Dynamic Pricing with Reference Price Dependence

Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand

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 effect, which implies that the underlying optimization problem is nonsmooth and no standard optimization methods can be applied. We identify a few key structural properties of the problem, 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 and provide an upper bound on the optimal objective value. Finally, we conduct numerical experiments to study the optimal price path and demonstrate the value of dynamic pricing when demands are seasonal. We further compare numerically one of the exact algorithms with the heuristic and offer managerial suggestions. This paper was accepted by Yinyu Ye, optimization.

Two-stage pricing strategies of a dual-channel supply chain considering public green preference

Social perception influences product and brand evaluations. Consumers are especially susceptible to reference price effects when they make purchase decisions for a certain product. Meanwhile, the advertising and pricing are the determinable factors that have impact on consumers’ reference price which also are fundamental marketing strategies. Therefore, how to dynamically set advertising and pricing to maximize firms’ profits are essential tasks. We investigate a duopoly market in a mature product category where two firms compete through time using advertising and pricing as their dominated marketing tools. The firms make the advertising and pricing decisions to maximize their own profits in the planning period. The main results of this paper include the following. (i) The optimal retail price and advertising effort are positively correlated to the initial reference price and basic market size. (ii) When the two firms’ initial basic market size are different, the retail price difference is positive correlated to the initial basic market size difference, so is the advertising effort difference. This conclusion will result to that the strong firm is getting stronger and stronger, however, the weak firm is getting weaker and weaker, this is the situation which happen in the e-commerce market, that is winners take all. (iii) The value of the initial reference price can also determines the reference price effect on the consumer demand rate, that is, when the initial reference price is relatively low, the reference price will have a negative effect on the consumers’ demand of both firms in the whole planning period; when the initial reference price is relatively high, the reference price will have a positive effect on the consumers’ demand of both firms in the whole planning period. Whereas, a moderate initial reference price may lead to different effects on demand.

A well-known concept in the marketing literature is that consumers use prices paid in the past or the current market price as a reference point to assess the value of the current price of an item. These 'reference prices' have been shown to have a significant impact on a consumer's purchase decision and a resulting influence on the selling firm's demand of the item. Recently, considerable research has been conducted on coordinated pricing and inventory decisions; however, very little has recognised the existence of reference prices. Thus, a single-period model is developed that incorporates the effects of reference prices, demand uncertainty and price elasticity on the solution. Numeric analysis indicates that accounting for reference prices can have a substantial impact on the pricing and inventory decisions and the resultant profitability, made by a firm.

Pharmaceutical policies: effects of reference pricing, other pricing, and purchasing policies.

Reference price effect of partially similar online products in the consideration stage

Reference pricing with endogenous generic entry

This study explores pricing and order interval decisions for competing manufacturers in a circular supply chain. With a growing focus on environmental sustainability in a business, two duopolistically competing manufacturers offer different products to their customers, involving both new and remanufactured products. This research investigates the impact of various products on market demand by considering selling prices and inventories. It highlights how the selling price in the demand function is influenced not only by its price but also by competitors’ prices. Additionally, it emphasizes the pivotal role of the order interval in determining product inventory. This research performs a quantitative method in the operational research. Initially, it structures the demand and profit functions for each manufacturer. By developing a Nash equilibrium model on the game theory, this study determines the optimal decisions for pricing and ordering interval to maximize profit for each manufacturer. The numerical results confirmed that the optimal prices and profits for the manufacturer selling new products (p1=67, π1=1443.96) are higher than the manufacturer (p2=65, π2=1435.22). Meanwhile, the comprehensive sensitivity analysis demonstrates that increasing the selling price (pi=45~85) and order interval (Ti=5.2~6) will reduce the manufacturer’s profit. On the other hand, increasing the competitors’ selling price and order interval will strategically increase the manufacturer’s profit. Furthermore, increasing the wholesale price (wi=35~70) and holding cost (hi=0~15) will decrease the order interval, increase the selling price, and ultimately reduce profit. The managerial insights summarize that it is imperative for a manufacturer to determine the optimal selling price and order interval to gain a competitive advantage when making pricing and ordering interval decisions

Accepted by: M. Zied Babai To investigate the pricing decisions in a dual-channel supply chain consisting of a manufacturer’s online sales channel and a retailer’s traditional offline channel, we establish fundamental models incorporating consumers’ reference price effect and the free-riding effect, and then, we design a two-part pricing coordination mechanism. The results indicate that as the proportion of consumers with a high reference price increases, supply chain members achieve higher profitability, and a continuous increase in the retailer’s promotional effort enables consumers to gain product knowledge more quickly and enhance overall market efficiency.

