Deceptive Equilibria in Markets: Firms Selling Inferior Products to Unknowing Consumers, Lecture notes of Literature

Download Deceptive Equilibria in Markets: Firms Selling Inferior Products to Unknowing Consumers and more Lecture notes Literature in PDF only on Docsity! Inferior Products and Profitable Deception∗ Paul Heidhues ESMT, CEPR and CESifo Botond Kőszegi Central European University, UC Berkeley, and CESifo Takeshi Murooka UC Berkeley November 2, 2012 Abstract We analyze conditions facilitating profitable deception in a simple model of a competitive retail market. Firms selling homogenous products simultaneously set a transparent up-front price and an additional price, and decide whether to unshroud the additional price to naive consumers. To model especially financial products such as banking services, credit cards, and mutual funds, we assume that there is a binding floor on the product’s up-front price. Our main results establish that “bad” products—ones that should not even be produced—tend to be more reliably profitable than “good” products. Specifically, (1) in a market with a single socially valuable product and sufficiently many firms, at least one firm is willing to unshroud, so a deceptive equilibrium does not exist and firms make zero profits. But perversely, (2) if the product is socially wasteful, a firm cannot profitably sell a transparent product, so there is no incentive to unshroud and hence a profitable deceptive equilibrium always exists. Furthermore, (3) in a market with multiple products, since a superior product both diverts sophisticated consumers and renders an inferior product socially wasteful in comparison, it guarantees that firms can profitably sell the inferior product by deceiving consumers. JEL Codes: D14, D18, D21 ∗The setup of this paper was developed in our earlier working paper “The Market for Deceptive Products,” which we split into two papers. The companion paper is titled “Exploitative Innovation.” Kőszegi thanks the Berkeley Center for Equitable Growth for financial support. We are grateful to Özlem Bedre-Defolie, Stefano DellaVigna, Drew Fudenberg, Michael Grajek, Michael Grubb, Péter Kondor, Nicolas Melissas, Antoinette Schoar, and seminar and conference audiences for useful comments and feedback. 1 1 Introduction In this paper, we investigate circumstances under which firms sell products by deceiving some consumers about the products’ full cost, focusing (in contrast to much of the literature) on deception that leads to positive equilibrium profits in seemingly competitive industries.1 We identify a novel, perverse aspect of profitable deception: products that generate lower social surplus than the best alternative facilitate deception precisely because they would not survive in the market if consumers understood hidden fees, and therefore firms often make profits on exactly such bad products but not on good products. Section 2 introduces our model, in which firms are engaged in simultaneous-move price compe- tition to sell a single homogenous product. Building on the seminal model of Gabaix and Laibson (2006), we assume that each firm charges a transparent up-front price as well as an additional price, and unless at least one firm decides to (costlessly) unshroud the additional prices, naive consumers ignore these prices when making purchase decisions. To capture the notion that in some markets, such as banking services, credit cards, and mutual funds, firms cannot return all profits from later charges by lowering initial charges, we deviate from most existing work and posit that there is a floor on the up-front price.2 We investigate conditions under which a profitable deceptive equilibrium—wherein all firms shroud additional prices—exists. Whenever such an equilibrium exists, it is the most plausible one: it is then the unique equilibrium in the variant of our model in 1 Hidden fees have often enabled firms to reap substantial profits despite seemingly considerable competition, at least at the price-competition stage when entry and marketing costs have been paid and customer bases have been identified and reached. Investigating trade and portfolio data from a large German bank, for example, Hackethal, Inderst and Meyer (2010) document that “bank revenues from security transactions amount to e2,560 per customer per year” (2.4 percent of mean portfolio value), a figure likely well above the marginal cost of serving a customer. Similarly, based on a number of measures, including the 20-percent average premium in interbank purchases of outstanding credit-card balances, Ausubel (1991) argues that credit-card companies make large profits. Ellison and Ellison (2009) describe a variety of obfuscation strategies online computer-parts retailers use, and document that such strategies can generate surprisingly large profits given the near homogeneity of products. These observations, however, do not mean that the net economic surplus taking all operating costs into account are large or even positive in these markets: for example, fixed entry costs can dissipate any profits from the later stage of serving consumers. 2 In Heidhues, Kőszegi and Murooka (2011), we provide a microfoundation for the price floor based on the presence of “arbitrageurs” who would take advantage of overly low prices. This microfoundation is an extreme variant of Ellison’s (2005) insight (developed in the context of add-on pricing) that firms may be reluctant to cut initial prices because these cuts disproportionately attract less profitable consumers. We briefly discuss these and other possible reasons for the price floor in Section 2.2 below. Ko (2011), Grubb (2012), and Armstrong and Vickers (2012) also analyze models with variants of our price-floor assumption. 1 investigates how firms exploit naive consumers by charging hidden or unexpected fees, previous work has not identified the central role of wasteful and inferior products in maintaining deception and generating profits. Indeed, in most previous models competition returns all of the profits from hidden fees to consumers, so that these models cannot investigate market conditions that facilitate profitable deception. We conclude in Section 7 with mentioning some policy implications of our findings, and by pointing out important further questions raised by our model. 