Asymptotically Tight Bounds on the Optimal Pricing Strategy with Patient Customers.

This article focuses on deriving asymptotically tight bounds for the optimal pricing strategy of a monopolist seller facing both patient and impatient customers, particularly when patient customers exhibit high patience levels. It establishes that the optimal pricing strategy is cyclic with a cycle period growing asymptotically as \( k + O(k^{1/4}) \), where \( k \) denotes the patience level, thereby resolving an open question about the cycle period posed in prior research. The study also shows that the optimal long-run average revenue approaches an upper bound \( R \) with a gap on the order of \( O(1/\sqrt{k}) \). These results are extended to settings with multiple patience levels, and numerical experiments confirm the theoretical findings and illustrate the pricing patterns. The work assumes continuous pricing flexibility and unlimited inventory, providing insights relevant to dynamic pricing and revenue management in monopolistic markets with heterogeneous customer patience.