Targeting elites or mass selling: How to signal quality?

Transcription

1 Targeting elites or mass selling: How to signal quality? Nada Ben Elhadj-Ben Brahim Didier Laussel May 17, 01 Abstract We investigate the best signalling strategy of a monopoly that introduces a new product with unobservable quality The firm signals its quality by informing only a selected subset of consumers of the existence of the new product We show that a high quality firm signals quality by selling less than would do a low quality one ( targeting elites ) when the cost/quality ratio is increasing or mildly decreasing but by selling more ( mass selling ) when it is strongly decreasing The complete information equilibrium is a least cost separating equilibrium (LCSE) when cost/ quality ratio is strongly increasing Otherwise, at a LSCE, a high quality firm always restricts its equilibrium sales below their full information level Finally, we show that, almost everywhere, no pooling equilibrium satisfies the Cho and Kreps Intuitive criterion when the ex ante probability that the firm is a high quality one is lower than a strictly positive critical value JEL Classification: L1, L15, M37 Keywords: Product quality, pricing and advertising signalling, separating equilibrium, asymmetric information, pooling equilibrium LIM-EPT, Ecole Polytechnique de Tunisie, Universit de Carthage and ISG, Universit de Tunis GREQAM, Université de la Méditerranée 1

2 1 Introduction In some product markets, such as experience goods, consumers are unable to observe quality directly prior to purchase, although a firm knows the quality of the product it provides Thus, in these markets, asymmetric information prevails Hence, firms that launch new products try in general to communicate and signal their product quality attributes to consumers through a variety of channels, such as pricing, advertising, demand rationing, releases of research reports and test results, or warranties and returns policies in order to entice consumers to try their new product We investigate in this paper, the signalling strategy of a monopoly firm that launches a new experience product The firm informs of the existence of the product only a subset of consumers, those who have a preference for quality higher than some chosen level By the extent to which it so limits information and, accordingly, the supply of the good, it sends a signal about its quality So limiting information is possible in various ways Targeted advertising is one of them Selecting distribution channels is one another: the firm may choose to sell the good only in some shops and/or in some areas Since a natural way of informing a subset of consumers is through targeted advertising, and since this accordingly restricts the supply of the good, the signalling strategy we study in our model is related to two other well-known types of signalling channels: advertising and rationing Firms are often willing to spend huge amounts of money in advertising or what is commonly called Burning money Once, consumers observe the important advertising level, they thereby, infer the expected high-quality of the product Advertising in this case has little or no obvious informational content The example cited by Milgrom and Roberts s (1986) is a perfect illustration of this kind of quality signalling: A relatively recent example is the ad that was shown when Diet Coca-Cola was introduced: a large concert hall full of people, a long chorus line kicking, a remarkable number of (high-priced) celebrities over whom the camera pans, and a simple announcement that Diet Coke is the reason for this assemblage This is an example, of dissipative advertising In our model, when targeted advertising is used to inform selected consumers, it directly affects demand Rationing supply is another way to signal quality by creating excess demand since consumers may believe in some cases that scarce products are of better quality Indeed, in some industries, firms rarely use advertising to signal their quality but rather create excess demand through some marketing practices such as limited edition, queuing, etc According to Bandyopadhyay et al (010) limiting supply is a commonly used business strategy For instance, the high quality handbag producer Louis Vuitton does

3 not advertise against a cheap handbag imitation producer but frequently produces limited edition bags Furthermore, Phil Patton (1999) reported that Luxury brands such as Ferrari, Patek Phillipe, Montblanc sometimes place limited editions on the market to signal the high-quality of their products Similarly, Hermès has a two-year waiting list for its Birkin handbag priced at more than $6,000 (Branch 004) Our approach is a bit different: by limiting to a subset of consumers the information about the existence of its product, the firm restricts its supply indirectly This may lead to some kind of virtual rationing, because there may be some some consumers who would have wanted to buy the good would have they been informed of its existence, but there is no open excess demand In our model, when supplying a high quality product, the firm has the possibility to inform, via targeted advertising and/or by reaching them selectively through the choice of an appropriate distribution network 1, only a fraction of consumers, with the highest preferences for quality, of the availability of the product and of its price Informed consumers may rationally infer the quality of the product from their number and from the price of the good But how high quality is to be signalled? As put by Amaldoss and Jain (008) when should a firm offer limited editions? While some manufacturers prefer to limit their sales, some fashion designers like Vera Wang, are selling their products through mass merchandisers like Kohl s In other words, when is it that firms choose to signal high quality by targeting elites an developing limited editions rather than by mass selling and vice versa? A main goal of this model is to provide a tentative answer to this question When it is proportionately more costly to produce higher quality products, it is natural to expect the high quality firm to signal quality by informing a smaller number of consumers than the number who buy the low quality good at the full information equilibrium and by a higher price, ie by targeting elites In that case, since this fraction of consumers notice that they are in a small number, they infer that the product is of a high-quality This kind of strategical behavior is observed and well-known in the world of designer fashion where the dressmakers organize a fashion show to launch the new wearing collection In general, only the high society, and VIP are invited to these types of events Also, Christian Louboutin the French designer of red bottom shoes, has a very selective advertising strategy that consists to be present its collection through the artists and actresses in high social events and movies festivals such as Canne festival A 1 By choosing for instance to sell its product only through luxury stores, a firm actually restricts its sales to the richest consumers who shop in these stores In this case, the high quality firm is, as put by Zhao (000) less efficient in producing quality We argue that this fits rather well the case of traditional products (clothes, watches, food) where manufacturing high quality products needs using expensive inputs 3

