How to make pricing strategy for the influencer’s self-operated products considering the content switching level?

We analyse a supply chain with a manufacturer and an influencer selling products by live streaming. The influencer can introduce self-operated products and choose between competitive pricing strategy (Scenario CM) and sequential pricing strategy (Scenario SQ). The content switching level and influencer power impact the demand, which are crucial factors influencing decisions. Additionally, we consider the long-term effect, where live streaming slices or product recommendation logos continue to drive sales after live streaming. We find that in Scenario CM, the manufacturer's optimal production quantity initially declines and then rises as the long-term effect, while in Scenario SQ, it consistently increases. The influencer's optimal profit initially declines and then rises as the long-term effect in both scenarios. When the influencer adopts Scenario SQ, introducing self-operated products always benefits her. However, in Scenario CM, the decision depends on the long-term effect. The manufacturer's and influencer's preferred pricing strategies vary based on the long-term effect. If the long-term effect is high (low), Scenario SQ is preferred to avoid direct competition, while if moderate, Scenario CM is preferred. This paper is the first examining pricing strategies for influencer-operated products, offering valuable insights on when to incorporate self-operated products into the live streaming.