ArticlePDF Available

Dynamic Pricing with Reference Price Effects in Integrated Online and Offline Retailing

International Journal of Production Research

July 2021
60(1)

DOI:10.1080/00207543.2021.1973136

Authors:

Régis Chenavaz

Kedge Business School

Walid Klibi

Kedge Business School

Rainer Schlosser

Hasso Plattner Institute

Omnichannel retailing is of growing importance. Yet, retailers lack knowledge about how to set prices over time in their different channels, that is, in-store and online, which gives rise to a dual channel pricing problem. The retailing issue is even more salient when consumers are prone to a psychological element, that is when a reference price exerts influence. This article fills the gap by offering an analytical model of intertemporal price setting for dual channel pricing problem. We present an optimal control framework of dynamic pricing when 1) consumer behavior is prone to a reference price and 2) the online channel is subject to the last-mile delivery cost. Analytical results, which hold for a general (non-linear) reference-dependent demand formulation, inform about the relationships between the store and online prices over time and also about the market power of the retailer in each channel. Numerical results describe the features of three different phases of the planning horizon. The managerial recommendations show how a retailer sets differentiated dynamic pricing policies when offline and online channels are integrated. Such recommendations pave the way to more profitable omnichannel management.

Dual Pricing and Delivery in Integrated Online and Offline Retailing.

…

Figures - uploaded by Régis Chenavaz

Content may be subject to copyright.

Content uploaded by Régis Chenavaz

Content may be subject to copyright.

Dynamic Pricing with Reference Price Eﬀects in Integrated Online and

Oﬄine Retailing

R´egis Chenavaza, Walid Klibib,∗, Rainer Schlosserc

aKEDGE Business School, Marseille, France

bCentre of Excellence for Supply Chain Innovation & Transportation, KEDGE Business School, Bordeaux, France

cHasso Plattner Institute, University of Potsdam, Potsdam, Germany

Abstract

Omnichannel retailing is of growing importance. Yet, retailers lack knowledge about how to set prices over

time in their diﬀerent channels, that is, in-store and online, which gives rise to a dual channel pricing problem.

The retailing issue is even more salient when consumers are prone to a psychological element, that is when

a reference price exerts inﬂuence. This article ﬁlls the gap by oﬀering an analytical model of intertemporal

price setting for dual channel pricing problem. We present an optimal control framework of dynamic pricing

when 1) consumer behavior is prone to a reference price and 2) the online channel is subject to the last-

mile delivery cost. Analytical results, which hold for a general (non-linear) reference-dependent demand

formulation, inform about the relationships between the store and online prices over time and also about

the market power of the retailer in each channel. Numerical results describe the features of three diﬀerent

phases of the planning horizon. The managerial recommendations show how a retailer sets diﬀerentiated

dynamic pricing policies when oﬄine and online channels are integrated. Such recommendations pave the

way to more proﬁtable omnichannel management.

Keywords:

Demand management, Retail supply chain, Dual channel operations, Dynamic pricing, Reference price

1. Introduction

1.1. Business Context and Motivation

It is well established that e-commerce accounts for a growing portion of retail sales. Currently, e-

commerce represents about 7 percent of total retail sales, with a projected growth of about 10 percent every

year, three times the rate of total retail sales (Tompkins and Ferrell, 2016). These pre-COVID estimates

are rather expected to grow in most of the post-covid businesses. The e-commerce channel is appreciated

by many customers because it is a convenient way of shopping through devices, the web, and interactive

terminals. Accordingly, to increase their customer base, several brick-and-mortar (B&M) retailers have

engaged in developing their online sales channels. This was traditionally operated with the “buy online

and ship to customer” (BOSC) option, which commits the retailer to deliver the ordered products to the

∗Corresponding author

Email addresses: regis.chenavaz@kedgebs.com (R´egis Chenavaz), walid.klibi@kedgebs.com (Walid Klibi),

rainer.schlosser@hpi.de (Rainer Schlosser)

Preprint submitted to International Journal of Production Research July 28, 2021

preferred location of the customer (home, oﬃce, or pick-up station) (Bell et al., 2014). Another online

practice that is gaining popularity is the “buy online and pick up in-store” (BOPS) option. The BOPS

option consists of the customer ordering online and going to the store to collect the order (Bell et al. 2014,

Gao and Su 2016). When a retailer engages in B&M, BOSC, and BOPS sales channels, we refer to that as

omnichannel retailing. The omnichannel business seeks a seamless shopping experience, allowing customers

to order anytime from anywhere, in person or through digital devices, and to have their purchase be delivered

at their preferred time and location (Bell et al. 2014, Cai and Lo 2020).

From a supply chain perspective, omnichannel retailing implies the integration of online and oﬄine

channels, which raises several key issues that are far from being resolved today. For instance, the in-

crease of customer bases and channels requires optimizing dynamically the product assortment, managing

cross-channel revenues, and oﬀering dynamically optimized online and oﬄine prices. Such dual-pricing op-

timization problem should be highly considered by supply chain analysts to manage demand streams on

multiple channels, which is a challenge that we address in this work. As highlighted in Harsha et al. (2019),

despite an omnichannel environment having many beneﬁts, it also introduces many new challenges for price

optimization. It is well established that price is a key revenue driver in value-creating supply chains because

it is considered as an order winner valued by customers, but it must be contrasted with the cost of goods

sold, including the delivery cost (Martel and Klibi, 2016). Savelsbergh and Van Woensel (2016), and Cai

and Lo (2020) underlined that despite its practical importance, there is little research on the implications

of the eﬀective management of omnichannel logistics.

Furthermore, Cavallo (2017) noticed that online prices are increasingly used for measurement and re-

search applications, yet little is known about their relation to prices collected oﬄine. Recently, the author

found that price levels between online and oﬄine are identical about 72 percent of the time when simultane-

ously comparing prices collected from the websites and physical stores of 56 large multichannel retailers in

10 countries. Brynjolfsson and Smith (2000) explained how and why online retailers can reduce online-oﬄine

frictions for consumers and deliver lower prices, but this was not done in cases where both channels are part

of the same company. As underlined in Chopra (2018) price-conscious customers will make the eﬀort to

select the channel that oﬀers the lowest price, even if the delivery service oﬀerings are weak. Some retailers

choose identical prices for the physical products and use additional delivery fees as the main steering element

of the online channel (Agatz et al., 2008). Finally, Fibich et al. (2003) and Popescu and Wu (2007) recalled

that the reference price is the price consumers have in mind and to which they compare the shelf price of a

speciﬁc product. The importance to consider the omnichannel customer journey between showrooming and

webrooming is key, as underlined in Bell et al. (2018) and Bijmolt et al. (2019), which also implies the eﬀect

of pricing. So far, the focus of the literature has been whether retailers match or set diﬀerent prices in the

store and online, but the explanation for the diﬀerence in pricing is missing.

Further, price update costs are constantly decreasing (online and oﬄine, cf. electronic price tags) allowing

for price updates with higher frequencies. Firms like, e.g., Amazon make millions of price changes every day

where the average product’s price will change once every 10 minutes (cf. Proﬁtero Price Intelligence 2014).

It is clear that the opportunities of dynamic pricing will lead several retailers to engage in the same path (cf.

Digiday 2021). Already, several solution providers to seek to develop specialized price management tools

(cf., e.g., competera.net, IBM, etc.). In this context, dynamic pricing strategies can also be of high interest

for retailers as they like to manage sales dispersion among channels and inﬂuence cross-channel sales, as,

Figure 1: Dual Pricing and Delivery in Integrated Online and Oﬄine Retailing.

e.g., for Macy’s retailer (Gao and Su, 2017), and when electronic price tags are deployed at point of sales

(Reinartz et al., 2019). For instance, a proprietary implementation of the Omnichannel Price optimization

tool (OCPX), proposed by (Harsha et al., 2019), is now commercially available as part of the IBM Commerce

Markdown Price Solution.

Figure 1 illustrates the dual-pricing problem investigated in this work where two prices are oﬀered by

the retailer when online (BOPS and BOSC) and oﬄine (BMS) channels are integrated. We note that in the

rest of the paper we refer to a dual-pricing problem rather that omnichannel, because BOPS and BOSC are

both sharing the price of the online channel. The case of a large European retailer in the cosmetics industry

that sells a broad catalog of products (care, makeup, accessories, ...) through physical stores and online

inspires our business context. The retailer has been operating its B&M sales (BMS) for a long time and

recently started oﬀering online sales. The retailer strategy builds on an integrated online and oﬄine customer

journey that starts from showrooming, which enhances experiencing product attributes. The historical sales

path relying on a BMS channel and the current showrooming strategy employed by the retailer conﬁrm

the interest to consider the store price a key referential. Accordingly, in this work, the retailer chooses

dynamic pricing by channel based on the associated demand in time and the delivery cost incurred by each

channel. Further, we focus on the case, in which customers are prone to reference dependence to oﬄine

prices, capturing an essential element of consumer behavior and linking the demand of both channels.

Figure 1 also illustrates the retailer’s downstream supply chain schema composed of a fulﬁllment center, a

store, and a set of customer locations. The ﬁgure describes product ﬂows with replenishment to the store and

ship-to for BOSC and BOPS options, and customers going to the store for BMS and for BOPS. The regular

option is the replenishment from the fulﬁllment center to the stores (bold arc), corresponding to scheduled

transportation using a contracted carrier. The second delivery option is the ship-to customer/store delivery

from the fulﬁllment center (regular arcs), corresponding to an on-demand transportation option contracted

with a parcel delivery provider. Dotted arcs correspond to customer moves (BMS and BOPS) and do not

incur costs for the retailer. Here, unitary delivery cost is considered lower with the scheduled replenishment

option (consolidation principle) than the on-demand online option, which is subject to unitary parcel tariﬀs.

This is congruent with the discussion on logistics costs in Chopra (2018), which underlines the high cost

incurred by home delivery for the online channel compared to the BOPS. Accordingly, a key question tackled

in this work is: What is the impact of the delivery cost on online and oﬄine channel prices and the dynamics

and market power by channel?

1.2. Contributions

This article highlights the importance of price optimization by channel and also the current need for

channel coordination. Most of the literature either tackled the dynamic pricing but without the notion of

the reference price (Harsha et al., 2019), or considered the reference price eﬀect but applied it in a static

setting (Wang et al., 2020). The emergent literature on omnichannel retailing is clearly at the marketing-

operations interface and started by investigating the price matching between channels (Kireyev et al., 2017),

but in a statically, whereas we investigate here how the prices dynamically evolve over time along the product

life cycle. The key notion of reference price (Fibich et al., 2003), captures the consumer behavior which

aﬀects the demand for a given product, but this was never investigated in a dual channel pricing setting.

Furthermore, some past work in dual-channel retailing assumed that both channels don’t belong to the

same retailer, and thus rely on a competition-based setting (Fisher et al., 2017; Zhang et al., 2014). Recent

research studies the centralized policy on managing both channels controlled by a partner, but without

accounting for the psychological element (Xiao and Shi, 2016; Radhi and Zhang, 2018). We follow on this

recent research stream where the same retailer is operating in a dual channel environment, by adding a

reference price eﬀect. For all these reasons, we believe that, to the best of our knowledge, this work is

among the ﬁrst to consider a dynamic price optimization with reference price eﬀects in a dual channel

context.

The contribution of this article is threefold. First, we develop an optimal control framework for the dual

pricing problem that maximizes the revenue of the retailer, and we characterize the relation between online

and store prices and their behavior in time. We integrate the role of a reference price, coming from past

store prices. Recall that this assumption is nurtured by our inspiring case where the retailer is traditionally

a B&M seller and moved recently to online sales but with keeping showrooming as a crucial part of the

omnichannel customer journey. We also consider the impact of the supply chain organization through the

introduction of a delivery cost for online sales. We focus here on the determinants of store and online prices

based on a general reference-dependent demand, which accounts for customer behavior.

Second, we deﬁne analytically the conditions under which the same product is more expensive in the

store or online. We formulate the conditions under which a higher reference price leads to higher or lower

channel prices. Our structural results, provide that for identical demand conditions, the unit markup in the

store is always greater than the unit markup online. Moreover, when the adjustment of the reference price

is fast enough, or the delivery cost is low enough, then the intertemporal element plays a major role, and

the store price is higher than the online price. We notice that our model is straightforwardly adaptable to

the case where the reference price would be coming from past online prices.

Third, we provide numerical results that illustrate how optimally controlled prices interact over time,

based on an approximate discrete-time model. These numerical results reveal that three phases emerge: (1)

an initial phase governed by the initial reference price, (2) a steady-state, and (3) a ﬁnal phase characterized

by decreasing prices. Our results help the retailer set better online and in-store prices over time. The results

also inform the retailer about when to set a higher or lower online price rather than the in-store price.

The rest of the article is organized as follows. Section 2 presents the literature review. Section 3

formulates the model, and Section 4 solves the model. Section 5 presents analytical results supported by

numerical evaluations given in Section 6 including more general (and analytical not tractable) examples

where the channels’ prices interact. In Section 7, we provide ﬁnal conclusions.

2. Literature Review

Hereafter, we divide the relevant literature into two categories: omnichannel retailing operations (cf. Sec-

tion 2.1) and dynamic pricing in retail (cf. Section 2.2).

