Competing for Consumer Inattention

Transcription

1 Competing for Consumer Inattention Geoffroy de Cippe Kfir Eiaz Kareen Rozen February 2014 Abstract Consumers purchase mutipe types of goods, but may be abe to examine ony a imited number of markets for the best price. We propose a simpe mode which captures these features, conveying new insights. A firm s price can defect or draw attention to its market, and consequenty, imited attention introduces a new dimension of cross-market competition. We characterize the equiibrium, and show that having partiay attentive consumers improves consumer wefare. With ess attention, consumers are more ikey to miss the best offers; but enhanced cross-market competition decreases average price paid, as eading firms try to stay under the consumers radar. We are gratefu to Phi Reny, Ran Spieger, Bruno Struovici, and three anonymous referees for vauabe suggestions, and Xiaosheng Mu for exceent research assistance. Brown University, Department of Economics. Te Aviv University and the University of Michigan - Ann Arbor, Departments of Economics. Yae University, Department of Economics and Cowes Foundation. Rozen thanks the NSF for financia support under grant SES

2 1 Introduction Cassic modes of price competition assume that consumers have unimited abiity to track down the best deas. The wide array of goods and services in the marketpace casts doubt that this is a faithfu description of the average consumer. With ony imited attention to devote to finding cheaper substitutes, consumers may pay cose attention to some purchases whie negecting to find the best price in others. This paper investigates the price and wefare impications of aocating imited attention across markets. Our simpe mode conveys some new insights: (i a firm s price can defect or draw attention to its market; and consequenty, (ii imited attention introduces a new dimension of competition across (even otherwise independent markets. We convey these insights in a simpe framework, but they shoud remain important considerations in more genera settings. Consumers in our mode have unit demand for each of M different goods. To make point (ii as starky as possibe, each consumer s utiity is separabe across goods, which ensures these markets woud be independent if attention were unimited. Reservation prices are assumed to be one for a consumers and a goods. Each good is offered by two seers whose constant margina cost is normaized to zero, and who set prices independenty. For each market, consumers have a defaut seer who is interpreted as the most visibe provider of that good or service. Consumers share the same defaut set of seers, who are thought of as the market eaders. Confronted with market eaders prices, consumers decide which markets to examine further, to see whether the competing firm (the market chaenger, whose identity and price they do not know offers a better dea. Consumers may have ony imited attention to devote to comparisonshopping, with the abiity to investigate at most k {0,..., M} markets. The distribution of attention in the popuation is captured by a probabiity distribution (α 0,..., α M. Our mode captures the view that imited attention introduces an auditing component into consumption decisions. Given his budget of attention, a consumer uses what he knows (in this case, the price offered by market ead- 1

3 ers to decide which dimensions of his consumption decision are worthiest of further investigation. For instance, when buying groceries onine, which items does a consumer buy from his saved ist, and which does he check for better bargains? In a sense, a consumer s probem under imited attention is akin to that of maintenance scheduing in operations research: ony a subset of items can be served, and those that are negected may suffer from poor performance. For a consumer with imited attention, inspecting one market means overooking another. The cost associated with this tradeoff is endogenous, equa to the expected equiibrium savings foregone by negecting that other market. Our setting is one of imperfect information, since consumers do not observe chaengers prices when aocating their attention. The anaysis focuses on partiay symmetric, perfect Bayesian Nash equiibria (henceforth equiibria. These preserve the symmetry of the mode, with firms in the same position (as eaders or chaengers using the same pricing strategy. In that case, consumers expect the most savings to be found in markets with the most expensive eaders. Hence firms profits may vary discontinuousy with the eaders prices, as consumers shift their attention between markets. A more standard form of discontinuity aso arises when firms in a market quote the same price. Despite these discontinuities, we constructivey estabish that a partiay symmetric equiibrium exists for any distribution of attention, and moreover, that ony one such equiibrium exists. In this equiibrium, a firms empoy atomess pricing strategies, but eaders systematicay charge a wider range of prices than chaengers. The support of the eaders strategy has no gap. However, depending on the distribution of attention, chaengers may avoid charging some intermediate prices. Constructing the unique equiibrium then requires an ironing procedure. What is the equiibrium effect of (inattention on consumer wefare? As might be expected, an increase in the proportion α 0 of fuy inattentive consumers is detrimenta. However, varying the distribution of partiay attentive consumers has perhaps surprising impications. Any change in the distribution of attention which decreases the average eve of attention (hoding α 0 constant is beneficia. This may seem unintuitive at first, since consumers 2

4 inspecting fewer markets are more ikey to miss the best deas. But this intuition does not take into account the countervaiing effect of partia inattention on firms behavior. 1 Consumers imited capacity to search for better deas induces cross-market competition for their inattention: by owering its price, a eader can increase the chance his market remains under the consumers radar. The overa effect coud, at east in theory, be determined by computing the consumer surpus directy using our expressions for the equiibrium strategies. Our argument foows a different route, taking advantage of the fact that tota surpus remains constant and that firms equiibrium profits turn out to be much simper to cacuate. We deve further into the mechanics of competition for inattention, exporing how the eaders pricing strategy adjusts. This paper proceeds as foows. In the next subsection we discuss how our paper fits within the iterature. Section 2 presents the mode. Section 3 presents the main resuts and their intuition, incuding how consumers aocate attention, the equiibrium characterization, and comparative statics with respect to partia attention. We aso iustrate these resuts for the specia case of two markets. The constructive proof of the unique equiibrium is presented in Section 4. Concuding remarks, and possibe directions for future research, are given in Section 5. Some proofs are reegated to the appendix. Reated iterature Our setting buids on the semina iterature on price dispersion (Saop and Stigitz, 1977; Rosentha, 1980; Varian, 1980, which expains observed variation in prices by introducing captive consumers who purchase from a randomy seected firm, without engaging in price comparisons. Among other differences with that iterature, we consider mutipe markets and introduce partiay attentive consumers, which are driving forces behind our resuts. These and other features of our framework, such as the endogenous cost of negecting a market and the asymmetric positions of firms, aso depart from 1 As an anaogy, think of auctions under asymmetric information. Fixing the bids, first price gives a stricty higher profit than second price. However, this does not mean that equiibrium profits are necessariy higher with a first-price auction, as individuas bidding behavior responds to the auction format. 3

