Quality differentiation and firms’ choices between online and physical markets

Accepted Manuscript

Quality differentiation and firms’ choices between online and physical markets Yijuan Chen, Xiangting Hu, Sanxi Li PII: DOI: Reference:

S0167-7187(17)30017-6 10.1016/j.ijindorg.2017.01.003 INDOR 2344

To appear in:

International Journal of Industrial Organization

Received date: Revised date: Accepted date:

29 February 2016 8 October 2016 5 January 2017

Please cite this article as: Yijuan Chen, Xiangting Hu, Sanxi Li, Quality differentiation and firms’ choices between online and physical markets, International Journal of Industrial Organization (2017), doi: 10.1016/j.ijindorg.2017.01.003

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Highlights • We develop a theoretical framework to study firms’ choices between online and physical markets with respect to product quality and competition.

• A pooling effect suggests that information asymmetry attracts low-quality products to the online market.

• A differentiation effect indicates that a firm with a lower-quality product may prefer to reprice competition.

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veal its product quality in the physical market because quality differentiation helps alleviate

• In an entrant-incumbent model, the entrant with product quality lower than that of the offline incumbent may choose the physical market.

• In a simultaneous-move model, the two contrasting effects give rise to a wide range of

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product quality—from low-end to high-end—in both the online and offline markets.

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Quality differentiation and firms’ choices between online and physical markets∗ Yijuan Chen†, Xiangting Hu‡, Sanxi Li§

Abstract

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January 18, 2017

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We study firms’ choices between online and physical markets with respect to product quality and competition, and examine consequences of transparency policies on price competition and market structure. We investigate two contrasting forces. First, since consumers cannot fully inspect an online product’s quality prior to purchase, conventional wisdom and some of the literature suggest that this attracts low-quality products to the online market (a pooling effect). On the other hand, the literature on vertical product differentiation indicates that a firm with a lower-quality product may prefer to reveal its product quality in the physical market because quality differentiation helps alleviate price competition (a differentiation effect). We show that an entrant firm with product quality lower than that of the offline incumbent may choose the physical market, whereas the entrant with a quality higher than the incumbent’s may sell online. More generally the two contrasting forces can give rise to a wide range of product quality—from low-end to high-end—in both markets.

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Keywords: online vs. offline competition; market choices with respect to quality and competition JEL Classification: L13

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We are grateful for comments from Martin Richardson, three anonymous reviewers and the Editor, and seminar participants at the La Trobe University, Huazhong University of Science and Technology, Southwestern University of Finance and Economics, Peking University, and Monash University. We are also obliged to comments from participants at the 6th Conference on Microeconomic Theory, Beijing, and the 2015 Conference on Industrial Organization, Jinan. We thank the financial support from the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (16XNB018). † Australian National University; E-mail: [email protected]; Corresponding author. ‡ Hanqing Advanced Institute of Economics and Finance, Renmin University of China; E-mail: [email protected]; Corresponding author. § School of Economics, Renmin University of China; E-mail: [email protected]; Corresponding author.

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Introduction

Growing with the prominence of e-commerce is the concern of researchers and policy makers regarding the quality of online products. Unlike at brick-and-mortar stores, where consumers easily obtain first-hand experience with product quality, an online buyer cannot physically inspect a product before placing an order. This suggests that online markets tend to drive away firms with high-quality products and accommodate those that offer inferior quality. For example, Jin and Kato (2006, 2007) show that ungraded sport cards sold online are of lower quality than those in the offline market. The rationale is akin to a benchmark result in the literature on ”voluntary

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disclosure” (Grossman and Hart, 1980; Jovanovic, 1982): since selling in a physical market is usually more costly than selling online, only sellers with higher-quality products will bear the cost and disclose their qualities via consumer inspection in the physical market.1 The sellers with lower-quality products pool themselves in the online market—a result that we call the ”pooling effect.”

However, there may also exist an opposite driving force: while the ”pooling effect” explains

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well why the online market is more attractive to bads—that is, products that consumers will not buy if they know the quality—in reality, there are many goods with qualities ranging from low to high. Taking into account quality differentiation of goods, the stream of literature following Shaked and Sutton (1982) shows that differentiation may help alleviate price competition, and, thus, a firm may voluntarily choose to provide a low-quality good if the competitor produces a high-quality product. In the context of market choice, this means that a low-quality firm may

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want to choose the physical market (and, thus, reveal its quality) if the competitor’s quality is high. In other words, one can argue that the physical market has a ”differentiation effect” that may also attract low-quality firms.

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Since the pooling effect and the differentiation effect act in opposite directions, it is not clear how they together affect firms’ market choices and the quality distribution between the markets. Moreover, how do transparency policies in online markets influence their structure and that of

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offline counterparts? We develop a theoretical framework to investigate these questions. The study is conducted in two steps. First, we investigate a situation in which an entrant chooses

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between online and offline markets to compete with an offline incumbent. Next, we examine the setting in which two firms simultaneously choose between the markets. To study the entrant’s market choice, we first analyze a baseline model where the entrant

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only produces goods. We show that, due to the differentiation effect, in the offline market, the entrant’s profit increases with the difference between the quality of its product and the incumbent’s. This drives the entrant with highly differentiated product qualities—which consist of not only the highest qualities, but also the lowest ones—to choose the offline market. However, the differentiation effect diminishes as the entrant’s product quality becomes closer to the incumbent’s. We assume that the offline market is more costly than the online market. Consequently, there is an interval of qualities such that at the two boundary points, the entrant will be indifferent between the two markets. For medium-differentiated qualities, which are slightly above the lower bound or below the upper bound of the aforementioned interval, the entrant will choose 1

In this paper, the words ”offline” and ”physical” are interchangeable.

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ACCEPTED MANUSCRIPT the online market mainly to save the costs of the offline market. The pooling effect then drives the low-differentiated qualities—namely, those near the incumbent’s quality—to pool with the medium-differentiated qualities in the online market. Moreover, the differentiation effect implies that, all else equal, the offline market will be more attractive to the entrant with higher qualities. As a result, the average quality of the online good is lower than the quality of the incumbent’s. However, the upper bound of the above-mentioned quality interval is above the incumbent’s quality. Hence the actual quality of the online good may be higher than the incumbent’s.

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The US mattress industry offers a good example for the above results. Although many brands of mattresses can be tested and bought in physical stores, some newer brands in the US sell almost completely online. Among these brands a famous one is Casper.com. According to third-party reviews,2 Casper sells a medium-quality mattress at an intermediate price: Its products are cheaper and less comfortable compared with the most expensive ones that are sold offline, but are also more expensive and better than the cheapest ones at physical stores.

We then analyze two extensions of the entrant-incumbent model. First, we expand the support

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of the entrant’s product quality to allow its product to be fraudulent. We compare the case where the entrant can only choose the physical market with that where it can select from the online and offline markets. In equilibrium, the lack of transparency of the online market causes it to attract fraudulent products. However, consistent with the baseline model, when the likelihood that the product is bad is low, the entrant with qualities close to the incumbent will continue to choose the online market, including those that are (modestly) higher than the incumbent’s.

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This result suggets a potential unintended consequence of the transparency policies: An overall more transparent online market will certainly prevent ”lemons”, however it will also dampen the

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pooling effect, costing the firms an avenue to alleviate price competition. The consequence is that some good products may also have to exit the markets, online and offline. Therefore it may be worth considering policies that target more specifically at fraudulent products while preserving

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the opaqueness of the online market.

In the second extension we show that, as the entry cost to the online market increases, the online market becomes less attractive to the entrant. Consequently, the range of qualities which

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goes online becomes narrower, thus the expected online quality becomes closer to the incumbent’s. This intensifies the price competition, and, as a result, the entrant’s online profit further deceases. We show that there exists a threshold of the online entry cost, beyond which the entrant will

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choose to either sell offline or remain inactive. While the entrant-incumbent model better fits the situation where the online business just

emerges and starts to compete with traditional brick-and-mortar retailers, the simultaneous-move model sheds light on the scenario where both e-business and physical shops are mature options for competing firms that offer new products. We conduct the analysis under two equally compelling, alternative assumptions. First, we assume that each firm only knows its own product quality before they simultaneously choose markets. In this case, a firm’s market choice can only depend on its own quality and the expected quality of the competitor. The situation for each firm, 2

Information about this industry, including product review, is available at http://www.sleeplikethedead.com/

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ACCEPTED MANUSCRIPT therefore, is similar as where the entrant chooses a market with the incumbent’s quality held as a constant. We show that each firm will choose the physical market when its quality is sufficiently different from the other’s expected quality, while it will choose the online market when its quality and the opponent’s expected quality are close. Alternatively, we assume that the firms know both qualities before they choose markets. We construct an equilibrium in which the market choices can be characterized by three areas of product qualities. In the first area, the firms’ qualities are low, and the pooling effect dominates the differentiation effect. As a result both firms choose the online market to avoid the higher

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costs of the physical market. Secondly, consistent with the entrant-incumbent model, when the firms’ qualities are sufficiently different, the differentiation effect outweighs the pooling effect. Consequently, both firms choose the physical market. Thirdly, when the firms’ qualities are sufficiently high (higher than when both are online) and sufficiently close (closer than when both are offline), we show that the firm with the higher quality will choose the physical market, while the lower-quality firm will sell online. A remarkable result here is that, when both firms have the highest quality, they will choose different markets. In this case, the online market is no longer a price competition in the physical market.

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shelter for low-quality products; rather, it is essentially used by high-quality firms to avoid direct Overall, we show that the pooling effect and the differentiation effect give rise to a large set of possible product qualities in each of the online and offline markets, ranging from low-end to high-end.

More generally, the physical and online markets can be regarded as two markets that differ

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in the extent to which they can reveal product qualities. This suggests another interpretation, and a policy implication, of our results: To tackle the potential adverse-selection problem, online

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markets have implemented various mechanisms, including consumer review, seller ranking, etc., to enhance the transparency of product quality. Our results suggest that firms prefer the more transparent market when their qualities sufficiently differ, and they will choose different markets

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when their qualities are close. Therefore, implementing a quality-disclosure mechanism in a market may result in more-differentiated qualities in the market, but not necessarily in better average product quality.

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The rest of the paper is organized as follows. In Section 2, we review the related literature. Section 3 is devoted to the entrant-incumbent model. In Section 4, we investigate the situation in which the firms simultaneously choose between the markets. We discuss practical implications

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and extensions of the model in Section 5. Section 6 concludes.