4 limited fraction of potential consumers are informed of its new collection This kind of market targeting and exclusiveness pushes these guests to infer the expected high quality of the wearing products and shoes This limited advertising investment discourages a low quality firm, that, when mimicking, has a larger profit margin and that may in addition possibly suffer from being limited in its subsequent sales, from mimicking the high quality one, and constitutes thereby a quality signal We show that this kind of signalling strategy is optimal when the cost/quality ratio is increasing or even slightly decreasing 3 We argue that this corresponds to rather traditional products for which a higher quality means more expansive inputs and higher manufacturing costs When it is proportionately much less costly to produce higher quality goods 4, the full information equilibrium sales level of the high quality firm turns out to be larger than the low quality firm s one This is because, in that case, the high quality firm is strongly constrained in its second period sales and, despite a lower first-period profit margin, benefits more than the mimicking low quality one, which is not constrained, from a market expansion We determine in this paper under what precise conditions, high quality becomes signalled by informing more consumers than would do a low quality firm In that case, high quality is signalled by mass selling 5 This signaling strategy is observed in the products markets using new technology such as video games where firms target a large fraction of potential consumers to signal the high quality of the product We indeed argue that it corresponds to the case of high technology products whose development is costly but manufacturing cost does not rise so much with quality Related literature There is a rich economic literature on signalling quality in asymmetric information situations The main result of the signalling games literature is that a high-quality firm (that wishes to reveal its identity) sends a signal that is stronger than the signal it would have sent, had there been no doubt as to its identity Thus, the high-quality firm widens the gap between its signal and the full-information signal of the low-quality type This relies on the common assumption that strengthening the signal is costlier for the low-quality firm than for the high-quality one This paper is related to two strands of signalling literature: 1 Advertising as a signal of unobservable product quality Most of these models deal with what is commonly known as dissipative advertising and are only very loosely related to our approach According to Nelson (1974), 3 Notice that when the cost/quality ratio is steeply increasing, the full information price and quantity are enough to signal quality 4 In which case the high quality firm is much more efficient in producing quality than the low quality one 5 Notice that this does not mean that the high quality firm s supply has to be larger than its own full information level 4

5 there need not be anything in the advertising that explicitly informs consumers Simply the fact that a firm is spending money on advertising is enough to tell the consumer that the product is of high quality, provided she can observe the amount of advertising expenditures An important contribution is made by Milgrom and Roberts (1986) who provide a two-type monopoly signalling model based on Nelson (1974) in which both the price and the level of advertising are choice variables in a repeat purchases context In many other papers the amount of dissipative advertising expenditures, in addition to the price level, signals high quality: Kihlstrom and Riordan (1984), Linnemer 6 (1998), Orzach et al 7 (00), Hertzendorf and Overgaard (001b) and Fluet and Garella (00) In all these models, contray to the present paper, advertising has no direct impact on demand or gross profits There are however several papers which consider, as we do here, non-dissipative advertising, informative and/or persuasive, which affects demand directly Most of them (Zhao (000),Orzach, Overgaard and Tauman(00),Bagwell and Overgaard (005)) conclude that the high quality firm signals quality by advertising less than the low quality one because the low quality firm, which has a strong incentive to advertise when mimicking, is thus deterred from mimicry 8 A first difference is that these papers consider random advertising whereas we choose to focus on targeted advertising 9 A second is that we obtain the equivalent of their modest advertising result for a given subset of parameter values but that the opposite result (say overadvertising ) holds for the other subset Rationing as a signal of product quality: Several papers focus on the importance of rationing quantity and creating shortages as a signal of high product quality, whether it is made or not in combination with price and/or advertising signalling (Franklin and Faulhaber, 1990; Bandyopadhyay et al, 010; Kim, 00, Stock and Balachander, 005) There is some partial concordance of the results In Bandyopadhyay et al, for instance, rationing makes mimicry more costly to a low quality firm since it has when mimicking a higher margin over cost 10 A somehow similar mechanism is at play in our paper but only 6 Linnemer (1998) considers signalling product quality to consumers and marginal cost to a potential entrant 7 Orzach et al (00), re-examine the role of prices and advertising expenditures as signals of quality with two different types of consumers (sensitive consumers to high quality and indifferent consumers) 8 Notice that Gonzales (1998) reaches a different conclusion since in its model, though advertising may be used to inform customers, only price does signal quality 9 Though the model could be formally written in this way because there is a one to one correspondence between the two, it s not in our framework the amount of advertising expenditures which signals quality but rather the number of targeted consumers 10 Stock and Balachander (005) model is different since they consider two types of audience: informed 5

