Optimal control over a continuous range of homogeneous and heterogeneous innovations with finite life-cycles.

The paper presents a fully dynamic framework for simultaneous analysis of variety enhancing and quality improving innovations for the case of a single agent. There is a continuous stream of variety enhancing innovations and every new product from this stream has its own quality dimension which also may be developed. In this framework the limited life-cycles of all these new products are assumed. Then the question of how longer or shorter life-cycles may influence the innovative activity of both types is analyzed. It is demonstrated, that in the case of homogeneous products the longer is the life-cycle the more intense are innovations of both types, while in heterogeneous case the effect is more complicated. At initial stage of industry development longer life-cycles (aka patents) stimulate innovative activity, while in mature stage of development they do not. This gives foundation for limited nature of optimal length of patents solely from the considerations of the type and nature of the industry itself, without usual reference to the competition effects. The model is developed using distributed optimal control technique.