2 Basic Model 2.1 Setup In this section, we introduce our basic model of a market for potentially deceptive products. N ≥ 2 firms compete for naive consumers who value each firm’s product at v > 0 and are looking to buy at most one item. Firms simultaneously set up-front prices fn and additional prices an, and decide whether to costlessly unshroud the additional prices. For simplicity, we assume that a consumer who buys must pay both prices—she cannot avoid the additional price. If all firms shroud, consumers make purchase decisions believing that the total price of product n is fn. If at least one firm unshrouds, all firms’ additional prices become known to all consumers, and consumers make purchase decisions based on the true total prices fn+an. We assume that the highest possible additional price firms can impose is a > 0. If consumers weakly prefer buying and are indifferent between a subset of firms, these firms split the market in proportion to exogenously given shares sn ∈ [0, 1).3 Firm n’s cost of providing the product is cn > 0. We let cmin = minn{cn}, and—to ensure that our industry is competitive in the corresponding classical Bertrand model—assume that there are at least two firms whose cost is equal to cmin. In addition, we assume that v + a > cn for all firms n; a firm with v + a < cn cannot profitably sell its product, so without loss of generality we can 3 In later sections, we consider several alternatives to and extensions of the above basic framework. In Section 4, we add sophisticated consumers, and also analyze multiproduct markets. In Section 5, we consider what happens when unshrouding is costly; discuss an alternative formulation of consumer naivete in which there is an add-on consumers can purchase after purchasing a base product, and consumers know the add-on price but mispredict their willingness to pay for it; and identify the limited ways in which heterogeneity in v affects our conclusions. 4 think of it as not participating in the market. We look for Nash equilibria of the game played between firms, where—deviating from much of the literature—we impose that firms face a floor on the up-front price: fn ≥ f . We assume that f ≤ v, so that consumers are willing to buy when they face an up-front price at the floor and believe that the additional price is zero. In stating our results, we focus on identifying conditions for and properties of deceptive equilibria—equilibria in which all firms shroud additional prices. Because no firm has an incentive to shroud if at least one firm unshrouds, there is always an unshrouded-prices equilibrium. When a deceptive equilibrium exists, however, it is more plausible than the unshrouded-prices equilibrium for a number of reasons. Most importantly, in Section 5.1 we show that in that case, the deceptive equilibrium is the unique equilibrium in the variant of our model in which unshrouding carries a positive cost, no matter how small the cost is. In addition, a positive-profit deceptive equilibrium is preferred by all firms to an unshrouded-prices equilibrium. Finally, for the lowest-priced firms the strategy they play in an unshrouded-prices equilibrium is weakly dominated by the strategy they play in a positive-profit deceptive equilibrium. 2.2 Motivation for and Interpretation of Key Assumptions Our model has three key assumptions: that naive consumers might ignore the additional prices when making purchase decisions, that there is a floor on the up-front price, and that firms can costlessly and fully educate all consumers. We discuss these assumptions in turn, and provide examples of how our model fits various real-world markets. The assumption that consumers might ignore some prices or fees is a simplified variant of many assumptions in behavioral industrial organization (DellaVigna and Malmendier 2004, Eliaz and Spiegler 2006, Grubb 2009, Heidhues and Kőszegi 2010, and others), and is consistent with observations in industries such as banking, credit-card, retail-investment, and mortgage services.4 Furthermore, while we interpret the additional prices primarily as financial prices, our model applies 4 For empirical evidence from a number of financial and other industries, see Hall (1997), Cruickshank (2000), Wilcox (2003), Barber, Odean and Zheng (2005), Agarwal, Driscoll, Gabaix and Laibson (2008), The Office of Fair Trading (2008), Stango and Zinman (2009), Federal Communication Commission (2010), and Woodward and Hall (2012). For some additional evidence from field experiments, see Hossain and Morgan (2006) and Chetty, Looney and Kroft (2009). 5 equally well to non-financial costs of owning a product that can be shrouded from consumers. For example, the product may be manufactured in disagreeable ways (e.g. in sweatshops or with environmentally unfriendly procedures), or it may be unhealthy or inconvenient to use.5 Our assumption of a price floor is based on a number of related arguments that have appeared in the literature. In Heidhues et al. (2011), we provide one microfoundation for the price floor based on the existence of “arbitrageurs” who would enter the market to make money off of a firm with overly low (for instance negative) prices, and who avoid the additional price because they are not interested in using the product itself. As a simple illustration in a specific case, consider the finding of Hackethal et al. (2010) that German bank revenues from security transactions amount to e2,560 per customer per year (2.43% of mean portfolio value). If a bank handed out such sums ex ante—even if it did so net of account maintenance costs—many individuals would sign up for (and then not use) bank accounts just to get the handouts. This threat creates a binding floor on banks’ up-front price. In related models, Ko (2011) derives a version of our price floor from the presence of sophisticated consumers, Grubb (2012) and Armstrong and Vickers (2012) impose a no-negative-prices constraint in the context of a market with naive consumers, and Farrell and Klemperer (2007) discuss the same constraint in the context of switching-cost models. Of course, in our model with a competitive market, we have imposed the price floor in an extreme form: there is a single number up to which firms are willing to lower the up-front price, but beyond which they cannot go. In reality, there is typically no such bright-line price floor. Generalizing from our setting, the intuitions for our main qualitative results on the role of wasteful and inferior products in generating profits require only that firms are less willing to cut the up- front price than a transparent total price, so that they make higher profits with shrouded than with unshrouded additional prices. This would be the case in any setting in which, similarly to our arbitrageurs model mentioned above and the model of add-on pricing by Ellison (2005), less 5 In contrast to our model, in most of the above examples a consumer has some control over how much of the additional price she pays. So long as consumers’ fundamental mistake is in underestimating additional prices, the logic of our model requires only that consumers cannot fully avoid these prices, so that firms can make profits on them. If consumers’ mistake is in mispredicting their own behavior rather than prices, the model of Section 5.2 applies. Similarly, in many markets naive consumers may (contrary to our model) underestimate but not fully ignore additional prices. In this case, we can apply our model by thinking of the extra fees consumers expect to pay as being included in fn, with the unexpected component of the extra fees being an. 6 By Proposition 1, therefore, deception can occur in