Abstract
We study the optimal pricing strategy for a privately informed monopolist in the presence of observational learning. Early adopters learn quality before purchasing the product. Late adopters learn quality from firstperiod price and early adopters’ purchase decisions. Prices generate revenues, signal quality, and determine information transmission through observational learning. Separation may occur through either high or low prices, depending on the elasticity of early adopters’ demand. When demand for goodquality products is less elastic, high prices are less costly for hightype firms due to static and dynamic effects. Hightype firms are marginally less affected by high prices, since they lose fewer consumers. Moreover, early sales at higher prices carry good news about quality to late adopters. The opposite occurs when the demand for goodquality products is more elastic.
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Notes
We interpret quality as the match between product features and consumers’ tastes.
Philips attempted to enter the videogame market in the late 80s with the release of the “Compact Disk Interactive” (CDi), a console that contained educational games and also played normal CDs. A high introductory price ultimately doomed the CDi, as consumers opted for Nintendo gaming systems that sold for half the price of a new CDi.
Restaurants with a considerable waiting list, bestselling books, and highclickvolume online offerings are usually perceived as highquality products. Similarly, a residential property that spends too much time on the market can start a bad news snowball.
Information transmission might also occur through wordofmouth communication (WOM) among consumer generations. We consider that observational learning fits better the class of products we have in mind. Since adopters are putting money behind their decisions, there is no concern about “fake reviews”; and information that is transmitted through observational learning is more credible and trustworthy than is WOM.
Market skimming refers to introducing the product at a high price for early adopters, then gradually lowering the price to attract thriftier consumers. Penetration pricing refers instead to an initial low price for a new product that increases after information spreads out (Kotler and Armstrong 2010).
We consider persistent types. A monopolist of type \(\theta\) produces a goodquality product with probability \(\theta\) in each period. Moreover quality is the same for all units produced in each period. Therefore, the monopolist perfectly learns quality after firstperiod production.
Moreover, this assumption fits well the notion of quality as the match between product features and consumers’ tastes.
Note that consumers might also be interested in buying badquality products.
It is inconsequential if the firm learns product quality at the end of the first period. Only consumers’ beliefs are relevant for the pricing decision. In fact, secondperiod prices maximize profits given beliefs with no rationale for signaling.
Specifically, \({\overline{D}}\left( P,\mu ,q\right) =\int _{s}{\widetilde{D}}\left( P,\mu ,s\right) h\left( s\mid q\right) ds\). With the standard assumption that f is increasing in both q and v, it is easy to show that the demand \({\tilde{D}}(P,\mu ,s)\) and its expectation \({\bar{D}}(P,\mu ,q)\) are increasing in \(q\,(s)\), and in \(\mu\) and are decreasing in P.
The hightype fullinformation monopoly price \(P^{H^{**}}\) is the maximizer of \(\Pi \left( H,P,\mu =H\right)\).
In Spence’s jobmarket signaling model, hightype workers signal their type via increased education, which low types are unable to replicate because education is more costly for them (SCP1). SCP2 is automatically satisfied due to the quasilinear structure of the worker’s utility function: \(w(\mu )c(e,\theta )\), where an increase in beliefs is equally valuable for all types.
Convexity of \(\pi\) implies that the effect on secondperiod profits of information transmission is higher when firstperiod prices are high.
In this case, it is also needed that the incentive of a hightype monopolist to extract rent from inframarginal consumers in the first period is small, which is easily satisfied for \(\lambda<<1\). The specific condition that \(\lambda\) must satisfy is given in condition 2 of Proposition 5.
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Acknowledgements
This research was partially funded by the Complex Engineering Systems Institute, ISCI (ICMFIC: P05004F, CONICYT: FB08016).
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Appendices
Appendix A
Lemma 8
There exists a separating equilibrium \(\left( P^{L^{*}},P^{H}\right)\) in which separation occurs through high (low) prices if

(SCP1)
\(\;\frac{\partial ^{2}\Pi \left( \theta ,P,\mu \right) }{\partial \theta \partial P}>\left( <\right) 0\)

(SCP2)
\(\;\frac{\partial ^{2}\Pi \left( \theta ,P,\mu \right) }{\partial \theta \partial \mu }>0\).