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Related Literature

Our paper is related to three strands of literature. First, Lieber and Syverson (2011) provides a comprehensive review of recent literature on online versus offline competition. In particular, our paper is related to the theoretical studies in this stream. Baye and Morgan (2001) examine how a market-maker (gatekeeper) set entry fees for firms and consumers to participate in the market and share price information, and how the fee-setting decisions impact, and are impacted by,

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ACCEPTED MANUSCRIPT the competitiveness of the product market it serves. Dinlersoz and Pereira (2007) study physical retailers’ adoption of e-commerce in a technology-adoption-race framework where some customers have loyalty for particular firms while others buy from the lowest-price firm. Using another model with loyal consumers versus price-sensitive ”switchers,” Kocas and Bohlmann (2008) study price dispersion between firms with homogeneous products. Loginova (2009) studies the strategic interactions between online and offline markets in a Salop (circular city) model. These prior studies assume firms’ marketplaces are exogenous to product quality.3 By contrast we endogenize the firm’s market choice with respect to product quality. The theme of this paper, therefore, is the differentiation effect is absent by assumption.

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in line with Chen et al. (2015). The latter, however, is abstracted from competition, and thus Secondly, the present study is related to the literature on information disclosure. In addition to the seminal papers such as Grossman and Hart (1980) and Jovanovic (1982), there is a stream of more recent papers that study information disclosure in a competitive environment. Cheong and Kim (2004) demonstrate a model, where, as the number of competing firms increases, in the limit no firm will disclose information. The market outcome thus converges to complete concealment

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of information. Levin, Peck and Ye (2009) analyze costly quality disclosure with horizontally differentiated products, and they compare the welfare under a duopoly with that under a cartel. Janssen and Roy (2014) investigate a situation where, despite very little disclosure costs, the competing firms choose not to use quality disclosure to communicate their private information before engaging in price competition. The previous studies normally assume that the competing firms are located in a single market, and to disclose information firms resort to mechanisms

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such as third-party certification, and advertisement (verifiable self-disclosure). In contrast, in this paper we take one step ”backward” and incorporate the stage at which the firms choose

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marketplaces. We show that with multiple marketplaces, the market choice per se can be used to (costly) disclose or hide product quality. In term of modelling approach, our paper is more related to Board (2009). However, he focuses on the welfare analysis of mandatory disclosure

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law on competing firms, and the equilibrium is such that the higher-quality firm always discloses, while the lower-quality firm discloses if its quality is neither too close to nor too far below that of the higher-quality firm. Thus, our paper differs from his in both its theme and results. In

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particular, a mandatory disclosure law, in the context of market choice, will require all firms to choose the physical market, which clearly is not feasible. Thirdly, our analytical framework is related to the literature on vertical product differenti-

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ation. Following the aforementioned seminal paper in 1982, Shaked and Sutton (1983) further characterize conditions in a vertical-differentiation model under which there is an upper bound on the number of active firms, and thus coin the concept ”natural oligopoly”. Sutton (1991) conducts an extensive study of industries where the relation between market size and concentration differs, and provides theoretical explanations within the vertical differentiation framework. In a more general model, Lahmandi-Ayed (2000) gives a more complete characterization of conditions 3

In Dinlersoz and Pereira (2007), the online good’s quality differs from the offline good’s by an exogenous constant. In Loginova (2009), firms’ choice of market type is also exogenous to the product quality, which is identical across firms. Consumers’ uncertainty about an online product’s quality is modeled via her uncertainty about her own type, which determines how well the product fits her and, before purchase, can be found out only by visiting a physical store.

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ACCEPTED MANUSCRIPT under which natural oligopolies exist. In this stream of literature, firms’ products are endogenously differentiated as a mean to alleviate price competition. In contrast, in the current paper we show that firms with exogenously differentiated products can take advantage of the (lack of) transparency of the markets, and use their market choices to alleviate price competition.

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Market Choice by the Entrant

3.1

Model setup

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There are two profit-maximizing firms i = A, B. Firm A is an entrant that chooses between an offline market and an online market, and firm B is an incumbent that sells in the offline market. Each firm’s product quality qi is an independent draw from a uniform distribution on [q, q¯] with 0 < q < q¯. A firm’s product quality is observable to the public in the offline market. But if firm A sells online, then qA will be unobservable to consumers prior to consumption and to the incumbent as well.4 After firm A chooses the market, the firms simultaneously post their prices

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pi , which are publicly observable. For ease of notation, denote the expected quality Q = (q + q¯)/2. We assume that the fixed cost of selling on the offline market, including rent and utility expenses, is larger than the fixed cost of selling online. For simplicity, we assume that the fixed cost to enter the offline market is F , and the entry cost for the online market is zero.5 The marginal cost of production is constant and normalized to 0.

The firms compete for a continuum of utility-maximizing consumers, indexed by j. Each

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consumer has a unitary demand and is characterized by a type λj , which measures the consumer’s ¯ with marginal utility of quality and is an independent draw from a uniform distribution on [λ, λ] ¯ Consumers are of mass M = λ ¯ − λ. For ease of illustrating the differentiation 0 < λ < λ.

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effect, we adopt a standard specification of consumer utility as in Belleflamme and Peitz (2010): a consumer’s (indirect) utility from buying firm i’s product is u(qi , pi , λj ) = r + λj qi − pi , where

r > 0 represents the basic willingness to pay for the product, and is assumed to be sufficiently

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large so that all consumers will buy from one firm in equilibrium. Denote firm i’s profit by π i . If a mass of m consumers buys from firm i, then π i = mpi − F

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if the firm sells in the offline market, and π i = mpi if it sells online.

Analysis

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A strategy of the entrant (firm A) is to choose a market and then a price, and a strategy of

the incumbent (firm B) is to choose a price given the entrant’s market choice. A consumer’s

strategy is to decide which firm to buy from given the prices and the observability of product quality. Because the entrant’s product quality will be unobservable in the online market prior to purchase, the game is essentially a dynamic game with incomplete information, and our solution concept is the perfect Bayesian equilibrium (PBE), which consists of a sequentially rational strategy profile and a consistent belief system. 4 We model uncertainty in the online market as consumers’ uncertainty about a firm’s product quality. In contrast, Loginova (2009) assumes that consumers are uncertain about their own types if they buy online. Therefore quality signalling by the firm’s market choice is abstracted from her model. 5 We will relax this assumption and consider positive online entry cost in Subsection 3.3.

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ACCEPTED MANUSCRIPT A PBE is separating if the entrant always chooses the offline market, as this will lead to perfect revelation of its product quality. An equilibrium is pooling if the entrant always chooses the online market, since the market choice will not reveal extra information about its quality. An equilibrium is partially pooling (partially separating) if the entrant chooses the offline market for some qualities and the online market for others. In equilibrium, the entrant will choose a market if the profit from that market is higher than that from the other. We first study the firms’ equilibrium behaviors in two situations: (i) the entrant has chosen the offline market; and (ii) the entrant has chosen the online market.

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Lemma 1 below shows the profits of the two firms if they compete in the offline market. Lemma 1 Suppose that both firms sell in the offline market. Then, given (qi , q−i ) with qi > q−i ,

in equilibrium the firms’ profits are π i = δ H (qi − q−i ) − F and π −i = δ L (qi − q−i ) − F , where ¯ − 2λ)2 and δ H = 1 (2λ ¯ − λ)2 . δ L = 1 (λ 9

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Proof. In the appendix.

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Lemma 1 is a standard result in the literature of vertical product differentiation (Belleflamme and Peitz, 2010): in the offline market, if the firms have the same quality, they will essentially be engaged in a cut-throat Bertrand competition. But when their qualities differ, their equilibrium profits both increase with the quality difference (qi − q−i ). Hence, quality differentiation helps

alleviate price competition and leads to higher profits for both firms, with the higher-quality firm making more profit than the other.

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Next, if firm A chooses the online market, then consumers have to base their purchase decision on the expected quality of the firm’s product. Suppose that there is a set QA ⊂ [q, q¯] such that firm

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A will choose the online market if qA ∈ QA . Denote QA = E[qA |qA ∈ QA ], which is the expected quality of firm A’s product from consumers’ perspective. The next corollary characterizes the firms’ equilibrium profits if the entrant chooses the online market:

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Corollary 1 If firm A chooses the online market, then in equilibrium π A = δ H (QA − qB ) and

π B = δ L (QA − qB ) − F if QA > qB ; π A = δ L (qB − QA ) and π B = δ H (qB − QA ) − F if QA < qB .

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Proof. The proof essentially replicates that of Lemma 1 by replacing qi and q−i with QA and qB , and thus is skipped.

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The proposition below characterizes a partial pooling equilibrium in which the entrant chooses

the offline market if its product quality is sufficiently high or sufficiently low, while for intermediate 2 qualities it chooses the online market. For ease of notation, denote q A = qB − δH +δ · δδHL · F , and L

q A = qB +

2 δ H +δ L

· F.

Proposition 1 If the support of the entrant’s product quality is sufficiently large, such that q < q A and q A < q, then there exists an equilibrium in which the entrant will choose the offline market if qA ∈ [q, q A ) ∪ (q A , q] and the online market if qA ∈ [q A , q A ]. Proof. In the appendix.

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Figure 1: Market choice by the entrant and average quality of the online good

The differentiation effect and the pooling effect, together, give rise to the entrant’s market choice in Proposition 1, the intuition for which has several layers. First, as Lemma 1 shows, the differentiation effect increases with the difference in the firms’ product qualities. Hence, the

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entrant with a quality close to the high-end q¯ or the low-end q has a stronger incentive to choose the offline market than does the entrant with a quality close to the incumbent’s. For the highest qualities [q A , q] as well as the lowest ones [q, q A ], the entrant has no incentive to pool with the qualities in the online market because, if it does, it will be regarded as having an intermediate quality, which, due to competition with the incumbent, will lead to a less profitable price than in the offline market. Consequently, as shown in Figure 1, the entrant will choose the offline market

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despite its higher fixed cost.

