Multiproduct Dynamic Pricing with Reference Effects Under Logit Demand

Problem definition: We consider the dynamic pricing problem of multiple products under (asymmetric) reference effects over an infinite horizon. Unlike existing literature, which is mostly focused on the single-product setting, our multiproduct setting takes into account the cross-product effects among substitutes and incorporates the memory-based reference prices into the multinomial logit (MNL) demand model. Even with the single-product logit demand, the structure of the optimal pricing policy is intractable. Therefore, we focus on the long-run patterns of the optimal pricing policy and also discuss the performance of the myopic pricing policy. Methodology/results: We first provide a comprehensive characterization of the myopic pricing policy, including its solution, long-run convergence behavior, and optimality gap. For the optimal pricing policy, we show an intricate connection between its long-run dynamics and types of reference effects. We demonstrate that the presence of any gain-seeking product renders a long-run constant pricing policy suboptimal. Conversely, the constant policy (or optimal steady state) can exist in both loss-neutral and loss-averse scenarios, where we provide a sufficient condition for such existence and give the analytical expression for the optimal steady state. We further show that when pricing perfect substitutes, the true optimal policy under the multiproduct framework is more likely to yield a long-run cyclic pattern than the policy derived from the single-product framework, a phenomenon that aligns well with the periodic discounts in real-world markets. Managerial implications: This discrepancy in the long-run behaviors between multi- and single-product-based policies highlights the importance of employing the multiproduct framework and addressing the cross-product effects, as sticking to the single-product framework while managing multiple substitutes can misrepresent long-run dynamics and result in suboptimality. In the multiproduct domain, our model suggests that retailers are more likely to benefit from appropriate price variations than maintaining a constant pricing policy. Funding: H. Jiang acknowledges support from the Natural Sciences and Engineering Research Council of Canada [NSERC Discovery Grant RGPIN 2024 05796]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.0801 .