Abstract
This paper presents a numerical method for the characterization of Markov-perfect equilibria of symmetric differential games exhibiting coexisting stable steady-states. The method relying on the calculation of 'local value functions' through collocation in overlapping parts of the state space, is applicable for games with multiple state variables. It is applied to analyze a piecewise deterministic game capturing the dynamic competition between two oligopolistic firms, which are active in an established market and invest in R&D. Both R&D investment and an evolving public knowledge stock positively influence a breakthrough probability, where the breakthrough generates the option to introduce an innovative product on the market. Additionally, firms engage in activities influencing the appeal of the established and new product to consumers. Markov-perfect equilibrium profiles are numerically determined for different parameter settings and it is shown that for certain constellations the new product is introduced with probability one if the initial strength of the established market is below a threshold, which depends on the initial level of public knowledge. In case the initial strength of the established market is above this threshold, the R&D effort of both firms quickly goes to zero and with a high probability the new product is never introduced. Furthermore, it is shown that after the introduction of the new product the innovator engages in activities weakining the established market, although it is still producing positive quantities of that product.
Highlights
Since the seminal contributions of Sethi [19], Skiba [20] and Dechert and Nishimura [9], it has been shown that rational planning over an infinite planning horizon can go along with outcomes that crucially depend on initial conditions
The most important contribution of this paper is technical in the sense that, to our belief, this is the first paper generating a history-dependent solution in the setting of a Markov-perfect equilibrium of a differential game with more than one state variable
In a model with two state variables, a Skiba curve separates the basins of attraction of the different locally stable steady states
Summary
Since the seminal contributions of Sethi [19], Skiba [20] and Dechert and Nishimura [9], it has been shown that rational planning over an infinite planning horizon can go along with outcomes that crucially depend on initial conditions. Such outcomes are denoted as history-dependent solutions, and the basins of attraction of the different long-run equilibria are separated by Skiba thresholds. These methods rely on the numerical calculation of stable paths in the state-costate system derived from the Maximum Principle [13,15], nonlinear model predictive control [14], or collocation [4]
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