Low quality as a signal of high quality

Clements, Matthew T. Article Low quality as a signal of high quality Economics: The Open-Access, Open-Assessment E-Journal Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Clements, Matthew T. (2011) : Low quality as a signal of high quality, Economics: The Open-Access, Open-Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 5, Iss. 2011-5, pp. 1-22, https://doi.org/10.5018/economics-ejournal.ja.2011-5 This Version is available at: https://hdl.handle.net/10419/45567 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Vol. 5, 2011-5 | March 7, 2011 | http://dx.doi.org/10.5018/economics-ejournal.ja.2011-5 Low Quality as a Signal of High Quality Matthew T. Clements St. Edward’s University, Austin, USA Abstract If a product has two dimensions of quality, one observable and one not, a firm can use observable quality as a signal of unobservable quality. The correlation between consumers’ valuation of high quality in each dimension is a key determinant of the feasibility of such signaling. A firm may use price alone as a signal, or price and quality together. Both signals tend to be used when the market is very uninformed, whereas price signaling alone tends to be used when the market is moderately informed. If high observable quality is inexpensive to provide, then it cannot signal high unobservable quality, and low observable quality is always an indication that unobservable quality is high. JEL D82, L15 Keywords Signaling; quality Correspondence Matthew T. Clements; St. Edward’s University, Austin, TX, USA. [email protected] Citation Matthew T. Clements (2011). Low Quality as a Signal of High Quality. Economics: The Open-Access, Open-Assessment E-Journal, Vol. 5, 2011-5. doi:10.5018/economics-ejournal.ja.2011-5. http://dx.doi.org/10.5018/economics-ejournal.ja.2011-5 © Author(s) 2011. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany conomics: The Open-Access, Open-Assessment E-Journal 1 Introduction Intuitively, it may seem that consumers who demand high quality will demand it in all dimensions. For example, a restaurant consumer who places a high value on high-quality food will probably also want high-quality service and a high-quality dining room. If quality is a normal good, differences in demand for quality of all kinds might be attributable to differences in income. We would then expect high-quality characteristics to be bundled together, as they very often are. However, there are notable exceptions. Wine may be packaged in a bottle or in a plastic bag inside a box. The box, or “cask,” prevents air from mixing with the wine and thus keeps the wine fresh longer, as well as being easier to open than the traditional bottle. Clearly any wine consumer would prefer more convenient packaging and longer-lasting wine (although the strength of the preference may vary); thus the cask can be regarded as the higher quality packaging. Only relatively low-quality wines are packaged in this manner; only the low-quality content is available in the high-quality (more convenient) packaging, and high-quality content is only available in the low-quality packaging. Consumers instinctively tend to associate traditional wine packaging with high-quality wine, probably because they have observed this association in the past. It would seem difcult to convince consumers that a brand of wine in a box is of high quality, even if consumers are well informed about the quality of the packaging itself. This paper provides a rational basis for such consumer instincts, for this as well as other examples. Lower-quality beer is usually sold in bottles with screw-off caps (which are easier to open), while premium beer is not. Newspapers can be sold in two formats, tabloid or broadsheet. The tabloid format is easier to handle, especially while riding a bus or a train. However, newspapers with high-quality content tend to be sold in the broadsheet format. Upscale hair-care products are only available at hair salons, while other brands can be purchased at grocery and drug stores. For many products and services, one must go to greater inconvenience to obtain higher quality. In the above