2.1. Omnichannel Retailing Operations

The work on omnichannel operations is still in its infancy. Chopra (2018) referred to omnichannel

retailing as the use of a variety of channels to interact with customers and fulﬁll their orders. Recently,

Melacini et al. (2018) provided a review on e-fulﬁllment and distribution in omnichannel retailing. The

authors noticed the increased interest in the omnichannel operations ﬁeld but underlined that many key

topics are still under-represented, including for instance the interplay among diﬀerent logistics aspects.

Gallino and Moreno (2014) are among the ﬁrst covering the integration of online and oﬄine channels

in retail. The authors discussed important eﬀects due to the implementation of a BOPS channel, which

shows the cross-selling eﬀect and channel-shift eﬀect. The authors underlined that integrated pricing strate-

gies represent important challenges in online-oﬄine integration eﬀorts, but their study was not covering it

explicitly. In the same way, Gao and Su (2016) studied an omnichannel environment with in-store fulﬁll-

ment, online fulﬁllment, and BOPS fulﬁllment channels. In the proposed model, the retailer decided the

store inventory to maximize revenue, and the customers decided the fulﬁllment channel based on utility,

including a hassle cost speciﬁc to each channel. The authors concluded that the BOPS channel proved

to increase traﬃc in the store by providing inventory information and increased convenience to customers.

Govindarajan et al. (2018) considered a retailer network including physical stores, online fulﬁllment centers,

and omnichannel physical stores facing two demands, online and in-store, for a single product. In this work,

inventory and fulﬁllment decisions at each facility were optimized, but the pricing strategy to maximize the

expected revenues was not linked to the fulﬁllment decision. Chopra (2018) analyzed transportation costs

for omnichannel alternatives, including BMS, BOSC, and BOPS, and concluded that, when transportation

is provided by a retailer, delivery cost in BMS is lower than for online sales with either pick-up or home

delivery. Recent work on designing omnichannel distribution networks underlined the challenge to minimize

replenishment, delivery and fulﬁllment costs when in-store and online retailing are integrated (Arslan et al.,

2020). Recently, Jasin et al. (2019) discussed the BOSC option in an omnichannel context and studied how

relying on a warehouse to fulﬁll customer online orders can reduce operational costs.

Moreover, Agatz et al. (2008) underlined that pricing models for a multichannel setting appeared to be

scarce at that time. They mentioned that the presence of a traditional sales channel adds further dimensions

to the pricing decision. The authors discussed in particular that retailers need to choose whether to oﬀer the

same prices or whether to price diﬀerentiate. A discussion on the importance of coordination between oﬄine

and online channels is provided in Cattani et al. (2004). The authors highlighted that the online channel

typically provides an alternative to an existing channel, so pricing decisions within the online channel often

must be made in the context of a multichannel scenario. Li et al. (2015) compared the optimal price of

the online and in-store retailing models using a price-setting newsvendor with assortment decisions. They

concluded that consumer patience and delivery cost are the two primary factors that inﬂuence an online

retailer’s pricing strategies. The authors noticed that there is a negative relationship between delivery

cost and time, and concluded that the faster the delivery, the higher the logistics costs. Using a static

game-modeling approach, Kireyev et al. (2017) introduced the concept of self-matching concerning pricing.

Self-matching refers to the possibility for the retailer to oﬀer the lowest of its online and in-store prices to

consumers. Assuming zero delivery cost in both channels, they ﬁnd that the price is never lower in the

store compared to online. Grewal et al. (2017), underlined that there are clear beneﬁts to an omnichannel

distributional structure from a consumer point of view, ranging from transparency to uniform policies.

Recently, He et al. (2020) explored store return strategy in an omnichannel retailing context. They used

a newsvendor model to explore the impacts of store return on the retailer’s pricing and ordering decisions

considering resale of returned products. Zhang and Zheng (2020) focused on the proﬁtability of customization

and product variety decisions for ﬁrms in online or oﬄine channels under a uniform pricing scheme. The

authors found that an omnichannel retailer should optimally oﬀer customized products in the online channel

and standard products in oﬄine stores. Chai et al. (2020) investigated the pricing strategy of B&M stores

when introducing store brands to combat showrooming. They considered a static dual-channel supply chain

model to analyze the equilibrium pricing strategies for the two retailers, but no reference price is considered.

On the practical side, H¨ubner et al. (2016) conducted an exploratory study with 33 retailers to identify the

forward and backward concepts in omnichannel retailing. They discussed the advantages and challenges of

centralized versus decentralized and of integrated versus dedicated distribution schema. They underlined

the lack of quantitative models to tackle the problem. To the best of our knowledge, only a few studies have

modeled delivery costs when online and oﬄine channels are integrated.

2.2. Dynamic Pricing in Retail

Another relevant research stream is related to dynamic pricing in the retail marketplace. In the revenue

management literature, research on dynamic pricing studies how a ﬁrm should dynamically change its price to

balance supply and demand (Talluri and Van Ryzin, 2006). Harsha et al. (2019) studied a price optimization

problem in the presence of cross-channel interactions in demand and supply. They proposed two pricing

policies that partition the inventory between channels using a dynamic stochastic program. Gao and Su

(2016) studied the impact of proﬁt sharing and product assortment for BOPS channels. However, the authors

assumed a ﬁxed price and didn’t discuss the notions of price matching or diﬀerentiating between channels.

From a sales perspective, Gallino et al. (2016), investigated how the ship-to-store strategy caused an increase

in sales dispersion generating a shift from high-selling products to medium- and low-selling products. They

assumed that the ship-to-store process is free of charge for the customers and thus is not explicitly reﬂected

on products’ cross-channel prices. Fibich et al. (2003) developed a new method for calculating optimal

strategies in the presence of reference price eﬀects, but this was not done in an dual channel setting. The

authors showed that the main concept behind reference price eﬀects is that diﬀerences between the reference

price and the shelf price aﬀect the demand for that product. Moreover, Tsay and Agrawal (2004) studied

in a competitive context the concept of channel conﬂict when a manufacturer uses direct sales and also any

existing reseller partners. The authors showed that pricing adjustment can be beneﬁcial for both parties. A

dynamic pricing approach, based on the consumer choice modeling, was recently used in a competition-based

setting in online retailing by Fisher et al. (2017). The authors considered that dynamic prices arise from

competitive multiple online retailers.

On the practical side, Cavallo (2017) studied prices collected from the websites and physical stores of

56 large multichannel retailers in 10 countries. The author observed three types of companies stand out:

those with nearly identical online and oﬄine prices, those with stable online markups (either positive or

negative), and those with diﬀerent prices that are not consistently higher or lower online. According to this

study, some of these patterns seem to be sector-level behaviors, and others are common for most retailers

within a speciﬁc country. In the same way, the Brynjolfsson and Smith (2000) study found that prices on the

Internet are 9 to 16 percent lower than prices in conventional stores, depending on whether taxes, shipping,

and shopping costs are included in the price. However, this study compared the prices of distinct retailers

on both channels and didn’t consider the case of integrated channels.

Furthermore, the literature on dynamic pricing focused on a single channel under a single retailer context.

Price and quality were jointly studied as follows: V¨or¨os (2006), Chenavaz (2012), and V¨or¨os (2013) consider

both price and productivity improvement. The conditions along which a product better quality causes a

lower selling price are studied by Chenavaz (2017), and generalized with the introduction of goodwill (Ni and

Li, 2018) and of a salvage value (V¨or¨os, 2019). The above research use a structural approach, oﬀering general

results. Stochastic demand and stochastic inventory are introduced in Cao et al. (2012) and Li et al. (2015),

respectively. Dye and Yang (2016) and Xue et al. (2016) study the optimal pricing of deteriorating items

in the presence of a reference price. Demand depending on inventory is analyzed by Lu et al. (2016) and

Hsieh and Dye (2017). The integration of dynamic advertising highlights marketing-mix concerns (Fruchter

and Sigue 2009, Karray and Mart´ın-Herr´an 2009, Fruchter and Van den Bulte 2011, Helmes and Schlosser

2013, Helmes et al. 2013, and Schlosser 2016, Lu et al. 2016), with stochastic generalization in Schlosser

(2017). To the best of our knowledge, dynamic pricing based on optimal control has not been applied to

dual channel retailing.

Behavioral economics, accounting for a descriptive element of consumer decision, points to the role

of the reference price. (See Arslan and Kachani 2011 for a comprehensive review of this literature.) A

reference price is a psychological construct, used as a benchmark against which the customer evaluate

the selling price (Kalyanaram and Winer, 1995; Mazumdar et al., 2005); a selling price higher than the

reference price seems too expensive, reducing demand, whereas a selling price lower than the reference price

appears cheap, stimulating demand. In optimal control applications, the ﬁrst formalization of the reference

price eﬀects appears in Sorger (1988). Modeling reference price has since then been popular in optimal

control applications (Kopalle et al., 1996; Fibich et al., 2003; Li et al., 2015; Lu et al., 2016; Bi et al.,

2017; Crettez et al., 2019). More precisely, Fibich et al. (2003) advocate the interest of continuous-time

to analytically understand reference eﬀects. Popescu and Wu (2007) oﬀer the seminal analysis based on a

general (non-linear) demand function and oﬀer structural results. There is further related work dealing with

additional aspects such as deteriorating items (Xue et al., 2016), supply chain (Zhang et al., 2014) inventory

systems (Hsieh and Dye, 2017; Chenavaz and Paraschiv, 2018; Cao and Duan, 2020; Dye, 2020), and duopoly

competition (Schlosser, 2019). Chen et al. (2019) investigate the role of stochasticity. Surprisingly, the above

research on optimal control considering reference price focused on lots of diﬀerent retailing contexts, but

never on the context of dual channel.

Table 1: Main Notations

T= ﬁxed terminal time of the planning horizon,

S= Salvage value, parameter,

ps(t) = store price at time t, control variable,

po(t) = online price at time t, control variable,

r(t) = reference price at time t, state variable,

β= reference price forget, parameter,

dt =β(ps−r) = reference point dynamics at time t,

λ(t) = current-value adjoint variable at time t,

cs= marginal cost for store channel,

co= marginal cost for online channel,

Ds(ps, r) = demand for in-store channel,

Do(po, r) = demand for online channel,

π(ps, po, r) = (ps−cs)Ds(ps, r)+(po−co)Do(po, r) = current revenue,

H(ps, po, λ) = π+λβ(ps−r) = current-value Hamiltonian,

ex=∂D

∂x

D= elasticity of the demand Dwith respect to the variable x.

3. Model Formulation

In this article, we apply an optimal control framework to model the dynamic pricing policies of a B&M

retailer that launched an online channel. The retailer sells an identical product through its two channels,

namely, in the store and online (Bell et al., 2014). We assume that the retailer manages the inventory of its

channels in a decentralized way. For simplicity, the retailer has a monopoly position in both channels.

The essential elements of the modeling are as follows. On the demand side, the retailer proposes diﬀer-

ent prices for the same product, depending on the retailing channel. To simplify the analysis and better

understand the eﬀects at play, the price in one channel exerts no direct inﬂuence on the demand in the other

channel. Yet, an indirect inﬂuence exists between the two channels coming from a psychological construct

of consumer behavior, namely, the reference price, which inﬂuences the demand in both channels. Because

the retailer was ﬁrst known to consumers as a B&M store, the reference price comes from past store pricing.

Based on the supply chain organization, the delivery cost is higher using the online channel than the in-store

channel (Chopra, 2018). More precisely, we consider a monopolistic retailer setting a store price, ps(t), and

an online price, po(t), over time, t. The store prices aﬀect the reference price, r(t), of consumers, which,

in turn, inﬂuences the demand both in the store and online. While setting the channel prices, the retailer

accounts for (1) the reference price for in-store pricing at the demand side and (2) the delivery cost aﬀecting

online revenue at the supply side.

The ﬁrm was a brick and mortar, selling ﬁrst oﬄine, and then moving online. Hence, the targeted

subclass of the problem is relevant for this kind of ﬁrm. Further, the assumption to focus on oﬄine prices

as the reference price (for a brick and mortar going online) is also in line with other works, such as, e.g.,

Fruchter and Tapiero (2005) or Wang et al. (2020), see page 8-9: “Consumers use historical prices, advertised

prices, suggested retail prices, or competitor prices to form “reference prices”. Note, for sake of simplicity,

we refrain from further aspects such as promotion campaigns or special discounts for bundles, etc.

Next, we develop an optimal control framework for our model analysis. The planning horizon is ﬁxed

and ﬁnite of length T, and time tin [0, T ] is continuous. Table 1 presents the notations used in this article.

3.1. Consumer Reference Price

To assess if a price represents a good deal, triggering a purchase decision at time t, a consumer benchmarks

the store and online prices, ps(t)≥0 and po(t)≥0, with a reference price r(t)≥0. This behavioral element

is used in optimal control applications since the seminal work of Sorger (1988) surveyed in Popescu and

Wu (2007). If the selling price is higher than the reference price, then the consumer senses a loss, reducing

its interest in the product. On the contrary, if the selling price is lower than the reference price, then the

consumers feel a gain, boosting its product interest (Cebiro˘glu and Horst, 2015; Janssen et al., 2020).

The reference price is usually modeled as a weighted average of past product prices (Sorger, 1988; Kopalle

et al., 1996; Kopalle and Winer, 1996; Fibich et al., 2003; Chenavaz, 2016; Liu et al., 2016; Lu et al., 2016;

Wang and Zhang, 2017; Hsieh and Dye, 2017; Chenavaz and Paraschiv, 2018; Crettez et al., 2019; Cao and

Duan, 2020; Dye, 2020). Here, since the retailer is applying a showrooming strategy to initiate the dual

channel customer journey, the consumers are primarily informed of the store prices, which is the basis for

their reference price (Bell et al., 2018). The assumption of a reference price made in-store is consistent with

the situation of a brick and mortar moving online (Fruchter and Tapiero, 2005) and in line with Wang et al.

