Signalling Rivalry and Quality Uncertainty in a Duopoly

Abstract

This paper considers price competition in a duopoly with quality uncertainty. The established firm (the ‘incumbent’) offers a quality that is publicly known; the other firm (the ‘entrant’) offers a new good whose quality is not known by some consumers. The incumbent is fully informed about the entrant’s quality. This leads to price signalling rivalry because the incumbent gains and the entrant loses if observed prices make the uninformed consumers more pessimistic about the entrant’s quality. When the uninformed consumers’ beliefs satisfy the ‘intuitive criterion’ and the ‘unprejudiced belief refinement’, prices signal the entrant’s quality only in a two–sided separating equilibrium and are identical to the full information outcome.

1 Introduction

It is well established that in markets with asymmetric information firms may use prices, possibly in conjunction with additional marketing devices, to signal quality information to uninformed market participants. In particular, if only some fraction of consumers is informed about quality, then firms may signal their qualities to the uninformed by setting prices higher than under perfect information. The idea is that high-quality firms suffer less from decreased sales to informed consumers due to price increases than low-quality firms. Therefore a high-quality firm can separate itself by setting a high price which is not profitable to imitate for the low-quality firm. Signalling thus leads to distorted pricing and an inefficient reduction in the supply of high-quality goods.

This paper studies an extension of the standard price signalling model to a durable goods duopoly. In this environment the informative signalling equilibrium is free of distortions and identical to the perfect information equilibrium. This observation has significant welfare implications. The existing literature concludes that under incomplete information about quality, prices are distorted upwards in order to signal private information. Therefore price competition is softened and the firms’ market power increases. This causes a welfare loss because buyer decisions are distorted and their choice of qualities may be socially inferior to the full information outcome. Our paper shows that at least for some class of markets, when signalling occurs it must be non–distortionary and does not lead to market power. Further, when signaling is possible, regulations that e.g. require mandatory disclosure of quality are not likely to lead to any welfare gain.

We obtain our results for a horizontally and vertically differentiated duopoly market with price-setting competitors engaging in a game of signalling rivalry: An established firm (“incumbent”), whose quality is known by all market participants, faces a competitor (“entrant”) who is either supplying the same quality as the incumbent or a superior quality acquired through some product innovation. Both firms and some fraction of consumers know the entrant’s quality. The uninformed consumers use prices set by both firms to infer quality information. An important feature of price competition is that the two firms have opposing interests in conveying information, because the incumbent gains and the entrant loses when observed prices make the uninformed consumers more pessimistic about the entrant’s quality.

In our model consumers are confronted with two price signals concerning a single uncertain variable, the entrant’s quality. For the analysis of equilibrium, we apply two standard refinements for the uninformed consumers’ out-of-equilibrium beliefs. First, we use the ‘intuitive criterion’ of Cho and Kreps (1987). Second, in situations where one of the firms’ pricing is informative we adopt the ‘unprejudiced belief criterion’ of Bagwell and Ramey (1991) to the pricing strategy of its competitor, because the intuitive criterion is no longer applicable. Under the unprejudiced belief criterion the consumers trust in the price signal of the non–deviating competitor whenever only one of the two firms selects an out-of-equilibrium price. This means that, given the other firm plays an equilibrium separating strategy, a deviating firm cannot influence beliefs by deviating to a non-equilibrium price and therefore always sets its best response price as under perfect information.

The unprejudiced belief criterion therefore excludes all separating equilibria with prices distorted from full-information prices. We show that these prices constitute the unique separating equilibrium outcome in our model as long as the fraction of informed consumers is not too small. If only rather few consumers are informed, there is no informative equilibrium. The reason is that either the low-type entrant could gain by deviating to the high-type equilibrium price or the incumbent playing against the high-type would deviate to the low-type equilibrium price. Thus the firms’ price signals would become contradictory: The entrant would signal that his quality is high and the incumbent that the entrant’s quality is low.

1.1 Related Literature

The standard prediction of the literature on price signalling is that quality uncertainty leads to distorted pricing for signalling purposes. The earliest contributions to this literature consider a market with a single seller. For example, Milgrom and Roberts (1986) show that a monopolist may use price and advertisement to convince consumers of the quality of a newly introduced product. In their model, which is based on repeat purchases of a non-durable good, prices can be distorted up– or downwards depending on expectations over future sales. Bagwell and Riordan (1991) consider a monopolist who produces a durable good whose quality may be high or low. The existence of informed consumers and cost differences between qualities allow the monopolist to signal high quality through an upward distorted price.¹ Basically, our model extends Bagwell and Riordan (1991) to a horizontally differentiated duopoly in which one of the two firms offers a quality that is known to the competitor but not to all consumers.

One strand of the literature extends the analysis of price signalling to oligopolistic markets under the assumption that firms have private information only about their own quality. They are not informed about the other firms and, therefore, have the same prior about their competitors’ qualities as the uninformed consumers. Daughety and Reinganum (2007) and Daughety and Reinganum (2008) examine a horizontally and vertically differentiated duopoly and n–firm oligopoly, respectively. Price setting takes into account the ex-ante probabilities of rivals to be high– or low–quality types. Separating equilibria imply upward distorted prices, increasing in the ex-ante probability of firms being high–types. Similarly, Janssen and Roy (2010) show for a homogenous oligopoly that fully revealing mixed strategy equilibria exist in which high–types distort prices upward and low–types randomize prices over an interval, thereby generating sufficient rents to avoid mimicry of the high–types.

Closer related to the information structure in our model is the other strand of the literature that assumes the oligopolists to be informed about their rivals’ qualities. Hertzendorf and Overgaard (2001a) analyze price setting and advertising in a duopoly. To keep the analysis tractable, they assume that qualities are perfectly negatively correlated and consumers only know that one firm offers high quality and the other low quality. They apply two refinements that lead to a unique separating and a unique pooling equilibrium. In the separating equilibrium, a high degree of vertical differentiation leads to upwards distorted prices and a low degree to downward distorted prices. Yehezkel (2008) introduces some informed consumers into a similar model and examines how pricing and advertising strategies depend on the fraction of informed consumers.

In Fluet and Garella (2002) the ex ante distribution of the firm’s qualities is such that either both firms offer low quality or one firm offers low and the other high quality. The authors avoid the use of selection criteria and find multiple separating and pooling equilibria. For small quality differences separation can only be achieved with a combination of upward distorted prices and advertisement. This result is similar to a finding by Hertzendorf and Overgaard (2001b), who show that fully revealing separating equilibria satisfying the unprejudiced belief condition do not exist.

These papers differ from our model in that they consider product differentiation only in the vertical dimension. This implies that the duopolists have a common interest in signalling different qualities since they earn zero profits if consumers believe that they both offer the same expected quality. In our model of signalling rivalry such a common interest does not exist because consumer preferences are differentiated horizontally between the firms, and in the vertical dimension all consumers have identical preferences. As a consequence, the incumbent always prefers the consumers to believe that the entrant’s quality is identical to his own quality, whereas the entrant gains by convincing the consumers that he offers a superior quality. Another feature that distinguishes our model from the above literature is that the duopolists are not in a symmetric position. In contrast with Hertzendorf and Overgaard (2001a, 2001b) and Fluet and Garella (2002), firms are not “anonymous” but consumers know which firm is quoting which price. The consumers are uninformed only about the entrant’s and not about the incumbent’s quality, and they interpret the prices of both firms as signals only about the entrant’s quality.

In our analysis, there is no role for expenditures on directly uninformative advertising as an additional signal. This observation differs from Barigozzi et al. (2006), who in a closely related framework show that the entrant may prefer advertising together with price over pure price signaling. In their model, however, all consumers are uninformed and so a separating equilibrium without distortions does not exist. In contrast, in our model there are some informed consumers and under our belief refinements only the full–information equilibrium without distortions survives. Thus, dissipative advertising cannot occur in equilibrium.

From a methodological perspective our analysis is closely related to Bagwell and Ramey (1991) and Schultz (1999). They study limit pricing by two incumbents to affect the entry decision of a third firm. The incumbents’ prices signal their information about an industry–wide parameter. The third firm enters the market only if it concludes that the probability of a favorable state is sufficiently high. In the paper by Bagwell and Ramey (1991) the competitors have a common interest, both want to signal an unfavorable state in order to prevent entry. Introducing the unprejudiced belief refinement, the authors find that only non–distorted separating equilibria exist. Further, under additional assumptions the intuitive criterion of Cho and Kreps (1987) eliminates all equilibria with pooling. By applying the same belief refinements to our context, we arrive at similar conclusions for the qualitative features of equilibrium. Schultz (1999) considers a variation of Bagwell and Ramey (1991) where the incumbents have conflicting interests, i.e. one incumbent prefers the entrant to stay out of the market, whereas its competitor profits from entry. Again, separating equilibrium prices are not distorted. But due to signalling rivalry these equilibria only exist if the effect of entry on the incumbent’s profits is relatively small. We obtain a related result in our model when the fraction of informed consumers is rather small.

