of Motion. Here is a discrete time exposition of Solow’s model [11] of economic growth with full employment. There is only one producible commodity which can be either consumed or used as an input along with labor to produce more of itself. When consumed, it simply disappears from the scene. Net output at the “end” of period t, denoted by Yt(= 0) is related to the input of the producible good Kt (called “capital”) and la- bor Lt employed “at the beginning of” period t according to the following technological rule (“production function”): Yt = F (Kt, Lt) (1.2) RANDOM DYNAMICAL SYSTEMS IN ECONOMICS 3 where Kt = 0, Lt = 0. The fraction of output saved (at the end of period t) is a constant s, so that total saving St in period t is given by St = sYt, 0 < s < 1. (1.3) Equilibrium of saving and investment plans requires St = It (1.4) where It is the net investment in period t. For simplicity, assume that capital stock does not depreciate over time, so that at the beginning of period t + 1, the capital stock Kt+1 is given by K K + I (1.5) t+1 ≡ t t Suppose that the total supply of labor in period t, denoted by Lˆt is determined completely exogeneously, according to a “natural” law: t Lˆt = Lˆ0(1 + η) , Lˆ0 > 0, η > 0. (1.6) Full employment of the labor force, requires that Lt = Lˆt (1.7) Hence, from (1.2) - (1.7), we have Kt+1 = Kt + sF (Kt, Lˆt) Assume that F is homogeneous of degree one. We then have K Lˆ K K t+1 t+1 = t + sF t , 1 Lˆt+1 · Lˆt Lˆt µ Lˆt ¶ Writing k K /Lˆ we get t ≡ t t kt+1(1 + η) = kt + sf(kt) (1.8) where f(k) F (K/L, 1). ≡ From (1.8) kt+1 = [kt/(1 + η)] + [sf(kt)/(1 + η)] or kt+1 = α(kt) (1.9) 4 MUKUL MAJUMDAR where α(k) [k/(1 + η)] + s[f(k)/(1 + η)] (1.10) ≡ Equation (1.9) is the fundamental dynamic equation describing the intertemporal behavior of kt when both the full employment condition and the condition of short run savings-investment equilibrium [see (1.4) and (1.7)] are satisfied. We shall refer to (1.9) as the law of motion of the Solow model in its reduced form. For any k > 0, the trajectory τ(k) from k is given j ∞ 0 1 j j−1 by τ(k) (α (k))j=0 where α (k) k, α (k) α(k), α (k) α(α (k)) for j = 2.≡ ≡ ≡ ≡ Assume that f(0) = 0, f 0(k) > 0, f 00(k) < 0 for k > 0; and lim f 0(k) = k↓0 , lim f 0(k) = 0. Then, using (1.10), we see that α(0) = 0; ∞ k↑∞ α0(k) = (1 + η)−1[1 + sf 0(k)] > 0 at k > 0; α00(k) = (1 + η)−1sf 00(k) < 0 at k > 0. (1.11) Also, verify the boundary conditions: limα0(k) = lim[(1 + η)−1 + (1 + η)−1sf 0(k)] = . k↓0 k↓0 ∞ lim α0(k) = (1 + η)−1 < 1. (1.12) k↑∞ The existence, uniqueness and stability of a steady state k∗ > 0 of the dynamical system (1.9) can be proved. Here is a summary of the results: Proposition 1.1. There is a unique k∗ > 0 such that k∗ = α(k∗); equivalently, k∗ = [k∗/(1 + η)] + s[f(k∗)/1 + η] (1.13) If k < k∗, the trajectory τ(k) from k is increasing and converges to k∗. If k > k∗, the trajectory τ(k) from k is decreasing and converges to k∗. 1.2. The Quadratic Family in Dynamic Optimization Prob- lems. We consider a family of economies indexed by a parameter µ, where µ²A = [1, 4]. Each economy in this family has the same production function, f : + + and the same discount factor δ²(0, 1). The economies in this < → < 2 family differ in the specification of their return functions, w : + A + [depending on the parameter value of µ²A that is picked]. < × → < The following assumptions of f are used: (F.1) f(0) = 0. (F.2) f is non-decreasing, continuous and concave on +. (F.3) There is K > 0, such that f(x) < x for all x >< K, and f(x) > x for all 0 <x<K. RANDOM DYNAMICAL SYSTEMS IN ECONOMICS 5 A program from an initial input x 0 is a sequence (x ) satisfying ≥ t x = x, 0 x f(x − ) fort 1 0 ≤ t ≤ t 1 ≥ We interpret xt as the input in period t, and this leads to the output f(xt) in the subsequent period.