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Viability theory is the study of dynamical systems that asks what set of initial conditions will generate evolutions which obey the laws of motion of a system and some state constraints, for the length of the evo- lution. We apply viability theory to Judd’s (JPE, 1987) dynamic tax model to identify which economic states today are sustainable under only slightly constrained tax-rate adjustments in the future, when the dynamic budget constraint and consumers’ transversality condition at infinity are satisfied. We call the set of such states the economic viability kernel. In broad terms, knowledge of the viability kernel can tell the planner what economic ob- jectives are achievable and assist in the choice of suitable controls to realise them. We observe, unsurprisingly, that a very high consumption economy lies outside such kernels, at least for annual tax-adjustment levels limited by 20%; higher consumption levels can only be sustained when capital is abundant. Furthermore, we notice that the sizes of the kernel slices for a given taxation level do not diminish as the tax rate rises, hence high taxation economies are not necessarily more prone to explode, or implode, than their low taxation counterparts. In fact, higher tax rates are neces- sary to keep many consumption choices viable, especially when capital approaches the constraint-set boundaries.

This paper is a successor of [AK08]. Both papers describe the same suite of MATLAB R° routines devised to provide an approximately optimal solution to an infinite horizon stochastic optimal control problem. The difference is that this paper explains how to allow for state and control constraints. The suite routines implement a policy improvement algorithm to optimise a Markov decision chain approximating the original control problem, as described in [Kra01c] and [Kra01b].

A new technology adoption problem can be modelled as a two-stage control problem, in which model parameters ("technology") might be altered at some time. An optimal solution to utility maximisation for this class of problems needs to contain information on the time, at which the change will take place (0, finite or never), along with the optimal control strategies before and after the change. For the change, or switch, to occur the "new technology" value function needs to dominate the "old technology" value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and we study when the graphs can intersect. If they do not, the switch cannot occur at a positive time. Using this characterisation we analyse a technology adoption problem and showmodels, for which the switch will occur at time zero or never.

We study a trade-off between economic and environmental indicators using a two-stage optimal control setting where the player can switch to a cleaner technology, that is environmentally ?effcient?, but economically less productive. We provide an analytical characterization of the solution paths for the case where the considered utility functions are increasing and strictly concave with respect to consumption and decreasing linearly with respect to the pollution stock. In this context, an isolated player will either immediately start using the environmentally effcient technology, or for ever continue applying the old and ?dirty? technology. In a two-player (say, two neighbor countries) dynamic game where the pollution results from a sum of two consumptions, we prove existence of a Nash (open-loop) equilibrium, in which each player chooses the technology sel?shly i.e., without considering the choice made by the other player. A Stackelberg game solution displays the same properties. Under cooperation, the country reluctant to adopt the technology as an equilibrium solution, chooses to switch to the cleaner technology provided it bene?ts from some ?transfer? from the environmentally effcient partner.

We consider a situation, in which a regulator believes that constraining a complex good created jointly by competitive agents, is socially desirable. Individual levels of outputs that generate the constrained amount of the externality can be computed as a Pareto efficient solution of the agents' joint utility maximisation problem. However, generically, a Pareto efficient solution is not an equilibrium. We suggest the regulator calculates a Nash-Rosen coupled-constraint equilibrium (or a "generalised" Nash equilibrium) and uses the coupled-constraint Lagrange multiplier to formulate a threat, under which the agents will play a decoupled Nash game. An equilibrium of this game will possibly coincide with the Pareto efficient solution. We focus on situations when the constraints are saturated and examine, under which conditions a match between an equilibrium and a Pareto solution is possible. We illustrate our findings using a model for a coordination problem, in which firms' outputs depend on each other and where the output levels are important for the regulator.

We consider a situation, in which a regulator believes that constraining a complex good created jointly by competitive agents, is socially desirable. Individual levels of outputs that generate the constrained amount of the externality can be computed as a Pareto efficient solution of the agents' joint utility maximisation problem. However, generically, a Pareto efficient solution is not an equilibrium. We suggest the regulator calculates a Nash-Rosen coupled-constraint equilibrium (or a "generalised" Nash equilibrium) and uses the coupled-constraint Lagrange multiplier to formulate a threat, under which the agents will play a decoupled Nash game. An equilibrium of this game will possibly coincide with the Pareto efficient solution. We focus on situations when the constraints are saturated and examine, under which conditions a match between an equilibrium and a Pareto solution is possible. We illustrate our findings using a model for a coordination problem, in which firms' outputs depend on each other and where the output levels are important for the regulator.

This article is a modified version of [AK06]. Both articles explain how a suite of MATLAB routines distributed under the generic name SOCSol can be used to obtain optimal solutions to continuous-time stochastic optimal control problems. The difference between the SOCSol suites described by the articles arises from the underlying computing platforms used. This article describes a beta version of SOCSol that utilises the MATLAB Parallel Computing Toolbox, while [AK06] describes a version of SOCSol that does not use parallel computing methods.

Consider a country with two regions that have developed differently so that their current levels of energy efficiency differ. Each region's production involves the emission of pollutants, on which a regulator might impose restrictions. The restrictions can be related to pollution standards that the regulator perceives as binding the whole country (e.g., enforced by international agreements like the Kyoto Protocol). We observe that the pollution standards define a common constraint upon the joint strategy space of the regions. We propose a game theoretic model with a coupled constraints equilibrium as a solution to the regulator's problem of avoiding excessive pollution. The regulator can direct the regions to implement the solution by using a political pressure, or compel them to employ it by using the coupled constraints' Lagrange multipliers as taxation coefficients. We specify a stylised model that possesses those characteristics, of the Belgian regions of Flanders and Wallonia. We analytically and numerically analyse the equilibrium regional production levels as a function of the pollution standards and of the sharing rules for the satisfaction of the constraint. For the computational results, we use NIRA, which is a piece of software designed to min-maximise the associated Nikaido-Isoda function.

We study a trade-off between economic and environmental indicators using a two-stage optimal control setting where the player can switch to a cleaner technology, that is environmentally “efficient”, but economically less productive. We provide an analytical characterization of the solution paths for the case where the considered utility functions are increasing and strictly concave with respect to consumption and decreasing linearly with respect to the pollution stock. In this context, an isolated player will either immediately start using the environmentally efficient technology, or for ever continue applying the old and “dirty” technology. In a two-player (say, two neighbor countries) dynamic game where the pollution results from a sum of two consumptions, we prove existence of a Nash (open-loop) equilibrium, in which each player chooses the technology selfish i.e., without considering the choice made by the other player. A Stackelberg game solution displays the same properties. Under cooperation, the country reluctant to adopt the technology as an equilibrium solution, chooses to switch to the cleaner technology provided it benefits from some “transfer” from the environmentally efficient partner

A two-stage control problem is one, in which model parameters (“technology”) might be changed at some time. An optimal solution to utility maximisation for this class of problems needs to thus contain information on the time, at which the change will take place (0, finite or never) as well as the optimal control strategies before and after the change. For the change, or switch, to occur the “new technology” value function needs to dominate the “old technology” value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and study when the graphs can intersect and hence whether the switch can occur. Using this characterisation we analyse a technology switching problem.