5 the standard approach taken in the search and rationa inattention iteratures. In the search iterature, consumers incur a fixed, exogenous cost of samping prices of a product sod by mutipe firms; cassic references incude Burdett and Judd (1983, where consumers decide in advance how many prices to simutaneousy sampe, or Stah (1989, where consumers search sequentiay. Recent papers aim to capture suggish price adjustments by introducing a cost for firms to review its price poicy or gather information about current market conditions. Some authors assume a fixed cost of review (e.g., Mankiw and Reis (2002 and more recenty, Avarez, Lippi and Pacieo (2011. Others, in the rationa inattention iterature, mode an exogenous cost of information processing using entropy measures (e.g., Sims (2003 and Woodford (2009. The decision makers diemma in that iterature is whether to obtain any information, and if so, how much. In our approach, prices serve as cues to determine which markets are worthiest of attention, which introduces an eement of competition across seers of different goods. Market interaction between profit-maximizing firms and consumers with imited attention is, of course, more intricate than the styized environment we anayze. Our mode isoates an aspect of the feedback between consumer attention and firm behavior that has not been studied in the iterature. One strand of this iterature has focused on a different aspect of attention: when firms offer a muti-dimensiona product, consumers may take ony a subset of these dimensions into consideration. This approach is exempified by Spieger (2006, where a consumer sampes one price dimension from each firm seing a product with a compicated pricing scheme (e.g., heath insurance pans; Gabaix and Laibson (2006, where some consumers do not observe the price of an add-on before choosing a firm; Armstrong and Chen (2009, who extend the notion of captive consumers to those who aways consider one dimension of a product but not another (say, price but not quaity; and Bordao, Gennaioi and Sheifer (2013, who study a duopoy mode where firms decide on price and quaity, taking into account that the reative weights consumers give to these attributes is determined endogenousy by the choices of both firms. The above works study symmetric pricing equiibria for firms in a singe market, 4

6 with some differing impications for wefare. In Gabaix and Laibson (2006, for instance, prices increase as more consumers notice add-ons; whie in Armstrong and Chen (2009, reducing the proportion of captive consumers reduces the incentive to offer ow quaity, but has an ambiguous effect on consumer wefare. Taking a different approach to attention, Eiaz and Spieger (2011a,b formaize a mode of competition over consumers who ony consider a subset of avaiabe products. They abstract from prices and anayze firms who compete over market share ony by offering a menu of products together with a payoff irreevant marketing device (e.g., packaging. Consumers in their mode are characterized by a preference reation and a consideration function, which determines, given firms choices, whether a consumer pays attention ony to its (exogenousy determined defaut firm or whether he aso considers the competitor. They show that consumer wefare need not be monotonic in the amount of attention impied by the consideration function. 2 The mode We propose a simpe mode capturing the feature that consumers purchase mutipe types of goods and services, but may have the capacity to examine ony a imited number of markets in search of the best price. The market for each good or service consists of two firms, a eader and a chaenger, who compete in prices. A consumers know the market eaders prices, but need to pay attention to a market to identify the chaenger and earn his offer. Consumers differ in the number of markets to which they can pay attention. The eader in a market is interpreted as the most visibe provider of the good or service, and is the defaut provider for a consumer who chooses not to aocate the time or capacity to search that market further. There is a unit mass of consumers, each of whom desires at most one unit of any given good. For simpicity, we assume that the consumers reservation price for each type of good is one. Letting M denote the number of markets (one per good, a consumer s utiity from purchasing the bunde (x 1, x 2,..., x M {0, 1} M at prices (p 1, p 2,..., p M is M m=1 (1 p mx m. 5

7 The distribution of attention in the consumer popuation is captured by a probabiity distribution α = (α 0, α 1,..., α M, where α k is the proportion of consumers who can inspect up to k markets to find the best price. Consumers optimay decide which markets to inspect. If a consumer inspects a market, then he can choose whether to purchase from the market eader, the chaenger, or not at a. If he does not inspect a market, then his ony decision for that market is whether to purchase from its eading firm. The distribution of attention is common knowedge among firms. We assume throughout a positive measure of fuy attentive consumers (α M > 0, inattentive consumers (α 0 > 0, and partiay attentive consumers (α 0 +α M < 1. We further assume that a positive fraction of fuy attentive consumers insist on inspecting a market when indifferent. 2 The game unfods over two periods. First, a firms independenty set prices to maximize (expected profit. We normaize margina costs to zero, so reaized profit is simpy the product of the firm s price and its market share. Upon observing a the eaders offers, consumers decide how to aocate their attention, and make their purchasing decisions, to maximize (expected utiity. Equiibrium. Because consumers have ony imperfect information when aocating their attention, the equiibrium notion appied is that of Perfect Bayesian equiibrium. We restrict attention throughout to partiay symmetric equiibria where market eaders foow a common pricing strategy, as do market chaengers. The eaders strategy may differ from that of the chaengers, and we do not impose restrictions on the consumers strategies. We note that equiibrium existence is nontrivia, since firms profits are discontinuous. 3 2 One coud instead assume any positive measure of consumers who are standard, that is, aware of a firms and prices. Without either assumption, the mode admits Diamond-type equiibria (Diamond, 1971, as in many search modes, in addition to the partiay symmetric equiibrium we characterize. Indeed, for any p [α 0, 1], there woud be an equiibrium where a firms charge p, and consumers inspect none of the markets on the equiibrium path. 3 Firms payoffs exhibit two forms of discontinuity. The first, reated to how a eader and a foower in a market share consumers when quoting the same price, appears in many modes of competition. Existence in such cases foows from resuts by Dasgupta and Maskin (1986 or Reny (1999. The second form of discontinuity is reated to how consumer attention is aocated across markets, and its impact on chaengers profits, when some eaders quote 6