Proof
We consider two candidates for separating equilibrium: The first one involves separation through high prices, and the second one separation through low prices. In both cases we define a price \(P^{H}={\overline{P}}\) (higher and lower than \(P^{H^{**}}\), respectively) such that the lowtype monopolist is indifferent between following the equilibrium strategy and mimicking the hightype one:
A separating equilibrium in which high prices signal high quality exists if at the price \({\overline{P}}\) the hightype monopolist has no incentive to deviate:
For 4 to be satisfied, the following two conditions are sufficient:
which are directly implied by SCP1 and SCP2.
SCP1 and SCP2 are sufficient conditions for the existence of separation through high prices. Note first that SCP1 implies directly that \(P^{H^{*}}>P^{L^{*}}\). When allowed to choose the optimal price, the hightype monopolist will prefer to set a higher price than the lowtype one, holding beliefs constant. This is so because higher prices are marginally less costly for the hightype monopolist. Since beliefs affect profits in a nonseparable way, an additional assumption is needed to guarantee separation. SCP2 implies that the shift from pessimistic to optimistic beliefs is more attractive to the hightype firm. Finally, note that these are sufficient conditions, and separation could still exist under less restrictive assumptions, even though it makes the economic analysis and interpretation more complex. The same reasoning applies for the case in which low prices signal high quality. \(\square\)
Lemma 9
The only equilibrium that satisfies the intuitive criterion is\(\left( P^{L*},{\overline{P}}\right)\), where the price charged by the highquality monopolist is the least costly among the ones that induce separation:\(P^{H}={\bar{P}}\)such that
Proof
A separating equilibrium \(\left( P^{L*},P^{H}\right)\) satisfies the intuitive criterion if there is no price \(P^{\text {'}}\) such that: a) \(\Pi \left( H,P^{'},\mu =H\right) \ge \Pi \left( H,P^{H},\mu =H\right)\); and b) \(\Pi \left( L,P^{'},\mu =H\right) <\Pi \left( L,P^{L*},\mu =L\right)\). If there exists a price \(P^{'}\) such that the high type prefers to deviate and the low type prefers to stick to the equilibrium strategy, consumers should interpret such a deviation as if coming from a high type, which would collapse the equilibrium in the first place. Then the only equilibrium that satisfies the intuitive criterion is the leastcostly for the high type: the one in which the hightype monopolist charges the lowest (highest) of the prices that the lowtype one would not find profitable to mimic. We now show that \(\left( P^{L^{*}},{\overline{P}}\right)\) is the only equilibrium that satisfies the intuitive criterion. We prove the result for the case in which separation occurs through high prices. We first show that there is no equilibrium price \(P>{\overline{P}}\) that satisfies the intuitive criterion. Consider the price \(P>{\overline{P}}\) such that \(\left( P^{L^{*}},P\right)\) is a separating equilibrium. Define \(P^{'}=P\varepsilon\). Then it is easy to see that: a) \(\Pi \left( H,P^{'},\mu =H\right) \ge \Pi \left( H,P,\mu =H\right)\); and b) \(\Pi \left( L,P^{'},\mu =H\right) <\Pi \left( L,P^{L^{*}},\mu =L\right)\). Noting that \(P^{H^{**}}<{\overline{P}}\) (signaling is costly), it follows that \(P^{H^{**}}<P^{'}<P\). Therefore \(\Pi \left( H,P^{'},\mu =H\right) \ge \Pi \left( H,P,\mu =H\right)\). Moreover we know by Lemma 4 that \(\Pi \left( L,P,\mu =H\right) <\Pi \left( L,P^{L^{*}},\mu =L\right)\). Then by continuity \(\Pi \left( L,P^{'},\mu =H\right) <\Pi \left( L,P^{L^{*}},\mu =L\right)\). Thus for any price \(P<{\overline{P}}\) condition a) is not satisfied, violating the intuitive criterion.
We now show that \(\left( P^{L^{*}},{\overline{P}}\right)\) is the only separating equilibrium that satisfies the intuitive criterion. If \(P^{'}>{\overline{P}}\), condition a) is not satisfied. Then, \(P^{'}\ge {\overline{P}}\). But if \(P^{'}<{\overline{P}}\) , there is no separating equilibrium, since any deviation at \(P^{'}<{\overline{P}}\) is profitable for the lowquality seller. Then it must be \(P^{'}={\overline{P}}\), and \(\left( P^{L^{*}},{\overline{P}}\right)\) is the only separating equilibrium that satisfies the intuitive criterion. \(\square\)
Corollary 10
Suppose that\({\bar{D}}(P,\mu ,1)\le \frac{1}{2}\). Then

\(D_{\theta P}>0\) implies \(\frac{\partial \mu ^{Y}\left( P,\mu \right) }{\partial P}\frac{\partial \mu ^{N}\left( P,\mu \right) }{\partial P}\ge 0\)

\(D_{\theta P}<\lambda \frac{\mu }{1\mu }\) and \(\frac{\partial }{\partial q}\left[P\frac{{\bar{D}}_{P}(P,\mu ,q)}{{\bar{D}}(P,\mu ,q)}\right]<0\) imply \(\frac{\partial \mu ^{Y}\left( P,\mu \right) }{\partial P}\frac{\partial \mu ^{N}\left( P,\mu \right) }{\partial P}\le \lambda\).