our model even if the price floor is not binding. Indeed, for some commonly invoked examples of deceptive products, such as hotel rooms and printers, the price floor does not seem to be binding. Nevertheless, for several reasons deception is likely to be less widespread and economically harmful in this case than in the case of a binding price floor below. First, while in the current case all consumers buy from a most-efficient firm and pay a total price equal to the firm’s cost, in the case of a binding price floor they pay a higher price and also buy from less efficient firms. Second, the profits firms earn when the price floor is binding can potentially can potentially attract entry of less efficient producers, exacerbating productive inefficiency. Third, we show in our companion paper (Heidhues, Kőszegi and Murooka 2012) that the incentive to come up with deceptive products in the first place is very different with and without the price floor. With a non-binding price floor, firms have no incentive to invent new ways of charging additional prices, so they limit themselves to obvious deception opportunities. With a binding price floor, in contrast, the incentive to invent new hidden fees—even ones competitors can easily copy—is often very strong. As a second benchmark, we consider equilibria when all consumers are sophisticated in that they observe the total prices fn + an and make purchase decisions based on these prices. Proposition 2 (Equilibrium with Sophisticated Consumers). Suppose all consumers are sophisti- cated, and consider any f . If v > cmin, then in any Nash equilibrium consumers buy the product at a total price of cmin from a most efficient firm. If v < cmin, then in any Nash equilibrium consumers do not buy the product. Firms earn zero profits in any equilibrium. Since sophisticated consumers understand the total price, firms cannot break even by selling a socially wasteful product to them. Furthermore, firms make zero profits in selling a socially valuable product as well: if there is a binding floor on the up-front price, firms simply switch to competing on the additional price, and—as there is no floor on this price—bid down the total price until profits are zero. 9 3.2 Naive Consumers with a Binding Price Floor Taken together, Propositions 1 and 2 imply that for profitable deception to occur, both naive consumers must be present and the price floor must be binding. We turn to analyzing our model when this is the case, assuming for the rest of this section that all consumers are naive and f > cn−a for all n. To identify when a deceptive equilibrium exists, first note that if additional prices remain shrouded, all firms set the maximum additional price a. Then, since firms are making positive profits and hence have an incentive to attract consumers, they bid down the up-front price to f . With consumers being indifferent between firms, firm n gets market share sn and therefore earns a profit of sn(f + a− cn). For this to be an equilibrium, no firm should want to unshroud additional prices. Once a firm unshrouds, consumers will be willing to pay exactly v for the product, so that firm n can make profits of at most v− cn by unshrouding and capturing the entire market. Hence, unshrouding is unprofitable for firm n if the following “Shrouding Condition” holds: sn(f + a− cn) ≥ v − cn. (SC) By extension, a deceptive equilibrium exists if (SC) holds for all n. Furthermore, since sn < 1, Condition (SC) implies that f+a > v, so that in a deceptive equilibrium consumers receive negative utility. Proposition 3 states this result, and also says that if some firm violates Condition (SC), there is no deception in equilibrium: Proposition 3 (Equilibrium with Binding Price Floor). Suppose f > cn−a for all n. If Inequality (SC) holds for all n, a deceptive equilibrium exists. In any deceptive equilibrium, fn = f and an = a for all n, consumers receive negative utility, and firms earn positive profits. If Inequality (SC) is violated for some n, in equilibrium prices are unshrouded with probability one, consumers buy from a most efficient firm at a total price of cmin, and firms earn zero profits. The intuition for why firms might earn positive profits despite facing Bertrand-type price com- petition is in two parts. First, as in previous models and in our model with a non-binding price floor, firms make positive profits from the additional price, and to obtain these ex-post profits each firm wants to compete for consumers by offering better up-front terms. But the price floor prevents firms from competing away all profits from the additional price by lowering the up-front price. 10 Second, since firms cannot compete for consumers by cutting their up-front price, there is pressure for competition to shift to the additional price—but because competition in the additional price requires unshrouding, it is an imperfect substitute for competition in the up-front price. When a firm unshrouds and cuts the additional price by a little bit, consumers learn not only that the firm’s product is the cheapest, but also that all products are more expensive than they thought. If f + a > v—that is, if consumers’ utility from buying at the current total price is negative—this surprise leads consumers not to buy, so that the firm can attract consumers by unshrouding only if it cuts the additional price by a discrete margin. Since this may be unprofitable, the firm may prefer to shroud. Importantly, if purchasing at the total market price is optimal (i.e., if f + a ≤ v), then despite their surprise consumers are willing to buy from a firm that unshrouds and undercuts competitors’ additional price by a little bit, so in this case a deceptive equilibrium does not exist. This logic indicates that deception in our model requires individual-welfare-reducing consumer purchases. And because firms can earn positive profits only through deception, our model also says that any profits must be associated with suboptimal consumer choices.7 Beyond showing that Condition (SC) is sufficient for profitable deception to occur, Proposition 3 establishes the (technically more difficult) converse that Condition (SC) is also necessary: if it is violated for some firm, then there is no deception in equilibrium, so that by classic Bertrand logic firms earn zero profits. The proof is by contradiction, and proceeds roughly as follows. If rivals shroud with positive probability, a firm can ensure positive profits by shrouding and choosing prices f, a. Since a firm that sets the highest total price when unshrouding has zero market share if some other firm unshrouds, to earn positive profits it must be that with positive probability all rivals set higher total prices when shrouding. For these high total prices, firms earn positive profits only when shrouding occurs, so that arguments akin to those above imply that they set prices f, a. But then a firm that violates Condition (SC) prefers to unshroud. 