8 Notation and definitions. The eaders and chaengers pricing strategies are described by (right-continuous cumuative distribution functions F : R [0, 1] and F c : R [0, 1], respectivey. A price p is said to be in the support of the pricing strategy F if F (p + ε > F (p ε, for a ε > 0. A price p is said to be an atom of the strategy F if F is discontinuous at p, that is, F (p > F (p, where F (p = im p p F (p. We do not put a priori restrictions on the presence of atoms or gaps in the support of the pricing strategies. 3 Main resuts and intuitions In this section, we first present our characterization of partiay symmetric equiibria and some of the intuitions behind it, eaving the compete equiibrium anaysis to Section 4. We then examine how the equiibrium and consumer wefare change with the distribution of attention among consumers. 3.1 Consumer attention and its impications Suppose the eading firm in market i quotes a price p i. The expected gain from inspecting market i is the expected savings from finding a cheaper price by the chaenger, i.e., min{pi,1} 0 (min{p i, 1} xdf c (x. (1 Note that the above expression reies on the symmetry in the chaengers pricing strategies. Optimaity requires the foowing. If a consumer inspects market i, and inspecting market j gives stricty higher expected savings, then he aso inspects market j. If a consumer inspects fewer markets than his capacity aows, then any market eft uninspected has zero expected savings. the same price. For each price he may quote, a chaenger s profit is discontinuous over a continuum of eaders prices, which prevents a direct appication of Dasgupta and Maskin (1986. It aso impies that chaengers cannot secure themseves a positive payoff in the sense of Reny (1999. Whie aternative methods may be used to show existence, we provide a constructive proof that aso estabishes uniqueness. 7

9 The next proposition formaizes these statements using (1. Proposition 1. Suppose market eaders post the prices p 1,..., p M, and et q i = min{p i, 1}, for each i. If a consumer inspects market i, q i < q j and F c (q j > 0, then he aso inspects market j. If a consumer inspects fewer markets than his capacity aows, then F c (q i = 0 for any uninspected market i. Proposition 1 takes a simpe form when eaders prices are a distinct, are no higher than the consumers reservation price, and there is positive probabiity that each market s chaenger posts a cheaper price than the eader: the consumer inspects the k markets with the highest eader prices. Through a series of resuts in Section 4, we show that these properties hod in equiibrium for amost a prices quoted by eaders. In the remainder of this section, we use this simpe characterization of attention aocation to express firms incentives. Market eaders. We begin by computing the probabiity that a eader s market is paid attention to by a consumer with k units of attention, assuming that eader charges the price p and that a other market eaders foow the pricing strategy F. Letting x = F (p, we denote this probabiity by π k (x. Observe that his market receives attention from such a consumer if there are no more than k 1 other markets whose price turns out to be higher than p. Since the probabiity that another eader charges above p is 1 x (which foows from the symmetry of eaders strategies, we find that 4 k 1 ( M 1 π k(x := x M 1 i (1 x i. (2 i i=0 As expected, π 0(x = 0 and π M (x = 1. In addition, the probabiity of being inspected by a given consumer is increasing in his capacity for attention k, and increasing with one s price (as captured by x. 4 This amounts to having at most k 1 successes in M 1 trias that are i.i.d., where the probabiity of success (which means finding a price higher than p is 1 F (p. 8

10 Market chaengers. Consider a chaenger s probabiity of seing to a consumer with k units of attention, assuming that he himsef charges the price p and that a market eaders foow the pricing strategy F. Letting x = F (p, we denote this probabiity by π c k (x. If a consumer is ony partiay attentive (that is, k < M, then π c k (x is not simpy 1 x, the ex-ante probabiity that the eader s price is higher than p. For the chaenger, seing requires the consumer to pay attention to the market, an event whose probabiity is itsef impacted by the eader s price. We may compute π c k (x as foows. The chaenger has zero probabiity of making a sae if the eader in his market quotes a price stricty ess than p. If the eader quotes a price q > p, then the consumer wi purchase from the chaenger so ong as he inspects the market, which occurs with probabiity π k (F (q. Integrating over the possibe prices of the market eader, the desired probabiity is given by π p k (F (qdf (q. This probabiity depends ony on x = F (p and not the entire distribution F, as can be seen using the change of variabes t = F (q: π c k(x := 1 x π k(tdt. (3 As expected, π c 0(x = 0 and π c M (x = 1 x. In addition, the probabiity of seing to a given consumer is increasing in his capacity for attention k, and decreasing with the probabiity x that the eader s price is better. 3.2 Equiibrium characterization It wi be hepfu to define the tota probabiity that a eader s market draws attention if he charges a price p, and the tota probabiity that a market chaenger ses if he charges a price p. Recaing that α is the distribution of attention among consumers, and etting x = F (p, those probabiities are Π (x := M α k π k(x and Π c (x := k=1 M α k π c k(x, k=1 9

