Asymmetric Reference Price Effects and Dynamic Pricing Policies
Authors: Kopalle Praveen K. , Rao Ambar G. , Assunção João L. 
Published: October 2008

We consider a group of frequently purchased consumer brands which are partial substitutes and examine two situations; the first where the group of brands is managed by a retailer, and second where the brands compete in an oligopoly. We assume that demand is a function of actual prices and reference prices, and develop optimal dynamic pricing policies for each situation. In addition to researchers studying pricing strategy, our results may interest retailers choosing between hi-lo pricing and an everyday low price, and manufacturers assessing whether to follow Procter & Gamble's lead and replace a policy of funding consumer price reductions through trade deals with a constant wholesale price. A reference price is an anchoring level formed by customers based on the pricing environment. The literature suggests that demand for a brand depends not only on the brand price, but also whether the brand price is greater than the reference price (a perceived loss) or is less than it (a perceived gain). The responses to gains and losses are asymmetric. Broadly speaking, we find that when enough consumers weigh gains more than losses, the optimal pricing policy is cyclical. Likewise, when they weigh losses more than gains, a constant price is optimal. Thus, we provide a rationale for dynamic pricing which is quite distinct from the three explanations previously offered: (1) decreasing unit variable costs due to learning effects, (2) the transfer of inventory to consumers who face lower inventory holding costs than do retailers, and (3) competitive effects. Our explanations apply even when the other explanations do not, i.e., in mature product categories where learning effects are minimal, when retailer inventories are minimized through the use of just-in-time policies and when competitive effects do not exist, as in a monopoly. Greenleaf (1995) has shown numerically that in the presence of reference price effects, the optimal pricing policy for a monopolist can be cyclical. We first analytically extend Greenleaf's result to a monopolist with a constant cost of goods, facing a homogeneous market where all customers either weigh gains more than losses or vice versa. Using this building block we examine a monopolist retailer managing multiple brands. We assume that demand is a linear function of prices of multiple brands, and together with an expression which reflects the reference price effect. Further, we assume that the retailer maximizes average profit per period. Next, we analyze a duopoly and extend the results to an oligopoly. We assume that the manufacturers are able to set the retail prices, as in an integrated channel. Here, we retain the same demand function as for the retailer and derive Markov Perfect Nash equilibria. We use two alternative processes of reference price formation: the exponential smoothed (ES) past price process which is frequently used in the literature, and for the multi-brand situations, the recently proposed reference brand (RB) process (Hardie, Johnson, and Fader 1993). In the latter, the reference price is the current price of the last brand bought—the reference brand. We adapt the individual level RB formulation in Hardie et al., to an aggregate demand specification. For the ES process, we obtain most results analytically; for the RB process we use simulation. Finally, we extend our results to a population with two customer segments: Segment 1 which weighs gains more than losses, and Segment 2 which does the opposite, i.e., is loss averse. When the market consists exclusively of Segment 1 customers and ES is the reference price process, we find that prices are cyclical in all cases analyzed, i.e., for a monopoly, a monopolist retailer managing multiple brands, a duopoly, and an oligopoly. If the RB formulation is the underlying process, a monopolist retailer managing two brands uses cyclical prices, but in a duopoly, the equilibrium solution is for the brands to maintain constant prices. When all customers belong to Segment 2 (i.e., they are loss averse) constant prices are optimal in all cases for both reference price formulations. When the population consists of both Segment 1 and Segment 2 and the ES process applies, we develop a sufficient condition for cyclical pricing policies to be optimal. The condition is expressed in terms of the proportion of the two segment sizes, the absolute difference between the gain and loss parameters of each segment, and their respective exponential smoothing constants. Interestingly, for reasonable values of the latter two factors, cyclical policies are optimal even when the proportion of Segment 1 is quite small. Similar magnitudes are obtained numerically for the RB case.