This paper is organized as follows. In Section 2 we describe the model and, as a reference point, we derive the equilibrium under full information. Section 3 defines the Perfect Bayesian Equilibrium and explains the belief refinements of our analysis. In Section 4 we show that under our refinements only the full information equilibrium prices can survive in an informative equilibrium and that such an equilibrium exists if the fraction of informed consumers is not too small. Section 5 provides concluding remarks. The proofs of all formal results are relegated to an appendix in Section 6.

2 The Model

We employ the demand structure of the standard Hotelling (1929) duopoly with the modification that the two firms may offer different qualities. While one of the firms offers a quality that is publicly known by all market participants, the quality of the other firm is not known to some share of consumers. This environment may describe a market in which a new firm has entered to compete against an already established firm. In such a market it is reasonable to assume that consumers are informed about the established firm’s quality through their prior experience, but that some consumers are uncertain about the new firm’s product quality. In the terminology of Nelson (1970), the new good is an experience good so that an uninformed consumer learns its true quality only after purchase. For convenience, in what follows we call the established firm the incumbent and the new firm the entrant. Our formal analysis focuses on post-entry price competition, but in the conclusions in Section 5 we indicate some implications for market entry decisions.

The incumbent firm is fully informed about the true quality of the entrant. The uninformed consumers use the firms’ prices to draw inferences about the entrant’s quality. Accordingly, the price setting behavior of both firms takes into account that prices are quality signals.

There is a unit mass of consumers whose preference characteristic x is uniformly distributed on the interval [0,1]. Each consumer purchases at most one unit of the good from either the incumbent I or the entrant E. Given the incumbent’s quality q _I and the entrant’s (expected) quality q _E , the valuation of a consumer with characteristic x∈[0,1] is

$$ v_{I}(x) = q_{I} - t x, \quad v_{E}(x) = q_{E} - t (1-x) $$

(1)

for the incumbent’s and the entrant’s good. The parameter t reflects the degree of horizontal product differentiation. The two firms are also vertically differentiated if q _I ≠q _E . But the quality differential between the two firms affects the taste of all consumers in the same way, independently of their characteristic x. This aspect distinguishes our model from the price signalling models of Hertzendorf and Overgaard (2001a) and Fluet and Garella (2002) who similarly to Shaked and Sutton (1982) assume that consumers differ in their valuation of quality and that the goods are not horizontally differentiated. In what follows, we assume that the firms’ qualities are sufficiently high so that each consumer buys one unit of the good.

All consumers observe the incumbent’s price p _I and the entrant’s price p _E . The critical consumer type $\tilde x$, who is indifferent between purchasing from firm I and firm E, is then determined by $v_{I}(\tilde x) - p_{I} = v_{E}(\tilde x) - p_{E}$, and by Eq. 1 we have

$$ \tilde x(p_{I}, p_{E}, q_{E} - q_{I}) =\max\left\{0,\min\left\{ \frac{p_{E} - p_{I} - (q_{E} - q_{I}) + t}{2 t},1\right\}\right\}. $$

(2)

All consumers with $x < \tilde x$ optimally buy the incumbent’s good, whereas consumers with $x > \tilde x$ purchase from the entrant.

There are two possible qualities, q _L and q _H , with 0<q _L <q _H . The incumbent’s quality is commonly known to be q _I =q _L . There is uncertainty, however, about the entrant’s quality. Its quality is q _E =q _H with ex ante probability λ and q _E =q _L with probability 1−λ. One interpretation is that with probability λ the entrant has realized a product innovation which increases the ‘standard’ quality q _L by the amount q _H −q _L . We normalize the unit cost of producing quality q _L to zero and assume that the unit cost of producing quality q _H is c>0.

By Eq. 2 only the quality differential between the two firms affects the consumers’ demand decisions. Therefore, we can simplify notation by defining

$$ {equation}\label{Delta} {\Delta} \equiv q_{H} - q_{L}. $$

(3)

We assume that the high quality entrant has a competitive advantage because cΔ.

Both firms observe the realization of q _E before setting prices. In addition some fraction γ∈(0,1) of consumers becomes informed about the entrant’s true quality before making demand decisions. Each consumer type x is equally likely to be informed. This means that in each subset of the consumers’ characteristic set [0,1] the fraction of informed consumers is identically equal to γ.

The uninformed consumers use the observed prices p _I and p _E to draw inferences about the entrant’s quality. We denote their posterior belief that the entrant’s quality is q _E =q _H by μ∈[0,1]. Thus the uninformed consumers anticipate that the entrant offers the expected quality μ q _H +(1−μ)q _L =q _L +μΔ. Since consumers are risk–neutral with respect to quality, for given prices p _I and p _E their demand decisions depend only on the expected quality difference between the two sellers.

In the uninformed consumers’ expectation the quality difference q _E −q _I is always equal to μΔ, independently of the entrant’s true quality. If the entrant’s quality is q _E =q _L , the informed consumers know that q _E −q _I =0. Therefore, by Eq. 2 the incumbent’s and the entrant’s market shares, D _{I L} and D _{E L} , are given by

$$\begin{array}{@{}rcl@{}} D_{IL}(p_{I}, p_{E}, \mu) &=& \gamma \tilde x(p_{I}, p_{E}, 0)+ (1-\gamma)\tilde x(p_{I}, p_{E}, \mu {\Delta}), \\ D_{EL}(p_{I}, p_{E}, \mu) &=& 1 - D_{IL}(p_{I}, p_{E}, \mu). \end{array} $$

(4)

If q _E =q _H , the informed consumers know that q _E −q _I =Δ. In this case, the two sellers’ market shares, D _{I H} and D _{E H} , are equal to

$$\begin{array}{@{}rcl@{}} D_{IH}(p_{I}, p_{E}, \mu) &=& \gamma \tilde x(p_{I}, p_{E}, {\Delta}) + (1-\gamma) \tilde x(p_{I}, p_{E}, \mu {\Delta}), \\ D_{EH}(p_{I}, p_{E}, \mu) &=& 1 - D_{IH}(p_{I}, p_{E}, \mu). \end{array} $$

(5)

Therefore, if the entrant’s quality is q _E =q _L , the incumbent’s profit is π _{I L} =p _I D _{I L} and the entrant’s profit π _{E L} =p _E D _{E L} . If q _E =q _H , the duopolists’ profits are π _{I H} =p _I D _{I H} and π _{E H} =(p _E −c)D _{E H} .

Note that for all γ∈(0,1) it is the case that

$$ \frac{\partial {\Pi}_{EL}}{\partial \mu} \geq 0, \quad \frac{\partial {\Pi}_{EH}}{\partial \mu} \geq 0; \quad \frac{\partial {\Pi}_{IL}}{\partial \mu} \leq 0, \quad \frac{\partial {\Pi}_{IH}}{\partial \mu} \leq 0. $$

(6)

Irrespective of the true quality, the entrant never loses and the incumbent never gains when the uninformed consumers raise their belief that the entrant offers high quality. Since these consumers interpret the firms’ prices as quality signals, price competition entails a signalling rivalry: The entrant has an incentive to choose a price that indicates high quality. This is in conflict with the incumbent’s interest to convince consumers that the entrant offers low quality.

Before analyzing how the duopolists’ signalling rivalry affects their price competition, we briefly describe the equilibrium under full information. The firms compete by simultaneously setting prices and their pricing strategies are contingent on the entrant’s quality. If q _E =q _L , we denote the incumbent’s and the entrant’s price by p _{I L} and p _{E L} , respectively; if q _E =q _H the firms’ prices are denoted by p _{I H} and p _{E H} . When all consumers know the entrant’s quality, the firms’ profits can be calculated by setting μ≡0 for q _E =q _L and μ≡1 for q _E =q _H .² The full information equilibrium prices ${\hat {p}} = (({\hat {p}}_{IL}, {\hat {p}}_{E L}), ({\hat {p}}_{IH}, {\hat {p}}_{EH}))$ are then defined by the conditions for profit maximization so that for all p≥0

$$\begin{array}{@{}rcl@{}} {\Pi}_{IL}({\hat{p}}_{IL}, {\hat{p}}_{EL}, 0) \ge {\Pi}_{IL}(p, {\hat{p}}_{EL}, 0),& {\Pi}_{EL}({\hat{p}}_{IL}, {\hat{p}}_{EL}, 0) \ge {\Pi}_{EL}({\hat{p}}_{IL}, p, 0),& \\ {\Pi}_{IH}({\hat{p}}_{IH}, {\hat{p}}_{EH}, 1) \ge {\Pi}_{IH}(p, {\hat{p}}_{IH}, 1),& {\Pi}_{EH}({\hat{p}}_{IH}, {\hat{p}}_{EH}, 1) \ge {\Pi}_{EH}({\hat{p}}_{IH}, p, 1).& \end{array} $$

(7)

From the corresponding first–order conditions one can easily derive the full information reaction functions

$$\begin{array}{@{}rcl@{}} R_{IL}({\hat{p}}_{EL})&=&\max\left\{\frac{{\hat{p}}_{EL}+t}{2},{\hat{p}}_{EL}-t,0\right\} \end{array} $$

