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A feedback Nash equilibrium solution for non-cooperative innovations in a stochastic differential game framework. (English) Zbl 0733.90088

The paper analyzes an N-firm stochastic differential game of technological innovation. The firms are identical and each firm chooses its rate of investment in innovation research. The state of the dynamical system is represented by a single, abstract variable “the state of technology” facing each firm. The evolution equation is a stochastic differential equation: increasing innovation efforts increase the state of technology but, in the absence of such efforts the state decays exponentially. The state is finally influenced by a random term (a Wiener process). The objective of a firm is to maximize the expected present value of profits being comprised of revenues (price \(\times\) production) minus linear costs of variable inputs and convex innovation costs. The game is played on a fixed and finite interval of time.
The solution follows from a two-step procedure. Invoking a Cournot-type assumption each firm determines its choice of productive inputs as a static optimization problem, taking the production decisions of other firms as fixed. In the next step each firm selects an innovation investment strategy to maximize present value profits subject to the dynamical equation and the Cournot input choices. The investment strategies are sought as feedback strategies. Due to the assumptions the Hamilton-Jacobi-Bellman equations can be explicitly solved and the feedback strategies determined. To illustrate the results obtained it turns out that a firm’s research intensity decreases with the level of the “state of technology”, the discount rate and the number of firms in the industry.

MSC:

91A23 Differential games (aspects of game theory)
91B62 Economic growth models
91A60 Probabilistic games; gambling
91A15 Stochastic games, stochastic differential games
91B24 Microeconomic theory (price theory and economic markets)
91A80 Applications of game theory
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