examples, high observable quality (e.g., packaging) is inexpensive and available to all rms, and it is not clear why some rms, particularly those that sell high unobservable quality (e.g., content), do not use it. In this paper, I examine the use of quality in an observable dimension to signal quality in an unwww.economics-ejournal.org 1 conomics: The Open-Access, Open-Assessment E-Journal observable dimension. Someone buying an unfamiliar brand of beer might choose a bottle with a pry-off cap over a bottle with a screw-off cap without any direct knowledge of the content. That is, the consumer might choose the product that is of low quality in the only observable characteristic. This is a rational choice for the consumer if the packaging is a signal of the quality of the content. In fact, for such examples, the only possible reason for a rm to produce low-quality packaging is to signal the quality of the content.1Furthermore, it is not clear how the use of observable quality as a signal corresponds to other signaling models. The basic intuition of signaling offered by Spence (1973) is that, in order to be an effective differentiation device, a signal must be more costly to one type of agent. If a rm with high-quality content does not use high-quality packaging, it must be that a rm with low-quality content would face a different cost of using low-quality packaging. That is, there must be some incentive for the rm with high-quality content to use low-quality packaging that a rm with low-quality content would not have.2 The quality signaling literature has examined price as a signal of quality, either alone or in conjunction with other signals such as advertising.3In Bagwell and Riordan (1991), price alone can be used to signal unobservable quality. The highquality rm's price is distorted upward from the full-information price in such a way that a low-quality rm would not mimic it. The lower quantity sold is more damaging to the low-quality rm (since the low-quality rm has a larger margin of price over marginal cost than the high-quality rm), and the low-quality rm loses sales to informed consumers by charging a higher price. Bagwell (1992) more generally considers the use of multiple signals to signal quality. The central 1This is true if, as in the examples, high-quality packaging is relatively inexpensive, in a sense stated explicitly in Section 2. 2Another reason for a rm to degrade quality is as a means of second-degree price discrimination, as in Deneckere and McAfee (1996). This is only possible if the rm offers at least two versions of its product. 3This literature includes Cooper and Ross (1985), Milgrom and Roberts (1986), Bagwell (1987), Bagwell and Ramey (1988), Wolinsky (1988), and Judd and Riordan (1994). www.economics-ejournal.org 2 conomics: The Open-Access, Open-Assessment E-Journal result is that, if marginal costs are increasing in quality, a signal that reduces demand is more attractive to a high-quality rm.4 The model presented here is related to that of Bagwell and Riordan (1991), and again the key issue is what kind of signal will not be mimicked by a low-quality rm. However, marginal production costs of high quality (in either dimension) have no direct bearing on the rm's choice of signaling regime. There is an opportunity cost if a rm lowers observable quality: consumers' willingness to pay for the product is lower. This cost is relatively low for a high-quality rm (a rm that produces high unobservable quality) if the correlation between consumers' valuations of high quality in each dimension is negative. For the beer example, this would mean that consumers who value high-quality content very highly are close to indifferent between the two kinds of packaging. Because this opportunity cost is higher for a low-quality rm, the high-quality rm does not have to distort price greatly to signal effectively. Quality signaling can then mitigate the distortion that arises from price signaling. There is the greatest incentive to use quality signaling if the market is very uninformed, when the distortion arising from price signaling is large. If the market is moderately informed, only price is used as a signal. When high observable quality is inexpensive to provide, only a high-quality rm will ever choose low observable quality. Thus, low observable quality is a sure sign that unobservable quality is high. Furthermore, high observable quality cannot be used to signal high unobservable quality. In the following section, I introduce the model. In Section 3, I derive conditions for separating equilibria in the price-quality signaling game. In Section 4, I present a numerical example in which quality signaling is protable. Section 5 concludes. 