7 Furthermore, if a deceptive equilibrium is played by firms, then productive efficiency also fails to hold: market shares are determined by how consumers happen to choose when indifferent. This contrasts sharply with natural specifications of classical Bertrand competition, where the market share of firms other than the most efficient ones is zero. 11 in Section 4.2. Throughout this section, we assume that the proportion of sophisticated consumers is κ ∈ (0, 1), and that the price floor is binding: f > cn − a for all n. 4.1 Sophisticated Consumers in Our Basic Model Notice that in any deceptive equilibrium sophisticated consumers do not buy the product: if they did, they would buy from a firm with the lowest total price, and such a firm would prefer to either undercut equal-priced competitors and attract sophisticated consumers, or (if there are no equal-priced competitors) to unshroud and attract all consumers. Furthermore, note that if firm n unshrouds, it attracts consumers if and only if it cuts the total price to at most v—but if it does so, it attracts all naive and sophisticated consumers. Combining these considerations, unshrouding is unprofitable if (1− κ)sn(f + a− cn) ≥ v − cn, (1) and a deceptive equilibrium exists if and only if Condition (1) holds for all n. Condition (1) for the existence of a deceptive equilibrium has two notable implications. First, if the product is socially wasteful, the presence of sophisticated consumers does not affect our results, as a profitable deceptive equilibrium always exists. Intuitively, sophisticated consumers do not buy a socially wasteful product in equilibrium, so their presence is irrelevant—firms just attempt to exploit naive consumers. But second, if the product is socially valuable, the condition for a deceptive equilibrium to exist is stricter in the presence of sophisticated consumers. Intuitively, while these consumers do not buy the product when the additional price is high, they can be attracted by a price cut, creating pressure to cut the additional price—and by implication also to unshroud. 4.2 Sophisticated Consumers with an Alternative Transparent Product We now move beyond Section 4.1 by assuming not only that there are sophisticated consumers, but also that there is another product in the market. Our analysis is motivated by the observation that in many markets, products that are more transparent than and seemingly superior to the deceptive products exist. Mutual-fund investors can choose low-cost index funds that will earn them higher 14 returns than most managed funds. Many credit-card consumers could use debit cards for the same set of basic services and avoid most fees and interest. And many mortgage borrowers would be better served by simple traditional mortgages than by the complicated exotic products that have gained significant market share recently. Formally, we modify our model above by assuming that each firm has an additional, transparent, product with value w > 0, where firm n’s cost of producing product w is cwn . We let minn{cwn } = cwmin > 0, and assume that there are at least two firms whose cost of producing product w is cwmin. Crucially, we posit that product w is socially valuable (w − cwmin > 0), and is not inferior to product v: w − cwmin ≥ v − cmin. Consumers are interested in buying at most one product. Firms simultaneously set the up-front and additional prices for product v, the single transparent price for product w, and decide whether to unshroud the additional price of product v. If consumers weakly prefer buying and are indifferent between a number of firms in the market for product v or w, firms split the respective market in proportion to sn and swn , respectively. Then: Proposition 4 (Profitability of Inferior Products). Suppose f > cn − a for all n. For any shares sn, s w n , there exists an equilibrium in which each firm shrouds the additional price of product v, naive consumers buy product v, and sophisticated consumers buy product w, if and only if v−f ≥ w−cwmin. In such an equilibrium, firms sell the superior product to sophisticated consumers and earn zero profits on it, while they sell the inferior product to naive consumers and earn positive profits on it. Quite in contrast to the message of Section 4.1 that sophisticated consumers increase the pres- sure to unshroud, Proposition 4 says that if v−f ≥ w−cwmin, a positive-profit equilibrium in which naive consumers are deceived always exists. This insight has a perverse implication: all it takes for profitable deception to occur is the availability of an inferior product that has a shroudable price component and a binding floor on the up-front price, and firms earn all their profits from selling the inferior product. The intuition for why the superior product guarantees a deceptive equilibrium with positive profits from the inferior product is in two parts. First, because sophisticated con- sumers realize that the deceptive product is costly but naive consumers believe it is a better deal, in equilibrium the two types of consumers separate. Second, if a firm unshrouded the additional 15 price of the inferior product, consumers would realize that the other product is better, and would buy that product. As a result, a firm cannot make positive profits by unshrouding the additional price of the inferior product. In a sense, the superior product serves as a barrier to unshrouding the inferior product by rendering the inferior product socially wasteful in comparison. The condition v − f ≥ w − cwmin for a positive-profit deceptive equilibrium to exist is a sorting condition: it implies that because they ignore its additional price, naive consumers mistakenly find the inferior product v more attractive than the superior product w. This condition holds if product w is not much better than product v or f is not too high. For instance, although a naive consumer may realize that a debit card fulfills the same functions that she uses in a credit card, she may still prefer a credit card because she falsely believes that its perks (e.g. cash-back bonuses) make it a better deal.11 As an example, consider the market for mutual funds. Although not everyone agrees with this view, many researchers believe that because few mutual-fund managers can persistently outperform the market by enough to make up for their high fees (Carhart 1997, Kosowski, Timmermann, Wermers and White 2006), most managed funds are inferior to index funds. As a result, the explosion of managed funds is often seen as a puzzle (Gruber 1996, French 2008, for example). Our model says that managed funds could have remained profitable (and hence have attracted a lot of entry) not despite, but exactly because index funds that are superior to them exist. The conclusions of Proposition 4 continue to hold if we assume that consumers misperceive the inferior product’s value rather than its price. Suppose that product v has no additional price, but consumers have false beliefs about its value: they believe the value is v, but it is actually