(2020). The reference price at time t,r(t), is deﬁned with an exponentially decaying function of past store

prices: r(t) = e−βt(r0+βRt

0eβups(u)du) where r0represents the initial reference price at time t= 0, and

β≥0 the continuous forgetting (or adjustment speed) parameter. The time diﬀerentiation of r(t) yields for

all tin (0, T ):

dr(t)

dt =β(ps(t)−r(t)),with r(0) = r0.(1)

Equation (1) informs that the dynamics of the reference price increases with the speed of adjustment

β. Also, if ps> r, then the reference price augments. Whereas if ps< r, then the reference price falls.

Following Sorger (1988), Fibich et al. (2003), Dye and Yang (2016), Hsieh and Dye (2017), and Chenavaz

and Paraschiv (2018), the interpretation of βis as follows: If β= 0, there is no adjustment of the reference

price, which remains constant and equal to the ﬁrst selling price. If β > 0, there is an adjustment of the

reference price adapts. It changes over time, accounting for all past prices, with recent selling prices weighing

more than former selling prices. Eventually, if βtends to +∞, the reference price adjusts instantaneously

to the last selling price.

3.2. Reference-Dependent Demand

The demands in the store, Ds≥0, and online, Do≥0, depend on the selling price set in the cor-

responding channel and on the reference price. The general reference-dependent demand functions write

Ds=Ds(ps(t), r(t)) and Do=Do(po(t), r(t)) for the store and online channels. Such general formulations

allow for nonlinearities when accounting for dynamics is tied to changes in the reference price (Popescu and

Wu, 2007). Hereafter and when there is no confusion, we omit any argument for ease of reading, especially

the temporal variable. For tractability, the demand functions are assumed twice continuously diﬀerentiable,

satisfying the following conditions, pi, r ≥0,

∂Di

∂pi

<0,∂Di

∂r >0,∂2Di

∂pi∂r ≥0,with i=s, o. (2)

The ﬁrst and second conditions state that the demand for each channel decreases with the selling price

in the corresponding channel and increases with the reference price. Conforming with behavioral literature,

the third condition imposes that demand is more price-sensitive for a higher reference price. In their survey,

Popescu and Wu (2007) showed that the supermodularity assumption ∂2Di/(∂pi∂r)≥0 is well documented

in consumer behavior.

Conditions (2) impose loose constraints on the impact of retail price and reference price on demand,

enabling much freedom in the formulation of the demand function. Note that Conditions (2) are compatible

with the popular linear demand function

D(p, r) = µ−θp −γ(p−r),

with strictly positive parameters µ, θ, γ and with psuch that Dis nonnegative.

The above demand function is widely used, for instance by Sorger (1988), Kopalle et al. (1996), Kopalle

and Winer (1996), Dye and Yang (2016); Dye (2020), Xue et al. (2016), Hsieh and Dye (2017), and Cao and

Duan (2020). The linear demand function, additively separable in pand r, imposes ∂2D/(∂p∂r) = 0, which

makes the model more tractable and oﬀers sharper results.

3.3. Revenue

The current revenue function π:R3+ →Ris assumed twice continuously diﬀerentiable. The revenue of

the retailer corresponds to the revenue of the store channel plus the revenue of the online channel. The store

and online marginal costs are respectively cs, co≥0. Thus, the total revenue net of the marginal costs writes

π(ps, po, r) = (ps−cs)Ds(ps, r)+(po−co)Do(po, r).1The marginal costs, csand co,include the marginal

production and retailing cost. The marginal production costs are identical for both channels, as it is the

same physical product made by the same manufacturer. Yet, the marginal retailing costs diﬀer, because of

the product delivery. Indeed, Chopra (2018) shows that the outbound transportation cost incurred by the

online retail tends to be higher than in-store because of the impact of aggregation of facilities and inventories.

For simplicity, we keep a positive online marginal cost, co≥0,and we normalize the store delivery cost to

zero, cs= 0.The revenue function of the retailer reads, ps, po, r ≥0,

π(ps, po, r) = psDs(ps, r)+(po−co)Do(po, r).(3)

The revenue function (3) accounts for dual channel characteristics both on the demand- and supply-sides.

On the demand side, we integrate the psychological element of consumer behavior: We point to the role of

the store channel in anchoring the reference price used by consumers both in the store and online channels.

On the supply side, we diﬀerentiate the marginal cost in the retailing channels: We account for a greater

delivery cost online than in the store channel. We discuss the possibility of a more general model including

arbitrary delivery costs and even price interactions across channels, cf. Footnote 2 below, which, however, is

analytically not tractable anymore. For this general model, Appendix A describes the optimality conditions

and Section 6.2.4 - 6.2.5 show numerical experiments, checking the robustness of our results.

1Appendix A proposes the optimality conditions for the general case, which is studied numerically in Section 6.2.4 - 6.2.5.

3.4. Dynamic Behavior of the Retailer

The retailer maximizes the intertemporal revenue (or present value of the revenue stream) over the

planning horizon, by simultaneously choosing the in-store and online pricing policies, accounting for the

dynamics of the reference price. For simplicity, the salvage value of the reference price is zero. Let the

interest rate be α≥0. The objective function of the retailer writes:

max

ps(u), po(u)≥0,∀u∈[0,T ]ZT

e−αtπ(ps(t), po(t), r(t))dt,

subject to dr(t)

dt =β(ps(t)−r(t)),for all tin (0, T ),with r(0) = r0.

(4)

4. Model Analysis

This section analyzes the model. As in Chenavaz and Paraschiv (2018), we follow the classical analysis

of Popescu and Wu (2007) considering a general demand function with a reference price eﬀect. Our analysis

diﬀers from the former, which focuses on the pricing of an inventory with a reference price eﬀect; they

have one control variable, the price, and two states variables, the reference price and the inventory level,

whereas we have two control variables, the price in each channel, and one state variable, the reference

price. We present the conditions from solving the dynamic optimization program (4), based on the dynamic

optimization methods exposed in Sethi and Thompson (2000) and Jørgensen and Zaccour (2012). Let λ(t)

be the shadow price (or current-value adjoint variable) of the reference price at time t. The current-value

Hamiltonian Hreads, ps, po, r ≥0, λin R,

H(ps, po, r, λ) = psDs(ps, r)+(po−co)Do(po, r) + λβ(ps−r).

The Hamiltonian, H, represents the intertemporal revenue, as the sum of the current revenue, psDs(ps, r)+

(po−co)Do(po, r), and the future revenue, λβ(ps−r). The future revenue, also noted λdr

dt ,represents the value

of the change in the reference price, created by the eﬀect of the store price decision (Sethi and Thompson,

2000; Jørgensen and Zaccour, 2012).

We conﬁne our interest to interior solutions for psand po, assuming that they exist for the considered

demand functions Dsand Do. The Hamiltonian, H, is assumed strictly concave in psand po. The necessary

and suﬃcient ﬁrst- and second-order conditions and the maximum principle for Hmaximization, provide

for all tin (0, T ): 2

2In the equations system (5a)-(5f), we are especially interested in understanding the role of the reference price and of the

last-mile delivery cost in the retailer policies. This model can be generalized to integrate a distribution cost incurring on both

channels and direct price interaction in both channels. In Appendix A, we write the Hamiltonian function and provide the

optimality conditions for this generalized model. Unfortunately, the complexity explodes and we are not able to derive formal

guarantees for analytical results. Yet, in Subsections 6.2.4 and 6.2.5, we provide numerical simulations of this generalized model

to assess the robustness of our results.

∂H

∂ps

= 0 =⇒Ds+ps

∂Ds

∂ps

+βλ = 0,(5a)

∂H

∂po

= 0 =⇒Do+ (po−co)∂Do

∂po

= 0,(5b)

∂2H

∂p2

<0 =⇒2∂Ds

∂ps

+ps

∂2Ds

∂p2

<0,(5c)

∂2H

∂p2

<0 =⇒2∂Do

∂po

+ (po−co)∂2Do

∂p2

<0,(5d)

∂2H

∂p2

∂2H

∂p2

−∂2H

∂ps∂po2

>0 =⇒∂2H

∂p2

∂2H

∂p2

>0,(5e)

dλ

dt =rλ −∂H

∂r , λ(T) = 0 =⇒dλ

dt = (r+β)λ−ps

∂Ds

∂r −(po−co)∂Do

∂r , λ(T)=0.(5f)

Revenue-maximizing retailers knows a speciﬁc relationship between the optimal store and online prices.

We now deﬁne this relationship. Let p∗

s(po) be the store price verifying (5a). This store price maximizes the

intertemporal revenue for any online price. Similarly, let p∗

o(ps) be the online price satisfying (5b), which

maximizes the intertemporal revenue for any store price. The intertemporal revenue is the maximum of

the store and online price pair such that (p∗

s, p∗

o)=(p∗

s(p∗

o), p∗

o(p∗

s)). In the remainder of this article, the

store and online pricing are said to be optimal in the sense of maximizing the intertemporal revenue. For

simplicity, we will now omit the ∗superscript notation when no confusion exists.

We obtain the concavity conditions on 1) on Dsby rewriting (5c) and substituting (5a) in it and 2) on

Doby substituting (5b) in (5d) and rearranging:

2−(Ds+βλ)

∂2Ds

∂p2

(∂Ds

∂ps)2>0,(6a)

2−Do

∂2Do

∂p2

(∂Do

∂po)2>0,(6b)

Equations (6a)-(6b) are tied to the convexity of the demand functions to the corresponding price. The

technical implication is that the demand functions cannot be “too” convex in the price. The assumption of

demand convexity is widely used in dynamic pricing literature that refers to structural demand functions,

as in Dockner et al. (2000), V¨or¨os (2006), Jørgensen and Zaccour (2012), Chenavaz (2012), Chenavaz and

Jasimuddin (2017), and Ni and Li (2018).

The second-order condition (5e) is a technicality and is not directly interpretable. It guarantees that the

solution is a maximum instead of a saddle point. Equation (5e) is satisﬁed because ∂2H

∂p2

s<0, ∂2H

∂p2

o<0, and

∂2H

∂ps∂po= 0.

Lemma 1. The value of λover time for all tin [0, T ]is measured by

λ(t) = ZT

e−(r+β)(u−t)ps

∂Ds

∂r + (po−co)∂Do

∂r du. (7)

Proof. Integrate equation (5f) with respect to time.

The value of λsums the intertemporal beneﬁt of a greater reference price for the store and online channel.

Lemma 2. The sign of λover time for all tin (0, T )is given by

λ(t)>0.(8)

Proof. The conditions on demand (2) imposes a positive eﬀect of the reference price on demand and the

ﬁrst order condition (5b) implies po−co>0.

Consequently, a greater reference price always increases the intertemporal revenue. Even with an arbi-

trary large co,λis positive because the markup of the online channel po−cois always positive.

5. Analytic Results

This section presents the diﬀerent analytic results, which cover the characteristics of the channel prices,

the market power of the retailer in each channel, and the relationships between the dynamics of the channel

and reference prices.

5.1. Channel Prices

We characterize here the dynamic and static prices by channel. A dynamic price maximizes the in-

tertemporal revenue, which sums current and future revenues. A static (also called myopic) price maximizes

the sole current revenue. Technically, the dynamic price incorporates λgiven in Lemma 2, which measures

how the reference price aﬀects future revenues. Because the static price disregards future revenues, for its

computation λis set to zero and we have λ(t) = 0 for all tin [0, T ].

Proposition 1 computes the channel prices for all tin [0, T ]. Recall that we omit the temporal argument

for readability.

Proposition 1. The intertemporal revenue maximizing prices in-store and online channels are such that

ps=Ds+βλ

−∂Ds

∂ps

,(9a)

po=co+Do

−∂Do

∂po

,(9b)

with λgiven by (7).

Proof. Rearrange the ﬁrst order conditions (5a)-(5b). Also, note that Lemma 2 imposes λ > 0, and that

the demand conditions (2) imply −∂Ds

∂ps>0 and −∂Do

∂po>0.

The store price, ps, is dynamic, as it integrates the eﬀect of the reference price on the future store and

also online demand. On the contrary, the online price, po, is static, as it disregards the eﬀect of the reference

price on the future store and even online demand.

The pricing rules of Proposition 1 show that the price in-store integrates the behavioral element and the

price online takes into account the delivery cost.

Deﬁnition 1. We deﬁne and explain here the dynamic and static prices.

•The dynamic prices for store and online channels, psand po, are the prices that maximize the in-

tertemporal revenue, which sums the current and future revenues. With a dynamic price, λis given

by Lemma 1, and Lemma 2 guarantees that λis positive over [0, T ). In the notation of our results, ps

and poalways refer to the dynamic prices.

•The static prices for store and online channels, ps/static and po/static, are the prices that maximize the

current revenue. With a static price, future revenues do not matter for all the planning horizon, and

we have λ= 0 for all tin [0, T ].

Corollary 1. The diﬀerences between the dynamic and static prices read:

ps−ps/static =βλ

−∂Ds

∂ps

≥0,(10a)

po−po/static = 0,(10b)

with λgiven by (7).