Secondly, the differentiation effect diminishes as the entrant’s quality approaches the incum-

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bent’s. For the entrant’s quality that is slightly above q A or below q A , if the two markets are equally costly the entrant will still make a higher profit from the offline market than from going online. However, the higher fixed cost F of the offline market now drives the entrant to choose the

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online market. Those medium-differentiated qualities then become the target for qualities around qB to pool with in the online market. It is worth noting that, in contrast to the conventional wisdom, in this case, the higher qualities (those close to qB ) want to pool with the lower qualities

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(those close to q A ), not the other way around. Moreover, q¯A > qB implies that the actual quality of the online product may be higher than the incumbent’s. Thirdly, Proposition 1 implies that in equilibrium the expected quality of the online product

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is QA = qB −

δ H −δ L δ L (δ H +δ L )

· F < qB , as shown in Figure 1. That is, the online product is, on

average, lower-quality than the offline product. To see the intuition of this result, note that 00

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0 and q such that q − q 0 = q 00 − q (i.e., q 0 and q are equally distant because δ H > δ L , for qA B B A A A A A

00 ’s profit from choosing the offline market is higher than that of from qB ), the higher quality qA 0 . So the offline market is more attractive to q 00 than to q 0 . Hence, there are the lower quality qA A A

more qA < qB than qA > qB to choose the online market. Consequently, the expected quality of the online product falls below the quality of the incumbent’s product in the offline market. It is worth noting that in Proposition 1 q A and q A are independent of q and q. Hence, the condition that q < q A and q A < q, under which the partial-pooling equilibrium exists, is fairly ¯ and F. general. Below is a numerical example in which [q , q ] is characterized given qB , λ, λ, A

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ACCEPTED MANUSCRIPT ¯ = 20, and F = 5. Then Proposition Numerical Example: Suppose that qB = 10, λ = 1, λ 1 implies that q A = 9.77, q A = 10.05, and, thus, QA = 9.91. When the entrant chooses the online market—i.e., qA ∈ [q A , q A ]—the profits of the entrant and the incumbent are π A = 3.24

and π B = 10.23.

3.3

Adverse selection with a fraudulent product

The baseline model illustrates how the pooling and differentiation effects together drive a firm

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to choose between the online and offline markets. In this subsection we extend it to incorporate the possibility that the entrant may produce a fraudulent product, which consumers will not buy if they know the true quality.6 We conduct the analysis in two steps. First, we consider the case with the absence of the online market. The entrant only chooses between entering the physical market and remaining inactive. Next we add the online market into the entrant’s options. Comparing the two cases then allows us to examine the impact of online business on the quality of products that appears in the markets.

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Specifically, to capture the possibility of fraudulent products, we extend the support of the entrant’s product quality to [q, q¯] such that q < 0 < q¯, and reduce the consumers’ reservation value r to zero. Moreover, assume that λj is uniformly distributed on [0, 1]. Thus a consumer’s (indirect) utility from buying firm i’s product is u(qi , pi , λj ) = λj qi − pi , which implies that

a consumer will not buy if she knows qA is negative. We maintain the assumption that F

is sufficiently small so that the physical market will always be chosen for some quality of the

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entrant.

The next proposition characterizes the entrant’s market choice with respect to its product

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quality when it only has the option to sell offline or to remain inactive.

Proposition 2 Suppose the entrant can only choose between entering the physical market and

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remaining inactive. Then there exists three threshold value qˆ1 , qˆ2 , qˆ3 , with 0 < qˆ1 < qˆ2 < qB < qˆ3 < q, such that firm A chooses to sell offline if qA ∈ (ˆ q1 , qˆ2 ) ∪ (ˆ q3 , q) and remains inactive if

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qA ∈ [q, qˆ1 ] ∪ [ˆ q2 , qˆ3 ].

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Proof. In the appendix.

Figure 2 gives a graphical illustration of Proposition 2. Intuitively, when the physical market

is the only entry option, the entrant will refrain from the market if its product is fraudulent, because otherwise consumers can verify the quality and refuse to buy. However, the entrant may still remain inactive even if it is producing a ”good”. This occurs when its quality is sufficiently low or sufficiently close to the incumbent’s. With a low quality, it can not attract enough demand in the physical market, and thus its profit cannot cover the entry cost. When the entrant’s product is sufficiently close to the incumbent’s, the fierce price competition with the incumbent will also 6

We are indebted to comments from the anonymous reviewers that lead to the results in this and the following subsections.

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Figure 2: When the entrant may produce a fraudulent product and can sell offline or remain inactive (Without online market)

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Figure 3: When the entrant may produce a fraudulent product and can sell online, offline, or remain inactive (With online market)

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lead to a profit below the entry cost. In contrast, when qA is sufficiently high and differentiated from qB , i.e. in (ˆ q1 , qˆ2 ) or (ˆ q3 , q), the differentiation effect attracts the entrant to the physical market.

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Next, suppose the online market is also an option for the entrant. The following proposition characterizes the equilibrium.

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Proposition 3 Suppose the entrant can choose between the online and offline markets. Then there is a threshold value qˆ < 0, such that

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(i) if q < qˆ, there exists an equilibrium in which firm A chooses to sell offline if qA ∈

(ˆ q1 , qˆ2 ) ∪ (ˆ q3 , q) and remains inactive if qA ∈ [q, qˆ1 ] ∪ [ˆ q2 , qˆ3 ].

(ii) if q ≥ qˆ, there exists an equilibrium in which there are three threshold value q˜1 , q˜2 , q˜3 , with

0 < q˜1 < q˜2 < qB < q˜3 < q, such that firm A chooses to sell offline if qA ∈ (˜ q1 , q˜2 ) ∪ (˜ q3 , q) and

online if qA ∈ [q, q˜1 ] ∪ [˜ q2 , q˜3 ]. Moreover, we have the order qˆ1 < q˜1 ; qˆ2 > q˜2 ; and qˆ3 < q˜3 . The

expected online quality QA is strictly lower than the incumbent’s quality qB . Proof. In the appendix.

In Proposition 3, the entrant’s choice in the case q < qˆ coincides with that in Proposition 2, that is, the entrant will shun the online market. Since quality is observable in the physical 11

ACCEPTED MANUSCRIPT market, the only possibility for the entrant to sell a fraudulent product is to go online. In term of the viability of the online market, qˆ can be interpreted as the bottom line for how low the entrant’s quality can be. Thus q < qˆ corresponds to the scenario where the the entrant’s product is so likely to be fraudulent that consumers completely refrain from buying online. Consequently, the entrant’s options are essentially reduced to the physical market and remaining inactive. Figure 3 illustrates the entrant’s market choice under q ≥ qˆ, as characterized by part (ii) of

Proposition 3. The condition q ≥ qˆ corresponds to the situation where the probability that the

entrant’s product is fraudulent is low. As a result, despite the possibility of buying a ”lemon”,

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consumers will still buy from the online market because the expected quality is high enough. Then the entrant who otherwise would have stayed out of business now enters the online market. This occurs when its quality is sufficiently low, i.e. in [q, q˜1 ], or sufficiently close to the product quality of the incumbent, i.e. in [˜ q2 , q˜3 ].

Following Proposition 2, the social welfare with only the physical market is: (Z

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λqB dλ dqA +

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q

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1

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1 q−q

λqB dλ dqA

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−F

(1)

Following Proposition 3, if q > qˆ, the social welfare with the online and physical markets is: (Z

qA ∈[q,˜ q1 ]∪[˜ q2 ,˜ q3 ]

+

Z

ˆ A ,qB ) λ(Q

ˆ A ,qB ) λ(q

˜ A ,qB ) λ(q

"Z

qA ∈(˜ q2 ,˜ q3 )

λqA dλ +

ˆ B ,qA ) λ(q

˜ B ,qA ) λ(q

λqA dλ +

˜ A ,qB ) λ(Q

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qA ∈(˜ q1 ,˜ q2 )

"Z

"Z

M

1 q−q Z +

λqB dλ +

Z

1

ˆ A ,qB ) λ(q

Z

#

1

ˆ A ,qB ) λ(Q

λqB dλ dqA

#

(2)

λqB dλ dqA

1

ˆ B ,qA ) λ(q

Z

#

λqA dλ dqA

)

−F

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˜ A , qB ) and λ(Q ˆ A , qB ) are defined in the proofs of In (1) and (2), the cut-off values such as λ(q Propositions 2 and 3. Although (1) and (2) indicate that it will be less tractable to directly compare the social welfare between with and without the online market, Propositions 2 and 3

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shed light on the impact of introducing the online market on social welfare. Recall the result in the baseline model that the actual quality of the online product may be higher than that in

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the physical market. Proposition 3 shows that this result still holds despite the possibility of a fraudulent product. While the conventional wisdom suggests that the online market provides a haven for low-quality products, from Proposition 3 we see that this view only depicts half of the picture. The accommodation of ”lemon” products certainly reduce the social welfare, but the online market also improves social welfare on two accounts: First, the online market allows consumers with low propensity of purchase (low λ) to buy products. As we show in the proof of Proposition 2, without the online market, when the entrant remains inactive only consumers with λ >

1 2

will buy from the incumbent. But with the online

market, as shown in Proposition 3, the entrant with a low-quality good is able to serve the low-λ consumers in the online market, improving their welfare.

12

ACCEPTED MANUSCRIPT Secondly, the online market provides an outlet for the entrant’s quality that is close to the incumbent’s. Without the online market, when the products of the entrant and the incumbent are of similar quality, the entrant will have to stay out of the business due to the entry cost and the fierce price competition. As we can see from Proposition 2 and Figure 2, this may happen even if the entrant’s quality is actually higher than the incumbent’s. But thanks to the pooling effect, such products can survive on the online market.

Positive entry costs in the online market

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3.4

In this subsection we extend the baseline model to include a positive entry cost in the online market: Assume that the entry cost of the physical market is F1 while the entry cost of the online market is F2 . The baseline model, therefore, is a special case with F2 = 0. Due to the entry costs, we further assume that the entrant also has the option to remain inactive, which leads to zero profit. In the following analysis, we will hold F1 fixed, and investigate how the change of F2 affects the entrant’s decisions. Consistent with the baseline model, we will focus on

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the situation where the support of product quality [q, q¯] is sufficiently large such that Proposition 1 holds.

Intuitively, the online market becomes less attractive to the entrant as its entry cost increases. This leads to a narrower range of qualities in the online market, which makes the expected quality in the online market closer to the incumbent’s quality. This intensifies the price competition

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between the online entrant and the offline incumbent, causing the online market to be even less attractive to the entrant. We can show that there exists a threshold online entry cost Fˆ2 . Below the threshold, the entrant’s market choice is similar as that in Proposition 1. Above Fˆ2

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the entrant will abandon the online market, and only choose between the physical market and staying inactive. Proposition 4 below formalizes this result, followed by a graphical illustration

PT

in Figure 4.

Proposition 4 If the support of product quality [q, q¯] is sufficiently large, there is a threshold Fˆ2 ∈ (0, F1 ), such that:

CE

i) For F2 < Fˆ2 , there exists an equilibrium where the entrant will choose the offline market if

AC

qA ∈ [q, q A ) ∪ (q A , q] and the online market if qA ∈ [q A , q A ]. ii) For F2 > Fˆ2 , there exists an equilibrium where the entrant will choose the offline market if qA ∈ [q, q 0A ) ∪ (q 0A , q] and remain inactive if qA ∈ [q 0A , q 0A ]. Proof. In the appendix.