v− a. Continuing with the mutual-fund example, naive investors might overestimate the ability of a manager to pick good investments rather than underestimate the fees she charges. Even then, if 11 Although we have exogenously imposed that product w is transparent, this will often arise endogenously even if firms make an unshrouding decision regarding both products. Clearly, under the condition of Proposition 4, an equilibrium in which product v is shrouded and product w is unshrouded exists in that case as well. If in addition w > v and there are sufficiently many firms in the market, the only profitable equilibrium is the one in which the superior product is unshrouded and the inferior product is shrouded. Consider, for example, a candidate equilibrium in which the superior product w is shrouded. Then, naive consumers must be buying product w; otherwise, a firm could attract all these naive consumers by setting prices f, a on product w, and for a low-profit firm this would be a profitable deviation. But if naive consumers are buying product w, a low-profit firm has an incentive to unshroud product w in order to capture this socially valuable market. 16 addition, we suppose that a firm can offer only one pricing scheme f, a.13 Then, the most tempting deviation from a candidate deceptive equilibrium with prices f, a is to unshroud, still charge fn = f to attract a share sn of the remaining uneducated consumers, and lower an to v − f to attract all educated consumers. This is unprofitable if the following variant of the Shrouding Condition (SC) holds: sn(f + a− cn) ≥ [λn + (1− λn)sn](v − cn)− η. (2) Costly unshrouding somewhat qualifies the distinction between socially wasteful and socially valu- able products we have emphasized in Section 3.3. By Condition (2), it is still true that if the product is socially wasteful, a deceptive equilibrium exists for any number of firms. It is no longer true, however, that with a socially valuable product and sufficiently many firms, a deceptive equilibrium does not exist: if unshrouding is expensive or very partial, no firm may prefer to do it even if it can sell the product at positive gross profits. Nevertheless, a version of our basic message continues to hold: the worse a product is, the more likely it is that a deceptive equilibrium exists. While we have not characterized equilibria in general when Condition (2) is violated for some firm, an interesting equilibrium arises in the special case in which this condition is violated for some firm n but (1−λn) sn′ 1− sn (f + a− cn′) ≥ max{[λn′ +(1−λn′)sn′ ](v− cn′)− η, [λn+(1−λn)sn′ ](v− cn′)} (3) for all n′ ̸= n. Then, there is an equilibrium in which firm n educates a fraction λn of consumers, while (by Condition (3)) all other firms sell to naive consumers. This equilibrium seems consistent with the observation that some firms have attempted to unshroud and sell transparent products to consumers, but they have often had only limited impact on their market. Our theory makes comparative-statics predictions on which firm is most likely to use a transparent strategy. Rewriting the reverse of Condition (2) as λn(1− sn)(v − cn)− sn(f + a− v)− η > 0, we obtain that (ceteris paribus) a firm is more likely to offer a transparent product if (i) it can educate more consumers 13 If a firm can offer multiple pricing schemes, from the perspective of deriving conditions for a deceptive equilibrium to exist we can think of educated and uneducated consumers as being in separate markets. Because deviating from deceptive pricing in the market for uneducatable consumers is obviously unprofitable, a firm’s incentives to change its prices and to unshroud derive entirely from the market for educatable consumers. Hence, this case is equivalent to that above for λn = 1. Incorporating the unshrouding cost η, the condition for an equilibrium in which all firms always shroud to exist becomes sn(f + a− cn) ≥ (v − cn)− η for any n. 19 (λn is high); (ii) it, such as a new entrant, has a smaller market share (sn is low); or (iii) it is more efficient (cn is lower).14 While it qualifies our results on single-product valuable industries, the addition of an unshroud- ing cost does not affect the main message of Section 4 that there is often an equilibrium in which naive consumers buy a deceptive inferior product: since a firm has no incentive to unshroud the inferior product even if this is free and full, it certainly does not have an incentive to do so if the same thing is costly and partial. Market power. We now consider the effect of market power of a very simple form—we assume that firm n has strictly lower cost than any other firm: cn < minn′ ̸=n cn′ . We also suppose first that η = 0. In this case, the condition for when a deceptive equilibrium exists in our basic model remains unchanged. Intuitively, since when prices are shrouded competitors’ total prices are not sensitive to their costs, these costs do not play a role in determining whether firm n wants to unshroud. Market power does have an interesting effect on outcomes in our multiple-products model. In particular, suppose that firm n has market power in the market for the superior product (cwmin−cwn ≡ M > 0 where cwmin ≡ minn′ ̸=n c w n′). For this market, we make the common assumption that no firm charges a price below cost, so that in equilibrium firm n charges cwmin and attracts all consumers. Then: Proposition 6 (Market Power in the Superior Product). Suppose f > cn′ − a for all n′, η = 0, and v − f ≥ w − cwn . I. If M > sn(f + a − cn), there is an equilibrium in which firm n unshrouds the additional price of product v and educates a fraction λn of naive consumers, other firms do not unshroud, sophisticated consumers and consumers educated by firm n buy product w, and uneducated naive consumers buy product v. II. If M ≤ sn(f + a− cn), there exists an equilibrium in which each firm shrouds the additional price of product v, sophisticated consumers buy the transparent product w, and naive consumers buy the shrouded product v. 14 If Condition (2) is violated for multiple firms, a free-riding issue arises: a firm prefers another firm to pay the unshrouding cost, even if it then prefers to compete in price for educated consumers. This implies that no pure-strategy equilibrium exists, and it is difficult to characterize the equilibrium in general. 