Generative Engines (GEs) such as ChatGPT and Google's AI Overviews are rapidly reshaping search economics by delivering synthesized responses that allow users to bypass third-party websites, cutting those sites' advertising revenue. Yet this shift also leaves GEs facing their own monetization problem: whether to insert ads into synthesized responses or keep them ad-free to drive subscription conversions. In this paper, we introduce a dynamic framework to study this problem, which captures how query-level design choices shape user engagement, retention, and subscription conversion over time. Using this framework, we show that the optimal policy follows a cutoff rule: ads should only be shown to users only when the immediate ad payoff exceeds the long-term value of providing ad-free responses. This cutoff shifts toward with-ad responses when i) ad revenue is high or ii) users are less sensitive to ads, and toward ad-free responses when iii) subscription conversion becomes relatively more valuable. In addition, the presence of rival GEs shifts the optimal policy further toward ad-free responses, as ad-heavy monetization becomes less sustainable when users can freely switch to alternatives. Our findings reveal incentives for real-life generative engine providers to adopt designs that enhance user experience and long-term sustainability.

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