11 respectivey. Since there is a positive measure of partiay attentive consumers, Π is stricty increasing and Π c is stricty decreasing; hence their inverses Π 1 and Π 1 c are we-defined. Deriving indifference conditions. Propositions 2 through 7 in Section 4 show that any equiibrium, if one exists, must satisfy the foowing properties. First, α 0 is the owest price in the support of both the eaders and chaengers strategies. Second, both eaders and chaengers pricing strategies must be atomess. Third, the eaders strategy has fu support over the interva [α 0, 1], whie the chaenger s highest price p c must be stricty smaer than 1. Given these properties, we can derive firms equiibrium profits. A eader is sure to se to captive consumers as ong as his price is ess than one. When charging arbitrariy cose to one, however, he is neary certain to ose a non-captive consumers to the chaenger (because F is atomess, and p c < 1. Hence a eader s equiibrium profit must be α 0. Since the eader ses at the price p either when a consumer does not pay attention, or when he pays attention but the chaenger s price is higher, we must have ( ( p 1 Π F (p ( + Π F (p ( 1 F c (p = α 0, (4 for prices p in the support of the eaders strategy. Next, a chaenger s profit from each price in its support must equa its profit from quoting α 0. This profit is given by α 0 Π c (0, which in turn equas α 0 EA(α/M, where EA(α := M α k k is the expected eve of attention in the consumer popuation. Indeed, because the eaders strategy is atomess and prescribes ony prices above α 0, the chaenger is sure to se to consumers who pay attention; and given that market eaders a use the pricing strategy F, there is a k out of M chance that his market eader s price wi be among the k-highest. 5 Therefore, for any price p k=1 5 This can aso be seen by appying the Euer integra 1 0 ta 1 (1 t b 1 dt = (a 1!(b 1! (a+b 1! 10

12 in the support of F c, it must be that pπ c ( F (p = α 0EA(α M. (5 For any price in the support of a chaenger s strategy, the eader s strategy is derived from the indifference condition (5; for a other prices, it is derived from the indifference condition (4, as a function of the (constant eve of F c. In other words, F (p = Π 1 c Π 1 ( α0 EA(α Mp ( p α 0 pf c(p for a p in the support of F c, for a other p [α 0, 1]. (6 The chaenger s strategy is aso derived from the indifference condition (4 for any price in its support. Soving for F c in (4 and appying the expression for F above, we see that for each price in the support of the chaengers pricing strategy, F c must coincide with the function F c defined by F c (p := pπ (Π 1 c p α 0 ( α0 EA(α Mp, for a p [α 0, 1]. (7 Which prices does a chaenger charge? The difficuty ies in knowing the support of the chaengers strategy, since F c may be nonmonotonic without further restrictions on the attention distribution. Such an exampe is iustrated in Figure 1. If an equiibrium exists, then any nonmonotonicity in F c must be ironed by introducing one or more gaps in the support of the chaengers strategy. Due to the absence of atoms, F c must be continuous. Hence any singe gap in F c must be an interva between two prices whose F c vaues coincide. In Figure 1, for instance, a gap cannot start at a price ower than p 1. On the other hand, there is a range of prices arger than p 1 which can serve as the eftmost endpoint of a gap. Remember that the eaders pricing strategy F is defined in the definition of π c k (0 to show that it simpifies to k/m. 11

13 piecewise in (6 according to the chaengers support. Can F c be ironed in a way that ensures F is increasing and atomess, as we know it must be? These requirements turn out to be unrestrictive: F satisfies them whenever F c is ironed in the continuous manner described above. Thus there are infinitey many ways to construct vaid distribution functions F and F c which eave the eaders and chaengers indifferent over a prices in their respective supports. However, there is a unique way to iron F c that yieds equiibrium pricing strategies F and F c. Using any other approach, the chaenger woud have a profitabe deviation to some price outside his support, as expained further beow. Theorem 1. For any distribution of attention α, there exists a unique partiay symmetric equiibrium. The chaengers pricing strategy F c is atomess and given by F c (p = min p [p,1] F c ( p, for a p [α 0, p c ], (8 where p c (α 0, 1 is the smaest price for which the above expression equas one, and F c is given by Equation (7. The eaders pricing strategy F has fu support on [α 0, 1], is atomess, and given by Equation (6. Theorem 1 is proved in Section 4. There we provide a compete equiibrium anaysis, covering some important steps (e.g., ruing out the presence of atoms, characterizing the support of the eader that have been gossed over in this section when deriving necessary equiibrium conditions. Moreover, we resove the question of existence by verifying that the construction indeed yieds an equiibrium. To state the characterization of F c a bit differenty, note that among a pricing strategies which ie beow the graph of F c, the chaengers strategy is the one which is pointwise highest. Hence it prescribes the cheapest price distribution among those, in the sense of first-order stochastic dominance. Graphicay, this means F c must be ironed as iustrated in Figure 1, by starting any gap at the smaest possibe price whie sti preserving continuity. understand why this must be the case, consider a price p which is in a gap of the chaenger s pricing strategy. In this case, F (p is found using the eaders 12 To