2 The Model A product consists of two characteristics, one observable and the other unobservable. Let qoand qube the respective qualities of the observable and unobservable 4Engers (1987) also considers the use of multiple signals, generalizing the results of Spence (1973). When there are many unobservable quality attributes and potentially many signals, the cost of a signal must be decreasing in quality in order for the signal to be used. www.economics-ejournal.org 3 conomics: The Open-Access, Open-Assessment E-Journal characteristics. For the beer example, qois the packaging (type of bottle) and qu is the content (the beer itself). In each dimension, the product may be of either high or low quality: qo2fL;Hg;qu2fL;Hg:All consumers observe qo, but only informed consumers know qu. There is a total mass of consumers M;and Xis the ratio of informed to uninformed consumers. All consumers have a reservation utility of zero. I assume that consumers are willing to pay αfor low observable quality and that willingness to pay (WTP) for high observable quality is uniformly distributed on [α;β]:Similarly, consumers are willing to pay γfor low unobservable quality, and WTP for high unobservable quality is uniformly distributed on [γ;δ]. Let LH be a consumer's WTP for the bundle (qo=L;qu=H)and HL a consumer's WTP for the bundle (qo=H;qu=L):Given the above uniformity assumptions, LH is uniformly distributed on [α+γ;α+δ]and HL is uniformly distributed on [γ+α;γ+β]: I further assume that HH, the WTP for the bundle (qo=H;qu=H);is uniformly distributed on U[A;B];where A2[α+γ;β+γ]and B2[α+δ;β+δ]: Note that this distribution is not taken to be the sum of the distributions for high quality in each dimension alone. For tractability, I am directly assuming that the distribution of WTP for HH is uniform, which is in effect an assumption about the nature of consumer utility.5The assumption coincides with additive separability in the cases of perfectly positive correlation of consumer WTP for high quality in each dimension, in which case the interval is [α+γ;β+δ], and of perfectly negative correlation, in which case the interval is [β+γ;α+δ]. Formally, we could arrive at this distribution by taking the sum of the distributions for high quality in each dimension and mapping it into a uniform. Clearly such a mapping exists, given that the distributions involved are well-behaved.6 5I also assume that utility is well-behaved in the following sense. Consider consumers iand j; with WTPs LHi;LHj;etc. If LHiLHjand HLiHLj;then HHiHHj: 6The assumptions regarding willingness to pay are quite strong. The upper and lower bounds of the distribution of the sum of WTPs do not change when the two WTPs are uncorrelated, but there is more mass near the middle of the distribution than in the tails (it is actually a triangle). The specication of the distribution of WTP for the bundle of high qualities in both dimensions preserves the characteristic that, as the correlation changes from 1 to -1, mass shifts toward the middle of the distribution from the tails. This is the key characteristic that makes the proofs go through. It seems www.economics-ejournal.org 4 conomics: The Open-Access, Open-Assessment E-Journal A single rm enters the market and observes qu:7If qu=H;the rm is called a “high-quality” rm. After observing qu;the rm chooses qoand sets price. All consumers observe price and qo, and informed consumers observe qu:Consumers then decide whether or not to buy. Uninformed consumers' belief that quality is high, given the price and observable quality, is b(P;qo);where 0 b1: Low quality in either