6 in the targeting elites case More generally, a first difference with our approach is that, in the above models 11, rationing is random, whereas, in ours, only the less valuable customers may be virtually rationed A second difference is that the restriction of supply occurs indirectly, through the limitation of demand 1 Main results We check the existence of a perfect Bayesian equilibrium in a twoperiod model, where a firm produces a good, the quality (low or high) of which is not observable to consumers The firm might signal its quality by targeting a given fraction of the consumers, that have the highest preferences for quality, with advertisements which inform them and only them of the existence of the product and/or by choosing, with the same goal, an adapted distribution network (mass merchandisers vs luxury shops for instance) We show that in such situation, the optimal choice of the firm depends on the quality ratio 13 and/or the cost ratio 14 We first characterize the set of separating equilibria It turns out that, depending on parameter values, a high quality firm can signal its quality either by a supply lower than the low quality firm s one and a higher price or, on the contrary, by a larger supply and a lower price The former is the case when the unit cost of quality is increasing or slightly decreasing, the latter when it is steeply decreasing Then, in the set of separating equilibria, we focus on the least cost separating equilibria (LCSE), the only ones to satisfy Cho and Kreps (1987) Intuitive Criterion We show that there exists almost everywhere a unique LCSE The full information strategy is shown to be a least cost separating equilibrium when the cost / quality ratio increases strongly with the quality of the product When it does not increase too strongly or even decreases, we find that the high quality firm ends-up limiting its informed market size in equilibrium relative to its own full information value, setting thus a higher price without inducing the low-quality firm to mimic it 15 Finally, we turn to the pooling equilibria and show that almost always there is no pooling equilibrium that satisfies Cho and Kreps (1987) Intuitive Criterion when the ex ante probability that the firm is a high quality is lower than some critical value and uninformed consumers, and assume a zero or small cost differential, thus eliminating the price margin effect Signaling the quality to the uninformed consumers is possible through the scarcity of the product or the price Scarcity hurts may signal quality because it hurts more the low quality firm which target the uninformed consumers 11 The probability of getting the good is uniform over consumers types 1 As noticed above, the firm informs only a fraction of the more valuable consumers 13 The quality ratio is the ratio between high and low product qualities 14 The cost ratio is the ratio between the high quality firm s cost and the low quality firm s cost 15 Remember however than, when the cost / quality ratio is steeply decreasing, the high quality firm s equilibrium market share is larger than the low quality one s 6

7 The remainder of the paper is organized as follows Section introduces the model Section 3, briefly presents the outcome of the second-period The benchmark fullinformation equilibrium is characterized in Section 4 Section 5, provides a detailed analysis of the equilibrium of the market with asymmetric information Finally, section 6 concludes the paper Proofs are relegated to the Appendix The Model We consider a monopolistic producer which has just developed a new product quality of the product may be high (q H ) or low (q L ) with corresponding marginal costs c H and c L per unit of output Since one consumer buys at most one unit of the good, c X (X = H, L) is the cost which the firm has to bear in order to manufacture and deliver one unit of the good of quality X to one (more) consumer It may include the cost of informing him/her of the existence of the good 16 Producing a high quality is assumed to be more costly than manufacturing a low quality one so that: c L c H The firm knows the true quality of its product but the potential consumers do not Their common prior is such that the product is a high quality one with probability µ and a low quality one with probability 1 µ This probability is common knowledge There is no direct way by which the firm could inform the consumers of the true quality of its product before they first purchase it There is a unit mass of consumers distributed uniformly on [0, 1] according to their marginal utility of quality θ In a given period, a type θ-consumer derives an utility U = { θq P if he buys one unit of the good 0 if he stays out of the market q and P being respectively the quality and the price of the good Assuming, similarly to Milgrom and Roberts (1986), that quality is not a choice variable but exogenously given, we consider the following two-period advertising and pricing game 16 If we assume, for instance, a unit advertising cost per buyer, there is so a one-to-one relationship between first-period market share and first-period advertising expenditures It follows that it would not be difficult, in order to allow easier comparison with the papers (Zhao (000),Orzach, Overgaard and Tauman(00),Bagwell and Overgaard (005)) in which non-dissipative advertising is a signal, to rephrase our subsequent analysis by considering advertising expenditures as the signal observed by the buyers, The 7

8 1 In the first period, nature selects quality, the firm observes it and decides simultaneously on its price P 1 and the market s subset [θ, 1] that will be informed, through advertising and/or possibly the choice of an adequate distribution network, about the existence of the product We assume that when informing only these customers of the existence of the product the firm sets a price such that these customers are indeed willing to buy if they derive a positive utility For customers beliefs p(p 1, θ ), the expected quality is q e = p(p 1, θ )q H + (1 p(p 1, θ ))q L and then the firm must account for the constraint, P 1 θ q e to ensure that all consumers in [θ, 1] are willing to buy What this assumption introduces is the strategic ability for a high quality firm to signal the quality of its product by voluntarily limiting information on the existence of the good to richer customers, even when some other consumers, would they be informed of the existence of the product, would choose to buy it at the posted price This is equivalent to rationing the potential demand (if totally informed) by selling only to those with a larger willingness to pay Informed customers, after observing P 1 and θ 17, revise their beliefs about the quality of the good, represented by the probability p(p 1, θ ) that the good is a high quality one, and make their initial purchase decisions Informing new consumers takes time so that, in the second period, only the consumers initially informed in the first period, are aware of the existence of the product and may buy it in the second period They know whether it is a high or a low quality one from direct usethere is no possible word of mouth communication between consumers The crucial feature here is that second period sales are constrained by first period ones 18 Given the second period price P, the demand for the good of true quality X at 17 We suppose that the informed buyers may observe how many they are, because they observe how selective are the advertisements which they receive and/or they see that only some specific shops distribute the good 18 Even if word of mouth communication and/or a new advertising campaign in the second period were possible, such a limitation of sales could alternatively follow from capacity constraints In which case an actual rationing could take place in the second period 8

