We develop a continuous-time Stackelberg game model to investigate the optimal timing of introducing a distributor into a dynamic cooperative advertising supply chain, where the timing influences the pricing and advertising decisions of each supply chain member. By defining the Hamiltonian functions for decision makers before and after the distributor's entry, we obtain the optimal entry timing, contingent on the savings in transportation costs, the distributor's advertising, and entry costs. Our study yields two noteworthy conclusions. First, despite potential profit margin losses from introducing a distributor, the manufacturer can still benefit in the long run by delaying the distributor's entry. Second, when the distributor delays entry, the manufacturer's dynamic advertising effort can be λ-shaped, that is, the manufacturer should initially accelerate the increase in her advertising investment to facilitate product diffusion. Subsequently, she can drive product diffusion by relying on the distributor's advertising efforts, enabling a gradual reduction in her advertising efforts at an increasing rate. Finally, we use a particle filtering (PF) method to estimate data from two retailers, achieving a strong level of fit. The comparisons between observed advertising expenditures and optimal values align closely with the previously discussed managerial insights.
Wang et al. (Mon,) studied this question.