(8)

$$\begin{array}{@{}rcl@{}} R_{EL}({\hat{p}}_{IL})&=&\max\left\{\frac{{\hat{p}}_{IL}+t}{2},{\hat{p}}_{IL}-t,0\right\} \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} R_{IH}({\hat{p}}_{EH})&=&\max\left\{\frac{{\hat{p}}_{EH}+t-{\Delta}}{2},{\hat{p}}_{EH}-t-{\Delta},0\right\} \end{array} $$

(10)

$$\begin{array}{@{}rcl@{}} R_{EH}({\hat{p}}_{IH})&=&\max\left\{\frac{{\hat{p}}_{IH}+t+{\Delta}+c}{2},{\hat{p}}_{IH}-t+{\Delta}, c\right\}. \end{array} $$

(11)

The solution of Eqs. 8–11 depends on the high quality entrant’s competitive advantage. If Δ−c<3t the innovation is non-drastic and the incumbent gets positive profits even when the entrant offers high quality. Otherwise, if Δ−c≥3t, the innovation is drastic and implies that the incumbent’s market share is zero. The full information equilibrium prices in these two cases are

$$\begin{array}{@{}rcl@{}} &{\hat{p}}_{IL} = t, \,\, {\hat{p}}_{EL} = t, \,\, {\hat{p}}_{IH} = t - {({\Delta} - c)}/{3}, \,\, {\hat{p}}_{EH} = t + {({\Delta} + 2 c)}/{3} &\quad\text{if}\quad {\Delta}-c < 3t, \\ &{\hat{p}}_{IL} = t,\,\, {\hat{p}}_{EL} = t, \,\, {\hat{p}}_{IH} = 0, \,\, {\hat{p}}_{EH} = {\Delta}-t &\quad\text{if}\quad {\Delta}-c \geq 3t. \end{array} $$

(12)

If q _E =q _L , both firms charge the same price and have the same market share D _{I L} =D _{E L} =1/2. If q _E =q _H , the incumbent is disadvantaged against the entrant and, even though he sets a lower price, his market share for non-drastic innovations D _{I H} =(3t−Δ+c)/(6t) is smaller than the entrant’s share D _{E H} =(3t+Δ−c)/(6t). For drastic innovations we have D _{I H} =0 and D _{E H} =1.

3 Equilibrium and Belief Restrictions

We envisage the market to operate in three stages. First, both firms and a fraction γ of consumers observe the realization of the entrant’s quality. Second, the firms compete by simultaneously setting prices. Finally, in the third stage the uninformed consumers use observed prices to update their beliefs about the entrant’s quality, and all consumers decide whether to buy from the incumbent or the entrant.

In what follows we study pricing strategies of the firms and consumer beliefs that constitute a Perfect Bayesian Equilibrium of this game. The firms choose their prices contingent on their information about the entrant’s quality, and the uninformed consumers’ posterior probability of facing the high quality entrant is a function of the firms’ prices. In equilibrium, each firm’s price maximizes its profit and the uninformed consumer’s posterior belief is consistent with Bayesian updating.³

More formally, $(p^{*}, \mu ^{*}(\cdot )) = ((p_{IL}^{*}, p_{EL}^{*}), (p_{IH}^{*}, p_{EH}^{*}), \mu ^{*}(\cdot ))$ with $\mu ^{*}\colon \text {I\kern -.25em R}_{+}^{2} \to [0, 1]$ is a Perfect Bayesian Equilibrium (PBE) if (a) for Q=L,H

$$\begin{array}{@{}rcl@{}} p_{IQ}^{*} &=& {\text{argmax}}_{p} \, {\Pi}_{IQ}(p, p_{EQ}^{*}, \mu^{*}(p, p_{EQ}^{*})), \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} p_{EQ}^{*} &=& {\text{argmax}}_{p} \, {\Pi}_{EQ}(p_{IQ}^{*}, p, \mu^{*}(p_{IQ}^{*}, p)), \end{array} $$

(14)

and (b)

$$\begin{array}{@{}rcl@{}} \mu^{*}(p_{IL}^{*}, p_{EL}^{*}) = 1 - \mu^{*}(p_{IH}^{*}, p_{EH}^{*}) = 0, &\quad\text{if}\quad p_{IL}^{*} \ne p_{IH}^{*} \quad\text{or}\quad p_{EL}^{*} \ne p_{EH}^{*}, \end{array} $$

(15)

$$\begin{array}{@{}rcl@{}} \mu^{*}(p_{IL}^{*}, p_{EL}^{*}) = \mu^{*}(p_{IH}^{*}, p_{EH}^{*}) = \lambda, &\quad\text{if}\quad p_{IL}^{*} = p_{IH}^{*} \quad\text{and}\quad p_{EL}^{*} = p_{EH}^{*}. \end{array} $$

(16)

Equilibrium conditions Eqs. 13 and 14 state that, for each quality q _E ∈{q _L ,q _H }, the incumbent and the entrant choose their prices to maximize profits, taking the competitor’s price and the uninformed consumers’ beliefs μ ^∗(⋅) as given. Equilibrium conditions Eqs. 15 and 16 require that on the equilibrium path the buyers’ belief is consistent with Bayes’ rule. The buyers become fully informed about the entrant’s true quality not only in a two–sided separating equilibrium, where $p_{iL}^{*} \ne p_{iH}^{*}$ for both i∈{I,E}, but also in a one–sided separating equilibrium, where $p_{iL}^{*} \ne p_{iH}^{*}$ for some i∈{I,E} and $p_{jL}^{*} = p_{jH}^{*}$ for j≠i. Prices remain uninformative only if $p_{iL}^{*} = p_{iH}^{*}$ for both i∈{I,E}. In such a pooling equilibrium the posterior belief is equal to the a priori probability λ. Since we are interested in the existence and properties of informative equilibria, we ignore pooling in our subsequent analysis.

By Eqs. 13 and 14, the uninformed consumers’ quality expectations affect the duopolists’ pricing decisions. But, conditions Eqs. 15 and 16 impose restrictions on expectations only for prices that are actually chosen in equilibrium. Since out–of–equilibrium beliefs remain arbitrary, there are multiple equilibria, which are a typical feature of signalling games. This is so because the profit of a deviation from the equilibrium price depends on the uninformed consumers’ interpretation of this deviation. For example, the incumbent may be deterred from changing its price simply because consumers would interpret this as a signal that the entrant’s quality is high. Similarly, the entrant may be kept from changing its price if consumers view this as a signal of low quality. Without restrictions on consumer beliefs multiple equilibria with both upward and downward distorted prices can be found .

To avoid this problem, the literature usually applies refinements that impose restrictions on out–of–equilibrium beliefs. A prominent refinement is the ‘intuitive criterion’ of Cho and Kreps (1987), which has been used in a variety of price signalling games.⁴ Unfortunately, this criterion is not generally applicable in the present context because it is defined for signalling games where each player has private information only about his own and not the other players’ characteristics. In our model, however, the duopolists have common private information and not only the entrant’s but also the incumbent’s price may signal the entrant’s quality. Therefore, the intuitive criterion cannot be used in our model if both firms’ prices are informative. Nonetheless, it remains applicable if one of the firms’ equilibrium prices are uninformative, i.e. if $p_{iL}^{*} = p_{iH}^{*}$ for some i∈{I,E}. In this case, the intuitive criterion can be used to refine beliefs for out–of–equilibrium prices of firm j≠i.

Consider the incumbent in a situation where the entrant charges $p_{EL}^{*} = p_{EH}^{*}$ and the incumbent knows that the entrant’s quality is low. Suppose the incumbent wishes to deviate to some price p _I if the uninformed consumers interpret p _I as a signal that indicates a low quality entrant. Then the idea of the intuitive criterion is that p _I should indeed convince the consumers that the entrant offers low quality if the following is true: If the incumbent knew that the entrant’s quality is high, he would not gain from deviating to p _I even if the consumers would respond favorably for the incumbent by believing that p _I indicates a low quality entrant.

An analogous argument applies to the high quality entrant in a situation where the incumbent’s pricing $p_{iL}^{*} = p_{iH}^{*}$ reveals no information. In this case, the intuitive criterion requires the uninformed consumers to believe that a price p _E signals high quality if for this belief deviating to p _E is profitable only for the high quality entrant and not for the low quality entrant.