9 price P in the second period is D = min{1 θ, 1 P q X } Note that P q X is the marginal consumer indifferent, when informed of the existence of the product, between buying product X or not buying at all Straightforwardly the second period equilibrium is such that { P = max θ q X, q } X(1 + b X ) π (q X ) = and { (1 b q X ) X if θ (1+b X) 4 q X (1 θ )(θ b X ) if θ (1+b X) denote Finally we assume that the firm discounts second-period profits by factor β and we Π(P 1, θ, X, p) the function giving the expected present value of a firm s profit of true quality X that sets an introductory price P 1, sells to consumers types larger than θ and is believed with probability p(p 1, θ ) of being of quality H In the following, we shall denote and assume that: b X = c X q X < 1 (with X = L, H) be the cost/quality ratio of the product, m = c H c L q H q L the marginal quality cost when switching from low quality to high quality product, the quality ratio and the cost ratio k = q H q L > 1 λ = c H c L > 1 We shall speak of increasing (resp decreasing) cost of quality when the cost/quality 9

10 ratio is larger (resp lower) for the product of the high quality One can expect the development and production of a high quality good to be more expensive than those of a low quality one At the risk of being a little bit simplistic, the total cost differential may come mainly from a difference in variable costs or from a difference in fixed R & D costs The first case corresponds to traditional goods like watches, dresses, alcohols, fine food: manufacturing a high quality, luxury, product requires using expensive inputs and therefore leads to much higher variable costs The second case concerns high technology goods like computers, smartphones and so on, where higher quality products do not necessarily mean higher manufacturing costs 19 but generally need large investments in R & D In the latter case, the cost/quality ratio is certainly steeply decreasing In the former one, it s, on the contrary, clearly increasing In the case of increasing cost of quality, we observe that b L b H m or, equivalently, λ k In the case of decreasing cost of quality, we have instead or, equivalently, m b H b L, Note that λ k quality ratio The following assumption will be useful to ensure the existence of non-trivial equilibria: = b H bl λ k so that, the higher is this ratio, the more increasing is the cost Assumption 1: m < β 1 + β (H1) 3 Full information equilibrium benchmark Consider now the full information case, in which consumers are aware of the existence of the product and know with certainty whether it is a high or low quality one In this 19 Srinavasan and Lovejoy (1997) show that a higher quality product needs not have a higher manufacturing cost 10

11 case, the firms is willing to sell to all consumers who are willing to buy at price P X and the demand for the good of quality X is which implies the following profit function D t = 1 θ X = 1 P Xt q X, t = 1, ( π t = (P Xt c X ) 1 P ) Xt, t = 1, q X We obtain the same equilibrium in both periods which is such that: θ = θ X = 1 + b X, (1) P 1 = P = P X = q X (1 + b X ) = q X + c X, Since we assumed that b X < 1, X = H, L, so that both the low and the high quality firm have a positive demand (θ 1) under full information On one hand, a high quality firm always benefits from a higher full information equilibrium price than the low quality firm On the other hand, the high quality firm has a lower market share (1 θ ) than the low quality one when the cost of quality is increasing (ie b H > b L ) and a larger share in the reverse case Accordingly, the total revenues of the high quality firm are, in the case of a decreasing marginal cost of quality, unambiguously more important than those of the low quality one 4 Perfect Bayesian Equilibria under quality uncertainty In this section, we investigate the firm s advertising and pricing strategies in a situation of asymmetric information, where consumers observe the advertising level and the product s price but are uncertain about the firm s quality even after exposure to its advertising We throughout employ the notion of Perfect Bayesian equilibrium As usual, it requires the monopolist s strategy to be sequentially rational 0 and consumers beliefs to conform 0 A strategy profile is sequentially rational at each information set, given a system of beliefs, if the action taken by the player must maximize his payoff given his belief and the subsequent strategy combination 11