4

Simultaneous Market Choices

Next, we investigate the situation in which the firms simultaneously choose between online and offline markets. While the entrant-incumbent model better fits situations where the online business 13

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Figure 4: Positive entry costs in the online market

started to develop and compete with traditional brick-and-mortar retailers, the simultaneousmove model will shed light on the scenario where both e-business and physical shops are mature

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options for firms that offer new products. Moreover, the simultaneous-move firms will be ex ante identical, and thus the model will be less biased than the baseline model. In addition, the simultaneous model offers a robustness check for the previous results.

The timeline involves three stages. At the first stage, nature independently draws each firm’s product quality qi , i = A, B, from a uniform distribution on [q, q¯] with 0 < q < q¯. At the next stage, the firms simultaneously choose between online and offline markets. At the third stage, the

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firms announce prices. The other settings replicate the baseline model in the previous section. We will divide our analysis into two cases, corresponding to two possibilities at the first stage. In the first case, we assume that each firm only observes its own product quality; in the

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second case the qualities qA and qB are observable to both firms, but not to the public, before they choose markets. As the analysis below will show, the intuition from the previous entrantincumbent model carries over to the first case, and thus despite both firms choosing markets

PT

simultaneously, we derive a result similar to Proposition 1. The analysis for the second case will be more intricate, but the interplay between the differentiation effect and the pooling effect is

4.1

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still at the core of the firms’ decisions.

The firms observe only their own qualities before choosing a market

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Suppose each firm observes only its own quality before they simultaneously choose between the online and offline markets. Following this specification, we assume that at the second stage a firm’s product quality will become common knowledge after it chooses the physical market but will remain unobservable to the competitor and the consumers if it goes online. After the firms choose the marketplaces, they simultaneously decide on their prices. Since the competitor’s product quality is unobservable at the beginning of the market-choice stage, a firm’s market choice can only depend on its own quality and the expected quality of the competitor, Q. The situation for each firm, therefore, is similar as the previous case where the entrant chooses the market with the incumbent’s quality held as a constant. Although the analysis will be more complex since now both firms choose markets, the intuition of the entrant14

ACCEPTED MANUSCRIPT incumbent model carries over. That is, a firm will choose the physical market when its product quality is sufficiently different from Q, while it will choose the online market when its quality and Q are close. The following proposition formalizes this result, and constructs an equilibrium that is characterized by two threshold quality levels q o and q o , determined by (3) and (4) below. Each firm will choose the offline market if its product quality is below the former or above the latter, and it will sell online if the quality falls in between them.

1 q−q =

1 q−q

qo

q

"Z

qo

q

"Z

q

qo

δ H (q o − qj )dqj +

δ H (Qo − qj ) dqj +

Z

δ H (q o − qj ) dqj +

q

qo

Z

qo

q

"Z

δ H (Qo − qj ) dqj +

Z

q

where Qo = (q o + q o )/2.

q

qo

δ L (qj − Qo ) dqj

q

qo

(3) #

δ L (qj − q o )dqj + (q o − q o )δ L (Qo − q o ) − F,

qo

Z

δ L (qj − Qo ) dqj

#

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=

1 q−q

"Z

#

(4)

#

δ L (qj − q o ) dqj + (q o − q o )δ H (q o − Qo ) − F,

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1 q−q

Proposition 5 There exist q o and q o with 0 < q < q o < q o < q¯, such that each firm i will choose

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the physical market if qi ∈ [q, q o ) ∪ (q o , q], and will choose the online market if qi ∈ [q o , q o ].

4.2

ED

Proof. In the appendix.

The firms observe both qualities before choosing a markets

PT

In this subsection, we analyze the case where the two qualities qA and qB are observable to both firms, but not to the consumers. After knowing the qualities, the firms simultaneously choose

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between online and offline market and simultaneously post their prices afterwards. In this case, a strategy of each firm is to choose a market and then a price for any (qA , qB ). The quality is not observable to the public prior to purchase if a firm chooses the online

AC

market, and, thus, the consumers have to base their purchase decision on the expected quality of the firm’s product. From the consumer’s perspective, an information set of the game consists of the market choices of both firms, together with the quality of any offline firm(s). Specifically, there are three cases. Firstly, if both firms choose the offline market, then the products’ qualities (qA , qB ) are fully revealed. We denote the information set as h = (qA , qB ) in this case. Second, if firm i with quality qi chooses the offline market while its competitor chooses the online market, we denote the information set as h = (qi , O−i ), where ”O” means ”online.” In addition, in this case,

we suppose that there is a set Q−i (qi ) ⊂ [q, q¯] such that firm −i will choose the online market

if q−i ∈ Q−i (qi ). We denote Q−i (qi ) = E[q−i |q−i ∈ Q−i (qi )], which is the expected quality of

firm −i’s product from the consumer’s perspective. Thirdly, we denote the information set where 15

ACCEPTED MANUSCRIPT both firms choose the online market as h = (OA , OB ). In this case, we suppose that there is a set Q ⊂ [q, q¯] × [q, q¯] such that both firms will choose the online market if (qA , qB ) ∈ Q; and we

let QA = E[qA |(qA , qB ) ∈ Q] and QB = E[qB |(qA , qB ) ∈ Q] be the expected quality of the online product from the consumer’s perspective.

As in the previous section, we use PBE as our solution concept. In equilibrium, an information set is either on or off the equilibrium path. If an information set is on the equilibrium path, then the belief Q−i (qi ) or Q should be consistent with the firm’s strategy profile. Proposition 2 below characterizes an equilibrium in which firms make entry decisions simultaneously.

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Proposition 6 If the firms observe their qualities before choosing the markets, there exists an equilibrium where:

(1) Both firms will choose the online market if their qualities are sufficiently low and sufficiently close, such that qi < q +

2F δH ,

i = A, B.

(2) One firm (i) will choose the offline market and the other (−i) will choose the online market if their qualities are sufficiently high and sufficiently close, such that |qi − q−i | < i = A, B.

and

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qi ≥ q +

2F δH ,

2F δL

(3) Both the firms will choose the offline market if their qualities are sufficiently high and

sufficiently differentiated, such that |qi − q−i | ≥ Proof. In the appendix.

2F δL .

Figure 5 illustrates the outcome of the equilibrium characterized by Proposition 2. When

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(qA , qB ) is in region I, both firms choose the online market. When (qA , qB ) is in region II, firm B chooses the offline market while firm A chooses the online market; and when (qA , qB ) is in region III, firm A chooses the offline market, while firm B chooses the online market. In regions IV and 7

ED

V, both firms choose the offline market.

Proposition 6 shows that, similar to Proposition 1, a firm’s market choice depends not only

PT

on its own quality but also on the quality and the market choice of its competitor. However, since the firms move simultaneously, the specific results of their market choices differ from those in the incumbent-entrant model. In particular, Proposition 6 implies that each market can

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accommodate the full range of product qualities. For example, consider the three polar cases in the above-discussed regions— namely, qi = q−i = q in region I (both online), qi = q−i = q¯ in

regions II/III (one online, one offline), and (qi = q¯, q−i = q) in regions IV/V (both offline). We

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see that both the offline and online markets can accommodate the highest and the lowest quality. 7

Since the fives regions of quality pairs in Figure 2 are mutually exclusive, consumers’ information sets that are on and off the equilibrium path can be completely characterized by these regions. Specifically, as described in detail in the proof of Proposition 2, the following information sets are on the equilibrium path: (i) h = (qA , qB ) when (qA , qB ) are in regions IV and V; (ii) h = (qi , O−i ), i = A, B, when qi ∈ [q + δ2F , q]; and (iii) h = (OA , OB ). If H a firm deviates from the prescribed equilibrium strategy, the game can go to an information set either on or off the equilibrium path. For example, consider a quality pair (qA , qB ) in region V (both firms choose the offline market in ˆ = (qA , OB ) the equilibrium), and suppose that firm B deviates to the online market such that an information set h 2F is reached. If qA < q + δ2F , then this information set is off the equilibrium path. However, if q A ≥ q + δ , the H H information set is still on the equilibrium path (region III), and, thus, the consistency between the belief system ˆ and the strategy profile in the PBE requires that the consumers still hold the on-the-equilibrium-path belief at h and the firms’ optimal prices after the deviation remain the same as the information set appears on the equilibrium path when there is no deviation.

16

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ACCEPTED MANUSCRIPT

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Figure 5: Market choices when the firms observe both qualities before choosing markets Specifically, the intuition of Proposition 6 is as follows. Firstly, in contrast to the previous model in which the incumbent is fixed at the offline market, in the current setting, both firms may choose the online market. According to Proposition 6, this occurs when both firms have low qualities (region I). When both firms’ qualities are unobservable, consumers hold the identical expectation about the firms’ quality. Consequently the firms are engaged in a cutthroat price

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competition, which leads to zero profit. Although the differentiation effect implies that a firm can make higher revenue by switching to the physical market, neither firm has the incentive to deviate: since the firms qualities are low in region I, the increase of revenue from the offline

ED

market will not be enough to cover the market’s higher cost. In other words, in region I, the pooling effect outweighs the differentiation effect. Secondly, regions II and III represent the area where the firms’ qualities are sufficiently high

PT

(compared with those in region I) and close (closer than those in regions IV and V). The firms’ strategies are symmetric in this area, with the higher-quality firm choosing the offline market

CE

and the lower-quality one going online. In comparison with region I, in this area, the firms will no longer stay together in the online market because the differentiation effect and the higher qualities imply that the revenue from switching to the offline market exceeds the market’s fixed

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cost.

On the other hand, since the qualities are close in regions II/III, the differentiation effect is not

large enough for the offline market to accommodate both firms. In this case, the online market essentially provides a haven that allows firms to avoid the price competition in one market and even more so when the online firm’s quality is closer to that of the offline firm. While the intuition is similar to that in the incumbent-entrant model, the current model depicts a more dramatic scenario: when both firms have the highest quality q¯, one will choose the offline market, whereas the other will go online. In this case, the online firm would prefer to be regarded as having a lower (expected) quality than to enter the profit-eroding price competition in the physical market. Lastly, in regions IV and V, the result is consistent with the entrant-incumbent model: when 17

ACCEPTED MANUSCRIPT the qualities are sufficiently apart, the differentiation effect drives both firms to choose the offline market. But probably more striking than the baseline model, we have the following observation: In the case qA = q¯ and qB = q in region IV, both firms choose the offline market because of the strong differentiation effect. In this case firm B essentially pays F to disclose its low quality, even though if it chose the free online market, the consumers would believe its quality is much higher - around q¯ −

F δL .