More formally, the PBE (p ^∗,μ ^∗(⋅)) satisfies the intuitive criterion if the following two conditions (a) and (b) are satisfied: (a) If $p_{EL}^{*} = p_{EH}^{*} = p_{E}^{*}$, then $\mu ^{*}(p_{I}, p_{E}^{*}) = 0$ for all p _I such that

$$ {\Pi}_{IH}(p_{I}, p_{E}^{*}, 0) \le {\Pi}_{IH}(p_{IH}^{*}, p_{E}^{*}, \mu^{*}(p_{IH}^{*}, p_{E}^{*})) $$

(17)

and

$$ {\Pi}_{IL}(p_{I}, p_{E}^{*}, 0) > {\Pi}_{IL}(p_{IL}^{*}, p_{E}^{*}, \mu^{*}(p_{IL}^{*}, p_{E}^{*})). $$

(18)

(b) If $p_{IL}^{*} = p_{IH}^{*} = p_{I}^{*}$, then $\mu ^{*}(p_{I}^{*}, p_{E}) = 1$ for all p _E such that

$$ {\Pi}_{EL}(p_{I}^{*}, p_{E}, 1) \le {\Pi}_{EL}(p_{I}^{*}, p_{EL}^{*}, \mu^{*}(p_{I}^{*}, p_{EL}^{*})) $$

(19)

and

$$ {\Pi}_{EH}(p_{I}^{*}, p_{E}, 1) > {\Pi}_{EH}(p_{I}^{*}, p_{EH}^{*}, \mu^{*}(p_{I}^{*}, p_{EH}^{*})). $$

(20)

As a refinement for situations where firm i∈{I,E} defects from the equilibrium and firm j≠i uses a separating strategy $p_{jL}^{*} \ne p_{jH}^{*}$, we employ the ‘unprejudiced belief criterion’ introduced by Bagwell and Ramey (1991). The basic idea of this criterion is that upon observing an out–of–equilibrium price pair (p _I ,p _E ) the uninformed consumers rationalize their observation with the fewest number of deviations from the equilibrium strategies. Therefore, if a price pair occurs where one of the prices is out–of–equilibrium while the other price belongs to the separating pricing strategy of the competitor, the consumers believe that the entrant’s quality is signaled by the competitor.

Actually, since there are only two types of the entrant, in our context it is sufficient to consider a simplified version of the unprejudiced belief criterion: If only the entrant chooses an out–of–equilibrium price p _E and the incumbent’s equilibrium price $p_{IH}^{*}$ indicates a high quality entrant, then the uninformed consumers should conclude that the entrant offers high quality; there are no belief restrictions if the incumbent’s price $p_{IL}^{*}$ signals low quality. Indeed, a high quality signal of the incumbent looks rather convincing since it is against his interest to admit that his competitor offers a superior good. An analogous reasoning applies when the uninformed consumers conjecture that the price p _I constitutes a unilateral deviation by the incumbent. In this situation, they should infer from the entrant’s price $p_{EL}^{*}$ that his quality is low; there are no belief restrictions if the entrant’s price is $p_{EH}^{*}$. Again, this seems plausible because expecting high quality makes little sense if the entrant acknowledges that his quality is low.

More formally, the PBE (p ^∗,μ ^∗(⋅)) satisfies the (simplified) unprejudiced belief criterion if the following two conditions (a) and (b) are satisfied: (a) If $p_{IL}^{*} \ne p_{IH}^{*}$, then $\mu ^{*}(p_{IH}^{*}, p_{E}) = 1$ for all $p_{E} \ne p_{EL}^{*}$. (b) If $p_{EL}^{*} \ne p_{EH}^{*}$, then $\mu ^{*}(p_{I}, p_{EL}^{*}) = 0$ for all $p_{I} \ne p_{IH}^{*}$.

Notice that in a two–sided separating equilibrium the criterion does not impose belief restrictions on the out–of–equilibrium price constellations $(p_{IH}^{*}, p_{EL}^{*})$ and $(p_{IL}^{*}, p_{EH}^{*}),$ under which the signals of the incumbent and the entrant appear contradictory. For these constellations it is not clear whether the incumbent or the entrant has deviated from his equilibrium strategy.

In what follows, we call a PBE (p ^∗,μ ^∗(⋅)) that satisfies the intuitive and the unprejudiced belief criterion a signalling equilibrium.⁵ In the following section, we investigate the existence and properties of equilibria where prices reveal the entrant’s quality.

4 Informative Equilibria

4.1 One–Sided Separating Equilibria

We start with the analysis of one–sided separating equilibria, in which one firm chooses a pooling and the other a separating pricing strategy. We will show that such equilibria typically do not exist, except for special parameter constellations. First, consider the case where the incumbent’s price $p^{*}_{I} = p_{IL}^{*} = p_{IH}^{*}$ is independent of the entrant’s quality, whereas the entrant chooses quality contingent prices $p_{EL}^{*}$ and $p_{EH}^{*}$ with $p_{EL}^{*} \ne p_{EH}^{*}$. Because in equilibrium the uninformed consumers infer the entrant’s quality from his price, their beliefs satisfy $\mu ^{*}(p_{I}^{*}, p_{EL}^{*}) = 0$ and $\mu ^{*}(p_{I}^{*}, p_{EH}^{*}) = 1$. The following lemma establishes necessary conditions for this type of equilibrium.

Lemma 1

Suppose that the prices p, with p _I =p _IL =p _IH ,p _EL ≠p _EH , can be supported as a signalling equilibrium (p,μ(⋅)) by some beliefs μ(⋅). Then p must satisfy

$$ p_{EL} ={\text{argmax}}_{p} \, {\Pi}_{EL}(p_{I}, p, 0), $$

(21)

$$ p_{I} ={\text{argmax}}_{p} \, {\Pi}_{IH}(p, p_{EH}, 1) = {\text{argmax}}_{p} \, {\Pi}_{IL}(p, p_{EL}, 0), $$

(22)

$$ p_{EH} \, \, \text{maximizes} \quad {\Pi}_{EH}(p_{I}, p, 1) \quad\text{subject to}\quad {\Pi}_{EL}(p_{I}, p, 1) \le {\Pi}_{EL}(p_{I}, p_{EL}, 0). $$

(23)

Condition Eq. 21 simply states that the low quality entrant’s price reaction against p _I is not distorted by signalling considerations. Indeed, some price p not satisfying Eq. 21 can maximize the low quality seller’s profit only if μ(p _I ,p)>0. But this is inconsistent with an equilibrium where prices reveal the true quality. The same argument underlies the first condition in Eq. 22 for the incumbent’s price when competing against the high quality entrant. The incumbent’s price reaction against p _{E H} cannot be distorted because the consumers’ belief that the entrant has high quality is already the worst possible belief from the incumbent’s perspective.

The second condition for p _I in Eq. 22 is implied by part (b) of the unprejudiced belief criterion. This criterion restricts the consumers’ belief to μ(p,p _{E L} )=0 for all p≠p _I . Further, Bayes’ rule in Eq. 15 requires that μ(p _I ,p _{E L} )=0. Thus, the incumbent’s pricing has no impact on consumer beliefs when facing the low quality entrant, and so in this situation there are also no signalling distortions.

Finally, the constraint in condition Eq. 23 has to be satisfied because otherwise the low quality entrant would gain by imitating the high quality entrant’s price. Further, the intuitive criterion implies that consumers infer high quality whenever the entrant gains by deviating to some price satisfying this constraint. Accordingly, the high quality entrant’s price p _{E H} must solve the constrained maximization problem in Eq. 23.

Lemma 1 allows us to show that a one–sided separating equilibrium with $p_{EL}^{*} \ne p_{EH}^{*}$ exists at most for a single value of the parameter γ. Since there is no reason for why the fraction of informed consumers should be identical to this value, an equilibrium of this type generically fails to exist.

Proposition 1

For all γ≠max{t/(t+Δ),Δ/(t+Δ)} there exists no signalling equilibrium (p ^∗ ,μ ^∗ (⋅)) such that $p_{iL}^{*} = p_{iH}^{*}$ and $p_{EL}^{*} \ne p_{EH}^{*}$.

The nonexistence result stated in Proposition 1 is a straightforward implication of Lemma 1. The lemma shows that prices in a one–sided separating equilibrium have to satisfy four conditions. Yet, such an equilibrium determines only three prices. This means that not all conditions can hold simultaneously, unless the exogenous parameters accidentally make one of the conditions redundant. The following lemma shows that a similar observation applies to the other type of one–sided separating equilibria, in which the entrant adopts a pooling strategy $p_{E}^{*} = p_{EL}^{*} = p_{EH}^{*}$ and only the incumbent’s prices $p_{IL}^{*}$ and $p_{EH}^{*}$ reveal the entrant’s quality so that $\mu ^{*}(p_{E}^{*}, p_{IL}^{*}) = 0$ and $\mu ^{*}(p_{E}^{*}, p_{IH}^{*}) = 1$.

Lemma 2

Suppose that the prices p, with p _IL ≠p _IH ,p _E =p _EL =p _EH , can be supported as a signalling equilibrium (p,μ(⋅)) by some beliefs μ(⋅). Then p must satisfy

$$ p_{IH} ={\text{argmax}}_{p} \, {\Pi}_{IH}(p, p_{E}, 1), $$

(24)

$$ p_{E} ={\text{argmax}}_{p} \, {\Pi}_{EL}(p_{IL}, p, 0) ={\text{argmax}}_{p} \, {\Pi}_{EH}(p_{IH}, p, 1) , $$

(25)

$$ p_{IL} \, \, \text{maximizes} \quad {\Pi}_{IL}(p, p_{E}, 0) \quad\text{subject to}\quad {\Pi}_{IH}(p, p_{E}, 0) \le {\Pi}_{IH}(p_{IH}, p_{E}, 1). $$

(26)

By our next proposition, the implications Lemmas 2 are similar to Lemma 1. In fact, Lemma 2 implies that a one–sided separating equilibrium with $p_{iL}^{*} \ne p_{iH}^{*}$ typically fails to exist.