Proof. Immediate from Proposition 1.

The dynamic price psis always higher than the static price ps/static. Indeed, the retailer invests in a

higher reference price by augmenting the dynamic price above the static price. In eﬀect, a greater current

store price increases the reference price, which in turn, boosts the future store and online revenues. Such

policy trades oﬀ current revenue, tied to less current store revenue, to future revenue, linked to more future

store and online revenues. Note that if the reference price does not adjust over time, that is β= 0, then the

dynamic price matches the static price and ps=ps/static, even if the dynamic element plays and λ > 0.

The online dynamic price pocorresponds to the online static price po/static. In eﬀect, the online pricing

does not aﬀect on future revenue, because it does not aﬀect the reference price for a retailer, which is ﬁrst

known as a B&M business. In the remainder of the article, if not otherwise stated, prices refer to dynamic

prices.

To better characterize the pricing insights, suppose for Corollary 2 identical conditions on demand in-

store and online, that is Ds=Doand ∂Ds

∂ps=∂Do

∂po.

Corollary 2. For identical demand conditions, the diﬀerence between the store and online prices writes:

ps−po=βλ

−∂Ds

∂ps

−co,(11)

with λgiven by (7).

Proof. Immediate from Proposition 1, subtracting Equation (9b) from (9a).

Corollary 2 shows that, for identical demand conditions, the store price may be higher or lower than

the online price, depending on the relative impact of the reference price, captured by βλ

−∂Ds

∂ps

>0, the online

delivery cost, known as co. The reference price and delivery cost represent two eﬀects pushing for a greater

store price and online price, respectively: The retailer may set a higher store price to beneﬁt from greater

demand later in the store and online, which is a dynamic eﬀect. When the delivery cost increases for online

shopping, the retailer echoes this cost augmentation on the online price, which is a static eﬀect. Plus,

rewriting equation (11), it appears that for identical demand conditions, the unit markup in-store is always

greater than the unit markup online:

ps−(po−co) = βλ

−∂Ds

∂ps

≥0.

Though, in general, the store or the online price can be the larger one, we have the following result.

Remark 1. When 1) the adjustment of the reference price is fast enough, that is βis arbitrarily high, or 2)

the delivery cost is low enough, for instance co= 0, then the intertemporal element plays a major role, and

the store price is higher than the online price. On the contrary, when 1) the delivery cost is high enough,

that is cois arbitrarily high, or 2) the adjustment of the reference price is slow enough, for instance β= 0,

then the cost borne for online retailing has a major inﬂuence, and the online price is higher than the in-store

price.

The delivery cost cois constant over the planning horizon, whereas the shadow price of the reference

price λchanges over time. Recall 1) from Lemma 2 that λis strictly positive over the product lifetime,

λ(t)>0 for each tin (0, T ),and 2) from the transversality condition (5f) that λis null at the end of the

product lifetime, λ(T)=0.Therefore, λis more likely to be higher at the beginning and lower at the end

of the planning horizon. Thus, we can express Remark 2:

Remark 2. The store price may, but must not, be higher than the online price at the beginning of the

planning horizon. The online price is higher than the store price at the end of the planning horizon, provided

the existence of a delivery cost, that is co>0.

5.2. Market Power by Channel

We investigate here the market power of the retailer by channel, using the Lerner index and the price

elasticity of demand. Deﬁne the Lerner index L=p−ci

pwith iin {s, o}, and note that 0 ≤L≤1. The

Lerner index measures the market power of a retailer: If a retailer has no market power, then p=ciwith i

in {s, o}, and L= 0. Conversely, if a retailer has a strong market power, then ciwith iin {s, o}, is negligible

compared to p, and L= 1. Proposition 2 calculates the Lerner indexes for all tin (0, T ). Recall that we

omit the temporal argument.

Proposition 2. The Lerner indexes for the store and online channels read

Ls= 1,0< Lo=

−Do

∂Do

∂po

co−Do

∂Do

∂po

<1,(12a)

Ls> Lo.(12b)

Proof. Substitute psand pogiven in (9a)-(9b) in the deﬁnition of the Lerner index. Also, recall from

condition (2) that ∂Do

∂po<0. Consequently, −∂Do

∂po>0 and −Do

∂Do

∂po

>0.

Proposition 2 informs that the retailer enjoys full market power for the store channel, but only partial

market power for the online channel. That is, due to the delivery cost co, the retailer always has more

market power in the store than online. Further, the higher the delivery cost, the lower the market power

for the online channel is. Note that λ, which account for future revenues, does not explicitly appear in

(12a)-(12b). In that sense, Proposition 2 refers to a static context.

To consider the dynamic element, we consider now the explicit role of λ. Thus, we examine the market

power by channel, looking at the sensitivity of consumers to price. Price sensitivity is captured by the price

elasticity of demand, e=−∂D

∂p

D, which measures the percentage of demand decrease after a 1 percent

of a price increase. Recall that a static monopoly charges so the demand is of unitary elasticity (e= 1),

whereas a static competitive ﬁrm charges so that the demand is elastic (e > 1). Note es=−∂ Ds

∂ps

Dsand

eo=−∂Do

∂po

Dothe price elasticities of demand for the store and online channels.

Proposition 3 informs about the price elasticity for all tin (0, T ). Again, we omit the temporal argument.

Proposition 3. Price elasticities of demand in the store and online channels are such that

es= 1 + βλ

>1,(13a)

eo=1

>1,(13b)

with λgiven by (7).

Proof. Rearrange the ﬁrst order conditions (5a)-(5b) and use the elasticity notations. Also, recall that

Equation (8) imposes λ > 0 and that Proposition 2 dictates 0 < Lo<1.

The main lessons derived from Proposition 3, which applies in a dynamic setting, are the following: When

the retailer behaves dynamically by also valuing future revenues, the store channel is deprived of part of the

static monopoly market power. Indeed, es>1, because the retailer integrates the impact of the reference

price on the future revenue, which is a dynamic motivation. Because of the lasting eﬀect of the reference

price, the retailer loses freedom in setting the store price, enjoying less market power in the store channel,

as shown in Chenavaz (2016) and Chenavaz and Paraschiv (2018). Along the same line, the online channel

provides no monopoly power, eo>1, because it bears the delivery cost, which is a static motivation. In a

nutshell, neither the store channel nor the online channel yields a (static) monopoly power, even though the

retailer is in a monopoly situation in both channels, but for distinct reasons.

Remark 3. Following Proposition 2, based on a static context, the retailer beneﬁts from full market power

in the store channel. Yet, Proposition 3, built on a dynamic context, informs that the retailer loses part

of the static market power in the store channel. There seems to be a puzzle between both propositions. We

resolve the puzzle by distinguishing the managerial implications in the static and dynamic context. Following

Proposition 2, the retailer may charge a high price in-store, compared to the marginal cost. In that sense,

the retailer beneﬁts from monopoly freedom when choosing the price. But, Proposition 3 points to the lasting

inﬂuence of the current price on the future reference price. Consequently, the price setting of the retailer

accounts for a dynamic constraint, losing some monopoly freedom. In fact, because of the dynamic element,

the retailer sets even a higher price to invest in a higher reference price, as claimed in Corollary 1.

Remark 4. The static case, with λ= 0, implies the monopoly market power in the store because es= 1;

the null delivery cost case, with co= 0, also oﬀers the monopoly market power online because eo= 1.

Next, Proposition 4 characterizes the channel sensitivity to price for all tin [0, T ]. Recall, we omit the

temporal argument.

Proposition 4. The ratio of price elasticities is such that:

=Ds+λβ

| {z }

≥1

po−co

| {z }

≥1

≥1,(14)

with λgiven by (7).

Proof. Compute the values of esand eoand simplify the ratio. Lemma 2 states that λ(t)>0 for all tin

[0, T ) and the transversality condition imposes λ(T) = 0. Consequently, we have λ(t)≥0 for all tin [0, T ].

Lemma 2 also shows that po−co>0.

The ratio of the price elasticities es

eois superior to 1, meaning that store consumers are more price-

sensitive than online consumers, es> eo. Consequently, the retailer beneﬁts from more market power online

than in the store, giving the retailer more freedom in price setting.

Remark 5. The static and null delivery cost case, with λ=co= 0, implies a unitary ratio of elasticities,

eo= 1. Plus, the retailer sets the static monopoly price in both channels. Yet, even if es=eo, demand

conditions may remain distinct, and Proposition 1 veriﬁes that the channel prices may diﬀer, i.e., ps6=po.

5.3. Dynamics of Channel and Reference Prices

The retailer may invest in the reference price, by improving the store price. In eﬀect, a greater reference

price increases the demand both in the store and online. Knowing the relationship between the dynamics

of the reference and channel prices is thus of managerial importance to set revenue enhancing prices over

time. So far, we obtained results concerning the optimal prices by channels and the relationships between

such prices. Yet, no causality was considered. We investigate here how the channel prices adapt to changes

in the reference price.

To derive the relationship between the channel selling prices and the reference price, we observe that the

optimality conditions (5a)-(5b) must hold over the whole planning period. That is, if the reference price

changes, then the channel prices, in turn, must adapt so that the optimality conditions always hold. Such

adaptation roots the links between the dynamics of the reference price and the selling price

The formal link between the reference and the channel price dynamics is obtained by the diﬀerentia-

tion of the ﬁrst-order conditions of the in-store price (5a) and online price (5b) with respect to time t.

Mathematically, we have for all tin (0, T ):

Ds+ps

∂Ds

∂ps

+βλ = 0 =⇒d

dt Ds+ps

∂Ds

∂ps

+βλ= 0,

Do+ (po−co)∂Do

∂po

= 0 =⇒d

dt Do+ (po−co)∂Do

∂po= 0.

Recall that Ds=Ds(ps(t), r(t)), Dp=Do(po(t), r(t)), ps=ps(t), po=po(t), r=r(t), λ=λ(t). The

chain rule, used to compute the derivative of the composition of functions, provides:

2∂Ds

∂ps

dps

dt +∂Ds

∂r

dt +ps∂2Ds

∂p2

dps

dt +∂2Ds

∂ps∂r

dt +dλ

dt β= 0,

2∂Do

∂po

dpo

dt +∂Do

∂r

dt + (po−co)∂2Do

∂p2

dpo

dt +∂2Do

∂po∂r

dt = 0.

Substitute psand po−cofrom (9a) and (9b) and rearrange. Divide the ﬁrst equation by −∂Ds

∂psand the

second equation by −∂Do

∂po:

dps

dt 

2−(Ds+λβ)

∂2Ds

∂p2

(∂Ds

∂ps)2

=dr

dt −

∂Ds

∂r

∂Ds

∂ps

+ (Ds+λβ)

∂2Ds

∂ps∂r

(∂Ds

∂ps)2!+dλ

∂Ds

∂ps

,(15a)

dpo

dt 

2−Do

∂2Do

∂p2

(∂Do

∂po)2

=dr

dt −

∂Do

∂r

∂Do

∂po

+Do

∂2Do

∂po∂r

(∂Do

∂po)2!.(15b)

Recall the price elasticities of demand es=−∂Ds

∂ps

Dsand eo=−∂Ds

∂ps

Dsfor the two channels. Introduce

the reference price elasticities of demand es/r =∂Ds

∂r

Dsand eo/r =∂Do

∂r

Dofor the two channels. We have

−

∂Ds

∂r

∂Ds

∂ps

=es/r

rand −

∂Do

∂r

∂Do

∂po

=eo/r

r. Proposition 5 informs about the prices dynamics over time.

Proposition 5. The dynamics of the channel and reference prices are such that, tin (0, T ):

dps

z }| {



2−(Ds+λβ)

∂2Ds

∂p2

(∂Ds

∂ps)2

=dr







z }| {

es/r

z }| {

(Ds+λβ)

∂2Ds

∂ps∂r

(∂Ds

∂ps)2







+/−

z}|{

dλ

z }| {

−∂Ds

∂ps

,(16a)

dpo

dt 

2−Do

∂2Do

∂p2

(∂Do

∂po)2



| {z }

=dr







eo/r

| {z }

+Do

∂2Do

∂po∂r

(∂Do

∂po)2

| {z }







(16b)

with λgiven by (7).

Proof. Substitute es/r

rand eo/r

rin (15a) and (15b).

Proposition 5 associates the dynamics of the channel prices, dps

dt and dpo

dt , to the dynamics of the reference

price, dr

dt , for general demand functions, Ds(ps, r) and Do(po, r). The dynamics of psand po, given by (16a)

and (16b), are tied to their sole corresponding ﬁrst order condition (5a) and (5b). As such, they are

independent of each other, oﬀering robust results.

On the left-hand side of (16a) and (16b), the second factors (2 −(Ds+λβ)∂2Ds

∂p2

s/(∂Ds

∂ps)2) and (2 −

Do∂2Do

∂p2

o/(∂Do

∂po)2) are positive because of the second order conditions (6a) and (6b). On the right-hand side

of (16a) and (16b), the second factors (under parentheses) sum two demand-side eﬀects, namely the markup

and sales eﬀects; see Chenavaz (2012, 2017); Ni and Li (2018) for more extensive discussions.

Deﬁnition 2. Here we deﬁne and explain the markup and sales eﬀects.

•The markup eﬀects,es/r

rand eo/r

r, quantify the increase in consumers’ willingness to pay in the

store and online, following an increase in the reference price. All elasticities being positive, the markup

eﬀects are positive.