dimension has a marginal cost of zero. The respective marginal costs of qo=Hand qu=Hare coand cu:I assume that co< min(α;βα)and cu<min(γ;δγ):This guarantees that, in a full-information setting, a rm will always sell high quality in both dimensions if it has the option of doing so. For the beer example, this means that any rm would package the beer in screw-top bottles if all consumers knew the quality of the content, because the benet of the packaging to any given consumer outweighs the cost to the rm. I further assume δγ>βα: the difference between high and low quality is greater in the unobservable dimension. Then, in a full-information setting, if a rm could only produce high quality in one dimension, it would choose the unobservable. Firm prot is a function of observable and unobservable quality, consumer beliefs, and price: π(qo;qu;b;P): I solve the model for perfect Bayesian equilibria, wherein all players' strategies are sequentially rational given their beliefs, and beliefs are consistent with strategies. The intuitive criterion is used to restrict the set of separating equilibria, and the divinity criterion to restrict the set of pooling equilibria. Both of these renements restrict player beliefs regarding out-of-equlibrium events in a way that is intuitively very appealing. Interpretation of the renements in the context of the model is discussed along with the results. I refer to an equilibrium in which the highand low-quality rms set different prices, but both rms set qo=H;as a “price separating equilibrium,” and an equilibrium in which rms set intuitive that the results would still hold for a the distribution of the sum of WTPs, although the math becomes intractable. 7If the rm chooses qu;there would be the usual trade-off between the cost of attaining high quality (e.g., research and development cost) and the value of high quality (the additional expected prot). The only way in which this decision interacts with the signaling issue presented here is that the value of being able to provide qu=Hdepends on whether the rm will be able to signal this quality credibly. www.economics-ejournal.org 5 conomics: The Open-Access, Open-Assessment E-Journal different prices and different observable qualities as a “price-quality separating equilibrium.” 3 Price and Quality Signaling First consider the effect of price signaling only: qu2fL;Hgand qo=H:Assume that both the high-quality and low-quality rms always produce high observable quality, as if low observable quality is not an option. The model then collapses to that of Bagwell and Riordan (1991). In any separating equilibrium, the low-quality rm sets PL=P(qo=H;qu=L) = β+γ+co 2;the price that maximizes π(qo=H;qu=L;b=0;P):This gives the low-quality rm the greatest prot conditional on consumers' belief that unobservable quality is low. If the high-quality rm sets price P;the low-quality rm will mimic if π(qo=H;qu=L;b=1;P)>π(qo=H;qu=L;b=0;PL);i.e. if the prot gained by inducing the belief that unobservable quality is high is greater than the prot at the low-quality price. Otherwise, the low-quality rm prefers to set price PL:The boundary of the region within which the low-quality rm will mimic the high-quality rm's price is the set of prices fPjπ(qo=H;qu=L;b=0;PL) = π(qo=H;qu=L;b=1;P)g;the larger parabola in Fig. 1. Looking at price as a function of X;the proportion of informed consumers, this set includes all P such that P(X) = B+co 2r(Bco)2 4(1+X)(BA) βαβ+γco 22:If the market is suf- ciently informed, then there is no possibility of mimicry, and the high-quality rm will simply set the full-information monopoly price. This price is Pm po = Pm(qo=H;qu=H) = B+co+cu 2:If the market is less informed, the high-quality rm differentiates itself from the low-quality rm by setting price equal to Ppo (X) =B+co 2+r(Bco)2 4(1+X)(BA) βαβ+γco 22:The intuitive criterion of Cho and Kreps (1987) restricts the set of equilibrium prices here and in the following proposition.8Details of the proof of the following proposition may be found in the appendix. 