:Using a novel scanner data set, I study price-setting decisions of major retailers in an emerging market economy. Pricing dynamics are heterogeneous across retail chains. Heterogeneity is especially pronounced in the case of posted (as opposed to reference) prices. Furthermore, retail chains appear to set prices in a centralized fashion: most barcode/store-level prices coincide with the intrachain modal price. The relationship between reference and chain-wide prices reveals that deviations from reference prices cannot be solely attributed to shocks to local market conditions. I find strong evidence of synchronization of price changes across stores within chains but weaker evidence of synchronization across retail chains.

We study a multi‐period, multi‐item dynamic pricing problem faced by a retailer. The objective is to maximize the total profit by choosing prices, while satisfying several business rules. The strength of our work lies in our graphical model reformulation, which allows us to use ideas from combinatorial optimization. We do not make any assumptions on the structure of the demand function. The complexity of our method depends linearly on the number of time periods but is exponential in the memory of the model (number of past prices that affect current demand) and in the number of items. We prove that the profit maximization problem is NP‐hard by showing an approximation preserving reduction from the weighted Max‐3‐SAT problem. We next introduce the discrete reference price model which is a discretized version of the reference price model, accounting for an exponentially smoothed contribution of all past prices. We show that our problem can be solved efficiently under this model. We then approximate common demand functions using the discrete reference price model. To handle cross‐item effects among multiple items, we propose to use a virtual reference price that assigns a reference price for each category of items (as opposed to a reference price for each item). To enhance the tractability of our approach, we cluster items into blocks and show how to adapt our method to include business constraints across blocks. Finally, we apply our solution approach using demand models calibrated with supermarket data and validate its practical performance.

An emerging consensus in marketing is that consumers respond to price relative to some standard or reference price. Most researchers modeling brand choice have reasoned that this standard is based on past prices of the brand. The authors argue that consumers do use reference prices, but one that is also based on context- other prices in the store-rather than on past prices alone. An analysis of households' brand choices in two subcategories and over three cities support this premise. Within context, the lowest price seems to be an important cue for reference price, whereas within time, a brand's own past prices seem to be the most important cue. Households' use of a contextual reference price also varies predictably across some consumer characteristics. Though their model can be applied to other categories, the findings have important managerial implications: Managerial focus on temporal reference prices could lead to an everyday high price, whereas focus on contextual reference prices could lead to an everyday low price. Only the inclusion of both contextual and temporal reference prices justifies variable pricing.

Existing revenue management methods and heuristics rely on specific demand-side assumptions such as customers’ independent decisions over time. We relax the assumption that purchasing decisions depend only on the current price and are independent of previous prices of the same or similar products. On the contrary, we assume that customers’ decisions depend on the product’s past prices through a reference price. With this new dimension, a firm needs not only to manage its remaining capacity but also to control the reference price to maximize its expected future profit. In this situation, we show that some of the main analytical properties such as monotonicity or modularity of the firm’s value function no longer hold. Consequently, the effectiveness of existing heuristics that have been developed based on these properties become limited and might generate poor results. Examples of such heuristics are the bid price and the booking limit, which are popular methods of revenue management in practice. For the new setting, incorporating a reference price, we update the existing revenue management heuristics and measure their updated effectiveness. We use a comprehensive simulation study to illustrate our results.

This study aimed to identify warning thresholds that can guide sellers in achieving reasonable profits while reducing buyer payment costs. Monthly price data from January 2009 to June 2023 and the two-regime threshold regression model were used for an empirical analysis of the asymmetric effects of the changes in household consumption under different pork price fluctuation levels to verify that the changes in household consumption have significant pork price fluctuation thresholds and threshold effects. It was found that as the fluctuation of pork prices approached the threshold, the temporary price-intervention policy (e.g., reference price) effect of adjusting prices diminished. An empirical analysis was conducted from the perspective of asymmetric effects of the upstream (feed and piglets), midstream (live pigs), and downstream (pork) of price transmission. The impacts of the products in the upstream and midstream of the industry chain on pork prices declined gradually, and among the products, the impact of the prices of feeds for fattening pigs on pork prices was found to be the weakest. Thus, the price intervention policy (e.g., reference price) based on the pork price threshold suggests that strengthening the supervision of the price fluctuations of midstream products (e.g., controlling the profit margin) when the pork price fluctuation is near the threshold (8.24%) would have more significant effects on all sides.