Price-taking behavior versus continuous dynamic optimizing

In a quantity-competition oligopoly, previous studies have shown that a price-taking firm can outperform any rival with identical technology that produces at a different output level at any intertemporal equilibrium. This research seeks to examine this seemingly counter-intuitive fact in a heterogeneous duopoly consisting of an adaptive price-taker and a dynamic optimizer who dynamically optimizes its either absolute or relative profit stream over an infinite planning horizon. With conventional economic assumptions, the Hamiltonian system always converges to a saddle point. If goal of the dynamic optimizer is absolute profit, the price-taker produces more than the dynamic optimizer at the equilibrium, around which there exists a long-run relative profitability portion for the price-taker. Consequently, the price-taker unconsciously makes higher relative profit than the dynamic optimizer in the long-run. In contrast, if the dynamic optimizer pursues the relative profit, both firms produce at the competitive level at the equilibrium while the whole optimal path lies in the relative profitability regime of the dynamic optimizer so that it will enjoy a higher profit than the price-taker before reaching equilibrium. These results provide economical justification for ever-growing evolutionary game theoretical literatures in appreciating price-taking behavior and additionally clarify possible misunderstandings in interpreting their conclusions.