•The sales eﬀects,(Ds+λβ)

∂2Ds

∂ps∂r

(∂Ds

∂ps)2and Do

∂2Do

∂po∂r

(∂Do

∂po)2, measure the increase in sales after higher price in

the store and online together with a greater reference price. Demand increases with the reference price,

and it would increase even more with a lower price. Because of the positivity of λ, proved in Lemma 2,

and the demand conditions (2), the sales eﬀects are positive.

When the reference price increases, then the retailer makes more revenues both in the store and online

thanks to two mechanisms. First, as consumer willingness to pay increases, the retailer may increase the

selling price, and thus the markup. Second, there is a synergy phenomenon between the reference price and

the selling price, which together boost the demand. That is, when the reference price increases, the retailer

has an incentive to increase the price, beneﬁting from greater sales.

There is now a diﬀerence in the dynamics of the in-store and online price, captured by the shadow

price λand its time derivative, which exert inﬂuence in the store, Equation (16a), but not online, Equation

(16b). This aspect is consonant with Proposition 1, which claims that the store price psis dynamic, in the

sense of maximizing both current and future revenues, whereas the online price pois static, in the sense of

maximizing the sole current revenue.

If one only considers the markup and sales eﬀect, then both the in-store and online prices increase with

greater reference price. Indeed, all associated factors are positive in (16a) and (16b). Yet, a greater reference

price may lower the store price because of the dynamic eﬀect dλ

−∂Ds/∂ps, which may be positive or negative.

In eﬀect, if λincreases, then dλ

dt is positive and so is the dynamic eﬀect. On the contrary, if λdecreases, then

dλ

dt is negative and so is the dynamic eﬀect. Recall that λrepresents the impact on the future revenue of a

higher reference price, which stimulates future store and online demand. Then, λis more likely to increase

at the beginning of the planning horizon, when the retailer invests in a higher reference price. Conversely, λ

may, but must not, decrease at the end of the planning horizon, when future revenues matter less; indeed,

there is no guarantee that the negative intertemporal eﬀect outweighs the positive markup and sales eﬀects.

The above ideas and their managerial implications for the retailer are summarized in Remark 6.

Remark 6. The relationship between the store price, ps, and the reference price, r, is positive at the

beginning of the planning horizon and can be positive or negative at the end of the planning horizon. The

relationship between the online price po, and the reference price, r, is positive over the whole horizon.

6. Numerical Evaluation

to study how optimally controlled prices interact over time, in this section, we provide numerical examples

of the model analyzed in the previous sections.

6.1. Solution of Generalized Discrete-Time Models

Due to the complexity of the model, an explicit solution is unlikely to be tractable. To numerically

approximate the model’s solution, we use a discrete-time version of the model with Tperiods, Tin N. For

each period, the store price psand the online price pohave to be chosen from a (discrete) set of admissible

prices P. For the length of one period, we use the discount factor δ=e−α. Note, the case of no discounting,

i.e., α= 0, corresponds to δ= 1 and that the case of discounting, i.e., α > 0, refers to 0 ≤δ < 1.

The state of the Markov decision process is characterized by the current reference price rand follows the

dynamics described in Section 3, cf. (1). The initial state is r0. The best discounted future revenues G(t)

from time ton (discounted on time t), cf. (4), t= 0,1, ..., T ,

G(t) =

T−1

k=t

δk−tπ(ps(k, r(k)) , po(k, r(k)) , r(k)),

optimized over all state-dependent Markovian feedback policies (psand po) are represented by the value

function V(t, r), t= 0,1, ..., T ,r≥0, which is determined by the terminal condition V(T, r) = 0, r≥0, and

the Bellman equation (Bertsekas, 2017), r≥0, t= 0,1, ..., T −1,

V(t, r) = max

ps∈P,po∈P{π(ps, po, r) + δV (t+ 1, r +β(ps−r))}.(17)

As (17) can be recursively solved, we are able to compute (approximated) optimal feedback controls,

i.e., store prices ps(t, r) and online prices po(t, r), r≥0, t= 0,1, ..., T −1, which are given by the argmax

of (17), i.e., the prices that maximize (17); they depend on time tand the current reference price r. If the

optimal control is not unique, we choose the largest one.

6.2. Numerical Examples

To exemplify the study of properties of optimal solutions, we consider the following example.

Example 1. As a reference case, (if not chosen diﬀerently) we let T= 40,δ= 0.99,r0= 12,β= 0.4, and

we consider the generalized revenue function, cp. Appendix A,

π(ps, po, r)=(ps−cs)Ds(ps, po, r)+(po−co)Do(po, ps, r)

with online unit cost are co= 5 and store unit costs are cs= 0. We use the demand functions Ds(ps, po, r) :=

D(ps, r)and Do(po, ps, r) := D(po, r), where D(p, r) = µ−θp −γ(p−r), see Section 3.2, is characterized

by µ= 1,θ= 0.1, and γ= 0.04 (for price interaction eﬀects see Section 6.2.5). Further, we consider the

state space, cf. rin P, and the control space, cf. psin P,poin P, to be discretized with granularity 0.02,

i.e., P={0.02,0.04, ..., 15}.

6.2.1. Feedback Policies

For our reference example, the optimal feedback prices ps(t, r) and po(t, r) are displayed in Figure 2, which

shows that both feedback prices increase with the current reference price. Further, we observe that although

the feedback online price is time-independent, the feedback store price decreases with the ﬁnal periods. The

observed structures are in line with Corollary 1. As the store price is coupled with the evolution of the

reference price, which aﬀects future revenues, the optimal store price to maximize intertemporal revenues is

time- and state-dependent. Instead, the online price is decoupled from the evolution of the state and aﬀects

neither the reference price nor future revenues. Thus, the optimal online price to maximize intertemporal

0 10 20 30 t

ps(t,r)

(a) Feedback store price ps(t, r)

0 10 20 30 t

po(t,r)

(b) Feedback online price po(t, r)

Figure 2: Illustration of feedback prices ps(t, r) and po(t, r) assuming diﬀerent reference prices r= 3 (blue), r= 6 (orange),

and r= 12 (green) at time t= 0,1, ..., 40; Example 1.

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(a) Reference case solution (Example 1 with

δ= 0.99, r0= 12, β= 0.4, co= 5, cs= 0),

steady-state: r∗=5.36, p∗

s=5.36, p∗

o=6.84

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(b) Initial reference price r0=3 (vs. r0=12),

steady-state: r∗=5.36, p∗

s=5.36, p∗

o=6.84

Figure 3: Evaluation of prices over time: store prices ps(t) (orange), online feedback prices po(t) (green), and reference prices

r(t) (blue) for diﬀerent r0,t= 0,1, ..., 40; Example 1.

revenues in a given state coincides with maximizing current revenues, which in the case of time-homogeneous

demand does not depend on time.

6.2.2. Evolution and Sensitivity Analysis

Further, we evaluate the optimal feedback prices of Example 1 to obtain the associated evolutions over

time, cf. open-loop paths. In this context, Figure 3 illustrates prices over time for diﬀerent initial reference

prices r0(the solution of the steady-state is indicated by ∗). The following remark summarizes structural

observations obtained.

Remark 7. The results reveal that the optimal evolution of the model consists of the following three phases:

•(i) In the ﬁrst phase, the initial state (cf. initial reference price r0) changes towards a steady-state.

•(ii) The second phase is characterized by the steady-state, i.e., all prices remain constant.

•(iii) At the end of the time horizon (third phase), the store price, the reference price, and the online

price decrease.

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(a) Reference price forget β=0.3 (vs. β=0.4),

steady-state: r∗=5.36, p∗

s=5.36, p∗

o=6.84

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(b) Online delivery cost co= 1 (vs. co= 5),

steady-state: r∗=5.76, p∗

s=5.76, p∗

o=4.90

Figure 4: Illustration of the impact of βand co: store price ps(t) (orange), online feedback prices po(t) (green), and reference

prices r(t) (blue) over time t= 0,1, ..., 40; Example 1.

Note, although phase (i) mainly depends on r0(cf. Figure 2, t << T), phase (ii) is independent of r0

and characterized by stable steady-state prices (denoted by r∗,p∗

s, and p∗

o), cf. Figure 3. In phase (i), the

correction towards the steady-state is obtained due to the (mean-reverting character of the) feedback prices

as for r > r∗we have r > ps(t, r)> r∗and for r < r∗we have r < ps(t, r)< r∗. At the end of the time

horizon, i.e., in phase (iii), it is optimal to decrease the store price (cf. Figure 2(a) and Figure 3) and, in

turn, the reference price – which can be seen as a demand potential – to generate ﬁnal revenues as future

revenues cannot be made anymore. Accordingly, the online price decreases due to lower reference prices, cf.

Figure 2(b).

Next, Figure 4 studies how the price paths of the reference case in Example 1, cf. Figure 3(a), change if

the reference price forget parameter βor the delivery costs coare chosen diﬀerently. Figure 4(a) illustrates

the impact of β. As expected, the adaption in phase (i) and (iii) takes longer as βis smaller compared to the

reference case, see Figure 3(a); the steady-state remains the same. Further, Figure 4(b) shows the evolution

of optimal price paths over time for a smaller markup co= 1. In contrast to the markup co= 5, cp. Figure

3(a), now the store price (orange) is higher than the online price (green) for most of the time. The switch

that the store price falls below the online price occurs in phase (iii). This phenomenon supports Remark 2.

Note, at the end of the time horizon, the store price is always lower than the online price. In the following

remark, we summarize how the optimal solution path is aﬀected by diﬀerent parameters of the model.

Remark 8. We obtain the following (numerical) sensitivity results:

•Initial reference price r0: The initial reference price aﬀects only phase (i), see paragraph following

Remark 7; i.e., if r0is larger (smaller) than the steady-state store price p∗

sthen the store and the

online price increase (decrease) to their steady-states. Consequently, an increase in r0, allowing for

greater rent extraction by the retailer, generates higher intertemporal (total) revenue.

•Reference price forget β: A larger β, i.e., a higher impact of the current store price pson the reference

price rdoes not aﬀect the steady-state, i.e., neither the reference price r∗nor the two channel prices

p∗

sand p∗

o. However, if βincreases, the system can faster converge to or leave the steady-state, and

in turn, the adjustment phase (i) and the ﬁnal phase (iii) become shorter, see Figure 4(a). Hence, the

(positive) impact of r0on intertemporal revenue in phase (i) decreases with βas the importance of the

initial state becomes less important.

•Length of time horizon T: A longer time horizon Tleads to a longer phase (ii), but does not aﬀect the

lengths of phase (i) and (iii). In other words, the length of the time horizon aﬀects only the duration

of the steady-state. The result directly follows from the Bellman principle, cf. the backward induction

used in (17).

•Online delivery cost co: A higher coleads to a lower reference price, a lower store price, and a higher

online price (in steady-state, cf. phase (ii)), see Figure 5(a). If cois small, the store price exceeds

the online price, i.e., we have ps> po(e.g., if co<2.5in Example 1). However, if cois suﬃciently

large, we have the opposite, i.e., ps< po, see Remark 1 (e.g., if co>2.5in Example 1). Further,

as expected, the intertemporal revenue decreases with co. For suﬃciently large co(e.g., co≥8.5in

Example 1), the online channel is turned oﬀ, i.e., all revenues come from the store, see Figure 5.

6.2.3. Steady-state Analysis

Moreover, the discrete-time model shows that the steady-state can be derived by value iteration (or

approximate dynamic programming). Hence, we can assume that the steady-state of the continuous time

model, cf. Section 3 - 5, coincides with the solution of the inﬁnite horizon model, cf. (4) for T=∞. Because

we do not have time-dependent model parameters, the (feedback) solution of this model will also not depend

on time, and the only state is the current reference price r. In this context, let p∗

s(r) and p∗

o(r), r≥0, denote

the feedback controls of the continuous time inﬁnite horizon model, then the steady-state is characterized

by the (constant) reference price r∗that satisﬁes r∗=p∗

s(r∗). In particular, the steady-state will neither

depend on r0nor on β, cf. (1) as well as Figure 3 and Figure 4(a).

0 2 4 6 8 10

prices

(a) Store price p∗

s(orange) and online

price p∗

o(green)

0 2 4 6 8 10

profits

*+Go

(b) Store demand D∗

s(orange), online

demand D∗

o(green), and the sum D∗

D∗

o(blue)

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

demands

*+Do

s(orange), online

revenue G∗

o(green), and the sum G∗

G∗

o(blue)

Figure 5: steady-state prices, demand, and revenues (within one period) of both channels for diﬀerent delivery cost co,0≤

co≤10; Example 1 with δ= 0.99, cs= 0.

The steady-state is, however, aﬀected by the online delivery cost, co. In the context of Example 1,

Figure 5(a) illustrates the in-store price and the online price in steady-state as a function of co. We observe

that the in-store price (p∗

s) is decreasing in cowhile the online price (p∗

o) is increasing in co, which is in

line with Corollary 2 and Remark 1. The corresponding demands and revenues of both channels (within

one unit of time in steady-state) are illustrated in Figure 5(b) and 5(c). Although in the in-store channel

the steady-state price, demand, and revenues are hardly aﬀected by co, their counterparts of the online

channel crucially depend on co. If cois suﬃciently high the online channel does not pay oﬀ, however, if co

is suﬃciently small the revenues generated through the online channel can even exceed those of the store.

Aggregated demand and aggregated revenues of the channels are both decreasing in co.

