Dynamics and implications on a cooperative advertising model in the supply chain

In this paper, the theory of nonlinear dynamic system is applied to a cooperative advertising model in the supply chain, where the decision-making mechanism of firms is modeled as a dynamic adjustment process and the advertising response is assumed to be an exponential function. We find four possible Nash equilibrium points in this dynamic system and provide the conditions for their existence. After that, we analyze the impacts of key parameters on the stability of the positive Nash equilibrium point and prove that two kinds of bifurcation may occur when this positive Nash equilibrium point becomes unstable. Finally, sensitivity analysis and robustness test are performed to obtain some managerial insights. This study provides some valuable insights both into the cooperative advertising programs and into the features of dynamic game with exponential demand function.