In summary, when the firms choose markets simultaneously with the knowledge of each other’s

product quality, we show that the pooling effect and the differentiation effect give rise to a large quality to the high-end.

5

Discussion

5.1

Market transparency and entry costs

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set of possible product qualities in each of the online and offline markets, ranging from the low-end

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In the main models we assume that offline product quality is more transparent than online and that the offline market has a higher entry cost. These conditions correspond to scenario 1 in Table 1. Our analytical framework, however, has the flexibility to investigate alternative situations. In particular, one may argue that with the implementation of quality disclosure mechanisms, consumers may learn more about a product from the online market than from visiting a physical shop. For example, for digital electronic products such as a GPS or camera, online forums allow

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consumers to share the sort of vivid post-purchase experience that is almost unattainable in a physical shop. Moreover, with a steady increase of online expenses, such as advertisement fees,

ED

the entry cost for the online market may also dwarf that of physical stores. These arguments correspond to scenario 4 in Table 1. But it is straightforward to see that scenarios 1 and 4 are symmetric, and, thus, we need only to switch the notations of ”online” and ”offline” in the main

PT

model, and the rest of the analysis carries over. It is also easy to see that scenarios 2 and 3 are symmetric. If the offline market is more transparent and the online market more costly (scenario 2), then the pooling effect will vanish. To

CE

see this, consider the entrant-incumbent model. The two cutoff levels q A and q A , as characterized in Proposition 1, will no longer exist: they were the points at which the differentiation effect decreased until that the entrant was indifferent between the costly physical market and the

AC

free online market. But now, with the online market being more costly, at each quality level the entrant will prefer the physical market. Consequently, the entrant will completely shun the online market. This is analogous to the fully-unravelling result under zero disclosure cost in the literature of voluntary disclosure. Next, in the simultaneous-move model, the prevailing differentiation effect implies that both firms will choose the physical market, with the equilibrium outcome characterized by Lemma 1.

18

ACCEPTED MANUSCRIPT Offline market more costly

Online market more costly

Offline market more transparent

1

2

Online market more transparent

3

4

Table-1: Alternative assumptions on market transparency and entry costs

5.2

Return policies

An online firm can implement a return policy, which allows a consumer to return the product for

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refund in case the product’s quality turns out to be below the consumer’s expectation. Hypothetically, a policy that fully refunds a consumer for the returned product can make the online market transparent. As discussed above, a fully transparent and less costly online market means that firms will abandon the more costly physical market. On the other hand, the improved transparency also means that the pooling effect vanishes, and this may not be socially desirable, because, as we showed in the first extension of the baseline model, a firm with quality that is

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close to the competitor’s may choose not to enter the markets at all.

In practice, for various reasons, we would like to argue that the results in the main models can still hold even with the implementation of return policies. First, for products such as furniture and electronic appliances, even if the return policy fully repays consumers for the price and the shipping cost, in order to test and to return a product, consumers may still have to incur the costs to unpack, assemble, re-pack, and transport. When such costs are sufficiently large, say, larger

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than q¯ − q, a return policy will essentially be ineffective. Secondly, in order to resell returned products, firms may impose stringent conditions on the eligibility for return. For example, firms

ED

may find it difficult, or even illegal, to resell used goods such as personal hygienic products. Thus they may require a product to be returned in the original sealed package or in a ”mint” condition, which makes it almost impossible for consumers to test and evaluate the product before returning.

PT

Thirdly, there may exist opportunistic consumers who will take advantage of a full-refund policy. Such consumers buy and use the products, and then return them for refund just before the policy expires. When the proportion of this kind of consumers is sufficiently large, in order to avoid loss,

CE

a firm will have to scale back the coverage of its return policy. For example, the return policy will only compensate the consumers who can prove that the products are fraudulent. Last, for products such as high-tech electronic appliances, the value may depreciate rapidly and firms can

AC

only offer partial refunds. It is worth noting that, if a return policy only covers fraudulent products, and thus effectively

prevent them from entering the markets, then the situation is essentially the same as the main models where the lower bound of the product quality q is above zero. Hence our analysis still applies.

5.3

Empirical implications: quality distribution vs. quality levels

To tackle the potential adverse selection problem, online markets have implemented various mechanisms, including consumer review, seller ranking, etc., to enhance the transparency of product 19

ACCEPTED MANUSCRIPT quality. More generally, the physical and online markets in the model can be regarded as two markets that differ in the magnitude to which product qualities can be revealed. Our results have empirical implications for evaluating the effects of such mechanisms: the above results suggest that firms prefer the more transparent market when their qualities are sufficiently differentiated, and that they will choose different markets when the qualities are close. Previous empirical studies on online product quality focus mainly on a single market, and thus ignore the possibility that sellers may choose among alternative markets (Hui et al. forthcoming, Saeedi 2014, Cabral and Hortacsu 2010). While Jin and Kato (2007) consider sellers’ choices

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between online and offline markets, their study is abstracted from strategic interactions among sellers. We show that the improved transparency of a market may result in more differentiated qualities but not necessarily in an increase in the quality levels in the market. Therefore, in addition to the impact on the quality levels, researchers can investigate how the quality distribution in the market changes with the implementation of a quality disclosure mechanism. A quantile regression may be utilized to examine whether the tails of the quality distribution become heavier

6

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as the market becomes more transparent.

Conclusion

Despite an increasing amount of literature on online and offline business, many previous studies assume that firms’ market places are exogenously given. In this paper we endogenize firms’ market choices with respect to product quality. We show that, because of the online market’s pooling

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effect and the offline market’s differentiation effect, firms’ market choices can help alleviate their price competition. In an entrant-incumbent model, the entrant with product quality lower than

ED

that of the offline incumbent may choose the physical market, whereas the entrant with a quality higher than the incumbent’s may sell online. In a simultaneous-move model, the two contrasting effects give rise to a wide range of product quality—from low-end to high-end—in both the online

PT

and offline markets.

The results provide a caveat to policy-makers that aims at improving the transparency of the online market. An overall more transparent online market will certainly discourage ”lemons”,

CE

but at the meantime it will also prohibit the pooling effect, deprive the firms of an avenue to alleviate price competition, and thus some good products may also have to disappear from the markets. It is therefore worth considering policies that specifically target at fraudulent products

AC

but preserve the opaqueness of the online market. Moreover, conventional wisdom suggests that the online market houses low-quality products. Following this view a naive firm may want to ”hide” its product in the online market when the quality is low, and rush to the physical market or run advertisement when its quality is high. But based on the analysis of the present paper, a wise manager can take advantage of the (lack of) transparency of a market. Even if the firm’s product quality is actually higher than the competitor’s, it may be better that consumers do not know it. By pooling its product with other quality in the online market, the firm may make a even higher profit than that from a costly disclosure. Our findings can be a starting point for several directions of future study. First, it remains a

20

ACCEPTED MANUSCRIPT future direction to investigate the more general situation with more than two firms. We should notice that the duopolistic framework in the present study imposes a logical limitation to the results. For example, if there are already many competitors in the offline market and they offer a wide range of differentiated products, then there may be little space for the entrant to further differentiate itself in the offline market. In this case, the entrant may still choose the online market even if its product is at the high-end or the low-end. Secondly, we have investigated a polar case in which the product quality can be perfectly disclosed in one market and fully concealed in the other. Hence the results can be used as a benchmark for investigating more-general cases in

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which the markets’ quality transparency is in between the extreme cases. Moreover, one may endogenize the firms’ quality choices prior to their market choices. Furthermore, it will also be interesting to investigate in more detail how other features of the online market, such as market

AC

CE

PT

ED

M

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size, transportation costs, and specific quality disclosure mechanisms, affect firms’ market choices.

21

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7

Appendix

7.1

Proof of Lemma 1

Proof. The proof is similar to that in Belleflamme and Peitz (2010) for a standard model of vertical product differentiation in an offline market: When both firms choose to sell in the offline market, (qB , pB ) and (qA , pA ) are common knowledge. Without loss of generality, given (qB , pB ) ˆ who will be indifferent and (qA , pA ) with qB < qA and pB < pA , there will be a consumer with λ

ˆ B − pB = λq ˆ A − pA , λq ˆ = pA − pB . ⇒λ qA − qB

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between the two firms:

ˆ − λ)pB − F = ( pA −pB − λ)pB − F , while firm A’s profit is Hence, firm B’s profit is π B = (λ qA −qB ¯ − λ)p ˆ A − F = (λ ¯ − pA −pB )pA − F . π A = (λ qA −qB

Given qB and qA , each firm chooses the price to maximize its profit. The first-order condition pA −pB qA −qB

−λ−

pB qA −qB

= 0, which leads to

pB =

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for firm B implies

1 [pA − λ(qA − qB )] . 2

¯− The first-order condition for firm A implies λ

−

pA qA −qB

= 0, which implies

 1 ¯ λ (qA − qB ) + pB . 2

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pA =

pA −pB qA −qB

ED

Solving the two first-order conditions for pB and pA , we have

PT

pB = pA =

1 ¯ (λ − 2λ)(qA − qB ), 3 1 ¯ (2λ − λ) (qA − qB ) . 3

Thus given qB < qA , in equilibrium  pA − pB π B (qB , qA ) = − λ pB − F qA − qB " # 1 ¯ 1 ¯ 1 ¯ 3 (2λ − λ) (qA − qB ) − 3 (λ − 2λ)(qA − qB ) = − λ (λ − 2λ)(qA − qB ) − F qA − qB 3

AC

CE



=

1 ¯ (λ − 2λ)2 (qA − qB ) − F 9

22

ACCEPTED MANUSCRIPT   pA − pB ¯ pA − F π A (qB , qA ) = λ− qA − qB " # 1 ¯ 1 ¯ (q − q ) − − q ) (2 λ − λ) ( λ − 2λ)(q 1 ¯ A B A B 3 3 ¯− = λ (2λ − λ) (qA − qB ) − F qA − qB 3 =

Proof of Proposition 1

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7.2

1 ¯ (2λ − λ)2 (qA − qB ) − F 9

Proof. First suppose that QA < qB . We consider two possible cases. Case 1: qA ∈ [q, qB ].

If firm A chooses the offline market, then by Lemma 1, π A = δ L (qB − qA ) − F . If firm A

chooses the online market, then, by Corollary 1, π A = δ L (qB − QA ). Firm A will choose the online market if δ L (qB − QA ) ≥ δ L (qB − qA ) − F ; that is, F . δL

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qA ≥ Q A − Case 2: qA ∈ (qB , q].