8The intuitive criterion species that if consumers observe a deviation from the equilibrium path that would benet only one type of rm, consumers should believe that it was that type of rm www.economics-ejournal.org 6 conomics: The Open-Access, Open-Assessment E-Journal Proposition 1 (Bagwell-Riordan) If only price may be used as a signal of unobservable quality, P(qo=H;qu=H) = maxPpo (X);Pm poand PL=β+γ+co 2are the only separating equilibrium prices satisfying the intuitive criterion. In a relatively uninformed market, the high-quality rm's price is distorted upward (from the full-information price). Uninformed consumers can infer from the higher price that qu=H:The magnitude of the price distortion is larger as the market is less informed. Next consider under what conditions the high-quality rm would signal using both price and observable quality. The rm now chooses qo(packaging) after learning qu(content). A high-quality rm might decide simply to set qo=Hand set price at the full-information level; or to set qo=Hand use price to signal the high unobservable quality; or to set qo=Land use both price and observable quality as signals of unobservable quality.9First I will establish necessary conditions for a separating equilibrium in which both price and quality are used as signals, and then I will examine the high-quality rm's incentive to use each signaling regime. The results will be proved in terms of the width of the interval [A;B], which is the range of consumer WTP for the bundle of high quality in both dimensions. The results can then be related to the correlation between consumer willingness to pay for high quality in each dimension, given the following lemma (proved in the appendix): Lemma 2 The greater the correlation of consumer WTP for high quality in each dimension, the wider the interval [A;B]: Finding necessary conditions for a price-quality separating equilibrium is very similar to the preceding analysis. As above, in any separating equilibrium, the low-quality rm sets price to maximize prot conditional on consumers' belief that unobservable quality is low. The boundary of the region within which the lowquality rm will not mimic the high-quality rm if it is using both price and observable quality as signals of unobservable quality is the smaller parabola in Fig. that deviated. E.g., if consumers observe an action by a rm that would only benet a high-quality rm, consumers believe that the rm is in fact high-quality. 9As I show below, there are no circumstances in which the rm would ever use observable quality, but not price, to signal unobservable quality. Also shown below, qo=Hcan never be used as a signal because the low-quality rm would certainly mimic it. www.economics-ejournal.org 7 conomics: The Open-Access, Open-Assessment E-Journal expressions in Section 6.6); i.e., prot for a high-quality beer producer is greater using price-quality signaling. If X> :61;neither price nor quality is used as a signal, and for :28 <X< :61;price alone is used as a signal. This means that a given variety of high-quality beer will be sold in pry-off bottles if less than 28% of the potential consumers of that variety are aware of the quality. Thus, where the market is sufciently uninformed, which is more likely to be the case for a microbrew or an import, high-quality beer is sold in a bottle with a pry-off cap. However, even for a high-quality beer, if the market is sufciently informed, the beer will be sold in a bottle with a screw-off cap. The assumption about correlation means that those consumers who place the highest value on the quality of the beer place the lowest value on the quality of the packaging, and vice versa. This would mean that those who value high-quality beer do not care about the benets of the screw-off bottle, and those who nd the screw-off bottle valuable are undiscriminating about the quality of the beer itself. A similar result holds for a less extreme correlation: if [A;B]is somewhat wider than [β+γ;α+δ], packaging is still used as a signal if the market is sufciently uninformed. For example, if A=5 and B=13;price-quality signaling is more protable if X< :19: 5 Conclusion For products that have an observable dimension of quality and an unobservable dimension of quality, a rm may lower the observable