14 F c F c α 0 p 1 p 2 p c 1 Figure 1: The construction of F c in an exampe where F c is not increasing. indifference condition (4. If the chaenger charges p, his expected profit is pπ c ( Π 1 ( p α0. (9 pf c (p Contrary to Theorem 1, suppose that F c (p > F c (p for the gap price p. Since Π c is decreasing and Π 1 is increasing, the expression in Equation (9 increases when repacing F c (p with the ower vaue F c (p, with the resuting expression simpifying to α 0EA(α, the chaengers equiibrium M profit.6 Hence the chaenger woud obtain stricty higher profit by charging the gap price p than any price in the support of his strategy. The presence of a gap in the chaengers strategy depends on the way attention is distributed among consumers. For any attention distribution α, the distribution of partia attention is ( α 1 α 1 α 0,..., M 1 α 0. This is simpy α condi- 6 For some intuition, note from indifference condition (4 that the more ikey are chaengers to be cheaper than p, the more ikey are eaders to be more expensive than p. Indeed, eaders prices increase so that a eader charging p is better shrouded from consumer attention, and can maintain its equiibrium profit against the more competitive chaenger. Hence F c (p > F c (p impies a chaenger s market share when charging p is arger than that yieding his equiibrium profit. 13

15 tioned on consumers being at east partiay attentive, that is, k 1. The ack of monotonicity in Figure 1 can be attributed to having mutipe peaks in the partia attention distribution. Gaps can be rued out when, given the proportion of consumers with attention span k and the proportion with attention span k + 2, there are sufficienty many consumers faing in between. More formay, the partia attention distribution is og-concave if α 2 k α k 1α k+1 for each k {2,..., M 1}, or equivaenty, the ikeihood ratio α k+1 /α k is decreasing in k. Note that this is triviay satisfied when there are ony two markets, and is impied whenever the entire attention distribution is og-concave. When partia attention has this feature, the form of the equiibrium pricing strategies simpifies. Theorem 2. When F c is stricty increasing, the chaengers pricing strategy F c has fu support on [α 0, p c ] and the eaders pricing strategy F simpifies to F (p = max { Π 1 c (α 0 EA(α Mp, Π 1 ( α 0 } 1 p for a p [α 0, 1]. A sufficient condition for F c to be stricty increasing is og-concavity of the partia attention distribution. Theorem 2 is proved in the appendix. Many distributions (and their truncations satisfy og-concavity. For exampe, the property is satisfied by a positive binomia distribution, where consumers start with M units of attention but can ose up to M 1 of them due to independent, exogenousy occurring emergencies (e.g., the consumer s washing machine breaks down, his chid gets the fu, his boss asks for overtime, etc.. We note that gaps can aso be rued out under other assumptions on partia attention, such as when the distribution is increasing (that is, α k α k+1 for each k We show in the appendix that gaps can be rued out when Π c (0 Π c (x is stricty og-concave. Whie Π c (0 Π c (x can be written as the sum of og-concave functions, og-concavity is not necessariy preserved by aggregation. We show that og-concavity is preserved if the sequence β 1 = α M, β k = β k 1 + k i=1 α M i+1 is og-concave. This is impied, for instance, by og-concavity of partia attention, or by increasingness. 14

16 3.3 The comparative statics of attention A natura question that arises from our anaysis is how partia attention affects consumer surpus. One can think of at east two reasons why ess attention coud be detrimenta for consumers as a whoe. First, eaders might have an incentive to take advantage of ess comparison shopping, thereby quoting higher prices. Second, fixing eaders prices, the chaengers reaize that partiay attentive consumers who approach them do so because their market s eader is expensive. However, these intuitions ignore a countervaiing effect: partia attention introduces a new form of cross-market competition, as each market eader has an incentive to ower its price in order to better defect consumer attention. Thus, the comparative statics of attention invove subte interactions between cross-market and within-market competition. In this section, we start by investigating the effects on consumer wefare, before examining the effects on equiibrium pricing. Theorem 3. Consider two distributions of attention α and ˆα which share the same proportion of fuy inattentive consumers (α 0 = ˆα 0. Then consumer wefare is higher under α than ˆα if, and ony if, the expected eve of attention under α is ower than under ˆα. Proof. As argued in Section 3.2, a eader s equiibrium expected profit is equa to the proportion of fuy inattentive consumers, and is thus the same under both α and ˆα. As aso argued there, a chaenger s equiibrium expected profit is equa to the proportion of fuy inattentive consumers, mutipied by the expected eve of attention, divided by M. Hence producer surpus is ower under α than ˆα if, and ony if, the expected eve of attention under α is ower than under ˆα. The resut then foows from the fact that tota surpus remains constant (equa to M. Neither fuy attentive nor fuy inattentive consumers generate competition for inattention. Whie fuy attentive consumers do generate within-market competition, fuy inattentive consumers are simpy captive to market eaders. As might be expected, increasing the proportion α 0 of captive consumers has 15