Proposition 2

For generic values of γ there exists no signalling equilibrium (p ^∗ ,μ ^∗ (⋅)) such that $p_{iL}^{*} \ne p_{iH}^{*}$ and $p_{EL}^{*} = p_{EH}^{*}$.

Our results so far show that in an informative equilibrium it cannot happen that one of the duopolists adopts a pooling strategy. Propositions 1 and 2 eliminate one–sided pooling by combining the intuitive and the unprejudiced belief criterion. This leaves a two–sided separating equilibrium as the remaining candidate for a signalling equilibrium.

4.2 Two–Sided Separating Equilibria

In a two–sided separating equilibrium the uninformed consumers’ equilibrium belief is $\mu ^{*}(p_{IL}^{*}, p_{EL}^{*}) = 0$ and $\mu ^{*}(p_{IH}^{*}, p_{EH}^{*}) = 1$ as $p_{iL}^{*} \ne p_{iH}^{*}$ and $p_{EL}^{*} \ne p_{EH}^{*}$. Since each firm’s price is informative, the intuitive criterion is no longer applicable. Therefore, only the unprejudiced belief criterion plays a role in the following lemma which provides necessary and sufficient conditions for a two–sided separating equilibrium.

Lemma 3

The prices p, with p _IL ≠p _IH ,p _EL ≠p _EH , can be supported as a signalling equilibrium (p,μ(⋅)) by some beliefs μ(⋅) if and only if (a) p is identical to the perfect information equilibrium ${\hat {p}}$ , and (b) there exists some $\bar {\mu } \in [0, 1]$ such that

$$ {\Pi}_{IH}(p_{IH}, p_{EH}, 1) \ge {\Pi}_{IH}(p_{IL}, p_{EH}, \bar{\mu}), \quad {\Pi}_{EL}(p_{IL}, p_{EL}, 0) \ge {\Pi}_{EL}(p_{IL}, p_{EH}, \bar{\mu}). $$

(27)

By statement (a) of Lemma 3, in a two–sided separating equilibrium the firms’ prices are identical to the outcome of price competition under full information of all market participants about the entrant’s quality. Thus, even though prices act as signals, they are not distorted by incentive restrictions. This observation is a well–known implication of the unprejudiced belief refinement (see Bagwell and Ramey (1991)).⁶ The idea is simply that the high quality entrant can ignore signalling effects when already the incumbent’s price convinces the uninformed consumers of high quality. Similarly, the incumbent does not have to resort to distorted pricing to indicate a low quality entrant, because the entrant himself already reveals his quality through his price setting strategy. In a two–sided separating equilibrium, therefore, the firms’ prices are determined as mutually undistorted best responses against the competitor and are thus identical to the full information equilibrium.

While prices are not distorted by signalling effects, statement (b) of Lemma 3 shows that they have to satisfy an incentive compatibility restriction, which is related to the signalling rivalry between the duopolists. The uninformed consumers will be perplexed when they observe the out–of-equilibrium price pair $({\hat {p}}_{IL}, {\hat {p}}_{EH})$. These prices are contradictory because the incumbent’s price signals a low quality entrant and the entrant’s price a high quality. Also, it is not clear which firm has deviated from its equilibrium strategy. The prices $({\hat {p}}_{IL}, {\hat {p}}_{EH})$ could originate from the equilibrium pair $({\hat {p}}_{IH}, {\hat {p}}_{EH})$ because the incumbent has deviated to ${\hat {p}}_{IL}$; or they could originate from the equilibrium pair $({\hat {p}}_{IL}, {\hat {p}}_{EL})$ because the entrant has deviated to ${\hat {p}}_{EH}$.

Condition Eq. 27 states that there must be some belief $\bar {\mu } = \mu ({\hat {p}}_{IL}, {\hat {p}}_{EH})$ that deters both kinds of deviations. On the one hand, by the first inequality in Eq. 27, $\bar {\mu }$ must be high enough so as to make it unattractive for the incumbent to deviate from ${\hat {p}}_{IH}$ to ${\hat {p}}_{IL}$. On the other hand, the second inequality in Eq. 27 requires that $\bar {\mu }$ is small enough so that the entrant cannot gain by deviating from ${\hat {p}}_{EL}$ to ${\hat {p}}_{EH}$. To examine whether both inequalities can be satisfied, we first consider the case of non-drastic innovations (Δ−c<3t) and then drastic innovations (Δ−c≥3t).

4.2.1 Non-Drastic Innovations

Whether condition (b) of Lemma 3 is satisfied for $p = {\hat {p}}$, depends on how large the fraction γ of informed consumers is. As the proof of our next result shows, Lemma 3 implies that a two–sided separating equilibrium exists if and only if γ is not too small.

Proposition 3

Let Δ−c<3t. There exists a ${\bar {\gamma }} = {\bar {\gamma }}(t, {\Delta }, c) \in (0,1)$ such that the following holds: (a) If $\gamma \ge {\bar {\gamma }},$ then there exists a signalling equilibrium (p ^∗ ,μ ^∗ (⋅)) with $p_{iL}^{*} \ne p_{iH}^{*}$ and $p_{EL}^{*} \ne p_{EH}^{*}$ . The prices p ^∗ in this equilibrium are identical to the perfect information equilibrium ${\hat {p}}$ . (b) If $\gamma < {\bar {\gamma }},$ there exists no signalling equilibrium (p ^∗ ,μ ^∗ (⋅)) such that $p_{iL}^{*} \ne p_{iH}^{*}$ and $p_{EL}^{*} \ne p_{EH}^{*}$.

In a two–sided separating equilibrium prices are not distorted by signalling. The incumbent or the entrant can gain by a unilateral deviation only because this changes the uninformed consumers’ beliefs. Therefore, a deviation is unprofitable as long as sufficiently many consumers are informed. This explains why $({\hat {p}}, \mu ^{*}(\cdot ))$ can constitute a signalling equilibrium for $\gamma \ge {\bar {\gamma }}$. If $\gamma < \bar {\gamma }$, then the firms’ signalling rivalry is too intense to prevent profitable deviations: Either the incumbent will defect from the equilibrium if q _E =q _H , or the entrant will defect if q _E =q _L . As observed by Schultz (1999) in a different context, conflicting interests may thus rule out the existence of a two–sided separating equilibrium for some parameter constellations.

In Table 1 some numerical calculations illustrate how ${\bar {\gamma }}$ depends on c and t if Δ=10. They reveal that ${\bar {\gamma }}$ does not depend monotonically on the parameters of our model. In particular, if the measure t of horizontal differentiation increases, ${\bar {\gamma }}$ first decreases and then increases. Our next result shows that this is true in general.

Table 1 Numerical values for ${\bar {\gamma }}$

Proposition 4

(a) For t sufficiently large the critical value $\bar {\gamma }(t, {\Delta }, c)$ , stated in Proposition 3, satisfies

$$ \frac{\partial\bar{\gamma}(t, {\Delta}, c)}{\partial t} > 0, \quad \left.\frac{\partial\bar{\gamma}(t, \lambda{\Delta}, \lambda c)}{\partial \lambda}\right\vert_{\lambda = 1} < 0. $$

(28)

(b) For t sufficiently close to (Δ−c)/3, ⁷

$$ \frac{\partial\bar{\gamma}(t, {\Delta}, c)}{\partial t} < 0, \quad \left.\frac{\partial\bar{\gamma}(t, \lambda{\Delta}, \lambda c)}{\partial \lambda}\right\vert_{\lambda = 1} > 0. $$

(29)

Prices can be used as credible signals because of their effect on the informed consumers’ demand. When the degree t of horizontal differentiation is sufficiently large, some of these consumers still purchase from the firm that has deviated to a higher price. In this situation, $\bar {\gamma }$ is increasing in t because the price sensitivity of demand is negatively related to the product differentiation parameter t. Conversely, fewer informed consumers are required for the two–sided separating equilibrium if Δ−c is increased to λΔ−λ c, with λ>1. Such an increase improves the competitiveness of the high quality entrant. It facilitates the requirements for an equilibrium because it raises the price differences $\vert {\hat {p}}_{IH} - {\hat {p}}_{IL}\vert$ and $\vert {\hat {p}}_{EL} - {\hat {p}}_{EH}\vert$. Therefore, a deviation of the incumbent from ${\hat {p}}_{IH}$ to ${\hat {p}}_{IL}$ or of the entrant from ${\hat {p}}_{EL}$ to ${\hat {p}}_{EH}$ becomes less profitable, and a smaller fraction $\bar {\gamma }$ of informed consumers suffices for existence of a signalling equilibrium.

As part (b) of Proposition 4 shows, the comparative statics properties of $\bar {\gamma }$ are reversed if t is relatively small. Then, by deviating from the full information price to a higher price, a firm loses the entire demand of the informed consumers. Therefore, the price sensitivity of the informed consumers’ demand plays no role. Instead, because equilibrium profits are increasing in t, the incentives for deviation become weaker if t increases. In particular, the incumbent is driven out of the market by the high quality entrant and receives zero profits in the limit as t→(Δ−c)/3. Therefore, when t becomes small more informed consumers are required to prevent the incumbent from deviating to ${\hat {p}}_{IL}$.