20 Part I of Proposition 6 says that if firm n’s market power is larger than its share of deceptive profits, it prefers to unshroud and attract the naive consumers it can educate. Such a situation could arise, for instance, if firm n sells only the superior product (sn = 0). This observation explains why some specialists in superior products—e.g., Vanguard in the market for mutual funds—have tried to educate consumers about the inferiority of alternative products. At the same time, Part II of Proposition 6 says that if firm n has a large share in the market for the inferior product or the margin on this product is large (sn(f + a − cn) is high), it prefers not to educate consumers even if it has market power in the superior product. Similarly to Section 4.2, the intuition is that deception creates a large profit margin in the inferior product despite competition, and unshrouding the additional price of this product only leads consumers to buy the less profitable superior one. Market power and costly unshrouding. Going further, the extent of unshrouding can be even more limited if unshrouding is costly (η > 0). Suppose, as often seems to be the case in reality, that competition in the superior product is relatively fierce (M is small). Then, firm n is unwilling to educate naive consumers unless η is also small. And going slightly beyond our model, if it is relatively cheap to educate a low fraction of consumers, but much more expensive to educate a significant fraction, firm n would choose limited education. These considerations may help explain why consumer education is often limited. 5.2 Misprediction of Add-On Demand As an alternative to our specification of consumer naivete above, in this section we analyze a model in which a consumer underestimates not the total price of the product, but her own demand for some add-on to the product. When getting a credit card, for example, a consumer may be aware that she will face a high interest rate on any long-term debt she carries, but incorrectly expect to pay off her outstanding debt within a short period.15 We use the same model as in Section 2, with the following modifications. Instead of assuming that fn and an are two components of a product’s price, we posit that fn is the price of a base 15 Consistent with this example, Ausubel (1991) finds that consumers are much less responsive to the post- introductory interest rate in credit-card solicitations than to the teaser rate, even though the former is more important in determining the amount of interest they will pay. 21 5.3 Further Extensions and Modifications For simplicity, our basic model assumes that all consumers have the same valuation v for the prod- uct, but our main insights survive when there is heterogeneity in v. As an analogue of Proposition 3, a deceptive equilibrium with prices f, a often exists because unshrouding would lead consumers with values between f and f + a not to buy, discretely reducing industry demand. The decep- tive equilibrium is more likely to exist when there are more such consumers—that is, when there are more consumers who are mistakenly buying the product. And a deceptive equilibrium exists whenever the product could not be profitably sold to consumers who understand its total price. As above, this is the case whenever the product is socially wasteful to produce, for example because no consumer values it above marginal cost, or (in a natural extension of our model) the number of such consumers is insufficient given some fixed costs of production. But if the product can be profitably sold in a transparent way, then with a sufficient number of firms at least one firm would choose to unshroud, eliminating the deceptive equilibrium. Consider also what happens when there are sophisticated consumers in the population who are not separated by a superior transparent product, and who are heterogeneous in v. So long as a positive fraction of sophisticated consumers buys the product despite their knowing about the high additional price, a cut in the additional price attracts all these sophisticated consumers, so that an arbitrarily small fraction of these consumers induces some competition in the additional price. Whenever shrouding can be maintained, however, firms’ profits are not driven to zero because— similarly to the “captive” consumers in Shilony (1977) and Varian (1980)—naive consumers provide a profit base that puts a lower bound on firms’ total profit. Furthermore, it is clear that these profits can be sufficient to deter unshrouding.18 18 The main results of our paper are also robust to allowing the maximum additional price to be different across firms, with firm n being able to set an; in fact, our proof of Propositions 3 and 5 in the appendix allows for this possibility. This assumption substantively modifies only Proposition 1 on equilibria with a non-binding price floor: because firms that are better at exploiting consumers can afford lower up-front prices, it is now not the firms with the lowest cn that sell to consumers in a deceptive equilibrium, but the firms with the lowest cn − an. This creates allocative inefficiency even though firms make zero profits. Finally, while in our basic analysis we assume that the price floor is binding (f > cn − a) for all firms, the same qualitative points regarding the existence and properties of a positive-profit deceptive equilibrium survive if the price floor is binding for at least two, but not necessarily all, firms. Consider a version of Condition (SC) in which shares are adjusted assuming that only firms with f > cn − a are in the market. If this modified condition holds for all firms with f > cn − a, then there is a positive-profit deceptive equilibrium in which the other firms do not sell: since 24 6 Related Theoretical Literature In this section, we discuss theories most closely related to our paper. Relative to this literature, our main contribution is to identify the central role of socially wasteful and inferior products in maintaining profitable deception. In Gabaix and Laibson’s (2006) model, consumers buy a base good and can then purchase an add-on whose price might be shrouded by firms, and can also take costly steps to avoid the add-on. Gabaix and Laibson’s main prediction is that unshrouding can be unattractive because it turns profitable naive consumers (who buy the expensive add-on) into unprofitable sophisticated consumers (who avoid the add-on). Although the precise trade-off determining a firm’s decision of whether to unshroud is different, we start from a similar insight, and draw out a number of new implications.19 In research complementary to ours, Grubb (2012) considers services, such as a mobile phone or a bank account with overdraft protection, for which consumers may not know the marginal price, and asks whether requiring firms to disclose this information at the point of sale increases welfare. If consumers correctly anticipate their probability of running into high fees, such price-posting regulation can actually hurt because it interferes with efficient screening by firms. If consumers underestimate their probability of running into fees, in contrast, fees allow firms to extract more rent from consumers, and price posting prevents such exploitation.20 Our result that firms sell profitable inferior products to unknowing consumers may seem rem- Condition (SC) can only be satisfied if f + a > v, which together with f ≤ cn − a implies that cn > v, a firm that is out of the market cannot profitably unshroud and undercut competitors. 