quality as a means of signaling unobservable quality. A rm may engage in quality signaling even when price alone can signal quality. When price alone is used as a signal, the highquality rm raises price above the full-information level. This distortion allows the high-quality rm to differentiate itself from the low-quality rm. If both price and quality are used as signals, the price is still distorted above the corresponding full-information price. However, the price-quality distortion may be lower than the price-only distortion, and price-quality signaling may be more protable. If quality signaling is used, it is used when the market is very uninformed. This is where the distortion of price away from the full-information level is greatest, and www.economics-ejournal.org 14 conomics: The Open-Access, Open-Assessment E-Journal the rm has the most to gain from adding quality as a signal. In a moderately informed market, price alone is used as a signal. Quality signaling is used only if the correlation between consumer valuations of high quality in each dimension is not strongly positive. In this case, if the high-quality rm lowers observable quality, the total value of the product does not change much. It is relatively costly for the low-quality rm to mimic the quality signal (by producing low quality in both dimensions) because of the loss of sales to informed consumers. In this case, quality signaling is attractive to the highquality rm. If there were no need to signal unobservable quality, observable quality would always be high, since it is relatively inexpensive for the rm to provide and is weakly preferred by consumers. In some cases, there are both observable and unobservable dimensions of quality, but all rms produce high observable quality. This will happen if the correlation is sufciently positive and the difference in cost for high and low observable quality is not drastic. Then, apart from signaling issues, it is always in a rm's interest to provide high observable quality. In other cases, when high observable quality is costly to provide, like qualities may be bundled. This would particularly be true if there is a positive correlation between consumer valuation of high quality in each dimension. Thus, the intuitive notion that the various attributes of a single product should be of similar quality holds in many cases but fails in others. 6 Appendix 6.1 Proof of Proposition 1 In any separating equilibrium, PLmust be such that b(PL) = 0:The greatest prot the low-quality rm can earn is at the price PL=maxPfπ(qo=H;qu=L;0;P)g: This price is PL=P(qo=H;qu=L) = β+γ+co 2:Any other price fails the intuitive criterion. The set fPjπ(qo=H;qu=L;0;PL) = π(qo=H;qu=L;1;P)g denes the boundary of the region within which the low-quality rm would nd it protable to mimic the high-quality rm's price. Given that consumer valuation of the bundle (qo=H;qu=L)is uniformly distributed on [α+γ;β+γ]; π(qo=H;qu=L;0;PL) = M(PLco)β+γPL βα=M βαβ+γco 22:When www.economics-ejournal.org 15 conomics: The Open-Access, Open-Assessment E-Journal qo=Hand b(P) = 1;uninformed consumers believe the bundle being sold is (qo=H;qu=H); consumer valuation of this bundle is uniformly distributed on [A;B]:The prot from selling only to uninformed consumers is π(qo=H;qu=L;1;P) = M 1+XBP BA(Pco):We can later verify that informed consumers will not buy from the low-quality rm at any price that is part of a separating equilibrium. Setting π(qo=H;qu=L;0;PL) = π(qo=H;qu=L;1;P);we obtain a quadratic expression in P;the solution of which is Ppo (X) = B+co 2r(Bco)2 4(1+X)(BA) βαβ+γco 22:Let Ppo (X)be the positive root of Ppo (X);and note that Ppo (X)only exists if X<Xpo;where Xpo =(Bco)2(βα) (BA)(β+γco)21:For X<Xpo;it is straightforward to establish that π(qo=H;qu=L;0;PL)<π(qo=H;qu=L;1;P)for prices inside the parabola dened by Ppo; this is the region in which the low-quality rm will mimic the high-quality rm. Dene Xm po by Ppo Xm po=Pm po. For X>Xm po;only Pmsatises the intuitive criterion. For X<Xm po;any price Psuch that P6=Ppo (X)fails the intuitive criterion. Furthermore, given cu>0;only Ppo (X)satises the intuitive criterion. The low-quality rm will