4.2.2 Drastic Innovations

Now we show that Proposition 3 holds also for drastic innovations: if the fraction γ of informed consumers is not too small, then the full information prices ${\hat {p}}$ in Eq. 12 can be supported as a signalling equilibrium by some beliefs μ ^∗(⋅). We begin with the following lemma:

Lemma 4

Let Δ−c>3t. If and only if $\bar {\mu } \ge ({\Delta } - t)/{\Delta },$ then

$$ {\Pi}_{IH}({\hat{p}}_{IH}, {\hat{p}}_{EH}, 1) \ge {\Pi}_{IH}({\hat{p}}_{IL}, {\hat{p}}_{EH}, \bar{\mu}), $$

(30)

i.e. the incumbent does not gain from deviating to ${\hat {p}}_{IL}$ if q _E =q _H .

Since the incumbent makes zero profit at the prices $({\hat {p}}_{IH}, {\hat {p}}_{EH})$, he could gain by deviating to ${\hat {p}}_{IL}$ only if this attracts some uninformed consumers. Lemma 4 shows that this is not possible as long as the uninformed consumers remain sufficiently optimistic about the entrant’s quality when observing the out–of–equilibrium prices $({\hat {p}}_{IL}, {\hat {p}}_{EH})$.

Lemma 5

Let Δ−c>3t and $\bar {\mu } \ge ({\Delta } - t)/{\Delta }$ . If and only if γ≥(2Δ−3t)/(2Δ−2t), then

$$ {\Pi}_{EL}({\hat{p}}_{IL}, {\hat{p}}_{EL}, 0) \ge {\Pi}_{EL}({\hat{p}}_{IL}, {\hat{p}}_{EH}, \bar{\mu}). $$

(31)

i.e. the entrant does not gain from deviating to ${\hat {p}}_{EH}$ if q _E =q _L .

By deviating to ${\hat {p}}_{EH}$ the low quality entrant loses the demand from the informed consumers. But, by Lemma 4 he attracts all uninformed consumers. A deviation is therefore not profitable only if the fraction of informed consumers is not too small.

Lemmas 3, 4 and 5 immediately imply the following extension of Proposition 3 to the case of drastic innovations:

Proposition 5

Let Δ−c>3t and define $\tilde \gamma \equiv (2 {\Delta } - 3 t)/(2 {\Delta } - 2 t)$ . (a) If $\gamma \ge \tilde \gamma ,$ there exists a signalling equilibrium (p ^∗ ,μ ^∗ (⋅)) with $p_{iL}^{*} \ne p_{iH}^{*}$ and $p_{EL}^{*} \ne p_{EH}^{*}$ . The prices p ^∗ in this equilibrium are identical to the perfect information equilibrium ${\hat {p}}$ . (b) If $\gamma < \tilde \gamma ,$ there exists no signalling equilibrium (p ^∗ ,μ ^∗ (⋅)) such that $p_{iL}^{*} \ne p_{iH}^{*}$ and $p_{EL}^{*} \ne p_{EH}^{*}$.

Since $\partial \tilde \gamma /\partial t < 0$ and $\partial \tilde \gamma /\partial {\Delta } > 0$, the comparative statics properties of $\tilde \gamma$ are in line with part (b) of Proposition 4. Interestingly, in the limit t→0 we obtain $\tilde \gamma \to 1$. This means that in a homogenous market a separating signalling equilibrium cannot exist with some uninformed consumers. The intuition is that the incumbent is driven out of the market by the high quality entrant, and for t→0 also the low quality entrant’s profits become negligible at the full information prices. Therefore, if q _E =q _H the incumbent will deviate from $p^{*}_{IH}$ to $p^{*}_{IL}$ if this attracts some uninformed consumers. Should no uninformed consumer purchase from the incumbent, this means that all uninformed consumers turn to the entrant when observing the prices $(p_{IL}^{*}, p_{EH}^{*})$. But this implies that the low quality entrant will gain by deviating from $p^{*}_{EL}$ to $p^{*}_{EH}$.

5 Conclusion

Our analysis shows that a firm may not have to resort to distorted pricing to signal its quality to the uninformed consumers. If its quality is known to a competitor, then the prices of both firms become quality signals and signalling competition may lead to non–distorted pricing in equilibrium. Indeed, under two belief refinements that have frequently been used in the literature, we show that prices convey quality information only if the are identical to the equilibrium under full information.

This finding has obvious implications for other strategic choices. For example, consider the market entry decision of a firm whose quality is not publicly observable. As long as sufficiently many consumers are informed, our analysis indicates that entry decisions are not distorted when at least one of the incumbent firms learns the new firm’s quality after it has entered the market. A similar conclusion obtains for R&D investments in product innovation when some consumers cannot observe whether the investment has been successful or not. If competing firms and a sufficient fraction of consumers become informed about the outcome, our results suggest that the incentives for product innovation are not distorted by the presence of uninformed consumers.

The main objective of our paper is a characterization of informative equilibria where prices fully reveal quality. Our analysis also identifies the parameter constellations under which this type of equilibrium does not exist. When this is the case, our arguments imply that only equilibria in which both firms choose some pooling strategy are consistent with our belief refinements. This means that with positive probability each firm has to charge a price independent of the entrant’s quality. In addition to full pooling, this leaves also the possibility that in equilibrium one or both firms adopt a partial pooling strategy by randomizing between a non–revealing and a revealing price. Investigating the properties of equilibria with such pricing strategies may be an interesting subject of further research.⁸

6 Appendix

Proof of Lemma 1:

Since p _{E L} ≠p _{E H} implies μ(p _I ,p _{E L} )=0 and ∂π _{E L} /∂ μ>0, it follows from equilibrium condition Eq. 14 that for all p≥0

$$ {\Pi}_{EL}(p_{I}, p_{EL}, 0) \ge {\Pi}_{EL}(p_{I}, p, \mu(p_{I}, p)) \ge {\Pi}_{EL}(p_{I}, p, 0). $$

(32)

This proves that Eq. 21 must hold. Analogously, μ(p _I ,p _{E H} )=1 and ∂π _{I H} /∂ μ<0 imply by Eq. 13 that for all p≥0

$$ {\Pi}_{IH}(p_{I}, p_{EH}, 1) \ge {\Pi}_{IH}(p, p_{EH}, \mu(p, p_{EH})) \ge {\Pi}_{IH}(p, p_{EH}, 1). $$

(33)

This proves that p _I must satisfy the first condition in Eq. 22.

Suppose that p _I does not satisfy the second condition in Eq. 22. Since part (b) of the unprejudiced belief criterion implies μ(p,p _{E L} )=0 for all p≠p _I , then there exist some p such that

$$\begin{array}{@{}rcl@{}} {\Pi}_{IL}(p_{I}, p_{EL}, \mu(p_{I}, p_{EL})) &=& {\Pi}_{IL}(p_{I}, p_{EL}, 0) < {\Pi}_{IL}(p, p_{EL}, 0) \\ &=& {\Pi}_{IL}(p, p_{EL}, \mu(p, p_{EL})). \end{array} $$

(34)

This is a contradiction to the condition that in equilibrium p _I has to satisfy Eq. 13 for Q=L.

Note that p _{E H} must satisfy the constraint in Eq. 23 because equilibrium condition Eq. 14 implies that

$$\begin{array}{@{}rcl@{}} {\Pi}_{EL}(p_{I}, p_{EL}, 0) &=& {\Pi}_{EL}(p_{I}, p_{EL}, \mu(p_{I}, p_{EL})) \ge {\Pi}_{EL}(p_{I}, p_{EH}, \mu(p_{I}, p_{EH})) \\ &=& {\Pi}_{EL}(p_{I}, p_{EH}, 1). \end{array} $$

(35)

Suppose that p _{E H} does not solve the maximization problem in Eq. 23. Then there exists some p that satisfies the constraint in Eq. 23 and π _{E H} (p _I ,p,1)>π _{E H} (p _I ,p _{E H} ,1). Because part (b) of the intuitive criterion then implies μ(p _I ,p)=1, this yields

$$\begin{array}{@{}rcl@{}} {\Pi}_{EH}(p_{I}, p, \mu(p_{I}, p)) &=& {\Pi}_{EH}(p_{I}, p, 1) > {\Pi}_{EH}(p_{I}, p_{EH}, 1) \\ &=& {\Pi}_{EH}(p_{I}, p_{EH}, \mu(p_{I}, p_{EH})), \end{array} $$

(36)

a contradiction to equilibrium condition Eq. 14 for Q=H.