19 Relatedly, Piccione and Spiegler (2010) characterize how firms’ ability to change the comparability of prices through “frames” affects profits in Bertrand-type competition. If a firm can make products fully comparable no matter what the other firm does—which is akin to unshrouding in our model and that of Gabaix and Laibson (2006)—the usual zero-profit outcome obtains. Otherwise, profits are positive. Piccione and Spiegler highlight that increasing the comparability of products under any frame through policy intervention will often induce firms to change their frames, which can decrease comparability, increase profits, and decrease consumer welfare. Investigating different forms of government interventions, Ko (2011) and Kosfeld and Schüwer (2011) demonstrate that educating naive consumers in a Gabaix-Laibson framework can decrease welfare because formerly naive consumers may engage in inefficient substitution of the add-on. 20 Our theory also builds on a growing literature in behavioral industrial organization that assumes consumers are not fully attentive, mispredict some aspects of products, or do not fully understand their own behavior. See for instance DellaVigna and Malmendier (2004), Eliaz and Spiegler (2006), Spiegler (2006a, 2006b), Laibson and Yariv (2007), Grubb (2009), and Heidhues and Kőszegi (2010). 25 iniscent of a similar potential implication in classical asymmetric-information models. But the mechanism in our model based on the incentive to unshroud is different, and hence generates different comparative statics. For instance, while in a rational asymmetric-information setting a lower-quality inferior product may be more profitable to sell than a superior product because it is cheaper to produce and consumers do not know its value, our theory predicts that a lower-quality inferior product may be more profitable than a superior product even if it is more expensive to produce and consumers do know its value. In assuming that consumers can be induced to pay high additional fees once they buy a product, our theory also shares a basic premise with the large literature on switching costs. But even if firms cannot commit to ex-post prices and there is a floor on ex-ante prices—so that positive profits obtain in equilibrium—our model’s main insights do not carry over to natural specifications of a rational switching-cost model. Most importantly, if consumers know or learn product attributes, a rational switching-cost model predicts that a firm is better off selling a superior rather than an inferior product, so that such a model does not predict the systematic sale of inferior products in competitive markets.21 In addition, this type of model does not predict a consideration analogous to the threat of unshrouding by competitors, and hence it does not generate our results regarding the effect of entry.22 7 Some Policy Implications and Conclusion While the main goal of this paper is to explore features of markets for deceptive products, our insights have some immediate policy implications—and call for exploring more of these implications. 21 This point is immediate if consumers know the products’ values at the time of original purchase. As an illustration of the same point when consumers learn product values only after initial purchase, suppose that there are two products with values vH and vL and costs cH and cL, respectively, product H is strictly superior, and the switching cost is k. Consider a firm who could sell either product to a consumer at an ex-ante price of p followed by an ex-post price of its choice. If the firm sells the superior product, it can charge an ex-post price of cH + k in a competitive market, so it makes profits of p + cH + k − cH . If the firm sells the inferior product, then (as the consumer realizes that she can switch to the superior product) it can charge an ex-post price of cH + k − (vH − vL), so it makes profits of p+ cH + k − (vH − vL)− cL. It is easy to check that the firm prefers to sell the superior product. 22 Similarly, when consumers must pay classical search costs to find out prices (or product features), at a broad level one can think of the prices as being partly shrouded. 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Wilcox, Ronald T., “Bargain Hunting or Star Gazing? Investors’ Preferences for Stock Mutual Funds,” The Journal of Business, 2003, 76 (4), 645–663. 30 Woodward, Susan E. and Robert E. Hall, “Diagnosing Consumer Confusion and Sub-Optimal Shopping Effort: Theory and Mortgage-Market Evidence,” American Economic Review, 2012, 100 (2), 511–515. Appendix: Proofs Proof of Proposition 1. First, we establish properties of any deceptive equilibrium. Note that in any such equilibrium, for almost any fn for which firm n has positive expected market share, an = a. Since the additional price of a firm that has zero market share does not affect the behavior and outcomes of consumers or any other firm, when solving for the prices at which consumers buy and profits we can assume without loss of generality that an = a also for other prices. Now consider a model of Bertrand competition in which firm n has cost cn − a and is choosing fn. For any strategy profile, this game generates the same profits as in a modification of our game in which firms are restricted to shrouding. Hence, the standard Bertrand proof implies that consumers buy the product from a most efficient firm and pay a = a, f = cmin − a; ex-post utility of consumers is v − cmin; and firms earn zero profits. Now each firm setting fn = cn − a and shrouding is clearly an equilibrium. Proof of Proposition 2. With consumers who observe and take into account the additional price, we have Bertrand competition in the total price. Proof of Proposition 3. We establish a slightly more general version of this proposition: we allow the maximum additional prices firms can impose to differ across firms. Let an be the maximum additional price firm n can impose. We prove the statement of Proposition 3 with Inequality (SC) replaced by sn(f + an − cn) ≥ v − cn. (5) We have argued in the text that in any deceptive equilibrium firm n sets its maximal additional price, which now is an. The same argument as in the text also establishes that if Inequality (6) holds for all n, then there is a deceptive equilibrium in which all firms set f, an. We now provide a formal argument for why firms set f in any deceptive equilibrium. The proof is akin to a standard 31 front prices in the interval (f̂ ′ − ϵ, f̂ ′) with positive probability. As ϵ → 0, the probability of firm m charging the highest up-front price conditional on shrouding and the total price being strictly higher than t̂ goes to one. Therefore, the profits go to zero with probability one when unshrouding occurs or when shrouding occurs and some other firm sets a total price strictly greater than t̂. Now follow the same steps as in the previous paragraph to derive a contradiction. We conclude that f̂ ′ = f . Because f̂ ′ = f , each firm n′ ̸= n sets an up-front price of f with probability one conditional on its total price being strictly higher than