17 a negative effect on consumer surpus. 8 At the opposite end of the attention spectrum, Theorem 3 means that making fuy attentive consumers ess attentive benefits consumers as a whoe. In particuar, the coser the attention distribution is to the imit distribution (α 0, 1 α 0, 0,..., 0, the better off consumers are; simiary, the coser is the distribution to the imit distribution (α 0, 0,..., 0, 1 α 0, the worse off consumers are. To gain some intuition for Theorem 3, remember that in equiibrium, eaders are wiing to quote prices that are more expensive than what a chaenger woud ever charge. When charging such a price p, a eader s profit, given by p(1 Π (F (p, reies on not drawing too much consumer attention. Suppose partia attention decreases. If the other eaders pricing strategy were to remain unchanged, then the eader s profit from quoting p woud rise above α 0. Yet competition impies that no eader can make a profit that arge. Hence the ikeihood of having other eaders quote prices smaer than p must go up, so that the eader quoting p sticks out with sufficient probabiity. The pricing effects of a change in partia attention may be more ambiguous for ower prices, as eaders become competitive against the chaengers. Buiding on the insight from Theorem 2, we focus on cases where F c is stricty increasing and show that the eaders pricing strategies are comparabe under first-order stochastic dominance when the change in partia attention can be ranked in the monotone ikeihood ratio order. Given two attention distributions α and ˆα, we say that the partia attention distribution under α dominates the partia attention distribution under ˆα in the monotone ikeihood ratio order (MLR if ˆα k /α k is increasing in k {1,..., M}, with at east one strict inequaity. The MLR ordering has a ong tradition in economics, starting with Migrom (1981, and is known to be stronger than first-order stochastic dominance. Theorem 4. Let α and ˆα be two attention distributions with α 0 = ˆα 0 and og-concave partia attention distributions. If the partia attention distribution under ˆα dominates that under α in the MLR order, then market eaders 8 Increasing α 0 at the expense of reducing (α 1,..., α M by the infinitesima amounts (ε 1,..., ε M has a tota effect on producer surpus of M i=1 ε i(m + EA(α α 0 i > 0. 16

18 equiibrium prices are first-order stochasticay higher under ˆα than under α. More generay, Theorem 4 remains true when repacing the og-concavity requirement with any conditions on α and ˆα guaranteeing that the chaengers strategy has no gap (e.g., as in footnote 7. For intuition on why the resut hods, remember that Π and Π c (the probabiities that a eader s market receives attention and that a chaenger makes a sae depend on the attention distribution. In what foows, Π and Π c correspond to the attention distribution α, whie ˆΠ and ˆΠ c correspond to the attention distribution ˆα. Reca from Theorem 2 that the probabiity F (p that a eader charges a price ower than p under attention distribution α is simpy { max Π 1 c (α 0 EA(α Mp, Π 1 ( 1 α 0 p }, (10 when the partia attention distribution is og-concave. An anaogous expression describes the probabiity ˆF (p that a eader charges a price ower than p under attention distribution ˆα. This is iustrated in Figure 2. The theorem is proved by showing that each of the two expressions on the right-hand side of (10 shifts downwards when consumer attention increases from α to ˆα. Consequenty, market eaders charge first-order stochasticay higher prices when attention increases. The downward shift for the second expression in (10, which reates to the intuition given earier for prices above the chaengers support, actuay hods for any first-order stochastic increase in partia attention. The downward shift in the first expression in (10 is ess obvious, and hods for MLR shifts. 9 Changes in partia attention have a more ambiguous effect on the chaengers pricing strategy. Since consumer wefare increases when there is ess attention, it is cear that chaengers cannot increase their prices by too much. As we next iustrate, when there are just two markets, og-concavity of the partia attention distribution is triviay satisfied, and MLR-dominance reduces to first-order stochastic dominance. In that case, one can show that both eaders and chaengers prices decrease when partia attention decreases. 9 This sufficient condition is not necessary, as the attention distributions used for Figure 2 do not have the MLR property but do have the critica feature that ˆΠ c /Π c decreases. 17

19 1 1.0 F Fˆ α 0 p c (α p c (α ˆ 1 Figure 2: Comparative Statics on F. The bod curves depict the market eaders pricing strategies, which are the upper enveope of the corresponding two functions from (10. F corresponds to attention distribution α, whie ˆF corresponds to ˆα. More generay, however, it is uncear whether the chaengers strategy shifts according to first-order stochastic dominance. 3.4 Iustration: the case of two markets We iustrate the equiibrium pricing strategies in the case M = 2. A eader s probabiity of drawing the attention of a consumer with k units of attention (defined in Equation (2, and a market chaenger s probabiity of seing to such a consumer (defined in Equation (3, take a simpe form: π 1(x = x, π 2(x = 1, π c 1(x = 1 x2 2 and π c 2(x = 1 x. The chaengers indifference condition (5 reduces to α 1 1 F (p α 2 (1 F (p = α 0EA(α, Mp 18

20 α 2 + or F (p = α α 1 EA(α(1 α 0 p for any price p in the support of F c. Simiary, pugging F (p into the eaders indifference condition (4 and soving for F c (p gives: α 1 F c (p = 1 α 0 p, α α 1 EA(α(1 α 0 p for any price p in the support of F c. It is easy to check that the right-hand side, which is F c, is increasing for any p greater than α 0. This is consistent with Theorem 2, since the og-concavity condition is satisfied for any distribution of attention when M = 2. The chaengers strategy therefore has no gap, and the maxima price in the support of F c is the price p c (α 0, 1 at which F c reaches Straightforward agebra gives p c = 2α 0 2 α 1 EA(α α 2 1EA(α 2 + 4α 2 2. A that remains is to find the eaders strategy F for prices between p c and 1. This foows from the eaders indifference condition (4, which gives: for each p [ p c, 1]. F (p = 1 α 0 p α 2 α 1 To perform comparative statics, note that when M = 2, increasing attention whie keeping the proportion of captive consumers fixed simpy amounts to shifting weight from α 1 to α 2. A such shifts are comparabe in the MLR order. By Theorem 4, F (p must decrease when shifting weight from α 1 to α 2. This can aso be checked directy given the expression of F in the previous paragraph. Whie the effect on F c is ambiguous for genera M, in the case M = 2 the chaengers prices aso first-order stochasticay increase when 10 Notice that F c is increasing, F c (α 0 = 0, and F c (1 = 1 α 0 α 2 2 +α 1EA(α(1 α 0 1 α 0 > 1, where the second equaity foows from (1 α0 2 α α2 2 +α 1 EA(α = (1 α α 1EA(α = 19