12 with Bayes rule 1 whenever it applies We start by characterizing the Separating Equilibria We check later if the full information equilibrium is a separating one Then, we investigate the existence of a least cost separating equilibrium Finally we study briefly the Pooling Equilibria and show that, almost everywhere, none of them satisfies Cho and Kreps (1987) Intuitive Criterion when the probability µ is below a critical value 41 Separating Equilibria In a separating equilibrium, the high-quality firm signals its quality by choosing a strategy (P 1, θ ) such that the low-quality firm prefers to reveal its true quality rather than to mimic the high-quality firm At the same time, (P 1, θ ) must be such that the highquality firm is better-off signalling its true quality than by being perceived as a low quality one For a given couple (P 1, θ ) such that P 1 θ q H, the profit of the high quality firm when p(p 1, θ ) = 1, ie when it is recognized as such, is given by two different expressions according as it is or not capacity constrained in the second period:: Π(P 1, θ, H, 1) = { (1 θ )(P 1 b H q H ) + β(1 θ )(θ q H b H q H ) if θ b H + 1 (1 θ (1 b )(P 1 b H q H ) + βq H ) H 4 if θ b H + 1 The profit of the low-quality firm when p(p 1, θ ) = 0, ie when it is recognized as such, is simply its complete information strategy one, ie Π( q L (1 + b L ), 1 + b L (1 b L ), L, 0) = (1 + β)q L 4 The profit of the low-quality firm, when it is mistaken for a high-quality one, is: Π(P 1, θ, L, 1) = { (1 θ )(P 1 b L q L ) + βq L (1 θ )(θ b L ) if θ b L + 1 (1 θ (1 b )(P 1 b L q L ) + βq L ) L 4 if θ b L + 1 Notice that for θ b L + 1, a low-quality firm which mimics initially a high-quality one is capacity constrained in the second period, in which its profit-maximizing price is 1 Bayes rule is the standard way to update beliefs in probability theory ie it explains how you should change your existing beliefs in the light of new evidence When the low quality firm does not mimic a high quality one, it advertises uniformly toward all its potential consumers to inform them about its existence and the true quality of its product 1

13 simply θ q L In the reverse case, it can freely choose its profit maximizing second period output (and price) Finally, the profit of a high-quality firm when it is mistaken for a low-quality one equals: Π(q L 1 + b L, 1 + b L, H, 0) = 1 b L b L + 1 (1 b H ) (q L b H q H ) + βq H 4 if b H b L, in the case of increasing cost of quality Indeed, in that case, when it mimics the low quality one in the initial period, it is not capacity constrained in the subsequent one given that its first period market share is 1 ( b L + ) 1 and its second period market share will be its complete information level 1 ( b H + ) 1 The profit of a high-quality firm when it is mistaken for a low-quality one equals: Π(q L 1 + b L, 1 + b L, H, 0) = 1 b L (q L b L + 1 b H q H )+β 1 b L 1 + b L (q H b H q H ) if b H b L, in the case of decreasing cost of quality Indeed, in this case, when it mimics the low quality one in the initial period, it is capacity constrained in the subsequent one since its unconstrained profit- maximizing output is larger than the low quality firm s In a separating equilibrium, the two types of sellers choose different prices and outputs Therefore, the initially uninformed consumers, after observing the advertising signal and the first period price, infer the true quality of the product Hence, a separating equilibrium is analytically defined as follows: Definition 1 A sequential separating equilibrium is a pair (P 1, θ ) (P L, θ L ) such that the incentive constraints and Π(P 1, θ, H, 1) Π(q L 1 + b L Π( q L (1 + b L ) are satisfied, as well as the participation constraint, 1 + b L, H, 0), (IC1), 1 + b L, L, 0) Π(P 1, θ, L, 1), (IC) P 1 θ q H (P C) Customers beliefs are given by p (P 1, θ ) = 1, 13

14 and p (P L, θ L ) = 0 and, more generally 3, p (P, θ) = 0, (P, θ) (P 1, θ ) Condition IC1 means that the profit of a high-quality firm is larger, when it is recognized as such, than when it is mistaken for a low-quality one, ie that it has no incentive to mimic a low quality firm Condition IC implies that a low-quality firm has no incentive to mimic a high-quality one The condition P C guarantees that all initial customers of types θ θ are indeed willing to buy the high quality good at price P 1 When these constraints are satisfied, consumers rationally infer that the firm that transmits a signal is the high-quality provider: the pair (P 1, θ ) is a separating equilibrium It is worth noticing that the assumption that there is no communication between consumers in the two periods plays a key role in our analysis and results According to this assumption, the initially uninformed consumers still remain unaware of the existence of the product in the second period If it wasn t the case and if word-of-mouth communication was possible at least on a very large scale, a low-quality firm would always mimic a high-quality one, since it would likely benefit from a high price in first period without being restricted to a second-period market-share below its full information level and we won t obtain consequently any separating equilibrium Remark 1 Notice that, at the strategy pair (P L, θ L ), the two incentive constraints are trivially satisfied with equality while the participation condition is satisfied with strict inequality However, it cannot be a separating equilibrium since the high-quality firm chooses exactly the same strategy as a low-quality one, what does not allow consumers to infer the quality of the firm s product The Lemma 1 is a technical one useful to comment the Lemma 1 Lemma 1 From the constraint IC and the Implicit Function Theorem the maximum possible initial price P 1 is an increasing function G of the critical consumer type θ Let us define λ = 1+kβ 1+kβ Notice then that 1+β 1+β values such that λ < λ = 1+kβ < k Clearly now, the set of parameters corresponds to a steeply decreasing cost of quality 1+β corresponds to an Conversely, the set of parameters values such that λ > λ = 1+kβ 1+β increasing or slightly decreasing cost of quality We are now ready to characterize the set of separating equilibria 3 As put by Milgrom and Roberts ((1986), page 104), p(p, θ) has simply to be sufficiently small (eg zero) that neither player wishes to deviate to p(p, θ) 14