not mimic the high-quality rm for any price P=Ppo (X);but because the cost of qu=His positive, the high-quality rm earns greater prot by setting P=Ppo (X): 6.2 Proof of Lemma 2 Let ρbe the correlation between LH and HL:Consider an observation afrom the distribution LH;where a>E(LH):Let b1=E(HLja)when ρ=ρ1;and let b2=E(HLja)when ρ=ρ1+ε;where ε>0:A straightforward implication of the properties of correlation is that b2>b1:If we repeat this exercise for some asuch that a<E(HL);then b2<b1:Now consider the effect on the distribution of HH of an increase in ρ:Since we know that E(HLja)increases for every a>E(LH) and E(HLja)decreases for every a<E(LH);and given the properties of the distributions described in footnote 5, an increase in correlation shifts mass in HH away from the mean. Given that this distribution is constrained to be uniform, the only way to accommodate mass shifting away from the mean of the distribution is for the range of the distribution to be wider. A similar argument demonstrates www.economics-ejournal.org 16 conomics: The Open-Access, Open-Assessment E-Journal that mass in HH shifts toward the mean when the correlation decreases, and thus the range of the distribution must be narrower. 6.3 Proof of Proposition 3 The proof proceeds similarly to that of Proposition 1. In a price-quality separating equilibrium, the low-quality rm's price is still PL=P(qo=H;qu=L) = β+γ+co 2: Given b(PL) = 0;the low-quality rm sets qo=H;since the value to consumers more than outweighs the additional cost incurred. Considering the set fPjπ(qo=H;qu=L;0;PL) = π(qo=L;qu=L;1;P)g;we nd that the boundary of the region within which the low-quality rm will not mimic the high-quality rm is given by Ppq (X) = α+δ 2r(α+δ)2 4(1+X)(δγ) (βα)β+γco 22:Let Ppq (X) be the positive root of Ppq (X);and note that Ppq (X)only exists if X<Xpq;where Xpq =(βα)(α+δ)2 (δγ)(β+γco)21:Dene Xm pq by Ppq Xm pq=Pm pq:Again, the high-quality rm prices at Ppq (X)for X<Xm pq;and at Pm po for XXm pq:These are the only prices satisfying the intuitive criterion, for exactly the same reasons as in Proposition 1. 6.4 Proof of Lemma 4 The prots from price-only and price-quality signaling are πpo = MBPpo BA(Ppo cocu)and πpq =Mα+δPpq δγ(Ppq cu);and the corresponding derivatives with respect to Xare ∂π po ∂X=M BA(B+co+cu2Ppo)∂Ppo ∂X and ∂πpq ∂X=M δγ(α+δ+cu2Ppq)∂Ppq ∂X:In order to have ∂πpo ∂X>∂πpq ∂Xfor some X;the following inequality must be true: (Bco)2(α+δ)2>(1+X)(BAδ+γ) βα(β+γco)2:(1) Rearranging (1), it can be written as X<[(Bco)2(α+δ)2](βα) (BAδ+γ)(β+γco)21 if BA δ+γ>0;and X>[(Bco)2(α+δ)2](βα) (BAδ+γ)(β+γco)21 if BAδ+γ<0:Let Y1= www.economics-ejournal.org 17 conomics: The Open-Access, Open-Assessment E-Journal [(Bco)2(α+δ)2](βα) (BAδ+γ)(β+γco)21 and Y2=[(Bco)2(α+δ)2](βα) (BAδ+γ)(β+γco)21:Then Y2is negative, and Y1may be positive or negative. Since X0 always, it is trivially true that X>Y2for every X:Also, if X<Y1;then it must be true that X<Y1for every Xless than X:Thus, whenever (1) is true for X;it is also true for X<X:This proves the lemma. 6.5 Proof of Lemma 5 The derivatives of prices with respect to Xunder each signaling regime are ∂Ppo ∂X=1 2h(Bco)2 4(1+X)(BA) βαZi1=2hBA βαZiand ∂Ppq ∂X= 1 2h(α+δ)2 4(1+X)(δγ) βαZi1=2hδγ βαZi;where Z=β+γco 22:Using these expressions along with the derivatives of ∂πpo ∂Xand ∂πpq ∂Xfrom the preceding proof and substituting the expressions for prices, we have ∂πpo ∂X= MZ βα"cu q(Bco)2(1+X)(BA) βα(β+γco)21#and ∂πpq ∂X=MZ βα"cu q(α+δ)2(1+X)(δγ) (βα)(β+γco)21#: Next, let B=β+δεand BA=βα+δγ2εfor some ε>0:Establishing that ∂πpq ∂X>∂πpo ∂Xfor εclose to zero proves that there can be no price-quality signaling if BAis sufciently large. Then ∂πpq ∂X>∂πpo ∂Xif and only if (α+δ)2(1+X)(δγ) (βα)(β+γco)2> (β+δεco)2(1+X)(βα+δγ2ε) βα(β+γco)2;which simplies (approximately) to (β+δco)2(α+δ)2<(1+X)(β+γco)2for εclose to zero. Noting that X0 and using assumptions from Section 2, it is straightforward to establish this inequality. This proves the lemma. 