21 shifting weight from α 1 to α 2. Indeed, one can check that df c (p dα 2 which is negative for p [α 0, 1]. df c(p = α 0α 2 1 α 0 p ( dα 1 p α α 1 EA(α(1 α 0 p, 3/2 4 Compete equiibrium anaysis Buiding on the characterization of consumer attention in Proposition 1, we first deveop a series of necessary conditions on firms equiibrium pricing strategies that uniquey pin down the equiibrium, if one exists. We then resove the matter of existence by checking that the construction works. 4.1 Necessary conditions We begin with a usefu observation about the supports of the chaengers and eaders strategies. Proposition 2. The owest price in the support of F and F c coincide, and is greater than or equa to α 0. The highest prices in the support of F and F c are both smaer than or equa to one. Proof. A market eader is sure to se to inattentive consumers, even when charging the reservation price of 1. He can thus guarantee himsef a profit of at east α 0. Any price beow α 0 or above 1 generates a profit stricty ess than α 0. Hence a eader woud not choose a strategy for which F (1 < 1 or F (p > 0, for some p < α 0. Let p be the owest price in the support of F and et p c be the owest price in the support of F c. Suppose p < p c. Consider a deviation by some eader to a pricing strategy F that puts an atom equa to F (p on some price p (p, p c and coincides with F for a p > p. To see that this deviation increases the eader s profit note that for each price p [p, p in the support of the origina strategy F the deviant eader wi now se at a higher price. This is true 20

22 whether or not its market is inspected, since at a price of p it sti undercuts the chaenger. Suppose next that p c < p and that a chaenger deviates to a strategy F c that puts an atom equa to F c (p on some price p (p c, p and coincides with F c for a p > p. Then conditiona on being inspected, for each price p [p c, p in the support of the origina strategy F c the chaenger woud se at a stricty higher price. Since there are fuy attentive consumers who wi inspect the market, this deviation raises the chaenger s expected profits. It remains to show that the argest price in the support of F c is smaer or equa to 1. Any price above 1 does not yied a sae, as it is higher than the consumers reservation price. In this case, as he can se to at east some fuy attentive consumers, any positive price beow α 0 constitutes a profitabe deviation for the chaenger. We next argue that F is atomess. If eaders have an atom at a price stricty above the owest price p in their support, then some eader coud profitaby deviate by moving mass from this price to one which is sighty beow it. This sma price decrease is more than compensated by the decreased attention to the eader s market. However, if the eaders atom is on p, we must distinguish between two cases. If the chaenger s strategy does not have an atom at p, or if some consumers favor the eader in case of a tie at p, then the chaenger coud profitaby deviate by shifting weight to prices sighty beow p. Otherwise, a eader can profitaby deviate for the same reasons as given above. Proposition 3. The eaders pricing strategy F is atomess. Proof. Let p be the smaest price in the support of F, and suppose F has an atom at p (p, 1]. For any sma ε > 0, consider the aternate pricing strategy for the eader which equas F (p for a q [p ε, p], and coincides with F esewhere. This deviation has two opposite effects on the eader s profit. There is a negative effect from seing at a price p ε compared to those prices q (p ε, p]. This oss is of order ε and can be made as sma as desired by decreasing ε. In view of Proposition 1, there is aso a positive effect from the decrease in attention when charging p ε rather than a price in 21

23 (p ε, p]. This gain admits a stricty positive ower bound that is independent of ε for the eader of at east one market. To see this, consider a eader whose market is inspected with positive probabiity when a eaders quote p. The probabiity that a other eaders charge p and that one s chaenger has a price stricty ess than p occurs with probabiity F c (p (F (p F (p M 1, which is positive since p is an atom and F c (p > 0 by Proposition 2. For the fraction 1 α 0 α M of partiay attentive consumers, if the deviator charges p ε they surey do not pay attention to his market, whie if he charges p the probabiity of drawing attention is stricty positive (and independent of ε. Hence this deviation is stricty profitabe for ε > 0 sma enough. Suppose now that F has an atom at p, which is aso the owest price in the support of F c by Proposition 2. For any sma ε > 0, consider the aternate pricing strategy for the chaenger which equas F c (p+ε for a q [p ε, p+ε], and coincides with F c esewhere. This deviation has two opposite effects on the chaenger s profit. There is a negative effect from seing at a ower price p ε compared to those q (p, p + ε]; this resuts in a decrease in profit of no more than 2εF c (p + ε. There is aso a positive effect occurring in the event that the market draws attention when the eader s price is p, which occurs with stricty positive probabiity (independent of ε. In this event, the deviation yieds a sae at the price p ε with probabiity F c (p + ε, whie the origina strategy yieds a sae at the price p with probabiity F c (pβ, where β [0, 1] is the proportion of consumers who purchase from the chaenger when there is a tie at p. The chaenger s aternate strategy is a profitabe deviation for ε > 0 sma enough if either F c (p = 0 (that is, the chaenger does not have an atom at p, or there is an atom at p and β < 1. To concude the proof, suppose that both F c and F have an atom at p and β = 1. In that case, consider the aternate pricing strategy for the eader which equas F (p for a q [p ε, p], and coincides with F for higher prices. The oss from seing at a ower price can be made arbitrariy sma, whie the gain in winning against the chaenger is bounded from zero. Indeed, there is positive probabiity that the market is inspected, and β = 1 impies the chaenger wins in the event of a tie at p, an event occurring with probabiity F (pf c (p > 0 under F. This 22

