Dynamic Model of Innovation: Optimal Investment, Optimal Timing, Market Competition

A dynamic model of investment process for a technology innovator in a market environment is designed. The "light" dynamics of the active innovator is described by the system of exponential trajectories in which one can quickly change growth parameters. It is assumed that the innovator operates in the inert market environment which can be presented by "heavy" exponential trajectories. The growth parameters of the market trajectories vary slowly and can be identified to some accuracy in the dynamic process of econometric observation basing on information of the current technology stock (the average market technology stock) and its rate (the average market technology rate). The model consists of three decision making levels for dynamical identification, optimization of the commercialization time and optimal control design. On the first level the innovator makes assessment for the current commercialization time using econometric characteristics of the current level of the market technology stock and the market technology rate. Since the market environment is inert and its acceleration (the second derivative) is small then information about the market technology stock (current position) and the market technology rate (current first derivative) gives an opportunity to estimate exponential parameters of the market growth trajectories, to forecast the market commercialization time and indicate its sensitivity. On the second level the innovator optimizes its commercialization time basing on its own current technology stock and taking into account the forecast of the market commercialization time. Two scenarios are possible for the innovator: the "slow" scenario with "large" time of innovation is oriented on the local extremum with usual level of sales of invented products, the "fast" scenario with "small" time of innovation maximizes the early level of innovation with bonus sales due to the market overtaking. On the third level the innovator solves an optimization problem for the investment policy basing on information about the chosen scenario, the commercialization time, and the difference between the achieved technology stock and the demanded technology stock for starting commercialization. Dynamical optimality principles for optimizing discounted innovation costs on investment trajectories are used for finding the optimal investment plan and designing optimal feedback for its realization. Properties of sensitivity and robustness are investigated for the optimal profit result and innovation feedbacks.