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G. Feichtinger:
"Multiple equilibria and Skiba sets in optimal control models";
Vortrag: wiss. Vorrtrag, Singapore (eingeladen); 31.03.2008.

Multiple equilibria and history dependence are important features of solutions in intertemporal optimization models. Multiplicity means, given the initial state of a problem, there exist multiple optimal solutions, i.e. the decision maker is indifferent which to choose. Referring to a threshold property, history dependence turns the attention to the fact that the optimal solution depends on its time history, i.e. starting position.
First a short historical introduction is given explaining the denotation DNSS sets for points of indifference dating from the work of Dechert, Nishimura, Sethi and Skiba. Then, some general results on infinite time, discounted, autonomous optimal control problems are presented. Here the possibility of multi-valued optimal vector fields is crucial. The concepts are exemplified by a paradigmatic one-state model. In particular the optimal costate rule and the existence of indifference points are illustrated. The analysis is visualized by means of bifurcation diagrams and phase portraits. Afterwards general definitions of indifference points and threshold points are given for the case of a finite number of solutions. Strong and weak DNSS points as well as the role of foci and unstable nodes are discussed.
The second part of the lecture provides economic examples of two-state dynamic optimization models exhibiting DNNS curves. Among the peculiar results is the non-uniqueness optimal policy for non-concave models.