Theory of Stochastic Optimal Economic Growth

This paper is a survey of the theory of stochastic optimal economic growth.International Development,

Idempotent structures in optimization

Consider the set A = R ∪ {+∞} with the binary operations o1 = max and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries in A. Let the generalised sum u o1 v of two vectors denote the vector with entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector v ∈ An denote the vector with the entries a o2 vj . With these operations, the set An provides the simplest example of an idempotent semimodule. The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation. Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces. We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics

Control of singularly perturbed hybrid stochastic systems

In this paper, we study a class of optimal stochastic control problems involving two different time scales. The fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process. Under a fundamental “ergodicity” property for a class of “infinitesimal control systems” associated with the fast mode, we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system. Both the finite- and infinite-discounted horizon cases are considered. We show how an approximate optimal control law can be constructed from the solution of the limit control problem. In the particular case where the infinitesimal control systems possess the so-called turnpike property, i.e., are characterized by the existence of global attractors, the limit control problem can be given an interpretation related to a decomposition approach

Using nonlinear model predictive control for dynamic decision problems in economics

Gruene L, Semmler W, Stieler M. Using nonlinear model predictive control for dynamic decision problems in economics. Journal of Economic Dynamics and Control. 2015;60:112-133.This paper presents a new approach to solve dynamic decision models in economics. The proposed procedure, called Nonlinear Model Predictive Control (NMPC), relies on the iterative solution of optimal control problems on finite time horizons and is well established in engineering applications for stabilization and tracking problems. Only quite recently, extensions to more general optimal control problems including those appearing in economic applications have been investigated. Like Dynamic Programming (DP), NMPC does not rely on linearization techniques but uses the full nonlinear model and in this sense provides a global solution to the problem. However, unlike DP, NMPC only computes one optimal trajectory at a time, thus avoids to grid the state space and for this reason the computational demand grows much more moderately with the space dimension than for DP. In this paper we explain the basic idea of NMPC, give a proof concerning the accuracy of NMPC for discounted optimal control problems, present implementational details, and demonstrate the ability of NMPC to solve dynamic decision problems in economics by solving low and high dimensional examples, including models with multiple equilibria, tracking and stochastic problems. (C) 2015 Elsevier B.V. All rights reserved

Discrete-continuous analysis of optimal equipment replacement

In Operations Research, the equipment replacement process is usually modeled in discrete time. The optimal replacement strategies are found from discrete (or integer) programming problems, well known for their analytic and computational complexity. An alternative approach is represented by continuous-time vintage capital models that explicitly involve the equipment lifetime and are described by nonlinear integral equations. Then the optimal replacement is determined via the optimal control of such equations. These two alternative techniques describe essentially the same controlled dynamic process. We introduce and analyze a model that unites both approaches. The obtained results allow us to explore such important effects in optimal asset replacement as the transition and long-term dynamics, clustering and splitting of replaced assets, and the impact of improving technology and discounting. In particular, we demonstrate that the cluster splitting is possible in our replacement model with given demand in the case of an increasinTheoretical findings are illustrated with numeric examples.vintage capital models, optimization, equipment lifetime, discrete-continuous models.

Measuring intertemporal substitution: The role of durable goods

As pointed out by Hall (1988), intertemporal substitution by consumers is a central element of many modern macroeconomic and international models. For example, many of the policy implications of an endogenous growth model studied by Barro (1990) depends on the assumption that the intertemporal elasticity of substitution is positive. In estimating the intertemporal elasticity of substitution (IES), however, Hall (1988) fmds that when time aggregation is taken into account, his point estimates are small and not significantly different from zero. Hall concludes that ti:e elasticity is unlikely to be much above 0.1 and may well be zero. We argue that Hall's estimator for the IES is downward biased because the intra-temporal substitution between nondurable consumption goods and durable consumption goods is ignored and because the changes in real interest rates affect user costs of durable goods. We use a two-step procedure that combines a cointegration approach to preference parameter estimation with Hansen and Singleton's (1982) Generalized Method of Moments approach in order to take these effects into account. In contrast to Hall's result, our estimates for the IES are positive and significantly different from zero even when time aggregation is taken into account.consumption durable goods real interest rates saving