6.6 Proof of Proposition 6 Consider the expressions for prot under each signaling regime: πpo =M BA (BPpo)(Ppo cocu)and πpq =M δγ(α+δPpq)(Ppq cu):Substitutwww.economics-ejournal.org 18 conomics: The Open-Access, Open-Assessment E-Journal ing the expressions for prices and simplifying, these expressions are πpo =M BA Bco 2r(Bco)2 4(1+X)(BA) βαZ! "B+co 2+r(Bco)2 4(1+X)(BA) βαZ#(co+cu)! =M BA2 6 6 4 (1+X)(BA) βαβ+γco 22 cu Bco 2r(Bco)2 4(1+X)(BA) βαZ!3 7 7 5 and πpq =M δγ α+δ 2r(α+δ)2 4(1+X)(δγ) (βα)Z! "α+δ 2+r(α+δ)2 4(1+X)(δγ) (βα)Z#cu! =M δγ2 6 6 4 (1+X)(δγ) (βα)β+γco 22 cu α+δ 2r(α+δ)2 4(1+X)(δγ) (βα)Z!3 7 7 5;where Z=β+γco 22: In order for the high-quality rm to use price-quality signaling, it must be that this leads to greater prot: πpq >πpo;which is true if and only if BA δγ"α+δ 2r(α+δ)2 4(1+X)(δγ) (βα)Z#<Bco 2r(Bco)2 4(1+X)(BA) βαZ:We cannot verify directly whether or when this expression is true. However, the expression becomes manageable under the following assumptions: B=α+δ+ε1(2) BA=αβ+δγ+2ε1(3) where ε1>0 is small, and δγ=βα+ε2(4) where ε2>0 is small. Note that (2) and (3) are true if BAis sufciently small, and (4) is true if βαis sufciently large and δγis sufciently small (but the assumption that δγ>βα is maintained). Then the condition for price-quality signaling prot www.economics-ejournal.org 19 conomics: The Open-Access, Open-Assessment E-Journal to be greater is ε2+2ε1 βα+ε2α+δ 2r(α+δ)2 4h(1+X)1+ε2 βαiZ <α+δ+ε1co 2r(α+δ+ε1co)2 4(1+X)(ε2+2ε1) βαZ:Now assume ε1= βα n1and ε2=βα n2;where n1and n2are large. Then the condition is 3 n2+1α+δ 2r(α+δ)2 4(1+X)1+1 n2Z<α+δ+βα n1co 2 sα+δ+βα n1co2 4(1+X)1 n2+2 n1Z;which is clearly true if we x n1and choose n2sufciently large (corresponding to δγand βαbeing close enough). Together with Lemma 4, this proves the proposition. Note also that raising comakes price-quality signaling relatively more attractive. However, for any value of comeeting the restrictions in Section 2, if the sufcient conditions are met, the high-quality rm prefers price-quality signaling. 6.7 Proof of Proposition 7 Consider a candidate equilibrium in which there is pooling at low observable quality. Each type of rm sets qo=Land price P;and consumers believe that the rm is high-quality with probability b:This would be the prior probability that a rm is high-quality. In order for this to be an equilibrium, consumers must believe that the low-quality rm is more likely than the high-quality rm to deviate to qo=H: That is, if consumers observe a deviation, they would update their belief of the probability that the rm is high-quality to b0, where b0<b:(If this is not true, both types of rm have incentive to deviate.) However, the highand low-quality rms have essentially the same incentive to deviate to qo=H:If beliefs are held xed, the deviation prot would be greater for either type of rm. Therefore, according to the divinity criterion of Banks and Sobel (1987), a reasonable restriction on consumers' beliefs would be that the probability that the rm is high-quality is the same whether or not there is a deviation to qo=H:Since both types of rm have incentive to deviate from the proposed equilibrium, the deviation itself conveys no information to consumers. Given this restriction on beliefs, either type of rm will in fact deviate to qo=H;and the equilibrium candidate fails. www.economics-ejournal.org 20 conomics: The Open-Access, Open-Assessment E-Journal References Bagwell, K. (1987). Introductory Price as a Signal of Cost in a Model of Repeat Business. Review of Economic Studies, 54, 365–384. http://ideas.repec.org/a/bla/restud/v54y1987i3p365-84.html Bagwell, K. (1992). Pricing to Signal Product Line Quality. Journal of Economics and Management Strategy, 1, 151–174. http://ideas.repec.org/a/bla/jemstr/v1y1992i1p151-74.html Bagwell, K., and G. Ramey (1988). 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Review of Economic Studies, 50, 647–658. http://ideas.repec.org/a/bla/restud/v50y1983i4p64758.html www.economics-ejournal.org 22 Please note: You are most sincerely encouraged to participate in the open assessment of this article. You can do so by either recommending the article or by posting your comments. Please go to: www.economics-ejournal.org/economics/journalarticles/2011-5 The Editor © Author(s) 2011. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany