Model predictive control, the economy, and the issue of global warming

This study is motivated by the evidence of global warming, which is caused by human activity but affects the efficiency of the economy. We employ the integrated assessment Nordhaus DICE-2007 model [16]. Generally speaking, the framework is that of dynamic optimization of the discounted inter-temporal utility of consumption, taking into account the economic and the environmental dynamics. The main novelty is that several reasonable types of behavior (policy) of the economic agents, which may be non-optimal from the point of view of the global performance but are reasonable form an individual point of view and exist in reality, are strictly defined and analyzed. These include the concepts of “business as usual”, in which an economic agent ignores her impact on the climate change (although adapting to it), and of “free riding with a perfect foresight”, where some economic agents optimize in an adaptive way their individual performance expecting that the others would perform in a collectively optimal way. These policies are defined in a formal and unified way modifying ideas from the so-called “model predictive control”. The introduced concepts are relevant to many other problems of dynamic optimization, especially in the context of resource economics. However, the numerical analysis in this paper is devoted to the evolution of the world economy and the average temperature in the next 150 years, depending on different scenarios for the behavior of the economic agents. In particular, the results show that the “business as usual”, although adaptive to the change of the atmospheric temperature, may lead within 150 years to increase of temperature by 2°C more than the collectively optimal policy.environmental economics, dynamic optimization, optimal control, global warming, model predictive control, integrated assessment

Age-structured optimal control in population economics

This paper brings both intertemporal and age-dependent features to a theory of population policy at the macro-level. A Lotkatype renewal model of population dynamics is combined with a Solow/Ramsey economy. By using a new maximum principle for distributed parameter control we derive meaningful qualitative results for the optimal migration path and the optimal saving rate.

Control Systems with Constraints and Uncertain Initial Conditions

AMS subject classification: 49N55, 93B52, 93C15, 93C10, 26E25.We study the problem of finding a control such that all solutions of a control systems, starting from a given set of initial conditions, satisfy a given constraint. This problem is an extension of the well-known Viability Problem when the initial condition is a set. The present paper is mainly a survey of results recently obtained by the authors, but some new results with proofs are also included

Workshop on "Age-structured models in population dynamics and economics at the Vienna Institute of Demography, October 27-28, 2003

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Markets for emission permits with free endowment: a vintage capital analysis

In this paper we develop a vintage capital model for a firm involved in a market for tradable emission permits. We analyze both the firm’s optimal investment plans and the market equilibrium. This allows us to scrutinize how firms use permits free endowment, and to highlight the implications of non-optimal uses both at the firm and at the market level. We provide a new rationale for the market of tradable permits not to be cost-efficient. The novel technical points in this context are the use a distributed (vintage) optimal control model of the firm, the use of optimality conditions for non-smooth problems, and the involvement of a nonlinear Fredholm integral equation of the first kind for the description of the equilibrium price of permits, and its practical meaning for market regularization.

Keeping a learned society young

Aging organizations - regardless of whether they are firms, faculties, societies, political bodies, teams, or national academies - seek ways to rejuvenate. This paper demonstrates that the best way to keep an organization young is through a mixed strategy of recruiting both young and old, and that, contrary to intuition, recruiting those of middle age is the least effective strategy for maintaining a younger age structure. The aging of learned societies is a problem in many national academies. Faced with rising life expectancy, particularly for older persons, the average age of academy members is increasing. Another reason for "overaging" is an increase in the age at election. In an organization with a fixed size, the annual intake is strictly determined by the number of deaths and the statutory retirement age. This can, among many learned societies, lead to a fundamental dilemma: the desire to maintain a young age structure, while still guaranteeing a high recruitment rate. We derive an optimal recruitment policy which is bimodal, i.e., it entails shifting recruitment partly to younger ages and partly to older ages, while decreasing the recruitment of middle-aged candidates. Although the optimization problem explicitly involves only the average age and the recruitment rate as objectives, the methodology implicitly allows us to take into consideration all other objectives (formal or informal) used in the actual election practice.age-specific recruitment policy, fixed-size population, optimal control

Adaptive Model-Predictive Climate Policies in a Multi-Country Setting

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2012.29 - ISSN : 1955-611XThe purpose of this paper is to extend the use of integrated assessment models by defining rational policies based on predictive control and adaptive behavior. The paper begins with an review of the main IAMs and their use. Then the concept of Model Predictive Nash Equilibrium (MPNE) is introduced within a general model involving heterogeneous economic agents operating in (and interfering with) a common environment. This concept captures the fact that agents do not have a perfect foresight for several ingredients of the model, including that of the environment. A version of the canonical IAM (DICE) is developed as a benchmark case. The concept of MPNE is then enhanced with adaptive learning about the environmental dynamics and the damages caused by global warming. The approach is illustrated by some numerical experiments in a two-region setting for several scenarios.L'objectif de cet article est d'inclure dans un modèle d'estimation intégrée (IAM) des politiques rationnelles basées sur les théories de contrôle optimal et de comportement adaptatif. L'article commence avec une revue des IAMs les plus importantes et leur utilisation dans la littérature. Par la suite, nous introduisons le concept d'Equilibre de Nash dans le cas d'un Modèle Prédictif (MPNE) et l'intégrons à un modèle général comprenant des agents économiques hétérogènes qui agissent (et qui interfèrent) dans un même milieu. Ce concept reprend des agents qui n'ont pas de "perfect foresight" par rapport à différents ingrédients du modèle, y compris l'environnement. Une version du DICE, le modèle IAM canonique, est développé comme modèle cadre. Le concept de MPNE est alors amélioré à travers un processus d'apprentissage adaptif concernant la dynamique de l'environnement et les dommages induits par le changement climatique. Notre approche est illustrée au moyen de plusieurs simulations numériques dans un cadre à deux régions

Adaptive model-predictive climate policies in a multi-country

The purpose of this paper is to extend the use of integrated assessment models by defining rational policies based on predictive control and adaptive behavior. The paper begins with an review of the main IAMs and their use. Then the concept of Model Predictive Nash Equilibrium (MPNE) is introduced within a general model involving heterogeneous economic agents operating in (and interfering with) a common environment. This concept captures the fact that agents do not have a perfect foresight for several ingredients of the model, including that of the environment. A version of the canonical IAM (DICE) is developed as a benchmark case. The concept of MPNE is then enhanced with adaptive learning about the environmental dynamics and the damages caused by global warming. The approach is illustrated by some numerical experiments in a two-region setting for several scenarios

Solution stability of parabolic optimal control problems with fixed state-distribution of the controls

The paper presents results about strong metric subregularity of the optimality mapping associated with the system of first-order necessary optimality conditions for a problem of optimal control of a semilinear parabolic equation. The control has a predefined spatial distribution and only the magnitude at any time is a subject of choice. The obtained conditions for subregularity imply, in particular, sufficient optimality conditions that extend the known ones. The paper is complementary to a companion one by the same authors, in which a distributed control is considered.Comment: arXiv admin note: substantial text overlap with arXiv:2209.0892

On the solution stability of parabolic optimal control problems

The paper investigates stability properties of solutions of optimal control problems for semilinear parabolic partial differential equations. H\"older or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may non-linearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper -- the proved metric subregularity of the mapping associated with the system of first-order necessary optimality conditions. This property also enables error estimates for approximation methods. Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product