Proof Proposition 1:

Using the full-information best-response functions to solve the conditions for Eqs. 21 and 22 in Lemma 1, we get the prices

$$ p_{I}^{*} = t, p_{EL}^{*} = t, \, p_{EH}^{*} = t + {\Delta}. $$

(37)

If the constraint in Eq. 23 is not binding, we obtain the high-quality entrant’s best-response

$$ p^{*}_{EH}=\max\left\{\frac{2t+{\Delta}+c}{2},{\Delta}\right\}. $$

(38)

This, however, is inconsistent with the last equation in Eq.37 as Δ>c and t>0. If the constraint in Eq. 23 is binding, then ${\Pi }_{EL}(p_{I}^{*}, p^{*}_{EH}, 1) = {\Pi }_{EL}(p_{I}^{*}, p_{EL}^{*}, 0)$. By Eq. 37 this equality is equivalent to

$$ \frac{({\Delta} + t)(t - \gamma {\Delta})}{2t} = \frac{t}{2} $$

(39)

for Δ≤t and

$$ \frac{({\Delta} + t)(1-\gamma)}{2} = \frac{t}{2} $$

(40)

for Δ>t. From these equations it follows that the conditions of Lemma 1 are satisfied only if γ=t/(t+Δ) if Δ≤t and γ=Δ/(t+Δ) if Δ>t.

Proof Lemma 2:

The argument is analogous to the proof of Lemma 1.

Proof Proposition 2:

From the full-information best responses for Eqs. 24 and 25 in Lemma 2, we obtain the solution

$$ p_{IL}^{*} = \frac{3 t + 2 {\Delta} + 4 c}{3}, \, p_{IH}^{*} = \frac{3 t - {\Delta} +c }{3}, \, p_{E}^{*} =\frac{3 t + {\Delta} + 2 c}{3} $$

(41)

for Δ−c≤3t and

$$ p_{IL}^{*} = 2{\Delta}-3t, \, p_{IH}^{*} = 0, \, p_{E}^{*} ={\Delta}-t $$

(42)

for Δ−c>3t.

If the constraint in Eq. 26 is not binding and Δ−c≤3t, the full-information best-response of IL is given by $p_{IL}^{*} =\max \left \{ (6 t + {\Delta } + 2 c)/6,({\Delta }+2c)/3\right \}$. This, however, is inconsistent with the first equation in Eq. 41. If the constraint in Eq. 26 is not binding and Δ−c>3t, the full-information best-response of IL is given by $p_{IL}^{*} =\max \left \{\Delta /2,{\Delta }-2t\right \}$. This, however, is inconsistent with the first equation in Eq. 42.

For Δ−c<3t, if the constraint in Eq. 26 is not binding, the full-information best-response of IL is given by $p_{IL}^{*} =\max \left \{ (6 t + {\Delta } + 2 c)/6,({\Delta }+2c)/3\right \}$. This, however, is inconsistent with the first equation in Eq. 41. If the constraint in Eq. 26 is binding, then ${\Pi }_{IH}(p_{IL}^{*}, p_{E}^{*}, 0) = {\Pi }_{IH}(p_{IH}^{*}, p_{E}^{*}, 1)$. If Δ−c<3t≤Δ+2c, this equality is by Eq. 41 equivalent to

$$ 0=\frac{(3 t - {\Delta} + c)^{2}}{18 t}. $$

(43)

If Δ+2c<3t≤4Δ+2c, then the equality ${\Pi }_{IH}(p_{IL}^{*}, p_{E}^{*}, 0) = {\Pi }_{IH}(p_{IH}^{*}, p_{E}^{*}, 1)$ by Eq. 41 is equivalent to

$$ \frac{(1-\gamma)(3 t+2 {\Delta} + 4 c)(3 t - {\Delta} - 2 c )}{18 t} = \frac{(3 t - {\Delta} + c)^{2}}{18 t}. $$

(44)

Finally, if 4Δ+2c≤3t, then by Eq. 41 then the equality ${\Pi }_{IH}(p_{IL}^{*}, p_{E}^{*}, 0) = {\Pi }_{IH}(p_{IH}^{*}, p_{E}^{*}, 1)$ is equivalent to

$$ \frac{(3 t+2 {\Delta} + 4 c )(3 t - 3 \gamma {\Delta} - {\Delta} - 2 c )}{18 t} = \frac{(3 t - {\Delta} + c)^{2}}{18 t}. $$

(45)

As Δ−c<3t, the Eqs. 43 – 45 have either no solution or a unique solution in γ Therefore, the conditions of Lemma 2 can hold at most for particular values of γ.

Proof Lemma 3:

We first show that (a) and (b) must hold in a signalling equilibrium (p,μ(⋅)). By Eq. 13

$$ {\Pi}_{IH}(p_{IH}, p_{EH}, 1) \ge {\Pi}_{IH}(p, p_{EH}, \mu(p, p_{EH})) \ge {\Pi}_{IH}(p, p_{EH}, 1) $$

(46)

for all p≥0, where the second inequality follows from ∂π _{I H} /∂ μ<0. Similarly, Eq. 13 and part (b) of the unprejudiced belief criterion imply

$$ {\Pi}_{IL}(p_{IL}, p_{EL}, 0) \ge {\Pi}_{IL}(p, p_{EL}, \mu(p, p_{EL})) = {\Pi}_{IL}(p, p_{EL}, 0) $$

(47)

for all p≠p _{I H} . By continuity of π _{I L} (⋅,p _{E L} ,0), therefore also

$$ {\Pi}_{IL}(p_{IL}, p_{EL}, 0) \ge {\Pi}_{IL}(p_{IH}, p_{EL}, 0). $$

(48)

By an analogous argument it follows from Eq. 13, ∂π _{E L} /∂ μ>0, and part (a) of the unprejudiced belief criterion that

$$ {\Pi}_{EL}(p_{IL}, p_{EL}, 0) \ge {\Pi}_{EL}(p_{IL}, p, 0), \quad {\Pi}_{EH}(p_{IH}, p_{EH}, 1) \ge {\Pi}_{EH}(p_{IH}, p, 1) $$

(49)

for all p≥0. By Eqs. 46–49, p satisfies the conditions that define ${\hat {p}}$ in Eq. 7. This proves that (p,μ(⋅)) must satisfy claim (a) that $p = {\hat {p}}$. Note that by Eqs. 13 and 14

$$\begin{array}{@{}rcl@{}} {\Pi}_{IH}(p_{IH}, p_{EH}, 1) &\ge& {\Pi}_{IH}(p_{IL}, p_{EH}, \mu(p_{IL}, p_{EH})), \\ {\Pi}_{EL}(p_{IL}, p_{EL}, 0) &\ge& {\Pi}_{EL}(p_{IL}, p_{EH}, \mu(p_{IL}, p_{EH})). \end{array} $$

(50)

This proves that statement (b) holds for $\bar {\mu } \equiv \mu (p_{IL}, p_{EH})$.

Next we show that $({\hat {p}}, \mu (\cdot ))$ is a signalling equilibrium for some μ(⋅) only if (b) holds. Note that the intuitive criterion does not apply to ${\hat {p}}$ because ${\hat {p}}_{IL} \ne {\hat {p}}_{IH}$ and ${\hat {p}}_{EL} \ne {\hat {p}}_{EH}$. In line with the unprejudiced belief criterion, define

$$ \mu({\hat{p}}_{IH}, p) \equiv 1 \,\,\text{for all}\,\, p \ne {\hat{p}}_{EL},\kern-3.4pt \quad \mu(p, {\hat{p}}_{EL}) \equiv 0 \,\,\text{for all}\,\, p \ne {\hat{p}}_{IH}, \quad \mu({\hat{p}}_{IH}, {\hat{p}}_{EL})\equiv \lambda. $$

(51)

Further, if Eq. 27 in part (b) of the lemma holds for $p = {\hat {p}}$ we can set

$$ \mu({\hat{p}}_{IL}, p) \equiv 0 \,\,\text{for all}\,\, p \ne {\hat{p}}_{EH},\kern-3.4pt \quad \mu(p, {\hat{p}}_{EH}) \equiv 1 \,\,\text{for all}\,\, p \ne {\hat{p}}_{IL}, \quad \mu({\hat{p}}_{IL}, {\hat{p}}_{EH})\equiv \bar{\mu}. $$

(52)

The beliefs for all other price pairs (p _I ,p _E ) play no role in the definition of a PBE and so they are arbitrary. Since $\mu ({\hat {p}}_{IL}, {\hat {p}}_{EL}) = 0$ and $\mu ({\hat {p}}_{IH}, {\hat {p}}_{EH}) = 1$ by Eqs. 51 and 52, these beliefs satisfy Bayes rule Eq. 15 in part (b) of the definition of a PBE. Further since ${\hat {p}}$ satisfies Eqs. 7 and 50 holds for $p = {\hat {p}}$, it is easily verified that $({\hat {p}}, \mu (\cdot ))$ satisfies also the conditions Eqs. 13 and 14 for profit maximization in part (a) of the definition of a PBE. This proves that ${\hat {p}}$ and the beliefs μ(⋅) in Eqs. 51 and 52 constitute a signalling equilibrium if Eq. 27 in part (b) of the lemma holds for $p = {\hat {p}}$. If the latter condition does not hold, then there is no belief μ(p _{I L} ,p _{E H} ) that satisfies both conditions in Eq. 50 for $p = {\hat {p}}$. In this case, there exists no PBE (p,μ(⋅)) with $p = {\hat {p}}$ because at least one of the conditions Eqs. 13 and 14 for profit maximization is violated.