15 Proposition 1 The set of separating equilibria is non-empty iff λ λ = 1+kβ 1+β At a separating equilibrium: whenever the cost of quality is increasing or slightly decreasing, ie λ > λ = 1+kβ 1+β (CASE A), the high-quality firm has always a smaller market-share than the lowquality full-information one, ie θ > θ L whenever the cost of quality is steeply decreasing, ie λ < λ = 1+kβ (CASE B), 1+β the high quality firm has always a larger market share than the low-quality fullinformation one, ie θ < θ L Proposition 1 allows to distinguish two rather different cases, corresponding respectively to Figures and 6 Notice that there is no separating equilibrium when λ = λ In that case, (P L, θ L ) is the only pair which satisfies the incentive and participation constraints but choosing this pair does not allow a high quality firm to distinguish itself from a low quality one The intuition behind Proposition 1 is quite simple Let us consider the effect of a first-period market expansion on respectively the profits of a high quality firm (H) which is recognized as such, namely Π(P 1, θ, H, 1), and the profits of a low quality firm (L) when mimicking the latter, ie Π(P 1, θ, L, 1) The first-period effect of a market share increase (ie a decrease in θ ) is unambiguously larger on L s mimicry profits since L having a lower marginal cost has a larger profit margin The second period effect on L s profits is larger than the effect on H profits in the case of increasing cost of quality since, in that case L would like to produce more and is constrained in its second period sales while H is not This is the reverse in the case of decreasing cost of quality, since then H is constrained while L is not So what about the aggregate effect? It is unambiguously larger on L s profits when the cost/quality ratio is increasing That remains true when this ratio is slightly decreasing But, when the cost of quality is decreasing enough (ie when λ < λ ), the second period sales constraint effect on H s profits is larger and large enough to dominate the first period profit margin effect Accordingly, when the cost of quality is increasing or slightly decreasing (resp steeply decreasing), a first period market share lower (resp larger) than (1 θ L ) signals a high quality firm Of course, when λ = λ, the two effects cancel exactly so that H cannot signal its quality 4 In all the figures below, we label for the sake of convience IC1, IC and PC, the frontiers of the sets of the couples (P 1, θ ) which are respectively defined by the incentive constraints (IC1 and IC) and the participation constraint (PC) 5 The figure is drawn for the following parameters values: c H = 4, c L = 1, q H = 4, q L = 7, β = 1 6 The figure is drawn for the following parameters values:: c H = 4, c L = 1, q H = 4, q L = 7, β =

16 15 IC1 IC PC 10 5 b L 1 Figure 1: The separating equilibria when λ > 1+kβ 1+β IC1 IC PC Figure : The separating equilibria when λ < 1+kβ 1+β When the cost of quality is either increasing or only slightly decreasing (CASE A), at any separating equilibrium, the high-quality firm signals its quality by choosing a market share smaller than the full information market share of the low-quality firm (ie by increasing θ and targeting only the consumers with higher preferences for quality) and a larger first period price (according to Lemma 1) without inducing the low-quality firm to mimic it Indeed, a smaller market share ( a higher θ ) signals a high quality firm: a low quality, when it mimics a high quality one, benefits more than the latter from a market expansion When the cost of quality is steeply decreasing (CASE B), at any separating equilibrium, the high-quality firm signals quality by choosing a market share larger than the full information market share of the low-quality firm and a first period price which is lower (according to Lemma 1) In this case, the high quality firm benefits more than the low quality which mimics it from a market expansion This is because, contrary to L, H is constrained in its second period sales This capacity constraint is here strong enough 16

17 to dominate the first period profit margin effect which works in the opposite direction It should finally be noticed that, in order to characterize separating equilibria, the relevant comparison is between H s first-period price and market share (P 1, 1 θ ) and L s ones But we can t at this stage draw any conclusion on the comparison with H s full information equilibrium values (p H, 1 θ H ) The first question which arises is precisely whether the full information strategies, as derived in Section 4, correspond to a separating equilibrium Does it happen that the low-quality firm does not gain from mimicking the full information choices of the high-quality one and reciprocally? The answer is stated in Proposition Proposition The complete information pair (P 1, θ ) = (P H, θ H ) is a separating equilibrium iff λ λ with λ = k b L+ (k 1)(k+β b L βb L)+βb L b L > k (k+β) A straightforward implication of Proposition is that the complete information pair is never a separating equilibrium when the cost of quality is decreasing Indeed, with the latter case, the high quality firm has both a higher price and a larger market share under full information than a low quality one, so that the latter would always benefit from mimicking the former Proposition 1 shows that the complete information pair is a separating equilibrium when the cost of quality increases strongly, ie the ratio λ = c H cl between the unit costs is large relative to the quality ratio k = q H q L Indeed, the lowquality firm faces two contradictory strategic effects when mimicking the high-quality firm The first positive effect is the benefit from a first period higher price which is the larger the larger is the quality differential 7 The second negative effect is the cost from being constrained in its sales level in both periods This negative effect is the stronger the larger is the differential between full information market shares, ie the larger the unit quality costs differential, since the difference between the full information equilibrium market shares is equal to b H b L As a result, the low-quality firm chooses to signal its true type Whenever λ λ, there is an infinite number of separating equilibria So it is interesting to focus now on a more restrictive equilibrium concept, the least cost separating equilibria 7 The price differential equals q H(1+b H ) q L(1+b L ) = 1 q L (k (1 + b H ) 1 b L ) 17