(5)

If firm A chooses the offline market, then π A = δ H (qA − qB ) − F . If firm A chooses the online

market, then π A = δ L (qB − QA ). Firm A will choose the online market when δ L (qB − QA ) ≥ qA ≤ qB + 1 2

h

ED

(5) and (6) imply that QA = for QA , we have

QA −

QA = qB −

PT

1 [F + δ L (qB − QA )]. δH

M

δ H (qA − qB ) − F ; that is,

F δL



+



1 δ H [F

(6)

i + δ L (qB − QA )] . Solving this equation

δH − δL · F < qB . δ L (δ H + δ L )

(7)

Note that QA < qB confirms our assumption.

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Since firm A will choose to sell online when qA ∈ [QA − δFL , qB + δ1H [F +δ L (qB −QA )]], plugging

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(7) into the lower bound and the upper bound of the set, we get 2 δH · · F, δH + δL δL 2 = qB + · F. δH + δL

q A = qB − qA

To complete the description of the PBE, we need to construct the consumers’ belief system. On

the equilibrium path, the belief system is straightforward: by assumption, if the entrant chooses the offline market, consumers can verify her quality. On the other hand, if the entrant chooses the online market, consumers believe that her quality is QA . Then, the characterization of q A and q A implies that an offline entrant has no incentive to deviate to the online market, and vice versa. A subtle issue remains in the off-equilibrium-path belief: since the entrant chooses a marketplace

23

ACCEPTED MANUSCRIPT and then sets the price, a quality in the online market may want to post a different price in order to separate itself from other online qualities. In particular, q A or q A has the strongest incentive to deviate in this way, because the deviation will allow them to reveal their types while not paying the offline-market cost F . To deter this kind of deviation, it suffices to let consumers hold the off-equilibrium-path belief that any online price other than the one in equilibrium comes from an entrant with quality QA because, then, the most profitable price-deviation in the online market will lead to the same profit as staying on the equilibrium path. Next, we will show that it cannot be QA > qB in equilibrium: Suppose, toward a contradiction, Case 1: qA ∈ [q, qB ].

CR IP T

that QA > qB . There are two cases: If firm A chooses the offline market, then π A = δ L (qB − qA ) − F . If firm A chooses the

online market, then π A = δ H (QA − qB ). Firm A will choose the online market if δ H (QA − qB ) ≥ δ L (qB − qA ) − F ; that is, qA ≥ qB −

1 δ L [F

Case 2: qA ∈ (qB , q].

+ δ H (QA − qB )].

If firm A chooses the offline market, then π A = δ H (qA − qB ) − F . If firm A chooses the

AN US

online market, then π A = δ H (QA − qB ). Firm A will choose the online market if δ H (QA − qB ) ≥ δ H (qA − qB ) − F ; that is, qA ≤ QA + δFH .   h i Hence, QA = 12 QA + δFH + qB − δ1L [F + δ H (QA − qB )] . Solving the equation for QA , δ H −δ L δL δH

equilibrium.

7.3

· F < qB , contradicting QA > qB . Therefore it cannot be QA > qB in

Proof of Proposition 2

M

we have QA = qB −

ED

Suppose that the entrant can only choose between entering the physical market and remaining inactive.

CE

two firms:

PT

First, suppose that both firms sell offline and that 0 ≤ qA < qB . Given (qB , pB ) and (qA , pA ) ˆ who will be indifferent between the with qA < qB and pA < pB , there will be a consumer with λ

ˆ B − pB = λq ˆ A − pA , λq ˆ = pB − pA . ⇒λ qB − qA

AC

ˆ will buy from firm B. Thus, firm B’s demand is DB (pA , pB ) = 1 − λ b= Consumers with λ > λ 1−

pB −pA qB −qA .

The market is partially covered, meaning that consumers with extremely low λ will not buy ˜ by the equation λq ˜ A − pA = 0, from which one can get from any firm. Define marginal type λ p ˜ = A . The consumer λ ˜ is indifferent between buying from firm A and buying nothing. Thus, λ qA

ˆ so he will not buy the product from firm a consumer buys from firm A if and only if (i) λ ≤ λ, ˜ , so he will get a positive utility from buying product from firm A. The demand of B; (ii) λ ≥ λ

firm A is given by

b−λ ˜ = pB − pA − pA . DA (pA , pB ) = λ qB − qA qA 24

ACCEPTED MANUSCRIPT Firm A chooses its price pA to maximize its profit max pA DA (pA , pB ) = pA pA





pB − pA pA − qB − qA qA

.

The first order condition with respect to pA yields pA =

qA p B . 2qB

(8)

Firm B’s optimization problem is

pA

pB − pA 1− qB − qA

The first order condition with respect to pB yields pB =

qB − qA + pA . 2



CR IP T

max pB DB (pA , pB ) = pB



(9)

pA =

(qB −qA )qA 4qB −qA

and pB =

AN US

Combining (8) and (9), we can solve for the equilibrium prices, which are then given by 2(qB −qA )qB 4qB −qA .

The equilibrium demands are then given by DA =

M

and

pB − pA pA qB − = qB − qA qA 4qB − qA

DB = 1 −

pB − pA 2qB = . qB − qA 4qB − qA

ED

The profit of firm A and firm B (gross of the fixed cost F ) are given by (qB − qA ) qA qB (4qB − qA )2

PT

π A (qA , qB ) = pA DA =

and

2 4 (qB − qA ) qB

CE

π B (qA , qB ) = pB DB =

(4qB − qA )2

.

Using similar calculation, one can derive the equilibrium results for qA > qB . The equilibrium

AC

profit of firm A are given by

π A (qA , qB ) =

 (qB −qA )qA qB , if q ≤ q A B (4q −q )2 B

A

2 4(qA −qB )qA 2 , (4qA −qB )

.

if qA ≥ qB

Secondly, suppose that qA < 0 < qB . Then, firm A must be inactive if it sells offline. In this case, firm B is a monopoly. It is easy to show that firm B’s optimal pricing strategy is to set pB =

qB 2 .

Consequently, only consumers with λ >

and profit for the incumbent are DB =

1 2

1 2

and π B =

will buy from the incumbent. The demand qB 4 .

Lemma 2 π A = 0 for qA ∈ [q, 0]. π A is first increasing and then decreasing in qA for qA ∈ [0, qB ]; 25

ACCEPTED MANUSCRIPT and π A is increasing in qA for qA ≥ qB . Proof. For qA ≤ qB , we have h i 1 2 (q − 2q ) q ∗ (4q − q ) + (q − q ) q q ∗ 2 (4q − q ) B A B B A B A A B B A (4qB − qA )4 1 [(qB − 2qA ) qB ∗ (4qB − qA ) + 2 (qB − qA ) qA qB ] (4qB − qA )3 2 (4q − 7q ) qB B A

= = =

(4qB − qA )3

CR IP T

∂π A ∂qA

A from which we have sign ∂π ∂qA = sign (4qB − 7qA ), which is positive if qA ≤

and negative

≤ qB .When qA ≤ qB , π A reaches the maximum value when qA =

∂π A ∂qA

= = =

where


2 2 − 2q q 2 4 3qA A B (4qA − qB ) − 4 (qA − qB ) qA ∗ 8 (4qA − qB ) (4qA − qB )4
2

AN US

For qA ≥ qB .

4 7 qB ,

2 − 3q q + 4q 4qA 2qB A B A

4qA

(4qA − qB )3
2 + 3q (q − q ) + q 2 2qB A A B A (4qA − qB )3

> 0.

M

if 47 qB ≤ qA 1 π A = 48 qB .

4 7 qB

It is then easy to see that Proposition 2 holds following the comparison between π A and F ,

7.4

ED

as Figure 2 illustrates.

Proof of Proposition 3

PT

We will first prove part (ii) of the proposition, and then prove part (i). First, suppose QA > 0 and that π A (QA , qB ) > 0. Suppose further that QA < qB , which is

CE

shown to be true later.

There are two cases. Case 1: qA ∈ [q, qB ].

AC

Firm A will choose offline market if and only if π A (qA , qB ) − F ≥ π A (QA , qB ) .

From the Lemma 2 in the proof of Proposition 2, we know that π A is first increasing and then
decreasing in qA for qA ∈ [q, qB ]. Hence, if π A (QA , qB ) < maxqA∈[0,q] π A (qA , qB ) = π A 74 qB , qB −

F (which holds if F is sufficiently small) then, there exists two threshold value q˜1 < q˜2 , with qˆ1 < q˜1 and qˆ2 > q˜2 , such that firm A chooses to sell offline if qA ∈ (˜ q1 , q˜2 ) and online if   qA ∈ q, q˜1 ∪ [˜ q2 , qB ], where q˜1 and q˜2 are the two solutions of the equation π A (qA , qB ) − F =

π A (QA , qB ).

26

ACCEPTED MANUSCRIPT Case 2: qA ∈ [qB , q].

From the Lemma 2, we know that π A is increasing in qA for qA ∈ [qB , q]. Hence, if π A (QA , qB ) <

π A (q, qB ) − F (which holds if F is sufficiently small), then there exists a threshold value qˆ3 <

q˜3 , such that firm A chooses to sell offline if qA ∈ (˜ q3 , q).

q˜3 is the solution of equation

π A (qA , qB ) − F = π A (QA , qB ) .

The expected online quality is given by q˜1 − q

· q˜1 − q + (˜ q3 − q˜2 )

q˜1 + q q˜3 − q˜2 q˜3 + q˜2 + · . 2 q˜1 − q + (˜ q3 − q˜2 ) 2

CR IP T

QA = 

To prove QA < qB , it suffices to prove qB − q˜2 > q˜3 − qB . To prove qB − q˜2 > q˜3 − qB , it suffices to prove π A (qB − δ, qB ) < π A (qB + δ, qB ) for any positive δ. From the expression of π A , we have

AN US

π A (qB + δ, qB ) − π A (qB − δ, qB )
δ 2 3 4 = 4δ 4 + 48δ 3 qB + 96δ 2 qB + 81δqB + 27qB
2 2 2 4δ + 15δqB + 9qB > 0.

Note that QA is a decreasing function of q. Moreover, limq→−∞ QA = −∞ and QA |q=0 > 0.

Hence, there exists −∞ < qˆ < 0 , such that QA > 0 if q > qˆ. It is then straightforward to see that part (ii) of Proposition 3 holds, as illustrated in Figure 3.

Secondly, If q < qˆ , then QA < 0 and hence the online market can no longer be active. Hence,

M

this case is equivalent to the case in which firm A can choose only between the offline market

7.5

ED

and being inactive.