6.2.4. Diﬀerent Unit Cost Structures of Both Channels

Recall the store and online marginal cost, csand co(where cswas previously normalized to zero). As

discussed in Footnote 2 and Appendix A, as a robustness check, we now evaluate cases with cs>0 and,

in particular, cs> co, i.e., when the costs in-store exceeds those online. For the setting of Example 1,

in Figure 6 we illustrate how the optimal solution of the model is aﬀected by the cost parameter cs(cf.

cs= 3,5,8, where co= 5). In the considered setting, we observe that – even if cs> co– the structure of

solutions characterized by three phases (cf. Remark 7) remains similar to the one derived for cs= 0, see the

reference case of Figure 2(a) with r∗=5.36, p∗

s=5.36, p∗

o=6.84.

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(a) Store unit costs cs= 3, steady-

state: r∗=6.92, p∗

s=6.92, p∗

o=7.06

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(b) Store unit costs cs= 5, steady-

state: r∗=7.90, p∗

s=7.90, p∗

o=7.20

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

state: r∗=9.48, p∗

s=9.48, p∗

o=7.42

Figure 6: Impact of unit costs in-store cs= 3, 5, and 8 (cp. cs= 0), where online unit costs are co= 5: store price ps(t)

(orange) and online feedback prices po(t) (green), and reference prices r(t) (blue) over time, t= 0,1,..., 40; Example 1.

We ﬁnd that the impact of csis as follows: The higher cs, the higher is the store price (in steady-state);

online prices are less aﬀected and slightly higher. The demand and revenue realized in-store decreases. Note,

if csis suﬃciently large the store demand decreases to zero, cf. (3). Hence, the transition from cs< coto

cs> cois not critical. Instead, the structure of the solution (with three phases and positive investments)

depends on whether the store marginal costs csare small enough such that the store business still pays oﬀ

and has positive demand.

The numerical insights support the analytic results and provide a more comprehensive understanding

of the solution of the model. In addition, it provides a viable approach to further study the model from a

qualitative as well as a quantitative perspective.

6.2.5. Direct Price Interaction between Channels

Finally, we check our results for the possibility of direct price interaction between the store and online

channels. Direct price interaction does not allow to obtain analytical results because of greater complexity.

Consequently, we perform here numerical experiments to obtain insights when prices in each channel directly

interact with the demand in the other channel. The discrete-time model can also be used to numerically

solve the more general model where psand pomutually inﬂuence each other’s demand, i.e., via substitution

eﬀects within Ds(r, ps, po) and Do(r, ps, po), cf. Appendix A.

In this context, we consider an example using Ds(ps, po, r) := D(ps, po, r) and Do(po, ps, r) := D(po, ps, r),

where D(p, k, r) = µ−θp −γ(p−r)−φ(p−k) is characterized by µ= 1, θ= 0.1, γ= 0.04, and φ≥0.

Further, we consider the setting of Example 1, i.e., co= 5 and cs= 0. Compared to the model without price

interaction, i.e., φ= 0, cf. Figure 3(a) with r∗=5.36, p∗

s=5.36, p∗

o=6.84, Figure 7 illustrates the impact of

φ= 0.04 and φ= 0.08 in the considered setup. We ﬁnd that due to the higher competition (cf. cannibal-

ization eﬀects) the store price (slightly) decreases while the online price increases. Note, if φis suﬃciently

large (φ≥0.1), we observe that the online demand decreases to zero (r∗=p∗

s= 5.00 and p∗

o= 7.08), i.e.,

the two channel solution collapses. Overall, although the third phase is less signiﬁcant, we obtain that the

structure of the solution remains similar to the model without price interaction.

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(a) Price interaction φ=0.04, steady-state:

r∗=5.18, p∗

s=5.18, p∗

o=7.00

0 10 20 30 40 t

prices

ps(t)

po(t)

r(t)

(b) Price interaction φ=0.08, steady-state:

r∗=5.06, p∗

s=5.06, p∗

o=7.06

Figure 7: Evaluation over time: store price ps(t) (orange), online feedback prices po(t) (green), and reference prices r(t) (blue)

for diﬀerent price interaction coeﬃcients φ,t= 0,1, ..., 40.

Based on the numerical experiment enriching our model with direct price interaction, we conclude that

our previous results, cf. Remark 7 - 8, hold for the chosen set of parameters. For other inputs, we observed

that the overall solution structure also remained. Note, besides other price interaction eﬀects also asymmetric

demand functions can be studied in the proposed framework. Further, in the numerical model the evolution

of the the reference price, cf. (1), could also be a function of both channel prices, psand po.

7. Conclusion

This article investigates the optimal relationships in dual channel pricing between in-store and online

prices. The optimal relationships account for the interplay with a consumer psychological construct, namely,

the reference price, and the last-mile delivery cost, a burden in e-commerce. The reference price represents

an asset in which the retailer invests. Supporting the reference price is costly today because it implies a

higher selling cost than the one maximizing the current store’s revenue. But, it pays oﬀ later, developing

future both in-store and online demand, and thus future in-store and online revenues. If a retailer ignores

the long-term implications of the store’s pricing strategy, then the model indicates that the retailer will

systematically set a too low price in the store, thereby losing revenues. Also, the store and online channels

diﬀer by the delivery cost for the last-mile, a salient feature in online shopping, which the retailer must take

into account to remain proﬁtable.

Our structural (opposing parametric) results are derived from an optimal control model and hold for

general (nonlinear) reference-dependent demand formulations. Such general formulations enable the rich

possibilities of price inﬂuences and capture the nonlinearities and dynamics of the channel prices, echoing

changes in the reference price. We examine the analytic conditions along which the in-store price is higher

or lower than the online price. We also show the diﬀerence between the dynamic and static prices for both

channels. We further expand on the market power of the retailer in both in-store and online channels. We

detail the causalities of the dynamics of the reference price on the dynamics of the two channels’ prices. Our

theoretical results are complemented by numerical evaluations, which support the analysis and provide a

deeper insight into the evolution of optimally controlled prices. Speciﬁcally, we reveal three phases: (1) an

initial phase governed by the initial reference price, (2) a steady-state, (3) and a ﬁnal phase characterized by

decreasing prices. Further, we study how the evolution path of the model is (qualitatively and quantitatively)

aﬀected by diﬀerent model parameters. Also, numerical experiments show that our results hold with a more

complex and realistic model with direct price interactions from both channels and a delivery cost for in-store

sales. Eventually, our numerical experiments also provide insights about the eﬀects at play when there is

a direct price interaction between channels. Our analysis favors a better understanding of the complex

interplay of dual channel pricing, a main issue for a B&M retailer launching an e-commerce channel.

A novelty of this research is to oﬀer analytical guarantees about the eﬀects at play in dual channel

dynamic pricing. More broadly, this article articulates in a new way the following eﬀects: On the demand

side, the reference price documented with consumer behavior exerts dynamic inﬂuences; on the supply

side, the delivery cost observed for e-commerce has static inﬂuences. This articulation provides a more

comprehensive understanding of the linkages between the channel prices in the store on the one hand and

online and the reference pricing on the other hand. Results oﬀer clear-cut managerial implications. Such

implications, in turn, call for further theoretical examinations and empirical validation.

A straightforward direction for future research oﬀering insights to an alternative case is the following.

In this research, we focused on a brick and mortar retailer moving online, but keeping a clear showrooming

strategy in the dual channel journey of its customers. Consequently, consumers are used to coming to the

store for experiencing product attributes. Thus, the reference price comes from the in-store price. Yet, in

other cases, the consumers follow a webrooming journey where they search for information online or checking

online reviews (Bell et al. 2018), and then come to the store and compares the in-store price with the online

price. In this alternative situation, the reference price comes from the online price, and it may aﬀect the

purchase behavior in the physical store. Further research investigating a reference price build online would

thus be complementary to our research.

Appendix A. Optimality Conditions for the General Model

To be able to discuss the analytical tractability of more general models with direct price interactions

and arbitrary unit delivery cost for both channels we here provide the associated generalized optimality

conditions, cp. (5a)-(5f). More precisely, the demand in each channel depends on the price in both channels

with for ps, po, r > 0, Ds=Ds(ps, po, r) and Do=Ds(po, ps, r), with Ds, Do≥0.Also, the delivery costs

in-store and online write cs≥0 and cr≥0, respectively. Consequently, the Hamiltonian becomes for

ps, po, r ≥0, λin R,

H(ps, po, r, λ)=(ps−cs)Ds(ps, po, r)+(po−co)Do(po, ps, r) + λβ(ps−r).

The necessary and suﬃcient optimality conditions read for all tin (0, T ) as follows:

∂H

∂ps

= 0 =⇒Ds+ (ps−cs)∂Ds

∂ps

+ (po−co)∂Do

∂po

+βλ = 0,

∂H

∂po

= 0 =⇒(ps−cs)∂Ds

∂po

+Do+ (po−co)∂Do

∂po

= 0,

∂2H

∂p2

<0 =⇒2∂Ds

∂ps

+ (ps−cs)∂2Ds

∂p2

+ (po−co)∂2Do

∂p2

<0,

∂2H

∂p2

<0 =⇒2∂Do

∂po

+ (ps−cs)∂2Ds

∂p2

+ (po−co)∂2Do

∂p2

<0,

∂2H

∂p2

∂2H

∂p2

−∂2H

∂ps∂po2

>0 =⇒∂2H

∂p2

∂2H

∂p2

−(ps−cs)∂2Ds

∂ps∂po

−(po−co)∂2Do

∂ps∂po2

>0,

dλ

dt =rλ −∂H

∂r , λ(T) = 0 =⇒dλ

dt = (r+β)λ−(ps−cs)∂Ds

∂r −(po−co)∂Do

∂r , λ(T) = 0.

As can be seen from the optimality conditions above, the complexity of the problem increases dramatically

compared to the system of equations (5a)-(5f). Unfortunately, clear-cut analytic results are out of reach.

Therefore, we develop a simpler analytic model in Section 3, directly tied to our research questions, to obtain

actionable insights. Further, to verify the robustness of the speciﬁc analytic model, we numerically simulate

the more complete model based on the above optimality conditions in Subsections 6.2.4 and 6.2.5.

References

Agatz, N. A., M. Fleischmann, and J. A. Van Nunen (2008). E-fulﬁllment and multi-channel distribution–a review. European

Journal of Operational Research 187(2), 339–356.

Arslan, A. N., W. Klibi, and B. Montreuil (2020). Distribution network deployment for omnichannel retailing. European

Journal of Operational Research.

Arslan, H. and S. Kachani (2011). Dynamic pricing under consumer reference-price eﬀects. Wiley Encyclopedia of Operations

Research and Management Science.

Bell, D. R., S. Gallino, and A. Moreno (2014). How to win in an omnichannel world. MIT Sloan Management Review 56 (1),

45.

Bell, D. R., S. Gallino, and A. Moreno (2018). Oﬄine showrooms in omnichannel retail: Demand and operational beneﬁts.

Management Science 64 (4), 1629–1651.

Bertsekas, D. (2017). Dynamic Programming and Optimal Control. Athena Scientiﬁc.

Bi, W., G. Li, and M. Liu (2017). Dynamic pricing with stochastic reference eﬀects based on a ﬁnite memory window.

International Journal of Production Research 55(12), 3331–3348.

Bijmolt, T. H., M. Broekhuis, S. De Leeuw, C. Hirche, R. P. Rooderkerk, R. Sousa, and S. X. Zhu (2019). Challenges at the

marketing–operations interface in omni-channel retail environments. Journal of Business Research.

Brynjolfsson, E. and M. D. Smith (2000). Frictionless commerce? A comparison of Internet and conventional retailers.

Management Science 46 (4), 563–585.

Cai, Y.-J. and C. K. Lo (2020). Omni-channel management in the sharing economy era: A systematic review and future

research agenda. International Journal of Production Economics 229, 107729.

Cao, P., J. Li, and H. Yan (2012). Optimal dynamic pricing of inventories with stochastic demand and discounted criterion.

European Journal of Operational Research 217 (3), 580–588.

Cao, Y. and Y. Duan (2020). Joint production and pricing inventory system under stochastic reference price eﬀect. Computers

& Industrial Engineering, 106411.

Cattani, K. D., W. G. Gilland, and J. M. Swaminathan (2004). Coordinating traditional and internet supply chains. In In

David Simchi-Levi, S. David Wu, and Zuo-Jun (Max) Shen (Eds.), Handbook of quantitative supply chain analysis, pp.

643–677. Boston: Kluwer Academic Publishers.

Cavallo, A. (2017). Are online and oﬄine prices similar? Evidence from large multi-channel retailers. American Economic

Review 107 (1), 283–303.

Cebiro˘glu, G. and U. Horst (2015). Optimal order display in limit order markets with liquidity competition. Journal of

Economic Dynamics and Control 58, 81–100.

Chai, L., D. D. Wu, A. Dolgui, and Y. Duan (2020). Pricing strategy for b&m store in a dual-channel supply chain based on

hotelling model. International Journal of Production Research, 1–14.

Chen, X., Z.-Y. Hu, and Y.-H. Zhang (2019). Dynamic pricing with stochastic reference price eﬀect. Journal of the Operations

Research Society of China 7 (1), 107–125.

Chenavaz, R. (2012). Dynamic pricing, product and process innovation. European Journal of Operational Research 222(3),

553–557.