18 4 Least-cost separating equilibrium In this section we characterize the most profitable separating equilibrium, called the least cost separating equilibrium (LCSE), that is the only separating equilibrium that satisfies the Cho-Kreps intuitive criterion (1987) According to the Intuitive criterion, if a firm deviates from a candidate equilibrium pair, it is intuitively recognized as a high quality one if a low quality firm would be worse off when implementing this deviation, whatever the consumers beliefs upon observing the deviation 8 Definition A least-cost separating equilibrium (LCSE) is a pair ( P 1, θ ) (P L, θ L ) which maximizes Π(P 1, θ, H, 1) subject to the incentive constraint IC and the participation constraint P C As appears from Lemma 1, there are two disjoint sets of separating equilibria in the (k, λ) space, corresponding to two opposite ways of signalling a high quality product The frontier λ = λ between these sets corresponds to a zero dimensional set for which no separating equilibrium, and of course no LCSE exists: for all points in this set, π (θ ) takes its maximum value at θ = θ L, implying that P 1 = P L, and the high quality firm cannot be distinguished from a low quality one The set of ( P 1, θ ) pairs satisfying the constraints of the optimization problem are depicted in Figure 3 and Figure 4 respectively for λ > λ and λ < λ 5 PC IC b L 1 Θ Figure 3: Feasible strategies when λ > 1+kβ 1+β Let θ and ˆθ, (such that θ < ˆθ), be the intersection [ points 9 between IC and P C Notice that (i) when λ > 1+kβ, IC is binding on b L +1, ˆθ ] ] and P C is binding on [ˆθ, 1+β 1, 8 This is always the case that occurs if, when deviating, the low quality firm is (mis)taken with probability 1 to be a high quality one 9 The analytical expression of ˆθ and θ are given in the Appendix respectively by Expressions 5 and 6 18

19 8 PC IC 6 4 Θ Figure 4: Feasible strategies when λ < 1+kβ 1+β while (ii) when λ < [0, 1+kβ, P C is binding on θ ] and IC is binding on 1+β Proposition, in the latter case, we always check that θ < b H+1 < b L+1 Notice that λ has already been defined and let us define in addition: λ = k(1 + β) b L + β (k 1) 3 (k + β (1 + β)b L ) (1 + β)(k + β)b L, ] b [ θ, L +1 From We prove in Appendix the following useful Claim, which will allow us to establish in the space (k, λ) a regioning of the four different types of LCSE Claim 1 λ > λ > λ Obviously, at a LCSE of the game, either the participation constraint(pc) is binding alone (in which case on gets the full information pair), or both constraints are simultaneously binding or, finally, the second incentive constraint (IC) is binding alone It is rather intuitive, and will be shown rigorously below, that the two first cases can occur only in the area of increasing costs of quality (λ > k) while the third subdivides into two different cases according as λ is larger or lower than λ In the latter (sub)case (λ < λ < k), the high quality firm signals quality by a supply larger and a price lower than those of the low quality one, while this is the contrary in the three other situations The two Propositions below characterize the LCSE of the game in the two opposite cases previously identified in Lemma 1, corresponding to the two qualitatively different sets of separating equilibria Figure 5 depicts the zones of the four possible LCSE depending on the quality ratio and cost ratio Proposition 3 When λ > λ (CASE A), ie when the cost / quality ratio is increasing or slightly decreasing with the quality of the product, there is a least cost separating equilibrium such that : 19

20 7 Λ 6 Zone I Λ 5 4 Zone II 3 Zone III Λ Zone IV 1 Figure 5: Regioning of LCSE in the (k, λ) space (i) Iff λ λ, only the participation constraint (PC) is binding (Zone I), with: ( P1, θ ) = (P H, θ H ), which corresponds to the full-information strategy (ii) Iff λ [ λ, λ], both constraints(pc) and (IC) are simultaneously binding (Zone II), with: ( P1, θ ) = ( θ q H, 1 ( 1 + b L (1 + β) + (k 1) (k + β b L βb L ) k + β )), and (iii) Iff λ (λ, λ], only the incentive constraint (IC) is binding (Zone III), with: ( P1, θ ) = (P A, θ A ), where θ A = (β+1)m β + 1, P A = q L m (β+1) mb L (β+1) +β(b L +βb L +1) β m(β+1) Notice that Assumption 1 (H1) ensures that θ A < 1 and P A > 0 When the cost ratio λ is large enough with respect to the quality ratio k, ie the cost/quality ratio is steeply increasing (Zone I in Fig 5), only the participation constraint is binding and the LCSE corresponds to the full information equilibrium In this 0