Proof of Proposition 4

Using previous notations, suppose there is a set QA ⊂ [q, q¯] such that the entrant will choose the

PT

online market if qA ∈ QA . Denote the expected quality QA = E[qA |qA ∈ QA ]. With the entry

costs F1 and F2 , the entrant will choose online market if the following two conditions hold: (i)

CE

π A (QA , qB ) − F2 ≥ π A (qA , qB ) − F1 , and (ii) π A (QA , qB ) − F2 ≥ 0. The function form of π A

in the two conditions remain the same as that in the baseline model. The first condition implies that the entrant prefers the online market to the offline market, and the second condition says

AC

that entering the online market is better than remaining inactive. Note that condition (i) can be rewritten as π A (QA , qB ) − π A (qA , qB ) ≥ F1 − F2 .

First, suppose F2 is sufficiently small such that condition (ii) holds. As a result, the entrant

will not choose to be inactive. Then the entrant’s market-choice problem is essentially the same as in the baseline model, except that F there should be replaced by F1 − F2 . That is, it is the

difference between the entry costs of the two markets that determines the range of qualities that

goes to online market. Following the proof of Proposition 1, it is easy to show that, firm A will choose to sell online when qA ∈ [q A , q A ], where q A and q A are two thresholds now determined by

27

ACCEPTED MANUSCRIPT

δH 2 · · (F1 − F2 ) , δH + δL δL 2 = qB + · (F1 − F2 ) . δH + δL

q A = qB − qA

L The expected quality of firm A’s product in the online market is: QA = qB − δLδ(δHH−δ ·(F1 − F2 ). +δ   L) δ H −δ L The profit from choosing the online market is π A qB − δL (δH +δL ) · (F1 − F2 ) , qB .

Next, as F2 increases, the online market becomes less attractive to the entrant. Consequently,

CR IP T

the range of qualities which goes to the online market becomes narrower, i.e., q A increases and q A decreases. The expected quality of firm A’s product in online market becomes closer to qB . As a result, π A (QA , qB ) decreases. Therefore, the entrant’s net profit π A (QA , qB ) − F2 is deceasing in F2 . Consequently, there exists a threshold Fˆ2 ∈ (0, F1 ), such that π A (QA , qB ) − F2 = 0.

For any online-entry cost higher than Fˆ2 , condition (ii) above fails, and thus the entrant will

abandon the online market. That is, it will either sell offline or remain inactive, which can be

AN US

characterized by two threshold value q 0A and q 0A that solve the equation π A (qA , qB ) − F1 = 0. If

qA ≤ q 0A or qA ≥ q 0A , the entrant will choose the physical market, while if qA ∈ [q 0A , q 0A ] it will 0 ˆ remain inactive. h The value of the i threshold cost, therefore, is determined by F2 = π A (QA , qB ) , with Q0A = E qA |qA ∈ [q 0A , q 0A ] .

Proof of Proposition 5

M

7.6

Suppose that such an equilibrium exists, then the expected quality of a firm who chooses online market is Qo =

q o +q o 2 .

Consider the market choice of firm i with quality qi . Given firm j’s

1 q−q

CE

=

qo

π i (Qo , qj ) dqj +

q

1 q−q

"Z

qo

"Z

qo

q

q

π i (Qo , qj ) dqj +

qo

π i (Qo , qj ) dqj +

q

Z Z

q

π i (Qo , qj ) dqj

qo

Z

δ H (Qo − qj ) dqj +

#

Z

qo

qo

#

q

qo

π i (Qo , Qo ) dqj

#

δ L (qj − Qo ) dqj ,

AC

=

"Z

PT

Eπ on =

1 q−q

ED

equilibrium strategy, if firm i chooses online market, then its expected profit is

where the second equality is due to the fact that π i (Qo , Qo ) = 0, i.e., both firms’ profit are zero if both of them choose online market. Notice that Eπ on is independent of firm i’s true quality qi . Firm i’s expected profit of choosing offline (gross of fixed cost) is 1 Eπ of f (qi ) = q−q

"Z

q

qo

π i (qi , qj ) dqj +

Z

q

qo

28

π i (qi , qj ) dqj +

Z

qo

qo

#

π i (qi , Qo ) dqj .

ACCEPTED MANUSCRIPT For qi > q o , we have Eπ of f (qi ) =

1 q−q

"Z

qo

q

δ H (qi − qj ) dqj +

i +(q o − q o )δ H (qi − Qo ) .

Z

qi

qo

δ H (qi − qj ) dqj +

Z

q

δ L (qj − qi ) dqj

qi

!

For Qo < qi < q o , we have Eπ of f (qi ) =

"Z

qo

q

δ H (qi − qj ) dqj + +

i +(q o − q o )δ H (qi − Qo ) .

Z

qo

For q o < qi < Qo , we have 1 q−q

Eπ of f (qi ) =

"Z

qo

q

δ H (qi − qj ) dqj +

For qi < q o , we have Eπ of f (qi ) =

1 q−q

("Z

qi

q

Z

δ L (qj − qi ) dqj

q

qo

δ L (qj − qi ) dqj

AN US

i +(q o − q o )δ L (Qo − qi ) .

q

CR IP T

1 q−q

δ H (qi − qj ) dqj +

M

o +(q o − q o )δ L (Qo − qi ) .

Z

qo

qi

#

δ L (qj − qi ) dqj +

Z

q

qo

δ L (qj − qi ) dqj

ED

Firm i chooses online market if and only if Eπ on ≥ Eπ of f (qi ) − F .

We first derive the following lemma, which shows some properties of the function Eπ of f (qi ).

∂ ∂qi Eπ of f

∂ ∂qi Eπ of f

(qi ) is continuous and increasing in qi on [q, Qo ) and on (Qo , q]. Moreover,

PT

Lemma 3

(qi ) |qi =q < 0 and that

∂ ∂qi Eπ of f

(qi ) |qi =q > 0.

CE

Proof. The result follows directly from the expression of Eπ of f (qi ). The above lemma suggests Eπ of f (qi ) is decreasing for qi close to q and increasing for qi close

AC

to q. The next lemma first assumes that Eπ on < Eπ of f (q) − F and Eπ on < Eπ of f (q) − F , which

we will show to be true later, and then derives firm i’s best strategy.

Lemma 4 Suppose that Eπ on < Eπ of f (q) − F and Eπ on < Eπ of f (q) − F . Then, there exists two threshold values q 0o and q 0o , such that firm i will choose offline market if qi ∈ [q, q 0o ) ∪ (q 0o , q] and the online market if qi ∈ [q 0o , q 0o ], where q 0o and q 0o are the two solutions of the equation Eπ of f (qi ) − F = Eπ on .

Proof. Note that firm i with true quality Qo always chooses online market because Eπ of f (Qo ) = Eπ on . Because Eπ of f (Qo ) − F < Eπ on and Eπ of f (q) − F > Eπ on , there exists a number q 0o such

that Eπ of f (q 0o ) − F = Eπ on .

29

ACCEPTED MANUSCRIPT Because

∂ ∂qi Eπ of f

(qi ) is continuous and increasing in qi on [q, Qo ), we have

    max Eπ of f (qi ) = max{Eπ of f q 0o , Eπ of f (Qo )} = Eπ of f q 0o . qi ∈[q 0o ,Qo ] i h Therefore, for qi ∈ q 0o , Qo , we have Eπ of f (qi ) − F ≤ Eπ of f (q 0o ) − F = Eπ on . We thus have i h shown that firm i will choose online market if qi ∈ q 0o , Qo . Now we will show that Eπ of f (qi ) is a decreasing function for qi ∈ [q, q 0o ]. Since

continuous and increasing in qi on [q, Qo ), it suffice to show that ∂ ∂qi Eπ of f ∂ ∂qi Eπ of f

∂ ∂qi Eπ of f

∂ ∂qi Eπ of f

∂ ∂qi Eπ of f

(qi ) is

(qi ) |qi =q0 < 0. We prove o

CR IP T

(qi ) |qi =q0 < 0 by contradiction. Suppose (qi ) |qi =q0 > 0, then we must have o o i h 0 (qi ) > 0 for qi ∈ q o , Qo . Therefore, we have Eπ of f (Qo ) − F > Eπ of f (q 0o ) − F = Eπ on ,

a contradiction with the fact that Eπ of f (Qo ) − F < Eπ on .

Because Eπ of f (qi ) is a decreasing function for qi ∈ [q, q 0o ] and that firm i with quality q 0o is

indifferent between online and offline market, firm i will choose offline market if qi < q 0o . offline market for qi ∈ (q 0o , q].

AN US

Using similar logic, we can prove that firm i will choose online market for qi ∈ [Qo , q 0o ] and In a symmetric equilibrium, we must have q 0o = q o and q 0o = q o . That is, q o and q o are

determined by the following two equations:

1 q−q

"Z

qo

q

"Z

qo

q

δ H (Qo − qj ) dqj +

"Z

q

qo

Z

q

qo

Z

δ L (qj − Qo ) dqj

#

M

q

δ H (q o − qj )dqj +

δ H (q o − qj ) dqj +

q

Z

q

qo

Z

δ L (qj − Qo ) dqj

#

q

qo

(10) #

δ L (qj − q o )dqj + (q o − q o )δ L (Qo − q o ) − F,

qo

δ H (Qo − qj ) dqj +

CE

=

1 q−q

qo

PT

=

1 q−q

"Z

ED

1 q−q

(11) #

δ L (qj − q o ) dqj + (q o − q o )δ H (q o − Qo ) − F.

AC

Equations (10) and (11) can be simplified as Eπ of f (q o ) − Eπ on =

  i qo − qo h 
δL q − q − δH q − q = F o o 2 q−q

Eπ of f (q o ) − Eπ on =

qo − qo


δ H q o − q − δ L (q − q o ) = F 2 q−q

Combing the above two equations, we have


2 δLq + δH q qo + qo = . δH + δL

30

(12)

(13)

ACCEPTED MANUSCRIPT Substitute this into (12) and (13), we have (δ H + δ L )





δLq + δH q − qo δH + δL

δLq + δH q (δ H + δ L ) q o − δH + δL

2

2

= =


q−q F
q−q F

such that

qo = Since

δ L q+δ H q δ H +δ L


q−q F δH + δL s
q−q F δLq + δH q 2 + δH + δL δH + δL δLq + δH q − δH + δL

2

CR IP T

qo =

s

∈ (q, q), as long as F is small enough, (12) and (13) have a solution (q o , q o ) with

Qo =

qo + qo 2

AN US

q < q o < q o < q. Moreover,

=

δLq + δH q ∈ (q o , q o ). δH + δL

Now suppose firm j adopts the equilibrium strategy and chooses offline market if qj ∈ [q, q o )∪

(q o , q] and the online market if qj ∈ [q o , q o ]. Then, from lemma above, we must have Eπ of f (qi ) −

F < Eπ on for qi ∈ [q o , q o ] and Eπ of f (qi ) − F > Eπ on for qi ∈ [q, q o ) ∪ (q o , q]. That is, firm i’s

M

best response is also to adopt the equilibrium strategy.

