Mukul Majumdar

Optimization and chaos

M. Majumdar, T. Mitra: Dynamical Systems: A Tutorial.- T. Mitra: Introduction to Dynamic Optimization Theory.- M. Majumdar, T. Mitra: Periodic and Chaotic Programs of Optimal Intertemporal Allocation in an Aggregative Model with Wealth Effects.- K. Nishimura, M. Yano: Optimal Chaos, Nonlinearity, and Feasibility Conditions.- R. Becker, C. Foias: The Local Bifurcation of Ramsey Equilibrium.- G. Sorger: On the Structure of Ramsey Equilibrium: Cycles, Indeterminacy and Sunspots.- M. Majumdar, T. Mitra: Robust Ergodic Chaos in Discounted Dynamic Optimization Models.- K. Nishimura, M. Yano: Non-Linear Dynamics and Chaos in Optimal Growth: A Constructive Exposition.- K. Nishimura, G. Sorger, M. Yano: Ergodic Chaos in Optimal Growth Models with Low Discount Rates.- G. Sorger: On the Minimum Rate of Impatience for Complicated Optimal Growth Paths.- T. Mitra: An Exact Discount Factor Restriction for Period-Three Cycles in Dynamic Optimization Models.- K. Nishimura, M. Yano: On the Least Upper Bound of Discount Factors that are Compatible with Optimal Period-Three Cycles.- T. Mitra: On the Relationship between Discounting and Entropy of Dynamic Optimization Models.

Periodic and chaotic programs of optimal intertemporal allocation in an aggregative model with wealth effects

SummaryWe examine a discrete-time aggregative model of discounted dynamic optimization where the felicity function depends on both consumption and capital stock. The need for studying such models has been stressed in the theory of optimal growth and also in the economics of natural resources. We identify conditions under which the optimal program is monotone. In our framework, the optimal program can exhibit cyclic behavior for all discount factors close to one. We also present an example to show that our model can exhibit optimal behavior which is chaotic in both topological and ergodic senses.

Dynamic Optimization Under Uncertainty: Non-convex Feasible Set

An editorial note in the Economic Journal (May 1930) reported the death of Frank Ramsey, and his 1928 paper was described as ‘one of the most remarkable contributions to mathematical economics ever made’. In the same issue the editor organized a symposium on increasing returns and the representative firm. This symposium seems to be a natural follow-up of a number of papers published by the Journal during 1926–8, including the well-known article of Allyn Young (1928) that is still available, and duly remembered. The problems of equilibrium of a firm under increasing returns, or more generally, of designing price-guided resource allocation processes to cope with increasing returns, has since been a topic of continuing interest. Ramsey’s contribution was enshrined as a durable piece with a resurgence of interest in intertemporal economics in the fifties. But neither John Keynes, the editor of the Economic Journal who was most appreciative of Ramsey’s talents, neither the subsequent writers on ‘growth theory’ in Cambridge, England (nor, for that matter, those in Cambridge, Massachusetts), have made any precise suggestion towards incorporating increasing returns in a Ramsey-type exercise.

On a Class of Stable Random Dynamical Systems: Theory and Applications

Abstract We consider a random dynamical system in which the state space is an interval, and possible laws of motion are monotone functions. It is shown that if the Markov process generated by this system satisfies a splitting condition, it converges to a unique invariant distribution exponentially fast in the Kolmogorov distance. A central limit theorem on the time-averages of observed values of the states is also proved. As an application we consider a system that captures an interaction of growth and cyclical forces: of two possible laws, one is monotone, but the other is unimodal with two periodic points. Journal of Economic Literature Classification Numbers: C6, D9.

A note on controlling a chaotic tatonnement

Abstract The paper studies the possibility of ‘targeting’ or ‘controlling’ a chaotic tatonnement by suitable perturbations of the law of motion. The analysis is in the context of a parametric class of exchange economies which are shown to constitute the logistic family of dynamical systems under the tatonnement process. Computer simulations suggest that the control method is effective in attaining neighborhoods of competitive equilibria for many members of this class of economies in a decentralized manner.

Controlled semi-Markov models, the discounted case

Abstract Let the state space S be a Borel subset of a complete separable metric space, the action space A compact metric. Existence of stationary optimal policies is proved for general semi-Markov models with possibly unbounded rewards. The corresponding dynamic programming equations are also derived. The paper presents a synthesis and extensions of earlier results.

Optimal growth in a stochastic environment: Some sensitivity and turnpike results

Abstract We present an aggregative model of dynamic optimization under uncertainty and seek to synthesize and extend results that apply to a number of well-studied stochastic optimization exercises. First, the question of existence of a unique finite horizon optimal program is considered. Optimality is characterized in terms of Ramsey-Euler conditions. Sensitivity of the optimal programs with respect to changes in initial and terminal stocks is explored. As the horizon expands, the optimal programs converge to a unique limit program, which is linked to the infinite horizon optimal program. A turnpike property of optimal programs is also derived.

Dynamic Optimization with a Non-Convex Technology: The Case of a Linear Objective Function

The paper studies the problem of optimal intertemporal allocation in an aggregative model with a non-convex technology set and a discounted sum of consumptions as the objective function. The study demonstrates the existence of a threshold initial stock such that the long-run behaviour of optimal programmes depends critically on whether the initial stock is, above or below the threshold. This is in contrast with the standard turnpike theory of convex models in which the long-run behaviour of optimal programmes is independent of the initial stock.

Random Dynamical Systems

This treatment provides an exposition of discrete time dynamic processes evolving over an infinite horizon. Chapter 1 reviews some mathematical results from the theory of deterministic dynamical systems, with particular emphasis on applications to economics. The theory of irreducible Markov processes, especially Markov chains, is surveyed in Chapter 2. Equilibrium and long run stability of a dynamical system in which the law of motion is subject to random perturbations is the central theme of Chapters 3-5. A unified account of relatively recent results, exploiting splitting and contractions, that have found applications in many contexts is presented in detail. Chapter 6 explains how a random dynamical system may emerge from a class of dynamic programming problems. With examples and exercises, readers are guided from basic theory to the frontier of applied mathematical research.

SYMMETRIC STOCHASTIC GAMES OF RESOURCE EXTRACTION: THE EXISTENCE OF NON- RANDOMIZED STATIONARY EQUILIBRIUM

A two-person discounted stochastic game (see, e.g., [15] and [17] for related references) is described by a tuple S,A 1 (s),A 2(s),q,r 1,r 2,β having the following interpretation: S, a non-empty Borel subset of a Polish space, is the set of all states of the system; A i (s), a non-empty Borel subset of a Polish space, is the set of actions available to player i(= 1,2), when the state is s ∈ S. It is typically assumed that for each i = 1,2, A i (s) ⊂ A i for all s ∈ S, where the A i ’s are themselves Borel subsets of Polish spaces. q defines the law of motion of the system by associating (Borel-measurably) with each triple (s, a 1, a 2 ) ∈ S x A 1 x A 2 a probability measure q[|s, a 1, a 2 ) on the Borel subsets of S. r 1 and r 2 are bounded measurable functions on S x A 1 xA 2; the function r i is the instantaneous reward function for player i. Lastly, β is the discount factor the players employ. Periodically, the players observe a state s ∈ S and pick actions a i ∈ A i (s), i = 1,2