Proof
Consider separation through high prices (\(D_{\theta P}>0\)). Condition \(\frac{\partial \mu ^{Y}\left( P,\mu \right) }{\partial P}\ge \frac{\partial \mu ^{N}\left( P,\mu \right) }{\partial P}\), is equivalent to:
Note that \(\left[ {\overline{D}}_{P}\left( P,\mu ,0\right) {\overline{D}}_{P}\left( P,\mu ,1\right) \right] <0\) since \(D_{\theta P}>0\). Therefore a sufficient condition is given by
Then consider separation through low prices (\(D_{\theta P}<0\)). Condition \(\frac{\partial \mu ^{Y}\left( P,\mu \right) }{\partial P}\frac{\partial \mu ^{N}\left( P,\mu \right) }{\partial P}\le \lambda\), is equivalent to:
Noting that
holds if \(\frac{\partial }{\partial q}[P\frac{{\bar{D}}_{P}(P,\mu ,q)}{{\bar{D}}(P,\mu ,q)}]<0\) (demand for good quality products is more elastic than demand for bad quality products), and \({\bar{D}}(P,\mu ,1)\le \frac{1}{2}\) (demand for good quality is sufficiently low). Then we just need to verify that
which is implied by
which in turn is implied by \(D_{\theta P}<\lambda \frac{\mu }{1\mu }\) .
\(\square\)
Appendix B
1.1 Proof of Lemma 2
A necessary condition for C1 to be satisfied is that the lowtype monopolist charges in equilibrium the fullinformation monopoly price \(P^{L^{*}}\): the maximizer of \(\Pi \left( L,P,\mu =L\right)\) . Moreover C3 requires that the hightype monopolist should not have any incentive to deviate from the equilibrium price, as such deviation implies pessimistic beliefs. Then it is sufficient to control for the best deviation, which occurs at \(P^{H^{*}}\): the maximizer of \(\Pi \left( H,P,\mu =L\right)\). Then, two conditions are sufficient for the existence of a separating equilibrium:
 1.
\(\Pi \left( L,P^{L^{*}},\mu =L\right) \ge \Pi \left( L,P^{H},\mu =H\right)\)
 2.
\(\Pi \left( H,P^{H},\mu =H\right) \ge \Pi \left( H,P^{H^{*}},\mu =L\right)\)\(\square\)
1.2 Proof of Proposition 4
We first analyze SCP1:
Since demand is increasing in quality \(D_{\theta }\left( \theta ,P,\mu \right) ={\overline{D}}\left( P,\mu ,1\right) {\overline{D}}\left( P,\mu ,0\right) >0\) , profits are increasing and convex in beliefs, then conditions 1 and 2 imply the result. Consider now SCP2:
Since \(D_{\theta }\left( \theta ,\mu ,P\right) >0\), and profits are convex in beliefs, then condition 3 is a sufficient condition for SCP2 to be satisfied. \(\square\)
1.3 Proof of Proposition 5
Analogous to Proof of Proposition 4. \(\square\)
Proof of Corollary 6
Condition 3 in Proposition 4 is equivalent to
which is implied by \(\left( \frac{H}{1H}\right) ^{2}\le \frac{{\bar{D}}(v,0)}{\gamma \left[ 1\gamma \left( 1{\bar{D}}(v,0)\right) \right] }\), where v is the maximum willingness to pay of ineslastic consumers. \(\square\)
1.4 Proof of Corollary 7
Given \(\frac{\partial \mu ^{Y}}{\partial P}=0\), condition 2 in Proposition 5 is equivalent to
which is implied by the parametric condition
On the other hand, it is easy to see that the condition 3 in Proposition 5 is implied by
\(\square\)
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Figueroa, N., Guadalupi, C. Signaling Quality in the Presence of Observational Learning. Rev Ind Organ 56, 515–534 (2020). https://doi.org/10.1007/s1115101909728z
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DOI: https://doi.org/10.1007/s1115101909728z