On Duopolistic Competition with the Gradient Adjustment Mechanism Under Constant and Decreasing Returns to Scale

This paper investigates duopoly competition under both constant and decreasing returns to scale in a market characterized by an isoelastic demand function, where firms adjust their strategies using a gradient adjustment mechanism. To establish the stability conditions of the model, we adopt different analytical approaches depending on the type of returns to scale. Under constant returns to scale, we employ a traditional approach by deriving the closed-form solution of the Nash equilibrium and analyzing the Jacobian matrix to verify whether the moduli of all eigenvalues are less than one. In contrast, under decreasing returns to scale, we analyze the local stability of the Nash equilibrium using symbolic computation methods without deriving a closed-form solution. The results show that when firms have heterogeneous costs, the model can exhibit both period-doubling and Neimark–Sacker bifurcations under both types of returns to scale. However, when costs are homogeneous, only period-doubling bifurcations occur. Numerical simulations support these analytical results and further demonstrate the emergence of complex dynamics, including chaotic behavior.