Proof Proposition 3:

By Lemma 3 it is sufficient to show that for $p = {\hat {p}}$ Eq. 27 has a solution $\bar {\mu } \in [0, 1]$ if and only if $\gamma \ge \bar {\gamma }$. Using ${\hat {p}}$ in Eq. 12, we have ${\Pi }_{IH}({\hat {p}}_{IH}, {\hat {p}}_{EH}, 1) = {(3 t - {\Delta } + c)^{2}}/{(18 t)}$ and

$$\begin{array}{@{}rcl@{}} {\Pi}_{IH}({\hat{p}}_{IL}, {\hat{p}}_{EH}, \bar{\mu}) &=& \frac{(1-\gamma)({\Delta}(1-3 \bar{\mu}) + 2 c + 3 t)}{6}\\ &&+ \min\left[0, \frac{\gamma(3 t - 2 {\Delta} - 2c)}{6}\right], \end{array} $$

(53)

because if q _E =q _H the incumbent’s demand from the informed consumers is zero at $({\hat {p}}_{IL}, {\hat {p}}_{EH})$ for 3t≤2(Δ−c). Therefore, solving the the first inequality in Eq. 27 for $\bar {\mu }$ yields

$$ \bar{\mu} \ge {\bar{\mu}}_{I}(\gamma) \equiv \frac{9 t {\Delta} (1- \gamma) -({\Delta} - c)^{2}}{9 t {\Delta} (1- \gamma)} + \max\left[0, \frac{\gamma}{1-\gamma} \frac{2 {\Delta} - 2c - 3 t}{3 {\Delta}}\right]. $$

(54)

By Eq. 12, we have ${\Pi }_{EL}({\hat {p}}_{IL}, {\hat {p}}_{EL}, 0) = t/2$ and

$$\begin{array}{@{}rcl@{}} {\Pi}_{EL}({\hat{p}}_{IL}, {\hat{p}}_{EH}, \bar{\mu}) &=& \frac{(1-\gamma)(3 t + 2 c + {\Delta})(3 t - 2 c + {\Delta} (3\bar{\mu} - 1))}{18 t} \\ & & + \min\left[0, \frac{\gamma[(3 t)^{2} -({\Delta} + 2 c)^{2} ]}{18 t}\right], \end{array} $$

(55)

because if q _E =q _L the entrant’s demand from the informed consumers is zero at $({\hat {p}}_{IL}, {\hat {p}}_{EH})$ for 3t≤Δ+2c. Therefore, solving the the second inequality in Eq. 27 for $\bar {\mu }$ yields

$$ \bar{\mu} \le {\bar{\mu}}_{E}(\gamma) \equiv \frac{({\Delta} + 2 c)^{2}}{3 {\Delta} (1-\gamma)(3 t + 2 c + {\Delta})} - \max\left[0, \frac{\gamma}{1-\gamma} \frac{\Delta + 2 c - 3 t}{3 {\Delta}}\right]. $$

(56)

Thus, the two inequalities in Eq. 27 admit a solution $\bar {\mu }$ if and only if ${\bar {\mu }}_{I}(\gamma ) \le {\bar {\mu }}_{E}(\gamma )$. It is easily verified that

$$ 0 < {\bar{\mu}}_{E}(0) <{\bar{\mu}}_{I}(0) <1, \,{\lim}_{\gamma\to 1}[{\bar{\mu}}_{I}(\gamma)- {\bar{\mu}}_{E}(\gamma)] < 0, \, \, {\bar{\mu}}_{I}^{\prime}(\gamma) < 0, \, {\bar{\mu}}_{E}^{\prime}(\gamma) > 0. $$

(57)

Therefore, by the intermediate value theorem there exist a (unique) $\bar {\gamma } \in (0,1)$ such that ${\bar {\mu }}_{I}(\bar {\gamma }) = {\bar {\mu }}_{E}(\bar {\gamma }) \in (0,1)$ and ${\bar {\mu }}_{I}(\gamma ) \le {\bar {\mu }}_{E}(\gamma )$ if and only if $\gamma \ge \bar {\gamma }$.

Proof Proposition 4:

By the proof of Proposition 3, $\bar {\gamma }$ is uniquely determined by the solution of ${\bar {\mu }}_{I}(\bar {\gamma } \vert t, {\Delta }, c) = {\bar {\mu }}_{E}(\bar {\gamma } \vert t, {\Delta }, c)$, where ${\bar {\mu }}_{I}$ and ${\bar {\mu }}_{E}$ are defined in Eqs. 54 and 56 and satisfy 57. For 3t> max[2(Δ−c),Δ+2c] one obtains

$$ \partial{{\bar{\mu}}_{I}(\bar{\gamma} \vert t, {\Delta}, c)}/{\partial t} > 0, \quad \partial{{\bar{\mu}}_{E}(\bar{\gamma} \vert t, {\Delta}, c)}/{\partial t} < 0, $$

(58)

which immediately implies the first inequality in Eq. 28. Since ${\bar {\mu }}_{I}(\bar {\gamma } \vert \lambda t, \lambda {\Delta }, \lambda c) = {\bar {\mu }}_{I}(\bar {\gamma } \vert t, {\Delta }, c)$ and ${\bar {\mu }}_{E}(\bar {\gamma } \vert \lambda t, \lambda {\Delta }, \lambda c) = {\bar {\mu }}_{E}(\bar {\gamma } \vert t, {\Delta }, c)$, the first inequality in Eq. 28 implies the second inequality in Eq. 28. This proves part (a).

To prove part (b), let Δ−c<3t < min[2(Δ−c),Δ+2c]. Then solving ${\bar {\mu }}_{I}(\bar {\gamma } \vert t, {\Delta }, c) = {\bar {\mu }}_{E}(\bar {\gamma } \vert t, {\Delta }, c)$ yields

$$ \bar{\gamma}=\frac{\Delta(27t^{2}+3t{\Delta}-{\Delta}^{2})+12c t{\Delta}+3c^{2}({\Delta}-5t)-2c^{3}}{18t^{2}(3t+2c+{\Delta})}. $$

(59)

Since

$$ \lim_{t \to ({\Delta} -c)/3} \,\frac{\partial \bar{\gamma}}{\partial t} = -\frac{3({\Delta} + 2 c)}{2(2 {\Delta} + c)^{2}} < 0, $$

(60)

this proves the first inequality in Eq. 29. As $\bar {\gamma }$ in Eq. 59 is homogenous of degree zero in (t,Δ,c) the first inequality implies the second inequality in Eq. 29.

Proof Lemma 4:

Note that ${\Pi }_{IH}({\hat {p}}_{IH}, {\hat {p}}_{EH}, 1)= 0$ because ${\hat {p}}_{IH} = 0$. Therefore Eq. 30 holds if and only if no consumer purchases from the incumbent upon observing $({\hat {p}}_{IL}, {\hat {p}}_{EH})$. The uninformed consumers’ demand is zero if and only if even the consumer with x=0 does not prefer to buy from the incumbent. This is the case if

$$ q_{L} - {\hat{p}}_{IL} \le \bar{\mu} q_{H} + (1-\bar{\mu})q_{L} - {\hat{p}}_{EH} - t. $$

(61)

Since ${\hat {p}}_{EH} - {\hat {p}}_{IL} ={\Delta } - 2t$, this is identical to $\bar {\mu } \ge ({\Delta } - t)/{\Delta }$. Clearly, if even the uninformed consumers do not purchase from the incumbent, then at the prices $({\hat {p}}_{IL}, {\hat {p}}_{EH})$ also the informed consumer do not buy from the incumbent because they know that q _E =q _H .

Proof Lemma 5:

We first show that no informed consumer purchases from the low quality entrant if he deviates to $p_{EH}^{*}$. This is the case if even the informed consumer with x=1 does not buy from the entrant, i.e. if

$$ q_{L} - {\hat{p}}_{IL} - t \ge q_{L} - {\hat{p}}_{EH}. $$

(62)

Since ${\hat {p}}_{EH} - {\hat {p}}_{IL} = {\Delta } - 2 t$, this is identical to Δ≥3t and is satisfied as Δ−c≥3t.

While a deviation to $p_{EH}^{*}$ leads to zero demand from the informed consumers, by the proof of Lemma 4 all uninformed consumers buy from the entrant at the prices $({\hat {p}}_{IL}, {\hat {p}}_{EH})$. Therefore,

$$ {\Pi}_{EL}({\hat{p}}_{IL}, {\hat{p}}_{EH}, \bar{\mu}) = (1-\gamma){\hat{p}}_{EH} = (1-\gamma)({\Delta} - t). $$

(63)

Since ${\Pi }_{EL}({\hat {p}}_{IL}, {\hat {p}}_{EL}, 0) = {\Pi }_{EL}(t, t, 0) = t/2$, the inequality in Eq. 31 is equivalent to γ≥(2Δ−3t)/(2Δ−2t).

Proof Proof of Proposition 4:

Statements (a) and (b) follow immediately from Lemmas 3, 4 and 5.