Chenavaz, R. (2016). Dynamic pricing with reference price dependence. Economics: The Open-Access, Open-Assessment

E-Journal 10(2016-22), 1–17.

Chenavaz, R. (2017). Better product quality may lead to lower product price. The BE Journal of Theoretical Economics 17 (1),

1–22.

Chenavaz, R. and C. Paraschiv (2018). Dynamic pricing for inventories with reference price eﬀects. Economics: The Open-

Access, Open-Assessment E-Journal 12 (2018-64), 1–16.

Chenavaz, R. Y. and S. M. Jasimuddin (2017). An analytical model of the relationship between product quality and advertising.

European Journal of Operational Research 263 (1), 295–307.

Chopra, S. (2018). The evolution of omni-channel retailing and its impact on supply chains. Transportation Research Proce-

dia 30, 4–13.

Crettez, B., N. Hayek, and G. Zaccour (2019). Existence and characterization of optimal dynamic pricing strategies with

reference-price eﬀects. Central European Journal of Operations Research, 1–19.

Digiday (2021). Pressured by amazon, retailers are experimenting with dynamic pricing. Retrieved from,https://digiday.

com/retail/amazon-retailers- experimenting-dynamic- pricing/, Accessed 04-June-2021.

Dockner, E., S. Jørgensen, V. N. Long, and S. Gerhard (2000). Diﬀerential Games in Economics and Management Science.

Cambridge University Press.

Dye, C.-Y. (2020). Optimal joint dynamic pricing, advertising and inventory control model for perishable items with psychic

stock eﬀect. European Journal of Operational Research 283(2), 576–587.

Dye, C.-Y. and C.-T. Yang (2016). Optimal dynamic pricing and preservation technology investment for deteriorating products

with reference price eﬀects. Omega 62, 52–67.

Fibich, G., A. Gavious, and O. Lowengart (2003). Explicit solutions of optimization models and diﬀerential games with

nonsmooth (asymmetric) reference-price eﬀects. Operations Research 51 (5), 721–734.

Fisher, M., S. Gallino, and J. Li (2017). Competition-based dynamic pricing in online retailing: A methodology validated with

ﬁeld experiments. Management Science 64 (6), 2496–2514.

Fruchter, G. E. and S.-P. Sigue (2009). Social relationship and transactional marketing policies—maximizing customer lifetime

value. Journal of Optimization Theory and Applications 142(3), 469–492.

Fruchter, G. E. and C. S. Tapiero (2005). Dynamic online and oﬄine channel pricing for heterogeneous customers in virtual

acceptance. International Game Theory Review 7 (02), 137–150.

Fruchter, G. E. and C. Van den Bulte (2011). Why the generalized bass model leads to odd optimal advertising policies.

International Journal of Research in Marketing 28 (3), 218–230.

Gallino, S. and A. Moreno (2014). Integration of online and oﬄine channels in retail: The impact of sharing reliable inventory

availability information. Management Science 60 (6), 1434–1451.

Gallino, S., A. Moreno, and I. Stamatopoulos (2016). Channel integration, sales dispersion, and inventory management.

Management Science 63 (9), 2813–2831.

Gao, F. and X. Su (2016). Omnichannel retail operations with buy-online-and-pick-up-in-store. Management Science 63 (8),

2478–2492.

Gao, F. and X. Su (2017). Omnichannel service operations with online and oﬄine self-order technologies. Management

Science 64 (8), 3595–3608.

Govindarajan, A., A. Sinha, and J. Uichanco (2018). Joint inventory and ful ﬁllment decisions for omnichannel retail networks.

Working Paper Ross School of Business (1341).

Grewal, D., A. L. Roggeveen, and J. Nordf¨alt (2017). The future of retailing. Journal of Retailing 93 (1), 1–6.

Harsha, P., S. Subramanian, and J. Uichanco (2019). Dynamic pricing of omnichannel inventories. Manufacturing & Service

Operations Management 21(1), 47–65.

He, Y., Q. Xu, and P. Wu (2020). Omnichannel retail operations with refurbished consumer returns. International Journal of

Production Research 58 (1), 271–290.

Helmes, K., R. Schlosser, and M. Weber (2013). Optimal advertising and pricing in a class of general new-product adoption

models. European Journal of Operational Research 229 (2), 433–443.

Helmes, K. L. and R. Schlosser (2013). Dynamic advertising and pricing with constant demand elasticities. Journal of Economic

Dynamics and Control 37 (12), 2814–2832.

Hsieh, T.-P. and C.-Y. Dye (2017). Optimal dynamic pricing for deteriorating items with reference price eﬀects when inventories

stimulate demand. European Journal of Operational Research 262 (1), 136–150.

H¨ubner, A., A. Holzapfel, and H. Kuhn (2016). Distribution systems in omni-channel retailing. Business Research 9 (2),

255–296.

Janssen, D.-J., J. Li, J. Qiu, and U. Weitzel (2020). The disposition eﬀect and underreaction to private information. Journal

of Economic Dynamics and Control 113, 103856.

Jasin, S., A. Sinha, and J. Uichanco (2019). Omnichannel operations: Challenges, opportunities, and models. In Operations

in an Omnichannel World, pp. 15–34. Springer.

Jørgensen, S. and G. Zaccour (2012). Diﬀerential Games in Marketing, Volume 15. Springer Science & Business Media.

Kalyanaram, G. and R. S. Winer (1995). Empirical generalizations from reference price research. Marketing Sci-

ence 14 (3 supplement), G161–G169.

Karray, S. and G. Mart´ın-Herr´an (2009). A dynamic model for advertising and pricing competition between national and store

brands. European Journal of Operational Research 193 (2), 451–467.

Kireyev, P., V. Kumar, and E. Ofek (2017). Match your own price? self-matching as a retailer’s multichannel pricing strategy.

Marketing Science 36 (6), 908–930.

Kopalle, P. K., A. G. Rao, and J. L. Assuncao (1996). Asymmetric reference price eﬀects and dynamic pricing policies.

Marketing Science 15 (1), 60–85.

Kopalle, P. K. and R. S. Winer (1996). A dynamic model of reference price and expected quality. Marketing Letters 7 (1),

41–52.

Li, S., J. Zhang, and W. Tang (2015). Joint dynamic pricing and inventory control policy for a stochastic inventory system

with perishable products. International Journal of Production Research 53 (10), 2937–2950.

Li, Z., Q. Lu, and M. Talebian (2015). Online versus bricks-and-mortar retailing: a comparison of price, assortment and

delivery time. International Journal of Production Research 53(13), 3823–3835.

Liu, G., S. P. Sethi, and J. Zhang (2016). Myopic vs. far-sighted behaviours in a revenue-sharing supply chain with reference

quality eﬀects. International Journal of Production Research 54(5), 1334–1357.

Lu, L., Q. Gou, W. Tang, and J. Zhang (2016). Joint pricing and advertising strategy with reference price eﬀect. International

Journal of Production Research 54(17), 5250–5270.

Lu, L., J. Zhang, and W. Tang (2016). Optimal dynamic pricing and replenishment policy for perishable items with inventory-

level-dependent demand. International Journal of Systems Science 47 (6), 1480–1494.

Martel, A. and W. Klibi (2016). Designing value-creating supply chain networks. Springer.

Mazumdar, T., S. Raj, and I. Sinha (2005). Reference price research: Review and propositions. Journal of Marketing 69(4),

84–102.

Melacini, M., S. Perotti, M. Rasini, and E. Tappia (2018). E-fulﬁlment and distribution in omni-channel retailing: a systematic

literature review. International Journal of Physical Distribution & Logistics Management 48 (4), 391–414.

Ni, J. and S. Li (2018). When better quality or higher goodwill can result in lower product price: A dynamic analysis. Journal

of the Operational Research Society 70 (5), 726–736.

Popescu, I. and Y. Wu (2007). Dynamic pricing strategies with reference eﬀects. Operations Research 55 (3), 413–429.

Proﬁtero Price Intelligence (2014). Amazon makes more than 2.5 million daily price changes. Retrieved from,https://www.

profitero.com/2013/12/profitero-reveals- that-amazon- com-makes- more-than- 2-5-million- price-changes- every-day/,

Accessed 04-June-2021.

Radhi, M. and G. Zhang (2018). Pricing policies for a dual-channel retailer with cross-channel returns. Computers & Industrial

Engineering 119, 63–75.

Reinartz, W., N. Wiegand, and M. Imschloss (2019). The impact of digital transformation on the retailing value chain.

International Journal of Research in Marketing 36 (3), 350–366.

Savelsbergh, M. and T. Van Woensel (2016). 50th anniversary invited article city logistics: Challenges and opportunities.

Transportation Science 50(2), 579–590.

Schlosser, R. (2016). Joint stochastic dynamic pricing and advertising with time-dependent demand. Journal of Economic

Dynamics and Control 73, 439–452.

Schlosser, R. (2017). Stochastic dynamic pricing and advertising in isoelastic oligopoly models. European Journal of Operational

Research 259 (3), 1144–1155.

Schlosser, R. (2019). Dynamic pricing under competition with data-driven price anticipations and endogenous reference price

eﬀects. Journal of Revenue and Pricing Management 18, 451–464.

Sethi, S. P. and G. L. Thompson (2000). What is optimal control theory? Springer.

Sorger, G. (1988). Reference price formation and optimal pricing strategies. In G. Feichtinger (Ed.), Optimal Control Theory

and Economic Analysis 3, pp. 97–120. Elsevier.

Talluri, K. T. and G. J. Van Ryzin (2006). The theory and practice of revenue management, Volume 68. Springer Science &

Business Media.

Tompkins, B. and C. Ferrell (2016). E-commerce supply chain: Survey results. Tompkins Supply Chain Consortium.

Tsay, A. A. and N. Agrawal (2004). Channel conﬂict and coordination in the e-commerce age. Production and operations

management 13(1), 93–110.

V¨or¨os, J. (2006). The dynamics of price, quality and productivity improvement decisions. European Journal of Operational

Research 170 (3), 809–823.

V¨or¨os, J. (2013). Multi-period models for analyzing the dynamics of process improvement activities. European Journal of

Operational Research 230 (3), 615–623.

V¨or¨os, J. (2019). An analysis of the dynamic price-quality relationship. European Journal of Operational Research 277 (3),

1037–1045.

Wang, J. and X. Zhang (2017). Optimal pricing in a service-inventory system with delay-sensitive customers and lost sales.

International Journal of Production Research 55(22), 6883–6902.

Wang, N., T. Zhang, X. Zhu, and P. Li (2020). Online-oﬄine competitive pricing with reference price eﬀect. Journal of the

Operational Research Society, 1–12.

Xiao, T. and J. J. Shi (2016). Pricing and supply priority in a dual-channel supply chain. European Journal of Operational

Research 254 (3), 813–823.

Xue, M., W. Tang, and J. Zhang (2016). Optimal dynamic pricing for deteriorating items with reference-price eﬀects. Inter-

national Journal of Systems Science 47(9), 2022–2031.

Zhang, C. and X. Zheng (2020). Customization strategies between online and oﬄine retailers. Omega, 102230.

Zhang, J., W.-y. Kevin Chiang, and L. Liang (2014). Strategic pricing with reference eﬀects in a competitive supply chain.

Omega 44, 126–135.

Conceptual Framework for Analysing Service Innovations, Resources, and Dynamic Capabilities in Omnichannel Retailing

Chapter

Mar 2025

To be successful in today’s competitive marketplace, retailers are required to utilise multiple channels of distribution, which is known as omnichannel retailing (OCR). This chapter identifies the necessary resources and capabilities, which retailers require to prepare an OCR strategy. Besides this, the chapter combines resources and capabilities with the notions of service innovations and distinguishes these between front- and back-end activities, which literature did not considered so far. By this, the framework integrates customer-facing interactions with operational processes with the goal of improving customer experience.

Physical and Digital Service Quality for Sustainable Customer Satisfaction in Lapo Tuak: Evidence from South Sumatra

Article

Full-text available

Jan 2025

This study investigates the dual impact of online and offline service quality on customer satisfaction at Lapo Tuak, a traditional Batak eatery that integrates physical and digital services. Utilizing purposive sampling, data were collected from 150 respondents through structured questionnaires and analyzed using Smart PLS for structural equation modeling. The findings demonstrate that both E-Service Quality and Service Quality significantly enhance customer satisfaction, emphasizing their critical roles in fostering loyalty and repeat business. While offline interactions remain essential for building trust, the growing significance of convenient digital services, such as smooth ordering and timely delivery, cannot be overlooked. Therefore, Lapo Tuak should focus on enhancing its digital platforms while maintaining strong offline service to improve customer experiences. From a marketing perspective, promoting the ease and reliability of online delivery alongside exceptional offline service can strengthen customer loyalty and broaden the business's digital presence. This research contributes valuable insights for SMEs in the culinary sector, highlighting the importance of service quality in achieving long-term sustainability in an increasingly competitive market. https://journal.binus.ac.id/index.php/BECOSS/article/view/12861

Consumer Behavior in Online Retail

Article

May 2025

Rapid technological advancements have led to significant growth in e-commerce, with more people purchasing products online. These online shopping activities are associated with multiple benefits, such as convenience, access to various products, and consumer ability to compare offers and select the most favorable deal. So, understanding consumer behavior in online retail is crucial for businesses to design effective marketing strategies, enhance the user experience, build trust, and drive sustainable growth in the digital marketplace. This paper aims to identify research trends in the field through a systematic bibliometric literature review of research on consumer behavior in online retail. The review includes 115 articles published in the Scopus™ and WoS™ databases, presenting up-to-date knowledge. The R-Tool “Biblioshiny for Bibliometrix” was used to perform a comprehensive bibliometrics analysis based on evaluative and relational bibliometrics techniques, which allow us to analyze the knowledge structure in terms of three different structures: conceptual, intellectual, and social. The findings indicate that the factors affecting consumer online behaviors in e-commerce can be categorized into seven significant classifications: demographic, psychographic, economic, technological, social, environmental, and cultural. The success of online retail businesses depends on their ability to cater to diverse customer needs and expectations to ensure that these factors are met. As a result, companies need to address potential challenges such as cybersecurity risks, data privacy concerns, cultural stereotypes, and misinterpretations that may hinder customers' willingness to participate in online shopping activities.