21 case indeed, it is much more costly to produce a high quality product, so that the full information output of a high quality is so much lower than the full equilibrium output of a low quality firm that the latter is naturally deterred from mimicking the former When the cost/quality ratio is moderately increasing or slightly decreasing with the quality of the product (Zones II and III in Fig 5), a high quality firm has to reduce its market share below its full information value in order to deter a low quality one from mimicking it At the same time it continues to signal that it is indeed a high-quality one by setting a higher price and a smaller supply than a low quality firm As we already proved in Lemma 1, when IC is saturated, the price P 1 is an increasing function of θ Thus, the firm has incentives to limit its informed market share (increase θ ) in order to be able to charge a higher price In the second place, in Zone II (Fig binding In this case, k θ has the same sign as 5), the two constraints are simultaneously ( k + kb L b L + β βb L) bl (k 1) (k + β b L βb L ) Evaluating the difference of the square of the two positive terms, we obtain (b L 1) (b L + 1) (k + β) > 0 Thus, the equilibrium value of θ increases with the quality ratio k from its full information value to (β+1)m β + 1 Proposition 4 When λ [1, λ) (CASE B), ie when the cost / quality ratio is steeply decreasing with the quality of the product, there is a least cost separating equilibrium such that only the incentive constraint (IC) is binding (Zone IV), and ( where θ B = 1 + b L λ(1+β) 1, P kβ B = q L ( P1, θ ) = (P B, θ B ), 1 ) kβ(b L 1) b L +kβ λb L βλb L + b L Notice that λ < λ together with b L < 1 is enough to ensure that θ B < 1 and P B > 0 In the case analyzed in Proposition 3 (Zone IV in Fig 5), the cost / quality ratio is steeply decreasing The firm signals its high quality by selecting a market share larger than the market share of the low quality one and, accordingly, a lower price This is quite obvious since θ B = 1 + b L λ(1+β) 1 < θ kβ L = 1+b L λ < λ The steeply decreasing costs of quality mean that the high quality firm is able to operate a technological breakthrough without increasing very much its variable manufac- 1

22 turing costs 30, which allows it to introduce its product at very low price relative to that of the low quality firm Lemma The high-quality firm s equilibrium market share (1 θ ) is lower than its full-information market share (1 θ H ) for all λ [1, ˆλ] There exists λ c ] λ, ˆλ[ such that the high-quality firm s equilibrium price P 1 is lower than its full-information price P H for all λ [1, λ c ], larger for λ [λ c, ˆλ] and equal to it for all λ ˆλ Λ Λ Λ Λ Λ Λ Figure 6: (1 θ ) (1 θ H ) wrt λ Figure 7: P1 P H wrt λ One should notice that the high-quality firm s price is higher than its full-information value in Zone II, where both constraints are binding, and by continuity, at least in part of Zone III In Zone IV, by reducing its supply under its full information level, the firm is indeed able to rise its price along the (IC) constraint without exceeding the price of the lowquality firm Most of the existing literature on dissipative advertising and price signalling, following Milgrom and Roberts (1986), has shown that a high introductory price and a high level of dissipative advertising may serve to reveal the high quality of a new experience good We find on the contrary that the relationship between advertising and product quality is negative at equilibrium ie a high-quality firm reduces its advertising investment and targets a limited consumers market with high preferences for quality But the mechanisms at play are quite different It is accordingly more interesting to compare our results with those obtained in the non-dissipative advertising literature (Zhao (000),Orzach et al (00), Bagwell and Overgaard (005)), with which there exist greater similarities All these papers conclude 30 Of course this may require important previous investments in R & D

23 that the high quality firm lowers its advertising spending with respect to its complete information level 31 We obtain in this paper the equivalent result that the high quality firm reduces its market share 3 But these papers also obtain the more striking conclusion that the high quality firm advertises less than the low quality one 33 By contrast, in our paper, the equivalent of this modest advertising outcome obtains if and only if λ > λ, ie in Case A of increasing or slightly decreasing cost of quality We get the opposite result iff λ < λ, ie in Case B of steeply decreasing cost of quality The reason underlying these divergent results lies in our two-period framework In the above models, the intuition behind the modest advertising result is that the low quality firm when mimicking the high quality one has a higher profit margin since it has a lower cost So it benefits more than the high quality firm from a market expansion due to increased advertising expenditures Reducing the latter is a way to deter mimicry and thus to be able to raise the price As we already noticed, the same mechanism is at work in our model in the first period In the second period, in the case of increasing cost of quality, the high quality is unconstrained while the low quality firm is, when mimicking, constrained by the firstperiod market share, so that, one again the latter benefits more from a market expansion than the former and this reinforces the first period effect In the case of decreasing cost of quality, on the contrary, it s the high quality firm which is capacity constrained in the second period while the mimicking low quality one is unconstrained Here the first period and second period effects work in opposite directions When the second period effect becomes larger, ie when λ < λ, choosing a first period market share larger than the low quality firm s full information output becomes the right way to deter mimicry In the rationing literature, the comparison with Bandyopadhyay et al (010) model, who conclude that the high quality firm creates shortages in order to signal quality, leads to similar conclusions The underlying mechanism is indeed basically the same as in the non-dissipative advertising literature, ie the price margin effect A supply restriction reduces more L s profits when it tries to mimics a high quality firm than H s profits simply because it has then a larger profit margin Accordingly, rationing signals high quality 34 As already noticed, this corresponds precisely to our results in the case when the cost/quality ratio does not decrease too steeply but opposite results are obtained in the reverse case 31 In Zhao this is obtained if, according to our notations, b H > b L 3 Except in Zone I where the full information equilibrium is the LCSE 33 When the high quality firm s full information advertising level is larger the low quality firm s one, this is a signal reversal 34 Kim (00) has a rather different model in which two buyers buy sequentially and the firm can produce one or two units The mechanism is however similar since the crucial feature that enables a high type to be separated from a low type is a cost difference It may be too costly for a high type to produce two units and not to sell them all, whereas it is less costly for a low type (page 10) 3