Changing market structures in intertemporal optimization

Optimal control (unilateral decision maker) and differential game theory (several decision makers) are established tools to analyze dynamic processes in economics and related fields. Both theories rely on the assumption that the model framework (i.e. number of players, form of collaboration between the players, etc.) does not change over time. However, many real phenomena demonstrate the opposite, e.g. 1. Monopolistic market changes to competition (and vice versa) 2. Entry/exit of competitors to/from the market 3. Begin/end of a collaboration of players. While such behaviour is extensively studied in static and two period models (see Tirole 1988 for an overview), this has not been done in continuous time. We plan to work on a systematic analysis of models of this type. For that reason we have to link an optimal control model and a differential game (example a), two differential games with a different number of players (example b) and a non-cooperative differential game with a differential game with cooperating players (example c). The models will be analyzed with various assumptions concerning the switching time (exogenous, endogenous, stochastic), the commitment structure (open-loop and feedback) and the initial states (e.g. different initial capital stocks of new market entrants). Since linking models in the above proposed way has not been done so far (except very few examples), we have to deal with new mathematical challenges. With the analysis of such models it is possible to learn the optimal behaviour before the change in the market structure. Comparing the results to the situation without switch, provides important economic insights that go beyond the knowledge from static and two-period models. It is possible to deal with questions like, weather it is optimal to increase (decrease) the capital stock before competition gets tougher (weaker)? How should a firm influence the competitors decisions already before the switching time? Should a firm influence the switching time or is this too costly?

Optimal control models and differential games are standard methods of Operations Research, Economy and related areas. They deal with optimal decision processes of one or several decision makers over time. The change of the state of the system is described by a system of differential equations. In reality it sometimes happens that the number of decision makers changes (e.g. market entry or exit of a firm), which is not included by the classical methods. The objective of the finished project was to investigate this feature. On the one hand it was necessary to connect the existing methods (optimal control and differential games) for theoretical models. The crucial question, which we discussed, was the change in the optimal behaviour due to a change of the decision framework (in our case the change of the number of decision makers). We demonstrate that this is one of the crucial points in the choice of the optimal policy. An important topic treated in this project was the optimal behaviour of a monopolist under the threat of the entrance of a potential competitor. We looked at different cases, which implied different optimal behaviour. I.e. the development of the optimal capital stock before the potential competitor enters the market has opposite directions depending on the information structure (open-loop, feedback). Furthermore we discuss under which condition the monopolist is able to deter the potential competitor from an entrance, and under which conditions this is optimal. Also here a different behaviour depending on the information structure is optimal for the monopolist. A further important topic discussed the question if a software developing firm should switch to an open-source product policy. With the closed-source product policy, the firm gets profits from selling the software and complimentary products, but does not get development impact from the software users. If the firm switches to the open-source product policy, there is no profit from selling the software, but the software can be easier improved because of the impact from the software users. Moreover, a competitor can also offer complementary products for the open-source software. The analysis shows that different optimal behaviour of the software firm and the potential competitor depending on the model parameters. An analysis of this type would not be possible by using the standard methods.