t̂. Hence f +an′ ≥ t̂ for any n′ ̸= n. We now argue that whenever shrouding, any firm n′ ̸= n does not set up-front prices strictly above f with positive probability. Suppose by contradiction that firm n′ sets prices above f with positive probability when shrouding. As n′ sets f with probability one when charging a total price strictly above t̂, the associated additional price must almost always satisfy an′ ≤ t̂ − fn′ when shrouding and setting the up-front price strictly above f . Since n′ sets up-front prices strictly above f with positive probability when shrouding, there exists an up-front price ĝ′ > f such that it sets prices above ĝ′ with positive probability. There cannot be a competitor whose up-front price when shrouding falls on the interval [f, ĝ′] with positive probability; if this was the case, firm n′ could increase its profits by decreasing all prices above ĝ′ to f and increasing its additional price holding the total price constant. But then, firm n′ could raise its up-front price from f to ĝ′ and increase profits—a contradiction. Thus, any firm n′ ̸= n sets the up-front price f with probability one when shrouding. Now suppose that firm n charges an up-front price strictly above f when shrouding with positive probability. Then it can only earn profits when unshrouding occurs and hence must almost always charge a total price less or equal to t̂ when shrouding. But if it unshrouds and sets the same prices, it would also earn profits when all rivals shroud and set a price above t̂, thereby strictly increasing its profits—a contradiction. Hence firm n also must set f with probability one when shrouding. Second, suppose not all firms other than n set a total price strictly above t̂. Hence, some firm n′ ̸= n sets its total price equal to t̂ with positive probability. Then, by the above argument no other firms set total price t̂ with positive probability. Take the supremum of firms’ up-front prices f̂ ′ conditional on the total price being greater than or equal to t̂. The remainder of the proof is the 34 same as above. (iii): Additional prices are unshrouded with probability one. Suppose not. Then, each firm chooses to shroud with positive probability. Take the infimum of total prices t̃ set by any firm when shrouding. We consider two cases. First, suppose t̃ ≤ v. Take a firm that achieves the infimum. By (i), this firm earns positive profits. For any ϵ > 0, take total prices below t̃+ ϵ of the firm. By unshrouding and setting t̃−ϵ, the firm decreases its profits by at most 2ϵ when one or more other firms unshroud, but discretely increases its market share if all other firms shroud. Hence, for a sufficiently small ϵ > 0 this is a profitable deviation—a contradiction. Second, suppose t̃ > v. Take firm n that violates Inequality (6). By (ii), firm n charges the up-front price f whenever it shrouds. Note that firm n’s profits are zero when a rival decides to unshroud, and its profits are at most sn(f + an − cn) when shrouding occurs. But then, deviating and setting a total price to v is profitable by Inequality (6), because conditional on others shrouding it would earn v − cn. Proof of Proposition 4. First, we show that if an equilibrium of the type identified in the proposition exists, then v−f ≥ w−cwmin. Since in such an equilibrium sophisticated consumers are buying the transparent product, standard Bertrand-competition logic implies that the total price of product w is cwmin and firms earn zero profits on w. For product v, in turn, the same argument as in Proposition 3 shows that in a deceptive equilibrium in which naive consumers buy this product firms choose the up-front price f and additional price a. Then, naive consumers’ ex-ante perceived utility from buying v is v − f and their ex-ante perceived utility of buying w is w − cwmin. Hence, naive consumers are willing to choose product v only if v − f ≥ w − cwmin. Second, we show that if v − f ≥ w− cwmin, then the above is actually an equilibrium. To do so, it is sufficient to show that no firm prefers to unshroud product v. If a firm unshrouds product v, to attract consumers it must provide consumer value of at least as much as they would get from product w. Hence, a firm that unshrouds must provide value of at least w−cwmin ≥ v−cmin ≥ v−cn, which it cannot profitably do. Proof of Proposition 5. Again, as in the proof of Proposition 3 we establish a slightly more general version of this proposition in which the maximum additional prices firms can impose differ across firms. Let an be the maximum additional price firm n can impose. We prove the statement of 35 Proposition 5 maintaining the assumption that the price floor is binding for all firms, i.e. f > cn−ān for all n. Recall from the proof of Proposition 3 that with different maximal additional prices, a deceptive equilibrium exists for η = 0 if and only if for all firms n, sn(f + an − cn) ≥ v − cn. (6) This proof has five steps. (i): No firm unshrouds the additional price with probability one. If a firm unshrouds with probability one, all consumers become sophisticated and hence buy from the firm with the lowest total price f + a. Hence by the exact same argument as in Proposition 2, all consumers buy at a total price f +a = c and no firm makes positive profits from selling to the consumers excluding the unshrouded cost. Then, the firm that chooses to unshroud makes negative profits—a contradiction. (ii): All firms earn positive profits. According to (i), in any equilibrium there is positive prob- ability that no firm unshrouds. Then, each firm n can earn positive profits by shrouding the additional prices and offering (f, an). (iii): The distributions of total prices are bounded from above. Suppose firm n sets the total price fn + an > v + a with positive probability in equilibrium. When the additional prices are shrouded, consumers never buy the product from firm n because this inequality implies fn > v. When the additional prices are unshrouded, consumers never buy from firm n because fn+an > v. Firm n’s profits in this case is at most zero, a contradiction with (ii). (iv): No firm unshrouds the additional price with positive probability. Let t̂n be the supremum of the equilibrium total-price distribution of firm n conditional on firm n unshrouding; set t̂n = 0 in case firm n does not unshroud. Let t̂ = maxn { t̂n } . Consider firm n that unshrouds and for whom t̂n = t̂. Note that in any equilibrium in which some firm unshrouds with positive probability, t̂ > cn by (ii). Also, by (iii), t̂ is bounded from above and hence well-defined !!! this sentence should be earlier. !!!. First, suppose that firm n charges the total price t̂ with positive probability. If some other firm n′ ̸= n also sets the total price t̂ with positive probability, then firm n has an incentive to slightly decrease its total price—a contradiction. Thus, only firm n charges the total price t̂ with positive probability. Because t̂ is the supremum of the total-price distribution conditional on unshrouding, 36