In this work the dynamical framework which combines different aspects of innovative activity is analyzed. First the basic model with finite time horizon is constructed where the single agent (planner) is optimizing his stream of investments into the process of creation of new products together with investments into the improvement of already invented products. The range of products which might be invented is given by the bounded real interval. Agent may develop quality of only those products which are already invented for any given time t. The role of heterogeneity of the investment characteristics of these new products is analyzed and it is demonstrated that this heterogeneity plays the essential role in the dynamics of the model. That is the considered special case of the model with homogeneous investment characteristics of products do not demonstrate any important dynamic link between two different types of innovations which are considered. Due to the linear-quadratic costs of investments rather then linear ones the investments into the development of quality of all invented products stop only at terminal time while leaving room for the gradual decline of the achieved quality level near the terminal time of the model. The basic model reacts in a rather natural way for changes in the exogenous parameters values and the dynamics replicates the empirical findings concerning the process of innovative activity.

In the second part of the thesis the analysis is extended to account for long-run behavior of the planner on the infinite-time horizon. This gives significant simplification to the basic model and allows to decompose the entire dynamical problem in two consecutive parts: growth of quality problem and variety expansion problems. The explicit formulation of optimal investment rules is given. Optimal investments into quality growth for every invented product is constant in time but differ from product to product, while investments into variety expansion follow the linear feedback rule and depend negatively on the level of variety already achieved. Variety expansion in infinite-time time horizon is faster then for the limited time basic model. Quality increases towards the maximal attainable quality level (different for each product) and then remains constant on this steady-state level. This also distinguishes the infinite-time extension from the basic model.

Two following chapters of the thesis deal with two different extensions of the basic model. Possibilities for further analysis of the given approach and the difference in conclusions and policy implications with earlier approaches to innovations' analysis are demonstrated. First the effect of patenting policy on the innovative activity is taken into account. Every product upon it’s invention is supposed to be finitely-living during some constant time which is then interpreted as the length of the patent granted for the products upon its invention. No costs of preserving the patent during its length are assumed. After the expiration of the patent the quality developed becomes common knowledge and the agent has no longer incentives for the future development of quality of this product. The diversity of possible outcomes with respect to the patent's length is demonstrated and it is argued that this effect may not be captured without the presence of heterogeneity of innovative products under analysis. In particular under the heterogeneity of products’ characteristics there are two opposite effects of the change in the patent’s length onto the variety expansion process, and this weakens the growth rates of variety. The previously considered infinite-time horizon extension is the limiting version of the dynamics of this model and demonstrates faster growth of both types of innovations. It is claimed that with longer length of the patent more resources are allocated in relative measure to the development of qualities of already invented products rather than to the variety expansion process and this establishes ground for the finite and limited length of patents completely from the technological point of view without any market mechanisms.

In the last chapter the extension which introduces several innovating agents into the framework is considered. There the subsequent optimal control problem is transformed into the differential game in infinite-dimensional space. The case of dynamic duopoly includes the possibility for any of the players to become a costless imitator in the process of the development of qualities while both players participate in the process of variety expansion. The resulting dynamic problem is decomposed in the set of quality growth games (the infinite-dimensional one) and the variety expansion game. The set of piecewise-constant strategies for every quality growth game is derived and it is shown to be the only one stable set in the class of at most linear feedback strategies. Every such set leads to the possibility of two symmetric outcomes (with zero imitation speed and non-zero imitation speed) to the situation of one player being the constant leader in quality growth while the other is imitating him and to the possibility of catching-up. This last happens when both players have sufficiently close to each other technological characteristics and thus depends on the technological distance. The set of conditions for this catching-up is derived and optimal strategies of players in all 3 possible cases are characterized. Variety expansion game demonstrates that very is a specialization of activities between players, with one of them investing more in quality growth for each product and the other one investing more into the variety expansion. It is demonstrated that this specialization of activities has a foundation in internal characteristics of these agents: the agent who is more effective in quality development is selected as a leader in quality growth games while the other player invests more into variety while being the imitator in quality of all invented products.

The suggested work provides several prospects for further enrichment and development in all the areas being considered.