24 is thus a profitabe deviation from F, a contradiction. As a consequence of the previous two propositions, Proposition 1 now takes the simper form, the consumer inspects the k markets with the highest eader prices. Our next resut is concerned with the highest prices firms coud charge. Chaengers, who make their profit by underbidding their market eader, certainy woud not charge more than a eader s highest price. We show, furthermore, that chaengers charge stricty ess. Since we have not yet rued out the possibiity that F c has an atom at its highest price (or esewhere, the strict ranking of highest prices is hepfu to derive the eaders highest price. Whenever a eader charges his highest price, any consumer who is at east partiay attentive wi inspect his market, and find a cheaper aternative, with probabiity one. As such, the eader may as we take fu advantage of the remaining consumers inattention, by charging a the way up to their reservation price. Proposition 4. The highest price in the support of F c is stricty smaer than the highest price in the support of F, which is one. Proof. Let p (p c be the highest price in the support of F (respectivey, F c. Since F is atomess, there exists ε > 0 sma enough that the probabiity a eader charges more than p ε is stricty smaer than α 0. Thus the chaenger s profit from charging any price above p ε is stricty smaer than the profit obtained by charging α 0, given that he cannot affect the attention to his market. Since the chaenger woud have a profitabe deviation if F c (p ε < 1, we concude that p c < p. We now show that the eaders highest price is one. If p < 1, then for each ε > 0, consider the aternate pricing strategy for a market eader which equas F (p ε for a q [p ε, 1 ε and coincides with F esewhere. For ε sma enough, p ε is arger than the highest price in the support of F c. By charging 1 ε instead of p [p ε, p, the eader has a gain of at east α 0 (1 ε p, since fuy inattentive consumers buy from the eader at any price beow their reservation eve. The eader s oss from this deviation is proportiona to the increase in probabiity of having partiay attentive consumers check his market (thereby finding a cheaper price. As F atomess, when ε is sma then it is 23

25 amost certain that when charging p [p ε, p his market was aready being checked by these consumers. Thus, there is ε sufficienty sma that for any p [p ε, p, the oss is stricty ess than the gain α 0 (1 ε p. The expected change in profit from deviating, obtained by integrating gains minus osses over p [p ε, p, is thus stricty positive. The above resuts aow us to derive the eaders equiibrium profit, as shown in Section 3.2. Coroary 1. The eaders equiibrium profit is α 0. With Coroary 1 in mind, it becomes possibe to identify the common owest price of chaengers and eaders. Proposition 5. The owest price in the support of both F and F c is α 0. Proof. We know that F and F c share a common owest price p α 0. Suppose by contradiction that p > α 0. Consider a deviation where the eader charges (p + α 0 /2 with probabiity one. In this case, the eader ses to a consumers, whether or not they pay attention to his market. This deivers a profit of (p + α 0 /2. Since equiibrium profit is α 0 by Coroary 1, the deviation is stricty profitabe. As expained in Section 3.2, the above resuts can be used to derive the equiibrium profit of chaengers. Coroary 2. The chaengers equiibrium profit is α 0 EA(α/M. The foowing resut rues out atoms for the chaenger. Of course, Propositions 3 and 6 impy that no firm can use a pure strategy in equiibrium. Proposition 6. The chaengers pricing strategy F c is atomess. Proof. Suppose that F c has an atom at some price p > α 0. We begin by pointing out that there cannot exist ε > 0 for which F (p + ε F (p = 0. Otherwise, F has a gap in its support to the right of p, and the chaenger coud profitaby deviate by shifting his atom from p to p + ε. 24

26 Consider an aternate strategy for the eader which equas F (p + ε for q [p ε, p+ε] and is given by F esewhere. For each q [p ε, p+ε], the ony oss associated with this deviation is the decrease in price, which is at most 2ε. Among the various gains in profit from switching is the increased probabiity of seing by underbidding the chaenger when the market is examined. Notice that the eader s market is examined with a probabiity bounded from beow by α M. Thus there is positive probabiity, bounded from zero, both that the market is inspected and that the chaenger quotes p. In this joint event, the gain by charging p ε instead of any q (p, p + ε is stricty positive, since the eader ses to an inspecting consumer when charging p ε, but not when charging q. Hence, for ε sma enough, this deviation is stricty profitabe for the eader, a contradiction. We concude F c is atomess for prices above α 0. Finay, suppose by contradiction that F c has an atom at α 0. Since α 0 aso beongs to the support of F, the eader must get a profit α 0 by charging any price p (α 0, α 0 + ε. However, for any such price there is probabiity arger than α M F c (α 0 > 0 that the eader does not se. Hence the profit from any such p is bounded away from α 0 for sma ε, a contradiction. We now examine whether firms necessariy use stricty increasing strategies. Whie for market eaders the answer is a cear yes, for market chaengers the answer depends on the distribution of consumer types. This contrasts with the previous iterature on competition with mixed strategies over prices, in which a firms use stricty increasing cumuative distribution functions. Proposition 7. The eaders strategy F cannot have any gaps in its support. Proof. Suppose that F has a gap in its support, that is, F is constant over an interva inside [α 0, 1]. Consider then p and p with F (p = F (p such that for a ε > 0, F (p > F (p ε and F (p + ε > F (p. In other words, p is the eft-most point of the gap, and p is the right-most point of the gap. We know that α 0 < p < p < 1 since α 0 and 1 beong to the support of F, which is atomess. Notice that F c must aso be constant on [p, p. Otherwise, any mass paced on that interva by F c can be moved to an atom at p. Indeed, this 25