We analyze the joint inventory and pricing decisions of a firm when demand depends on not only the current selling price but also a memory-based reference price and customers are loss averse. The presence of reference price effect leads to a non-concave one-period expected revenue in price and reference price. We introduce a transformation technique that allows us to prove under some mild assumptions the optimality of a reference-price-dependent base-stock list-price policy, which is characterized by a base-stock level and a target reference price. In addition, we show that the target reference price is increasing in the reference price but except in the loss neutral case the base-stock level is not monotone in the reference price. We also show that in the steady state of the model with the reference price effect the optimal price is lower while the optimal base-stock level is higher than their counterparts in the model without the reference price effect.

We study a dynamic pricing problem of a firm facing reference price effects at an aggregate demand level, where demand is more sensitive to gains than losses. We find that even the myopic pricing strategy belongs to one type of discontinuous maps, which can exhibit complex dynamics over time. Our numerical examples show that, in general, the optimal pricing strategies may not admit any simple characterizations and the resulting reference price/price dynamics can be very complicated. We then show for a special case that a cyclic skimming pricing strategy is optimal, and we provide conditions to guarantee the optimality of high-low pricing strategies.

We analyze a finite horizon dynamic pricing model in which demand at each period depends on not only the current price but also past prices through reference prices. A unique feature but also a significant challenge in this model is the asymmetry in reference price effects which implies the underlying optimization problem is non-smooth and no standard optimization methods can be applied. We identify a few key structural properties on the problems, which enable us to develop strongly polynomial time algorithms to compute the optimal prices for several plausible scenarios. We complement our exact algorithms by proposing an approximation heuristic which also provides bounds on the optimal objective value. Finally, we demonstrate the efficiency and robustness of our algorithms by applying them to a practical problem with real data.

In this paper we formulate and analyze a novel model on a firm’s dynamic inventory and markdown decisions for perishable goods. We consider a dynamic stochastic setting, where every period consists of two phases, clearance phase and regular-sales phase. In the clearance phase, the firm decides how much to order for regular sales, as well as whether to markdown some (or all) of the leftover inventory from the previous period that will be disposed otherwise. Since strategic consumers may buy the product during clearance sales for future consumption, markdown may cannibalize future sales at regular price. Hence, the firm needs to make a trade-off between product spoilage and intertemporal demand substitution. We show that the firm should either put all of the leftover inventory on discount or dispose all of it, and the choice depends on the amount of leftover inventory from the previous period. In particular, the firm should introduce markdown when the amount of leftover inventory is higher than a certain threshold, and dispose all otherwise. We also conduct numerical studies to further characterize the optimal policy, and to evaluate the loss of efficiency under static policies when compared to the optimal dynamic policy.

This paper establishes a new preservation property of supermodularity in a class of two-dimensional parametric optimization problems, where the constraint sets may not be lattices. This property and its extensions unify several results in the literature and provide powerful tools to analyze a variety of operations models including a two-product coordinated pricing and inventory control problem with cross-price effects that we use as an illustrative example.