Abstract
In this paper we do a comprehensive analysis of the model of oligopoly with sticky prices with full analysis of behaviour of prices outside its steady state level in the infinite horizon case. An exhaustive proof of optimality is presented in both open loop and closed loop cases.
Highlights
Dynamic oligopoly models have a long history, starting from Clemhout et al [12] and encompassing many issues, including, among other things, advertisement (e.g. Cellini and Lambertini [8]), adjustment costs (e.g. Kamp and Perloff [24], Jun and Viwes [23]), goodwill (e.g. Benchekroun [7]), pricing, hierarchical structures (e.g. Chutani and Sethi [11]), nonstandard demand structure (e.g. Wiszniewska-Matyszkiel [29] with demand derived from dynamic optimization of consumers at a specific market) or a combination ofA
In the open-loop case, applying an appropriate form of Pontryagin maximum principle, is nontrivial, even using the latest findings in infinite horizon optimal control theory
As we prove, applying these necessary conditions to the optimal control arising from calculation of the open-loop Nash equilibrium, and given the initial price, restricts the set of possible solutions to a singleton, so it is enough to check that the optimal solution exists to prove sufficiency of the condition
Summary
Dynamic oligopoly models have a long history, starting from Clemhout et al [12] and encompassing many issues, including, among other things, advertisement (e.g. Cellini and Lambertini [8]), adjustment costs (e.g. Kamp and Perloff [24], Jun and Viwes [23]), goodwill (e.g. Benchekroun [7]), pricing (see e.g. Jørgensen [22]), hierarchical structures (e.g. Chutani and Sethi [11]), nonstandard demand structure (e.g. Wiszniewska-Matyszkiel [29] with demand derived from dynamic optimization of consumers at a specific market) or a combination of. Comprehensive reviews of differential games modelling oligopolies including models with sticky prices can be found in, among others, Dockner et al [13] and Esfahani [18] Both open loop and feedback information structures in the infinite horizon case were considered in [19] and [9]. In the open-loop case, applying an appropriate form of Pontryagin maximum principle (as it is well known, the standard maximum principle does not have to be fulfilled in the infinite horizon case), is nontrivial, even using the latest findings in infinite horizon optimal control theory To address these two issues, in this paper we concentrate on the off-steady-state analysis of the model, both in the open loop and the feedback information structure cases, and we give a rigorous proof, including applicability of a generalisation of a Pontryagin maximum principle and checking terminal conditions in both. When the feedback case is considered, we use the standard Bellman equation stated in e.g. Zabczyk [30], since the value function is proved to be smooth.
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