ED

The proof of Proposition 5 will then be completed by showing that the assumptions for Lemma
4, i.e., Eπ on < Eπ of f (q)−F and Eπ on < Eπ of f q −F actually hold. In the above discussion we

have shown that when F is sufficiently small, there exists (q o , q o ) satisfying q < q o < q o < q, such   q +q that Eπ on = Eπ of f q o − F and Eπ on = Eπ of f (q o ) − F . Note that for Qo = o 2 o ∈ (q o , q o ),

PT

Eπ of f (Qo ) = Eπ on such that Eπ on > Eπ of f (Qo ) − F . Since from Lemma 3,

∂ ∂qi Eπ of f

(qi ) is

7.7

CE

continuous and increasing in qi on [q, Qo ) and on (Qo , q], it must hold that Eπ on < Eπ of f (q) − F
and Eπ on < Eπ of f q − F .

Proof of Proposition 6

AC

We first consider the pricing strategy. If both firms choose to sell on the online market, due to the symmetry, consumers will hold the expectation that both firms’ product quality is Q = QA = QB . Then the firms will be essentially engaged in a standard Bertrand competition with identical products, and, thus, undercut each other’s price until it reaches 0. Apparently, both firms get zero profit in this scenario. When both firms choose the offline market, or when one firm chooses the online market while the other firm chooses the offline market, the proof for the pricing strategy part is basically the same as that in the proof of Lemma 1 and Corollary 1, and thus is omitted here. To sum up, the firms’ pricing strategies are: (1) If both firms choose the online market, then pi = 0 for i = A, B. (2) If firm i chooses the online market, while firm −i chooses the offline market, then pi = 31

ACCEPTED MANUSCRIPT 1 ¯ 3 (2λ

¯ − 2λ)(Qi (q−i ) − q−i ) if Qi (q−i ) > q−i ; and pi = 1 (λ ¯− − λ)(Qi (q−i ) − q−i ) and p−i = 13 (λ 3 ¯ − λ)(q−i − Qi (q−i )) if Qi (q−i ) < q−i . 2λ)(q−i − Qi (q−i )) and p−i = 1 (2λ 3

¯− (3) If both firms sell in the offline market, then, given (qi , q−i ) with qi > q−i , pi = 13 (2λ ¯ − 2λ)(qi − q−i ). λ) (qi − q−i ) and p−i = 1 (λ 3

Given the consumer’s belief, firms’ profit is characterized by Lemma 1 and Corollary 1 in

these two scenarios, as well. Next, consider the following belief system of the consumers: (2) If h = (qi , O−i ), then consumers hold the belief that  2F   [q, q + δH ] if Q−i (qi ) = if [q, qi ]   2F [qi − δL , qi ] if

and

qi ∈ [q + qi ∈ [q +

F δH .

2F δH )

2F 2F δH , q + δL ) 2F δ L , q]

qi ∈ [q, q +

qi ∈ [q +

and QA = QB = q +

2F δH )

2F 2F δH , q + δL ) 2F δ L , q]

AN US

 F if   q + δH 1 Q−i (qi ) = 2 (q + qi ) if   qi − δFL if

qi ∈ [q, q +

2F 2 δH ]

CR IP T

(1) If h = (OA , OB ), consumers hold the belief that Q =[q, q +

qi ∈ [q +

(3) At h = (qA , qB ), consumers just fully observe the true qualities. For ease of notation in the rest of the proof, denote the above belief system as (BS). We will show that given (BS), it is optimal for both firms to choose the marketplaces as described, and

M

(BS) is consistent with the strategy profile. This part of the proof consists of several lemmas below, which verify the rationality of the strategy profile and the consistency of the belief system

ED

in different areas of the [q, q¯] × [q, q¯] space.

Lemma 5 Given (BS), when qi ∈ [q, q + δ2F ), i = A, B (Region (I) in Figure 5), firm i optimally H

chooses the online market given that firm −i is online.

PT

Proof. Given that firm −i is online, if firm i chooses the online market, it gets zero profit. If

CE

firm i chooses the offline market, then its profit is no greater than

AC

When qi ∈ [q, q +

2F δH )

π i = δ H |Q−i (qi ) − qi | − F.

and Q−i (qi ) = q +

F δH ,

|Q−i (qi ) − qi | ≤

F δH ,

and thus,

δ H |Q−i (qi ) − qi | − F ≤ 0.

Hence, firm i will get no benefit by deviating to the offline market. Lemma 6 Given (BS), for i = A, B, when qi ∈ [q +

2F δH , q

+

2F δL )

and q−i ∈ [q, qi ] (the shaded

parts of regions II and III in Figure 6), firm −i will optimally choose the online market given that firm i is offline, and firm i will optimally choose the offline market when firm −i is online. Proof. We first show that given (BS), when qi ∈ [q +

2F δH , q

+

2F δL )

and q−i ∈ [q, qi ], firm −i

will optimally choose the online market given that firm i is offline. When firm i is offline, if 32

Figure 6: qi ∈ [q +

2F δH , q

+

2F δL )

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ACCEPTED MANUSCRIPT

and q−i ∈ [q, qi ]

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firm −i chooses the offline market, then, by Lemma 1, π −i = δ L (qi − q−i ) − F since qi ≥ q−i .

If firm −i chooses the online market, then, since Q−i (qi ) =

1 2 (q

+ qi ) < qi , by Corollary 1,

π −i = δ L (qi −Q−i (qi )). Firm −i will choose the online market if δ L (qi −Q−i (qi )) ≥ δ L (qi −q−i )−F ;

i.e.,

q−i ≥ Q−i (qi ) − 2F 1 δ L , 2 (q

+ qi ) −

F δL

market when firm i is offline.

< q. This implies that any q−i ∈ [q, qi ] will choose the online

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When qi < q +

1 F F = (q + qi ) − . δL 2 δL

To see why it is also optimal for firm i to choose the offline market given that firm −i is online,

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first note that both firms get zero profit if they compete in the online market. Hence, when firm −i is online, firm i optimally chooses the offline market when π i = δ H (qi − Q−i (qi )) − F ≥ 0, 2F δH .

PT

which is true when qi ≥ q +

Lemma 7 Given (BS), for i = A, B, when qi ∈ [q +

2F δ L , q]

and q−i ∈ [qi −

2F δ L , qi ]

(the un-shaded

CE

parts of regions II and III in Figure 3), firm −i will optimally choose the online market given that firm i is offline, and firm i will optimally choose the offline market when firm −i is online. Proof. We first show that given (BS), when qi ∈ [q +

2F δ L , q]

and q−i ∈ [qi −

2F δ L , qi ],

firm −i

AC

will optimally choose the online market given that firm i is offline. When firm i is offline, if firm

−i chooses the offline market, then, by Lemma 1, π −i = δ L (qi − q−i ) − F since qi ≥ q−i . If firm −i chooses the online market, then, since Q−i (qi ) = qi −

F δL

< qi , by Corollary 1, π −i =

δ L (qi − Q−i (qi )). Firm −i will choose the online market if δ L (qi − Q−i (qi )) ≥ δ L (qi − q−i ) − F ; i.e.,

q−i ≥ Q−i (qi ) − This implies that any q−i ∈ [qi −

2F δ L , qi ]

F 2F = qi − . δL δL

will choose the online market when firm i is offline.

To see why it is also optimal for firm i to choose the offline market given that firm −i is online,

first note that both firms get zero profit if they compete in the online market. Hence, when firm 33

ACCEPTED MANUSCRIPT −i is online, firm i optimally chooses the offline market when π i = δ H (qi − Q−i (qi )) − F ≥ 0, which is true when qi ≥ q +

2F δH .

2F δL

Lemma 8 Given (BS), when |qi − q−i | ≥

(Regions IV and V in Figure 5), firm −i will

optimally choose the offline market given that firm i is offline, i = A, B. Proof. We consider the area in which qB − qA ≥

for the area in which qA − qB ≥

when qB − qA ≥

(Region (V) in Figure 5). The discussion

(Region (IV) in Figure 5) follows similar arguments. First,

we have qB ≥ q +

2F δL ,

so firm A will optimally choose the offline market if

according to the proof of the previous lemma. Hence, all we need to show is that

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qA ≤ qB −

2F δL ,

2F δL ,

2F δL

2F δL

firm B will optimally choose the offline market given that firm A is offline when qB − qA ≥

2F δL .

When firm A is offline, if firm B chooses the offline market, then its profit is π B = δ H (qB − 2δ H −δ L δL

qA ) − F , which is no less than

· F since

2δ H − δ L 2F −F = · F. δL δL

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δ H (qB − qA ) − F ≥ δ H ·

If firm B chooses the online market, its profit can be one of the three cases below: Case 1: qA ∈ [q, q +

2F δH )

In this case, if firm B chooses the online market, then QB (qA ) = q +

F δH .

Thus, π B =

δ L (qA − QB (qA )) if qA > QB (qA ), and π B = δ H (QB (qA ) − qA ) if qA ≤ QB (qA ). But whether qA > QB (qA ) or qA ≤ QB (qA ), π B ≤ δ H · |QB (qA ) − qA | ≤ δ H · +

2F δL )

= F.

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Case 2: qA ∈ [q +

2F δH , q

F δH

This case can happen when q − q − δ2F > q + δ2F (see Figure 6). In this case, if firm i chooses H H

the online market, then QB (qA ) = 12 (q + qA ) and π B = δ L (qA − QB (qA )) = δ L · 21 (qA − q). Note Case 3: qA ∈ [q +

2F δL

ED

that π B < F since qA − q < 2F δL , q

in this case.

−q−

2F δH ]

PT

This case can happen when q − q − δ2F > q + 2F δ L (see Figure 6). In this case, if firm i chooses H

the online market, then QB (qA ) = qA − In sum, since F <

2δ H −δ L δL

F δL

and π B = δ L (qA − QB (qA )) = F .

· F , in any of the three cases, the profit will be less than

2δ H −δ L δL

·F

CE

and, thus, will be less the profit from choosing the offline market. Therefore, firm B will optimally choose the offline market when firm A is offline.

AC

So far, we have shown that given the belief system, the strategy profile satisfies rationality.

With this strategy profile, the information set (qi , O−i ) is on the equilibrium path when qi ≥ q+ δ2F H and off the equilibrium path when qi < q +

2F δH .

When (qi , O−i ) is on the equilibrium path, it is

straightforward to check that the belief system is consistent with the strategy profile. Therefore, the equilibrium characterized by Proposition 6 turns out to be a PBE.

34

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