Quality management and feedback operation for user-generated content considering dynamic value belief

Article

Mar 2025
EUR J OPER RES

Dynamic policies with reference-price eﬀects and reference-quality eﬀects

Article

Nov 2024

Bo Chen

Dynamic pricing and inventory model for perishable products with reference price and reference quality

Article

Nov 2024
J COMPUT APPL MATH

Protecting labor rights: Contract design and coordination between brand firms and suppliers

Article

Jul 2024
EUR J OPER RES

How to leverage blockchain to react to consumer reference effects and develop a distribution strategy in online retailing?

Article

Jul 2024
TRANSPORT RES E-LOG

REFERANS FİYAT

Chapter

Full-text available

Feb 2022

Continuous dynamic pricing strategy under competition for deteriorating products in a dual-channel supply chain

Article

Apr 2024
SADHANA-ACAD P ENG S

Anand Ranjan

This study delves into the challenges and economic implications of managing deteriorating products subject to deterioration over time. It emphasises the limitations of conventional inventory models and introduces the Weibull distribution as an effective model for understanding the deterioration rate of such products under uncertainty. The economic impact of deterioration is explored, highlighting the potential for increased profit by reducing perishable waste. The necessity for proficient inventory management systems in the face of a growing market for deteriorating products is discussed. Preservation technologies are proposed as a solution to reduce waste, but their use raises environmental concerns, necessitating appropriate carbon taxes or cap-and-trade policies. The study also emphasises the benefits of dynamic pricing in navigating the fluctuating utility and demand of decaying products. Competition in the market for deteriorating products is analysed for a two-echelon supply chain scenario, revealing insights into the impact of revenue-sharing contracts on retail prices and overall supply chain profitability. We provide a comprehensive overview of the intricate considerations involved in managing deteriorating products in the supply chain, addressing economic, environmental, and competitive aspects.

Distribution Network Deployment for Omnichannel Retailing

Article

Full-text available

Apr 2020
EUR J OPER RES

This article studies the distribution problem of brick-and-mortar retailers aiming to integrate the online channel into their operations. The article presents an integrated modeling approach addressing the online channel-driven distribution network deployment (e-DND) problem under uncertainty. The e-DND involves decisions on operating fulfillment platforms and on assigning a fulfillment mission to those platforms, while anticipating the revenues and costs induced by order fulfillment, replenishment, delivery, and inventory holding. To model this problem while taking into account the uncertain nature of multi-item online orders, store sales, and capacities, a two-stage stochastic program with mixed-integer recourse is developed. Two alternative deployment strategies, characterized by allocation of orders, inventory positioning, delivery schema, and inbound flow pattern decisions, are investigated using this model. The first deployment strategy investigates the ship-from stores practice where the on-hand inventory is used for all sales channels. The second deployment strategy additionally considers the advanced positioning of inventory at a fulfillment center in the urban area where the online orders are requested. To solve the two-stage stochastic model with integer recourse, an exact solution approach combining scenario sampling and the integer L-shaped method is proposed. Numerical results, inspired by the case of a European retailer, are provided to evaluate the performance of the deployment strategies and the efficiency of the proposed solution approach.

Optimal joint dynamic pricing, advertising and inventory control model for perishable items with psychic stock effect

Article

Full-text available

Nov 2019
EUR J OPER RES

Chung-Yuan Dye

In this study, a joint pricing, advertising and inventory control problem will be investigated for a firm selling perishable products with psychic stock effect. The proposed problem is analyzed in the deterministic multi-period setting in which demand at each period depends on not only the amount of inventory displayed and advertising goodwill affected by the firm’s current and past advertising efforts but also selling price and freshness index. The objective is to determine the optimal pricing, advertising, and psychic stock strategies maximizing the discount total profit over the infinite planning horizon. Our theoretical results prove that the firm would adopt a static pricing strategy, and then demonstrate that the optimal paths of the advertising goodwill and psychic stock can be determined uniquely. We also show that the convergence of the advertising goodwill towards its equilibrium is from above or below, depending on the relative location of the initial stock of advertising goodwill with respect to the unique equilibrium stock of advertising goodwill. Moreover, a set of structural properties is developed to explore the relationships among the optimal decisions. Finally, the concluding remarks and suggestions will be provided for future studies.

What Is Optimal Control Theory?

Chapter

Nov 2021

Suresh Sethi

Optimal control theory is a branch of mathematics developed to find optimal ways to control a dynamic system. Thus the theory applies to many management science and economics problems that involve systems evolving over time. The chapter begins with basic concepts and definitions in optimal control, formulates simple optimal control problems in inventory management, marketing, and finance as examples, provides a history of the optimal control theory, defines some concepts in calculus, linear algebra, convex analysis, and notations used in the book, and concludes with a plan of the book. There are many exercises at the end of the chapter.

Joint inventory and fulfillment decisions for omnichannel retail networks

Article

Dec 2020

An omnichannel retailer with a network of physical stores and online fulfillment centers facing two demands (online and in‐store) has to make important, interlinked decisions—how much inventory to keep at each location and where to fulfill each online order from, as online demand can be fulfilled from any location with available inventory. We consider inventory decisions at the start of the selling horizon for a seasonal product, with online fulfillment decisions made multiple times over the horizon. To address the intractability in considering inventory and fulfillment decisions together, we relax the problem using a hindsight‐optimal bound, for which the inventory decision can be made independent of the optimal fulfillment decisions, while still incorporating virtual pooling of online demands across locations. We develop a computationally fast and scalable inventory heuristic for the multilocation problem based on the two‐store analysis. The inventory heuristic directly informs dynamic fulfillment decisions that guide online demand fulfillment from stores. Using a numerical study based on a fictitious network embedded in the United States, we show that our heuristic significantly outperforms traditional strategies. The value of centralized inventory planning is highest when there is a moderate mix of online and in‐store demands leading to synergies between pooling within and across locations, and this value increases with the size of the network. The inventory‐aware fulfillment heuristic considerably outperforms myopic policies seen in practice, and is found to be near‐optimal under a wide range of problem parameters.

Pricing strategy for B&M store in a dual-channel supply chain based on hotelling model

Article

Sep 2020

We use hotelling model to analyse store brands as a strategy for B&M (brick-and-mortar) retailers to combat showrooming. We investigate how national-brand product mismatch and store-brand awareness affect supply chain’s performance. We reach four major conclusions. First, store-brand strategy may be an effective means for B&M stores to mitigate showrooming. However, it’s better to introduce premium store brands. Second, the B&M store’s profit grows – and the online store’s profit declines – as national-brand product mismatch increases in breadth. When many consumers feel the national-brand product does not match their needs, a product positioning strategy for the store brand can help B&M retailers improve profit margins. Third, as national-brand product mismatch increases in depth, the B&M store’s profit rises and online store’s profit falls. If national-brand products lack many features that consumers need, a product differentiation strategy can be implemented to use store brands to fill in the gaps left by national brands. Finally, the growth of store-brand awareness will not necessarily benefit the B&M store. The impact of store-brand awareness on the B&M store’s profit depends on the hassle cost factor t, and a brand promotion strategy will reduce the loss of B&M retailer’s profit.

Omni-channel Management in the New Retailing Era: A Systematic Review and Future Research Agenda

Article

Apr 2020
INT J PROD ECON

Joint production and pricing inventory system under stochastic reference price effect

Article

Mar 2020
COMPUT IND ENG

We investigate a manufacturer’s joint dynamic production and pricing problem under stochastic reference price effect over an infinite horizon in a continuous-time framework. The demand for products, which depends on their price and a memory-based reference price, is uncertain and characterized by a stochastic process. The reference price effect varies across consumers and subjects to some randomness characterized by a stochastic process. A joint dynamic production and pricing problem to maximize the total expected profit is modeled as a stochastic optimal control problem. We give the sufficient conditions of the existence of the optimal solutions and characterize the solutions for the general convex cost structure. In addition, we show that the strategies in a linear feedback form of the state variables (i.e., inventory and reference price)are optimal for a strictly convex cost structure, and provide sufficient conditions of stability and monotone convergence properties for the expected steady-state inventory and reference price. Numerical examples and sensitivity analysis are performed to provide several important managerial insights. We find that the optimal production rate and sales price are each negatively related to the inventory stock while positively to the reference price level. In expected steady-state, demand uncertainty has no influence on optimal strategies while a higher reference price uncertainty would result in a lower production rate as well as shortage level but a higher (reference) price. Moreover, the manufacturer is worse off when demand uncertainty is increased, whereas hecan generate more profit by taking advantage of the randomness of the reference price. Finally, we show that the expected steady-state production rate decreases, but price increases as the reference effect factor or customers’ memory parameter increases.

Customization Strategies between Online and Offline Retailers

Article

Feb 2020
OMEGA-INT J MANAGE S

In this paper, we investigate the optimal customization strategies and product variety decisions for firms in different channels (online or offline). Based on an e-tailer’s and the retailers’ strategies of whether to adopt customization, we analyze four different scenarios and highlight the impact of customization on firms’ pricing decisions, profits, and consumer welfare. Furthermore, we relax the assumptions on firms’ product line design, production flexibility, as well as channel structure, and identify several managerial insights. First, with a uniform pricing scheme, the only factor that affects single-channel e-tailer/retailers’ variety decision is their cost efficiency, irrespective of other firms’ strategies. In equilibrium, the e-tailer and retailers will adopt customization and launch the customization scopes in the middle of the product line to capture as much demand as possible. Interestingly, because of the trade-off between profit margin and market share, greater variety may correspond to a lower price when the customization cost is relatively small. Moreover, we show that customization is not the only way for firms to achieve strategic excellence. Offering a limited variety of standard products, a firm still can stand out from its mass customization competitors by continuously lowering the marginal cost. Also, we claim that customization strategy may play a strategic role in directing consumer “traffic”. Therefore, adding an extra standard product line may result in e-tailer’s profit increase with customization cost when firms have symmetric cost. For similar reasons, an omni-channel e-tailer should optimally offer customized products in the online channel and standard products in offline stores.

The Disposition Effect and Underreaction to Private Information

Article

Feb 2020
J ECON DYN CONTROL

We examine the role of the disposition effect in market efficiency following the arrival of private signals to a small group of informed traders. Subjects trade an ambiguous asset via a computer-based double auction. Using a 2 × 2 × 2 design, we endow two types of signal, i.e., positive vs. negative, to informed traders with two different levels of the disposition effect, i.e., high vs. low, that are measured in two domains, i.e., gain vs. loss. We find that (1) the disposition effect measured in the gain domain has qualitatively different implications from the disposition effect measured in the loss domain; (2) following a favorable signal, informed traders with high disposition effect levels are more likely to sell and less likely to hold the asset while following an unfavorable signal, the opposite is true; (3) there is some evidence of stronger price underreaction in markets with informed traders with high disposition effect levels than in markets with informed traders with low disposition effect levels, but the effect is overall relatively weak; and finally and most importantly (4) the above results hold only when the sign of the signal matches the domain that the disposition effect levels of the informed traders are measured in.

Online-offline competitive pricing with reference price effect

Article

Feb 2020
J OPER RES SOC

Nowadays, consumers can easily compare the prices of the online and offline channels before making purchase decisions, which might arouse the reference price effect among consumers. Considering the reference price effect, how should online and offline retailers set prices? To answer it, we incorporate the reference price effect into a Hotelling model to formulate the online-offline price competition. We find that, when the reference price effect is low (high), the offline (online) retailer monopolizes the market; when the reference price effect is moderate, the retailers co-exist. Interestingly, a first mover disadvantage and a second mover advantage exist in terms of market share, but they do not necessarily exist in terms of profit. However, both the online and offline retailers can be better off by negotiating on the game sequence. Furthermore, we find that each retailer prefers its own price to be regarded as the reference price. A lower reference price, although benefiting the e-retailing in the short run, might compromise the product image and hurt both retailers in the long run.

Dynamic Pricing with Reference Price Effects in Integrated Online and Offline Retailing

Abstract and Figures

Recommended publications

Competitive Dual-Collecting Regarding Consumer Behavior and Coordination in Closed-Loop Supply Chain

The impact of consumer behavior on preannounced pricing for a dual‐channel supply chain

Impacts of Online and Offline Channel Structures on Two-Period Supply Chains with Strategic Consumer...

Joint Dynamic Pricing and Marketing-Mix Strategies for Revenue Management Applications with Stochast...

A Joint Dynamic Pricing and Multi-Channel Advertising Model with Stochastic Demand

Dynamic Pricing, Reference Price, and Price-Quality Relationship ROUND 3

Dynamic pricing for inventories with reference price effects